· functional differential equations editor-in-chief elena litsyn, department of mathematics,...

FUNCTIONAL DIFFERENTIAL EQUATIONS

Editor-in-Chief

Elena Litsyn, Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653, Beer-Sheva, ISRAEL; [email protected]

R.P. Agarwal, Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901, USA Ya. Alber, Department of Mathematics,Technion-Israel Institute of Technology, Haifa 32000, ISRAEL; L. Berezansky, Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653, Beer-Sheva, ISRAEL; E. Braverman, Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, Alberta T2N 1N4, CANADA; A. Chentsov, Institute of Math. and Mech., S.Kovalevskoi Str., Ekaterinburg 620066, RUSSIA; S.-N. Chow, School of Mathematics, Georgia Institute of Mathematics, Atlanta, GA 30332, USA C. Corduneanu, Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019, USA; A. Domoshnitsky, Department of mathematics and Computer Science, Ariel University Center of Samaria, Ariel, ISRAEL; K.-H Foerster, Fachbereich 3 - Mathematik, Sekr. MA 6-4, Technische Universitat Berlin, Strabe des 17. Juni 135, D-10623, Berlin, GERMANY; E. Galperin, Department of Mathematics, Quebec University of Montreal, C.P.8888, succ. Centre Ville, Montreal, QB H3C 3P8, CANADA; Ya. Goltser, Department of mathematics and Computer Science, Ariel University Center of Samaria, Ariel, ISRAEL; I. Gyori, Department of Mathematics and Computing University of Veszprem P.O. Box 158, 8201 Veszprem, HUNGARY; Y. Kannai, Department of Theoretical Mathematics,The Weizmann Institute of Science, Rehovot 76100, ISRAEL; V. Katsnel'son, Department of Theoretical Mathematics,T he Weizmann Institute of Science, Rehovot 76100, ISRAEL; A. Kartsatos, Department of Mathematics, University of South Florida, PHY 114, Tampa, FL 33620-5700, USA; I Kiguradze, Razmadze Institute of Mathematics, Z.Rukhadze Str. 1, Tbilisi 380093, GEORGIA; N. Krasovskii, Institute of Math. and Mech., Ural Branch of RAS, 16' S.Kovalevskoi Str., Ekaterinburg GSP-384, 620219, RUSSIA; G. Ladas, Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA; V. Lakshmikantham, Department of Appl. Mathematics, Florida Institute of Technology, 150 W.University Blvd, Melbourne, FL 32901, USA; A. Lambert, Department of Mathematics, University of North Carolina in Charlotte, Charlotte, N.C. 28223, USA; V. Maksimov, Department of Economics, Perm State University, Bukirev Str. 15, Perm 614600, RUSSIA; J. Mallet-Paret, Dept. of Applied Mathematic, Brown University, 182 George Street, Providence, RI 02912-900, USA; V. Maz'ya, Department of Mathematics, Linkoping University, S-581 83, Linkoping, SWEDEN; E. Merzbach, Department of Mathematics and CS, Bar-Ilan University, Ramat-Gan 52900, ISRAEL; R. Nussbaum, Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA; A. Ponosov, Institutt for Matematiske Fag, Postboks 5035, N-1432 As-NLH, NORWAY; B. Shklyar - TeX Specialist, Department of Science, Holon Academic Institute of Technology, 52 Golomb St, P.O.B. 305,Holon 58102, ISRAEL; A. Skubachevskii, Department of Differential Equations and Mathematical Physics,Peoples' Friendship University of Russia Miklukho-Maklaya str., 6, Moscow, GSP, Russia, 117198; P. Sobolevskii, Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram Campus, Jerusalem 91904, ISRAEL; M. Sonis, Faculty of Social Sciences, Bar-Ilan University, Ramat-Gan 52900, ISRAEL; S. Verduyn-Lunel, Fac. Wiskunde and Informatica, University of Amsterdam, Plantage Muidergracht 24, Amsterdam, NETHERLANDS; H-O. Walther, Mathematical Institute, Universitat Giessen, Arndstr.2, D 35392 Giessen, GERMANY; J. Wu, Department of Mathematics and Statistics York University Toronto, Ontario M3J 1P3, CANADA Y. Yomdin, Department of Theoretical Mathematics, The Weizmann Institute of Science, 76100 Rehovot, ISRAEL; P. Zecca, Dip. di Energetica "S. Stecco", Universit di Firenze, v. S. Marta 3, 50139 Firenze, ITALY

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FUNCTIONAL

DIFFERENTIAL

EQUATIONS

Volume 19, 2012

No. 1-2

In Memory of Professor A.D. Myshkis

Ariel University Center of Samariya

Ariel University Center of Samariya

© All Right Reserved, 2012 Printed in Israel ISSN 0793-1786

TABLE OF CONTENTS

N. Kopachevskii, A. Skubachevskii, L. Rosovskii. Anatoly Dmitrievich Myshkis 3

D. Batenkov, N. Sarig, Y. Yomdin. An “algebraic” reconstruction of 13piecewise-smooth functions from integral measurements

L. Berezansky, E. Braverman. Stability and linearization 31for differential equations with a distributed delay

A. Chadov, V. Maksimov. Linear boundary 49value problems and control problems for a class offunctional differential equations with continuous and discrete times

C. Corduneanu. APr-almost periodic solutions to 63functional differential equations with deviated argument

G. Derfel, B. van Brunt, G. Wake. A cell growth model revisited 75

B. Dhage. Basic results in the theory of hybrid differential equations 87with mixed perturbations of the second type

A. Filinovskiy. Solutions of hyperbolic equation with 107growing coefficient in unbounded domains

Chengjun Guo, Donal O’Regan, Yuantong Xu, Ravi P. Agarwal. 121Existence of infinite periodic solutions for a class of first-orderfirst-order delay differential equations

N. Karazeeva. Correct solvability of integro-differential equations 131in classes of generalized solutions of the boundary-value problemfor differential-difference equations with incommensurable shifts

Tongxing Li, Ethiraju Thandapani. Oscillation theorems for odd-order 147neutral differential equations

1

D. Neverova. Smoothness of generalized solutions of the boundary-value problem 157for differential-difference equations with incommensurable shifts

G. Osharovich, M. Ayzenberg-Stepanenko. On resonant waves in lattices 169

S. Panigrahi, R. Basu. Oscillation results for fourth order 195non-linear mixed neutral differential equations with quasi-derivatives

V. Vlasov, N. Rautian. Correct solvability of integro-differential equations 213arising in the theory of heat transfer and acoustics

2

FUNCTIONALDIFFERENTIALEQUATIONS

VOLUME 19

2012, NO 1–2

PP. 3–11

ANATOLY DMITRIEVICH MYSHKIS(ON THE 90-TH ANNIVERSARY )

N.D. KOPACHEVSKY∗, L.E. ROSOVSKII† AND

A.L. SKUBACHEVSKII‡

Childhood and school years. Anatoly Dmitrievich Myshkis was bothan outstanding mathematician theorist and an applied mathematician as wellas a wonderful teacher.

A.D. Myshkis was born on the 13th of April 1920 in the town of Spassk ofRyazan Region. When he was a year and a half, the family moved to Kharkivand had lived there until the fall of 1932. His father Dmitrii SemyonovichErmakov, wounded in the First World War, was a worker at the Putilov plant.Anatoly Dmitrievich’s mother, Haya Samuilovna Myshkis, knew the Germanand the French languages and taught at the Communist International School.

In 1932, the father of Anatoly Dmitrievich was transfered to another jobin Moscow. Thereby A.D. began studying in the sixth form at the forestboarding school in Sokol’niki. He was then admited to the seventh form ofexemplary school number 25 of Moscow in September 1933. The childrenof Stalin, Molotov, Kuybyshev and some other well-known political figureswent to that school at different times.

Since September 1935, among the classmates of A.D. were L. Ovsyan-nikov, who became an academician afterwards, and T. Schneider. Three ofthem were much higher in mathematics than the rest of the class. Opening ofthe mathematical circle for schoolchildren at the mechanical-mathematicalfaculty (mechmath) of the Moscow State University (MSU) became an ex-tremely important event in the life of the young men. The first lesson at the

∗ Department of Mathematical Analysis, Taurida National V.I. Vernadsky University,4 Academician Vernadsky Ave., Simferopol, Ukraine

† Department of Differential Equations and Mathematical Physics, Peoples’ FriendshipUniversity of Russia, 6 Mikhlukho-Maklaya Str., Moscow, Russia

‡ Department of Differential Equations and Mathematical Physics, Peoples’ FriendshipUniversity of Russia, 6 Mikhlukho-Maklaya Str., Moscow, Russia

3

4 N.D. KOPACHEVSKY, A.L. SKUBACHEVSKII, L.E. ROSOVSKII

mathematical circle, where A.D. came with his friends, was conducted byI.M. Gel’fand, who was also very young. From that moment, I.M. Gelfandhad played the role of preceptor for A.D.

A.D. familiarized himself with the notions of higher mathematics alreadyin the eighth form. So he studied the last years at school without makingbig efforts.

Studying at the mechmath of the MSU. After finishing school withexcellent marks all of the three classmates were admitted to the mechmathwithout examinations. Here again A.D. met I.M. Gel’fand among the otheroutstanding mathematicians and teachers. I.M. Gel’fand conducted practi-cals on mathematical analysis and A.D. received a special care from him upto the fourth year at the university. It should be noted that A.D. regarded thefollowing prominent scientists as his teachers: I.M. Gel’fand, I.G. Petrovskii,and physicist theorist Ya.B. Zel’dovich.

A.D.’s lecturer on differential equations on the second year at the univer-sity was I.G. Petrovskii. During the third year S.L. Sobolev lectured on math-ematical physics, while the practical training was conducted by I.G. Petro-vskii. In the sequel, A.D. attended the lectures on complex analysis byI.I. Privalov, the lectures on probability theory by A.N. Kolmogorov, andthe lectures on functional analysis by I.M. Gel’fand. A.D. however preferedreading textbooks and monographs devoting all of his spare time to math-ematics. Besides that, A.D. practised music and spent about three hours aday on the exercises on the violin.

According to A.D., he read the manuscript of Petrovskii’s lectures ondifferential equations at that time and, having found several inaccuracies,told I.G. Petrovskii his reasons, with which I.G. immediately agreed. As thecourse of lectures came later off the press, the author acknowledged A.D. inthe preface. This kind of attitude had a strong effect on the young student.

The War period. The war began when A.D. finished his fourth year atthe MSU. All of the students of the mechmath were thrown to the vicinitiesof Moscow for digging of trenches and anti-tank ditches. Those who finishedthe fourth year were returned to Moscow in the beginning of September 1941,as the front came close to the city. They were then sent to Sverdlovsk tostudy at the N.E. Zhukovsky Military-Air Engineering Academy (MAEA).

A.D. Myshkis succeeded to take an external degree at the MSU, and thethree-year study at the aircraft armament faculty of the MAEA began. Thoseyears were very difficult. Intensive studies combined with the cold, hunger,harassing physical training and drill resulted in tiredness and torpor.

A.D. met I.M. Gel’fand again in September 1942 in Sverdlovsk where the

ANATOLY DMITRIEVICH MYSHKIS 5

mechmath of the MSU was moved to from Ashkhabad. Following Gel’fand’sadvice, he decided to enter the postgaduate study at the mechmath underthe supervision of I.G. Petrovskii. By that time A.D. graduated from theMAEA with honours. The title of his thesis at the department of air shootingwas ”‘Riffle and cannon armament of the multipurpose aircraft TU–2”’. Inthis work, A.D. suggested a plan of deployment of the weapons basing on thecriterion of the maximal probability of defeat of the attacking enemy. Thiscriterion would be widely adopted afterwards.

A.D. stayed at the MAEA where he had worked as a lecturer at thedepartment of higher mathematics for three years. In this period he commu-nicated closely with academics from engineering departments, acquiring thecorresponding, ”‘applied”’, way of thinking.

Teaching at the mechmath of the MSU, living in Riga andMinsk. A year after graduating from the MAEA A.D. began to work parttime at the department of differential equations of the mechmath. A num-ber of well-known mathematicians such as V.V. Stepanov, who was the headof the department, V.V. Nemytsky, I.G. Petrovskii, and S.L. Sobolev alsoworked there. A.D. was responsible for practical training on ordinary dif-ferential equations and mathematical physics. Among his students wereO.A. Ladyzhenskaya and O.A. Oleinik, subsequently outstanding women-mathematicians and academicians of the Soviet Academy of Sciences. Thiswas the time when A.D. published his first scientific papers. Interestingly,the areas were very various: there were papers on ordinary differential equa-tions, partial differential equations, stability theory etc. The candidate the-sis written by A.D. under the supervision of I.G. Petrovskii and defended inJune 1946 was devoted to the so-called modifyed Dirichlet problem for theLaplace equation in a general n-dimensional domain. During this period,A.D. published totally 10 papers and was awarded the prize of the MoscowMathematical Society for his article in the journal ”‘Soviet MathematicalSurveys.”’

Since then, the mechmath seminars on ordinary and partial differentialequations had played a very significant role in the life of A.D. These semi-nars were leaded by such mathematicians as V.V. Stepanov, I.G. Petrovskii,A.N. Tikhonov, I.N. Vekua, L.A. Lyusternik, and later by O.A. Ladyzhen-skaya, O.A. Oleinik, M.I. Vishik etc. A.D. recalled that the famous Interna-tional Conference ”‘Petrovskii Seminar”’ had originated apparently from theseminar on partial differential equations. A.D. mentions this period as oneof the most vivid in his life.

Heavy teaching load at the MAEA forced A.D. to think about changinghis place of work. Thus he accepted the proposal of the Air Forces Personnel


Department to take an academic position at the Aviation Engineering Mili-tary School in Riga in June 1947. Acting at the School as a director of thedepartment of higher mathematics, A.D. also worked part time at the Lat-vian State University where he organized an educational scientific seminarfollowed the pattern of the MSU.

Working on an applied problem, A.D. familiarized himself with a dif-ferential equation containing the retarded argument. It became clear laterthat equations of this type emerged in control theory, biology, economics,medicine, etc., while had not been systematically studied before. The opencountry appeared in front of A.D., and the results, as he wrote, fell thick andfast.

The first article by A.D. devoted to delay differential equations was pub-lished in ”‘Soviet Mathematical Surveys”’ in 1949. In December 1949, A.D.finished his doctoral thesis on this topic, making a scientific breakthrough.The corresponding monograph [1] was released in 1951. It was the firstmonograph in the field of functional differential equations to appear in themathematical literature.

Later on, the theory of functional differential equations would be furtherdeveloped in the works of many noted mathematicians such as N.N. Krasovskii,Yu.S. Osipov, Yu.A. Mitropolskii, L.E. Elsgolts, R. Bellman, K. Cooke,J. Hale, T. Kato and others. Hundreds of papers would have been pub-lished, and international conferences would be organized in this field everyyear.

Since February 1950, A.D. had worked full time at the Latvian StateUniversity (LSU) as a head of the department of general mathematics. Here,in particular, A.D. began studying the hyperbolic systems of one spatialvariable. His educational scientific seminar served as an important form ofcommunication with students and academic personnel. There appeared thefirst issues of the Proceedings of the fizmat of the LSU. For students, A.D.wrote the manual ”‘How to get ready to research”’ and organized a competi-tion in solving ε− δ problems from the beginning of mathematical analysis.The brochure based on this material became subsequently a big success notonly in Riga but even in the USA. The flat of A.D. in Riga was open toeveryone for tutorials and scientific conversations. A.D. gradually becamewell-known in Latvia. However, the situation was overshadowed by a con-flict between A.D. and the faculty administration. A.D. put the research atthe departments in the first place while the Dean of the faculty consideredpublic work of paramount importance. Therefore the administration ham-pered to nominate A.D. for professorship. There was also the family reason,as A.D.’s second son, Mitya, suffered from bronchial asthma and pneumonia


in the dewy climate of Riga.Thus in autumn 1953 A.D. changed his job in the LSU to a position in the

Byelorussian State University (BSU). But rooted in Riga A.D. had continuedhis collaboration with Lettish mathematicians long since then. During hisRiga period A.D. published about 30 papers in the fields of boundary valueproblems in domains with complex boundaries, delay differential equations,potential theory, mixed problems for linear systems of partial differentialequations, fixed points theorems for multivalued mappings etc.

A.D. worked in Minsk as a head of the department of differential equa-tions of the BSU. He was the only doctor of science in mathematics in Minsk.A.D. described Byelorussians as balanced, open-hearted, and well-wishingpeople favourable for any kind of collaboration.

The educational and research seminar for students and postgraduatesbegan its work in the lent term of 1953/54 under the direction of A.D.,who supervised 8 postgraduate students from Riga and Minsk at that time.His own scientific interests were then connected with the investigation ofmixed problems for hyperbolic systems of one spatial variable, the studyof stationary points of a general autonomous system with switching on theplane etc. A.D. published 13 papers in Minsk, he also translated and editedthe monograph ”‘Stability theory of differential equations”’ by R. Bellman.By the end of his Minsk period, A.D. met such outstanding mathematiciansas M.A. Krasnosel’skii, S.G. Krein, N.I. Ahiezer, and others. Later, theseconnections would play an important role in the life of A.D.

Unfortunately, despite of the good relations with students and aca-demics from the BSU, A.D. had to leave Minsk as his family could not geta proper flat there. N.I. Ahiezer knew that the Kharkiv Aviation Institute(KhAI) had a good vacant flat that could have been immediately movedinto. So N.I. Ahiezer helped A.D. to move to Kharkiv. A.D. had a clearview of the high scientific level in Kharkiv where N.I. Ahiezer worked withthe other distinguished mathematicians B.Ya. Levin, V.A. Marchenko, andA.V. Pogorelov. Finally, Kharkiv was the home town for A.D. since A.D.spent his childhood there.

Therefore in September 1956 the second part of the life in Kharkivstarted for A.D. after 24-year break. This period would last for 18 years.

The Kharkiv period. The family of A.D. moved to Kharkiv in au-tumn 1956, where A.D. headed the department of higher mathematics atthe Kharkiv Aviation Institute. Within the next few years he had wrotehis famous textbook on higher mathematics for technical colleges [2], whichA.D. considered as one of his major contributions. The textbook was basedon the proposition that the statement of higher mathematics for technical


colleges should be focused on applications. A.D. thought it was necessary toreduce the gap between the mathematics taught and the mathematics ap-plied. These ideas were later developed in his lecture course [3], books [4, 5],and the collaboration with academician Ya.B. Zel’dovich.

Formation of ”‘Physical-technical institute for low temperatures”’ of theUkrainian Academy of Sciences (PTILT) in Kharkiv gave rise to a new areaof research. A.D. had to organize and head the department of applied math-ematics at the PTILT. Graduates from KhAI, the engineers with a math-ematical bias, constituted the core of this department. Among them weresuch pupils of A.D. as V.G. Babskii, N.D. Kopachevskii, L.A. Slobozhanin,and A.D. Tyuptsov. All of them would later become well-known scientists.

Shortly after its foundation, the department began working on the fun-damental problem of fluid oscillation in low gravity fields where the capillaryforces and self-gravitation had to be taken into account. These investigationsmet the needs of the famous organization ”‘Energiya”’ led by S.P. Korolyovand interested in behaviour of liquid fuel in a rocket tank under the con-ditions close to zero-gravity. Soon enough, the following three directionsof research became clear: problems of hydrostatics (nonlinear problems ofequilibrium surfaces of capillary liquid, the stability of equilibrium states,their bifurcation, and stability margin); small oscillations of fluid approach-ing equilibrium with a free surface crooked by capillary forces; convectionof weightless fluid caused by self-gravitation and the thermo-capillary effect.At that time the department of A.D. collaborated actively with the scien-tific schools of S.G. Krein in Voronezh and V.I Yudovich in Rostov-on-Don,which was reflected in wide application of functional analysis to the prob-lems of fluid mechanics. The study of these problems and related topics tookquite a long time and resulted in the monograph on the low-gravity fluidmechanics [6], the first of its kind to appear in the mathematical literature.The monograph was written by A.D. in co-authorship with his four pupils.Its revised version was published in English in 1987 [7] and in Russian in1990.

According to N.D. Kopachevskii, the supervision of A.D. was unobtrusiveand very delicate, and created favourable conditions for fast scientific growthof his pupils. N.D. Kopachevskii recalls also that A.D. considered his work inthe department of applied mathematics of FTILT as the most efficient andinteresting in his scientific career. The cultural life in the department wasvery rich too. A.D. used to take the group to hiking tours. Some of thosetours, in Crimean mountains and in Ala Tau, would always remain in thememory of his pupils.

Being a multi-aspect researcher A.D. did not restrict himself to studying


low-gravity fluid mechanics only. Parallel to this subject he had continuedhis work on the mathematical problems close to his mind. In particular,he published two surveys on the then current state of the theory of func-tional differential equations and had studied the mixed problem for semilin-ear systems of equations of hyperbolic type. There were also a few parers onimpulse systems excited at prescribed times. Several articles were devotedto methodological questions concerning the essence of applied mathematics.A.D. translated and edited the interesting course on ordinary differentialequations by F. Trikomi. In addition, the book ”‘Elements of Applied Math-ematics”’ in co-authorship with Ya.B. Zel’dovich came off the press in 1965(the second edition of the book was already released in 1967). In collabora-tion with the collegues from Voronezh (Yu.G. Borisovich and pupils) A.D.began studying multivalued mappings and differential inclusions.

Those years A.D. took an active part in the Committee on mathe-matical education under the Academy of Sciences of the USSR and in theMethodological Council in mathematics under the Ministry of Education ofthe USSR. Together with Ya.B. Zel’dovich, A.D. spoke in the press aboutthe need of modernization of the curriculum in mathematics for secondaryschools.

Working at the MIIT. In September 1974, A.D. took the position ofa professor at the Moscow Institute of Transport Engineers (MIIT). He wasfinishing the book ”‘Low–Gravity Fluid Mechanics”’ and writing papers onfunctional differential equations at that time.

Working at the department of applied mathematics of MIIT, A.D. or-ganized a permanent seminar on differential equations and related topics.He participated actively in scientific conferences, both all-USSR and inter-national. After the English edition (essentially revised and supplemented) of”‘Low–Gravity Fluid Mechanics”’ was out in 1987, A.D. switched to anothersubject and wrote ”‘Elements of the theory of mathematical models”’ by1991.

During the period from 1974 to 1991, A.D. published totally 93 scientificpapers and was the author of seven books. The following topics lied in therange of his interests at that time: mechanics of capillary fluid, functionaldifferential equations, variational and boundary value problems for ellipticpartial differential equations, multivalued mappings, asymptotic and oscilla-tion properties of operator differential equations, Volterra integral equationsin a metric space with measure, extremum conditions in spectral isoperi-metric problems with variable boundaries, the problem of rolling of a solidalong two tracks (the last one arose in MIIT), the phenomena of stabilizationand destabilization under small dissipative forces in nonconservative systems,


new solution properties in the problem of transverse vibration of a threadwith beads, and others.

After 1991, A.D. periodically went on academic trips to the USA, Israel,Brasil etc., where he met his foreign colleagues. He had continued to workintensively in different directions as before.

The International conference on nonlinear analysis and functional dif-ferential equations ”ADM-2000” dedicated to the 80-th anniversary of A.D.Myshkis took place in Voronezh in May 2000. From 2002 through 2008A.D. visited regularly the famous Crimean Autumn Mathematical School-Symposium organized in the village of Laspi by N.D. Kopachevskii and thedepartment of mathematical analysis of the Taurida National University.There A.D. lectured mathematics, debated methodological questions, andwalked a lot in the environs. He was in a strong mathematical shape andenjoyed meeting his collegues and pupils very much.

During his last years A.D. published more then 80 papers including thefundamental one on the mixed functional differential equations [8].

Summarize the activities of the eminent scientist and educator, a verykind and interesting man, A.D. Myshkis. He may be regarded by right asthe founder of a number of scientific schools. He was the official supervisorfor 36 candidates of sciences. Seven of them subsequently gained doctoraldegrees. A.D. was the author and co-author of 17 books run into 43 editionsin 10 languages and 332 scientific articles, translated and edited 16 books.

Unfortunately, A.D., as well as his outstanding friends M.A. Krasnosel-skii and S.G. Krein, had not become a member of the Academy of Sciences ofthe USSR (Russian Academy of Sciences) yet undoubtedly deserved it. Theteacher of A.D. Myshkis, academician I.G. Petrovskii, regarded A.D. as oneof his best pupils.

REFERENCES

[1] Myshkis A.D. Linear differential equations with retarded argument. - Gostehizdat,1951 [in Russian]; German transl.: Veb. Deutsch. Verl. Der Wiss., 1955.

[2] Myshkis A.D. Lectures on higher mathematics. — M.: Nauka, 1964 [in Russian];English transl.: Mir Publishing House, 1972.

[3] Myshkis A.D. Mathematics for students of higher technical institutions: specialcourses. — M.: Nauka, 1971 [in Russian]; English transl.: Mir Publishing House,1975.

[4] Zel’dovich Ya.B., Myshkis A.D. Elements of applied mathematics. — M.: Nauka,1965 [in Russian]; English transl.: Mir Publishing House, 1976.

[5] Blekhman I.I, Myshkis A.D., Panovko Ya.G. Applied mathematics: object, logic, anddetails of approaches. — Kiev: Naukova Dumka, 1976 [in Russian]; Germantransl.: Veb. Deutsch. Verl. Der Wiss., 1984.


[6] Babskii V.G., Kopachevskii N.D., Myshkis A.D., Slobozhanin L.A., Tyuptsov A.D.Hydromechanics in zero gravity. — M.: Nauka, 1976 [in Russian].

[7] A. D. Myshkis, V.G. Babskii, N.D. Kopachevskii, L.A. Slobozhanin, A.D. TyuptsovLow–Gravity Fluid Mechanics. Mathematical theory of capillary phenomena. —Berlin: Springer-Verlag, 1987.

[8] A. D. Myshkis. Mixed functional differential equations // Contemporary Mathematics.Fundamental Directions. — M.: MAI Press, 4(2003), p. 5–120; English transl.:Journal of Mathematical Sciences, 129:5(2005), p. 4111–4226.


VOLUME 19

2012, NO 1–2

PP. 13–30

AN “ALGEBRAIC” RECONSTRUCTION OFPIECEWISE-SMOOTH FUNCTIONS FROM INTEGRAL

MEASUREMENTS ∗

D. BATENKOV † , N. SARIG† AND Y. YOMDIN†

Abstract. This paper presents some results on a well-known problem in AlgebraicSignal Sampling and in other areas of applied mathematics: reconstruction of piecewise-smooth functions from their integral measurements (like moments, Fourier coefficients,Radon transform, etc.). Our results concern reconstruction (from the moments or Fouriercoefficients) of signals in two specific classes: linear combinations of shifts of a given func-tion, and “piecewise D-finite functions” which satisfy on each continuity interval a lineardifferential equation with polynomial coefficients. In each case the problem is reduced toa solution of a certain type of non-linear algebraic system of equations (“Prony-type sys-tem”). We recall some known methods for explicitly solving such systems in one variable,and provide extensions to some multi-dimensional cases. Finally, we investigate the localstability of solving the Prony-type systems.

Key Words. nonlinear moment inversion, Algebraic Sampling, Prony system

AMS(MOS) subject classification. 94A12, 94A20

1. Introduction. It is well known that the error in the best approx-imation of a Ck-function f by an N -th degree Fourier polynomial is of orderCNk . The same holds for algebraic polynomial approximation and for otherbasic approximation tools (see e.g. [21, Chapters IV, VI], and [32, Vol.I,Chapter 3, Theorem 13.6]). However, for f with singularities, in particular,with discontinuities, the error is much larger: its order is C√

N. Considering

the so-called Kolmogorow N -width of families of signals with moving dis-continuities one can show that any linear approximation method provides thesame order of error, if we do not fix a priori the discontinuities’ position

∗ Supported by ISF and the Minerva Foundation† Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

13

14 D. BATENKOV, N. SARIG AND Y. YOMDIN

(see [11, Theorem 2.10]). Another manifestation of the same problem is the“Gibbs effect” - a relatively strong oscillation of the approximating functionnear the discontinuities. Practically important signals usually do have dis-continuities, so the above feature of linear representation methods presentsa serious problem in signal reconstruction. In particular, it visibly appearsnear the edges of images compressed by JPEG, as well as in the noise andlow resolution of the CT and MRI images.

Recent non-linear reconstruction methods, in particular, CompressedSensing ([6, 7]) and Algebraic Sampling ([8, 10, 15, 19, 23]), address thisproblem in many cases. Both approaches appeal to an a priori informationon the character of the signals to be reconstructed, assuming their “sim-plicity” in one or another sense. Compressed sensing assumes only a sparserepresentation in a certain (wavelets) basis, and thus it presents a rathergeneral and “universal” approach. Algebraic Sampling usually requires morespecific a priori assumptions on the structure of the signals, but it promises abetter reconstruction accuracy. In fact, we believe that ultimately the Alge-braic Sampling approach has a potential to reconstruct “simple signals withsingularities” as good as smooth ones. The most difficult problem in thisapproach seems to be estimating the accuracy of solution of the nonlinearsystems arising.

Our purpose in this paper is to further substantiate the Algebraic Sam-pling approach. On one hand, we present two algebraic reconstruction meth-ods for generic classes of signals. The first one is reconstruction of combina-tions of shifts of a given function from the moments and Fourier coefficients(Section 2). The second one concerns piecewise D-finite moment inversion(Section 4). On the other hand, we consider some typical nonlinear systemsarising in these reconstruction schemes. We describe the methods of theirsolution (Section 3), and provide some explicit bounds on their local stabil-ity (Section 5). We also present results of some numerical experiments inSection 6.

Our ultimate goal may be stated in terms of the following conjecture(which seems to be supported also by the results of [9, 14, 18, 23, 30]):

There is a non-linear algebraic procedure reconstructing any signal in aclass of piecewise Ck-functions (of one or several variables) from its first NFourier coefficients, with the overall accuracy of order C

Nk . This includes thediscontinuities’ positions, as well as the smooth pieces over the continuitydomains.

Recently [4] we have shown that at least half the conjectured accuracycan be achieved. However, the question of maximal possible accuracy remainsopen. Our results presented in this paper can be considered as an additional

ALGEBR.RECONSTR. OF PIECEWISE-SMOOTH FUNCTIONS 15

step in this direction.

2. Linear combinations of shifts of a given function. Recon-struction of this class of signals from sampling has been described in [8, 19].We study a rather similar problem of reconstruction from the moments. Ourmethod is based on the following approach: we construct convolution kernelsdual to the monomials. Applying these kernels, we get a Prony-type systemof equations on the shifts and amplitudes.

Let us restate a general reconstruction problem, as it appears in ourspecific setting. We want to reconstruct signals of the form

(1) F (x) =N∑j=1

ajf(x− xj)

where f is a known function of x = (x1, . . . , xd) ∈ Rd, while the number N ofthe shifts and the form (1) of the signal are known a priori. The parametersaj, x

j = (xj1, . . . , xjd), j = 1, . . . N are to be found from a finite number of

“measurements”, i.e. of linear (usually integral) functionals like polynomialmoments, Fourier moments, shifted kernels, evaluation over some grid etc.

In this paper we consider only linear combinations of shifts of one knownfunction f . Reconstruction of shifts of several functions based on “decou-pling” via sampling at zeroes of the Fourier transforms of the shifted functionsis presented in [29].

In what follows x = (x1, . . . , xd), t = (t1, . . . , td) ∈ Rd, j is a scalarindex, while k = (k1, . . . , kd), i = (i1, . . . , id) and n = (n1, . . . , nd) are multi-indices. Partial ordering of multi-indices is given by k ≤ k′ ⇔ kp ≤ k′p, p =1, . . . , d.

Assume that a multi-sequence of functions φ = φk(t), t ∈ Rd, k ≥(0, . . . , 0) is fixed. We consider the measurements µk(F ) provided by

(2) µk(F ) =

∫F (t)φk(t) d t, k ≥ (0, . . . , 0).

Our approach now works as follows: given f and φ = φk(t) we now tryto find an “f -convolution dual” system of functions ψ = ψk(t) in a form ofcertain “triangular” linear combinations

(3) ψk(t) =∑0≤i≤k

Ci,kφi(t), k ≥ (0, . . . , 0).


More accurately, we try to find the coefficients Ci,k in (3) in such a way that

(4)

∫f(t− x)ψk(t) d t = φk(x).

We shall call a sequence ψ = ψk(t) satisfying (3), (4) f - convolution dualto φ. Below we shall find explicitly convolution dual systems to the usualand exponential monomials.

We consider a general problem of finding convolution dual sequences toa given sequence of measurements as an important step in the reconstructionproblem. Notice that it can be generalized by dropping the requirement ofa specific representation (3): ψk(t) =

∑ki=0Ci,kφi(t). Instead we can require

only that∫f(t)ψk(t) d t be expressible in terms of the measurements sequence

µk. Also φk in (4) can be replaced by another a priori chosen sequence ηk.This problem leads, in particular, to certain functional equations, satisfied bypolynomials and exponential functions, as well as exponential polynomialsand some kinds of elliptic functions (see [28]).

Now we have the following result:Theorem 1. Let a sequence ψ = ψk(t) be f -convolution dual to φ. DefineMk by Mk =

∑0≤i≤k Ci,kµi. Then the parameters aj and x

j in (1) satisfy thefollowing system of equations (“generalized Prony system”):

(5)N∑j=1

ajφk(xj) =Mk, k ≥ (0, . . . , 0).

Proof. By definition ofMk and via (3) and (4) we have for each k ≥ (0, . . . , 0)

Mk =∑0≤i≤k

Ci,kµi =

∫F (t)

∑0≤i≤k

Ci,kφi(t) d t =

=

∫F (t)ψk(t) d t =

N∑j=1

aj

∫f(t− xj)ψk(t) d t =

N∑j=1

ajφk(xj).

This completes the proof.

According to Theorem 1, in order to reduce the reconstruction problemwith the measurements (2) and for signals of the a priori known form (1) toa solution of the generalized Prony system we have to find an f - convolutiondual system ψ = ψk(t) to the measurements kernels φ. In fact we needonly the coefficients Ci,k. Having these coefficients, we compute Mk and getsystem (5).

Solvability of system (5) and robustness of its solution depend on themeasurements kernels φ. Specific examples are considered below.


2.1. Reconstruction from moments. We are given a finite numberof moments of a signal F (x) =

∑Nj=1 ajf(x− xj) as in (1) in the form

(6) mk =

∫F (t)tk d t, k = (k1, . . . , kd) ≥ (0, . . . , 0).

So here φk(x) = xk11 · · · xkdd for each multi-index k. We look for the dualfunctions ψk satisfying the convolution equation

(7)

∫f(t− x)ψk(t) d t = xk

for each multi-index k. To solve this equation we apply Fourier transform toboth sides of (7). Assuming that f(0) = 0 and that f(ω) has the derivativesof all the orders at 0 we find (see [28]) that there is a unique solution to (7)provided by

(8) φk(x) =∑l≤k

Ck,lxl,

where

Ck,l =1

(√2π)d

(k

l

)(−i)k+l

[∂k−l

∂ωk−l

∣∣∣∣ω=0

1

f(ω)

].

The assumption f(0) = 0 is essential in the construction of the f -convolutiondual system for the monomials as well as for other measurement kernels(as well as in the study of the shifts of a given function in general). Theabove calculation can be applied, with proper modifications, in more generalsituations (see [28]). On the other hand, the assumption of differentiabilityof f at zero is not very restrictive, in particular, if we work with signals withfinite support.

Returning to the moments reconstruction, we set the generalized poly-nomial moments Mk as

(9) Mk =∑l≤k

Ck,lml

and obtain, as in Theorem 1, the following system of equations:

(10)N∑j=1

aj(xj)k =

N∑j=1

aj(xj1)

k1 · · · (xjd)kd =Mk, k ≥ (0, . . . , 0).

This system is called “multidimensional Prony system”. It appears in nu-merous problems of theoretical and applied mathematics. In Section 3 below


we recall one of the classical methods for its solution in one-dimensional case,and describe, under certain assumptions, a method for its solution in severaldimensions. Local stability of the solutions of the one-dimensional Pronysystem is discussed in Section 5.

2.2. Fourier case. The signal has the same form as in section 2.1:F (x) =

∑Nj=1 ajf(x−xj). The measurement kernels φ we now choose as the

harmonics φk(x) = eikx. So our measurements are the Fourier coefficientsck(F ) = f(k) =

∫F (x)eikx d x.

Let us assume now that f satisfies the condition f(k) = 0 for all integer k.Then the Fourier harmonics turn out to form essentially f -self-dual system:indeed, taking ψk(x) =

1

f(k)eikx we get immediately that

∫f(t−x)ψk(t) d t =∫

f(t− x) 1

f(k)eikt d t = f(k)

f(k)eikx = φk(x).

According to our general scheme we put now Mk = 1

f(k)ck(F ) and get a

system of the form

(11)1

f(k)ck(F ) =Mk =

N∑j=1

ajeikxj

=N∑j=1

aj(ρj)k, k ≥ (0, . . . , 0),

where ρj = eixj. This is once more a multidimensional Prony system as in

(10), with the nodes on the complex unit circle.

2.3. Further extensions. The approach above can be extended inthe following directions:

1. Reconstruction of signals built from several functions or with the ad-dition of dilations also can be investigated (a perturbation approachwhere the dilations are approximately 1 is studied in [26]).

2. Further study of “convolution duality” in [26] provides a certainextension of the class of signals and measurements allowing for aclosed-form reconstruction.

3. Prony system.

3.1. One-dimensional case. Let us start with the classical case ofone-dimensional Prony system:

(12)N∑j=1

aj(xj)k = mk, x

j ∈ R, k = 0, 1, . . . .


This system appears in many branches of mathematics (see [20]). There areseveral solution methods available, for instance direct nonlinear minimization(see e.g. [9]), the polynomial realization (original Prony method, [24]) or thestate-space approach ([25]). Let us give a sketch of a method based on Padeapproximation techniques which is rather close to the original Prony method.To simplify the presentation we shall assume that we know a priori that allthe nodes xj are pairwise distinct. General case is treated similarly ([22], seealso [31]).

Consider a “moments generating function” I(z) =∑∞

k=0mkzk, z ∈ C.

Assuming the equations (12) are satisfied for all k = 0, 1, . . . and summingup the geometric progressions we get

(13) I(z) =N∑j=1

aj1− xjz

.

So I(z) is a rational function of degree N tending to zero as z → ∞. Theunknowns aj and

1xj in (12) are the poles and the residues of I(z), respectively.

In order to find I(z) explicitly from the first 2N moments m0, . . . ,m2N

we use the Pade approximation approach (see [22]): write I(z) as P (z)Q(z)

with

polynomials P (z) = A0+A1z+ · · ·+AN−1zN−1 and Q(z) = B0+B1z+ · · ·+

BNzN of degrees N − 1 and N , respectively.Multiplying by Q we have I(z)Q(z) = P (z). Now equating the coeffi-

cients on both sides we get the following system of linear equations:

m0B0 = A0

m0B1 +m1B0 = A1

.............................

m0BN−1 +m1BN−2 + · · ·+mN−1B0 = AN−1

m0BN +m1BN−1 + · · ·+mN−1B1 +mNB0 = 0

m1BN +m2BN−1 + · · ·+mNB1 +mN+1B0 = 0

..............................

The rest of the equations in this system are obtained by further shifts ofthe indices of the moments, and so they form a Hankel-type matrix.

Now, being a rational function of degree N , I(z) is uniquely defined byits first 2N Taylor coefficients (the difference of two such functions cannotvanish at zero with the order higher than 2N − 1). We conclude that thelinear system consisting of the first 2N equations as above is uniquely solvableup to a common scaling of P and Q (of course, this fact follows also form ageneral Pade approximation theory - see [22]).


Now a solution procedure for the Prony system can be described asfollows:

1. Solve a linear system of the first 2N equations as above (with the co-efficients - the known moments mk) to find the moments generating function

I(t) in the form I(z) = P (z)Q(z)

.

2. Represent I(z) in a standard way as the sum of elementary fractionsI(z) =

∑Nj=1

aj1−xjz

. (Equivalently, find poles and residues of I(z)). Besidesalgebraic operations, this requires just finding the roots of the polynomialQ(z). Then (aj, x

j), j = 1, . . . , N form the unique solution of the Pronysystem (12).

3.2. Multi-dimensional case. Let us recall our multi-dimensionalnotations: x = (x1, . . . , xd) ∈ Rd, j is a scalar index, while k = (k1, . . . , kd),i = (i1, . . . , id) and n = (n1, . . . , nd) are multi-indices. Partial ordering ofmulti-indices is given by k ≤ k′ ⇔ kp ≤ k′p, p = 1, . . . , d.

Let xj = (xj1, . . . , xjd), j = 1, . . . , N. With the above notation, the

multidimensional Prony system has exactly the same form as in the one-dimensional case:

(14)N∑j=1

aj(xj)k = mk, x

j ∈ Rd, k ≥ (0, . . . , 0).

Exactly as above we get that for z = (z1, . . . , zd) ∈ Cd the momentsgenerating function I(z) =

∑k∈Nd mkz

k is a rational function of degree Ndof the form

(15) I(z) =N∑j=1

aj

d∏l=1

1

1− xjl zl.

Representing I(z) as P (z)Q(z)

we get exactly in the same way as above an infinitesystem of linear equations for the coefficients of P and Q, with a Hankel-type matrix formed by the momentsmk. By the same consideration as above,after we take enough equations in this system the solution is unique up to arescaling (see [1, 22] and references therein).

However, from this point the multi-dimensional situation becomes essen-tially more complicated. While in dimension one I(z) can be, essentially, anyrational function of degree N (naturally represented as the sum of elemen-tary fractions), in several variables I(z) has a very special form given by (15).


This fact can be easily understood via counting the degrees of freedom: thenumber of unknowns in (14) is N(d + 1) while a rational function of degreeNd in d variables has much more coefficients than that, for d > 1. It wouldbe desirable to use as many equations from (14) as the number of unknowns,but the method outlined above ignores a specific structure of I(z) and re-quires as many equations as in a Pade reconstruction for a general rationalfunction of degree Nd.

We treat this problem in [28] analyzing the structure of singularitiesof I(z) and on this base proposing a robust reconstruction algorithm. Letus give here one simple special case of this algorithm, which separates thevariables in the problem.

3.2.1. Separation of variables in the multi-dimensional Pronysystem. Let us assume that we know a priori that the solution (aj, x

j), j =1, . . . , N of multi-dimensional Prony system (14) is such that all the coor-dinates xjl , l = 1, . . . , d of the points xj, j = 1, . . . N are pairwise distinct.Moreover, we assume that aj1 = aj2 for j1 = j2. Under these assumptions weproceed as follows:

Consider “partial moment generating functions” Im(t), t ∈ C, m =1, . . . , d, defined by

(16) Im(t) =∞∑r=1

mremtr,

where em is a multi-index with (em)j = 0 for m = j and 1 otherwise. Wehave the following simple fact:Proposition 2. Im(t) is a one-dimensional moments generating function ofthe form

(17) Im(t) =N∑j=1

aj1

1− xjmt.

It coincides with the restriction of I(z) to the m-th coordinate axis in Cd.

Proof. Let us evaluate I(z) along the m-th coordinate axis, that is on theline z = tem with em as above and t ∈ C. We get

I(tem) =N∑j=1

aj

d∏l=1

1

1− xjl t(em)l=

N∑j=1

aj1

1− xjmt


which is the moments generating function (17). Now, to express Im(t)through the multi-dimensional moments mk we notice that any monomialxk, with k = pem vanishes identically on the m-th coordinate axis. Hence

I(tem) =∞∑r=0

mremtr.

This shows that Im(t) ≡ I(tem) and completes the proof of the proposition.

Now applying the method described in Section (3.1) above we find foreach m = 1, . . . , d the coordinates x1m, . . . , x

Nm and (repeatedly) the coeffi-

cients a1, . . . , aN . It remains to arrange these coordinates into the pointsxj = (xj1, . . . , x

jd). This presents a certain combinatorial problem, since

Prony system (14) is invariant under permutations of the index j. Underthe assumptions above we proceed as follows: for each m = 1, . . . , d wehave obtained the (unordered) collection of the pairs (aj, x

jm), j = 1, . . . , N .

By assumptions aj1 = aj2 for j1 = j2. Hence we can arrange in a uniqueway all the pairs (aj, x

jm), j = 1, . . . , N, m = 1, . . . , d into the string

[(a1, x11), . . . , (a1, x

1d)], . . . , [(aN , x

N1 ), . . . , (aN , x

Nd )]. This gives us the desired

solution of multi-dimensional Prony system (14).

Notice that the assumption aj1 = aj2 for j1 = j2 is essential here. Indeed,for x1 = x2 and x1 = (x1, x2), x

2 = (x2, x1), x1 = (x1, x1), x

2 = (x2, x2)we have mk = (x1)k + (x2)k ≡ (x1)k + (x2)k for k on each of the coordinateaxes. So the (unique up to permutations of the index j) solution of the Pronysystem cannot be reconstructed from these moments only.

Another remark is that the separation of variables as described aboverequires knowledge of 2dN moments mk (2N on each of the coordinate axes).This is almost twice more than N(d + 1) unknowns. This number can besignificantly reduced in some cases. See [28] for further investigation in bothof these directions.

Stability estimates for the solution of one-dimensional Prony system(Section 5) can be easily extended to the case considered in the presentsubsection. We do not give here explicit statement of this result.

4. Reconstruction of piecewise D-finite functions from moments.In this section we present an overview of our previous findings on algebraicreconstruction of a certain general class of signals. See [3] for the completedetails.

Let g : [a, b] → R consist of N + 1 “pieces” g0, . . . gN with N ≥ 0 jumppoints

a = x0 < x1 . . . < xN < xN+1 = b.

Furthermore, let g satisfy on each continuity interval some linear homo-geneous differential equation with polynomial coefficients: D gn ≡ 0, n =0, . . . , N where

(18) D =r∑

j=0

( kj∑i=0

bi,jxi

)dj

dxj(bij ∈ R)

Each gn may be therefore written as a linear combination of functions uiri=1

which are a basis for the space ND = f : D f ≡ 0:

(19) gn(x) =r∑

i=1

αi,nui(x), n = 0, 1, . . . , N.

We term such functions g “piecewise D-finite”. Many “simple” functionsmay be represented in this framework, such as polynomials, trigonometricpolynomials and algebraic functions.

The sequence mk = mk(g) is given by the usual moments

mk(g) =

∫ b

a

xkg(x) d x.

Piecewise D-finite Reconstruction Problem. Given r, ki, N, a, band the moment sequence mk of a piecewise D-finite function g, reconstructall the parameters bi,j, xi, αi,n.

The above problem can be solved as follows.

1. It turns out that the moment sequence of every piecewise D-finitefunction g satisfies a linear recurrence relation, such that the co-efficients of D annihilating every piece of g and the discontinuitylocations xj can be recovered from the moments by solving a lin-ear systems of equations plus a nonlinear step of polynomial rootsfinding. As we shall explain below, in many cases this step is equiva-lent to solving a certain generalized form of the previously mentionedProny system (12).

2. The function g itself can be subsequently reconstructed by numeri-cally calculating the basis for the spaceND and solving an additionallinear system of equations to recover the coefficients αi,n.


The above algorithm has been tested on reconstruction of piecewise poly-nomials, piecewise sinusoids and rational functions - see [3] for details. Theresults reported there were relatively accurate for low noise levels. In thispaper we continue to explore the numerical stability of the method - seeSections 5 and 6.

Now let us show how the Prony system arises in the piecewise D-finitereconstruction method. Consider the distribution D g. It is of the form (see[3, Theorem 2.12])

(20) D g =N∑j=1

lj−1∑i=0

ai,jδ(i) (x− xj)

where xj ∈ R are the discontinuity locations, ai,j ∈ R are the associated“jump magnitudes” which depend on the values g(i)(x±j ), and δ(x) is theDirac δ-function. In particular, when g is just a piecewise constant function,we have D = d

dxand so

(21) D g =N∑j=1

ajδ(x− xj)

where aj = g(x+j ) − g(x−j ). Let us now apply the moment transform to theequations (21) and (20). We get, correspondingly,

mk (D g) =N∑j=1

ajxkj ;(22)

mk (D g) =N∑j=1

lj−1∑i=0

ai,jk(k − 1)× · · · × (k − i+ 1)xk−ij .(23)

The system (22) is of course identical to the previously considered (12).However, the following question arises: how are the numbers in the left-handside of (22) and (23) related to the known quantities mk (g)? It turns outthat the numbers mk (D g) are certain linear combinations of these knownmoments, with coefficients which are determined by D in a well-defined way.The conclusion is as follows: if the operator D which annihilates every pieceof g is known (but the other parameters are not1), then one can recover thediscontinuity locations of g by solving the Prony-like system (23). We term

1 This assumption is realistic, for instance when reconstructing piecewise-polynomialsor piecewise-sinusoids.

the system (23) “confluent Prony system”. It can be solved in a similarmanner to the standard Prony system. A unique solution exists wheneverall the xj’s are distinct and the highest coefficients alj−1,j are nonzero. Weprovide the details in [2].

5. Accuracy of solving Prony-like systems. In the precedingsections we have shown that several algebraic methods for nonlinear recon-struction can be reduced to solving certain types of systems of equations, themost basic one of which is the Prony system (12), (22). A crucial factor forthe eventual applicability of the reconstruction methods is the sensitivity ofsolving these systems to measurement errors.

In this section we consider two such systems - (22) and (23) and providesome theoretical results regarding the local sensitivity of their solution tonoise. These results will hopefully enable further “global” analysis.

We consider the following question: if the measurements in the left-handside of (22) are known with error at most ε, how well can one recover theunknowns aj, xj? We regard this stability problem to be absolutely centralin Algebraic Sampling. To our best knowledge, no general treatment of thisproblem exists, therefore we consider our results below to be a step in thisdirection.

For simplicity, let us assume that the number of equations in (22) equalsthe number of unknowns, which in this case is P = 2N . Let us furtherconsider the “measurement map” P : RP → RP given by (22) (we call it the“Prony map”):

P(aj, xj

) def= mkP−1

k=0

Then, one possible answer (however by no means a complete one) to thequestion posed above can be given in terms of the local Lipschitz constantof the “solution map” P−1, whenever this inverse is defined. We then obtainthe following result.Theorem 3. Let mkk=0,...,P−1 be the exact unperturbed moments of themodel (21). Assume that all the xj’s are distinct and also aj = 0 forj = 1, . . . , N . Now let mk be perturbations of the above moments such thatmaxk |mk − mk| < ε. Then, for sufficiently small ε, the perturbed Pronysystem has a unique solution which satisfies:

|xj − xj| ≤ C1ε|aj|−1

|aj − aj| ≤ C1ε

where C1 is an explicit constant depending only on the geometry of x1, . . . , xN .

Proof. Consider the Jacobian determinant of the map P . It is easily factor-ized as follows:

dP =

1 0 . . . 1 0x1 1 . . . xN 1...xP1 PxP−1

1 . . . xPN PxP−1N

︸︷︷︸

def= V (x1,...,xN )

× diagD1, . . . , DN

where Dj is a 2× 2 block

Djdef=

[1 00 aj

].

The matrix V = V (x1, . . . , xN) is a special case of the so-called confluentVandermonde matrix, well-known in numerical analysis ([5, 12, 13, 16]). Inparticular, the paper [12] contains the following estimate (Theorem 3) forthe norm of V −1:

∥V −1∥∞ ≤ max1≤i≤N

bi

N∏j=1,j =i

(1 + |xj||xi − xj|

)2

where

bidef= max

(1 + |xi|, 1 + 2(1 + |xi|)

∑j =i

1

|xj − xi|

).

Now since the xj’s are distinct and aj = 0, the Jacobian is non-singular andso in a sufficiently small neighborhood of the exact solution, the map P isapproximately linear. By the inverse function theorem

dP−1 = diagD−11 , . . . , D−1

N × V −1

and so taking C1def= ∥V −1∥∞ completes the proof.

By a similar technique with slightly more involved calculations we obtainthe following result for the system (23).Theorem 4. Let mkk=0,...,P−1 be the exact unperturbed moments of the

model (20), where P =∑N

j=1 lj + N . Assume that all the xj’s are distinctand also alj−1,j = 0 for j = 1, . . . , N . Now let mk be perturbations of theabove moments such that maxk |mk−mk| < ε. Then, for sufficiently small ε,


the perturbed confluent Prony system has a unique solution which satisfies:

|ai,j − ai,j| ≤

C2ε i = 0

C2ε

(1 +

|ai−1,j ||alj−1,j |

)1 ≤ i ≤ lj − 1

|xj − xj| ≤ C2ε1

|alj−1,j|

where C2 is a constant depending only on the nodes x1, . . . , xN and the mul-tiplicities l1, . . . , lN .

Proof outline. As before, we obtain a factorization of the Jacobian determi-nant dP as a product of a confluent Vandermonde matrix V (x1, l1, . . . , xN , lN)(defined in a similar manner but with each column having lj − 2 “confluen-cies”) and the block diagonal matrix D = diagD1, . . . , DN where

Dj =

1 0 · · · 00 1 · · · a0,j...

.... . .

...0 0 · · · alj−1,j

.

Inverting this expression and taking C2 = ∥V −1∥∞ completes the proof.

An important generalization would be to consider the mappings

Ps :(ai,j, xj

) def= mkP+s−1

k=s

and investigate the reconstruction error as s→ ∞. Such a formulation makessense in the Fourier reconstruction problem ([4, 9, 11]). This is a work inprogress and we plan to present the results separately ([2]).

6. Numerical experiments. We have tested the piecewise D-finitereconstruction method on a simple case of a piecewise-constant signal. Theimplementation details are identical to those used in [3, Appendix]. As canbe seen from Figure 1, the reconstruction is accurate even in the presence ofmedium-level noise.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Reconstructing piecewise polynomial signal, degree=0SNR=19.9763, MSE=0.00946708

Noisy signalReconstructed signalOriginalSignal

Fig. 1. Reconstructing a piecewise-constant signal with two jumps. The reconstructed andthe original are very close.

We have already mentioned that for a piecewise constant function f ,the distribution f ′ is of the form (21). Therefore, the piecewise D-finitereconstruction problem for f essentially reduces to solving the Prony system(22), and so the local estimates of Theorem 3 should apply in the case of avery small noise level. This prediction is partially confirmed by the results ofour second experiment, presented in Figure 2. In particular, it can be seenthat indeed |∆xj| ∼ |aj|−1, while the accuracy of other jump points |∆xi|does not depend on |aj| for j = i. While this is certainly an encouragingresult, more investigation is clearly needed in order to fully understand thedependence of the error on all the parameters of the problem. Such aninvestigation should, in our opinion, concentrate on the global structure ofthe mapping P .


10−7

10−6

10−5

10−4

10−3

10−2

10−6

10−5

10−4

10−3

10−2

10−1

100

|a0,1

|

Accuracy of reconstrcting a piecewise polynomial signal, degree=0K=2, SNR=80

|∆ x

1|

|∆ x2|

Fig. 2. Dependence of the reconstruction accuracy on the magnitude of the jump.

REFERENCES

[1] G. A. Baker, P. Graves-Morris, Pade Approximants. Cambridge U.P., 1996.[2] D.Batenkov, Y.Yomdin, On the accuracy of solving confluent Prony systems, sub-

mitted.[3] D.Batenkov, Moment inversion problem for piecewise D-finite functions, Inverse

Problems, 25(10):105001, October 2009.[4] Batenkov, D., Yomdin, Y., Algebraic Fourier reconstruction of piecewise smooth

functions, Mathematics of Computation, 81(2012), 277–318.[5] A. Bjorck and T. Elfving. Algorithms for confluent Vandermonde systems. Nu-

merische Mathematik, 21(2) (1973), 130–137.[6] E. J. Candes. Compressive sampling. Proceedings of the International Congress of

Mathematicians, Madrid, Spain, 2006, III(2006), 1433–1452, Eur. Math. Soc.,Zurich.

[7] D. Donoho, Compressed sensing. IEEE Trans. Inform. Theory, 52 (2006), 1289–1306.

[8] P.L. Dragotti, M. Vetterli and T. Blu, Sampling Moments and Reconstructing Signalsof Finite Rate of Innovation: Shannon Meets Strang-Fix, IEEE Transactions onSignal Processing, Vol. 55 (2007), Part 1, 1741-1757.

[9] K. Eckhoff, Accurate reconstructions of functions of finite regularity from truncatedFourier series expansions, Mathematics of Computation, 64 (1995), 671–690.

[10] M. Elad, P. Milanfar, G. H. Golub, Shape from moments—an estimation theoryperspective, IEEE Transactions on Signal Processing, 52 (2004),1814–1829.

[11] B. Ettinger, N. Sarig. Y. Yomdin, Linear versus non-linear acqusition of step-


functions, J. of Geom. Analysis, 18 (2008), 369-399.[12] W. Gautschi. On inverses of Vandermonde and confluent Vandermonde matrices.

Numerische Mathematik, 4(1)(1962), 117-123.[13] W. Gautschi. On inverses of Vandermonde and confluent Vandermonde matrices

III. Numerische Mathematik, 29(4) (1978), 445-450.[14] A. Gelb, E. Tadmor, Detection of edges in spectral data II. Nonlinear enhancement,

SIAM J. Numer. Anal., 38 (2000), 1389-1408.[15] B. Gustafsson, Ch. He, P. Milanfar, M. Putinar, Reconstructing planar domains

from their moments. Inverse Problems 16 (2000), 1053-1070.[16] D. Kalman. The generalized Vandermonde matrix. Mathematics Magazine, pages

15–21, 1984.[17] V. Kisunko, Cauchy type integrals and a D-moment problem. C.R. Math. Acad. Sci.

Soc. R. Can. 29 (2007), 115–122.[18] G. Kvernadze, T. Hagstrom, H. Shapiro, Locating discontinuities of a bounded

function by the partial sums of its Fourier series., J. Sci. Comput. 14 (1999),301–327.

[19] I. Maravic and M. Vetterli, Exact Sampling Results for Some Classes of Paramet-ric Non-Bandlimited 2-D Signals, IEEE Transactions on Signal Processing, 52(2004), 175-189,

[20] Y.I. Lyubich. The Sylvester-Ramanujan System of Equations and The ComplexPower Moment Problem. The Ramanujan Journal, 8(1) (2004), 23–45,

[21] Natanson, I., Constructive Function Theory (in Russian), Gostekhizdat, 1949.[22] E. M. Nikishin, V. N. Sorokin, Rational Approximations and Orthogonality, Trans-

lations of Mathematical Monographs, 92, AMS, 1991.[23] P. Prandoni, M. Vetterli, Approximation and compression of piecewise smooth func-

tions, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999),2573-2591.

[24] R. Prony, Essai experimental et analytique. J. Ec. Polytech.(Paris), 2:24–76, 1795.[25] B.D. Rao and K.S. Arun, Model-based processing of signals: a state-space approach.

Proceedings of the IEEE, 80(2) (1992), 283–309.[26] N. Sarig, Signal Acquisition from Measurements via Convolution Duals, preprint,

Weizmann Institute, Dec. 2010.[27] N. Sarig, Y. Yomdin, Signal Acquisition from Measurements via Non-Linear Models,

C. R. Math. Rep. Acad. Sci. Canada 29(4) (2007), 97-114.[28] N. Sarig and Y. Yomdin, Multi-dimensional Prony systems for uniform and non-

uniform sampling, in preparation.[29] N. Sarig and Y. Yomdin, Decoupling of a Reconstruction Problem for Shifts of

Several Signals via Non -Uniform Sampling, in preparation.[30] E. Tadmor, High resolution methods for time dependent problems with piecewise

smooth solutions. Proceedings of the International Congress of Mathematicians,III (2002), 747–757, Higher Ed. Press, Beijing, 2002.

[31] Y. Yomdin, Singularities in Algebraic Data Acquisition, Real and Complex Singular-ities, M. Manoel, M. C. Romero Fuster, C. T. C. Wall, Editors, London Mathe-matical Society Lecture Note Series, No. 380, 2010, 378-394.

[32] Zygmund, A., Trigonometric Series. Vols. I, II, Cambridge University Press, NewYork, 1959.


VOLUME 19

2012, NO 1–2

PP. 31–47

STABILITY AND LINEARIZATION FOR DIFFERENTIALEQUATIONS WITH A DISTRIBUTED DELAY∗

L. BEREZANSKY † AND E. BRAVERMAN ‡

Dedicated to the memory of Anatolii Dmitrievich Myshkis

Abstract. We present some new stability results for nonlinear equations with a dis-tributed delay

x(t) +m∑

k=1

∫ t

hk(t)

fk(s, x(s)) dsRk(t, s) = 0.

The results are applied to establish local stability of some models of population dynamics:the Nicholson’s blowflies equation and the Mackey-Glass equation describing white bloodcells production.

Key Words. Distributed delay, stability, linearization, Nicholson’s blowflies equation,Mackey-Glass equation

AMS(MOS) subject classification. 34K20, 92D25

1. Introduction. Equations with a distributed delay provide a moreflexible and realistic description for real world phenomena than ordinary dif-ferential equations or equations with concentrated delays [1]. If a maturationdelay is incorporated in the equation, then the maturation time is, generally,not constant, but is distributed around its expectancy value. The same isvalid for the digestion delay, as well as for the latency period for most infec-tious diseases.

∗ The first author was partially supported by the Israeli Ministry of Absorption, thesecond author was partially supported by NSERC Research Grant

† Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105,Israel

‡ Department of Mathematics and Statistics, University of Calgary, 2500 UniversityDrive N.W., Calgary, AB, T2N 1N4, Canada

31

32 L. BEREZANSKY AND E. BRAVERMAN

Historically, equations with a distributed delay were studied even be-fore relevant models with concentrated delays appeared. To the best of ourknowledge the first systematic study of equations with a distributed delay canbe found in the monograph of Myshkis [2], the results obtained by 1993 aresummarized in the book of Kuang [1]. Presently equations with a distributeddelay are intensively studied. For various models of mathematical biologywith distributed and concentrated delays see the monographs [1, 3, 4, 5]. Dis-tributed delays mainly arise when modeling real world phenomena, such asdriver’s behaviour, optic systems, tumor-immune system interaction, infec-tious diseases treatment, see, for example, recent papers [6]-[9] and referencestherein. Here we do not mention extensive literature on neural networks andcontrol theory for equations with distributed delays, as well as partial differ-ential equations incorporating this type of delays.

In most publications integrodifferential equations are studied, see, forexample, [10]; however sometimes applied models are assumed to incorpo-rate both integral terms and equations with concentrated delays (see, forexample, [11, 12]); such equations are considered in the present paper. Mostof previously obtained results are not relevant for non-autonomous modelsand do not involve equations with a concentrated delay as a special case.The present paper fills up this gap.

The paper is organized as follows. In Section 2 we present stability re-sults for linear equations. Theorem 1 is a new result, and the Mean ValueTheorem for the equation with distributed delays (Lemma 3) is for the firsttime formulated in the form accounting for each delay. Further, we applythese results to nonlinear equations. Section 3 is concerned with the theo-retical justification of the linearization process presented in Theorem 2 andits corollary. There are many results of this type for nonlinear delay differen-tial equations. However to the best of our knowledge, none of the linearizedstability theorems can be applied to nonlinear differential equations withdistributed delays and measurable parameters which are considered in thispaper. Finally, in Section 4 local stability of the unique positive equilibriumis studied for some applied models, such as the Nicholson’s blowflies equa-tion and the Mackey-Glass equation describing the production of white bloodcells.

2. Preliminaries and Results for Linear Equations. We con-sider linear and nonlinear scalar differential equations with a distributeddelay

(1) x(t) +m∑k=1

∫ t

hk(t)

x(s)dsRk(t, s) = 0, t ≥ t0 ≥ 0,

LINEARIZATION OF EQUATIONS WITH DISTRIBUTED DELAY 33

(2)

x(t)+m∑k=1

∫ t

hk(t)

x(s)dsRk(t, s)+

q∑j=1

∫ t

gj(t)

fj(s, x(s)) dsGj(t, s) = 0, t ≥ t0 ≥ 0,

as well as both equations with a nondelay term, which for (1) becomes

(3) x(t) + b(t)x(t) +m∑k=1

∫ t

hk(t)

x(s)dsRk(t, s) = 0, t ≥ t0 ≥ 0.

Here we assume that for each t the memory is finite and define

(4) hk(t) = inf s ≤ t|Rk(t, s) = 0 , gj(t) = inf s ≤ t|Gj(t, s) = 0 .

We consider equations (1), (2) and (3) for a fixed t0 ≥ 0 with the initialcondition

(5) x(t) = φ(t), t ≤ t0

under the following assumptions:(a1) Rk(t, ·), Gj(t, ·) are left continuous functions of bounded variation

for any t; Rk(·, s), Gj(·, s) are locally integrable for any s, Rk(t, hk(t)) = 0,Gj(t, gj(t)) = 0 and Rk(t, s), Gj(t, s) are constant for s > t and coincide withthe right limits Rk(t, t

+), Gj(t, t+); the functions

(6) αk(t) =∨

τ∈[hk(t),t+]

Rk(t, τ), βj(t) =∨

τ∈[gj(t),t+]

Gj(t, τ)

are Lebesgue measurable and bounded on [0,∞), k = 1, · · · ,m, j = 1, · · · , q,where

∨τ∈I f(τ) denotes the variation of the function f on segment I;

(a2) hk, gj : [0,∞) → IR, k = 1, · · · ,m, j = 1, · · · , q are Lebesguemeasurable functions, hk(t) ≤ t, gj ≤ t,

lim supt→∞

[t− hk(t)] <∞, lim supt→∞

[t− gj(t)] <∞.

The integral

∫ t

hk(t)

x(s) dsRk(t, s) is understood as

∫ t+

hk(t)

x(s) dsRk(t, s),

we have ∫ t

hk(t)

x(s) dsRk(t, s) =

∫ t+ε

hk(t)

x(s) dsRk(t, s)

for any ε > 0.


For some particular choices of Rk(t, s) we obtain the following equationsas special cases of (1):

(7) x(t) +m∑k=1

ak(t)x(hk(t)) = 0,

(8) x(t) +

∫ t

h(t)

K(t, s)x(s) ds = 0,

(9) x(t) +m∑k=1

ak(t)x(hk(t)) +

∫ t

h(t)

K(t, s)x(s) ds = 0,

where

(10) Rk(t, s) = ak(t)χ(hk(t),∞)(s),

χI is the characteristic function of interval I,

(11) R(t, s) =

∫ s

h(s)

K(t, τ)dτ,

and(12)

Rk(t, s) = ak(t)χ(hk(t),∞)(s), k = 1, · · · ,m, Rm+1(t, s) =

∫ s

h(s)

K(t, τ) dτ,

respectively. Here K(t, s) = 0, s ∈ [h(t), t],∫ t

h(t)|K(t, s)| ds is a Lebesgue

measurable function bounded on [0,∞), ak(t) are Lebesgue measurable func-tions bounded on [0,∞), k = 1, · · · ,m. Then the relevant functions Rk(t, s)for (7)-(9) satisfy (a1); we also assume that (a2) holds for h(t), hk(t), k =1, · · · ,m.

Now let us proceed to the initial function φ. This function should satisfysuch conditions that the integral in the left hand side of (1) exists almosteverywhere. In particular, if R(t, ·) is absolutely continuous for any t (whichallows us to write (1) as an integrodifferential equation), then φ can be cho-sen as a Lebesgue measurable essentially bounded function. If Rk(t, ·) is acombination of step functions (which corresponds to an equation with con-centrated delays) then φ should be a Borel measurable bounded function.For any choice of R the integral exists if φ is bounded and continuous. Thus,we assume

(a3) φ : (−∞, 0] → IR is a bounded continuous function;the following hypothesis should be satisfied for functions fj in (2):(a4) fj(t, u) are Lebesgue measurable essentially bounded in [0,∞) func-tions of the first argument and are continuous and bounded in the secondargument, fj(t, 0) = 0, j = 1, · · · , q.

Everywhere below we will assume that for all equations and initial con-ditions hypotheses (a1)-(a4) hold.

Definition. An absolutely continuous function x : R → R is called a solutionof the problem (2),(5) if it satisfies equation (2) for almost all t ∈ [t0,∞) andconditions (5) for t ≤ t0.

In addition to linear equation (1) we will also consider the followingnon-homogeneous equation

(13) x(t) +m∑k=1

∫ t

hk(t)

x(s) dsRk(t, s) = f(t),

where f(t) is a Lebesgue measurable locally essentially bounded function.Definition. For each s ≥ t0 and t ≥ s the solution X(t, s) of the problem

(14)x(t) +

m∑k=1

∫ t

hk(t)

x(τ) dτRk(t, τ) = 0, t ≥ s,

x(t) = 0, t < s, x(s) = 1,

is called the fundamental function of equation (13). Here X(t, s) = 0, 0 ≤t < s.Lemma 1. [13, 14] The solution of the initial value problem (13),(5) can bepresented as

(15)

x(t) = X(t, t0)φ(t0) −∫ t

t0

X(t, s)

[m∑k=1

∫ s

hk(s)

φ(ζ)dζRk(s, ζ)

]ds

+

∫ t

t0

X(t, s)f(s)ds,

where φ(t) = 0, t > t0.Definition. Equation (1) is (uniformly) exponentially stable, if there existK > 0 and λ > 0 such that the fundamental function X(t, s) defined by (14)has the estimate

(16) |X(t, s)| ≤ K e−λ(t−s), t ≥ s ≥ 0.


Denote similar to (6)

(17) ak(t) = Rk(t, t+), αk(t) =

∨τ∈[hk(t),t+]

Rk(t, τ) =

∫ t

hk(t)

|dτRk(t, τ)| .

For (7) functions ak(t) coincide with the coefficients, αk(t) = |ak(t)|; in (8)we have

(18) a(t) =

∫ t

h(t)

K(t, s) ds, α(t) =

∫ t

h(t)

|K(t, s)| ds.

Let us introduce some functional spaces on a halfline.Denote by L∞[t0,∞) the space of all essentially bounded on [t0,∞) func-

tions with the essential supremum norm

∥y∥L∞ = ess supt≥t0

|y(t)|,

by C[t0,∞) the space of all continuous bounded on [t0,∞) functions withthe sup-norm.Lemma 2. [15, 16] Suppose that for any f ∈ L∞[t0,∞) the solution of (13)with the zero initial conditions belongs to C[t0,∞). Then (1) is exponentiallystable.Lemma 3. Suppose Rk(t, ·) are nondecreasing functions. Then for any solu-tion x(t) of (13) there exist gk(t) such that

(19) hk(t) ≤ gk(t) ≤ t

and x(t) is also a solution of the equation

(20) x(t) +m∑k=1

ak(t)x(gk(t)) = f(t),

where ak(t) = Rk(t, t+) = αk(t) was introduced in (17).

Proof. By the Mean Value Theorem for the Lebesgue Stiltjes integral, forany k = 1, · · · ,m and t ≥ t0 there exists gk(t) ∈ [hk(t), t] such that for thecontinuous function x(t) we have∫ t

hk(t)

x(s) dsRk(t, s) = x(gk(t))

∫ t

hk(t)

dsRk(t, s) = ak(t)x(gk(t)).

Thus x(t) satisfying (13) is also a solution of (20).

Lemma 4. [17] Suppose

ak(t) ≥ a0 > 0, supt≥t0

ak(t) <∞, supt≥t0

(t− hk(t)) <∞, k = 1, 2, · · · ,

and

(21) lim supt→∞

m∑k=1

ak(t)∑mi=1 ai(t)

∫ t

hk(t)

m∑i=1

ai(s)ds < 1 +1

e.

Then equation (7) is exponentially stable.Theorem 1. Suppose Rk(t, ·), k = 1, · · · ,m, are nondecreasing functions,ak(t) ≥ a0 > 0 and condition (21) holds, where ak(t) = Rk(t, t

+). Thenequation (1) is exponentially stable.

Proof. Suppose x(t) is a solution of equation (13) with the zero initial condi-tions. By Lemma 3 there exist functions gk(t), where hk(t) ≤ gk(t) ≤ t, suchthat x(t) also satisfies the linear equation with variable concentrated delays

(22) x(t) +m∑k=1

ak(t)x(gk(t)) = f(t).

Since hk(t) ≤ gk(t) then supt≥t0(t − gk(t)) < ∞ and condition (21) alsoholds if hk(t) are replaced by gk(t). Then by Lemma 4 equation (22) isexponentially stable.

For f ∈ L∞[t0,∞) the solution y(t) of (22) with the zero initial conditionshas the form (15):

y(t) =

∫ t

t0

Y (t, s)f(s)ds,

where Y (t, s) is the fundamental function of equation (22). Since Y (t, s) hasan exponential estimate in the form (16), then the solution y(t) of equation(22) belongs to C[t0,∞). It means that the same is true for equation (13).By Lemma 2 this equation is exponentially stable.

In Theorem 1 we suppose that Rk(t, ·) are nondecreasing functions. Inthe following results we can omit this restriction.Lemma 5. [18] Suppose that there exist a set of indices I ⊂ 1, 2, · · · , l andnumbers β > 0, 1 > γ > 0, such that

(23)∑k∈I

ak(t) ≥ β > 0,

(24)∑k∈I

∫ t

hk(t)

(∫ t

s

m∑j=1

αj(τ) dτ

)|dsRk(t, s)|+

∑k ∈I

αk(t) ≤ γ∑k∈I

ak(t)

for sufficiently large t > 0. Then equation (1) is exponentially stable.

Consider the equation with one delayed and one nondelayed terms

(25) x(t) + b(t)x(t) +

∫ t

h(t)

x(s) dsR(t, s) = 0.

Choosing I = 1 and I = 1, 2 and applying Lemma 5 we obtain thefollowing result.Corollary 1. [18] Suppose there exist β > 0 and γ ∈ (0, 1) such that atleast one of the following conditions holds:1) b(t) ≥ β > 0,

∨τ∈[h(t),t+]R(t, τ) ≤ γb(t);

2) R(t, t+) + b(t) ≥ β > 0,

|b(t)|∫ t

h(t)

∫ t

s

∨τ∈[h(s),s+]

R(s, τ) + |b(τ)|

dτ

|dsR(t, s)|

≤ γ[R(t, t+) + b(t)].Then equation (25) is exponentially stable.Now consider the equation with a nondelay term and proportional coef-

ficients

(26) x(t) + r(t)

[Bx(t) + A

∫ t

h(t)

x(s) dsR(t, s)

]= 0.

Corollary 2. [18] Let r(t) ≥ r0 > 0 and R(t, ·) be a nondecreasing functionsatisfying R(t, t+) =

∫ t

h(t)dsR(t, s) = 1. Suppose that at least one of the

following conditions holds:1) B > 0, |A| < B;

2) A+B > 0, |A|∫ t

h(t)

(∫ t

s

r(τ) dτ

)dsR(t, s) <

A+B

|A|+ |B|.

Then (26) is exponentially stable.Consider the function

ω(σ) =

∫ ∞

0

|Uσ(s)|ds, 0 < σ <π

2,

where Uσ(t) is the solution of the following initial value problem for theautonomous delay equation

x(t) + x(t− σ) = 0, x(t) = 0, t < 0, x(0) = 1.

In [19, 20] properties of ω(σ) were obtained and the values of ω(σ) weretabulated. In particular, it was shown that the constant

(27) U = supσω(σ)<1

(σ +

1

ω(σ)

)

is approximately

(28) U ≈ 1.425.

Lemma 6. [18] Suppose that Rk(t, ·) are nondecreasing functions,m∑k=1

Rk(t, t+) = 0 almost everywhere and

(29) a(t) :=m∑k=1

Rk(t, t+) ≥ 0,

∫ ∞

0

m∑k=1

Rk(t, t+) dt = ∞,

(30) lim supt→∞

∫ t

mink hk(t)

a(s)ds < U ≈ 1.425,

Then (1) is exponentially stable.

3. Stability by the First Approximation. In order to introducestability by the first approximation, we consider the initial value problem (2)with an arbitrary initial point and initial conditions (5).

We assume that (a1)-(a4) hold, which means that the initial function iscontinuous. However for some particular cases (for example, the equationwith several concentrated delays and the integrodifferential equation) we canomit this restriction and assume that φ is a Borel measurable bounded func-tion.

We present the following local stability definitions (see [13]).

Definition. We will say that the zero solution of (2) is (locally) uniformlystable if for any ε > 0 and t0 ≥ 0 there exists δ > 0 such that for anyinitial conditions |φ(t)| < δ, t ≤ t0, for the solution x(t) of (2)-(5) we have|x(t)| < ε, t ≥ t0, and the number δ does not depend on initial point t0.

The zero solution of equation (2) is (locally) uniformly asymptoticallystable, if it is uniformly stable and there exists δ > 0 such that for everyη > 0, there is a t1(η) such that |φ(t)| < δ for t ≤ t0 implies |x(t)| < η fort ≥ t0 + t1(η).Theorem 2. Suppose that for any sufficiently small A > 0 there exists B >0, where lim

A→0B(A)/A = 0, such that inequality |u| < A implies |fj(t, u)| < B,

j = 1, · · · , q, t ≥ t0.

In addition, suppose that the linear equation (1) is exponentially stablewith estimation (16) for its fundamental function X(t, s). Then the zerosolution of equation (2) is locally uniformly asymptotically stable.

Proof. Denote by x(t) the solution of (2) with initial conditions (5), by y(t)the solution of linear equation (1) with the same initial conditions,

mφ = supt≤t0

|φ(t)|, a = supt≥0

m∑k=1

∫ t

hk(t)

|dsRk(t, s)| , r = supt≥0

q∑j=1

∫ t

gj(t)

|dsGj(t, s)|,

H = maxk,j

supt≥t0

(t− hk(t)), supt≥t0

(t− gj(t))

.

Solution representation formula (15) implies

y(t) = X(t, t0)φ(t0)−∫ t

t0

X(t, s)

[m∑k=1

∫ s

hk(s)

φ(τ)dτRk(s, τ)

]ds,

where φ(s) = 0, if s > t0. Hence φ(τ) = 0, τ ∈ [hk(t), t] for t > t0 +H and

|y(t)| ≤ |X(t, t0)||φ(t0)|+∫ t

t0

|X(t, s)|

[m∑k=1

∫ s

hk(s)

|φ(τ)| |dτRk(s, τ)|

]ds

≤ Ke−λ(t−t0)mφ +Kamφ

∫ t0+H

t0

e−λ(t−s)ds ≤Mmφe−λ(t−t0),

where M = K

(1 +

a(eλH−1)λ

).

Thus for the solution of (1),(5) we have

(31) |y(t)| ≤Mmφe−λ(t−t0), t ≥ t0,

where the constant M does not depend on the initial data function φ(t).Without loss of generality we can assume M > 1.

By the assumptions of the theorem, fj is continuous in u and

limu→0

∣∣∣∣fj(t, u)u

∣∣∣∣ = 0

for any j uniformly on t ∈ [t0,∞), so there exists δ > 0 such that for anyt ≥ t0, u = 0 the inequality |u| < δ implies

(32) |fj(t, u)| <δλ

2M(r + 1), |fj(t, u)| < µ|u|,

where µ :=λe−λH/2

4rK, j = 1, · · · , q.

First, let us prove that as far as

(33) mφ <δ

2M,

we have |x(t)| < δ for any t. In fact, assuming there are points where|x(t)| = δ and taking t which is the minimal among all such points we obtainby solution representation formula (15) and later applying (31), (32) and(16):

|x(t)| =∣∣∣y(t)− ∫ t

t0X(t, s)

∑qj=1

[∫ s

gj(s)fj(τ, x(τ))dτGj(s, τ)

]ds∣∣∣

≤ |y(t)|+∫ t

t0|X(t, s)|

∑qj=1

[∫ s

gj(s)|fj(τ, x(τ))| |dτGj(s, τ)|

]ds

≤ Mδ2M

+ Krδ2M(r+1)

∫ t

t0λe−λ(t−t0) ds

< δ2+ rδ

2(r+1)

∫∞t0λe−λ(t−t0) ds

< δ2+ δ

2= δ,

since K < M ; the contradiction obtained proves the fact that |x(t)| < δ forany t.

Further, we will prove that for the solution x(t) of problem (2),(5) thefollowing estimation holds: if mφ > 0, then

(34) |x(t)| < 2Mmφe−λ(t−t0)/2, t ≥ t0.

This inequality is obviously satisfied for t = t0 and in [t0, t0 + ε) for someε > 0. Assume the contrary; let t be the first point where this inequalitybecomes an equality. By (31) and (32)

|x(t)| ≤ |y(t)|+

∣∣∣∣∣∫ t

t0

X(t, s)

q∑j=1

[∫ s

gj(s)

fj(τ, x(τ))dτGj(s, τ)

]ds

∣∣∣∣∣≤ Mmφe

−λ(t−t0) + r

∫ t

t0

Ke−λ(t−s)µ maxτ∈[s−H,s]

|x(τ)| ds

< Mmφe−λ(t−t0)/2 + rKµ

∫ t

t0

e−λ(t−s)mφ2Me−λ(s−H−t0)/2 ds

= Mmφe−λ(t−t0)/2 +

4rKM

λµeλH/2mφe

−λ(t−t0)/2

∫ t

t0

λ

2e−λ(t−s)/2 ds

< Mmφe−λ(t−t0)/2 +

4rKeλH/2µ

λ2Mmφe

−λ(t−t0)/2

∫ t

−∞

λ

2e−λ(t−s)/2 ds

= 2Mmφe−λ(t−t0)/2,

the contradiction proves (34).Since M and µ do not depend on the initial conditions, as far as they

do not exceed the value defined in (33), inequality (34) implies local uniformasymptotic stability of equation (2), which completes the proof.

Consider the nonlinear equation

(35) x(t) +

q∑j=1

Aj(t)

∫ t

gj(t)

fj(s, x(s)) dsGj(t, s) = 0, t ≥ 0,

where fk(t, u) are differentiable in u in some neighbourhood of u = 0,fk(t, 0) = 0. Then the following corollary is valid.Corollary 3. If fj(t, u) are differentiable in u, j = 1, · · · , q and the linearequation

y(t) +

q∑j=1

Aj(t)

∫ t

gj(t)

∂fj∂u

(s, 0)y(s) dsGj(t, s) = 0

is exponentially stable, then the zero solution of (35) is locally uniformlyasymptotically stable.

4. Applications to Equations of Population Dynamics. In thissection we apply the previous results to equations of mathematical biology.

First we consider the Nicholson’s blowflies equation. Its version with aconstant delay

(36)dN

dt= PN(t− τ)e−αN(t−τ) − δN(t)

was introduced by Nicholson [21] to model the laboratory fly population; itsdynamics was later investigated in [22] and [23]. We consider the Nicholson’sblowflies equation with a distributed delay

(37)dN

dt= r(t)

[P

∫ t

h(t)

N(s)e−N(s)dsR(t, s)− δN(t)

],

where R(t, ·) is a nondecreasing function satisfying R(t, t+) = 1 (the delaydistribution function has a probabilistic meaning), P > 0, δ > 0, r(t) ≥ 0,supt≥t0(t− h(t)) <∞.

Some aspects of global stability and oscillation of (37) were studied in[24], see also the recent review [25].

We can also consider (37) where the coefficients are not necessarily pro-portional

(38)dN

dt= P (t)

∫ t

h(t)

N(s)e−N(s)dsR(t, s)− δ(t)N(t).

Then Corollary 1, Part 1 and Corollary 3 imply the following result.Theorem 3. If there exists β > 0, γ ∈ (0, 1) such that δ(t) ≥ β, P (t) ≤γδ(t), then the zero solution of (38) is locally uniformly asymptotically stable.

If P > δ, then, in addition to the zero equilibrium, equation (37) also hasa positive equilibrium N∗ = ln(P/δ). After the substitution x(t) = N −N∗

equation (37) takes the form

(39)dx

dt= δr(t)

[P

∫ t

h(t)

(x(s) +N∗)e−x(s)dsR(t, s)−N∗ − x(t)

],

its linearized version is

(40)dy

dt= −δr(t)

[(lnP

δ− 1

)∫ t

h(t)

y(s) dsR(t, s) + y(t)

].

Applying Corollary 2, Corollary 3, Lemma 6 and Theorem 1 we obtainlocal stability results.Theorem 4. Suppose there exists α > 0 such that r(t) ≥ α > 0 and at leastone of the following conditions holds:

1) 1 e, δ

(lnP

δ− 1

)∫ t

h(t)

(∫ t

s

r(τ) dτ

)dsR(t, s) < 1;

3)P

δ> e, ln

P

δlim supt→∞

∫ t

h(t)

r(s)ds e,

(lnP

δ− 1

)lim supt→∞

∫ t

h(t)

r(s)ds < 1 +1

e.

Then the positive equilibrium of equation (37) is locally uniformly asymp-totically stable.

Evidently conditions 1) and 2) are independent of 3) and 4) (the firstcondition is delay-independent, the second test only depends on the distri-bution of R(t, s)), while 4) is sharper than 3) whenever

lim supt→∞

∫ t

h(t)

r(s)ds > U − 1− 1

e≈ 0.057.

Next, consider the Mackey-Glass Equation with variable coefficients anda distributed delay:

(41) N(t) = A(t)

∫ t

h(t)

N(s)

1 + [N(s)]γdsR(t, s)−B(t)N(t), t ≥ 0,

where R(t, ·) is a nondecreasing function satisfying R(t, t+) = 1, A(t) ≥0, B(t) ≥ 0, γ > 0, supt≥t0(t− h(t)) <∞.

This equation with a constant concentrated delay and constant coeffi-cients

(42)dN

dt=

rNτ

1 +Nγτ− bN,

for the first time was developed in [26] to model white blood cells production.Here N(t) is the density of mature cells in blood circulation, the functionrNτ

1 +Nγτ

models the blood cell reproduction, the time lag Nτ = N(t − τ)

described the maturational phase before blood cells are released into circula-tion, the mortality rate bN was assumed to be proportional to the circulation.Equation (42) was introduced to explain the oscillations in numbers of neu-trophils observed in some cases of chronic myelogenous leukemia [26, 27].Here we assume that the maturation time is not constant but is distributedaround some expectancy value.

For a modification of equation (42) to variable delays and coefficients,positivity of solutions and global asymptotic stability was recently studied in[28]; local stability conditions for its modification with the delay in the mor-tality term, as well as in the production term, were presented in [29]. Variousaspects of equations of type (42) with variable parameters were discussed in[30, 31]. Stability of equations of population dynamics with integral typedelays (both finite and infinite) was studied in [32].

The reproduction function can differ from one in (42): for instance,r

Kγ +Nγdescribes the red blood cells production rate [33], where three

parameters r,K, γ are chosen to match experimental data.The linearized equation for (41) has the form

(43) y(t) = A(t)

∫ t

h(t)

y(s) dsR(t, s)−B(t)y(t).

Thus Part 1 of Corollary 1 yields the following statement.Theorem 5. If there exists β > 0, γ ∈ (0, 1) such that B(t) ≥ β,A(t) ≤γA(t) then the zero solution of (41) is locally uniformly asymptotically stable.

Consider now equation (41) with proportional coefficients

(44) N(t) = r(t)

[∫ t

h(t)

AN(s)

1 + [N(s)]γdsR(t, s)−BN(t)

], t ≥ 0,

where A > 0, B > 0 and r(t) ≥ α > 0.Applying Theorem 5 to equation (44) gives the following result.

Corollary 4. If B > A then the zero solution of (44) is locally uniformlyasymptotically stable.

If A > B then equation (44) has the positive equilibrium

(45) N∗ =

(A

B− 1

) 1γ

.

Let us now derive local stability conditions of the positive steady state N∗.To this end substitute N(t) = N∗(1 + x(t)) in (44); we obtain the followingequation

(46) x(t) = r(t)

[∫ t

h(t)

A(1 + x(s))

1 + [N∗(1 + x(s))]γdsR(t, s)−B(1 + x(t))

], t ≥ 0.

Since the derivative of the function

z(u) =A(1 + u)

1 + [N∗(1 + u)]γ

satisfies

z′(0) = −(γ(A−B)− A)B

A,

then by Corollaries 3, 2, Lemma 6 and Theorem 1 we obtain the followingresult.Theorem 6. Suppose that A > B > 0 and at least one of the followingconditions holds:

1)|γ(A−B)− A|

A< 1;

2)|γ(A−B)− A|B

Alim supt→∞

∫ t

h(t)

(∫ t

s

r(τ) dτ

)dsR(t, s)

<γ(A−B)

A+ |γ(A−B)− A|;

3) γ >A

A−B,γ(A−B)B

Alim supt→∞

∫ t

h(t)

r(s)ds A

A−B,(γ(A−B)− A)B

Alim supt→∞

∫ t

h(t)

r(s)ds < 1 +1

e.

Then for equation (44) the positive steady state N∗ defined by (45) is lo-cally uniformly asymptotically stable.


REFERENCES

[1] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics.Mathematics in Science and Engineering, 191. Academic Press, Boston, MA,1993.

[2] A.D. Myshkis, Linear Differential Equations with Delayed Argument, Nauka,Moscow, 1972.

[3] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology andEpidemiology, Springer-Verlag New York, 2001.

[4] M. Kot, Elements of Mathematical Ecology, Cambridge Univ. Press, 2001.[5] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population

Dynamics, Kluwer Academic Publishers, Dordrecht, Boston, London, 1992.[6] R. Sipahi, F.M. Atay and S. I. Niculescu, Stability of traffic flow behavior with

distributed delays modeling the memory effects of the drivers, SIAM J. Appl.Math. 68 (2007/08), no. 3, 738–759.

[7] U. Meyer, J. Shao, S. Chakrabarty, S. F. Brandt, H. Luksch and R. Wessel, Dis-tributed delays stabilize neural feedback systems, Biol. Cybernet. 99 (2008), no.1, 79–87.

[8] B. Mukhopadhyay and R. Bhattacharyya, Temporal and spatiotemporal variations ina mathematical model of macrophage-tumor interaction, Nonlinear Anal. HybridSyst. 2 (2008), no. 3, 819–831.

[9] M.E. Alexander, S.M. Moghadas, G. Rost and J. Wu, A delay differential model forpandemic influenza with antiviral treatment, Bull. Math. Biol. 70 (2008), no. 2,382–397.

[10] S. Bernard, J. Belair and M.C. Mackey, Sufficient conditions for stability of lineardifferential equations with distributed delay, Discrete and Continuous DynamicalSystems - Series B 1 (2001), no. 2, 233–256.

[11] L. Berezansky and E. Braverman, Oscillation properties of a logistic equation withdistributed delay, Nonlinear Anal. Real World Appl. 4 (2003), 1-19.

[12] L. L. Wang and W.–T. Li, Periodic solutions and permanence for a delayed nonau-tonomous ratio-dependent predator-prey model with Holling type functional re-sponse, J. Comput. Appl. Math. 162 (2004), no. 2, 341–357.

[13] J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equa-tions, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.

[14] N.V. Azbelev, L. Berezansky, L. F. Rahmatullina, A linear functional-differentialequation of evolution type, Differential Equations 13 (1977), no. 11, 1331–1339.

[15] N.V. Azbelev, L. Berezansky, P.M. Simonov, A.V. Chistyakov, The stability oflinear systems with aftereffect. I. Differential Equations 23 (1987), no. 5, 493–500,II. Differential Equations 27 (1991), no. 4, 383–388, III. Differential Equations 27(1991), no. 10, 1165–1172, IV. Differential Equations 29 (1993), no. 2, 153–160.

[16] N.V. Azbelev and P.M. Simonov, Stability of differential equations with aftereffect.Stability and Control: Theory, Methods and Applications, 20. Taylor & Francis,London, 2003.

[17] L. Berezansky, E. Braverman, New stability conditions for linear differential equa-tions with several delays, Abstract and Applied Analysis 2011 (2011), Article178568, 19p.

[18] L. Berezansky and E. Braverman, Stability of linear differential equations with adistributed delay, Communications on Pure and Applied Analysis 10 (2011), no.5, 1361–1375.


[19] S.A. Gusarenko, Conditions for the solvability of problems on the accumula-tion of perturbations for functional-differential equations. (Russian) Functional-differential equations, 30–40, Perm. Politekh. Inst., Perm, 1987.

[20] I. Gyori, F. Hartung and J. Turi, Preservation of stability in delay equations underdelay perturbations, J. Math. Anal. Appl. 220 (1998), 290–312.

[21] A. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zool. 2(1954), 9–65.

[22] W. Gurney, S. Blythe and R. Nisbet, Nicholson’s blowflies revisited, Nature 287(1980), 17–21.

[23] R. Nisbet and W. Gurney, Modelling fluctuating populations, John Wiley and Sons,NY, 1982.

[24] E. Braverman and D. Kinzebulatov, Nicholson’s blowflies equation with a distributeddelay, Can. Appl. Math. Q. 14 (2006), no. 2, 107–128.

[25] L. Berezansky, E. Braverman and L. Idels, Nicholson’s blowflies differential equationsrevisited: main results and open problems, Applied Mathematical Modelling 34(2010), 1405–1417.

[26] M.C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,Science 197 (1977), 287–289.

[27] J. Losson, M.C. Mackey, and A. Longtin, Solution multistability in first order non-linear differential delay equations, Chaos 3 (2), (1993), 167–176.

[28] M. Fan and X. Zou, Global asymptotic stability of a class of nonautonomous integro-differential systems and applications, Nonlinear Anal. 57 (2004), no. 1, 111–135.

[29] L. Berezansky and E. Braverman, On stability of some linear and nonlinear delaydifferential equations, J. Math. Anal. Appl., 314 (2006), no. 2, 391-411.

[30] L. Berezansky and E. Braverman, On existence and attractivity of periodic solutionsfor the hematopoiesis equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math.Anal. 13B (2006), suppl., 103–116.

[31] L. Berezansky and E. Braverman, Mackey-Glass equation with variable coefficients,Comput. Math. Appl. 51 (2006), no. 1, 1-16.

[32] E. Liz, C. Martınez and S. Trofimchuk, Attractivity properties of infinite delayMackey-Glass type equations, Differential and Integral Equations 15 (2002), 875–896.

[33] M.C. Mackey, Mathematical models of hematopoietic cell replication and control,The Art of Mathematical Modelling: Case Studies in Ecology, Physiology andBiofluids (Eds. H.G. Othmer, F.R. Adler, M.A. Lewis, and J.C. Dallon), PrenticeHall (1997), 149–178.


VOLUME 19

2012, NO 1–2

PP. 49–62

LINEAR BOUNDARY VALUE PROBLEMS AND CONTROLPROBLEMS FOR A CLASS OF FUNCTIONAL

DIFFERENTIAL EQUATIONS WITH CONTINUOUS ANDDISCRETE TIMES ∗

A. CHADOV † AND V. MAKSIMOV ‡

Abstract. For a functional differential system with continuous and discrete times,the general linear boundary value problem and the problem of control with respect to anon-target vector-functional are considered. Conditions for the solvability of the problemsare obtained. Questions of computer-aided techniques for studying these problems arediscussed.

Key Words. Abstract functional differential equations, discrete-continuous systems,boundary value problems, control problems, hybrid controlled systems with aftereffect.

AMS(MOS) subject classification. 34K10, 34K30, 34K35, 91B74

1. Introduction. We consider here a system of functional differen-tial equations (FDE, FDS) that, formally speaking, is a concrete realizationof the so-called abstract functional differential equation (AFDE). Theory ofAFDE is thoroughly treated in [7, 9]. On the other hand, the system underconsideration is a typical one met with in mathematical modeling economicdynamic processes and covers many kinds of dynamic models with after-effect (integro-differential, delayed differential, differential difference, differ-ence) and impulsive perturbations resulting in system’s state jumps at pre-scribed time moments [13, 3, 4, 14, 19]. The equations of the system contain

∗ Supported by Grant 10-01-96054 of the Russian Foundation for Basic Research† Department of Information System and Mathematical Methods in Economics, Perm

State University, Bukirev Str. 15 , Perm, 614990, Russia, e-mail: [email protected]‡ Department of Information System and Mathematical Methods in Economics, Perm

State University, Bukirev Str. 15 , Perm, 614990, Russia, e-mail: [email protected]

49

50 A. CHADOV AND V. MAKSIMOV

simultaneously terms depending on continuous time, t ∈ [0, T ], and discrete,t ∈ 0, t1, ..., tN , T, this is why the term ”hybrid” seems to be suitable. Asthis term is deeply embedded in the literature in different senses, we willfollow the authors employing the more definite name ”continuous-discretesystems” (CDS), see, for instance, [1, 2, 16, 17, 18] and references therein.Notice that in [1, 2] a detailed motivation for studying CDS and examplesof applications can be found together with results on stabilization, observ-ability and controllabality for a class of linear CDS with continuous-timedynamics described by ordinary differential equations. To finish with theterminology, it is pertinent to note that the name ”concrete systems” couldbe used for short (as it was interpreted by V.I. Arnold, ”concrete” meanscon(tinuous-dis)crete).

First we descript in detail a class of continuous-discrete functional dif-ferential equations (CDFDE) with linear Volterra operators and appropriatespaces where those are considered. We are concerned with the representationof general solution to the system and derive basic relationships for the Cauchyoperator and the fundamental matrix. Next the setting of the general lin-ear boundary value problem (BVP) for CDFDE is given, and conditions forthe solvability of BVP are obtained. The control problem (CP) for CDFDEis set up and considered then. Here we give conditions for the solvabilityof CP and propose a technique of constructing the solutions to the prob-lem.Questions of computer-aided techniques for studying these problems arediscussed. Conclusively, we give a remark concerning CDFDE as an AFDE.

2. A class of continuous-discrete functional differential systems.First, let us introduce the Banach spaces where operators and equations areconsidered.

Fix a segment [0, T ] ⊂ R. By Ln = Ln[0, T ] we donote the space of

summable functions v : [0, T ] → Rn under the norm ||v||Ln =∫ T

0|v(s)|n ds,

where | · |n (| · | for short if the value of dimension is clear) stands for thenorm of Rn.

Given set τ1, ..., τm, 0 < τ1 < ... < τm < T , the space DSn(m) =DSn[0, τ1, ..., τm, T ] is defined (see [5, 8, 9]) as the space of piecewise abso-lutely continuous functions y : [0, T ] → Rn representable in the form

y(t) =

∫ t

0

v(s) ds + y(0) +m∑k=1

χ[τk,T ](t)∆y(τk),

where v ∈ Ln, ∆y(τk) = y(τk) − y(τk − 0), χ[τk,T ](t) is the characteristicfunction of the segment [τk, T ]: χ[τk,T ](t) = 1 if t ∈ [τk, T ] and χ[τk,T ](t) =0, t /∈ [τk, T ]. Thus the elements ofDSn(m) are the functions being absolutely

FDE WITH CONTINUOUS AND DISCRETE TIMES 51

continuous on each [0, τ1), [τ1, τ2), ..., [τm, T ] and continuous from the right atthe points τ1, ..., τm. Under the norm

||y||DSn(m) = ||y||Ln + |y(0)|n +m∑k=1

|∆y(τk)|n

the space DSn(m) is Banach.Let us give some remarks concerning the approach to the impulse sys-

tems based on the use of the space DSn(m). An approach to the study ofdifferential equations with discontinuous solutions is associated with the socalled ”generalized ordinary differential equations” whose theory was initi-ated by J.Kurzweil [10]. Nowadays this theory is highly developed (see, forinstance, [21, 6]). According to the accepted approaches impulsive equationsare considered within the class of functions of bounded variation. In this casethe solution is understood as a function of bounded variation satisfying anintegral equation with the Lebesgue-Stiltjes integral or Perron-Stiltjes one.Integral equations in the space of functions of bounded variation became tobe the subject of its own interest and are studied in detail in [22]. Recallthat the function of bounded variation is representable in the form of thesum of an absolutely continuous function, a break function, and a singularcomponent (a continuous function with the derivative being equal zero al-most everywhere). The solutions of equations with impulse impact, whichare considered below, do not contain the singular component and may havediscontinuity only at finite number of prescribed points. We consider theseequations on a finite-dimensional extension DSn(m) of the traditional spaceof absolutely continuous functions. This approach to the equations with im-pulsive impact was offered in [5]. It does not use the complicated theoryof generalized functions, turned out to be rich in content and finds manyapplications in the cases where the question about the singular componentdoes not arise.

Let us fix a set J = t0, t1, ..., tµ, 0 = t0 < t1 < ... < tµ = T.FDν(µ) = FDνt0, t1, ..., tµ denotes the space of functions z : J → Rν

under the norm

||z||FDν(µ) =

µ∑i=0

|z(ti)|ν .

We consider the system

(1)y = T11y + T12z + f,

z = T21y + T22z + g,

where the linear operators Tij, i, j = 1, 2, are defined as follows.

(T11) T11 : DSn(m) → Ln;

(T11y)(t) =

∫ t

0

K1(t, s)y(s) ds + A10(t)y(0) +

m∑k=1

A1k(t)∆y(τk) , t ∈ [0, T ].

Here the elements k1ij(t, s) of the kernel K(t, s) are measurable on the set0 ≤ s ≤ t ≤ T and such that |k1ij(t, s)| ≤ κ(t), i, j = 1, ..., n, κ(·) issummable on [0, T ] , (n × n)-matrices A1

0, ..., A1m have elements summable

on [0, T ]. Recall [8, 9] that such a form of T11 covers many kinds of linearoperators with concentrated and distributed delays including the so-calledinner superposition operator.

(T12) T12 : FDν(µ) → Ln; (T12z)(t) =

∑j:tj≤t−∆1

B1j (t)z(tj), t ∈ [0, T ],

where elements of matrices B1j , j = 0, , ..., µ, are summable on [0, T ], ∆1 ≥ 0.

As is it usually is, here and in the sequel∑l

i=k Fi = 0 for any Fi if l < k.

(T21) T21 : DSn(m) → FDν(µ);

(T21y)(ti) =

∫ ti−∆2

0

K2i (s)y(s)ds + A2

i0y(0) +m∑k=1

A2ik∆y(τk), i = 0, 1, ..., µ,

with measurable and essentially bounded on [0, T ] elements of matrices K2i

and constant (ν × n)-matrices A2ik , i = 0, 1, ..., µ, k = 0, 1, ...,m; ∆2 ≥ 0. .

(T22) T22 : FDν(µ) → FDν(µ); (T22z)(ti) =

i−1∑j=0

B2ijz(tj), i = 1, ..., µ,

with constant (ν × ν)-matrices B2ij.

In what follows we will use some results from [8, 9] concerning the equa-tion

(2) y = T11y + f

and the results of [3] concerning the equation

(3) z = T22z + g.


Recall that the homogeneous equation (2) (f(t) = 0, t ∈ [0, T ]) has thefundamental matrix Y (t) of dimension n× (n+mn):

(4) Y (t) = Θ(t) + X(t),

whereΘ(t) = (En, χ[τ1,T ]En, ..., χ[τm,T ]En),

En is the identity (n× n)-matrix, each column xi(t) of the (n× (n +mn))-matrix X(t) is a unique solution to the Cauchy problem

(5) x(t) =

∫ t

0

K1(t, s)x(s) ds + a1i (t), x(0) = 0 , t ∈ [0, T ].

Here a1i (t) is the i-th column of the matrix A1 = (A10, A

11, ..., A

1m).

The solution of (2) with the initial condition y(0) = 0 has the represen-tation

(6) y(t) = (C1f)(t) =

∫ t

0

C1(t, s)f(s) ds,

where C1(t, s) is the Cauchy matrix [11, 12] of the operator d/dt − T11. Thismatrix can be defined (and constructed) as the solution to

(7)∂

∂tC1(t, s) =

∫ t

s

K1(t, τ)∂

∂τC1(τ, s) dτ + K1(t, s), 0 ≤ s ≤ t ≤ T,

under the condition C1(s, s) = En.The matrix C1(t, s) is expressed in terms of the resolvent kernel R(t, s)

of the kernel K1(t, s). Namely,

C1(t, s) = En +

∫ t

s

R(τ, s) dτ.

The general solution of (2) has the form

(8) y(t) = Y (t)α +

∫ t

0

C1(t, s)f(s) ds,

with arbitrary α ∈ Rn+mn.As for equation (3), it has the immediate analogs of the above terms.

Namely, the fundamental matrix Z(ti), i = 0, ..., µ, of the homogeneous equa-tion (3):

z(ti) =i−1∑j=0

B2ijz(tj), i = 1, 2, ..., µ,


is defined as the solution of the initial problem

(9) Z(ti) =i−1∑j=0

B2ijZ(tj), i = 1, 2, . . . , µ, Z(t0) = Eν .

The Cauchy matrix C2(i, j) is defined by the recurrent relationships

(10) C2(i, j) = Eν +i−1∑k=j

B2ikC2(k, j), 1 6 j 6 i 6 µ,

and gives the solution to (3) under the condition z(t0) = 0:

z(ti) = (C2g) (ti) =i∑

j=1

C2(i, j)g(tj), i = 0, 1, . . . , µ.

Thus, the general solution of (3) has a representation

(11) z(ti) = Z(ti)β + (C2g) (ti), i = 0, 1, . . . , µ,

with arbitrary β ∈ Rν .Now we can apply (8) and (11) to the first equation and the second one

of (1) respectively. Thus we obtain in operator form

(12)y = Y α + C1 T12z + C1 f,

z = Z β + C2 T 21y + C2 g,

or

(13)

(I −C1 T12

−C2 T21 I

)·(yz

)=

(Y 00 Z

)·(αβ

)+

(C1 00 C2

)·(fg

),

where I is the identity operator in a proper space.To obtain a representation of the general solution to (1) and derive the

key relationships for the fundamental matrix and the Cauchy operator ofCDS (1), we shall solve (13) with respect to x = col (y, z). This will be donemaking use of the following Lemma.Lemma 1. Let ∆1 and ∆2 in definition of T12 and T21 be such that thecondition

(14) ∆1 +∆2 = 0


holds. Then the operator

P =

(I −C1 T12

−C2 T21 I

): DSn(m)× FDν(µ) → DSn(m)× FDν(µ)

is invertible.

Proof. It is easy to verify that a linear operator M =

(I AB I

)with linear

operators A : Z → Y and B : Y → Z (Y , Z are Banach spaces) is invertibleif (I−BA) : Z → Z has the inverse (I−BA)−1 : Z → Z. In such a situation,the inverse (I − AB)−1 exists too and

M−1 =

((I − AB)−1 −(I − AB)−1A

−B(I − AB)−1 (I −BA)−1

).

In the case under consideration, BA = C2T21C1T12 : FDν(µ) → FDν(µ)

is a τ -Volterra [12, p.106] operator with τ = ∆1 + ∆2 and, therefore, is anilpotent operator. In such a case, the spectral radius of BA equals zero.

In the sequel we assume that (14) holds. Thus, it is follows from (13)that

(15)

(yz

)=

(H11 H12

H21 H22

)·(Y 00 Z

)·(αβ

)+

(H11 H12

H21 H22

)·(C1 00 C2

)·(fg

),

where

(16) H11 = (I − C1 T12C2 T21)−1 ; H12 = −(I − C1 T12C2 T21)

−1C1 T12 ;

H21 = C2 T21(I − C1 T12C2 T21)−1 ; H22 = (I − C2 T21C1 T12)

−1.

Finally, the general solution x =

(yz

)∈ DSn(m)× FDν(µ) of (1) has

the form

(17) x = X(αβ

)+ C

(fg

),

where the fundamental matrix X is expressed in terms of the fundamentalmatrices Y and Z by the equality

(18) X =

(H11Y H12ZH21Y H22Z

)=

(X11 X12

X21 X22

)and the Cauchy operator C is expressed in terms of the Cauchy operators C1

and C2:

(19) C =

(H11C1 H12C2

H21C1 H22C2

)=

(C11 C12C21 C22

).


3. General linear boundary value problem. The general linearBVP is the system (1) supplemented by the linear boundary conditions

(20) ℓx = ℓ

(yz

)= γ, γ ∈ RN ,

where ℓ : DSn(m) × FDν(µ) → RN is a linear bounded vector functional.Let us give the representation of ℓ:

(21) ℓ

(yz

)=

∫ T

0

Φ(s)y(s) ds+Ψ0y(0) +m∑k=1

Ψk∆y(τk) +

µ∑j=0

Γjz(tj).

Here Ψk, k = 0, 1, . . . ,m, are constant (N × n)-matrices, Γj, j = 0, 1, . . . , µare constant (N × ν)-matrices, Φ is (N × n)-matrix with measurable andessentially bounded on [0, T ] elements. We assume that the componentsℓi : DS

n(m)×FDν(µ) → R, i = 1, . . . , N , of ℓ = col (ℓ1, . . . , ℓN) are linearlyindependent.

BVP (1),(20) is well-defined if N = n + mn + ν. In such a situation,BVP (1),(20) is uniquely solvable for any f, g if and only if the matrix

(22) ℓX =(ℓX 1, . . . , ℓX n+mn+ν

),

where X j is the j-th column of X is nonsingular, i.e.

(23) det ℓX = 0.

Hence the result may be summarized up as the following theorem.Theorem 1. Suppose that N = n+mn+ ν. Then BVP (1),(20) is uniquelysolvable for any f, g if and only if (23) holds, where (N × N)-matrix ℓX isdefined by (22),(21),(18),(16).

4. Problem of control with respect to a system of linear on-target functionals. Let us write CDS (1) in the form

(24) δx = Θx+ φ,

where x =

(yz

)∈ DSn(m)× FDν(µ), φ =

(fg

)∈ Ln(m)× FDν(µ),

Θ =

(T11 T12

T21 T22

),

and consider the system under control

(25) δx = Θx+ Fu+ φ.


Here u ∈ H is a control, H is a Hilbert space with the inner product ⟨·, ·⟩,F : H → Ln×FDν(µ) is a liner bounded operator responsible for realizationof control actions. To formulate the control problem for (25), we introducean on-target vector-functional ℓ : DSn(m) × FDν(µ) → RN of the generalform (21). The control problem (CP) with respect to a given finite systemof functionals ℓj, col (ℓ1, . . . , ℓN) = ℓ, for CDS (25) is the problem

(26)δx = Θx+ Fu+ φ,

x(0) =

(y(0)z(0)

)=

(αβ

)∈ Rn+ν ; ℓx = γ ∈ RN

as the problem of the existense of a control u ∈ H such that BVP

(27)δx = Θx+ Fu+ φ,

x(0) =

(αβ

); ℓx = γ

is solvable. If such a control exists for any φ ∈ Ln × FDν(µ), α ∈ Rn,β ∈ Rν , γ ∈ RN , then the CDS under control (25) is said to be controllablewith respect to the vector-functional ℓ.

We shall obtain conditions of the solvability to (26) on the base of therepresentation (17) which gives the description of all solutions to (25) underthe initial conditions y(0) = α ∈ Rn, z(0) = β ∈ Rν .

(28) x = X

ασβ

+ Cφ+ CFu.

Here σ = col (∆y(τ1), . . . ,∆y(τm)) ∈ Rmn is arbitrary. Applying the vector-functional ℓ to both sides of (28) and taking into account the goal of control-ling as reaching the given value γ ∈ RN for ℓx along the trajectories of (26),we arrived at the equation

(29) ℓX

ασβ

+ ℓCφ+ ℓCFu = γ

with respect to σ ∈ Rmn and u ∈ H.We shall reduce (29) to a linear algebraic system. Note that λj = ℓjCF is

a linear bounded functional defined on the Hilbert space H, this is why thereexists vj ∈ H such that λj = ⟨vj, u⟩ (vj = (CF )∗ ℓj, ·∗ stands for notation ofadjoint operator).


Let us seek the control u in the form of the linear span

u =N∑i=1

divi

(recall that the space H can be represented as the direct sumspan(v1, . . . , vN)⊕ [span(v1, . . . , vN)]

⊥).Thus, we have

(30) ℓCFu = V d,

where V = ⟨vi, vj⟩i,j=1,...,N is the Gram (N × N)-matrix for the systemv1, . . . , vN ∈ H.

Let us write the matrix ℓX in the form

(31) ℓX = (Ξy|, Ξ∆|, Ξz) ,

where the matrices Ξy, Ξ∆, Ξz have dimensions N × n, N × (mn), N × ν,respectively.

Now we arrive at the system

(32) Ξ∆σ + V d = γ − ℓCφ− Ξyα− Ξzβ

and formulate the result as the following theorem.Theorem 2. (cf. Theorem 2 [13]) The control problem (26) for CDS (25) issolvable if and only if the linear algebraic system (32) is solvable in (mn+N)-vector col(σ, d). Each solution col(σ0, d0), σ0 = col(σ1

0, . . . , σm0 ), of the system

(32) defines the control that solves CP (26) including the impulses σk0 =

∆y(τk), k = 1, . . . ,m, and the control u ∈ H, u =∑N

j=1 d0jvj.

5. Reliable computing experiment. The effective study of theoriginal problem, (BVP (1),(20) or CP (26)) is based on the use of the cor-responding linear algebraic system (LAS), ℓX · c = γ for BVP and (32) forCP. In doing so we have to understand that all parameters of such a systemcan be only approximately calculated. Thus the study of LAS for solvabil-ity requests a special technique with use of the so-called reliable computingexperiment (RCE) [9, 20]. Both the theoretical background and practicalimplementation of RCE need the elaboration of some specific constructivemethods of investigation based on the fundamental statements of the generaltheory with making use of contemporary software. It is relevant to noticethat the main destination of such methods is reliable establishing the factof the solvability of the problem. If it is done, the next task is to construct

an approximate solution in common with an error bound of quite high qual-ity. RCE as a tool for the study of differential and integral models is veryactively developing during last 20 years. There are some main directions inthis field: the study of the Cauchy problem for ordinary differential equa-tions (ODE) as well as for certain classes of partial DE (PDE) (H.Bauch,M.Berz, G.Corliss, B.Dobronetz, E.Kaucher and W.Miranker); the study ofboundary value problems (BVP) for ODE and PDE (S.Godunov, M.Plum,N.Ronto and A.Samoilenko); the study of integral equations (E.Kaucher andW.Miranker, C.Kennedy, R.Wang); the study of nonlinear operator equa-tions (S.Kalmykov, R.Moor, Yu.Shockin, Z.Yuldashev). A common idea inthis studies is the interval calculus in finite-dimensional and functional spacesand, as a consequence, the special techniques of rounding off when calcula-tions are produced by real computer. Our approach allows us to consider es-sentially more wide class of problems that are complicated by such propertiesas the property of not being a local operator, the presence of discontinuoussolutions, the presence of the inner superposition operator, as well as the gen-eral form of boundary conditions. In addition we do not use interval calcula-tions, which are characterized by high speed of the accumulation of roundingerrors, but make use of the rational numbers arithmetics with a specific tech-nique of definitely oriented rounding. The key idea of the constructive studyis as follows: by the original problem there is being constructed an auxiliaryproblem with reliably computable parameters, which allows one to producethe efficient computer-assisted testing for the solvability. If such the problemis solvable, the final result depends on the closeness of the original problemand the auxiliary one (recall that the inequality ||ℓX − ℓX || < 1/||[ℓX ]−1|| forapproximations ℓ, X to ℓ, X , implies that ℓX is nonsingular). The theorems,which stand for a background of RCE, give efficiently testable (by means ofcomputer) conditions of the solvability for the original problem. In the casethese conditions are failed one has to construct a new (and more close to theoriginal problem) auxiliary problem and then to test the conditions again.The implementation of the constructive methods in the form of a computerprogram (of course, it must be oriented to quite definite class of problems)allows one to study a concrete problem by a many-times repeated RCE. Atheoretical background and some details of the practical implementation ofRCE for the study of functional differential systems are presented in [20]. Itis clear that RCE includes the construction and the successive refinement ofapproximation to the key parameters of LAS with reliable error bounds. Anefficient computer-aided technique of such the construction for certain classesof FDE under some natural conditions is proposed in [15] (see also [9]).


6. Conclusive remark. CDFDS as an AFDE. First, recall thedefinition of AFDE. LetD and B be Banach spaces such thatD is isomorphicto the direct product B ×Rp ( D=B ×Rp for short).

The equation

(33) Lx = φ

with a linear bounded operator L : D → B is called the linear abstractfunctional differential equation (AFDE). The theory of the equation (33)was thoroughly treated in [7, 9]. Let us fix an isomorphism J = Λ, Y :B × Rp → D and denote the inverse J−1 = [δ, r]. Here Λ : B → D,Y : Rp → D and δ : D → B, r : D → Rp are the corresponding componentsof J and J−1:

Jz, α = Λz + Y α ∈ D, z ∈ B, α ∈ Rp,

J−1x = δx, rx ∈ B ×Rp, x ∈ D.

The system

(34) δx = z, rx = α

is called the principal boundary value problem (PBVP). Thus, for anyz, α ∈ B ×Rp,

(35) x = Λz + Y α

is the solution of (34). The representation (35) gives the representationof L: Lx = L(Λz + Y α) = LΛz + LY α = Qz + Aα, where the so-called principal part of L, Q : B → B, and the finite-dimensional operatorA : Rp → D are defined by Q = LΛ and A = LY. The general theory of(33) assumes Q to be a Fredholm operator (i.e. a Noether one with the zeroindex).

The system (1) written in the form (24) is an AFDE with Lx = δx−Θxconsidered as a linear bounded operator from the space D to the space B,where B = Ln × FDν(µ), D = [Ln × FDν(µ)] × [Rn+mn × Rν ]. In the caseunder consideration the principal part, Q, is invertible.

REFERENCES

[1] G.A. Agranovich, Some problems of discrete/continuous systems stabilization, Func-tional Differential Equations, 10 (2003), 5-17.

[2] G.A. Agranovich, Observability criteria of linear discrete-continuous sytems, Func-tional Differential Equations, 16 (2009), 35-51.


[3] D. L. Andrianov, Boundary value problems and control problems for linear defferencesystems with aftereffect, Russian Mathematics (Izv. VUZ), 37 (1993), 3-12.

[4] D. L. Andrianov, Difference equations and the elaboration of computer systems formonitoring and forecasting socioeconomic development of the country and ter-ritories, Proceedings of the Conference on Differential and Difference Equationsand Applications, Hindawi Publishing Corporation, New York-Cairo, 2006, 1231-1237.

[5] A.V. Anokhin, On linear impulse system for functional differential equations, Dok-lady Akademii Nauk USSR, 286 (1986), 1037-1040. (Russian)

[6] M. Ashordia, On the stability of solutions of the multipoint boundary value problemfor the system of generalized ordinary differential equations, Memoirs on Diff.Equations and Math. Phys., 6 (1995), 1-57.

[7] N.V. Azbelev and L. F. Rakhmatullina, Theory of linear abstract functional differ-ential equations and applications, Memoirs on Diff. Equations and Math. Phys.8 (1996), 1-102.

[8] N.V. Azbelev, V. P. Maksimov and L. F. Rakhmatullina, Introduction to the theoryof functional differential equations, Nauka, Moscow, 1991. (Russian)

[9] N.V. Azbelev, V. P. Maksimov and L. F. Rakhmatullina, Introduction to the theory offunctional differential equations: methods and applications, Hindawi PublishingCorporation, New York – Cairo, 2007.

[10] Ja. Kurzweil, Generalized ordinary differential equations and continuous dependenceon a parameter, Czechoslovak Math.J., 7 (1957), 418-449.

[11] V. P. Maksimov, The Cauchy formula for a functianal-differential equation, Differ-ential’nye Uravneniya, 13 (1977), 601-606. (Russian)

[12] V .P. Maksimov, Questions of the general theory of functional differential equations,Perm State University, Perm, 2003. (Russian)

[13] V. P. Maksimov, Theory of Functional Differential Equations and Some Problems inEconomic Dynamics, Proceedings of the Conference on Differential and Differ-ence Equations and Applications, Hindawi Publishing Corporation, New York-Cairo, 2006, 74-82.

[14] V. P. Maksimov and A.N. Rumyantsev, Boundary value problems and problems ofpulse control in economic dynamics: constructive study, Russian Mathematics(Izv. VUZ), 37 (1993), 48-62.

[15] V .P. Maksimov and A.N. Rumyantsev, Reliable computing experiment in thestudy of generalized controllability of linear functional differential systems, in:Mathematical modelling. Problems, methods, applications. Ed. by L.Uvarova andA.Latyshev, Kluwer Academic / Plenum Publishers, 2002, 91-98.

[16] V.M. Marchenko, O.N. Poddubnaya, Representation of solutions and relative con-trollability linear differential-algebraic systems with many aftereffects, DokladyAkademii Nauk, 404 (2005), 465–469. (Russian)

[17] V.M. Marchenko, O.N. Poddubnaya. Linear stationary differential-algebraic sys-tems. I. Representation of solutions, Izvestiya RAN. Teoriya i sistemy up-ravleniya, (2006), 24–38. (Russian)

[18] V.M. Marchenko, O.N. Poddubnaya, Representation of solutions and relative con-trollability of linear differential-algebraic systems with many delays, Differen-tial’nye Uravneniya, 42 (2006), 741-755. (Russian)

[19] A.D. Myshkis, A.M. Samoilenko, Systems with shocks at prescribed time moments,Matem. Sbornik, 74 (1967), 202-208 (Russian).

[20] A.N. Rumyantsev, Reliable computing experiment in the study of boundary value


problems, Perm State University, Perm, 1999. (Russian)[21] S. Schwabik, Generalized ordinary differential equations, World Scientific , Singapore,

1992.[22] S. Schwabik, M. Tvrdy and O. Veivoda, Differential and integral equations. Boundary

value problems and adjoints, Academia, Prague, 1979.


VOLUME 19

2012, NO 1–2

PP. 63–73

APR-ALMOST PERIODIC SOLUTIONS TO FUNCTIONALDIFFERENTIAL EQUATIONS WITH DEVIATED

ARGUMENT

C. CORDUNEANU ∗

Abstract. This paper, dedicated to the Memory of A.D. Myskis, is concerned withthe proof of existence of almost periodic solutions to some classes of functional equationsinvolving delays, or even delayed and advanced arguments. The classes of almost periodicfunctions are of a new type than those that are of a classical nature (Bohr, Besicovitch,Stepanov), representing a scale of spaces, starting with the space of Bohr almost periodicfunctions with absolutely convergent Fourier series, up to Besicovitch space with index 2.

Key Words. functional equations, almost periodic solutions, deviated argument.

AMS(MOS) subject classification. 45I05, 34K05

1. Introduction. The existence of almost periodic solutions, inclassical sense (Bohr), has been widely investigated by many authors, forvarious classes of functional equations. For results in book form, sources are:A. Halanay [7], J.K. Hale [8], C. Corduneanu [3], G. Gripenberg et al. [6],I.G. Malkin [10].

In our paper [4], we dealt with the case of APr-almost periodic solutions,1 ≤ r ≤ 2, to some of the simplest types of functional equations. Also, a casewhich will be amply generalized in this paper has been recently investigatedby M. Mahdavi [9]. We will introduce the delayed, as well as advancedarguments, which means that tj’s in (1.1) can be of any sign.

The aim of this paper is to investigate functional differential equationsof the form

(1.1) x(t) =n∑

j=1

ajx(t− tj) + (k ∗ x)(t) + (fx)(t),

∗ University of Texas, Arlington

63

64 C. CORDUNEANU

with linear associate

(1.2) x(t) =n∑

j=1

ajx(t− tj) + (k ∗ x)(t) + f(t),

with aj ∈ C, tj ∈ R, 1 ≤ j ≤ n, k ∈ L1(R, C), and x, f ∈ APr(R, C). Thenotations are those in Corduneanu [4], where the convolution product k ∗ x,in a generalized sense, is also defined, as follows:

(1.3) (k ∗ x)(t) ∼∞∑j=1

xj

(∫R

k(s)e−iλjsds

)eiλjt,

with

(1.4) x ∼∞∑j=1

xjeiλjt,

where xj ∈ C, λj ∈ R, j ∈ N.It has been shown in [4] that the convolution operator, with k integrable

on R, is continuous on any APr(R, C), 1 ≤ r ≤ 2, and

(1.5) |k ∗ x|r ≤ |k|L1 |x|r,

where r ∈ [1, 2], the norm | · |r being the Minkowski’s norm of index r:

(1.6) |x|r =[ ∞∑

j=1

|xj|r]1/r

,

under the assumption of convergence of the series in (1.6).

2. Existence for equation (1.2). We shall now define a functionthat plays an important role in formulating the results concerning the exis-tence of APr-almost periodic solutions to the (scalar) equation (1.1) or (1.2).Namely, we shall denote

(2.1) A(iω) =n∑

j=1

aje−iωtj +

∫R

k(s)e−iωsds, ω ∈ R.

The last term in (2.1) is generated by the convolution appearing in (1.1)and represents the Fourier transform of k ∈ L1(R, C). It will be also denoted

by k(iω).

APr-ALMOST PERIODIC SOLUTIONS 65

Let us consider now the linear case of equation (1.2), and substitute xby the series given by (1.4). We shall consider that

x(t− tj) ∼∞∑

m=1

xme−iλmtjeiλmt

as well as the definition of the convolution product given in formula (1.3).After equating the coefficients of the exponentials in left and right side ofthe relation, one obtains the following infinite system of linear (algebraic)equations, which must be satisfied by the xj, j ∈ N.

Namely, by using the function A(iω) defined by (2.1), we can write

(2.2) iλjxj = A(iλj)xj + fj, j ∈ N,

which allows to determine uniquely each xj, j ∈ N , provided

(2.3) iλj −A(iλj) = 0, j ∈ N.

We can write

(2.4) xj = [iλj −A(iλj)]−1fj, j ∈ N.

The answer is not complete, because for f ∈ APr(R, C), we expect to obtainx ∈ APr(R, C). Certainly, it would be interesting to examine the case whenf and x may belong to APr-spaces with different r. We shall pursue onlythe case when APr(R, C) is the same space for both f and x.

The condition that first comes to our mind is

(2.5) |iλj −A(iλj)| ≥ m > 0, j ∈ N,

for some m > 0.A somewhat stronger condition is

(2.6) |iω −A(iω)| ≥ m > 0, ω ∈ R,

which, in turn, assures the unique solvability of system (2.2), regardless ofthe choice of f ∈ APr(R, C).

Now, (2.4) and (2.5) imply

|xj| = |iλj −A(iλj)|−1|fj|, j ∈ N,

hence

(2.7) |xj| ≤ m−1|fj|, j ∈ N.

66 C. CORDUNEANU

But

(2.8) |xj|r ≤ m−r|fj|r, j ∈ N, 1 ≤ r ≤ 2,

implies

(2.9) x ∈ APr(R, C).

Therefore, under the above mentioned conditions, there exists a uniquesolution of the equation (1.2) in APr(R, C) such that (2.9) holds.

3. An equivalent form for condition (2.6) and statement of themain result. Apparently, the condition

(3.1) |iω −A(iω)| > 0 for ω ∈ R,

is weaker than (2.6). Actually, we can easily show that (2.6) and (3.1) areequivalent conditions.

Indeed, since A(iω) is uniformly bounded on R, more precisely,

(3.2) |A(iω)| ≤n∑

j=1

|aj|+ |k|L1 , ω ∈ R,

we see that

(3.3) |iω −A(iω)| → ∞ as |ω| → ∞.

Therefore, for any R0 > 0, there exists M > 0, such that

(3.4) |iω −A(iω) > R0 for |ω| > M.

This means that for finding min |iω − A(iω)|, for ω ∈ R, we can limitour considerations to the interval |ω| ≤ R0. But in the interval |ω| ≤ R0,(3.1) takes place and, in view of the continuity of the function iω − A(iω),one derives the inequality (2.6), for some m > 0.

The equivalence of conditions (2.6) and (3.1) is thus established. Thisfact allows us to state the existence result, for equation (1.2), as follows.

Theorem 3.1. Assume that the following conditions are satisfied by equa-tion (1.2):

(i) aj ∈ C, tj ∈ R, 1 ≤ j ≤ n;(ii) there exists m > 0, such that inequality (3.1), or equivalently (2.6),

holds true.


Then equation (1.2) has a unique solution x ∈ APr(R, C), for any f ∈APr(R, C), and r ∈ [1, 2].

The series characterizing the solutions is

(3.5) x ∼∞∑j=1

[iλj −A(iλj)]−1fje

iλjt,

with λj ∈ R, fj ∈ C, as given by

(3.6) f ∼∞∑j=1

fjeiλjt ∈ APr(R, C),

and A(iω) defined by (2.1).

4. The quasilinear system (1.1). It is now possible to prove theexistence and uniqueness of the APr-solutions to system (1.1), relying on thesimilar result for the linear counterpart (1.2).

First, a lemma will be proven, in which an estimate is found for thesolution of (1.2). Namely, one can prove that between the term f in (1.2),and the corresponding solution x in (3.5), a relation is given by

(4.1) |x|r ≤ K|f |r,

where K > 0 is a constant independent of f ∈ APr(R, C).Obviously, (4.1) proves that the linear operator f → x is continuous on

APr(R, C).Lemma 1. The operator f → x, where f is determined by (3.6), while x is theunique solution x ∈ APr(R, C), given by (3.5), is continuous from APr(R, C)into itself.

The proof of Lemma 1, i.e., to obtain (4.1), is obvious if we rely on theinequalities (2.8).

Next, we will state and prove the existence and uniqueness theorem forsystem (1.1).

Theorem 4.1. Consider equation (1.1), under the following assumptions:(i) and (ii) as in Theorem 3.1;(iii) f : APr(R, C) → APr(R, C), for a fixed r ∈ [1, 2], satisfies a Lipschitz

condition of the form

(4.2) |fx− fy|r ≤ γ|x− y|r, x, y ∈ APr(R, C),

with a sufficiently small γ.

68 C. CORDUNEANU

Then, there exists a unique solution x ∈ APr(R, C) of equation (1.1).Moreover, it can be obtained by the iteration process defined by

(4.3) xm(t) =n∑

j=1

ajxm(t− tj) + (k ∗ xm)(t) + (fxm−1)(t),

starting, for instance, with x0(t) ≡ f(t), t ∈ R.

Proof. We proceed by the standard method of fixed point. If one rewritesequation (1.1) as x = Tx, and apply (4.1), one obtains, if we take (4.2) intoaccount,

(4.4) |Tx− Ty|r ≤ Kγ|x− y|r, x, y ∈ APr.

Hence, accepting the inequality

(4.5) γ < K−1,

we obtain the contraction condition for T .This ends the proof of Theorem 4.1.

Remark 1. Following an argument as in our paper [4], one can find a moregeneral result than Theorem 4.1.

Namely, if one deals with the equations of the form

(4.6) x(t) = (Lx)(t) + (fx)(t), t ∈ R,

where L : APr → APr is linear and continuous, and one assume that thelinear equation

(4.7) x(t) = (Lx)(t) + f(t), t ∈ R,

is uniquely solvable, for any f ∈ APr, then the result of Theorem 4.1 remainsin force for equation (4.6).

Remark 2. Another generalization of Theorem 4.1, to the case of multi-dimensional systems, can be stated and proven following the same lines en-countered in case of Theorem 4.1. We shall state the result in case the spaceof values for x and f is the space APr(R, Cn), n > 1.

Theorem 4.2. Consider the system of functional differential equations

(4.8) x(t) =n∑

j=1

Ajx(t− tj) + (k ∗ x)(t) + (fx)(t),

where Aj ∈ L(Cn, Cn), j = 1, 2, ..., n, k(t) = (kij(t)), i, j = 1, 2, ..., n, issuch that |k(t)| ∈ L1(R,R), while f : APr(R, Cn) → APr(R, Cn) satisfies aLipschitz condition, with sufficiently small constant. Moreover, we assumethat

(4.9) det[iωI −A(iω)] = 0, ω ∈ R,

with

(4.10) A(iω) =n∑

j=1

Aje−iωtj +

∫R

k(s)e−iωsds.

Then, there exists a unique solution x ∈ APr(R, Cn) of equation (4.8).

Remark 3. The proof of the equivalence of condition 4.9, with a strongerform of the inequality, similar to condition (2.6), results from examining atranscendental equation of the form

ωn + α1ωn−1 + · · ·+ αn = 0,

with αj, j = 1, 2, ..., n, depending on ω, but such that |αj| ≤ M < ∞,j = 1, 2, ..., n, for ω ∈ R.

Remark 4. The generalization can be taken even further, to the case ofequations of the form

(4.11) x(t) =∞∑j=1

Ajx(t− tj) + (k ∗ x)(t) + (fx)(t),

under the assumption

(4.12)∞∑j=1

|Aj| <∞, |k| ∈ L1(R,R).

We leave to the reader the task to carry out the details of the proof, relyingon properties of the matrix-function algebra whose elements are of the form(4.10), with the summation extended to ∞. See I.C. Gokhberg et al. [5] forbackground. This approach will allow the consideration of problems of almostperiodicity, as those we’ve dealt with above, but also looking at classical types(Bohr, for instance).

70 C. CORDUNEANU

5. Equation (4.11) on the semiaxis R+. In the past (see Cor-duneanu [1], [2]), we have investigated equation (4.11), in which the productk ∗ x is meant in the classical sense, i.e., expressed by an integral. Since wewant to discuss equation (4.11) on the semiaxis R+ = [0,∞), we shall rewriteit in the form

(5.1) x(t) =∞∑j=1

Ajx(t− tj) +

∫ t

0

k(t− s)x(s)ds+ (fx)(t),

and in order to assure the fact we deal with a delay-functional equation, theassumption

(5.2) tj ≥ 0, j ∈ N.

Usually, one takes t1 = 0, and tj > 0, j ≥ 2.Since boundedness and stability were our concern, we have relied on the

following result.

Theorem 5.1. Consider the fundamental matrix X(t), associated to equation(5.1), and defined by

X(t) = (AX)(t), X(0) = I, X(t) = 0, t < 0,

where (AX)(t) is defined by the linear part of equation (5.1). Let us denote

A(s) =∞∑j=1

Aje−tjs +

∫ ∞

0

k(s)e−tsds, Re s ≥ 0,

which makes sense on behalf of (4.12).Then, the following conditions are equivalent:(1) det(sI −A(s)) = 0 for Re s ≥ 0;(2) |X(t)| ∈ L1(R+, R);(3) |X(t)| ∈ Lp(R+, R), 1 ≤ p ≤ ∞.

The proof of Theorem 5.1 is given in our book [2]. As shown there, thecondition (1), which is of the same nature as condition (2) in Theorem 3.1,is actually implying the asymptotic stability of the zero solution x = θ, ofthe homogeneous equation attached to (5.1).

We shall provide below an application of Theorem 5.1, in order to obtainanother type of asymptotic behavior for the solutions of (5.1).

Under our assumptions, stated above, we need an initial condition as-sociated to (5.1) which, in case tj unbounded, represents an example of afunctional differential equation with infinite delay is:

(5.3) x(t) = φ(t), t < 0, x(0) = x0 ∈ Rn,


where φ(t) ∈ Lloc(R−, Rn) can be subject to further restrictions. For in-

stance,

(5.4) φ ∈ L1(R−, Rn).

Let us consider now the linear equation

(5.5) x(t) =∞∑j=1

Ajx(t− tj) +

∫ t

0

k(t− s)x(s)ds+ f(t),

under initial condition (5.3), with φ satisfying (5.4). As shown in our book[2], p. 300, the unique solution of (5.5), (5.3) is given by

(5.6) x(t) = X(t)x0 +

∫ t

0

X(t− s)f(s)ds+

∫ t

0

X(t− s)(Bφ)(s)ds,

where

(5.7) (Bφ)(t) =∞∑j=1

Ajφ(t− tj), t ∈ R+.

The formula (5.6) makes sense for any f ∈ Cℓ(R+, Cn), and φ satisfying(5.4), with Cℓ the subspace of BC(R+, Cn), such that lim x(t) = x(∞) ∈ Cn,as t→ ∞, exists for x ∈ Cℓ.

Moreover, as noticed above, |X(t)| → 0 as t → ∞, which implies x(t),given by (5.7), to be in Cℓ. Indeed, the first term tends to θ ∈ Cℓ(R+, Cn), thesecond is in Cℓ because this space is invariant with respect to convolutionoperator with integrable kernel. Since Bφ ∈ L1(R+, Cn) for φ satisfying(5.4), the third term in (5.6) also has the limit θ ∈ Cℓ(R+, Cn) at infinity.All these facts lead to the conclusion that x(t), given by (5.6), belongs toCℓ(R+, Cn).

Based on this finding, in regard to the solution of equation (5.5), underconditions (5.3) and the assumption

(5.8) f ∈ Cℓ(R+, Rn),

we can state the following existence result for the equation (5.1), in the spaceCℓ(R+, Cn).

Theorem 5.2. Consider equation (5.1) in the space Cℓ(R+, Cn), under theassumptions:

(1) Aj, j ∈ N , and k ∈ L1(R+,L(Cn, Cn)) satisfy (4.12);(2) the condition (1) in Theorem 5.1 holds;

72 C. CORDUNEANU

(3) f : Cℓ(R+, Cn) → Cℓ(R+, Cn) is an operator satisfying a Lipschitzcondition on Cℓ,

(5.9) |fx− fy|BC ≤ γ|x− y|BC ,

with γ a small enough constant.Then, there exists a unique solution of the equation (5.1), satisfying the

initial conditions (5.3), (5.4), such that

x(t) ∈ Cℓ(R+, Cn).

Proof. We shall get first an estimate for the solution of the linear equation(5.5), corresponding to the case x0 = θ ∈ Cn.

This estimate is, obviously,

(5.10) |x(t)|BC ≤∣∣∣∣∫ t

0

X(t− s)f(s)ds

∣∣∣∣BC

+B0,

where B0 is a positive constant, such that

(5.11)

∣∣∣∣∫ t

0

X(t− s)(Bφ)(s)ds

∣∣∣∣ ≤ B0, t ∈ R+.

We shall proceed now by successive approximations, applied to equation(5.1), by letting x0(t) = f(t), t ∈ R+, and for γ < (|X(t)|L1)−1

xm+1(t) = X(t)x0 +

∫ t

0

X(t− s)(fxm)(s)ds+

∫ t

0

X(t− s)(Bφ)(s)ds,

for m ≥ 1. One obtains, by subtraction, for m ≥ 1,

d

dt[xm+1(t)− xm(t)] =

∫ t

0

X(t− s)[(fxm)(s)− (fxm−1)(s)]ds,

because the last term in (5.6) does not depend on m ≥ 1. Hence, whenapplying (5.10) to xm+1(t)−xm(t), we obtain, because xm+1(0)−xm(0) = θ,

|xm+1(t)− xm(t)| ≤∣∣∣∣∫ t

0

X(t− s)[(fxm)(s)− (fxm−1)(s)]ds

∣∣∣∣ ,which leads immediately to the estimate

|xm+1(t)− xm(t)|BC ≤ γK sup |xm(t)− xm−1(t)|,

for t ∈ R+, with K ≥ |X(t)|L1 . Further, we have for m ≥ 1,

(5.12) |xm+1(t)− xm(t)|BC ≤ γK|xm(t)− xm−1(t)|BC ,

which leads, by standard procedure, to the uniform convergence of xm(t);m ≥ 0 on R+, for γ < (|X(t)|L1)−1, and

(5.13) x(t) = lim xm(t), as m→ ∞,

is the solution we are searching for.This ends the proof of Theorem 5.2.Similar results can be obtained for other function spaces on R+, such as

boundedness (BC(R+, Cn)), the Cg-spaces. The fact that a solution belongsto one of the spaces provides information about its asymptotic behavior atinfinity.

REFERENCES

[1] C. Corduneanu, Integral Equations and Stability of Feedback Systems, AcademicPress, New York, 1973.

[2] C. Corduneanu, Integral Equations and Applications, Cambridge University Press,1991.

[3] C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, 2009.[4] C. Corduneanu, A scale of almost periodic function spaces, Differential and Integral

Equations, 24 (2011), 1–27.[5] I.C. Gokhberg and I.A. Feldman, Convolution Equations and Methods of Solution

(Russian), Nauka, Moscow, 1971.[6] G. Gripenberg, S.O. Londen, O. Staffans, Volterra Integral and Functional Equa-

tions. Encyclopedia of Mathematics and Its Applications (Cambridge UniversityPress, 1990).

[7] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, AcademicPress, New York, 1966.

[8] J.K. Hale, Theory of Functional Differential Equations, Springer, 1977.[9] M. Mahdavi, APr-almost periodic solutions to the equation x(t) = Ax(t) +

(k ∗ x)(t) + f(t). (to appear)[10] I.G. Malkin, Some Problems of the Theory of Nonlinear Oscillations (Russian),

Moscow, 1956.

VOLUME 19

2012, NO 1–2

PP. 75–85

A CELL GROWTH MODEL REVISITED

G. DERFEL ∗, B. VAN BRUNT † AND G. WAKE ‡

Dedicated to the memory of Anatolii Dmitrievich Myshkis

Abstract. In this paper a stochastic model for the simultaneous growth and divisionof a cell-population cohort structured by size is formulated. This probabilistic approachgives straightforward proof of the existence of the steady-size distribution and a simplederivation of the functional-differential equation for it. The latter one is the celebratedpantograph equation (of advanced type). This firmly establishes the existence of thesteady-size distribution and gives a form for it in terms of a sequence of probability distri-bution functions. Also it shows that the pantograph equation is a key equation for othersituations where there is a distinct stochastic framework.

Key Words. Steady-size distribution; Asymptotic behavior; Poisson process; Panto-graph equation

AMS(MOS) subject classification. 35B40, 35B41, 35 L45, 92C17

1. Introduction. In this paper we revisit a cell growth modeldeveloped by [7]. This model was originally developed to model plant cells[8], however, it has found applications in tumour growth in humans [2]. Afeature of this model is that a well-known functional differential equation, thepantograph equation (see [5], [13] for background), arises from a separation ofvariables solution to a Fokker-Planck equation. Specifically, let n(x, t) denotethe number density functions of cells of size x at time t i.e.,for 0 ≤ a < bthe quantity

∫ b

an(x, t) dx is the number of cells of size between a and b at

time t, x is “a variable size” of the cells in the cohort, often taken as “DNAcontent.” The cell growth process is modelled by a modified Fokker-Planck

∗ Department of Mathematics, Ben-Gurion University, Beer-Sheva, Israel† Institute of Fundamental Sciences, Massey University, Palmerston North, New

Zealand‡ Institute of Information and Mathematical Sciences, Massey University, Auckland,

New Zealand

75

76 G. DERFEL, B. VAN BRUNT, AND G. WAKE

equation, setting the dispersion term to zero for simplicity, of the form

∂

∂tn(x, t) = − ∂

∂x(gn(x, t)) + α2Bn(αx, t)

− (B + µ)n(x, t),(1)

where g is the rate of growth, µ is the rate of death, and B is the rate atwhich cells divide into α equally sized daughter cells. Here, α > 1 is the“multiplicity of division”, that is cells of size x divide to give α cells of sizex/α. The first term on the right hand side of (1) is the growth term; thesecond is the addition to the cohort at size x from division of bigger sizeαx with frequency B; and the last is the loss term from this cohort due todivision to cells of size x/α (also with frequency B), and the death of cellswith a per capita death rate of µ. For the original model that we study hereg, µ and B are positive constants. It is conceded that the assumption thatB, in particular, is constant is not in fact biologically realistic, see sectionsI.4 and III.4.2 in [12]. However, we made this assumption so as to explorethe deeper connections with the classical pantograph equation. The partialdifferential equation (1) is supplemented by the boundary conditions

limx→∞

n(x, t) = 0;(2)

limx→∞

∂

∂xn(x, t) = 0;(3)

n(0, t) = 0.(4)

In fact we need only the boundary condition (4): as (2) and (3) follow asconsequences when n(x, t = 0) = n0(x) satisfy these conditions. The steadysize distributions (SSDs) for the number density function correspond to so-lutions of the form n(x, t) = N(t)y(x) (i.e., separable solutions). Solutionsof this form yield

N ′(t)

N(t)= −gy

′(x)

y(x)+ α2By(αx)

y(x)− (B + µ)

= Λ,

where Λ is a constant of separation and ′ denotes differentiation with respectto the indicated argument. The above relation yields

N(t) = N0eΛt,(5)

where N0 is a constant, and the equation

−gy′(x) + α2By(αx)− (B + µ) y(x) = Λy(x).(6)

A CELL GROWTH MODEL REVISITED 77

Under suitable scaling (e.g. by choosing N0 =∫∞0n0(x)dx) a solution y ∈

L1[0,∞) to equation (6) corresponds to a probability density function in themodel. The boundary conditions (2)- (4) imply that

limx→∞

y(x) = 0,(7)

limx→∞

y′(x) = 0,(8)

y(0) = 0,(9)

and requirement that y be a probability density function leads to the condi-tions y(x) ≥ 0 for all x ∈ [0,∞) and∫ ∞

0

y(x) dx = 1.(10)

Integrating equation (6) from 0 to ∞ gives

Λ = (α− 1)B − µ,

and equation (6) reduces to

gy′(x) + αBy(x)− α2By(αx) = 0.(11)

Equation (11) is a special case of the pantograph equation, which has beenstudied extensively. A detailed analysis can be found in [11]. The pantographequation has found applications ranging from a partition problem in numbertheory to the collection of current in an electric train. The reader is directedto [10] for an overview of the literature and further analysis of the equation.

There are two other problems where the pantograph equation plays acentral role that have a distinct statistical flavour, viz. the absorption oflight in the Milky Way, [1] and a ruin problem in risk theory, [6]. Althoughthese problems seem distant from the cell growth model, there is nonetheless aconcrete link: all these models are based on the same type of pseudo Poissonprocess; consequently, they have the same limit distribution. The link ismore transparent using an approach of [4] that is based on a probabilistictechnique (see [3]). In the next section we detail this approach and recoversome known results about the cell growth model in a fundamentally differentframework. Specifically, we give a straightforward proof of the existenceof an SSD, the derivation of the pantograph equation for this distribution(invariant measure) and a solution in the form of a sequence of probabilitydistribution functions.

2. Limit Distribution for Cell Growth. Consider a spatiallyhomogeneous population of cells and suppose that the size x of a cell growslinearly as a function of time. Suppose further that at random momentsdefined by a Poisson process a cell of size x splits into α new cells of size x/α.Again we concede that in reality, for the most part, cells only divide in twoand the resulting daughter cells are not exactly the same size. Asymmetricaldivision of cells is currently under investigation. Here, α ≥ 1. Specifically,we suppose that the jumps x→ x/α occur in random moments

0 = t0 < t1 < · · · < tn < · · · ,

where the sequence τn defined by

τn = tn+1 − tn,

for n = 1, 2, . . . consists of independently and exponentially distributed ran-dom variables, i.e., for t > 0,

Pτn > t = e−t.

For simplicity, we assume that the between jumps the cell size x has a unitrate of growth so that after ∆t time a cell of size x grows to size x + ∆t.This assumption corresponds to choosing B and g such that αB/g = 1 in theFokker-Planck equation. The results detailed below follow mutatis mutandisfor a more general choice of constants.Lemma 1. There exists a limit distribution (invariant measure) for the sizeof a cell. This distribution is independent of the initial cell size x0.

Proof. Consider a cell of initial size x0 that splits (jumps) at randommomentst1, t2, . . . , tn, . . .. Let tn− denote the moment immediately before the nthsplitting. The size of a cell at t1− is

x(t1−) = x0 + τ1.

The cell then splits into α equal parts at t1 so that at t2− the size is

x(t2−) =1

α(x0 + τ1) + τ2.

Similarly, at t3−

x(t3−) =1

α

(1

α(x0 + τ1) + τ2

)+ τ3

=x0α2

+ τ3 +τ2α

+τ1α2,

and in general

x(tn−) =x0αn−1

+ τn +τn−1

α+ · · ·+ τ1

αn−1.(12)

Equation (12) shows that there exists a limit distribution for cell size x asn→ ∞ and that this distribution is independent of the initial cell size x0. In-deed, the limit distribution function coincides with a probability distributionfunction of the random variable

Z = η0 +η1α

+η2α2

+ · · ·+ ηnαn

+ · · · ,(13)

where the ηk are independently, exponentially distributed random variables.

Let

F (x) = Fτn(x) = 1− Pτn > x =

1− e−x, x > 00, x ≤ 0,

(14)

p(x) = pτn(x) = F ′(x) =

e−x, x > 00, x < 0.

(15)

Denote the probability distribution function (pdf) for (13) by z(x) and lety(x) = z′(x). The function y thus corresponds to the probability densityfunction for (13). The next theorem shows that the probability density func-tion defined by (13) satisfies the pantograph equation (11) with αB/g = 1and y(0) = 0.Theorem 1. The probability distribution function for (13) satisfies

z′(x) + z(x) = z(αx)(16)

z(0) = 0;

the probability density function for (13) satisfies

y′(x) + y(x) = αy(αx)(17)

y(0) = 0.

Proof. The proof follows a method developed by [4], which is based on theself-similarity of Z. In particular, equation (13) can be recast

Z = η0 +1

α

(η1 +

η2α

+η3α2

+ · · ·)= η0 +

1

αZ1,(18)

where Z1 has the same distribution as Z.


Given random independent variables w1 and w2 with pdfs G1(x) andG2(x) respectively, the pdf of their sum w1 + w2 is given by the Stieltjesconvolution (see [9])

(G1#G2)(x) =

∫ ∞

−∞G1(x− t) dG2(t).

In addition, for any β > 0 the pdf of βG1 is G1(x/β). The pdf for Z1/α istherefore z(αx) and the pdf for η0 is F (x). Equation (18) thus implies

z(x) = z(αx)#F (x).(19)

The Stieltjes convolution (19) can be expressed as a Laplace convolution.Since z(x) = 0 for x ≤ 0 and dF (x) = p(x) dx,

z(αx)#F (x) =

∫ x

0

z(α(x− t))e−t dt;

consequently,

z(x) = z(αx) ∗ p(x),(20)

where ∗ denotes the Laplace convolution. Now,

z′(x) = z(0)e−x + α

∫ x

0

z′(α(x− t))e−t dt,

and noting that z(0) = 0 integration by parts yields

z′(x) = α

−e

−t

αz(α(x− t))

∣∣∣∣x0

− 1

α

∫ x

0

z(α(x− t))e−t dt

= z(αx)− z(x).

We thus see that z satisfies (16). Equation (17) follows immediately from(16) by differentiation noting that z′(0) = 0.

3. A Solution Method. [11] showed that problems such as (16) and(17) do not have unique solutions. Indeed, there are an infinite number ofsolutions to these problems. The requirement that solutions to (16) are alsoprobability distribution functions, however, resolves this uniqueness problem.In essence, there is only one solution to (16) such that

limx→∞

z(x) = 1.(21)


A similar comment applies to problem (17) if condition (10) is imposed.These uniqueness results can be found in [6] and [7].

Problems such as (16) and (17) can be solved using Dirichlet series or,what leads to the same thing, Laplace transforms (cf. [6], [10] and [11]).Solutions to these problems can thus be expressed in the form

z(x) =∞∑n=1

ane−αnx,

y(x) =∞∑n=1

−αnane−αnx.

The probabilistic interpretation detailed in Section 2, however, brings tothe fore a different solution method. This method entails a sequence gener-ated by convolutions. We focus exclusively on the probability distributionfunction.

Let zn be the sequence defined by

z0(x) = 1− e−x

zn+1(x) = zn(αx)#F (x),(22)

where n ≥ 0 and x ≥ 0. We show that this sequence converges to a probabil-ity distribution function z that is a solution to problem (16). One advantageof this method is that the approximations to the solution preserve the statis-tical structure of the problem. Each term of the sequence is a pdf, and it isclear from the definition of the sequence that zn corresponds to the pdf forthe random variable at the nth splitting.Lemma 2. The sequence zn converges uniformly on intervals of the formI = [0, a], where a > 0.

Proof. We note first that

zn+1(x) = zn(αx)#F (x)

=

∫ x

0

zn(α(x− ξ))e−ξ dξ.(23)

Since z0 is continuous on [0,∞) it is clear that zn is also continuous on thisinterval for all n ≥ 1. For any continuous function f : [0,∞) → R and b > 0let

∥f∥b = supξ∈[0,b]

|f(ξ)|.

It is sufficient to show that the series

∞∑n=0

(zn+1(x)− zn(x))(24)

is uniformly convergent on I. For x ∈ I,

|zn+1(x)− zn(x)| ≤∫ x

0

|zn(α(x− ξ))− zn−1(α(x− ξ))| e−ξ dξ

≤ ∥zn − zn−1∥αxΛ,

where

Λ = 1− e−a.

The above calculation can be repeated to show that

|zn+1(x)− zn(x)| ≤ ∥z1 − z0∥αnaΛn.(25)

We have

z1(x) = z0(x)−e−x − e−αx

α− 1,(26)

so that for all x ∈ [0,∞)

|z1(x)− z0(x)| ≤1

α− 1.(27)

Inequalities (25) and (27) thus give

∥zn+1 − zn∥αna ≤1

α− 1Λn.(28)

Since 0 < Λ < 1, the Weierstrass M test can be used to show that the seriesconverges uniformly on I.

Lemma 3. Each term of the sequence zn is a pdf that is differentiable on[0,∞). The limit of the sequence is also a pdf.

Proof. The sequence zn is defined by a convolution with a pdf F (x). Sincez0(x) is a pdf, z0(αx) is also a pdf. Now, z1(x) = z0(αx)#F (x). Since z1is defined by the convolution of two pdfs, z1 must also be a pdf (cf. [14,p. 37]). The argument can be repeated to show that zn must be a pdf for all


n ≥ 1. Each zn is continuous on [0,∞). Equation (23) and the FundamentalTheorem of Calculus therefore imply that the zn are differentiable and

z′n+1(x) = α

∫ x

0

z′n(α(x− ξ))e−ξ dξ.(29)

Lemma 2 shows that there is a z such that zn(x) → z(x) as n → ∞for x ∈ [0,∞). To show that z must be a pdf we study the characteristicfunctions associated with the zn. The characteristic function of zn is givenby

ϕn(t) =

∫ ∞

0

eitξz′n(ξ) dξ.

Equation (22) implies that

ϕn+1(t) = Qn(t)ψ(t),

where Qn is the characteristic function for zn(αx) and

ψ =1

1− it

is the characteristic function for F . Now,

Qn(t) =

∫ ∞

0

eitξdzn(αξ)

=

∫ ∞

0

eitξ/αz′n(ξ) dξ

= ϕn(t/α);

therefore,

ϕn+1 =1

1− itϕn(t/α),

and consequently

ϕn+1 =n+1∏k=0

(1− it

αk

)−1

.

The product

∞∏k=0

(1− it

αk

)−1

converges uniformly on all compact intervals of R;hence,

ϕn(t) → ϕ(t) =∞∏k=0

(1− it

αk

)−1

,

where ϕ is continuous on R. We can now appeal to a standard result inprobability theory (cf. [14, p. 42]) that guarantees the existence of a uniquepdf z corresponding to ϕ; moreover, zn → z as n→ ∞.

Theorem 2. Let z denote the limit of the sequence defined by (22). Then zis the unique solution to equation (16).

Proof. Integrating the right hand side of equation (29) by parts gives

z′n+1(x) = zn(αx)− zn+1(x).(30)

Now,

|z′n+1(x)− z′n(x)| = |zn(αx)− zn+1(x)− (zn−1(αx)− zn(x))|≤ |zn(αx)− zn+1(x)|+ |zn−1(αx)− zn(x)|,

and since zn is uniformly convergent on I, the above inequality shows thatz′n is uniformly convergent on I. We thus have that z is differentiable onI and that z′n → z′ as n → ∞. Equation (30) therefore implies that z is asolution to equation (16).

The uniqueness of solutions to equation (16) satisfying the given bound-ary conditions has been established in [6] and [7]. For completeness, however,we give a proof.

Suppose that there are two distinct solutions z and w to the boundary-value problem. Let δ = z − w. Then

δ′(x) = δ(αx)− δ(x)(31)

δ(0) = 0(32)

limx→∞

δ(x) = 0.(33)

By hypothesis, the solutions are distinct and therefore δ(x) = 0 for some x >0. Without loss of generality we can assume that δ(x) > 0 for some x > 0.Now, δ is differentiable, a fortiori, continuous for x ≥ 0, and the boundaryconditions (32) and (33) imply that there exists a global maximumM at somepoint 0 < xm <∞. We thus have M = δ(xm) > 0 and δ′(xm) = 0. Equation(33) implies that δ(αxm) = δ(xm); therefore, the global maximum must alsobe achieved at αxm. The arguments can be repeated to show that the globalmaximum is achieved at αnxm for all n ≥ 0; consequently, δ(αnxm) = M .Since M = 0, we have the contradiction that limx→∞ δ(x) = 0. We thusconclude that δ(x) = 0 for all x > 0.


Acknowledgements. This research was started when the first author(GD) visited Massey University and the publication was completed whenthe third author (GCW) was a Visiting Fellow at OCCAM (Oxford Centrefor Collaborative Applied Mathematics) under support provided by AwardNo. KUK-C1-013-04, made by King Abdullah University of Science andTechnology (KAUST). Both authors thank them for their support.

REFERENCES

[1] V.A. Ambartsumyan,, On the fluctuation of the brightness of the Milky Way, Dokl.Akad. Nauk SSSR,44 (1944), 223–226.

[2] B. Basse, B. Baguley, E. Marshall, W. Joseph, B. van Brunt, G. Wake, D. Wall,Modelling Cell Death in Human Tumour Cell Lines Exposed to the AnticancerDrug Paclitaxel, J. Math. Bio., 49 (2004), 329–357.

[3] L. Bogachev, G. Derfel, S. Molchanov, and J. Ockendon, On bounded solutionsof the balanced generalized pantograph equation,IMA , 145 (2008),Topics instochastic analysis and nonparametric estimation, 29-49.

[4] G. Derfel, Probabilistic method for a class of functional differential equations,Ukrain. Mat. Zh. vol. 41 (1989), 1322-1327. English translation Ukrainian Math.J.,41 (1990), 1137–1141.

[5] L. Fox, D. F. Mayers, J. A.Ockendon, ,A.B. Tayler, On a functional differentialequation, J. Ins. Math. Appl. 8 (1971), 271–307.

[6] D. P. Gaver, An absorption probablility problem, J. Math. Anal. Appl., 9 (1964),384–393.

[7] A. J. Hall, & , G.C. Wake, A functional differential equation arising in themodelling of cell-growth, J. Aust. Math. Soc. Ser. B, 30 (1989), 424–435.

[8] A. J. Hall, G. C. Wake, & P. W. Gandar, Steady size distributions for cells in onedimensional plant tissues, J. Math. Bio., 30 (1991), 101–123.

[9] I. I. Hirshman, & D. V. Widder, The Convolution Transform, Princeton UniversityPress, Princeton, 1955.

[10] A. Iserles, On the generalized pantograph functional differential equation, Euro.Jour. Appl. Math., 4, (1993), 1–38.

[11] T. Kato & J.B. McLeod, The functional-differential equation y′(x) = ay(λx) +by(x), Bull. Amer. Math. Soc., 77 (1971), 891–937.

[12] J.A. J. Metz and O. Diekmann, editors. The dynamics of physiologically structuredpopulations. Springer-Verlag, 1980.

[13] J. R. Ockendon and A.B. Tayler. The dynamics of a current collection system foran electric locomotive, Proc. Royal Soc. London A, 332, (1971),447-468.

[14] H.R. Pitt. Integration Measure and Probability. Oliver & Boyd, 1963.


VOLUME 19

2012, NO 1–2

PP. 87–106

BASIC RESULTS IN THE THEORY OF HYBRIDDIFFERENTIAL EQUATIONS WITH MIXED

PERTURBATIONS OF SECOND TYPE

B. DHAGE ∗

Abstract. In this paper, some basic results concerning the existence, strict and non-strict inequalities and existence of the maximal and minimal solutions are proved for ahybrid differential equation of mixed perturbations of second type.

Key Words. Hybrid differential equation, Existence theorem, Differential inequalities,Comparison result, Extremal solutions.

AMS(MOS) subject classification. 434K10

1. Introduction. Given a bounded interval J = [t0, t0 + a) in Rfor some fixed t0, a ∈ R with a > 0, consider the initial value problems forhybrid differential equation (in short HDE),

(1)

d

dt

[x(t)− k(t, x(t))

f(t, x(t))

]= g(t, x(t)) a.e. t ∈ J

x(t0) = x0 ∈ R.

where, f : J × R → R+ − 0 and k, g : J × R → R.

By a solution of the HDE (1) we mean a function x ∈ C(J,R) such that

(i) the function t 7→ x− k(t, x)

f(t, x)is absolutely continuous for each x ∈ R,

and(ii) x satisfies the equations in (1).

∗ Kasubai, Gurukul Colony, Ahmedpur-413 515, Dist: Latur, Maharashtra, India

87

88 B. DHAGE

The importance of the investigations of hybrid differential equations liesin the fact that they include several dynamic systems as special cases. Theconsideration of hybrid differential equations is implicit in the works of Kras-noselskii [2] and extensively treated in the several papers on hybrid differ-ential equations with different perturbations. See Burton [3], Dhage [5] andthe references therein. This class of hybrid differential equations includesthe perturbations of original differential equations in different ways. A sharpclassification of different types of perturbations of differential equations ap-pears in Dhage [7] which can be treated with hybrid fixed point theory (seeDhage [4, 6]). In this paper, we initiate the basic theory of hybrid differentialequations of mixed perturbations of second type involving three nonlinear-ities and prove the basic result such as differential inequalities, existencetheorem and maximal and minimal solutions etc. We claim that the resultsof this paper are basic and important contribution to the theory of nonlinearordinary differential equations.

2. Strict and nonstrict inequalities. We need frequently thefollowing hypothesis in what follows.

(A0) The function x→ x− k(t, x)

f(t, x)is increasing in R for all t ∈ J .

We begin by proving the basic results dealing with hybrid differentialinequalities.

Theorem 1. Assume that hypothesis (A0) hold. Suppose that there existy, z ∈ C(J,R) such that

(2)d

dt

[y(t)− k(t, y(t))

f(t, y(t))

]≤ g(t, y(t)) a.e. t ∈ J

and

(3)d

dt

[z(t)− k(t, z(t))

f(t, z(t))

]≥ g(t, z(t)) a.e. t ∈ J.

If one of the inequalities (2) and (3) is strict and

(4) y(t0) < z(t0),

then

(5) y(t) < z(t)

for all t ∈ J .

BASIC RESULTS OF HYBRID DIFFERENTIAL EQUATIONS 89

Proof. Suppose that the inequality (5) is false. Then the set Z∗ defined by

(6) Z∗ = t ∈ J \ Z | y(t) ≥ z(t)

is non-empty, where Z is a set of measure zero in J . Denote t1 = inf Z∗.Without loss of generality, we may assume that y(t1) = z(t1) and y(t) < z(t)for all t < t1.

Assume thatd

dt

[z(t)− k(t, z(t))

f(t, z(t))

]> g(t, z(t))

for t ∈ J .Denote

Y (t) =y(t)− k(t, y(t))

f(t, y(t))and Z(t) =

z(t)− k(t, z(t))

f(t, z(t))

for t ∈ J .Now continuity of y and z together with the inequality (4) implies that

there exists a t1 > t0 such that

(7) y(t1) = z(t1) and y(t) < z(t)

for all t0 ≤ t < t1.As hypothesis (A0) holds, it follows from (6) that

(8) Y (t1) = Z(t1) and Y (t) < Z(t)

for all t0 ≤ t < t1. The above relation (8) further yields

Y (t1 + h)− Y (t1)

h>Z(t1 + h)− Z(t1)

h

for small h < 0. Taking the limit as h→ 0, we obtain

(9) Y ′(t1) ≥ Z ′(t1).

Hence, from (8) and (9), we get

g(t1, y(t1)) ≥ Y ′(t1) ≥ Z ′(t1) > g(t1, z(t1)).

This is a contradiction and the proof is complete. The next result is about the nonstrict inequality for the HDE (1) on J

which requires a one-sided Lipschitz condition.

90 B. DHAGE

Theorem 2. Assume that the hypotheses of Theorem 1 hold. Suppose thatthere exists a real number L > 0 such that

(10) g(t, y(t))− g(t, z(t)) ≤ L supt0≤s≤t

[y(s)− k(s, y(s))

f(s, y(s))− z(s)− k(s, z(s))

f(s, z(s))

]whenever y(s) ≥ z(s), t0 ≤ s ≤ t. Then,

(11) y(t0) ≤ z(t0)

implies

(12) y(t) ≤ z(t)

for all t ∈ J .

Proof. Let ϵ > 0 and let a real number L > 0 be given. Set

(13)zϵ(t)− k(t, zϵ(t))

f(t, zϵ(t))=z(t)− k(t, x(t))

f(t, z(t))+ ϵe2L(t−t0)

so thatzϵ(t)− k(t, zϵ(t))

f(t, zϵ(t))>z(t)− k(t, x(t))

f(t, z(t)).

Define

Zϵ(t) =z(t)− k(t, z(t))

f(t, z(t))and Z(t) =

z(t)− k(t, z(t))

f(t, z(t))

for t ∈ J .Now using the one-sided Lipschitz condition (10), we obtain

g(t, zϵ(t))− g(t, z(t)) ≤ L supt0≤s≤t

[Zϵ(s)− Z(s)] = Lϵe2L(t−t0).

Now,

Z ′ϵ(t) = Z ′(t) + 2Lϵe2L(t−t0)

≥ g(t, z(t)) + 2Lϵe2L(t−t0)

≥ g(t, zϵ(t)) + 2Lϵe2L(t−t0) − Lϵe2L(t−t0)

= (g(t, zϵ(t)) + Lϵe2L(t−t0)

> g(t, zϵ(t))

for all t ∈ J . Also, we have

Zϵ(t0) > Z(t0) ≥ Y (t0).

Now we apply Theorem 1 with z = zϵ to yield

Y (t) < Zϵ(t)

for all t ∈ J . On taking ϵ→ 0 in the above inequality, we get

Y (t) ≤ Z(t)

which further in view of hypothesis (A0) implies that (12) holds on J . Thiscompletes the proof. Remark 1. The conclusion of Theorems 1 and 2 also remains true if wereplace the derivative in the inequalities (2) and (3) by Dini-derivative D−

of the functionx(t)− k(t, x(t))

f(t, x(t))on the bounded interval J .

3. Existence result. In this section, we prove an existence resultfor the HDE (1) on a closed and bounded interval J = [t0, t0 + a] undermixed Lipschitz and compactness conditions on the nonlinearities involvedin it. We place the HDE (1) in the space C(J,R) of continuous real-valuedfunctions defined on J . Define a supremum norm ∥ · ∥ in C(J,R) defined by

∥x∥ = supt∈J

|x(t)|

and a multiplication “ · ” in C(J,R) by

(x · y)(t) = (xy)(t) = x(t)y(t)

for x, y ∈ C(J,R). Clearly C(J,R) is a Banach algebra with respect tothe above norm and multiplication in it. By L1(J,R) we denote the spaceof Lebesgue integrable real-valued functions on J equipped with the norm∥ · ∥L1 defined by

∥x∥L1 =

∫ t0+a

t0

|x(s)| ds.

We prove the existence of solution for the HDE (1) via a hybrid fixedpoint theorem in Banach algebras due to Dhage [4].

Theorem 3 (Dhage [4]). Let S be a closed convex and bounded subset of theBanach algebra E and let A,C : E → E and B : S → E be three operatorssuch that

(a) A and C are Lipschitz with the Lipschitz constants α and β respec-tively,

(b) B is compact and continuous,

92 B. DHAGE

(c) x = AxBy + Cx for all y ∈ S =⇒ x ∈ S , and(d) αM + β < 1, where M = ∥B(S)∥ = sup∥Bx∥ : x ∈ S.

Then the operator equation AxBx+ Cx = x has a solution in S.

We consider the following hypotheses in what follows.

(A1) There exist constants L1 > 0 and L2 > 0 such that

|f(t, x)− f(t, y)| ≤ L1|x− y|

and|k(t, x)− k(t, y)| ≤ L2|x− y|

for all t ∈ J and x, y ∈ R.(A2) There exists a function h ∈ L1(J,R) such that

|g(t, x)| ≤ h(t), t ∈ J

for all x ∈ R.

The following lemma is useful in the sequel.

Lemma 1. Assume that hypothesis (A0) holds. Then for any h ∈ L1(J,R+),the function x ∈ C(J,R+) is a solution of the HDE

(14)

d

dt

[x(t)− k(t, x(t))

f(t, x(t))

]= h(t) a.e. t ∈ J

x(0) = x0 ∈ R

if and only if x satisfies the hybrid integral equation (HIE)

(15) x(t) = k(t, x(t)) +[f(t, x(t))

](x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

h(s) ds

), t ∈ J.

Proof. Let h ∈ L1(J,R+). Assume first that x is a solution of the HDE (14).

By definition,x(t)− k(t, x(t))

f(t, x(t))is almost everywhere continuous on J , and

so, almost everywhere differentiable there, whenced

dt

[x(t)− k(t, x(t))

f(t, x(t))

]is

integrable on J . Applying integration to (14) from t0 to t, we obtain the HIE(15) on J .

Conversely, assume that x satisfies the HIE (15). Then by direct differ-entiation we obtain the first equation in (14). Again, substituting t = t0 in(15) yields

x(t0)− k(t0, x(t0))

f(t0, x(t0))=x0 − k(t0, x0)

f(t0, x0).

Since the mapping x 7→ x− k(t, x)

f(t, x)is increasing in R almost everywhere for

t ∈ J , the mapping x 7→ x− k(t0, x)

f(t0, x)is injective in R, whence x(t0) = x0.

Hence the proof of the lemma is complete. Now we are in a position to prove the following existence theorem for

HDE (1).

Theorem 4. Assume that the hypotheses (A0)-(A2) hold. Further, if

(16) L1

(∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

)+ L2 < 1

then the HDE (1) has a solution defined on J .

Proof. Set E = C(J,R) and define a subset S of E defined by

(17) S = x ∈ E | ∥x∥ ≤ N

where,

N =

F0

(∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

)+K0

1− L1

(∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

)− L2

,

andF0 = sup

t∈J|f(t, 0)| and K0 = sup

t∈J|k(t, 0)|.

Clearly S is a closed, convex and bounded subset of the Banach algebra E.Now, using the hypotheses (A0) and (A2) it can be shown by an applicationof Lemma 1 that the HDE (1) is equivalent to the nonlinear HIE

(18) x(t) = k(t, x(t)) +[f(t, x(t))

](x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, x(s)) ds

)for t ∈ J .

Define three operators A,C : E → E and B : S → E by

(19) Ax(t) = f(t, x(t)), t ∈ J,

(20) Bx(t) =x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, x(s)) ds, t ∈ J.

94 B. DHAGE

and

(21) Cx(t) = k(t, x(t)), t ∈ J.

Then, the HIE (18) is transformed into an operator equation as

(22) Ax(t)Bx(t) + Cx(t) = x(t), t ∈ J.

We shall show that the operators A, B and C satisfy all the conditionsof Theorem 3.

First, we show that A is a Lipschitz operator on E with the Lipschitzconstant L1. Let x, y ∈ E. Then, by hypothesis (A1),

|Ax(t)− Ay(t)| = |f(t, x(t))− f(t, y(t))| ≤ L1|x(t)− y(t)| ≤ L1∥x− y∥

for all t ∈ J . Taking supremum over t, we obtain

∥Ax− Ay∥ ≤ L1∥x− y∥

for all x, y ∈ E. This shows that A is a Lipschitz operator on E with theLipschitz constant L1. Similarly, it can be shown that C is also a Lipschitzoperator on E with the Lipschitz constant L2.

Next, we show that B is a compact and continuous operator on S intoE. First we show that B is continuous on S. Let xn be a sequence in Sconverging to a point x ∈ S. Then by dominated convergence theorem forintegration, we obtain

limn→∞

Bxn(t) = limn→∞

(x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, xn(s)) ds

)

=x0 − k(t0, x0)

f(t0, x0)+ lim

n→∞

∫ t

t0

g(s, xn(s)) ds

=x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

[limn→∞

g(s, xn(s))]ds

=x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, x(s)) ds

= Bx(t)

for all t ∈ J . Moreover, it can be shown as below that Bxn is an equicon-tinuous sequence of functions in X. Now, following the arguments similar tothat given in Granas et al. [1], it is proved that B is a a continuous operatoron S.

Next, we show that B is compact operator on S. It is enough to showthat B(S) is a uniformly bounded and equi-continuous set in E. Let x ∈ Sbe arbitrary. Then by hypothesis (A2),

|Bx(t)| ≤∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∫ t

t0

|g(s, x(s))| ds

≤∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∫ t

t0

h(s) ds

≤∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

for all t ∈ J . Taking supremum over t,

∥Bx∥ ≤∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

for all x ∈ S. This shows that B is uniformly bounded on S.

Again, let t1, t2 ∈ J . Then for any x ∈ S, one has

|Bx(t1)−Bx(t2)| =∣∣∣∣∫ t1

0

g(s, x(s)) ds−∫ t2

0

g(s, x(s)) ds

∣∣∣∣≤∣∣∣∣∫ t1

t2

|g(s, x(s))| ds∣∣∣∣

≤ |p(t1)− p(t2)|

where p(t) =

∫ t

t0

h(s) ds. Since the function p is continuous on compact J , it

is uniformly continuous. Hence, for ϵ > 0, there exists a δ > 0 such that

|t1 − t2| < δ =⇒ |Bx(t1)−Bx(t2)| < ϵ

for all t1, t2 ∈ J and for all x ∈ S. This shows that B(S) is an equi-continuousset in E. Now the set B(S) is uniformly bounded and equicontinuous set inE, so it is compact by Arzela-Ascoli theorem. As a result, B is a continuousand compact operator on S.

Next, we show that hypothesis (c) of Theorem 3 is satisfied. Let x ∈ Eand y ∈ S be arbitrary such that x = AxBy + Cx. Then, by assumption(A1), we have

|x(t)| ≤ |Ax(t)| |By(t)|+ |Cx(t)|

96 B. DHAGE

≤∣∣f(t, x(t))∣∣ ∣∣∣ (x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, y(s)) ds

) ∣∣∣+ ∣∣k(t, x(t))∣∣≤[|f(t, x(t))− f(t, 0)|+ |f(t, 0)|

]×(∣∣∣x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

|g(s, y(s))| ds)

+ |k(t, x(t))− k(t, 0)|+ |k(t, 0)|

≤ [L1|x(t)|+ F0]

(∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∫ t

t0

h(s) ds

)+ L2|x(t)|+K0

≤F0

(∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

)+K0

1− L1

(∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

)− L2

.

Taking supremum over t,

∥x∥ ≤F0

(∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

)+K0

1− L1

(∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

)− L2

= N.

This shows that hypothesis (c) of Theorem 3 is satisfied. Finally, wehave

M = ∥B(S)∥ = sup∥Bx∥ : x ∈ S ≤∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

and so,

L1M + L2 ≤ L1

(∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

)+ L2 < 1.

Thus, all the conditions of Theorem 3 are satisfied and hence the operatorequation AxBx+Cx = x has a solution in S. As a result, the HDE (1) hasa solution defined on J . This completes the proof.

4. Maximal and minimal solutions. In this section, we shallprove the existence of maximal and minimal solutions for the HDE (1) onJ = [t0, t0 + a]. We need the following definition in what follows.

Definition 1. A solution r of the HDE (1) is said to be maximal if for anyother solution x to the HDE (1) one has x(t) ≤ r(t), for all t ∈ J. Again, asolution ρ of the HDE (1) is said to be minimal if ρ(t) ≤ x(t), for all t ∈ J,where x is any solution of the HDE (1) existing on J.

We discuss the case of maximal solution only, as the case of minimalsolution is similar and can be obtained with the similar arguments with ap-propriate modifications. Given a arbitrary small real number ϵ > 0, considerthe the following initial value problem of HDE,

(23)

d

dt

[x(t)− k(t, x(t))

f(t, x(t))

]= g(t, x(t)) + ϵ a.e. t ∈ J

x(t0) = x0 + ϵ

where, f ∈ C(J × R,R \ 0), k ∈ C(J × R,R) and g ∈ C(J × R,R).

An existence theorem for the HDE (23) can be stated as follows:

Theorem 5. Assume that the hypotheses (A0)-(A2) hold. Suppose also thatthe inequality (16) holds. Then for every small number ϵ > 0, the HDE (23)has a solution defined on J .

Proof. By hypothesis, since

L1

(∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣+ ∥h∥L1

)+ L2 < 1,

there exists an ϵ0 > 0 such that

(24) L1

(∣∣∣x0 + ϵ− k(t0, x0 + ϵ)

f(t0, x0 + ϵ)

∣∣∣+ ∥h∥L1 + ϵa

)+ L2 < 1

for all 0 < ϵ ≤ ϵ0. Now the rest of the proof is similar to Theorem 4.

Our main existence theorem for maximal solution for the HDE (1) is

Theorem 6. Assume that the hypotheses (A0)-(A2) hold. Further, if thecondition (16) holds, then the HDE (1) has a maximal solution defined on J .

Proof. Letϵn∞0

be a decreasing sequence of positive real numbers suchthatlimn→∞

ϵn = 0, where ϵ0 is a positive real number satisfying the inequality

(25) L1

(∣∣∣x0 + ϵ0 − k(t0, x0 + ϵ0)

f(t0, x0 + ϵ0)

∣∣∣+ ∥h∥L1 + ϵ0a

)+ L2 < 1.

98 B. DHAGE

The number ϵ0 exists in view of the inequality (16). Then for any solution uof the HDE (1), by Theorem 1, one has

(26) u(t) < r(t, ϵn)

for all t ∈ J and n ∈ N ∪ 0, where r(t, ϵn) is a solution of the HDE,

(27)

d

dt

[x(t)− k(t, x(t))

f(t, x(t))

]= g(t, x(t)) + ϵn a.e. t ∈ J

x(t0) = x0 + ϵn ∈ R

defined on J .

Since, by Theorems 3 and 4, r(t, ϵn) is a decreasing sequence of positivereal numbers, the limit

(28) r(t) = limn→∞

r(t, ϵn)

exists. We show that the convergence in (28) is uniform on J . To finish, itis enough to prove that the sequence r(t, ϵn) is equi-continuous in C(J,R).Let t1, t2 ∈ J be arbitrary. Then,

|r(t1, ϵn)− r(t2, ϵn)|

=∣∣k(t1, r(t1, ϵn))− k(t2, r(t2, ϵn))

∣∣+

∣∣∣∣[f(t1, r(t1, ϵn))]( x0 + ϵnf(t0, x0 + ϵn)

+

∫ t1

t0

g(s, rϵn(s)) ds+

∫ t1

t0

ϵn ds)

−[f(t2, r(t2, ϵn))

]( x0 + ϵnf(t0, x0 + ϵn)

+

∫ t2

t0

g(s, rϵn(s)) ds+

∫ t2

t0

ϵn ds)∣∣∣∣

≤∣∣k(t1, r(t1, ϵn))− k(t2, r(t2, ϵn))

∣∣+

∣∣∣∣[f(t1, r(t1, ϵn))]( x0 + ϵnf(t0, x0 + ϵn)

+

∫ t1

t0

g(s, rϵn(s)) ds+

∫ t1

t0

ϵn ds)

−[f(t2, r(t2, ϵn))

]( x0 + ϵnf(t0, x0 + ϵn)

+

∫ t1

t0

g(s, rϵn(s)) ds+

∫ t1

t0

ϵn ds)∣∣∣∣

+

∣∣∣∣[f(t2, r(t2, ϵn))]( x0 + ϵnf(t0, x0 + ϵn)

+

∫ t1

t0

g(s, rϵn(s)) ds+

∫ t2

t0

ϵn ds)

−[f(t2, r(t2, ϵn))

]( x0 + ϵnf(t0, x0 + ϵn)

+

∫ t2

t0

g(s, rϵn(s)) ds+

∫ t2

t0

ϵn ds)∣∣∣∣


≤∣∣k(t1, r(t1, ϵn))− k(t2, r(t2, ϵn))

∣∣+∣∣f(t1, r(t1, ϵn))− f(t2, r(t2, ϵn))

∣∣ (∣∣∣ x0 + ϵnf(t0, x0 + ϵn)

∣∣∣+ ∥h∥L1 + ϵna)

+ F[∣∣p(t1)− p(t2)

∣∣+ |t1 − t2|ϵn](29)

where, F = sup(t,x)∈J×[−N,N ]

|f(t, x)| and p(t) =∫ t

t0

h(s) ds.

Since f and k are continuous on compact set J × [−N,N ], they areuniformly continuous there. Hence,∣∣f(t1, r(t1, ϵn))− f(t2, r(t2, ϵn))

∣∣→ 0 as t1 → t2

and ∣∣k(t1, r(t1, ϵn))− k(t2, r(t2, ϵn))∣∣→ 0 as t1 → t2

uniformly for all n ∈ N. Similarly, since the function p is continuous oncompact set J , it is uniformly continuous and hence∣∣p(t1)− p(t2)

∣∣→ 0 as t1 → t2

uniformly for all t1, t2 ∈ J .Therefore, from the above inequality (29), it follows that

|r(t1, ϵn)− r(t1, ϵn)| → 0 as t1 → t2

uniformly for all n ∈ N. Therefore,r(t, ϵn) → r(t) as n→ ∞

for all t ∈ J . Next, we show that the function r(t) is a solution of the HDE(14) defined on J . Now, since r(t, ϵn) is a solution of the HDE (27), we have(30)

r(t, ϵn) =[f(t, r(t, ϵn))

]( x0 + ϵnf(t0, x0 + ϵn)

+

∫ t

t0

g(s, rϵn(s)) ds+

∫ t

t0

ϵn ds

)+ k(t, r(t, ϵn))

for all t ∈ J. Taking the limit as n→ ∞ in the above equation (30) yields

r(t) = k(t, r(t)) +[f(t, r(t))

](x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, r(s)) ds

)for t ∈ J . Thus, the function r is a solution of the HDE (1) on J . Finally,form the inequality (26) it follows that

u(t) ≤ r(t)

for all t ∈ J . Hence the HDE (1) has a maximal solution on J . Thiscompletes the proof.

100 B. DHAGE

5. Comparison theorems. The main problem of the differentialinequalities is to estimate a bound for the solution set for the differentialinequality related to the HDE (1). In this section we prove that the maxi-mal and minimal solutions serve the bounds for the solutions of the relateddifferential inequality to HDE (1) on J = [t0, t0 + a].

Theorem 7. Assume that the hypotheses (A0)-(A2) hold. Suppose that thecondition (16) holds. Further, if there exists a function u ∈ C(J,R) suchthat

(31)

d

dt

[u(t)− k(t, u(t))

f(t, u(t))

]≤ g(t, u(t)) a.e. t ∈ J

u(t0) ≤ x0,

Then,

(32) u(t) ≤ r(t)

for all t ∈ J , where r is a maximal solution of the HDE (1) on J .

Proof. Let ϵ > 0 be arbitrary small. Then, by Theorem 6, r(t, ϵ) is a maximalsolution of the HDE (23) and that the limit

(33) r(t) = limϵ→0

r(t, ϵ)

is uniform on J and the function r is a maximal solution of the HDE (1) onJ . Hence, we obtain

(34)

d

dt

[r(t, ϵ)− k(t, r(t, ϵ))

f(t, r(t, ϵ))

]= g(t, r(t, ϵ)) + ϵ a.e. t ∈ J

r(t0, ϵ) = x0 + ϵ.

From above inequality it follows that

(35)

d

dt

[r(t, ϵ)− k(t, r(t, ϵ))

f(t, r(t, ϵ))

]> g(t, r(t, ϵ)) a.e. t ∈ J

r(t0, ϵ) > x0.

Now we apply Theorem 1 to the inequalities (31) and (35) and conclude that

(36) u(t) < r(t, ϵ)

for all t ∈ J . This further in view of limit (33) implies that inequality (32)holds on J . This completes the proof.


Theorem 8. Assume that the hypotheses (A0)-(A2) hold. Suppose that thecondition (16) holds. Further, if there exists a function v ∈ C(J,R) suchthat

(37)

d

dt

[v(t)− k(t, v(t))

f(t, v(t))

]≥ g(t, v(t)) a.e. t ∈ J

v(t0) ≥ x0,

Then,

(38) ρ(t) ≤ v(t)

for all t ∈ J , where ρ is a minimal solution of the HDE (1) on J .Note that Theorem 7 is useful to prove the boundedness and uniqueness

of the solutions for the HDE (1) on J . A result in this direction is

Theorem 9. Assume that the hypotheses (A0)-(A2) hold and let the condi-tion (16) be satisfied. Suppose that there exists a function G : J ×R+ → R+

such that∣∣g(t, x1(t))− g(t, x2(t))∣∣

≤ G

(t, max

s∈[t0,t]

∣∣∣∣x1(s)− k(s, x1(s))

f(s, x1(s))− x2(s)− k(s, x2(s))

f(s, x2(s))

∣∣∣∣)(39)

for all t ∈ J and x1, x2 ∈ E. If identically zero function is the only solutionof the differential equation

(40) m′(t) = G(t,m(t)) a.e. t ∈ J, m(t0) = 0,

then the HDE (1) has a unique solution defined on J .

Proof. By Theorem 4, the HDE (1) has a solution defined on J . Supposethat there are two solutions u1 and u2 of the HDE (1) existing on J . Definea function m : J → R+ by

(41) m(t) =

∣∣∣∣u1(t)− k(t, u1(t))

f(t, u1(t))− u2(t)− k(t, u2(t))

f(t, u2(t))

∣∣∣∣ .As (|x(t)|)′ ≤ |x′(t)| for t ∈ J , we have that

m′(t) ≤∣∣∣∣ ddt[u1(t)− k(t, u1(t))

f(t, u1(t))

]− d

dt

[u2(t)− k(t, u2(t))

f(t, u2(t))

]∣∣∣∣≤ |(Qx1)(t)− (Qx2)(t)|

102 B. DHAGE

≤ G

(t,

∣∣∣∣u1(t)− k(t, u1(t))

f(t, u1(t))− u2(t)− k(t, u2(t))

f(t, u2(t))

∣∣∣∣)= G(t,m(t))

for all t ∈ J ; and that m(t0) = 0.

Now, we apply Theorem 7 with k ≡ 0, f ≡ 1 to get that m(t) = 0 forall t ∈ J . This gives

u1(t)− k(t, u1(t))

f(t, u1(t))=u2(t)− k(t, u2(t))

f(t, u2(t))

for all t ∈ J . Finally, in view of hypothesis (A0) we conclude that u1(t) =u2(t) on J . This completes the proof.

6. Extremal solutions in vector segment. Sometimes it is desir-able to have knowledge of existence of extremal solutions for the HDE (1) ina vector segment defined on J . Therefore, in this section we shall prove theexistence of maximal and minimal solutions for HDE (1) between the givenupper and lower solutions on J = [t0, t0 + a]. We use a hybrid fixed pointtheorem of Dhage [5] in ordered Banach algebras for establishing our results.We need the following preliminaries in the sequel.

A non-empty closed set K in a Banach algebra E is called a cone withvertex 0, if (i) K + K ⊆ K, (ii) λK ⊆ K for λ ∈ R, λ ≥ 0 and (iii)−K ∩K = 0, where 0 is the zero element of E. A cone K is called to bepositive if (iv) K K ⊆ K, where “ ” is a multiplication composition in E.We introduce an order relation ≤ in E as follows. Let x, y ∈ E. Then x ≤ yif and only if y − x ∈ K. A cone K is called to be normal if the norm ∥ · ∥is semi-monotone increasing on K, that is, there is a constant N > 0 suchthat ∥x∥ ≤ N∥y∥ for all x, y ∈ K with x ≤ y. It is known that if the coneK is normal in E, then every order-bounded set in E is norm-bounded. Thedetails of cones and their properties appear in Heikkila and Lakshmikantham[9].

Lemma 2 (Dhage [5]). Let K be a positive cone in a real Banach algebra Eand let u1, u2, v1, v2 ∈ K be such that u1 ≤ v1 and u2 ≤ v2. Then u1u2 ≤ v1v2.

For any a, b ∈ E, a ≤ b, the order interval [a, b] is a set in E given by

[a, b] = x ∈ E : a ≤ x ≤ b.

Definition 2. A mapping T : [a, b] → E is said to be nondecreasing ormonotone increasing if x ≤ y implies Tx ≤ Ty for all x, y ∈ [a, b].

We use the following fixed point theorems of Dhage [6] for proving theexistence of extremal solutions for the BVP (1) under certain monotonicityconditions.

Theorem 10. (Dhage [6]). Let K be a cone in a Banach algebra E andlet a, b ∈ E. Suppose that A,B : [a, b] → K and C : [a, b] → E are threenondecreasing operators such that

(a) A and C are Lipschitz with the Lipschitz constants α and β respec-tively,

(b) B is completely continuous, and(c) AxBx+ Cx ∈ [a, b] for each x ∈ [a, b].

Further, if the cone K is positive and normal, then the operator equationAxBx+Cx = x has a least and a greatest solution in [a, b], whenever αM +β < 1, where M = ∥B([a, b])∥ := sup∥Bx∥ : x ∈ [a, b].

We equip the space C(J,R) with the order relation ≤ with the help ofthe cone K in it defined by

(42) K =x ∈ C(J,R) : x(t) ≥ 0 for all t ∈ J

.

It is well known that the cone K is positive and normal in C(J,R). Weneed the following definitions in the sequel.Definition 3. A function a ∈ C(J,R) is called a lower solution of the HDE(1) defined on J if it satisfies

d

dt

[a(t)− k(t, a(t))

f(t, a(t))

]≤ g(s, a(s)) a.e. t ∈ J

a(t0) ≤ x0.

Similarly, a function b ∈ C(J,R) is called an upper solution of the HDE (1)defined on J if it satisfies

d

dt

[b(t)− k(t, b(t))

f(t, b(t))

]≥ g(s, b(s)) a.e. t ∈ J

b(t0) ≥ x0.

A solution to the HDE (1) is a lower as well as an upper solution for theHDE (1) defined on J and vice versa.

We consider the following set of assumptions:

(B0) f : J × R → R+ − 0 , g : J × R → R+.(B1) The HDE (1) has a lower solution a and an upper solution b defined

on J with a ≤ b.

104 B. DHAGE

(B2) The function x 7→ x− k(t, x)

f(t, x)is increasing in the closed interval[

mint∈J

a(t),maxt∈J

b(t)]almost everywhere for t ∈ J .

(B3) The functions f(t, x), g(t, x) and k(t, x) are nondecreasing in x forall t ∈ J.

(B4) There exists a function h ∈ L1(J,R+) such that

g(s, b(s)) ≤ h(t).

for all t ∈ J .

Theorem 11. Suppose that the assumptions (A1) and (B0) through (B4 )hold. Furthermore, if

(43) L1

(∣∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣∣+ ∥k∥L1

)+ L2 < 1,

then the HDE (1) has a minimal and a maximal solution in [a, b] defined onJ .

Proof. Now, the HDE (1) is equivalent to hybrid integral equation (18) de-fined on J . Let E = C(J,R). Define three operators A, B and C on [a, b]by (19), (20) and (21) respectively. Then the integral equation (18) is trans-formed into an operator equation as Ax(t)Bx(t)+Cx(t) = x(t) in the orderedBanach algebra E. Notice that hypothesis (B0) implies A,B : [a, b] → K. andC : [a, b] → E.. Since the cone K in E is normal, [a, b] is a norm-boundedset in E. Now it is shown, as in the proof of Theorem 4, that the operatorsA and C are Lipschitz with the Lipschitz constant L1 and L2. Similarly,B is completely continuous operator on [a, b] into E. Again, the hypothesis(B3) implies that A, B and C are nondecreasing on [a, b]. To see this, letx, y ∈ [a, b] be such that x ≤ y. Then, by hypothesis (B3),

Ax(t) = f(t, x(t)) ≤ f(t, y(t)) = Ay(t)

for all t ∈ J. Similarly, we have

Cx(t) = k(t, x(t)) ≤ k(t, y(t)) = Cy(t)

for all t ∈ J. Again,

Bx(t) =

(x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, x(s)) ds

)≤(x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, y(s)) ds

)

= By(t)

for all t ∈ J . So A, B and C are nondecreasing operators on [a, b]. Again,Lemma 2 and hypotheses (B0),(B3) together imply that

a(t) ≤ k(t, a(t)) + [f(t, a(t))]

(x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, a(s)) ds

)≤ k(t, x(t)) + [f(t, x(t))]

(x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, x(s)) ds

)≤ k(t, b(t)) + [f(t, b(t))]

(x0 − k(t0, x0)

f(t0, x0)+

∫ t

t0

g(s, a(s)) ds

)≤ b(t),

for all t ∈ J and x ∈ [a, b]. As a result a(t) ≤ Ax(t)Bx(t) + Cx(t) ≤ b(t) forall t ∈ J and x ∈ [a, b]. Hence, AxBx+ Cx ∈ [a, b] for all x ∈ [a, b]. Again,

M = ∥B([a, b])∥ = sup∥Bx∥ : x ∈ [a, b] ≤∣∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣∣+ ∥k∥L1

and so,

L1M + L2 ≤ L1

(∣∣∣∣x0 − k(t0, x0)

f(t0, x0)

∣∣∣∣+ ∥k∥L1

)+ L2 < 1.

Now, we apply Theorem 10 to the operator equation AxBx + Cx = x toyield that the HDE (1) has a minimal and a maximal solution in [a, b] definedon J . This completes the proof. Remark 2. The hybrid differential equations is a rich area for variety ofnonlinear ordinary as well as partial differential equations. Here, we haveconsidered a very simple hybrid differential equation involving three nonlin-earities, however, a more complex hybrid differential equation can also bestudied on similar lines with appropriate modifications. Again, the resultsproved in this paper are very fundamental and therefore, all other problemsfor the hybrid differential equation in question are still open. In a forthcomingpaper we plan to prove some approximation results for the hybrid differentialequation considered in this paper.

REFERENCES

[1] A.Granas, R. B. Guenther and J. W. Lee, Some general existence principles forCaratheodory theory of nonlinear differential equations, J. Math. Pures etAppl.70 (1991), 153–196.

106 B. DHAGE

[2] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equa-tions, Pergamon Press 1964.

[3] T. A. Burton, A fixed point theorem of Krasnoselskii, Appl. Math. Lett. 11 (1998),83-88.

[4] B. C. Dhage, A fixed point theorem in Banach algebras with applications to func-tional integral equations, Kyungpook Math. J. 44 (2004), 145-155.

[5] B. C. Dhage, A nonlinear alternative in Banach algebras with applications to func-tional differential equations, Nonlinear Funct. Anal. & Appl, 8 (2004), 563–575.

[6] B. C. Dhage, Fixed point theorems in ordered Banach algebras and applications,PanAmer. Math. J, 9(4) (1999), 93–102.

[7] B. C. Dhage, Quadratic perturbations of periodic boundary value problems of secondorder ordinary differential equations, Diff. Equ. & Appl, 2 (2010), 465–486.

[8] B. C. Dhage and V. Lakshmikantham, Basic results on hybrid differential equations,Nonlinear Analysis: Hybrid Systems, 4 (2010), 414–424.

[9] S. Heikkila and V. Lakshmikantham, Monotone Iterative Technique for NonlinearDiscontinues Differential Equations, Marcel Dekker Inc., New York, 1994.

[10] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, AcademicPress, New York, 1969.


VOLUME 19

2012, NO 1–2

PP. 107–119

STABILIZATION OF SOLUTIONS OF HYPERBOLICEQUATION WITH GROWING COEFFICIENT IN

UNBOUNDED DOMAINS ∗

A. FILINOVSKIY †

Abstract. Let Ω ⊂ Rn, n ≥ 2, be an unbounded domain, whose closure does not con-tain the origin with smooth, probably unbounded, star-shaped boundary Γ. We considerthe first mixed problem for the hyperbolic equation with growing coefficient and study thebehavior of solutions to the problem at large times. We also obtain the estimates to theresolvent of spectral problem for different values of the parameters

Key Words. hyperbolic equation, unbounded domain, stabilization of solutions,weighted Laplace operator, spectral problem

AMS(MOS) subject classification. 35L20, 35J05

Let Ω ⊂ Rn, n ≥ 2, be an unbounded domain whose closure does notcontain the origin with smooth boundary Γ. Let us consider the first mixedproblem for the weighted wave equation

utt + Lu = 0, t > 0, x ∈ Ω,(1)

u(0, x) = f(x) , ut(0, x) = g(x), x ∈ Ω,(2)

u|Γ = 0, t > 0,(3)

where Lu = −rs∆u, r = |x|, s ≥ 0, is the weighted Laplace operator. It isassumed that the initial functions f , g are real-valued, smooth on Ω = Ω∪Γ,compatible with the boundary condition (3) and boundedly supported.

∗ The research was supported by the grant DSP N 2.1.1/7161† Moscow State Technical University, 105005, 2nd Baumanskaya, 5, Moscow, Russian

Federation

107

108 A. FILINOVSKIY

The solution of problem (1) – (3) in the domain Ω of an arbitrary geom-etry satisfiy the energy conservation law

(4) E(t) = ∥∇u∥2L2(Ω) + ∥r−s/2ut∥2L2(Ω) = E(0), t > 0.

In this paper we study the behavior of solutions to the problem (1) – (3)for large values of time.

The estimates to solutions of the mixed problem for the wave equa-tion (Lu = −∆u) for large t were studied for various classes of unboundeddomains ([3]-[7],[9], [10], [13]-[18]). In [11] were investigated some prob-lems of asymptotic behavior for solutions of the automorphic wave equation(Lu = −x22∆u+ 1

4u) in some special unbounded domain in R2.

We will consider the domains with boundary satisfying star-shapenesscondition

(5) (ν, x) ≤ 0, x ∈ Γ,

where ν is the outward unit normal vector to Γ. Without loss of generalitywe can assume that ln r > 1 in Ω.Theorem 1. Let n ≥ 2 and 0 ≤ s < 2. Then for any domain Ω ⊂ Rn

satisfying condition (5) the solution of problem (1) – (3) satisfy an estimate

(6)

∫ ∞

0

∥r−1(ln r)−qu∥2L2(Ω) dt ≤ C,

where q = 1 for n = 2, q = 0 for n ≥ 3, and the constant C depend on f andg.

Let us define the Hilbert space of integrable functions L2,s(Ω) with thenorm ∥u∥L2,s(Ω) = ∥r−s/2u∥L2(Ω). We also define the space of functionsH1

s (Ω) =u : u ∈ L2,s(Ω), uxj

∈ L2(Ω), j = 1, . . . , n

with the norm

∥u∥2H1s (Ω) = ∥∇u∥2L2(Ω) + ∥u∥2L2,s(Ω). By

o

H1s(Ω) denote the subspace of H1

s (Ω)

which is closure of the set of functions u ∈ H1s (Ω) vanishing in some neigh-

borhood of Γ.

Operator Lu = −rs∆u can be treated as a self-adjoint operator in thespace L2,s(Ω). Spectral properties of elliptic operators with growing coef-ficients in weighted spaces were studied in [1], [2], [11], [12]. By [6], thespectrum of operator L for star-shaped Γ and 0 ≤ s ≤ 2 fills an interval[λ0,∞) where λ0(Ω) ≥ 0 for s = 2 and λ0 = 0 for 0 ≤ s < 2. Moreover([6]), the spectrum of L is continuous for s = 2 and absolutely continuousfor 0 ≤ s < 2.

STABILIZATION OF SOLUTIONS OF HYPERBOLIC EQUATION 109

Consider the first boundary value problem for the weighted Helmholtzequation

(L− k2)v = h, x ∈ Ω,(7)

v|Γ = 0.(8)

For any k = ω + iµ ∈ Im k > 0 the problem (7), (8) has a unique

solution v(x, k) ∈o

H1s(Ω).

The proof of Theorem 1 is based on the properties of the solution of prob-lem (7), (8) in the half-plane Im k > 0, which are formulated in Theorem2. Estimates (9) and (11) were announced in [6].Theorem 2. Let a domain Ω satisfy condition (5) and 0 ≤ s < 2. Then,for any −∞ < ω < +∞, µ > 0,

(2− s)−1/2∥|(ν, x)|1/2∇v∥L2(Γ) + ∥vr − iωr−s/2v∥L2(Ω) +

+∥r−1∇Θv∥L2(Ω) + ∥r−1(ln r)−qv∥L2(Ω) ≤C

2− s∥r1−s(ln r)qh∥L2(Ω),(9)

∥rs/2−1(ln r)−q∇v∥L2(Ω) ≤C(|ω|+ 1)

2− s∥r1−s(ln r)qh∥L2(Ω).(10)

Here r−2|∇Θv|2 = |∇v|2 − |vr|2, q = 1 for n = 2 and q = 0 for n ≥ 3.Theorem 3. Let a domain Ω satisfy condition (5) and 0 ≤ s < 2. Then,for any −∞ < ω < +∞, µ > 0

(11)

∥∥∥∥rs/2−2−δ du

dk

∥∥∥∥L2(Ω)

≤ C|ω|+ 1

(2− s)2|k|∥∥r2−3s/2+δ f

∥∥L2(Ω)

,

where δ = 0 for n ≥ 3 and δ > 0 for n = 2.Note that generally the factor 1/|k| on the right-hand side of inequality

(11) cannot be replaced by 1/|k|1−σ, where σ > 0. Let n = 2, s = 0, Ω be

the exterior of the circle of radius r0, let h(x) = ξ(r), ξ(r) ∈o

C∞(r0,+∞),ξ(r) ≥ 0 and

∫ +∞r0

ξ(r) dr > 0. In this case, for any R > r0, there are positiveconstants N , C ′, and C ′′ such that the following estimates hold for Im k > 0and |k| < N :

C ′|k|−1 ln−2 |k| ≤∥∥∥∥dvdk

∥∥∥∥L2(ΩR)

≤ C ′′|k|−1 ln−2 |k|,

ΩR = Ω ∩ r < R.

The paper consists of three parts.In first part we establish uniform estimates for the solutions of the sta-

tionary problem.

110 A. FILINOVSKIY

In second part we study the behavior of solutions of the stationary prob-lem for large values of the parameter.

In third part we establish estimates for the solutions of the non-stationarymixed problem.

1. Estimates for the solutions of the first boundary value prob-lem for the weighted Helmholtz equation in the upper half-plane.Let us consider a generalized solution (in the sense of distributions) of

problem (1) – (3) belonging to the energy class (see [8]), i.e., a function

u ∈ C([0,+∞);

o

H 1s(Ω)

)such that ut ∈ C ([0,+∞);L2,s(Ω)) which satisfies

the initial conditions u(0, x) = f(x) and ut(0, x) = g(x), the integral identity∫QT

((∇u,∇χ)− r−sutχt

)dtdx =

∫Ω

r−sg(x)χ(0, x) dx,(12)

QT = (0, T )× Ω, T > 0,

for any χ(t, x) ∈ H1s (QT ) such that χ|(0,T )×Γ = 0 and χ|t=T = 0, and also

the energy relation (4).The uniqueness of such solution can be proved by methods of [8], Chapter

4. Also, use the finite speed of propagation for the equation (1), by themethods of [8] we can prove the existence of such solution for the initialfunctions ηRf , ηRg, R > 0, where ηR(r) = η(r−R), η(z) ∈ C∞(R), 0 ≤ η ≤1, η = 1 for z < 0 and η = 0 for z > 1. Therefore we can prove that for thesequence Rj → ∞, j → ∞, the corresponding sequence of solutions uj have

a limit function u ∈ C([0,+∞);

o

H1s(Ω)

)such that ut ∈ C

([0,+∞);L2,s(Ω)

)and u satisfy the integral identity (12) and the energy relation (4).

Use the same methods as in paper [3], one can prove that for any valueof a parameter k = ω + iµ, −∞ < ω <∞, µ > 0, the function

(13) v(x, k) =

∫ ∞

0

eiktu(t, x) dt

exists. The function v belong to the spaceo

H1s(Ω), it is analytic with respect

to k ∈ Imk > 0 relative to the norm of H1(ΩR), R > 0, and satisfies theintegral identity

(14)

∫Ω

((∇v,∇ψ)− k2r−svψ

)dx =

∫Ω

r−s(g − ikf)ψ dx

for an arbitrary ψ(x) ∈o

H1s(Ω), i.e., v is a generalized solution of problem (7),

(8) with h = g − ikf . By setting ψ = v in (14), we have the relation

(15)

∫Ω

(|∇v|2 + (µ2 − ω2 − 2iωµ)r−s|v|2

)dx =

∫Ω

r−shv dx.

Singling out the real and imaginary parts in (15), we obtain

(16)

∫Ω

(|∇v|2 + (µ2 − ω2)r−s|v|2

)dx = Re

(∫Ω

r−shv dx

),

(17) 2ωµ

∫Ω

r−s|v|2 dx = −Im

(∫Ω

r−shv dx

).

In this case, for µ > 0 and ω = 0, the uniqueness of a solution of problem

(7), (8) in the spaceo

H 1s(Ω) follows from relation (17) and, for µ > 0 and

ω = 0, from relation (16).

Thus, for any −∞ < ω < ∞, µ > 0, the function v ∈o

H1s(Ω) is a unique

solution of problem (7), (8). Moreover, for any R > 0, the function v belongsto the space H2(ΩR) and is analytic with respect to k relative to the normof H2(ΩR).

Let w(x, k) = ve−iωr1−s/2

1−s/2 . The function w is the solution of the boundaryvalue problem

∆w + iω(2r−s/2wr +

((n− s

2− 1)r−1−s/2 +

+2µr−s)w)− µ2r−sw = hr−se−

iωr1−s/2

1−s/2 , x ∈ Ω,(18)

(19) w|Γ = 0.

Lemma 1. For any −∞ < ω <∞, µ > 0 function w satisfy the relation

−∫Γ

(ν, x) |wν |2 ds+∫Ω

((1− s

2+ 2µr1−s/2

)|∇w|2 +

+µ2(s2r−s + 2µr1−s/2

)|w|2

)dx =

=

∫Ω

(µ2r−s|w|2 − 2

(1− s

2

)µr−s/2Re (wwr)

)dx−

−Re

∫Ω

r−s(2rwr +

(n− s

2− 1 + 2µr1−s/2

)w)he−

iωr1−s/2

1−s/2 dx.(20)

Proof of Lemma 1. Let us multiply equation (18) by

(21) 2rwr +(n− s

2− 1 + 2µr1−s/2

)w

112 A. FILINOVSKIY

and take the real part of the relation thus obtained. This gives

div(Re((

2rwr +(n− s

2− 1 + 2µr1−s/2

)w)∇w)−

−(|∇w|2 + µ2r−s|w|2

)x)−(1− s

2+ 2µr1−s/2

)|∇w|2 +

+µ2((

1− s

2

)r−s − 2µr1−3s/2

)|w|2 − 2µ

(1− s

2

)r−s/2Re (wwr) =

= r−sRe

((2rwr +

(n− s

2− 1 + 2µr1−s/2

)w)he−

iωr1−s/2

1−s/2

).(22)

Integrating relation (22) over the domain ΩR and taking into account thatthe terms with surface integrals vanish at R → ∞, we obtain the relation(20).

Lemma 2. Let the domain Ω satisfy condition (5) and 0 ≤ s < 2. Then,for any −∞ < ω < +∞, µ > 0

(2− s)

∫Γ

|(ν, x)||∇w|2 ds+

+(2− s)2∫Ω

(|∇w|2 + (r−2 ln−2q r)|w|2

)dx ≤

≤ C

∫Ω

r2(1−s) ln2q r|h|2 dx.(23)

Proof of Lemma 2. It follows from relation (20) and condition (5) that∫Γ

|(ν, x)||wν |2 ds+ (2− s)

∫Ω

|∇w|2 dx ≤

≤ C(µ2

∫Ω

r−s|wr|2 dx+ µ(2− s)

∫Ω

r−s/2|w||wr| dx+

+

∫Ω

r−s(r|wr|+

(1 + µr1−s/2

)|w|)|h| dx

).(24)

Then, apply the inequalities (see [3])∫Ω

r−2 ln−2q r |w|2 dx ≤ C

∫Ω

|wr|2 dx,(25)

µ2

∫Ω

r−s|w|2 dx ≤ C

∫Ω

r2(1−s) ln2q |h|2 dx,(26)

and Young inequality, we obtain the estimate (23). This proves Lemma 4.


Proof of Theorem 2. Since |w| = |v|, |∇w|2 = |vr−iωr−s/2v|2+r−2|∇Θv|2and |wν ||Γ = |vν |Γ = |∇v||Γ = |∇w||Γ, the inequality (23) can be transformedto

1

2− s

∫Γ

|(ν, x)||∇v|2 ds+∫Ω

(|vr − iωr−s/2v|2 +

+r−2(|∇Θv|2 + ln−2q r |v|2))dx ≤ C

(2− s)2

∫Ω

r2(1−s) ln2q r |h|2 dx,

and implies the estimate (9).Now, we have

|∇v|2 = |wr + iωr−s/2w|2 + r−2|∇Θw|2 ≤≤ C

(|∇w|2 + ω2r−s|w|2

).(27)

So, use the relation (27), we obtain the required estimate (10):∫Ω

rs−2 ln−2q r |∇v|2 dx ≤

≤ C

(∫Ω

rs−2 ln−2q r |∇w|2 dx+ ω2

∫Ω

r−2 ln−2q r |w|2 dx)

≤

≤ C

(∫Ω

|∇w|2 dx+ ω2

∫Ω

r−2 ln−2q r |w|2 dx)

≤

≤ C(ω2 + 1)

(2− s)2

∫Ω

r2 ln2q r |h|2 dx.

Theorem 2 is proved.Proof of Theorem 3. Let us prove inequality (11). We can differentiate

equation (7) and the boundary condition (8) with respect to k and conclude

that the function dvdk

∈o

H1s(Ω) is a generalized solution of the boundary value

problem

(L− k2)dv

dk= 2kv, x ∈ Ω,(28)

dv

dk

∣∣∣∣Γ

= 0.(29)

Let γ > 0. Consider the boundary value problem

(L− k2)y = r−γ dv

dk, x ∈ Ω,(30)

y|Γ = 0.(31)

114 A. FILINOVSKIY

Since the right-hand side of equation (30) belong to L2,s(Ω), it follows

that problem (30), (31) has a generalized solution y ∈o

H1s(Ω), and this solution

is unique. By virtue of (9), (10), the estimates

1

2− s

∫Γ

|(ν, x)||∇y|2 ds+∫Ω

r−2 ln−2q r |y|2 dx ≤

≤ C

(2− s)2

∫Ω

r2(1−s−γ) ln2q r

∣∣∣∣dvdk∣∣∣∣2 dx,(32)

(33)

∫Ω

rs−2 ln−2q r |∇y|2 dx ≤ C(ω2 + 1)

(2− s)2

∫Ω

r2−2s−2γ ln2q r

∣∣∣∣dvdk∣∣∣∣2 dx

holds for γ > 1 − s2and for any −∞ < ω < +∞, µ > 0. Let us multiply

equation (28) by r−sy and integrate over the domain Ω considering that the

functions dvdk

and y belongs to the spaceo

H1s(Ω). This gives the relation

(34)

∫Ω

r−(s+γ)

∣∣∣∣dvdk∣∣∣∣2 dx = 2k

∫Ω

r−svy dx.

It follows from equations (7) and (30) that

−r−s(((x,∇y) + (n− 2)y)h+ r−γ(x,∇v)dv

dk

)=

= ((x,∇y) + (n− 2)y)(∆v + k2r−sv) + (x,∇v)(∆y + k2r−sy) =

= div((x,∇y)∇v + (x,∇v)∇y − (∇v,∇y)x+ (n− 2)y∇v + k2r−svyx

)+

+(s− 2)r−sk2vy.(35)

Integrating relation (35) over the domain Ω with regard to the boundaryconditions (8) and (31), we obtain the relation

∫Ω

r−(s+γ)

∣∣∣∣dvdk∣∣∣∣2 dx = 2k

∫Ω

r−svy dx =1

(2− s)k

(∫Γ

(ν, x)vνyν ds+

+

∫Ω

r−s

(((x,∇y) + (n− 2)y)h+ r−γ(x,∇v)dv

dk

)dx

).(36)


Relation (36) imply the estimates∫Ω

r−(s+γ)

∣∣∣∣dvdk∣∣∣∣2 dx ≤ C

(2− s)|k|

(∫Γ

|(ν, x)||vν ||yν | ds+∫Ω

r1−s|∇y||h| dx+

+

∫Ω

r−s|y||h| dx+∫Ω

r1−s−γ|∇v|∣∣∣∣dvdk

∣∣∣∣ dx) ≤

≤ C

(2− s)|k|

((∫Γ

|(ν, x)||vν |2 ds) 1

2(∫

Γ

|(ν, x)||yν |2 ds) 1

2

+

+

(∫Ω

rs−2 ln−2q r |∇y|2 dx) 1

2(∫

Ω

r4−3s ln2q r|h|2 dx) 1

2

+

+

(∫Ω

r−2 ln−2q r|y|2 dx) 1

2(∫

Ω


2

+

+

(∫Ω

rs−2 ln−2q r |∇v|2 dx) 1

2

(∫Ω

r4−3s−2γ ln−2q r

∣∣∣∣dvdk∣∣∣∣2 dx

) 12

.(37)

Applying inequalities (9), (10), (32) and (33) to (37), we see that∫Ω

r−(s+γ)

∣∣∣∣dvdk∣∣∣∣2 dx ≤

≤ C

(2− s)2|k|

(∫Ω


∣∣∣∣dvdk∣∣∣∣2 dx

) 12 (∫

Ω

r2−2s ln2q r |h|2 dx) 1

2

+

+(ω2 + 1)1/2

(∫Ω


∣∣∣∣dvdk∣∣∣∣2 dx

) 12 (∫

Ω


2

+

+

(∫Ω


∣∣∣∣dvdk∣∣∣∣2 dx

) 12 (∫

Ω


2

+

+(ω2 + 1)1/2

(∫Ω


∣∣∣∣dvdk∣∣∣∣2 dx

) 12

×

×(∫

Ω


2

).(38)

Thus, let n ≥ 3. For γ = 4− 2s it follows from (38) that

(39)

∫Ω

rs−4

∣∣∣∣dvdk∣∣∣∣2 dx ≤ C(ω2 + 1)

(2− s)4|k|2

∫Ω

r4−3s|h|2 dx.

116 A. FILINOVSKIY

Now, for n = 2 let γ = 4 − 2s + δ1, δ1 > 0. Therefore we obtain from (38)the relation ∫

Ω

rs−4−δ1

∣∣∣∣dvdk∣∣∣∣2 dx ≤

≤ C(ω2 + 1)1/2

(2− s)2|k|

(∫Ω

rs−4−2δ1 ln2 r

∣∣∣∣dvdk∣∣∣∣2 dx

) 12

×

×(∫

Ω

r4−3s ln2 r |h|2 dx) 1

2

.(40)

The relation (40) means that for any δ > 0 we have an estimate

(41)

∫Ω

rs−4−2δ

∣∣∣∣dvdk∣∣∣∣2 dx ≤ C(ω2 + 1)

(2− s)4|k|2

∫Ω

r4−3s+2δ|h|2 dx.

The inequalities (39), (41) completes the proof of Theorem 3.

2. Behavior of solutions of the stationary problem for largevalues of the parameter. Let u be a solution of problem (1) – (3).Consider stationary boundary value problem

Lv − k2v = g − ikf, x ∈ Ω,(42)

v|Γ = 0.(43)

By (9), the inequality

∥r−1 ln−q r v∥2L2(Ω) ≤≤ C

(∥r1−s lnq r g∥2L2(Ω) + |k|2∥r1−s lnq r f∥2L2(Ω)

)(44)

hold for Im k > 0. It follows from equation (42) that

(45) v = k−2(Lv − g + ikf),

so

∥r−1 ln−q r v∥2L2(Ω) ≤ C|k|−4(∥r−1 ln−q r Lv∥2L2(Ω)+

+∥r−1 ln−q r g∥2L2(Ω) + |k|2∥r−1 ln−q r f∥2L2(Ω)

).(46)

Relations (42), (43) shows that the function Lv is the solution of the problem

L2v − k2Lv = Lg − ikLf, x ∈ Ω,

Lv|Γ = 0,

and hence, by (9), the inequality

∥r−1 ln−q r Lv∥2L2(Ω) ≤≤ C

(∥r1−s lnq r Lg∥2L2(Ω) + |k|2∥r1−s lnq r Lf∥2L2(Ω)

)(47)

holds. With regard to the estimates (46) and (47) we conclude that

∥r−1 ln−q r v∥2L2(Ω) ≤ C|k|−4(∥r1−s lnq r Lg∥2L2(Ω) + |k|2∥r1−s lnq r Lf∥2L2(Ω)+

+ ∥r−1 ln−q r g∥2L2(Ω) + |k|2∥r−1 ln−q r f∥2L2(Ω)

)≤ C1(|k|2 + 1)

|k|4,(48)

where the constant C1 depends on f and g.Estimating the function v by use inequality (44) in the domain KN =

Im k > 0, |k| < N, N > 0, and by inequality (48) in the domain Im k >0 \KN , we obtain the estimate

(49) ∥r−1 ln−q r v∥2L2(Ω) ≤C

|k|2 + 1, Im k > 0, β > 1,

where the constant C depend on f , g, s and β.

3. Estimates of the local energy of solutions of the mixed prob-lem for the wave equation for large values of time. Proof of Theorem

1. Let v ∈o

H1s(Ω), Im k > 0, be the solution of problem (42), (43). For almost

all t > 0 and for x ∈ Ω, the following inversion formula holds:

(50) e−µtu(t, x) =1

2π

∫ ∞

−∞e−iωtv(x, ω + iµ) dω.

With regard to inequality (49) and the Plancherel theorem, we derive theestimate ∫ ∞

0

e−2µt∥r−1 ln−q r u∥2L2(Ω) dt =

=1

2π

∫ ∞

−∞∥r−1 ln−q r v(x, ω + iµ)∥2L2(Ω) dω ≤ C,(51)

where the constant C does not depend on µ. Thus,

(52)

∫ ∞

0

e−2µt∥r−1 ln−q r u∥2L2(Ω) dt ≤ C, µ > 0.

Since the functions e−2µt∥r−1 ln−q r u∥2L2(Ω) satisfy the condition

e−2µt∥r−1 ln−q r u∥2L2(Ω) ∥r−1 ln−q r u∥2L2(Ω) as µ→ +0, t ∈ [0,+∞),

118 A. FILINOVSKIY

it follows from inequality (52) and from the monotone convergence theoremthat

(53)

∫ ∞

0

∥r−1 ln−q r u∥2L2(Ω) dt ≤ C.

The inequality (53) completes the proof of Theorem 1.

REFERENCES

[1] D.M. Eidus, The perturbed Laplace operator in a weighted L2-space, Journal ofFunctional Analysis, 100 (1991), 400 – 410.

[2] D.M. Eidus, The limiting absorbtion principle for an acoustic equation with a re-fraction coefficient vanishing at infinity, Asymptotic Analysis, 63 (2009), 143 –150.

[3] A.V. Filinovskiy, Stabilization of solutions of the wave equation in unbounded do-mains, Matematicheskij Sbornik, 187 (1996), 131 – 160.

[4] A.V. Filinovskiy, Stabilization of solutions of the first mixed problem for the waveequation in domains with non-compact boundaries,Matematicheskij Sbornik, 193(2002), 107 – 138.

[5] A.V. Filinovskiy, On the decay rate of solutions of the wave equation in domainswith star-shaped boundaries, Journal of Matematical Sciences, 143 (2007), 3429– 3440.

[6] A.V. Filinovskiy, On the spectral properties of the weighted Laplace operator inunbounded domains, Proceedings of the International Conference dedicated to85 anniversary of L.D. Kudrjavcev, Russian Peoples Friendship University,Moscow, March 2008, 344 – 346.

[7] V.Ya. Ivrii, Exponential decay of the solution of the wave equation outside an almoststar-shaped region, Doklady Akademii Nauk SSSR, 189 (1969), 938 – 940.

[8] O.A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Nauka,Moscow, 1973, [in Russian].

[9] P.D. Lax, C. S. Morawetz, R. S. Phillips, Exponential decay of solutions of the waveequation in the exterior of a star-shaped obstacle, Communications on Pure andApplied Mathematics, 16 (1963), 477 – 486.

[10] P.D. Lax, R. S. Phillips, Scattering Theory, Academic Press, 1967.[11] P.D. Lax, R. S. Phillips, Scattering Theory for Automorphic Functions, Princeton

University Press, 1976.[12] R.T. Lewis, Singular elliptic operators of second order with purely discrete spectra,

Transactions of American Mathematical Society, 271 (1982), 653 – 666.[13] V. P. Mihailov, On the principle of limiting amplitude, Doklady Akademii Nauk

SSSR, 159 (1964), 750 – 752.[14] C. S. Morawetz, The decay of solutions of the exterior initial-boundary value problem

for the wave equation, Communications on Pure and Applied Mathematics, 14(1961), 561 – 568.

[15] C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation,Communications on Pure and Applied Mathematics, 28 (1975), 229 – 264.

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[17] L.A. Muravei, Asymptotic behavior, for large time values, of the solutions of ex-terior boundary value problems for a wave equation with two space variables,Matematicheskij Sbornik, 107 (1978), 84 – 133.

[18] L.A. Muravei, The wave equation and the Helmholtz equation in an unboundeddomain with star-shaped boundary, Trudy Matematicheskogo instituta im. V.A.Steklova, 185 (1988), 171–180.


VOLUME 19

2012, NO 1–2

PP. 121–129

EXISTENCE OF INFINITE PERIODIC SOLUTIONS FOR ACLASS OF FIRST-ORDER DELAY DIFFERENTIAL

EQUATIONS ∗

CHENGJUN GUO † , DONAL O’REGAN‡ , YUANTONG XU§ AND

RAVI P. AGARWAL¶

Abstract. This paper is concerned with the existence of infinitely many periodicsolutions for a class of first-order delay differential equation. Using a generalized ver-sion of the Poincare-Birkhoff fixed point theorem, the authors establish conditions on fwhich guarantee that a first-order delay differential equation has infinitely many periodicsolutions.

1. Introduction. The purpose of this paper is to discuss the exis-tence of infinitely many periodic solutions for the following equation

x′(t) = f(t, x(t− τ)), (1.1)

where τ > 0 is a constant, f(t, x) ∈ C(R ×R,R) is τ−periodic in t and islocally Lipschitzian.

Existence of multiple periodic solutions for differential equations wasstudied in [2-17] and we note that there are some results on the existenceof multiple periodic solutions to delay differential equations[6-8,11,12,15-17].However only a few results are available on the existence of infinitely manyperiodic solutions for delay differential equations [15]. In this paper, by using

∗ This project is supported by grant 10871213 from NNSF of China, by grant 06021578from NSF of Guangdong.

† School of Applied Mathematics, Guangdong University of Technology 510006,P.R.China,

‡ Department of Mathematics, National University of Ireland, Galway, Ireland§ Department of Mathematics, Sun Yat-sen University, Guangzhou Guangdong

510275,P.R China¶ Department of Mathematics, Texas A&M University -Kingsville, Kingsville, TX

78363, USA

121

122 C. GUO, D. O’REGAN, Y. XU, R. AGARWAL

a generalized version of the Poincare-Birkhoff fixed point theorem, we showthat there are infinitely many periodic solutions of Eq. (1.1).

Let (r, θ) be a polar coordinate expression and O be the correspondingpolar point. Let A be an annular region in R2 : R1 ≤ r ≤ R2, (0 < R1 < R2).Definition 1. A map T : A → R2\0 is called torsional if

(i)

r∗ = g(r, θ), θ∗ = θ + h(r, θ),

where (r∗, θ∗) denotes the image of (r, θ) under T , and g and h are continuousand 2π−periodic in θ.

(ii) the twist condition:

h(R1, θ) · h(R2, θ) < 0

is satisfied.Let A be an annular region in R2 bounded by two disjoint simple closed

curves Γ1 and Γ2. Let Di denote the open set bounded by Γi,(i=1,2). Assumethat 0 ∈ D1 ⊂ D1 ⊂ D2. For the sake of completeness, we first state thefollowing generalized version of the Poincare-Birkhoff fixed point theorem.

Theorem A. (See [1,5,10]). Suppose T : A → T (A) ⊂ R2\0 is anarea-perserving homeomorphism. Suppose

(1) T is torsional;

(2) there exists an area-preserving homeomorphism T1 : D2 → R2, suchthat T1|A = T and 0 ∈ T1(D1).

Then T has at least two fixed points in A.

2. Main result. We now assume that f satisfies the following:

(H1) f(t, ·) is odd, i.e. for any x ∈ R, f(t,−x) = −f(t, x);(H2)

lim|x|→∞

f(t, x)

x= +∞;

(H3) there exist two positive constants ρ0, R0, such that

yf(t, y) + xf(t, x) ≥ ρ0(x2 + y2), x2 + y2 ≥ R2

0.

Now let us state our main result.

EXISTENCE OF INFINITE PERIODIC SOLUTION 123

Theorem 1. Suppose that f satisfies (H1)− (H3) and(H4) Every solution of Eq.(1.1) exists on the whole line.Then Eq.(1.1) has infinitely many nonconstant 4τ−periodic solutions.In order to prove the main theorem we need some preliminaries. Set

E := x|x(t− τ) = −x(t+ τ), ∀t ∈ R.

This implies that x(t) = x(t+ 4τ).Let x(t− τ) = y(t), x ∈ E, so we have from (H1)

x′(t) = f(t, y(t)),

y′(t) = −f(t, x(t)), x ∈ E.

(2.1)

It is easy to see that, under the assumption of locally Lipschitzian on f , (2.1)together with initial conditions x(0) = x0, y(0) = y0 has a unique solutionand through any point (x0, y0) ∈ R2 has a unique orbit[9].

Under the assumption of (H4), define a continuous mapping Tt : R2 →

R2 by

Tt(x, y) = (φ(t, x, y), ψ(t, x, y)).

We have

(i) T4τ is an area-preserving mapping (for details see [10]);The space E is the space of 4τ−periodic functions. Then T4τ is a

Poincare mapping. This means that if (x0, y0) is a fixed point of T4τ , thenx = φ(t, x0, y0), y = ψ(t, x0, y0) is a 4τ−periodic solution of ( 2.1). Hence T4τis an area-preserving mapping.

(ii) For each fixed t, from (H4), Tt is a homeomorphism.Lemma 1. Suppose that Eq.(1.1) satisfies the condition (H4). Then thereexists a constant α > 0 for arbitrary β ≥ 0, such that

|Tt(u)| > β, |u| > α,

where | · | denotes the the Euclidean norm of R2 and u ∈ R2.

Proof. Fix β ≥ 0. Arguing indirectly, we suppose that the lemma is not true.Then there exist two sequences un ⊂ R2, tn ⊂ [0, 4τ ] such that

|Ttn(un)| ≤ β, |un| → ∞.

Hence we can extract subsequences unk, tnk

respectively with

limk→∞ tnk= t0 ∈ [0, 4τ ]

and

limk→∞ qnk= limk→∞ Ttnk

(unk) = q0, (|q0| ≤ β).

Noting that unk= T−1

tnk(qnk

), we have

unk→ limk→∞ T−1

tnk(qnk

) = T−1t0 (q0).

This contradicts |un| → ∞. Therefore Lemma 2.2 is true.

2.1. Proof of Theorem 2.1.

Proof. From Lemma 2.2, these exists a R1 > 0, such that

|Tt(u)| > 0, t ∈ [0, 4τ ] (2.2)

with |u| ≥ R1 and u ∈ R2.Suppose A is an annular region in R2 with R1 ≤ r ≤ R2(0 < R1 < R2)

such that 0 ∈ Tt(A) for t ∈ [0, 4τ ], where R2 will be defined later.Let

x = r cos θ,

y = r sin θ,(2.3)

where x and y are defined in (2.1).From (2.1) and (2.3), we have

x′= r

′cos θ − r sin θθ

′= f(t, r sin θ),

y′= r

′sin θ + r cos θθ

′= −f(t, r cos θ).

Hence we have r′= f(t, r sin θ) cos θ − sin θf(t, r cos θ),

θ′= −f(t,r sin θ) sin θ

r− f(t,r cos θ) cos2 θ

r cos θ.

(2.4)

Note (2.4) shows that r and θ depend on t. Let r = r(t, r0, θ0), θ = θ(t, r0, θ0)be a solution of (2.4) which satisfies r(0) = r0, θ(0) = θ0. Obviously, if r = 0and r(t, r, θ) = 0, r and θ are continuous in (t, r, θ).

For r ≥ R1, we have

r(t, r, θ) = |Tt(P )| > 0, t ∈ [0, 4τ ],

where P = (r, θ).From Lemma 2.2 and the property of polar coordinates, we have

r(t, r, θ + 2π) = r(t, r, θ) (2.5)

andθ(t, r, θ + 2π)− θ(t, r, θ) = 2kπ, (2.6)

where k is a integer and (t, r, θ) ∈ [0, 4τ ]× [R1,∞)×R. Moreover, from thecontinuity of θ(t, r, θ + 2π) − θ(t, r, θ), we see that k is a constant. If t = 0we have k = 1. From (2.6), we have θ(4τ, r, θ) − θ is 2π−periodic in θ. Wewill call (r, θ) the polar coordinate expression for the solution (x, y) of (2.1).Using these two functions we obtain a polar coordinate expression for T4τ asfollows

r∗ = r(4τ, r, θ), θ∗ = θ + [θ(4τ, r, θ)− θ + 2mπ], (2.7)

where m is a arbitrary integer.It is clear that r(4τ, r, θ) and θ(4τ, r, θ) − θ + 2mπ are continuous and

are 2π−periodic in θ when r ≥ R1. Let m be large enough such that

h(R1, θ) ≡ θ(4τ, R1, θ)− θ + 2mπ > 0 ∀θ. (2.8)

In order to prove that T4τ is torsional on A, it suffices to prove

h(R2, θ) = θ(4τ, R2, θ)− θ + 2mπ < 0. (2.9)

To establish this we need the following Lemma.Lemma 2. For any positive integer m, there exists a constant R2 > 0 suchthat

θ(4τ, R2, θ)− θ < −2mπ, r ≥ R2, (2.10)

where r is given by (2.3)

Proof. Choose a small enough positive constant µ < π4and large enough

positive constant M , such that

µρ0< τ

2m, π−2µ

M sin2 µ< τ

m, (2.11)

where ρ0 is a constant defined in (H3).From (H2), there exists a constant M1 > 0 such that

x−1f(t, x) > M, |x| ≥M1. (2.12)

Let R2 be large enough (see Lemma 2.2) such that

r(t, r, θ) ≥ max R0,M1

sinµ, t ∈ [0, 4τ ], r ≥ R2, (2.13)

where R0 is the constant defined in (H3).Obviously we have from (2.13)

r(t) ≥ R0 t ∈ [0, 4τ ]. (2.14)

This implies that (H3) holds, i.e.

f(t,y)yr2

+ f(t,x)xr2

≥ ρ0, r ≥ R0.

From the second equation of (2.4), we have

θ′

= −f(t,r sin θ) sin θr

− f(t,r cos θ) cos2 θr cos θ

= −f(t,r sin θ)r sin θr2

− f(t,r cos θ)r cos θr2

= −f(t,y)yr2

− f(t,x)xr2

.

(2.15)

From (H3), (2.14) and (2.15), we have

θ′(t) ≤ −ρ0 < 0, t ∈ [0, 4τ ]. (2.16)

This shows that θ(t) is monotone decreasing on [0, 4τ ].Let [t0, t4] be a subinterval of [0, 4τ ] and

θ(t0)− θ(t4) = 2π. (2.17)

For convenience, let θ(t0) =π2− µ + 2kπ, where k is a integer. Since θ(t) is

strictly monotone decreasing on [0, 4τ ], we can find t0 < t1 < t2 < t3 < t4which are uniquely determined by θ(t), such that

θ(t1) = −π2+ µ+ 2kπ, (2.18)

θ(t2) = −π2− µ+ 2kπ (2.19)

andθ(t3) = −3π

2+ µ+ 2kπ. (2.20)

By (2.13), we have

|x(t)| = |r(t) cos θ(t)| ≥ M1

sinµcos(π

2− µ) =M1, t ∈ [t0, t1]. (2.21)

Now (2.21) implies that |x(t)| ≥M1 and so we can apply (2.12).By the second equation of (2.4), (2.12) and (2.16), we have

θ′(t) < −M cos2 θ(t) < −M sin2 µ, t ∈ [t0, t1]. (2.22)

Integrate (2.22) over [t0, t1] and we have

−π + 2µ = θ(t1)− θ(t0) < −M sin2 µ(t1 − t0), (2.23)

or(t1 − t0) <

π−2µM sin2 µ

. (2.24)

Similarly, by (2.19) and (2.20), we have

(t3 − t2) <π−2µ

M sin2 µ. (2.25)

Since θ′(t) ≤ −ρ0 on [0, 4τ ] by (2.18)− (2.20) and the definition of θ(t4), we

obtain(t2 − t1) <

2µρ0

(2.26)

and(t4 − t3) <

2µρ0. (2.27)

From (2.11), (2.24)− (2.27), we have

(t4 − t0) <4τm. (2.28)

Letθ(4τ)− θ(0) = θ(4τ)− θ(0) = −2nπ + φ, (2.29)

where θ(4τ) = θ(4τ, r0, θ0), n is a positive integer, 0 ≤ φ ≤ 2π.Hence we have from (2.17), (2.28) and (2.29)

n× 4τm> 4τ =⇒ n > m

and

θ(4τ, r, θ)− θ = −2nπ + φ ≤ −2mπ − 2π + φ.

(Since n and m all are integers, then n > m =⇒ n ≥ m+ 1.)This shows that there exists a large enough constant R2 > 0 such that

θ(4τ, R2, θ)− θ + 2mπ ≤ 0.

The proof is complete.

Lemma 2.3 shows that if R2 large enough, (2.7) holds. By applyingTheorem A, T4τ has at least two fixed points in D2.

On the other hand, by the above construction of the annular region A, R1

is arbitrary, so as a result we can construct infinitely many disjoint annularregion in R2 such that T4τ has at least two fixed points in every one annularregion. Obviously, the infinitely many 4τ−periodic solutions x(t) for Eq.(1.1) satisfying x ∈ E, is equivalent to the infinitely many fixed points ofT4τ in R2. The proof of Theorem 2.1 is complete.

Finally we note that our paper was motivated by some ideas in [3].


REFERENCES

[1] G. D. Birkhoff, Proof of Poincare’s geometric theorem, Trans. Amer.Math. Soc, 14 (1913), 14–22.

[2] T. R. Ding, An infinite class of periodic solutions of periodically Duff-ing’s equations at resonance, Proc. Amer. Math. Soc, 86 (1982),47–54.

[3] T. R. Ding, W. Y. Ding, Resonance problem for a class of Duffing’sequations, J. Math. Ann. China. 6B(4) (1985), 427–432.

[4] W. Y. Ding, Fixed points of twist mapppings and periodic solutionsof ordinary differential equations, Acta. Math. Sinica, 25 (1982),227–235.

[5] W. Y. Ding, A generalization of the Poincare-Birkhoff theorem, Proc.Amer. Math. Soc, 88 (1983), 341–346.

[6] Z. M. Guo, J. S. Yu, Multiplicity results for periodic solutions to de-lay differential difference equations via critical point theory, J. Diff.Eqns, 218 (2005), 15–35.

[7] C. J. Guo, Z. M. Guo, Existence of multiple periodic solutions for aclass of second-order delay differential equations, Nonlinear.Anal- B:Real World Applications, 10(5) (2009), 3825–3972.

[8] C. J. Guo, Z. M. Guo, Existence of multiple periodic solutions fora class of three-order neutral differential equations, Acta. Math.Sinica, 52(4) (2009), 737–751.

[9] J. K. Hale, Theory of functional differential equations, Springer-Verlag,1977.

[10] H. Jacobowitz, Periodic solutions of x′′(t) + f(x, t) = 0 via Poincare-

Birkhoff theorem, J. Diff. Eqns, 20 (1976), 37–52.[11] J. L. Kaplan, J. A. Yorke, On the nonlinear differential delay equation

x′(t) = −f(x(t), x(t− 1), J. Diff. Eqns, 23 (1977), 293–314.

[12] J. B. Li, X. Z. He, Proof and generalization of Kaplan-Yorke’s conjec-ture on periodic solution of differential delay equations, Sci China,Ser. A, 42(9) (1999), 957–964.

[13] J. Moser, Stable and radom motions in dynamical systems, PrincetonUniversity Press, 1973.

[14] P. H. Rabinowitz, Minimax methods in critical point theory with ap-plications to differential equations, Amer. Math. Soc., Providence,RI, 1986.

[15] X. B. Shu, Y. T. Xu, L. H. Huang,, Infinite periodic solutions to aclass of second-order Sturm-Liouville neutral differential equations,Nonlinear Anal. 68(4) (2008), 905–911.

[16] Y. T. Xu, Z. M. Guo, Applications of a Zp index theory to periodic


solutions for a class of functional differential equations, J. Math.Anal. Appl, 257(1) (2001), 189–205.

[17] Y. T. Xu, Z. M. Guo, Applications of a geometrical index theory tofunctional differential equations, Acta. Math. Sinica. 44(6) (2001),1027–1036.


VOLUME 19

2012, NO 1–2

PP. 131–145

CORRECT SOLVABILITY OF INTEGRO-DIFFERENTIALEQUATIONS IN CLASSES OF GENERALIZED SOLUTIONS

N. KARAZEEVA∗

Abstract. The Cauchy problem for integro-differential equation

dv

dt+

∫ t

0

K(t− τ)A2v(τ)dτ = q(t), v(+0) = ψ0

is investigated. Here K(t) is exponential series K(t) =∑∞

k=1ckγke−γkt The problem is

considered in abstract separable Hilbert space H. A is self-adjoint operator in H. Theexistence and uniqueness of the generalized solution in Sobolev spaces on the semiaxis R+

and existence and uniqueness of the smooth solution in Sobolev spaces for finite times isproved.

Key Words. Functional differential equations, Integrodifferential equations, Sobolevspace, Gurtin-Pipkin heat equation, Generalized solutions .

AMS(MOS) subject classification. 34D05, 34C23

1. Introduction. The paper is devoted to studies of integro-differential equations with unbounded operator coefficients in a Hilbert space.These equations are abstract forms of the Gurtin-Pipkin integro-differentialequations (see [17], [18] for more details), which describes heat propagationin media; it also arises in homogenization problems in porous media (Darcylaw) and in hydrodynamics of viscoelastic fluids. The flows of Maxwell fluidsare described by similar systems. (These systems were investigated in papers[27]-[31] .) After projecting onto the space of solenoidal vectors and lineariza-tion the equations of Maxwell flows may be reduced to integro-differential

∗ Steklov Mathematical Institute, S.-Petersburg Department, Fontanka 27, S.-Petersburg, 191011,Russia, [email protected]

S-Petersburg State University, Universitetskaya nab. 7-9, 199034, Russia

131

132 N. KARAZEEVA

equations under consideration (with finite sums instead of series in the kernelof operator). The methods, presented in these papers, can not be extended onthe case of exponential series becouse the authors used n-time differentiation( n is a number of summands in the finite sum) or introdused n additionalvariables.

The abstract Gurtin-Pipkin equations were investigated in papers [22],[23],[24]. The research is carried out in weighted Sobolev spaces. The authorsused Laplace transform and Paley-Wiener theorems and then studied thesolvability and spectral properties for equation under consideration.

In our paper we use the method of a-priori estimates and show, thatthe initial boundary value problems for these equations are well-solvable inSobolev spaces on the positive semi-axis. The results are formulated andproved at first in the class of generalized solutions. When we impose addi-tional conditions on the data we obtain more smooth solutions.

The main attention is paid to the asymptotic behavior of solutions toevolution equations. In this connection, it is natural and convenient toconsider integro-differential equations with unbounded operator coefficients(abstract integro-differential equations), which can be realized as integro-differential partial differential equations with respect to spatial variableswhen necessary. For the self-adjoin operator A considered in what followswe can take, in particular, the operator A2y = −y′′, where x ∈ (0, π),y(0) = y(π) = 0, or the operator A2y = −∆y satisfying the Dirichletconditions on a bounded domain with sufficiently smooth boundary. Atpresent, there is an extensive literature on abstract integro-differentialequations (see, e.g.,[1]-[13] and references therein).

2. Statement of the problem. Let H be a separable Hilbertspace, and let A be a self-adjoint positive operator with bounded inverseacting onH. We embed the domain Dom(Aβ) of the operator Aβ, β > 0 intoa Hilbert space Hβ by equipping Dom(Aβ) with the norm ∥ · ∥β = ∥Aβ · ∥,which is equivalent to the graph norm of the operator Aβ.

Consider the following problem for a first-order integro-differential equa-tion on R+ = (0,∞):

(1)dv(t)

dt+

∫ t

0

K(t− s)A2v(s)ds = q(t), t ∈ R+,

(2) v(+0) = ψ0.

CORRECT SOLVABILITY OF INTEGRO-DIFFERENTIAL EQUATIONS 133

It is assumed that the scalar function K(t) admits the representation

(3) K(t) =∞∑j=1

cjγje−γjt,

where cj > 0, γj+1 > γj > 0, j ∈ N, γj → +∞ (j → +∞) and

(4) K(0) =∞∑j=1

cjγj<∞.

In the case under consideration, condition (4) means that K ′(t) ∈L1(R+). If, in addition to (4), condition

(5)∞∑j=1

cj < +∞,

holds, then K ′(t) belongs to the space W 11 (R+).

At first we suppose that T ≤ ∞. Later on we use the notation. We write∥u∥H,∞ for the norm

(6) ∥u∥H,∞ = supt∈[0,T ]

∥u(t)∥H

The space of the functions with finite norm (6) will be denotedL∞(0, T ;H).The norms ∥u∥H,1, ∥u∥H,2 in the spaces L1(0, T ;H) and L2(0, T ;H) are de-fined as follows

(7) ∥u∥H,1 =

∫ T

0

∥u(t)∥Hdt

(8) ∥u∥2H,2 =

∫ T

0

∥u(t)∥2Hdt.

We write ut for the derivative ddtu. The notation K is used for the

operator of Volterra type

(9) Kv =

∫ t

0

K(t− τ)v(τ)dτ.

134 N. KARAZEEVA

For the sace of simplicity we denote dj =cjγj.

We need in just one more condition imposed on the coefficients of theoperator K in what follows.

(10)∞∑s=1

dsγs<∞.

3. A priori estimates. In order to obtain the results about solv-ability in a class of generalized solutions we need some a-priori estimates.At first we formulate the definition of generalized solution.

Definition 1. By the generalized solution of problem (1), (2) we mean afunction with finite norms

(11) supt∈[0,T ]

∥v(t)∥H <∞

(12) supt∈[0,T ]

∥Av(t)∥H <∞

and this function v satisfies the integral identity

(13) −∫ T

0

(v,Φt)Hdt+

∫ T

0

(KAv,AΦ)Hdt+

+ (v(T ),Φ(T ))H − (ψ0,Φ(0))H =

∫ T

0

(q,Φ)Hdt

for any function Φ ∈ W 11 (0, T ;H) ∩ L∞(0, T ;H) such that AΦ ∈

L1(0, T ;H).The correctness of the definition arises from the fact that all the integrals

in formula (13) are finite. Indeed we have

(14)

∣∣∣∣∫ T

0

(KAv,AΦ)Hdt

∣∣∣∣ ≤ ∫ T

0

∥KAv∥H∥AΦ∥Hdt ≤

ess sup ∥KAv∥H∥AΦ∥H,1 ≤ C∞,Kess sup ∥Av∥H∥AΦ∥H,1 ≤C∞,K∥Av∥H,∞∥AΦ∥H,1


The last inequality in (14) is true because of the continuity of the inte-gral operator K in the space L∞(0, T ;H) (This proposition is formulated inLemma 2, which is presented at the end of the paper).

To get a priory estimates for solutions we need some auxiliary results.These results are gathered in section 6. At first we multiply equation (1) byv in the sense of scalar product in the space H:

(15)1

2

d

dt∥v∥2H + (KA2v, v)H = (q, v)H .

Integration with respect to t ∈ [0, T ] gives

(16)1

2(∥v(T )∥2H − ∥v(0)∥2H) +

∫ T

0

(KAv,Av)Hdt

=

∫ T

0

(q, v)Hdt ≤∫ T

0

∥q∥H∥v∥Hdt ≤ ∥v∥H,∞∥q∥H,1.

By the nonnegativity of the third summand in the left-hand side of (16)(Lemma 1 from section 6) the following estimate is true

(17) ∥v(T )∥2H ≤ ∥v(0)∥2H + 2∥v∥H,∞∥q∥H,1.

The passage to the supremum in the left-hand side of (17) yields

(18) ∥v∥2H,∞ ≤ ∥v(0)∥2H + 2∥v∥H,∞∥q∥H,1.

Solving this quadratic inequality we get the following estimate for thenorm ∥v∥H,∞

(19) ∥v∥H,∞ ≤ ∥q∥H,1 +√

∥q∥2H,1 + ∥ψ0∥2H = C1.

Substitution of (19) into (16) leads to the inequality

(20)

∫ T

0

(KAv,Av)Hdt ≤1

2∥ψ0∥2H + ∥v∥H,∞∥q∥H,1 ≤

≤ 1

2∥ψ0∥2H + C1∥q∥H,1.

136 N. KARAZEEVA

To get the following estimate we apply the operator A to both sides ofequation (1) and multiply the obtained relation by Av in the space H.

(21)1

2

d

dt∥Av∥2H + (KA2v,A2v)H = (Aq,Av)H .

Then we integrate the obtained identity with respect to t ∈ [0, T ] andomit the positive term. Reasoning in similar way we find the upper boundfor the norm ∥Av∥H,∞

(22) ∥Av∥H,∞ ≤ ∥Aq∥H,1 +√

∥Aq∥2H,1 + ∥Aψ0∥2H = C2.

Furthermore we can get estimates for ∥A2v∥H,∞ and ∥Anv∥H,∞. To dothis we proceed exactly as in the case of estimate (21)

(23) ∥A2v∥H,∞ ≤ ∥A2q∥H,1 +√∥A2q∥2H,1 + ∥A2ψ0∥2H .

Estimates (21), (22) are sufficient for existence of generalized solutiondescribed in (13) for any T ≤ ∞.

4. Existence of generalized solution. To find a generalized solu-tion we apply the Galerkin method. Let φl, l = 1, 2, . . . be orthonormalbasic functions in the space H. The approximations vn of the solution v forthe Cauchy problem (1), (2) will be constructed in the form

(24) vn =n∑

l=1

Cln(t)φl, n = 1, 2, . . .

Let the initial function ψ0 be presented as a limit (in the topology of H)of the sums

(25) ψn =n∑

l=1

C0lnφl → ψ0, n→ ∞

The coefficients Cln(t) should satisfy the initial conditions

(26) Cln(0) = C0ln, l = 1, . . . , n.


We shall try to find vn which satisfy the following integral identity

(27) (vnt , φl)H + (KA2vn, φl)H = (q, φl)H , l = 1, . . . , n.

Substitution (24) into (27) yields

(28)d

dtCln(t) +

n∑s=1

(K[Csn(t)Aφs], Aφl)H = (q, φl)H .

System (28) is the system of linear ordinary integro-differential equa-tions. It can be solved locally by method of successive approximationswith Euler piecewise linear functions. The global existence is deduced fromboundedness of Cln. Indeed the functions Cln(t) are bounded because vn

are bounded. The functions vn satisfy (27) so vn satisfy integral identity(13). Thus (19) is true for vn and the functions vn can be estimated by theconstant C1.

The solution of problem (1), (2) will be constructed as a weak limit of thesequence vn. Indeed vn is a bounded set in the space L∞(0, T ;H). Nowwe show that there is a subsequence vnk such that the integrals converge

(29)

∫ T

0

(vnk ,Φ)Hdt→∫ T

0

(v,Φ)Hdt

for any function Φ ∈ L1(0, T ;H). Let us choose a dense set Φl in thespace L1(0, T ;H). For every Φl we extract a subsequence vnl

l such that

(30)

∫ T

0

(vnll ,Φ

l)Hdt→∫ T

0

(vl,Φl)Hdt.

(This can be done because the integrals in (30) are bounded numbersfor fixed Φl.) Then we apply the diagonal process and choose the desiredsubsequence. Later on we use the notation vn for this sequence. It con-verges weakly to some function v. In similar way we extract the subsequencevnk such that Avnk converges weakly to Av. (We note that sequence by vn).Moreover since KAvn is bounded then we choose the subsequence convergingweakly to KAv. This function v is a solution of problem (1), (2). To provethis fact we make a passage to the limit for n→ ∞ in the identity

138 N. KARAZEEVA

(31) −∫ T

0

(vn,Φt)Hdt+

∫ T

0

(KAvn, AΦ)Hdt

+ (vn(T ),Φ(T ))H − (ψn,Φ(0))H =

∫ T

0

(q,Φ)Hdt.

Thus we have proved the theoremTheorem 1. Let the function q belong to the space L1(0, T ;H) and Aq ∈L1(0, T ;H). The function ψ0 ∈ H and Aψ0 ∈ H. The operator K is oper-ator of Volterra type with the kernel K which is exponential series of form(3) with coefficients satisfying condition (10). Then the problem (1), (2)has a unique generalized solution v which satisfies integral identity (13) andpossesses estimates (19) and (22).

Proof. The existence of such a solution is already proved. To obtain theuniqueness we suppose that there is two solutions v1 and v2. Then u = v1−v2satisfies equation (1) with zero right-hand side and zero initial data. Estimate(19) yields u = 0.

Furthermore we can prove the theorem of increasing of smoothness.Theorem 2. Let the function q belong to the space L1(0, T ;H) and Anq ∈L1(0, T ;H). The initial function ψ0 ∈ H and Anψ0 ∈ H. The coefficients ofthe operator K are under hypothesis of Theorem 1. Then problem (1), (2)has a unique generalized solution v such that Anv ∈ L∞(0, T ;H).Remark 1. If we choose the eigenfunctions of the operator A as a funda-mental system φl, then we get a finite dimensional system of equationsinstead of (28). The solution of this system exists.

5. Smooth solutions. In this section we deduce the smoothnessof generalized solution in the case when T < ∞ and when the initial dataψ0 and the function q from the right-hand side of (1) are from appropriatespaces. At first we obtain another estimate for ∥v∥H,∞. We consider identity(15) and apply Gronwall lemma.

(32)1

2

d

dt∥v∥2H + (KAv,Av)H = (q, v)H ≤ ∥q∥H∥v∥H .

(33)d

dt∥v∥2H ≤ −2(KAv,Av)H + 2∥q∥H∥v∥H

≤ −2(KAv,Av)H + ∥q∥2H + ∥v∥2H .


Then

(34) ∥v(t)∥2H ≤ et[∥ψ0∥2H +

∫ T

0

(−2(KAv,Av)H + ∥q∥2H)dτ]

≤ eT[∥ψ0∥2H + ∥q∥2H,2

]= C3,

because∫ t

0(KAv,Av)Hdτ ≥ 0.

In similar way we get the upper bound for Av

(35) ∥Av(t)∥2H ≤ et[∥Aψ0∥2H +

∫ t

0

(−2(KA2v,A2v)H + ∥Aq∥2H)dτ]

≤ eT[∥Aψ0∥2H + ∥Aq∥2H,2

]= C4.

Moreover we can estimate A2v and Anv

(36) ∥A2v(t)∥2H ≤ eT[∥A2ψ0∥2H + ∥A2q∥2H,2

]≤ C5.

Now we try to estimate the norm vt. To do this we differentiate equation(1) with respect to t and multiply the obtained equation by vt in Hilbert spaceH. We have

(37)1

2

d

dt∥vt∥2H+

+

([(K(0)A2v(t)) +

∫ t

0

K ′(t− τ)A2v(τ)dτ

], vt

)H

=

= (qt, vt)H ,

where the notation K ′ is used for the derivative ddtK. Let us denote by

K′ the integral operator with the kernel K ′

(38) K′ =

∫ t

0

K ′(t− τ)u(τ)dτ.

Transform the second summand in the left-hand side of (37)

(39)1

2

d

dt∥vt∥2H +K(0)(Av,Avt)H + (K′Av,Avt)H = (qt, vt)H .

140 N. KARAZEEVA

Then we obtain

(40)1

2

d

dt∥vt∥2H +

1

2K(0)

d

dt∥Av∥2H ≤

≤ ∥K′Av∥H∥Avt∥H + ∥qt∥H∥vt∥H .

We integrate inequality (40) with respect to t ∈ [0, t].

(41)1

2∥vt(t)∥2H +

1

2K(0)∥Av(t)∥2H ≤ 1

2∥vt(0)∥2H+

+1

2K(0)∥Aψ0∥2H +

∫ t

0

(∥K′A2v∥H∥vt∥H

)dτ+

+

∫ t

0

(∥qt∥H∥vt∥H) dτ ≤ 1

2∥vt(0)∥2H+

+1

2K(0)∥Aψ0∥2H + ∥vt∥H,∞∥K′A2v∥H,1+

+ ∥vt∥H,∞∥qt∥H,1.

After rejection of the second term in the left-hand side of (41) we passto the supremum with respect to t

(42) ∥vt∥2H,∞ − 2∥vt∥H,∞(∥K′A2v∥H,1 + ∥qt∥H,1)−− (∥vt(0)∥2H +K(0)∥Aψ0∥2H) ≤ 0.

Solving this quadratic inequality we get the upper bound for ∥vt∥H,∞

(43) ∥vt∥H,∞ ≤(∥K′A2v∥H,1 + ∥qt∥H,1

)+

+

√(∥K′A2v∥H,1 + ∥qt∥H,1)

2 + ∥vt(0)∥2H +K(0)∥Aψ0∥2H .

The boundedness of the right-hand side of (43) is true by the fact thatthe operator K′ is continuous in the space L2(0, T ;H) (Lemma 2) and byestimate A2v (36). Indeed when T <∞

(44) ∥K′A2v∥H,1 ≤ ∥K′A2v∥H,2 ≤ C2,K′∥A2v∥H,2 ≤ C2,K′∥A2v∥H,∞,

where C2,K′ is the norm of operator K′ in the space L2(0, T ;H).

The value ∥vt(0)∥H may be found from equation (1) when q(0) ∈ H

(45) vt(0) = q(0)−KA2v0|t=0 = q(0).

Thus

(46) ∥vt∥H,∞ ≤ (C2,K′C5 + ∥qt∥H,1)+

+√(C2,K′C5 + ∥qt∥H,1)2 + ∥q(0)∥2H +K(0)∥Aψ0∥2H = C6.

Estimates (36) and (43) lead to the following theoremTheorem 3. Let T < ∞, the function q belong to the space L2(0, T ;H),moreover Aq and A2q belong to L2(0, T ;H). And let q(0) ∈ H, ψ0 ∈ H,Aψ0 ∈ H and A2ψ0 ∈ H. The operator K is operator of Volterra typewith the kernel K which is exponential series of form (3) with coefficientssatisfying (4) and(10). Furthermore the coefficients of the operator K shouldsatisfy the following condition

(47)∞∑s=1

dsγs <∞.

Then problem (1) (2) has unique smooth solution v ∈ L2(0, T ;H) suchthat Av ∈ L2(0, T ;H), A2v ∈ L2(0, T ;H) and vt ∈ L∞(0, T ;H) ( and auto-matically vt ∈ L2(0, T ;H) ). This solution v satisfies estimates (34), (35),(36), (46).

The conclusion of Theorem 3 follows from Theorem 1 and estimates (34),(35), (36), (46). The proof may be carried out independently in the spaceL2(0, T ;H).

The theorem of increasing of smoothness is also true.Theorem 4. Let T < ∞, the function q belong to the space L1(0, T ;H),furthermore Aq, A2q and Anq belong to the space L2(0, T ;H). And let q(0) ∈H, ψ0 ∈ H, Aψ0 ∈ H, A2ψ0 ∈ H and Anψ0 ∈ H. The coefficients ofthe operator K are under hypothesis of Theorem 3. Then problem (1), (2)has a unique smooth solution v ∈ L2(0, T ;H) such that Av ∈ L2(0, T ;H),Anv ∈ L2(0, T ;H) and vt ∈ L∞(0, T ;H).

6. Auxiliary results. In this section we present some auxiliarypropositions.

142 N. KARAZEEVA

Lemma 1. Let the operator K be operator of Volterra type (9) with the kernelK of form (3). The coefficients ds, γs satisfy condition

∞∑s=1

dsγs<∞.

Then the quadratic form, corresponding to this operator, is nonnegativedefinite

(48) I =

∫ T

0

(Ku, u)Hdσ ≥ 0

for such u for which this integral is finite (for instence u ∈ L∞(0, T ;H)and u ∈ L2(0, T ;H)).

Proof. At first we assume that T < ∞ and that u ∈ L2(0, T ;H). Passageto the limit T → ∞ yields the truth of the Lemma 1 . Suppose that thefunction K is continued on the negative semiaxis like an even function. ThenI is equal to the half of integral on [0, T ]× [0, T ].

(49) I =

∫ T

0

(∫ σ

0

K(σ − τ)u(τ)dτ, u(σ)

)H

dσ =

=

∫ T

0

(∫ σ

0

K(σ − τ)(u(τ), u(σ))Hdτ

)dσ =

=1

2

∫ T

0

∫ T

0

K(σ − τ)(u(τ), u(σ))Hdτdσ.

If we extend u by zero for t ∈ R and denote the obtained function by u,we may rewrite I in the form

(50) I =1

2

∫ ∞

−∞

∫ ∞

−∞K(σ − τ)(u(τ), u(σ))Hdτdσ.

To prove that integral (48) is positive we calculate at first the inverseFourier transform of the function K

(51) F−1(K) = L(p) = (2π)−12

∞∑s=1

2dsγsp2 + γ2s

≥ 0.


By (10) the series in (51) converges everywhere. Then we substitute (51)into (50)

(52) I =1

2

∫ ∞

−∞

∫ ∞

−∞F (L)(σ − τ)(u(τ), u(σ))Hdτdσ =

=1

2(2π)−

12

∫ ∞

−∞

∫ ∞

−∞

(∫ ∞

−∞e−i(σ−τ)pL(p)dp

)(u(τ), u(σ))Hdτdσ

Thus we have

(53) I =1

2(2π)−

12

∫ ∞

−∞

∫ ∞

−∞eiτp

(∫ ∞

−∞e−iσpu(σ)dσ, u(τ)

)H

dτL(p)dp =

=1

2(2π)−

12

∫ ∞

−∞

∣∣∣∣∣∣∣∣∫ ∞

−∞e−iσpu(σ)dσ

∣∣∣∣∣∣∣∣2H

L(p)dp ≥ 0,

because L(p) ≥ 0.

The last formula in (53) yields that the quadratic form under consider-ation is a norm.Lemma 1 was proved in [27] for the case when H = L2(Rn).

Lemma 2. The operator K with the coefficients ds, γs satisfying conditions(4),(10) is continuous in the space L2(0, T ;H) and in the space L∞(0, T ;H).

Proof of Lemma 2 is presented in [25] for the case when the operatorK is a convolution. The operator of Volterra type may be presented in thatform assuming that the function u = 0 for τ > t. The norm of the operatorK in both cases L∞(0, T : H) and L2(0, T ;H) may be chosen

(54)

∫ T

0

K(τ)dτ

7. Remarks. In conclusion, it is pertinent to mention that integro-differential equations considered in this paper were derived in [17] and afterthat, apparently, acquired the name Gurtin-Pipkin equations.

In [1]–[10], [14], [15] (see also the references therein), integro-differentialequations with principal part being an abstract parabolic equation were stud-ied. Equations with principal part being an abstract hyperbolic equation aresignificantly less studied. The works most closely related to this questionsare [13], [15], [17]–[19], [22]–[24].

144 N. KARAZEEVA

The main difference between the solvability results obtained in this paperand those already known (see, e.g., [18]–[19]) is that we allow the presenceof (integrable) singularities in the Volterra kernela of integral operators. Inworks on hyperbolic-type equations mentioned above, requirements on thekernels of integral operators are more severe. The situation with parabolic-type equations is fundamentally different because in this case, the conditionson the kernels of integral operators are substantially weaker (see [1]–[10], [14],[15]).

REFERENCES

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[2] G. Di Blasio, K. Kunisch , E. Sinestari , L2–regularity for parabolic partial inte-grodifferential equations with delays in the highest order derivatives, J. Math.Anal. Appl., 102 (1984), 38-57.

[3] G. Di Blasio, K. Kunisch, E. Sinestari, Stability for abstract linear functional differ-ential equations, Izrael. J. Mathematics, 50 ( 1985), 231-263.

[4] K. Kunisch, M. Mastinsek, Dual semigroups and structual operators for partialdifferential equations with unbounded operators acting on the delays, Differ.Integral Equations, 3 (1990), 733-756.

[5] K. Kunisch, W. Shappacher, Necessary conditions for partial differential equationswith delay to generate l0–semigroup, J. Differ. Equations, 50 (1983), 49-79.

[6] J. Wu, Semigroup and integral form of class of partial differential equations withinfinite delay, Differ. Integr. Equations, 4 (1991), 1325-1351.

[7] J. Wu, Theory and applications of partial functional differential equations, Appl.Math. Sci, 119, New York: Springer-Verlag, 1996.

[8] V. V. Vlasov, On solvability and onproperties of solutions for functional-differentialequations in Hilbert space, Mat. Sbornik, 186 ( 1995), 67–92.

[9] V. V. Vlasov, On solvability and on estimates of solutions of functional differentialequations in Sobolev spaces, Trudy Steklov Math. Inst., 227 (1999), 109-121.

[10] V. V. Vlasov, On correct solvability of abstract parabolic equations with afteref-fect, Doklady RAS, 415 (2007), 151-152.

[11] R. K. Miller, An integrodifferential equation for rigid heat conductors with mem-ory, J. Math. Anal. Appl., 66 (1978), 313-332.

[12] R. K. Miller, R. L. Wheeler, Well-posedness and stability of linear Volterra interod-ifferential equations in abstract spaces, Funkcialaj Ekvac., 21 ( 1978), 279-305.

[13] V. V. Vlasov, K. I. Shmatov, Correct Solvability of equations of hyperbolic type withdelay in Hilbert space, Trudy Steklov Math. Inst., 243 (2003), 127-137.

[14] D. A. Medvedev, V. V. Vlasov, J. Wu, Solvability and structural properties of ab-stract neutral functional differential equations, J. Functional Differential Equa-tions, 15 (2008), 249-272.

[15] V. V. Vlasov, D. A. Medvedev, Functional differential equations in Sobolev spacesand related problems of spectral theory, Journal of Mathematical Science, 164(2010), 659-841.

[16] J.-L. Lions, E. Magenes, Problemes aux limites non homogenes et applica-tions, Dunod, Paris, 1968.


[17] M. E. Gurtin, A. C. Pipkin, Theory of heat conduction with finite wave speed, Arch.Rat. Mech. Anal., 31 (1968), 113-126.

[18] L. Pandolfi, The controllability of the Gurtin-Pipkin equations: a cosine operatorapproach, Appl. Math. Optim., 52 (2005), 143-165.

[19] W. Desch, R. K. Miller, Exponential stabilization of Volterra Integrodifferentialequations in Hilbert space, J. Differential Equations, 70 (1987), 366-389.

[20] V. V. Zhikov, On extension and on application of method of two-scale conver-gence, Mat. sbornik, 191 (2000), 31-72.

[21] Sanchez, E. Palencia, Nonhomogeneous Media and Theory of Oscillation, Moscow,”Mir”, 1984.

[22] V. V. Vlasov, N. A. Rautian, Correct solvability and spectral analysis of abstract hy-perbolic integral differential equations, Trudy Semin. Im. Petrovskogo (in print).

[23] V. V. Vlasov,N. A. Rautian, A. S. Shamaev, Solvability and spectral analysis of inte-gral differential equations arising in heat conductivity and in acoustics, DokladyRAS, 434 (2010), 12-15.

[24] V. V. Vlasov, N. A. Rautian, A. S. Shamaev, Solvability and spectral analysis ofintegro-differential equations arising in the theory of heat transfer and acous-tics, Sovrem. Mat., Fundam.Napr. -(in print)

[25] E. M. Stein, Singular Integrals and Differential Properties of Functions, PrincetonUniv. Press, New Jersey, 1970.

[26] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous IncompressibleFlow, Mathematics and its Applications, 2, Gordon and Breach Science Pub-lishers, New York, 1969.

[27] N. A. Karazeeva, About the solvability in large of the main initial boundary valueproblem for 2D equations of Oldroid fluids, Zap. Nauch. Semin. LOMI, SteklovMath. Inst., 156 (1986), 69-72 (Russian).

[28] A. P. Oskolkov, On the theory of Maxwell fluids, Zap. Nauch. Semin. LOMI, SteklovMath. Inst., 101 (1981), 119-127 (Russian).

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[32] M. Matei, V. V. Montreanu, M. Sofonea, On the Signorini frictionless contact prob-lem for linear viscoelastic materials,Applied Anal., 80 (2001), 177-199.

[33] M. Fabrizio, J. M Golden, Maximum and minimum free energies for a linear vis-coelastic matherial, Quart. Appl. Math., 60 (2002), 341-381.

[34] J. M. Golden, A. C Graham, Boundary Value Problems in Linear Viscoelastic-ity, Springer, Berlin, 1988.

VOLUME 19

2012, NO 1–2

PP. 147–155

OSCILLATION THEOREMS FOR ODD-ORDER NEUTRALDIFFERENTIAL EQUATIONS ∗

TONGXING LI † AND ETHIRAJU THANDAPANI ‡

Abstract. In this paper, new oscillation criteria for the odd-order neutral differentialequations

(E) [x(t) + p(t)x(τ(t))](n)

+ q(t)x(σ(t)) + v(t)x(η(t)) = 0

are presented. An example is given to illustrate an application of our results.

Key Words. Odd-order neutral differential equation, comparison theorem, oscillationbehavior, asymptotic behavior.

AMS(MOS) subject classification. 34C10, 34K11

1. Introduction. This paper is concerned with the oscillation andasymptotic behavior of the solutions of odd-order neutral differential equa-tion

(E) [x(t) + p(t)x(τ(t))](n) + q(t)x(σ(t)) + v(t)x(η(t)) = 0,

where n ≥ 3 is an odd integer, p(t), q(t), v(t), τ(t), σ(t), η(t) ∈ C([t0,∞)) and(H1) q(t) > 0, v(t) ≥ 0, 0 ≤ p(t) ≤ p0 <∞;(H2) limt→∞ σ(t) = limt→∞ η(t) = ∞;(H3) τ(t) ∈ C1([t0,∞)), τ(t) ≤ t, τ ′(t) = τ0 > 0, τ σ = στ , τ η = ητ .

∗ Supported by Natural Science Foundation of Shandong (Z2007F08).† School of Control Science and Engineering, Shandong University, Jinan, Shandong

250061, P R China‡ Ramanujan Institute for Advanced Study in Mathematics, University of Madras,

Chennai 600 005, India

147

148 TONGXING LI AND ETHIRAJU THANDAPANI

We set z(t) = x(t)+p(t)x(τ(t)). By a solution of (E) we mean a functionx(t) ∈ C([Tx,∞)), Tx ≥ t0, which has the property z(t) ∈ Cn([Tx,∞)) andsatisfies (E) on [Tx,∞). We consider only those solutions x(t) of (E) whichsatisfy sup|x(t)| : t ≥ T > 0 for all T ≥ Tx. We assume that (E) possessessuch a solution. A solution of (E) is called oscillatory if it has arbitrarilylarge zeros on [Tx,∞) and otherwise, it is said to be non-oscillatory. Equation(E) itself is said to be almost oscillatory if all its solutions are oscillatory orconvergent to zero asymptotically.

Since the neutral differential equations have the applied applicationsin natural sciences, technology and population dynamics, there is the per-manent interest in obtaining new sufficient conditions for the oscillation ornon-oscillation of the solutions of varietal types of the even-order neutraldifferential equations [1,2,4,10,14–17,26,28,30] and odd-order neutral differ-ential equations [3,5–9,11–13,18–25,27,29].

Li et al. [17], Zafer [28] and Zhang et al. [30] studied the oscillation ofeven-order neutral differential equation

[x(t) + p(t)x(τ(t))](n) + q(t)x(σ(t)) = 0, n ≥ 2 is even.

Especially, Li et al. established some new oscillation results for the casewhen 0 ≤ p(t) ≤ p0 < ∞, τ ′(t) = τ0 > 0 and τ σ = σ τ . By employingcomparison technique, Baculıkova and Dzurina [2] considered the oscillationof second-order neutral differential equation

(r(t)[x(t) + p(t)x(τ(t))]′)′+ q(t)x(σ(t)) = 0

under the case when 0 ≤ p(t) ≤ p0 < ∞, τ ′(t) ≥ τ0 > 0 and τ σ = σ τ .Li [15] investigated the oscillation of second-order mixed neutral differentialequation(r(t)[x(t)+p1(t)x(t−σ1)+p2(t)x(t+σ2)]′

)′+q1(t)x(t−σ3)+q2(t)x(t+σ4) = 0,

where 0 ≤ pi(t) ≤ ai <∞ for i = 1, 2.

Regarding the oscillation of neutral differential equations of odd-order,some authors examined the neutral differential equation

[x(t) + p(t)x(τ(t))](n) + q(t)x(σ(t)) = 0, n is odd.

They considered the cases 0 ≤ p(t) ≤ 1 or −1 ≤ p(t) ≤ 0; see [13]. To thebest of our knowledge, there are few papers has been published for the casewhen p(t) > 1, see e.g., [18, 25].

OSCILLATION THEOREMS 149

Motivated by recent papers [2, 13, 15, 17], in this paper we try to derivesome new oscillation results for (E). In order to illustrate our results, wegive the following remarks.

Remark 1. The conditions τ σ = σ τ and τ η = η τ contained in thehypothesis (H3) mean that the deviating arguments τ(t), σ(t), and η(t) areof the same form, that is, if τ(t) = αt, then at the same time σ(t) = βt, andη(t) = γt.

Remark 2. All functional inequalities considered in this paper are assumedto hold eventually, that is, they are satisfied for all t large enough.

Remark 3. Without loss of generality we can deal only with the positivesolutions of (E).

2. The main results. The Kneser’s theorem is stated below, thereader may find this result in [1, Lemma 2.2.1], which plays an importantrole in the oscillation of higher-order differential equations.Lemma 1. (Kneser’s theorem) Let f ∈ Cn([t0,∞),R) be a function of fixedsign such that f (n) is of fixed sign and not identically zero on a sub-ray of[t0,∞). Then there exist m ∈ Z and t1 ∈ [t0,∞) such that 0 ≤ m ≤ n − 1,and (−1)n+mff (n) ≥ 0,

ff (j) > 0 for j = 0, 1, · · · ,m− 1 when m ≥ 1

and

(−1)m+jff (j) > 0 for j = m,m+ 1, · · · , n− 1 when m ≤ n− 1

hold on [t1,∞).

Lemma 2. [1, Lemma 2.2.3] Let f be a function as in Kneser’s theorem. Iff (n)(t)f (n−1)(t) ≤ 0 and limt→∞ f(t) = 0, then for every λ ∈ (0, 1), thereexists tλ ∈ [t1,∞) such that

|f(t)| ≥ λ

(n− 1)!tn−1|f (n−1)(t)|

holds on [tλ,∞).

Lemma 3. [13, Lemma 3] Let f and g ∈ C([t0,∞),R) and α ∈ C([t0,∞),R)with limt→∞ α(t) = ∞ and α(t) ≤ t for all t ∈ [t0,∞); further suppose thatthere exists h ∈ C([t−1,∞),R+), where t−1 := mint∈[t0,∞)α(t) such thatf(t) = h(t) + g(t)h(α(t)) holds for all t ∈ [t0,∞). If limt→∞ f(t) exists andlim inft→∞ g(t) > −1, then lim supt→∞ h(t) > 0 implies that limt→∞ f(t) > 0.

Lemma 4. If x is a positive solution of (E), then the corresponding functionz(t) = x(t) + p(t)x(τ(t)) satisfies

(1) z(t) > 0, z(n−1)(t) > 0, z(n)(t) < 0

eventually.

Proof. Due to Lemma 1, the proof is simple and so is omitted.

For our intended references, let us denote

(2) Q(t) := minq(t), q(τ(t)), V (t) := minv(t), v(τ(t)).

Theorem 1. Assume that

(3)

∫ ∞

t0

tn−1[Q(t) + V (t)

]dt = ∞.

Further, assume that the first-order neutral differential inequality(y(t) +

p0τ0y(τ(t))

)′

+λ

(n− 1)!

[Q(t)σn−1(t)y(σ(t)) + V (t)ηn−1(t)y(η(t))

]≤ 0(E1)

has no positive solution for some λ ∈ (0, 1). Then (E) is almost oscillatory.

Proof. Assume that x is a positive solution of (E), which does not tend tozero asymptotically. Then the corresponding function z satisfies

z(σ(t)) = x(σ(t)) + p(σ(t))x(τ(σ(t)))

≤ x(σ(t)) + p0x(σ(τ(t))),(4)

where we have used the hypothesis (H3) and similarly

(5) z(η(t)) ≤ x(η(t)) + p0x(η(τ(t))).

On the other hand, it follows from (E) that

(6) z(n)(t) + q(t)x(σ(t)) + v(t)x(η(t)) = 0

and moreover taking (H1) and (H3) into account, we have

0 =p0τ ′(t)

(z(n−1)(τ(t)))′ + p0q(τ(t))x(σ(τ(t))) + p0v(τ(t))x(η(τ(t)))

=p0τ0(z(n−1)(τ(t)))′ + p0q(τ(t))x(σ(τ(t))) + p0v(τ(t))x(η(τ(t))).(7)

Combining (6) and (7), we are led to[z(n−1)(t) +

p0τ0z(n−1)(τ(t))

]′+ q(t)x(σ(t)) + p0q(τ(t))x(σ(τ(t)))

+ v(t)x(η(t)) + p0v(τ(t))x(η(τ(t))) ≤ 0,

from which follows by (2), (4) and (5) that

(8)

[z(n−1)(t) +

p0τ0z(n−1)(τ(t))

]′+Q(t)z(σ(t)) + V (t)z(η(t)) ≤ 0.

Next, we prove that

z′(t) > 0

eventually. If not, it follows from Lemma 3 that limt→∞ z(t) = a > 0 andthen limt→∞ z(k)(t) = 0 for k = 1, 2, · · · , n. Integrating (8) from t to ∞for a total of (n − 1) times and integrating the resulting inequality from t1(t1 is large enough) to ∞, we obtain∫ ∞

t1

(s− t1)n−1

(n− 1)!

[Q(s)z(σ(s)) + V (s)z(η(s))

]ds <∞,

which yields ∫ ∞

t1

sn−1[Q(s) + V (s)

]ds <∞.

This is a contradiction to (3). Hence by Lemma 2 and Lemma 4, we get

z(t) ≥ λ

(n− 1)!tn−1z(n−1)(t) for every λ ∈ (0, 1).

By virtue of (8), we have[z(n−1)(t) +

p0τ0z(n−1)(τ(t))

]′+

λ

(n− 1)!

[Q(t)σn−1(t)z(n−1)(σ(t)) + V (t)ηn−1(t)z(n−1)(η(t))

]≤ 0.(9)

Therefore, setting z(n−1)(t) = y(t) in (9), one can see that y is a positivesolution of (E1). This contradicts our assumptions and the proof is complete.

Remark 4. In the comparison principle in Theorem 1 we do not stipulatewhether (E) is equation with delayed, advanced or mixed arguments, so thatthe obtained results are applicable to all three types of equations. On the otherhand, the comparison theorem established in Theorem 1 reduces oscillationof (E) to the research of the first-order neutral differential inequality (E1).Therefore, applying the conditions for (E1) to have no positive solution, weimmediately get oscillation criteria for (E).

Theorem 2. Assume that (3) holds. If the first-order differential inequality

w′(t) +λτ0

(n− 1)!(τ0 + p0)

×(Q(t)σn−1(t)w(τ−1(σ(t))) + V (t)ηn−1(t)w(τ−1(η(t)))

)≤ 0(E2)

has no positive solution for some 0 < λ < 1, then (E) is almost oscillatory.

Proof. We assume that x is a positive solution of (E), which does not tendto zero asymptotically. Then y(t) = z(n−1)(t) > 0 is a decreasing solution of(E1). We denote

w(t) = y(t) +p0τ0y(τ(t)).

Then

w(t) ≤ y(τ(t))

(1 +

p0τ0

).

Substituting this into (E1), we get that w is a positive solution of (E2). Thisis a contradiction, and hence the proof is complete.

Corollary 1. Assume that (3) holds and

(10) σ(t) < τ(t), η(t) < τ(t).

If σ(t) ≤ η(t) and also

(11) lim inft→∞

∫ t

τ−1(η(t))

(σn−1(s)Q(s) + ηn−1(s)V (s)

)ds >

(n− 1)!(τ0 + p0)

τ0 e,

or σ(t) ≥ η(t) and also

(12) lim inft→∞

∫ t

τ−1(σ(t))

(σn−1(s)Q(s) + ηn−1(s)V (s)

)ds >

(n− 1)!(τ0 + p0)

τ0 e,

then (E) is almost oscillatory.

Proof. We admit that w is a positive solution of (E2) for some 0 < λ < 1.If σ(t) ≤ η(t), then w(τ−1(σ(t))) ≥ w(τ−1(η(t))) and (E2) gives that w is asolution of the differential inequality

(E3) w′(t)+

λτ0(n− 1)!(τ0 + p0)

(σn−1(t)Q(t)+ηn−1(t)V (t)

)w(τ−1(η(t))) ≤ 0.

But according to [14, Theorem 2.1.1] the condition (11) guarantees that(E3) has no positive solution. Therefore (E2) has no positive solution andTheorem 2 provides the oscillation of (E). The case σ(t) ≥ η(t) can betreated similarly. This completes the proof.

As an application we give the following example.

Example 1. Consider the odd-order delay differential equation

(13)

[x(t) + p0x

(t

τ

)](n)+q0tnx

(t

σ

)= 0, t ≥ 1,

where p0 ∈ [0,∞), q0 ∈ (0,∞) and σ > τ ≥ 1.Let q(t) = q0/t

n and v(t) = 0. Then Q(t) = q0/tn and V (t) = 0.

Moreover, we have∫ ∞

t0

sn−1[Q(s) + V (s)

]ds = q0

∫ ∞

1

1

sds = ∞.

Hence by Corollary 1, equation (13) is almost oscillatory if

q0 >(n− 1)!(1 + τp0)σ

n−1

e ln στ

.

If p0 ∈ [0, 1), then by [13, Example 1], equation (13) is almost oscillatoryprovided that

q0 >(n− 1)!σn−1

e(1− p0) ln σ.

We find that our result improves [13] in some cases. For example, we letσ = e2 and τ = e. It is easy to verify that there exists a p0 ∈ (0, 1) such that

1

(1− p0) ln σ>

1 + τp0ln σ

τ

,

that is,

−ep02 + (e− 1)p0 +

1

2< 0.


3. Summary. In this paper we have introduced new comparison the-orems for investigation of the oscillation of (E). The established comparisonprinciples reduce oscillation of odd-order neutral equations to studying prop-erties of various types of first-order differential inequalities, which essentiallysimplifies examination of (E). Our technique permits to relax restrictionsusually imposed on the coefficients of (E). So that our results are of highgenerality. Obtained results are easily applicable and are illustrated on asuitable example.

REFERENCES

[1] R. P. Agarwal, S. R Grace and D. O’Regan. Oscillation Theory for Difference andFunctional Differential Equations, Kluwer Academic, Dordrecht, 2000.

[2] B. Baculıkova and J. Dzurina, Oscillation theorems for second order neutral dif-ferential equations, Computers and Mathematics with Applications, 61 (2011),94–99.

[3] B. Baculıkova and J. Dzurina, Oscillation of third-order neutral differential equa-tions, Mathematical and Computer Modelling, 52 (2010), 215–226.

[4] B. Baculıkova, T. Li and J. Dzurina, Oscillation theorems for second order neutraldifferential equations, Electronic Journal of Qualitative Theory of DifferentialEquations, 74 (2011), 1–13.

[5] T. Candan and R. S. Dahiya, Oscillatory and asymptotic behavior of odd orderneutral differential equations, Dynamics of Continuous, Discrete and ImpulsiveSystems Series A: Mathematical Analysis, 14 (2007), 767–774.

[6] P. Das, Oscillation in odd-order neutral delay differential equations, Proceedings ofthe Indian Academy of Sciences Mathematical Sciences, 105 (1995), 219–225.

[7] P. Das, B.B. Mishra and C.R. Dash, Oscillation theorems for neutral delay differ-ential equations of odd order, Bulletin of the Institute of Mathematics AcademiaSinica, 1 (2006), 557–568.

[8] J. Dzurina, Oscillation theorems for neutral differential equations of higher order,Czechoslovak Mathematical Journal, 54 (2004), 185–195.

[9] K. Gopalsamy, B. S. Lalli and B. G. Zhang, Oscillation of odd order neutral differ-ential equations, Czechoslovak Mathematical Journal, 42 (1992), 313–323.

[10] I. Gyori and G. Ladas. Oscillatory Theory of Delay Differential Equations with Ap-plications, Oxford University Press, New York, 1991.

[11] Z. Han, T. Li, S. Sun and C. Zhang, Oscillation behavior of third-order neutralEmden-Fowler delay dynamic equations on time scales, Advances in DifferenceEquations, 2010 (2010), 1–23.

[12] Z. Han, T. Li, S. Sun and C. Zhang, An oscillation criterion for third order neutraldelay differential equations, Journal of Applied Analysis, 16 (2010), 1–9.

[13] B. Karpuz, O. Ocalan and S. Ozturk, Comparison theorems on the oscillation andasymptotic behavior of higher-order neutral differential equations, Glasgow Math-ematical Journal, 52 (2010), 107–114.

[14] G. S. Ladde, V. Lakshmikantham and B. G. Zhang. Oscillatory Theory of Differ-ential Equations with Deviating Arguments, Marcel Dekker, New York, 1987.

[15] T. Li, Comparison theorems for second-order neutral differential equations of mixedtype, Electronic Journal of Differential Equations, 167 (2010), 1–7.


[16] T. Li, Z. Han, C. Zhang and S. Sun, On the oscillation of second-order Emden-Fowlerneutral differential equations, Journal of Applied Mathematics and Computing,37 (2011), 601–610.

[17] T. Li, Z. Han, P. Zhao and S. Sun, Oscillation of even-order neutral delay differentialequations, Advances in Difference Equations, 2010 (2010), 1–9.

[18] T. Li and E. Thandapani, Oscillation of solutions to odd-order nonlinear neutralfunctional differential equations, Electronic Journal of Differential Equations,23 (2011), 1–12.

[19] N. Parhi and R. N. Rath, On oscillation of solutions of forced nonlinear neutraldifferential equations of higher order, Czechoslovak Mathematical Journal, 53(2003), 805–825.

[20] N. Parhi and R. N. Rath, On osillation of solutions of forced nonlinear neutraldifferential equations of higher order II,Annales Polonici Mathematici, 81 (2003),101–110.

[21] R. N. Rath, Oscillatory and asymptotic behavior of solutions of higher order neutralequations, Bulletin of the Institute of Mathematics Academia Sinica, 30 (2002),219–228.

[22] R. N. Rath, L. N. Padhy and N. Misra, Oscillation of solutions of non-linear neutraldelay differentialequations of higher order for p(t) = 1, Archivum Mathematicum,40 (2004), 359–366.

[23] Y. Sahiner and A. Zafer, Bounded oscillation of nonlinear neutral differential equa-tions of neutral type, Czechoslovak Mathematical Journal, 51 (2001), 185–195.

[24] J.H. Shen and X.H. Tang, New oscillation criteria for odd order neutral equtions,Journal of Mathematical Analysis and Applications, 201 (1996), 387–395.

[25] E. Thandapani and T. Li, On the oscillation of third-order quasi-linear neutral func-tional differential equations, Archivum Mathematicum, 47 (2011), 181–199.

[26] G. Xing, T. Li and C. Zhang, Oscillation of higher-order quasi-linear neutral differ-ential equations, Advances in Difference Equations, 45 (2011), doi:10.1186/1687-1847-2011-45.

[27] G. Xing, T. Li and C. Zhang, Oscillation of third-order neutral delay differentialequations, Abstract and Applied Analysis, (in press)

[28] A. Zafer, Oscillation criteria for even order neutral differential equations, AppliedMathematics Letters, 11 (1998), 21–25.

[29] B.G. Zhang and W.T. Li, On the oscillation of odd order neutral differential equa-tions, Fasciculi Mathematici, 29 (1999), 167–183.

[30] Q. X. Zhang, J. R. Yan and L. Gao, Oscillation behavior of even-order nonlinear neu-tral differential equations with variable coefficients, Computers and Mathematicswith Applications, 59 (2010), 426–430.


VOLUME 19

2012, NO 1–2

PP. 157–167

SMOOTHNESS OF GENERALIZED SOLUTIONS OF THEBOUNDARY-VALUE PROBLEM FOR

DIFFERENTIAL-DIFFERENCE EQUATIONS WITHINCOMMENSURABLE SHIFTS

D. NEVEROVA∗

Abstract. It was proved that the set of discountity points of the rst derivative ofgeneralized solution of boundary value problem for dierential-dierence equations withseveral incommensurable shifts is dense in the nite interval.

KeyWords. Dierential-dierence equations, incommensurable shifts, nonsmoothness.

AMS(MOS) subject classication. 49N10, 49K27, 34B27, 34K10

1. Introduction. For the rst time generalized solutions of boundaryvalue problem for dierential-dierence equations were considered by G.A.Kamenskii, A.D. Myshkis in [1]. With the help of self-adjoint extensions ofdierential-dierence operators A.G. Kamenskii laid in [2] the foundationfor theory of boundary value problem for symmetric dierential-dierenceequations. Boundary value problem for dierential-dierence equations fornonself-adjoint operator was studied in the papers of A.L.Skubachevskii [3].He considered solvability of boundary value problem for dierential-dierenceequations, properties of spectrum of dierential-dierence operators, andsmoothness of the generalized solutions. But in papers [4] it was assumedthat the shifts of the dierence operator are integers or commensurablenumbers. Using this condition, it was shown that smoothness of generalizedsolutions could be broken in a bounded domain and was preserved only insome subdomains. Smoothness of generalized solutions of boundary valueproblem for dierential-dierence equations with two incommensurable shiftswas studied in [3]. It was proved that the set of discountity points of the rstderivative of generalized solution is dense in the interval (0, d).

∗ People Friendship University of Russia, Moscow, Russia

157

158 D. NEVEROVA

In this paper we investigate the boundary-value problem for dierential-dierence equation with incommensurable shifts and prove that the set ofdiscontinuity points of the rst derivative v′ is dense on the interval (0, d).

We consider the boundary value problem

(1) −(v − εRv)′′(t) = f0(t) (t ∈ (0, d))

(2) v(t) = 0 (t ∈ [−T, 0] ∪ [d, d+ T ]),

where f0 ∈ L2(0, d), (Rv)(t) =m∑j=1

ajv(t + τj), aj = 0, (j = 1, . . . ,m),

a =m∑j=1

|aj|, 0 < ε <1

2ais suciently small, τj are incommensurable numbers

among which there are both positive and negative numbers, T = max |τj|,T 6 1, for all q ∈ Zm and the following condition holds

(3)m∑j=1

qjτj = d.

Definition 1. A system of numbers τ1, . . . , τm is called incommensurable,

if this does not exist α1, . . . , αm ∈ Z such as α1τ1 + . . . + αmτm = 0 and

α21 + . . .+ α2

m = 0.We will assume that system of shifts τ1, . . . , τm is incommensurable.

2. Dierence Operators. We consider the dierence operator R :L2(R) → L2(R) dened by the formula

(4) (Rv)(t) =m∑j=1

ajv(t+ τj),

aj ∈ C, τj ∈ R.We introduce the operators:• IQ : L2(0, d) → L2(R) operator of extension of functions fromL2(0, d) by zero above (0, d), i. e.

(IQv)(t) = v(t), (t ∈ (0, d)), (IQv)(t) = 0, (t /∈ (0, d));

• PQ : L2(R) → L2(0, d) operator of restriction of functions fromL2(0, d) to (0, d)

(PQv)(t) = v(t), (t ∈ (0, d));

SMOOTHNESS OF GENERALIZED SOLUTIONS... 159

• RQ : L2(0, d) → L2(0, d) operator dened by formula

(5) RQ = PQRIQ,

where Q = (0, d).Operators PQ, IQ will be used in the study of boundary value problems

for dierential-dierence equations. The shifts τj ∈ [−T, T ] can map thepoints of [0, d] into the set [−T, 0]

∪[d, d + T ]. Hence we must consider the

boundary conditions for dierential-dierence equation not only at the points0, d, but also on the set [−T, 0]

∪[d, d+ T ]. In order to satisfy homogeneous

boundary conditions, we introduce the operator IQ. The employment ofoperator PQ is necessary to consider the equation not on the whole axisR, but on the interval (0, d) only.

We will need some properties of dierence operators (for proof see[3,Section 2]).Lemma 1. I∗Q = PQ, P

∗Q = IQ, i.e., for all u ∈ L2(0, d), v ∈ L2(R),

(IQu, v)L2(R) = (u, PQv)L2(0,d).

Lemma 2. The operators R : L2(R) → L2(R), RQ : L2(0, d) → L2(0, d) are

bounded,

R∗v(t) =m∑

j=−m

ajv(t− τj), R∗Q = PQR

∗IQ.

Lemma 3. The operator RQ maps continuously W k(0, d) into W k(0, d), and,

for all v ∈ W k(0, d)

(RQv)(j) = RQv

(j), (j 6 k).

3. Generalized Solutions. A function v ∈ W 1(0, d) is called ageneralized solution of the problem (1), (2), if v − εRQv ∈ W 2(0, d) and

(6) −(v − εRQv)′′(t) = f0(t) (t ∈ (0, d)).

This denition is equivalent to the following:A function v ∈ W 1(0, d) is called a generalized solution of the problem

(1), (2), if

(7) (v′ − ε(Rv)′, φ′)L2(0,d) = (f0, φ)L2(0,d)

for all φ ∈ W 1(0, d).Since the operator RQ = PQRIQ : L2(0, d) → L2(0, d) is bounded, by

virtue of the Riesz theorem concerning a general form of linear functional in

160 D. NEVEROVA

a Hilbert space, there exist bounded operators B : W 1(0, d) → W 1(0, d) andG : L2(0, d) → W 1(0, d) such that

((RQv)′, φ′)L2(0,d) = (Bv, φ)′

W 1(0,d),

(f0, φ)L2(0,d) = (Gf0, φ)′W 1(0,d)

.

Here (u,w)′W 1(0,d)

= (u′, w′)L2(0,d) is the equivalent inner product W 1(0, d).

Hence the integral identity (7) will have the form

(v − εBv, φ)′W 1(0,d)

= (Gf0, φ)′W 1(0,d)

.

From this we obtain

(8) (v − εBv) = Gf0.

If ε < ∥B∥−1, then the operator I − εB has a bounded inverse. Thusthe boundary value problem (1), (2) has a unique generalized solution v =(I − εB)−1Gf0.

4. Nonsmoothness of generalized solutions.Theorem 1. Let v ∈ W 1(0, d) be a generalized solution of the boundary

value problem (1), (2) and f0 ≡ 1. Then there exists a set, which is dense in

[0, d] and the function v′ is not continuous at the points of this set.

Äîêàçàòåëüñòâî. 1. Let v ∈ W 1(0, d) be a generalized solution of theboundary value problem (1), (2), where f0(t) ≡ 1. Analogously to Lemma 3,(RQv)

′ = RQv′. Therefore from (6) it follows that

(9) (I − εRQ)v′(t) = Φε(t) (t ∈ (0, d)),

where Φε(t) = −t+ cε, for certain constant cε.

The operator RQ : L → L is bounded and ∥RQ∥L 6 a, where L =

L2(0, d) or L = L∞(0, d). Therefore ε∥RQ∥ 6 1

2and

(10) v′ =∞∑j=0

(εRQ)jΦε,

where the series (10) converges in L.


-

6

t0d

−d2

d

2

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

r

r

Pic. 4.1

Evidently, one of the inequalities either |Φε(0+0)| > d

2, or |Φε(d−0)| > d

2holds (see Pic. 4.1). Without loss of generality, we assume that

|Φε(0 + 0)| > d

2.

In consideration of (IQΦε)(0− 0) = 0, we obtain that

(11) |(IQΦε)(0 + 0)− (IQΦε)(0− 0)| > d

2.

We denote h(q) = −m∑i=1

qiτi, where qi ∈ N0, i = 1, . . . ,m (q = (q1, . . . , qm)).

We dene the set M consisting of 0 and numbers h(q) ∈ (0, d) such thath(q) = l − τi for some l ∈M , i = 1, . . . ,m.

Since h(q)−h(q′) = −m∑i=1

(qi−q′i)τi andm∑i=1

(qi−q′i)τi = 0 (incommensurable

shifts), we have

(12) h(q) = h(q′) (q = q′).

2. Now we will show that derivative v′(t) is not continuous at the pointsof the set M .

By denition of the setM , the shift of argumentm∑i=1

qiτi

(−

m∑i=1

qiτi ∈M,

|q| = j)transforms the break of the function IQΦε at the point 0 into the

162 D. NEVEROVA

break of the function RjQΦε at the point −

m∑i=1

qiτi ∈ (0, d). By virtue of (12),

the shift of argumentm∑i=1

q′iτi of the function RjQΦε (|q′| = j, q = q′) cannot

annihilate the discontinuity of this function at the point −m∑i=1

qiτi. At the

same time the shifts of the point d do not belong to the set M and cannotdestroy the breaks of the function Rj

QΦε at the points of the set M (seecondition (3)).

Nevertheless the possibility to obtain the breaks of function also exists

RiQΦε, (i > j) at the point−

m∑i=1

qiτi ∈M . Generally speaking,(IQR

i−jQ Φε

)(0+

0) = 0, i. e.(IQR

i−jQ Φε

)(0+0)−

(IQR

i−jQ Φε

)(0−0) = 0. The shift of argument

m∑i=1

qiτi

(−

m∑i=1

qiτi ∈M, |q| = j

)transforms the break of the function IQR

i−jQ Φε

at the point 0 into the break of the functionRiQΦε at the point−

m∑i=1

qiτi ∈ (0, d).

Let us consider an arbitrary point −m∑i=1

qiτi ∈M . We shall estimate the

breaks of the functions (εRQ)kΦε at this point. We can rewrite (10) in the

following form

(13) v′ =J∑

k=0

(εRQ)kΦε +

∞∑k=J+1

(εRQ)kΦε,

where |q| =m∑i=1

qi, J > |q| will be chosen later. Denote

∆qF = F

(−

m∑i=1

qiτi + 0

)− F

(−

m∑i=1

qiτi − 0

).

Then

(14) ∆q

(J∑

k=0

(εRQ)kΦε

)=

J−|q|∑k=0

ε|q|+knq∆0(IQRkQΦε) = S1 + S2,

where S1 = ε|q|nq∆0(IQΦε), S2 =J−|q|∑k=1

ε|q|+knq∆0(IQRkQΦε), nq is a number

of all sequences rl = (rl1, . . . , rlm) (l = 1, . . . , |q|) such that r|q| = q and

−m∑i=1

rliτi ∈M (l = 1, . . . , |q|) and |rl+1 − rl| = 1.

By virtue of (11), we have

|S1| = |ε|q|nq∆0(IQΦε)| = ε|q|nq|IQΦε(0 + 0)− IQΦε(0− 0)| > ε|q|nqd

2.

We obtain the following estimate for S2, since |IQRkQΦε| 6 ∥IQ∥ · ∥Rk

Q∥ ·∥Φε∥L∞(0,d) 6 ak · ∥Φε∥L∞(0,d)

|S2| 6 ε|q|nq∥Φε∥L∞(0,d)

∞∑k=1

akεk 6

6 ε|q|nq∥Φε∥L∞(0,d)2εa = 2aε|q|+1nq∥Φε∥L∞(0,d).

We show that ∥Φε∥L∞(0,d) 6 c for all 0 < ε < min∥B∥−1

2, 12a. To obtain this

relation, we will prove that ∥Φε∥L2(0,d) 6 c1. Therefore, from (8) and (9) itfollows

∥Φε∥L2(0,d) 6 ∥I − εRQ∥ · ∥v′∥L2(0,d) 6 (∥I∥+ |ε| · ∥RQ∥)∥v′∥L2(0,d) 6

6(1 +

1

2

)∥v′∥L2(0,d) 6

3

2∥(I−εB)−1∥ ·∥Gf∥′

W 1(0,d)6 3

22∥G∥ ·∥f∥L2(0,d),

i. e. ∥(I − εB)−1∥ 6∞∑k=0

∥εB∥k < 2 that provides ∥εB∥ < 1

2.

We have following estimates

|S1| > ε|q|nqd

2, |S2| 6 ε|q|nq

(2mε∥Φε∥L∞(0,d)

).

Thus we can choose ε > 0 such that 2aε∥Φε∥L∞(0,d) 6d

4. Then

(15)

∣∣∣∣∣∆q

(J∑

k=0

(εRQ)kΦε

)∣∣∣∣∣ > ε|q|nqd

4.

In addition, we have

(16)

∥∥∥∥∥∞∑

k=J+1

(εRQ)kΦε

∥∥∥∥∥L∞(0,d)

6 (εa)J+1

1− εa∥Φε∥L∞(0,d) =

= (εa)J(εa)

1− εa∥Φε∥L∞(0,d) 6 2aε(εa)J ∥Φε∥L∞(0,d) 6

d

4(εa)J

164 D. NEVEROVA

Since the right-hand side of (15) does not depend on J , we can chooseJ such that

(17) (εa)J 6 ε|q|nq

2.

Hence, the function v′ is not continuous at the point −m∑j=1

qjτj ∈M .

3. We shall prove that the set M is dense in [0, d].Denote by Mτi,...,τn a set, which can be obtained by the similar rule as

the set M and generated by the shifts on τi, . . . , τn.a) Let us consider Mτ1,τ2 è Mτ3,τ4 , such that each of them is generated

by two dierent shifts.The setsMτ3,τ4 andMτ1,τ2 do not have common points, because τ1, τ2, τ3, τ4

are mutually incommensurable shifts. Thus, Mτi,τj

∩Mτk,τl = ∅ if the shifts,

which are generated these sets, are dierent.

b) For Mτ1,...τs , s = 2, . . . ,m we get C2s =

s!

2!(s− 2)!sets, generated by

two shifts and crossed at nite number of points , i. e

mes ∩

i, j = 1, . . . , si < j

Mτi,τj = 0.

Let us show it. We consider the sets Mτ1,τ2 and Mτ1,τ3 crossing at thepoints that satises the equality aτ1 + bτ2 − a′τ1 − b′τ3 = 0, a, a′, b, b′ ∈ N0:

τ3 =(a− a′)

b′τ1 +

b

b′τ2 or

a = a′;b = b′ = 0,

i. e. one of the shifts is a linear combination of the rest with special coecients.Thus, we can prove density of the set M =

∪i<j

Mτi,τj in [0, d] if we prove

density sets Mτi,τj . Denote h(p, q) = −pτi − qτj, where p, q ∈ N0.First we shall prove that for any δ > 0 there exists a point h(p, q) ∈Mτi,τj

such that 0 < h(p, q) < δ. It can be proved by the following theorem (see [5]).

Theorem 2. For any irrational number µ there are exist integers a, b suchthat a 0 we can nd a, b > 0 such that

0 <a

b− µ < b−2, b−1 < δ ⇒ 0 < a− bµ < δ.

We consider some cases for Mτ1,τ2 :

1. τ1 > 0, τ2 > 0. The set Mτ1,τ2 = 0, because the rest of the pointsdon't belong to [0, d].

2. τ1 < 0, τ2 < 0. The set Mτ1,τ2 consists of the points −pτ1 − qτ2,p, q ∈ N such that −pτ1 − qτ2 < d (nite number of them).

3. τ1 < 0, τ2 > 0. In this case for h(p, q) we have:

h(p, q) = −pτ1 − qτ2 = pτ1 − qτ2 = τ1

(p− q

τ2τ1

)= τ1

(p− qτ2/1

),

where τ1 = −τ1 > 0, τ2/1 = τ2/τ1.

Hence, according to Theorem 2, 0 0 we dene r = r(s) in the following way:1) r(1) = 1;

if |τ1| > |τ2|2) for h(s, r(s)) ∈Mτ1,τ2 such that 0 < h(s, r(s)) < τ2, we set

r(s+ 1) = r(s) + 1, åñëè h(s+ 1, r(s) + 1) < τ2;

r(s+ 1) = r(s) + 2, åñëè h(s+ 1, r(s) + 1) > τ2.

Clearly, h(1, 1) ∈ Mτ1,τ2 . Hence h(s, r(s)) ∈ Mτ1,τ2 for every s.Therefore, by virtue of denition of r(s) and inequality (18),r(p) = q.if |τ1| < |τ2|

2) for h(r(s), s) ∈Mτ1,τ2 such that 0 < h(r(s), s) < |τ1|, we set

r(s+ 1) = r(s) + 1, åñëè h(r(s) + 1, s+ 1) < |τ1|;r(s+ 1) = r(s) + 2, åñëè h(r(s) + 1, s+ 1) > |τ1|.

By the same way we get h(1, 1) ∈ Mτ1,τ2 . Hence h(r(s), s) ∈Mτ1,τ2 for every s. By virtue of denition of r(s) and inequality(18), r(p) = q. Thus, h(q, p) ∈Mτ1,τ2 and 0 < h(q, p) < δ.

From this it follows that for all natural k è n such that nh(p, q) <maxτ1, τ2, k ·maxτ1, τ2+ nh(p, q) < d, we have k ·maxτ1, τ2+nh(p, q) ∈Mτ1,τ2 . Thus the set Mτ1,τ2 is dense in [0, d].

Elements of the vector τ are positive and negative, that's why we willface the third case (τi < 0, τj > 0), hence one of the sets Mτi,τj will be densein [0, d]. Therefore, M =

∪i<j

Mτi,τj is dense in [0, d].

166 D. NEVEROVA

Ðèñ. 1. m = 6

5. Numerical calculation. Along with the theoretical analysis ofthe problem, numerical calculations are done, which allows one to visualizethe structure of the set of discontinuity points for dierent ε. It turns outthat the perturbations are small for small ε and cannot be distinguished onthe graph. The reason is that the Neumann series for the rst derivativedecreases exponentially. For dierent big ε we can see on the graphs or thedense set of discontinuity points on the hole interval, or the dense set onsubinterval. It was not investigate theoretically. A net method is used.

N - quantity of steps. h = d/Nm - number of shifts.t ∼ ih, v(t) ∼ vi,v(t+ τj) ∼ kjvi+[τj/h] + (1− kj)vi+[τj/h]−1 if τj < 0 andv(t+ τj) ∼ kjvi+[τj/h] + (1− kj)vi+[τj/h]+1 if τj > 0,

where kj coecients of linear interpolation, j = 1, . . . ,m, i = 1, . . . , N .Using this approximation we can rewrite (1), (2).


REFERENCES

[1] Kamenskii G.A., and Myshkis A.D. Formulation of boundary-value problems fordierential equations with deviating arguments containing highest-order terms,Dierentsialnye Uravneniya, 10 (1974) [in Russian]; Dierential Equations, 10(1975)[in English].

[2] Kamenskii A.G. Boundary value problems for equations with formally symmetricdierential-dierence operators, Dierentsialnye Uravneniya, 12 (1976) [inRussian]; Dierential Equations, 12 (1977) [in English].

[3] Skubachevskii A. Elliptic functional dierential equations and applications,Birkhauser, BaselBostonBerlin, 1997.

[4] Skubachevskii A. Generalized and classical solutions of boundary value problems fordierential-dierence equations, Dokl. Akad. Nauk Ross., 334 (1994), 433436[in Russian].

[5] Arnold V. I. Geometrical Methods in the Theory of Ordinary Dierential

Equations, Nauka, Moscow, 1978 [in Russian]; Springer-Verlag, NewYorkHeidelbergBerlin, 1983 [in English].


VOLUME 19

2012, NO 1–2

PP. 169–193

ON RESONANT WAVES IN LATTICES

G. OSHAROVICH ∗ AND M. AYZENBERG-STEPANENKO †

Abstract. Mathematical modeling of resonant waves propagating in 2D periodic infi-nite lattices is conducted. Rectangular-cell, triangular-cell and hexagonal-cell lattices areconsidered. Eigenvalues (here eigenfrequencies) of steady-state problems are determined,and dispersion properties of free waves are described. We show that the frequency spectraof the models possess several resonant points located both at the boundary of pass/stopbands and in the interior of a pass band. Time-dependent initial-boundary value prob-lems with a local monochromatic source are explored, and peculiarities of resonant wavesare revealed. Asymptotic solutions are compared with the results of computer simulation.Special attention is given to line-localized primitive waveforms at the resonance frequenciesand to the wave beaming phenomena at a resonant excitation.

Key Words. Lattice dynamics, Dispersion pattern, Transient response, Resonantwave, Wave beaming

AMS(MOS) subject classification. 33L05, 33L20, 65M06, 70B99

1. Introduction. Nowadays, nanotechnologies are rapidly develop-ing, and it has become necessary to obtain predictable properties of nanopar-ticles, nanofibers and nanotubes using various mechanical actions exerted onstarting systems. Among various physical characteristics of nanostructures,their waveguide properties are becoming increasingly important allowing the-oretical prediction of dynamic stress state and fracture propagation [16].

Mathematical models of nanostructure dynamics are, as a rule, basedon periodic lattices of diverse structures. A well-studied wave phenomenoninherent to periodic lattices is that free wave propagation takes place onlywithin certain discrete bands of frequencies known also as pass-bands alter-nated with stop-bands, where the steady-state wave propagation is forbidden.

∗ Department of Mathematics, Bar Ilan University Ramat Gan 52900, Israel† Department of Mathematics, Ben Gurion University, Beer-Sheva 84105, Israel

169

170 G. OSHAROVICH AND M. AYZENBERG-STEPANENKO

In the field waves in lattices or, more generally, waves in periodic media, themonograph by Brillouin [3] has been basic for subsequent investigations ofvarious theoretical and engineering aspects [15].

During recent decades, this topic got a second wind, when so-called“artificial crystals” used as band-gap materials were discovered. Artificialphononic (or zonic) crystals are periodic lattices or composite structures de-signed to control sound and vibration waves. Some important results relatedto band-gaps in phononic crystals can be found, e.g., in [7-8].

In the frequency spectrum of band-gap materials, there exist resonantfrequencies, which usually demarcate pass- and stop-bands. In the 1D case,the group velocity of the wave equals zero at these frequencies: there isno steady-state solution corresponding to an external non-self-equilibratedexcitation, and the wave energy flows from a source decelerating with time,like heat, and not as a wave [13]. It was shown in [2] that in 2D/3D cases,resonant frequencies also exist in the interior of pass bands. Such frequenciesdiffer from those in the 1D case, since the group velocity is zero only forsome special wave orientations. Corresponding resonance processes excitedin infinite rectangular-cell and triangular-cell lattices by harmonic sourcespossessing these frequencies were studied in [10-11].

This paper is aimed at obtaining a comparative description of the fre-quency spectrum of lattices of different structures, proving the existenceof so-called Localized Primitive Waveforms (LPWs) in these lattices anddescribing the peculiarities of resonant waves excited by a local monochro-matic source. The LPWs, initially discovered in [2] for square-cell lattices,are “self-equilibrated” standing waves strictly localized on a line of a certainorientation in the lattice, which do not appear along other orientations. TheLPWs extracted from the steady-state problem predict a pronounced beameffect obtained from the unsteady-state problem at the resonant excitationin a square-cell lattice which was analytically described [2, 11, 12].

Below we explore free- and time-dependent elastic wave propagation pro-cesses in rectangular-cell and hexagonal-cell lattices. The dispersion proper-ties of lattice structures are analytically studied, and computer simulationsare conducted in order to reveal the development of the unsteady state pat-tern. To compare waveguide properties in these two cases, we use knownresults for rectangular-cell lattices obtained in [12].

The presented numerical solutions (which are of independent signifi-cance) are complementary to analytical results and, together with the latter,provide a fairly complete picture of nanostructures dynamics to be analyzed.Note that results of pure computer simulations of wave processes in latticesobtained on the basis of the classical molecular dynamic models and corre-

ON RESONANT WAVES IN LATTICES 171

sponding advanced methods and tools can be found, for example, in [1, 4-6],but an analysis of specific effects for frequency regimes resulting in resonantwaves has not been performed.

To explain the term ‘resonant wave’, we compare frequencies in both1D finite and infinite simplest uniform mass-spring lattices (MSLs): pointparticles of mass M are linked by massless elastic bonds of unit length withthe stiffness g. In Fig. 1,(a) and (b), such systems are schematically depicted.In the case of a finite system of N oscillators, we have a finite spectrum of

( )kω

( )gc k

2

1

00 2π π

g

1n− n 1n+

M

( )a ( )c

1 2 1 N N−⋯

1 1m m m− +

( )a

( )b

M g

M g

Fig. 1. Finite (a) and infinite (b) MSLs and dispersion spectra of infinite MSL (c)

N eigenvalues (eigenfrequencies): ω1 < ωm < ωN , (m = 2, . . . , N − 1). Forphysical reasons, the minimal frequency is ωmin =

√2g/NM − all particles

perform in-phase motion, so only two boundary springs are subjected todeformation. We can also say that lim

N→∞ωN =

√4g/M , while all particles

tend to oscillate in anti-phase motion.Note that finite MSCs are usually used for description of the behavior

of locally interconnected dynamical cellular neural networks (see e.g. [9]).For an infinite MSC we have a continuous frequency spectrum, which can

be found using the Floquet approach. The system of homogeneous equationsof free waves propagating in the infinite MSL is:

(1) um − c20L2(um) = 0, c0 =√g/M, L2 (um) = um+1 − 2um + um−1

where z = ∂z/∂t, c0 is the sound velocity in an effective homogeneous spring,and L2 (um) is the difference operator of the second order.

We seek a solution of the homogeneous system (1) in the form of atraveling wave:

(2) um(t) = Uei(ωt±km)

where ω, k and λ = 2π/k are, respectively, the temporal frequency, the spacefrequency and the space wavelength.


Substituting (2) into (1), we obtain the frequency and the group velocityspectra

(3) ω = ±2c0 sin (k/2) , cg = dω/dk = ±c0 cos (k/2) .

This result has a simple physical sense: the longer the waves (k de-creases), the lesser the influence of the waveguide discreteness on wave prop-agation: cg → ±c0 (k → 0). The frequency ω = 2

√g/M at k = π (the edge

of the Brillouin zone) demarcates pass and stop bands, while the free waveswith frequencies ω > 2

√g/M do not propagate. Below, natural parameters

g and M are taken as measurement units, and due to symmetry, only theinterval [0, π] is considered.

In Fig. 1(c), dependencies ω(k) and cg(k) are depicted.Spatial forms of the MSL motion depending on the wave length are

determined by the ratio of displacements of neighboring particles. One cansee from (2) that in the case of the minimal wavelength, λ = 2 (k = π), thisratio equals –1: the shortest waves perform anti-phase oscillations.

Differences in resonant patterns for finite and infinite 1D mass-springlattices are distinct. In the N -DOF MSL, each nth eigenfrequency is theresonant one. An external excitation of the system with such a frequencyresults in a linear growth of the solution with time.

In the case of an infinite MSL, we have obtained (see [11]) the followingasymptotic solution for problem (1) with zero initial conditions and the actionof a local sine source of resonant frequency u0 (t) = sinωrt (ωr = 2):

(4)um(t) ∼

√t [F2 (λ) sin (2t− πm)− F1 (λ) cos (2t− πm)] ,

λ = 2 |m|/√

t (t→ ∞, |m| ≪ t)

where the well-known functions in the theory of unsteady waves, F1(λ) andF2(λ) (see, e.g., [11, 13, 14]), are oscillating and spreading with the growthof λ. The resonant wave (4) propagates along the lattice with the velocity∼ t−1/2 that corresponds to the heat propagation law. Within this process,amplitudes um(t) increase with time as t1/2 contrary to a finite MSL wherethe linear growth is detected.

In addition to the analytical solution, we present an example of a reso-nant wave numerically calculated in the infinite MSL subjected to a mono-chromatic force applied in the zero cross-section of the waveguide: Q =H(t)δ(x) sinωrt, where ωr = 2, while H(t) and δ(x) are Heaviside and Diracfunctions. We show how resonant waves are developing under such excitationand whether there is a correspondence between numerical and analytical so-lutions. Recall that this frequency determines zero group velocity (i.e. zero


energy flux) and, for this reason, the steady-state solution is absent. Someresults of computer simulations are presented in Fig. 2 [11]. Envelopes ofum(t) for a set of nodes can be seen in Fig. 2(a). Dependence u0(t) is accu-rately described by asymptotic solution (4) starting virtually from the verybeginning. The larger the distance from the considered node to the loadingpoint, the longer the time period needed to reach a good coincidence of thenumerical solution with the solution (4).

Distributions um(t) along the n-axis at t = 300, 600, and 900 are pre-sented in Fig. 2(b). In accordance with (4), the main perturbations move atthe velocity proportional to t−1/2, while their amplitudes increase with timeas t1/2. Calculations show that beginning with t ≈ 200, solution (4) can beapproximated by numerical results with a good accuracy.

( )a

0 250 500 750 1000 1250

-20

-10

0

10

20

m=100

m=50

m=25

m=0m=10

t

2r

ω =

0 40 80 120-7

0

8

-10

0

11-13

0

14

t=300

m

t=600

t=900

( )b

Fig. 2. Resonant process in a MSL: (a) envelopes of displacement oscillations vs. timein nodes m =0, 10, 25, 50, 100; (b) distributions along m-axis at t =300, 600 and 900

In 2D cases, in addition to resonances excited by the frequency demar-cating pass/stop bands, there exist resonances excited by frequencies locatedin the interior of a pass band. Below we determine such frequencies andcorresponding waveforms and calculate properties of resonant waves in rect-angular, hexagonal and triangular lattices.

2. Rectangular-cell lattice. In this section, we analyze dispersionpatterns and anti-plane wave localization processes in a uniform Rectangular-Cell Lattice (RCL). Emphasis is put on the dispersion analysis of wave propa-gation processes emerging in the lattices under the action of a point harmonicexcitation with a resonant frequency.

2.1. Dispersion pattern. Material particles of a lattice are locatedin nodes (m,n), m,n = 0,±1,±2, . . ., linked by elastic massless bonds – seeFig. 3. We use discrete coordinates m and ntogether with continuous coordi-nates x and y. The following nomenclature is used: M is the particle mass,


gx and gy are out-plane normalized stiffnesses of x- and y-bond, respectively,lx and ly are their lengths. We explore transversal oscillations of the lattice(see [12]). First, to explore the dispersion pattern, we are going to obtain adispersion equation of φ (ω, kx, ky) = 0 type, where ω is frequency, kx and kyare wave numbers or projections of the wave vector k(kx, ky). Then we an-alyze dispersion surfaces ω = ω (kx, ky) and calculate group velocity vectorscg (kx, ky), [(cg)x, (cg)y] = (ω′

kx, ω′

ky).

y

x

xg

M

yg

0

2

1−

1

m

n

01− 21

Fig. 3. Rectangular-cell lattice

In a linear approximation, systems of homogeneous equations of the RCLdynamics can be written as follows:

(5) Mum,n = gx (um,n+1 + um−1,n − 2um,n) + gy (um+1,n + um,n−1 − 2um,n)

where um,n is the out-plane displacement of the (m,n)-particle.We represent a general solution of the system (5) by a superposition of

sinusoidal waves

(6) um,n(t) = Um,neiωt, Um,n = Uei(kxx+kyy) = Uei(kxlx·m+kyly ·n),

where U is constant and |kxlx| ≤ π, |kyly| ≤ π. Substituting (6) into (5), weobtain the dispersion surface

(7) ω =√

2 [gx (1− cos kxlx) + gy (1− cos kyly)]/M.

If x- and y-bonds differ only in their lengths (lx = ly), then, putting l ≡ lyand taking lx and M as measurement units, we obtain a dispersion surfacefor a RCL with a single free parameter l,

(8) ω =√2 [1− cos kx + (1− cos kyl)/l]/M , (|kx| ≤ π, |kyl| ≤ π) .

This surface has a pass-band ω ∈ [0, ωr) and stop-band ω > ωr, whereωr =

√2 (2 + 2/l)/M is the resonance frequency at the bands interface.


In the case of a lattice with the same bonds (l = 1), Eqn. (8) becomesa well-known dispersion relation for a square-cell lattice (SCL),

(9) ω =√

2 (1− cos kx) + 2 (1− cos ky) (|kx| ≤ π, |ky| ≤ π) ,

with ωr =√8 at the pass/stop-bands interface.

Together with the RCL, we introduce its special kind adopted as a Sim-plified Rectangular-Cell Lattice (SRCL), in which gx = gy, l = 1. For aSRCL, the dispersion equation (8) acquires the following form:

(10) ω =√2 (1− cos kx) + 2 (1− cos kyl) , (|kx| ≤ π, |kyl| ≤ π) .

Then we have revealed similarities and differences inherent to dispersionproperties of the considered SCLs and RCLs. For a sinusoidal wave, thegroup velocity vector cg (as well as the energy flux) is oriented along anexternal normal to the equifrequency contour ω = const. As shown in [2],the contour ω = 2 is resonant for SCLs. Below we show that this contour isalso resonant for SRCL.

Fig. 4 depicts dispersion surfaces (a) and resonance contours at ω = 2(b) obtained from ((6)). The contour is rhombic: kx ± lky = ±π, in contrastto the square one, kx±ky = ±π, in the SCL [2]. From (10) we have obtainedx- and y-projections of the group velocity cg at ω = 2:

(11) (cg)x =sin kx2

, (cg)y =l sin kyl

2,

Here the energy flux along the axes x and y is absent: (cg)x = 0 (ky = 0)and (cg)y = 0 (kx = 0). As we show below, the group velocity orientation inthe kx, ky-plane coincides with the orientation of Localized Primitive Wave-forms (LPWs) originally discovered in [2]. It follows from (11) that for ω = 2the group wave velocity value |cg| and its orientation β are

(12)|cg| =

√1 + l2

2sin

π

1 + |tanα|(α = arctan (lky/kx)) ,

β = arctan((cg)y

/(cg)x

)= ± arctan l,

where α and β are the phase and group velocity orientations, respectively. Ifl = 1, we obtain results corresponding to a SCL [2]:

(13)|cg| =

√2

2sin

π

1 + |tanα|(α = arctan ky/kx) ,

β = arctan

(∂ω

∂ky

/∂ω

∂kx

)= ±π

4.


2ω=

2.5ω=

1.5ω=

xklπ

yk

0

1

2

8

lπ−

0π−

ω

π0

2ω =

( )b

π− π

lπ−

lπ

yk

xk

++++ββββ

−β−β−β−β++++ββββ

−β−β−β−β

0

( )c

LPW2ω =

1:x yg g M= = =

1 1

n

ml

l

++++ββββ( )a

Fig. 4. Dispersion pattern in SRCL: (a) dispersion surface (10), (b) equifrequency contourfor ω = 2 in plane kx, ky (arrows show the energy flux orientation), (c) three-particle-width band. Particles involved in anti-phase oscillations are marked by open circles with’+’ or ’- ’, and immobile ones - by black circles

As one can see from (12), the group velocity is zero only in the followingfour directions, α = ±0 and α = ±π/2. Directions determining LPW (diag-onal) orientations are associated with the angles ±β and shown in Fig. 4(b)by arrows. In Fig. 4(c), for one of the above-mentioned angles, +β, a three-particle-width band is shown, in which we consider the diagonal (m,m) anda neighboring particle (black circle) connected with two diagonal particlesinvolved in anti-phase oscillations. Their actions on a near-diagonal particleare self-equilibrated, and thus, black particles can be at rest. So the existenceof the LPW is a consequence of certain symmetry of the lattice structure.One can see that ω =

√2 (gx + gy)/M = 2 is the LPW frequency as in a

SCL (recall that lengths of x- and y-bonds in a SRCL are different, but theirstiffnesses are equal, gx = gy, and measurement units are gx = gy =M = 1).

Consider now a RCL possessing bonds of different lengths. The disper-sion relation for such a lattice is expressed by (8). Note that LPWs areabsent here, but, as shown below, the dispersion pattern has some commonpoints with the above-considered one in SCL (and SRCL).

Projections of the group velocity vector, its module and energy fluxdirections βobtained from (8) are

(14)(cg)x =

sin kxω

, (cg)y =sin(lky)

ω,

|cg| =1

ω

√sin2 kx + sin2(lkx), β = arctan

(sin(lky)

sin kx

).

Below we also use the expression obtained from (8) for β in terms of ω and


kx:

(15) β = arctan

√1− [1− l (ω2/2 + cos(kx)− 1)]

2

l sin(kx).

There exist four specific angular points in the dispersion surface (kx =

±π, ky = 0), where ω = 2 and (kx = 0, ky = ±π/l) –ω = 2/√

l , in which

the group velocity is equal to zero and, similarly to SCL (SRCL) cases, thedirections coinciding with the axes x and y are forbidden for the energy flux.

( ) a SRCL

2ω=

2.4ω=1.63ω=

1.8ω=

1C

2C

0.8ω=

π

yk

xk0

l

π

0

( ) , 1.5b RCL l =

2C

1C 1C

1.63ω=

0.8ω=

2.4ω=2ω=

xk ππ−l

π−

l

π

yk

0

0

2ω=2.4ω=

0.8ω=

xkππ−

0

l

π−

l

π

yk

0

0 πxk

l

π

yk

0

2.4ω=

0.8ω=

2ω=

Fig. 5. Group velocity pattern (upper figures) and equifrequency contours (lower figures)in square-cell (a) and rectangular-cell (b) lattices. Dotted straight lines in the lower pictureof (b) are tangents to contours at angular points

In two upper pictures in Fig. 5, the first quarter of the plane kx, ky isshown with the value and direction of the group velocity expressed by thelength and direction of the respective arrow. Besides, some other equifre-quency contours are depicted. These contours in the entire Brillouin zoneare presented in the lower row of Fig. 5. We compare results for SRL, column(a), and RCL (l = 1.5), column (b), in order to reveal their similarities anddifferences. Recall that in a SRCL, resonance frequency, ω = 2, determines a

rhombic contour kx± lky = ±π. In the RCL case, equifrequency contours, C1

and C2, are curvilinear, corresponding to the frequencies ω1 = 2/√

1.5 ≈ 1.63and ω2 = 2, respectively.

The analysis of relations (14) and (15) shows that for relatively low fre-quencies, there are no preferable directions of cg. If the frequency increases,the dependence of |cg| and β on coordinates kx and ky becomes sensible.

Consider the SRL case, Fig. 5(a). If the frequency tends to a resonantone, ω = 2, values of |cg| tend to zero in the vicinity of the above-mentionedangular points of the contour, kx = ±π, ky = 0 and kx = 0, ky = ±π/l.When ω0 = 2, the directions of cg with wave numbers kx ∼ 0 (ky ∼ 0) turnto the right (left) at an angle π/2. Such behavior of the free wave pattern isrelated to the formation of caustics. In the RCL case, the caustic appears inthe same angular points, which are located now in different contours, C1 andC2.

As shown in [12], sources with the frequencies ω0 = ω1 and ω0 = ω2 exciteresonance phenomena of a pronounced beaming character of the spatial wavepattern. With increasing ω0, the process of cg transition consists of two parts:

(i) If ω0 tends to ω1 = 2/√

l = 1.63, the value of |cg| in the vicinity

of kx ∼ 0 decreases with increasing ky. It tends to zero if ω0 = ω1 andky → π/l, and cgorientation (angle β) sharply turns fromβ = π/2 to β = 0.If kx increases, the value of |cg| in the contour C1 also increases, while thechange in β orientation obeys the requirement for the vector cg to be normalto C1.

(ii) With further increase in ω while approaching to ω2 = 2 (contour C2)and passing through it, the group velocity pattern is similar to that describedabove in case (i), but it is realized now near the domain (ω = ω2, kx → π),and the vector cg turns from β = 0 to β = π/2.

The analysis of the dispersion pattern shows that most parts of thecontours C1 and C2 can be approximated by tangents (which are mutuallyparallel) at angular points. From (15) we have obtained cg orientation cor-responding to these tangents:

(16) β = β∗ = arctan(1/√

l).

This angle is prevalent for wave propagation from a local source with thefrequencies ω1 and ω2 to the periphery. A certain part of the wave is scatteredinside the interval 0 < β < β∗, while the interval β∗ < β ≤ π/2 determinesforbidden directions for the energy flux.

2.2. Unsteady-state dynamics. We have analyzed wave patternsat kinematic excitation, u0,0 = U sinω0t, and/or force excitation, F0,0 =

F sinω0t, applied to a particle m = n = 0 at t = 0. Here U and F are excitedamplitudes (below we set U = 1 and F = 1). In computer simulations, weuse Eqn. (5) with zero initial conditions and above-mentioned loadings. Anexplicit finite-difference scheme has been used in a finite domain of latticenodes, while the domain boundaries are chosen at such a long distance fromthe source that their influence on the space region of interest has not beendetected. The time step is taken two orders less than the time measurementunit, which (as calculations show) allows the accuracy level ∼ 10−4.

First we consider a SCL as a partial case of the RCL. At the resonantexcitation, ω0 = 2, a phenomenon of “spatial wave separation” is detectedin Fig. 6. As one can see, a sufficiently large part of the source energy iscaptured by the diagonal line n = m (β = π/4). In accordance with thedispersion analysis and relations (13), cg =

√2 for ω = 2. At the same time,

oscillations along the directions n = m/2 and n = 0 are practically locked inthe source vicinity.

0

n = 0

n = m/2

0

0

n = m

t = 500

0.2

0 100 200 300m0 20 40 60

0

n = 0

n = m /2

0

0

n = m

t = 100

m

0.2

,m nu

Fig. 6. Distributions of displacements along three directions: n = m, n = m/2 and n = 0at the force excitation with ω0 = 2. Pronounced beaming pattern of wave propagation alongthe direction n = m is detected

Note that there are no noticeable differences in kinematic and force exci-tations (this conclusion remains valid in the examples of calculation presentedbelow).

Some results of the star-beaming phenomenon are presented in Fig. 7(a)and (b) for SCL and SRCL (with l = 1.5), respectively. The structures aresubjected to a local kinematic excitation with resonance frequency ω0 = 2.The star-like wave contours are plotted in such a way that the envelopesof oscillations in the outer area, |Um,n|, are lower than 10% of the sourceamplitude - |Um,n| < 0.1.

In Fig. 8, we present examples of the same perturbation pattern in aRCL (the parameter of the cell form is l = 1.5) at t = 250. Kinematicexcitation with several values of ω0 was applied. Here spatial wave patterns


are more saturated than in the SCL-SRCL case (compare with Fig. 7). First,in accordance with the orientation (16), a star-like pattern appears for twofrequencies ω0 = ω1 = 1.63 and ω0 = ω2 = 2. Calculations show that forthese frequencies a resonant character of oscillations is realized, similar tothat in SCLs.

( ) b SRCL

116116

116 116

( ) a SCL

116116

116 116

Fig. 7. Star-like beaming pattern at the resonance frequency ω0 = 2 in square-cell (a)and simplified rectangular-cell (b) lattices at t = 250. Outside the stars, displacements ofnodes remain less than 10% of the maximal value in the source

m0 8080−

n

0

80−

80−

01.63ω =

n

0

80−

80−

m0 8080−

01.8ω =

n

0

80−

80−

m0 8080−

02.0ω =

Fig. 8. Beaming wave patterns in the rectangular-cell lattice at t = 250. Points representparticles with |Um,n| ≥ 0.1

The spatial character of wave propagation with frequencies within theinterval (ω1, ω2) is restricted by forbidden directions in compliance with (15)and the above-presented dispersion analysis.


Finally, we have compared the results of computer simulations conductedin the case of the resonance frequency located at the interface of pass- andstop-bands: ω0 = ωr =

√2 (2 + 2/l). In Fig. 9, modules of envelopes, |Um,n|,

in diverse nodes (m,n) are depicted as functions of time for several values ofl (force source functions). The main difference of the wave response in theconsidered case from that discussed above, where exciting frequencies arewithin the pass-band, is that in the former there are no preferable directionsfor wave propagation. Besides, resonant growth of perturbations with timeis relatively slower.

00

0.2

200 400

,m nU

0.4

0.6

0.8

t

( )0,0

( )1,0

( )2,2

( )10,10

( )0,2

( )0,0

( )1,0

( )2, 2

( )10,10

( )0,2

00

0.2

200 400

0.4

0.6

0.8

t 00

0.2

200 400

0.4

0.6

0.8

t

( )0,0

( )1,0

( )2,2

( )10,10

( )0,2

( )c( )b( )a

Fig. 9. Envelopes |Um,n| vs. time in RCLs at excitations with frequency ω0 demarcating

pass and stop bands: (a) l = 0.5, ω0 =√12; (b) l = 1, ω0 =

√8; (c) l = 1.5, ω0 =

√20/3;

3. Hexagonal-cell lattice.

3.1. Dispersion pattern. Consider a uniform hexagonal-cell latticeHCL of material particles at nodes (m,n),m,n ∈ Z, linked by elastic masslessbonds - see Fig. 10. We use discrete rhombic Cartesian coordinates m andn together with continuous rectangular coordinates x and y. As above, weconsider out-plane oscillations of the lattice. The generating element of thelattice, rhombic cell [m,n], is bounded by coordinate lines m, n, m + 1 andn + 1. It consists of two nodes, whose displacements are denoted u andv. The v-nodes are located at the intersection of coordinate lines m andn, while u-nodes are located leftward of the correspondent v-node. Each v-node is connected to the three nearest u-nodes, and vice versa, each u-nodeis connected to the three nearest v-nodes. Such a consideration results in theexistence of two oscillating modes.

Let the distance between two neighboring nodes, particle masses andstiffnesses of connecting bonds be measurement units: M = g = a = 1.Then, with the cell geometry in mind, transverse motion of particles arranged

ay

x1

a

2a

n

1n−

1n+

1m+

m

1m−

,m nv

,m nu

+1 ,m nv

1 , 1m nu

+ +

, 1m nu

+

a

Fig. 10. Hexagonal-cell lattice

in a hexagonal lattice is described by the following system:

(17)um,n = vm,n + vm,n−1 + vm−1,n − 3um,n

vm,n = um,n + um,n+1 + um+1,n − 3vm,n.

As in [16], we assume harmonic solutions of (17) in the conventional form ofa plane wave (in the xy- plane)

(18) (um,n, vm,n) = (U, V ) · exp(i(ωt+ nk · a1 +mk · a2

)),

where a1 =√3/2x+ 1/2y, a2 =

√3/2x− 1/2y (x and y are unit vectors).

After substituting (18) into (17), we obtain a linear system, whose non-trivial solution determines a dispersion equation:

(19) ωI,II =

√3∓

√1 + 4 cos (ky/2)

(cos(kx√3/2)+ cos (ky/2)

)where signs ‘–’ and ‘+’ correspond to the first (acoustical) and second (opti-cal) modes.

Eqn.(19) shows that a complete stop-band between modes is absent:their dispersion surfaces are connected in four so-called conical points (CP),which are obtained by equating ωI = ωII . Coordinates of CPs are [kx, ky] =[±2π

/√3,±2π/3

], and ωCP =

√3 is the CP frequency. The point kx = ky =

0 determines resonant frequencyω =√6 demarcating pass band, ω <

√6,

and stop band, ω >√6.

Below we show that LPWs are realized at the same single contour ky =±π ∪ ky = ±2π ±

√3kx corresponding to two resonant frequencies ω =

√2

(mode I) and ω = 2 (mode II).


In Fig. 11, dispersion surfaces of modes I and II are depicted togetherwith sets of equifrequency contours for each of the modes.

( )a

xkyk

π−2 3π

0

π2 3π− 0

3

ω

0

1

2

( )b

6

ω

0

3

1

2

xkyk

π−2 3π

0

π2 3π− 0

Fig. 11. Dispersion surfaces and equifrequency contours for modes I - (a) and II - (b).Black circles are angular points of contours, in which group velocities are zero. Opencircles are CPs, in which surfaces of modes I and II are connected

To evaluate parameters of the energy flux depending on the frequencyand the wave vector, we have obtained from (19) group velocities (absolutevalues, x- and y-projections and directions, β): (cg,x)I,II =

√3 cos(ky/2) sin(

√3kx/2)

2ωI,IIϕ(kx,ky)

(cg,y)I,II =sin(ky/2) cos(

√3kx/2)+sin(ky)

2ωI,IIϕ(kx,ky)

ϕ (kx, ky) =

√1 + 4 cos (ky/2)

[cos(kx√3/2)+ cos (ky/2)

],

(20)

|cg|I,II =√(cg,x)

2I,II + (cg,x)

2I,II,

βI,II = arctan3cos2 (ky/2) sin

2(√

3kx/2)

sin (ky/2) cos(√

3kx/2)+ sin (ky)

.

Consider the equifrequency contour ky = ±π ∪ ky = ±2π ±√3kx while

−π ≤ kx ≤ π, which is of special interest:

(21)

ky = ±π ⇒ |cg|I,II =αI,II|cos(√3kx/2)|

4, β = ±π

2

ky = ±2π ∓√3kx ⇒ |cg|I,II =

αI,II|sin√3kx|

4, β = ±π

6(αI =

√2,αII = 1

).


The group velocity orientation for mode II is opposite to that for mode Iwhile their absolute values differ by a factor of

√2. As mentioned above, the

group velocity orientation in kx, ky-plane coincides with the orientation ofLPWs.

In the upper row of Fig. 12, first quarters of the xy-plane are shown;the value and direction of the group velocity are expressed by the length anddirection of the respective arrow; also parts of equifrequency contours can beseen. These contours in the entire Brillouin zone are presented in the lowerrow of Fig. 12.

( ) b( ) a

2ω =

2.15ω =

2.2ω =

2.4ω =

1.85ω =

1.98ω =2.02ω =

xk

yk

00

π

2 3πxk

yk

00

π

2 3π

2ω =

0.5ω =

1.2ω =

1.5ω =

1.7ω =

1.4ω = 1.43ω =

0.9ω =

xk

yk

2 3π− 0π−

0

π

2 3π

1.2ω =

2ω =1.5ω =

0.9ω =

0.5ω =

xk

yk

2 3π− 0π−

0

π

2 3π

2.4ω =

2.2ω =

2.15ω =

2ω =

1.85ω =

2.02ω =

Fig. 12. Dispersion pattern in the HCL. Group velocities (the upper row) in the firstquarters of the plane kx, ky and equifrequency contours in the Brillouin zone (the lowerrow): (a) - mode I, (b) - mode II

We compare the results for the acoustic mode, column (a), and the opticmode, column (b), in order to reveal their differences. In the case of longwaves kx, ky → 0, the mode I has a maximal value cg = 1/2, while the mode


II determines cg = 0; the opposite directions of velocities in these two casesare detected. The group velocity in CP is cg = 1/4 in both modes.

The LPW frequencies in HCLs can be obtained using considerationssimilar to those used above in the SRCL case. In Fig. 13, two-particle-widthbands are depicted allowing LPW frequencies for mode I (a) and mode II (b)to be evaluated; direction β corresponds to one of three directions in whichthe energy flux possesses the maximal value. First, consider a band shownin Fig. 13(a). The particles of u- and v- families of each cell move in-phase(this is denoted by the same signs), according to the oscillation form of modeI. The motion of particles u and v is realized in a neighboring cell also in-phase, but with the opposite sign. The actions of two neighboring pairs onintermediate black particles are self-equilibrated, and thus, the latter can beat rest. The oscillation frequency of the cell is ωI =

√4g/2M =

√2in the

considered case (recall that g and M are measurement units).

( )b

( )

Mode II

2ω =

-

- +

+

-

- +

+

- +β

u v

( )a

( )Mode I

2ω =--

--

++

++

++β

u v

( )c y

x

6β π=

2β π=

6β π= −

Fig. 13. LPWs of mode I (a) and mode II (b); (c) – energy flux orientations β

A similar consideration applied to anti-phase oscillations inherent tomode II, Fig. 13(b), allows to obtain the corresponding LPW frequency asωII =

√4g/M = 2: the total stiffness in this case is equal to 4g (2g is the

sum of stiffnesses of two bonds linking a moving particle to immobile ones,and 2g is the stiffness of the half-bonds linking moving particles with eachother). The LPW orientations are shown in Fig. 13(c).

In addition to the LPWs, a special oscillation form exists, in which par-ticles of u(v)-family are immobile, while neighboring particles of v(u)-familyoscillate in anti-phase. This form has the CP frequency ω =

√3. Finally,

note a special form in which neighboring particles oscillate in anti-phase. Thefrequency of this form turns out be the pass/stop-bands interface ω =

√6.

The latter case is an analogue of the simplest anti-phase resonant oscillations


in a one-dimensional mass-spring chain with the frequency ω = 2 demarcat-ing pass- and stop-bands.

3.2. Unsteady-state dynamics. Some results of computer simu-lations with HCLs are presented below in Fig. 14 and Fig. 15. Calculationswere conducted in the case of force excitation, and the location of particleu0,0 was associated with the origin of rectangular coordinates x = y = 0.

06ω =

,m nV

0.3

0

0.1

12

12006000 t

,m nU

0.3

0

0.1

12

12006000 t

02ω =

,m nV

0.4

0

0.2

1

2

12006000 t

,m nU

0.4

0

0.2

1

2

12006000 t

02ω =

,m nV

0.6

0

0.2

1

2

12006000 t

1

2

,m nU

0.6

0

0.2

12006000 t

Fig. 14. Envelopes of displacements in diverse nodes at a set of frequencies ω0

( )b

82

82

82

( )a

95

95

95

Fig. 15. Star-like beaming pattern in the hexagonal-cell lattice at the resonant frequencies(kinematic excitation): (a) mode I, ω0 =

√2 and (b) mode II, ω0 = 2 at t = 400. Outside

the contours, maximal displacements of nodes remain less than 10% of the maximal valuein the source

Envelopes of um,n and vm,n depicted in Fig. 14 correspond to two cells:1 – (m,n) = (23, 0) and 2 – (m,n) = (−15, 26). They are obtained atthe following frequencies ω0: ω0 =

√2 and ω0 = 2 (note that those are

resonant frequencies which excite LPWs). In rectangular coordinates, cell 1– (x, y) ≈ (35, 20) and cell 2 – (x, y) ≈ (16.5,−35.5) are located at practicallythe same distance, d ≈ 40, from the source, and cell 1 is located on theresonant ray (β = π/6), while cell 2 is located approximately in the middlebetween two resonant rays β = π/6 and β = π/2. In the case of ω0 =

√6

(recall that this frequency is located at pass/stop-band intersection), cell1 :(m,n) = (7, 0) and cell 2 : (m,n) = (−5, 8) were chosen. They are locatedat practically the same distance, d ≈ 12, from the source.

The pattern described above is also realized in symmetric directions(with respect to lattice structure). The presented results, together withthe results of additional calculations, allow the formulation of the followingconclusions:

1. Relatively long waves have no preferred directions of propagation (upto values close to the first resonance, ω0 =

√2). The influence of lattice struc-

ture does not appear, and waves propagate as in a corresponding effectivehomogeneous solid. Finally, the steady-state solution is reached relativelyrapidly.

2. The wave pattern is drastically changed at the resonant frequencyω0 =

√2 associated with the LPW for mode I – a detectable growth of

amplitudes in the star rays and a weak response in intermediate regions aredetected.

3. If the exciting frequency coincides with ω =√3, a surprising wave

pattern is detected at first glance: while oscillations of u-particles of thefamily, to which the excited particle u0,0 belongs, tend to zero with time,oscillations of v-particle tend to the steady state limit. On the other hand,it can be seen from the system for U and V , whose nontrivial solution de-termines dispersion equation (14), that the ratio U/V → 0 (or V /U → 0) ifω →

√3. So such a pattern obtained in the unsteady state solution could be

expected.

4. The LPW resonant frequency for mode II, ω0 = 2, results in prac-tically the same wave pattern as the one considered above for mode I. Thedifference is that in accordance with the dispersion pattern presented in (21)the main perturbations arrive at the observation points with a delay (com-pare with the results for ω0 =

√2).

5. Frequency ω0 = 2.2 located between two resonances 2 < ω <√6

results in the same qualitative process as in the case a low-frequency excita-tion ω0 = 1.2: the wave has no preferred directions of propagation, and the


steady state regime is reached relatively quickly.

6. Amplitudes of low-frequency long-wavelength resonance excited bythe frequency demarcating pass- and stop-bands, ω =

√6, increase relatively

slowly, and the resonant pattern is detected as soon as the perturbationsarrive at the reference point. As in the 1D case (see [11, 13]), the groupvelocity is equal to zero at this frequency – there is no steady-state solutioncorresponding to an external non-self-equilibrated excitation, and the waveenergy flows from a source decelerating with time, like heat, and not as awave.

In Fig. 15, star-like beaming patterns of wave propagation process areshown at source frequencies ω0 =

√2 and ω0 = 2. It is of interest that the

beaming is more pronounced in the case of mode II. Such a phenomenon willbe the point of further research.

4. Triangular-cell lattice. Two considered kinds of triangular-celllattices are shown in Fig. 16(a) and (b), respectively, a lattice generated by anequilateral triangle (ETL), and a lattice generated by a right-angled triangle(RTL).

gg

M

γγγγ

n

m0

1−

2

1

1− 0 1 2

( ) RTLb( ) ETLa1

3

Mg

g

g0

1−

2

1

1− 0 1 2 m

n

Fig. 16. Triangular-cell lattices: (a) ETL – a lattice of equilateral triangles, (b) RTL – alattice of right-angled triangles

4.1. Waves in ETLs. We consider a transverse motion of an ETLconsisting of particles (with the mass M) located at the points x = m +n/2, y =

√3n/2 (m,n ∈ Z) and connected by elastic massless springs of the

stiffness g. Let these natural parameters be measurement units: M = g = 1.Then a non-dimensional homogeneous system of dynamic equations for sucha lattice (m,n ∈ Z) is

(22) um,n = um,n+1+um+1,n+um−1,n+um,n−1+um−1,n+1+um+1,n−1−6um,n.

From (22) we have obtained the dispersion surface

(23)ω =

√√√√8− 4 cos

(kx2

)[cos

(kx2

)+ cos

(√3ky2

)],

−π < kx ≤ π, − 2π√3< ky ≤

2π√3.

Here, the (resonance) frequency demarcating pass- and stop-bands isωr = 3. Four resonant points of this frequency have y-coordinates at the endof the Brillouin zone qy = ±2π

/√3, whilex-coordinates, qx = ±2π/3 (which

is of interest) are found inside it [10]. The LPW frequency located inside thepass-band is ω =

√8 [2].

( )a

ω = 2

ω = 8

ω = 2.9

π

yk

2 3π−

xkπ− 0

0

2 3π−

( )b

11

1221

Fig. 17. Dispersion pattern in ETLs: (a) equifrequency contours and energy flux orien-tations (bold arrows) in the resonant case ω =

√8, (b) LPWs in a three-particle-width

band.

In Fig. 17(a), a resonant equifrequency contour at ω =√8 is plotted

along with two others contours corresponding to ω = 2.0 and 2.9. The groupvelocity obtained from (23) at ω =

√8 is

(24)|cg| =

1

2√8cos

(√3π

2tanα

),

β = 0, − π/6 < α = arctan (ky/kx) < π/6

and the energy flux in the sector |α| < π/6 is directed along ±x. For theremaining five sectors, 2k − 1 < 6α/π < 2k + 1 (k = 1, . . . , 5), we obtainthe same value for |cg| for five orientations βk = kπ/3. Thus, just as forrectangular lattices, the group velocity has the orientation of the nearest


LPW, and cg = 0 at the vertices of the hexagon, where cg direction isdiscontinuously changing.

The existing LPWs, whose orientations are associated with each of thethree bond lines, can be detected in a three-particle-width band presented inFig. 17(b) for β = π/3. Here, as in the case of the SRL, Fig. 4(c), particlesinside the band involved in anti-phase oscillations are shown by open circleswith signs ‘+’ and ‘–’, while black circles correspond to immobile particles.The action of oscillating particles on a neighboring one is self-equilibrated.The eigenfrequency of an oscillating particle is ω =

√GΣ/M , where M = 1

and GΣ is the total stiffness of bond system equal to 8 (it consists of fourbonds with unit stiffness connecting an oscillating particle with neighboringimmobile ones, and two half-bonds with the stiffness equal to 2 each). Thus,the system eigenfrequency is ω =

√8.

4.2. Waves in RTLs. Now we examine the transverse motion of aRTL. In this case, the system of homogeneous dynamic equations and thecorresponding dispersion relation are as follows (parameters M and g beingtaken as measurement units: M = g = 1):

(25)um,n = um,n+1 + um+1,n + um−1,n + um,n−1+

+γ (um+1,n+1 + um−1,n−1)− 6um,n,

(26) ω =√4− 2 cos kx − 2 cos ky + 2γ [1− cos (kx + ky)]

In contrast to the square-cell lattices and ETLs analyzed above, equa-tions (25) and (26) contain an additional parameter γ (stiffness of the diag-onal bond). Here, the obtained resonance frequencies that demarcate pass-and stop-bands are

(27)γ ∈ [0, 0.5] : ωr =

√8 (kx = ±ky = ±π) ,

γ > 0.5 : ωr = (2γ + 1)/√γ (kx = ±ky = ± arccos (−1/2γ)) .

A simple LPW can be detected in RTLs by referring to a tri-diagonal stripconstructed in such a manner that particles in the interior diagonal are inanti-phase oscillation and connected (i) with each other by bonds of thestiffness γ, and (ii) with immobile nodes in two outer diagonals by bondswith g = 1. Then the total dimensionless stiffness of the RTL partial systemis 4(1+γ), and the LPW frequency is ω = 2

√1 + γ. This resonant frequency

can be explicitly obtained from (26). The corresponding contours partiallyconsist of straight lines ky = ±π− kx, while other parts (if they are real) arecurvilinear.


Thus, in the directions coinciding with the diagonal y = x, there existsa LPW with the same anti-phase oscillation form as in the square-cell MSL.The relations for the energy flux obtained for the RCL (with l = 1) is alsopreserved for the considered RTL, if we set resonance frequency ω = 2

√1 + γ

instead of ω = 2 (corresponding to the SL) in (11) and (12). Then we obtainsimilar expressions for group velocity projections and for the energy fluxoriented along two directions β = ±π/4:

(28) (cg)x =sin kx

2√1 + γ

, (cg)y =sin ky

2√1 + γ

=sin kx

2√1 + γ

= (cg)x

and

(29) |cg| =√2

2√1 + γ

sinπ

1 + |tanα|

(α = arctan

kykx

), β = ±π

4.

In Fig. 18, the resonant contours for three different values of γ frequen-cies, γ = 0.44, 1 and 2.0625, are shown by solid lines (corresponding resonantfrequencies are ω = 2.4,

√8 and 3.5).

The main difference between the resonant LPW in RTL and RCL (γ = 0)is that in the former case LPWs exist only along the diagonals with additionalbonds (γ = 0). Two symmetric straight sides ky = ±π + kx of the squarecontour inherent to the MSL are as if decomposed in the RTL case intoparts, whose forms essentially depend on γ. For example, in the case ofrelatively small γ = 0.44, these parts are represented by sloping arcs inthe second and forth quadrants of kx, ky-plane. The contour correspondingto ω = 3.5 (γ = 2.0625) consists of the two above-mentioned straight linesky = ±π − kx and two sloping arcs located within the same, first and third,quadrants.

An intriguing pattern has been detected in the case of ω =√8 (γ =

1): the mentioned additional part of the contour consists of straight lineskx = ±π, ky = ±π. Thus, in this case the resonant equifrequency contourdetermines six preferable directions of wave propagation, two of them alongdiagonals with additional bonds and four along the coordinate axes. Follow-ing (29), we obtain maximal velocities of energy flux along the six mentioneddirections:

(30) |cg|max =1

2

(β = ±π

4

), |cg|max =

1√8

(β = ±0 andβ = ±π

2

).

To establish the average orientation of the group velocities in the consid-ered cases ω = 2.4 and ω = 3.5, we approximate the above-mentioned arcsby secants (see dotted straight lines). In the mentioned cases, the energy flux


orientations are determined by normals to these secants. Here, as in case ofγ = 1, six preferable orientations of energy flux shown by bold arrows areobserved.

0.44, 2.4γ ω= = 1, 8γ ω= = 2.0625, 3.5γ ω= =

yk

π− πxk

π−

π

0

0

yk

π− πxk

π−

π

0

0

yk

π− πxk

0

0π−

π

Fig. 18. Resonant contours. Arrows show energy flux orientations. Dotted straight linesare secants approximating curvilinear parts of contours.

5. Conclusions. Studies of mathematical models of wave propaga-tion in 2D periodic infinite rectangular, triangular and hexagonal-cell latticeshave been conducted. Eigenfrequencies of steady-state problems have beendetermined, and dispersion properties of free waves are described. We haveshown that the frequency spectra of the models possess several resonantpoints located both at the boundary of pass/stop bands, and in the interiorof a pass band. Special attention was given to the LPWs which exist atthe resonance frequencies in the interior of pass-bands. As shown above, theexistence of the LPW is a consequence of a certain symmetry of the latticestructure. It could be concluded that the existence of LPWs can be expectedin more complicated 2D/3D lattices.

The time-dependent initial-boundary value problems with a local mono-chromatic source have been explored, and resonant patterns were revealed.The most important results include the analysis of particularities of transientwaves in the above-mentioned resonant cases, and the correspondence of theobtained wave beaming phenomena to the LPW analysis.

Acknowledgement. This work was supported by The Israel ScienceFoundation, Grant 504/08


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[11] G. Osharovich, M. Ayzenberg-Stepanenko and O. Tsareva, Wave propagation inelastic lattices subjected to a local harmonic loading. I. A quasi-one-dimensionalproblem, Journal of Continuum Mechanics and Thermodynamics, 22 (2010),581–597.

[12] G. Osharovich, M. Ayzenberg-Stepanenko and O. Tsareva, Wave propagation in elas-tic lattices subjected to a local harmonic loading. II. Two-dimensional problems,Journal of Continuum Mechanics and Thermodynamics, 22 (2010), 599–616.

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[16] S. Seal. Functional nanostructures. Processing, Characterization, and Applications.Nanostructure Science and Technology, Springer, 2008.


VOLUME 19

2012, NO 1–2

PP. 195–211

OSCILLATION RESULTS FOR FOURTH ORDERNON-LINEAR MIXED NEUTRAL DIFFERENTIAL

EQUATIONS WITH QUASI-DERIVATIVES ∗

S. PANIGRAHI † AND R. BASU ‡

Abstract. Oscillatory and asymptotic behaviour of a class of nonlinear fourth orderneutral equation with quasi-derivatives of the form

L4(y(t) + p(t)y(t− τ)) + q(t)G(y(t− α))− h(t)H(y(t− β)) = 0,

and

L4(y(t) + p(t)y(t− τ)) + q(t)G(y(t− α))− h(t)H(y(t− β)) = f(t)

have been studied under the assumption∫ ∞

0

t

rn(t)dt = ∞, n = 1, 2, 3.

for various ranges of p(t), where Lnu(t) = rnddtLn−1u(t), n = 0, 1, 2, 3.

Key Words. Functional differential equations, neutral, nonlinear, oscillation, positiveand negative coefficients.

AMS(MOS) subject classification. 34 C 10, 34 C 15

∗ First author supported by Department of Science and Technology SERC Division,New Delhi, India, through the letter No: SR/S4/MS: 541/08, dated 30th September 2008.Second author supported by CSIR-New Delhi, through the Letter no 09/414 (0876)/2009-EMR-I dated October 20, 2009, India.

† Department of Mathematics and Statistics, University of Hyderabad, Hyderabad - 500046, India. Email: [email protected]

‡ Department of Mathematics and Statistics, University of Hyderabad, Hyderabad - 500046, India. Email: [email protected]

195

196 S. PANIGRAHI AND R. BASU

1. Introduction. In [3, 4] Kusano and Naito have studied oscillatorybehaviour of solutions of fourth order non-linear differential equation of theform

(r(t)y′′(t))′′ + y(t)F (y2(t), t) = 0,

where r and F are continuous and positive on [0,∞) and (0,∞) × [0,∞)respectively, under the assumption

(H)

∫ ∞

0

t

r(t)dt = ∞.

Moreover, in [5] Parhi and Tripathy have studied the oscillatory andasymptotic behaviour of solutions of a class of non-linear fourth order neutraldelay differential equations of the form

(1) (r(t)(y(t) + p(t)y(t− τ))′′)′′ + q(t)G(y(t− α)) = 0,

and

(2) (r(t)(y(t) + p(t)y(t− τ))′′)′′ + q(t)G(y(t− α)) = f(t)

under the assumption (H), where r ∈ C([0,∞), [0,∞)), p ∈ C([0,∞),R),q ∈ C([0,∞), [0,∞)), f ∈ C([0,∞),R), G ∈ C(R,R) is non-decreasing anduG(u) > 0 for u = 0, τ > 0 and α > 0.

If q(t) changes sign, that is, q(t) = q+(t)−q−(t), where q+(t) = max0, q(t)and q−(t) = max0,−q(t), then (1) and (2) can be viewed as

(3) (r(t)(y(t) + p(t)y(t− τ))′′)′′ + q+(t)G(y(t−α))− q−(t)G(y(t−α)) = 0,

and

(4) (r(t)(y(t)+p(t)y(t−τ))′′)′′+q+(t)G(y(t−α))−q−(t)G(y(t−α)) = f(t).

In a recent paper [6], we have studied the fourth order nonlinear neutraldelay differential equations of the form

(5) (r(t)(y(t) + p(t)y(t− τ))′′)′′ + q(t)G(y(t− α))− h(t)H(y(t− β)) = 0,

and its associated forced equations

(6) (r(t)(y(t)+ p(t)y(t− τ))′′)′′ + q(t)G(y(t−α))−h(t)H(y(t− β)) = f(t),

NEUTRAL DIFFERENTIAL EQUATIONS WITH QUASI-DERIVATIVES 197

where r ∈ C([0,∞), [0,∞)), p ∈ C([0,∞),R), q and h ∈ C([0,∞), [0,∞)),f ∈ C([0,∞),R), G and H ∈ C(R,R), G is nondecreasing and H is boundedand monotonic with uG(u) > 0, vH(v) > 0, for u, v = 0, τ > 0, α > 0 andβ > 0. Clearly, (3)/(4) is a particular case of (5)/(6).

Consider the nonlinear neutral equations with quasi-derivatives of theform

(7) L4(y(t) + p(t)y(t− τ)) + q(t)G(y(t− α))− h(t)H(y(t− β)) = 0,

and

(8) L4(y(t) + p(t)y(t− τ)) + q(t)G(y(t− α))− h(t)H(y(t− β)) = f(t)

for various ranges of p(t), where p ∈ C([0,∞),R), q ∈ C([0,∞), [0,∞)),h ∈ C([0,∞), [0,∞)), f ∈ C([0,∞),R), G ∈ C(R,R) and H ∈ C(R,R), G isnondecreasing and H is bounded and monotonic with uG(u) > 0, vH(v) > 0for u, v = 0, τ > 0 and α, β > 0 . In this work, an attempt is made to study(7) and (8) under the assumption

(H0)

∫ ∞

0

1

rn(t)dt = ∞, n = 1, 2, 3,

where rn ∈ C([0,∞), [0,∞)) for various ranges of p(t). For equations (7) and(8) we define quasi-derivative as follows:Let z(t) = y(t)+ p(t)y(t− τ), L0z(t) = z(t), L1z(t) = r1(t)

ddtL0z(t), L2z(t) =

r2(t)ddtL1z(t), L3z(t) = r3(t)

ddtL2z(t), L4z(t) =

ddtL3z(t).

Indeed equations (5) and (6) are particular case of (7) and (8).

The motivation of the paper is due to the work of [5]. It is interestingto observe that the nature of the functions rn(t), n = 1, 2, 3 influences thebehaviour of solutions of (7) and (8). The present work generalizes some ofthe existing results in the literature.

By a solution of (7)/(8) we understand a function y ∈ C([−ρ,∞),R)such that (y(t)+p(t)y(t−τ)) is continuously differentiable, L1, L2, L3 are dif-ferentiable operator and (7)/(8) is satisfied for t ≥ 0, where ρ = maxτ, α, β,and sup|y(t)| : t ≥ t0 > 0 for every t ≥ t0. A solution of (7)/(8) is said tobe oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscil-latory.

2. Oscillation Properties of Homogeneous Equations. Inthis section, sufficient conditions are obtained for oscillatory and asymptoticbehaviour of all solutions or bounded solutions of (7) under the assumption(H0). We need the following lemmas for our use in the sequel.

Lemma 1. [7] Let (H0) hold. Let u be a continuously differentiable functionon [0,∞) such that L1u, L2u, L3u are continuously differentiable functionsand L4u ≤ 0, ≡ 0 for large t. If u(t) > 0 ultimately, then one of the cases(a) and (b) holds for large t, and if u(t) < 0 ultimately, then one of the cases(b), (c), (d) and (e) holds for large t, where(a) L1u(t) > 0, L2u(t) > 0 and L3u(t) > 0(b) L1u(t) > 0, L2u(t) < 0 and L3u(t) > 0(c) L1u(t) < 0, L2u(t) < 0 and L3u(t) > 0(d) L1u(t) < 0, L2u(t) < 0 and L3u(t) < 0(e) L1u(t) < 0, L2u(t) > 0 and L3u(t) > 0.

Proof. The proof of lemma follows from [2] for n = 4. Hence details areomitted.

Lemma 2. [7] Let the conditions of Lemma 1 hold. Assume that r′1(t) ≥ 0 andr′3(t) ≥ 0. If u(t) > 0 ultimately, then for t ≥ T ≥ 0, u(t) ≥ R(t, T )L3u(t),where,

R(t, T ) =

∫ t

T

1

(r1(θ)r3(θ))(

∫ θ

T

(s− T )

r2(s)ds)dθ.

Lemma 3. [1] If q ∈ C([0,∞), [0,∞)) and

lim inft→∞

∫ t

t−τ

q(s)ds >1

e,

then x′(t)+q(t)x(t−τ) ≤ 0, t ≥ 0 cannot have an eventually positive solution.Lemma 4. [1] Let F,G, P : [t0,∞) → R and c ∈ R be such that F (t) =G(t)+P (t)G(t−c), for t ≥ t0+max0, c. Assume that there exists numbersP1, P2, P3, P4 ∈ R such that P (t) is one of the following ranges:

(1) P1 ≤ P (t) ≤ 0, (2) 0 ≤ P (t) ≤ P2 < 1, (3) 1 < P3 ≤ P (t) ≤ P4.

Suppose that G(t) > 0 for t ≥ t0, lim inft→∞G(t) = 0 and that limt→∞ F (t) =L ∈ R exists. Then L = 0.Theorem 1. Let 0 ≤ p(t) ≤ a < 1 or 1 < p(t) ≤ a < ∞, τ ≤ α,r′1(t), r

′3(t) > 0 and (H0) hold. If

(H1)∫∞0

1r1(s2)

∫∞s2

1r2(s1)

∫∞s1

1r3(s)

∫∞sh(θ)dθdsds1ds2 <∞,

(H2) there exists λ > 0 such that G(u) +G(v) ≥ λG(u+ v), u > 0, v > 0,(H3) G(u)G(v) = G(uv) and H(u)H(v) = H(uv) for u, v ∈ R,(H4) G is sublinear and

∫ c

0du

G(u)<∞ for all c > 0,

(H5)∫∞τQ(t)dt = ∞, Q(t) = minq(t), q(t− τ) for t ≥ τ

hold, then every solution of (7) either oscillates or converges to zero ast→ ∞.

Proof. Assume that (7) has a non-oscillatory solution on [t0,∞), t0 ≥ 0 andlet it be y(t). Hence y(t) > 0 or < 0 for t ≥ t0. Suppose that y(t) > 0 fort ≥ t0. Setting

(9) z(t) = y(t) + p(t)y(t− τ),

(10)

K(t) =

∫ ∞

t

1

r1(s2)

∫ ∞

s2

1

r2(s1)

∫ ∞

s1

1

r3(s)

∫ ∞

s

h(θ)H(y(θ − β))dθdsds1ds2

and

(11) w(t) = z(t)−K(t) = y(t) + p(t)y(t− τ)−K(t)

we obtain

(12) L4w(t) = −q(t)G(y(t− α)) ≤ 0, ≡ 0

for t ≥ t0 + ρ. Consequently, w(t), L1w(t), L2w(t), L3w(t) are monotonicfunctions on [t1,∞), t1 ≥ t0 + ρ. In what follows, we have two cases, viz.w(t) > 0 or w(t) < 0 for t ≥ t1. Suppose the former holds. By the Lemma1, any one of the cases (a) or (b) holds. Upon using (H2) and (H3), Eq. (7)can viewed as

0 = L4w(t) + q(t)G(y(t− α)) +G(a)L4w(t− τ) +G(a)q(t− τ)G(y(t− τ − α))

≥ L4w(t) +G(a)L4w(t− τ) + λQ(t)G(y(t− α) + ay(t− α− τ))

≥ L4w(t) +G(a)L4w(t− τ) + λQ(t)G(z(t− α))

for t ≥ t2 > t1. From (10), it follows that K(t) > 0 and K ′(t) < 0 and hencew(t) > 0 for t ≥ t1 implies that w(t) < z(t) for t ≥ t2. Therefore, the lastinequality yields that

L4w(t) +G(a)L4w(t− τ) + λQ(t)G(w(t− α)) ≤ 0,

for t ≥ t2, that is,

0 ≥ L4w(t) +G(a)L4w(t− τ) + λQ(t)G(R(t− α, T )L3w(t− α))

due to Lemma 2, for t ≥ T + ρ > t2. Hence

0 ≥ L4w(t) +G(a)L4w(t− τ) + λQ(t)G(R(t− α, T ))G(L3w(t− α))

that is,

λQ(t)G(R(t− α, T )) ≤ −[G(L3w(t− α))]−1L4w(t)

− G(a)[G(L3w(t− α))]−1L4w(t− τ)

≤ −[G(L3w(t))]−1L4w(t)

− G(a)[G(L3w(t− τ))]−1L4w(t− τ).

Because limt→∞ L3w(t) <∞, then using (H4), the above inequality becomes∫ ∞

T+ρ

Q(t)G(R(t− α, T ))dt <∞,

which contradicts (H5), where we have used the fact that R(t, T ) is monotonicincreasing function.

Next, we suppose that the later holds. Then

y(t) ≤ z(t) = y(t) + p(t)y(t− τ) < K(t),

that is, y(t) is bounded (because K(t) is bounded and monotonic). By theLemma 1, any one of the cases (b), (c), (d) or (e) hold.

Consider the case (b).Since limt→∞K(t) exists and limt→∞w(t) exists, then limt→∞ z(t) exists.

Further, limt→∞ L3w(t) exists implies that

∫ ∞

t1

Q(t)G(y(t− α))dt <∞

and hence it is easy to verify that lim inft→∞ y(t) = 0 due to (H5). Conse-quently, it follows from Lemma 4 that limt→∞ z(t) = 0. Since z(t) ≥ y(t),then limt→∞ y(t) = 0.

Consider the case (c) and (d).These two cases are not possible due to the fact that w(t) < 0, y(t) is

bounded, limt→∞K(t) exists and hence limt→∞w(t) exists. On the other-hand, integrating successively w′′(t) < 0 for t ≥ t1, we get limt→∞w(t) =−∞, a contradiction.

Consider the case (e).L2w(t) is nondecreasing on [t1,∞). Hence for t ≥ t1, L2w(t) ≥ L2w(t1),

that is,

r2(t)d

dtL1w(t) ≥ L2w(t1).(13)

Integrating (13) from t1 to t, we obtain

L1w(t) ≥ L1w(t1) + L2w(t1)

∫ t

t1

1

r2(s)ds

that is, L1w(t) > 0 for large t due to (H0), a contradiction.Finally, we suppose that y(t) < 0 for t ≥ t0. From (H3), we note that

G(−u) = −G(u) and H(−u) = −H(u), u ∈ R. Hence putting, x(t) = −y(t)for t ≥ t0, we obtain x(t) > 0 and

L4(x(t) + p(t)x(t− τ)) + q(t)G(x(t− α))− h(t)H(x(t− β)) = 0.

Proceeding as above, we can show that every solution of (7) oscillates orconverges to zero as t→ ∞. This completes the proof of the theorem.

Theorem 2. If all the conditions of Theorem 1 are satisfied, then everyunbounded solution of (7) oscillates.

The proof of the theorem follows from the proof of the Theorem 1 andhence the details are omitted.Theorem 3. Let 0 ≤ p(t) ≤ a < 1. Suppose that (H0), (H1) and (H3) hold.If(H6) lim inf |x|→0

G(x)x

≥ γ > 0,

(H7) lim inft→∞∫ t

t−αG(R(s− α, T ))q(s)ds > (eγG(1− a))−1,

and(H8)

∫∞0q(t)dt = ∞

hold, then every solution of (7) either oscillates or converges to zero ast→ ∞.

Remark 1. (H7) implies that(H9)

∫∞T+α

G(R(s− α, T ))q(s)ds = ∞.Indeed, if ∫ ∞

T+α

G(R(s− α, T ))q(s)ds = b <∞,

then for t > T + 2α,∫ t

t−α

G(R(s− α, T ))q(s)ds = (

∫ t

T+α

−∫ t−α

T+α

)G(R(s− α, T ))q(s)ds,

implies that

lim inft→∞

∫ t

t−α

G(R(s− α, T ))q(s)ds ≤ b− b = 0,

which contradicts (H7).

Proof. Suppose on the contrary that y(t) is a non-oscillatory solution of (7)such that y(t) > 0 for t ≥ t0. The case y(t) < 0 for t ≥ t0 is similar. Using(9), (10) and (11) we obtain (12). In what follows we suppose to considertwo cases viz. w(t) > 0 and w(t) < 0 for for t ≥ t1 > t0 + ρ. Let w(t) > 0on [t1,∞). Then any one of the cases (a) or (b) of Lemma 1 holds. In eachcase, w(t) is nondecreasing. We may note that K(t) > 0 and L1K(t) < 0.Hence

0 < L1w(t) = L1z(t)− L1K(t)

implies that L1z(t) > 0 or L1z(t) < 0 for t ≥ t2 > t1. If L1z(t) > 0, thenz(t) is nondecreasing as r1(t) > 0 and

(1− p(t))z(t) < z(t)− p(t)z(t− τ)

= y(t)− p(t)p(t− τ)y(t− 2τ) < y(t)

for t ≥ t2, that is,

y(t) > (1− a)z(t) > (1− a)w(t).

Thus (12) yields that

G((1− a)w(t− α))q(t) ≤ −L4w(t).

Using Lemma 2 and (H3), the above inequality becomes

G(1− a)q(t)G(R(t− α, T ))G(L3w(t− α)) ≤ −L4w(t)(14)

for t ≥ T + α > t2. Let limt→∞ L3w(t) = c, c ∈ [0,∞). If 0 < c < ∞, thenthere exists c1 > 0 such that L3w(t) > c1 for t ≥ t3 > T + α. From (14) weobtain for t ≥ t4 > t3 + α,

G(1− a)q(t)G(R(t− α, T ))G(c1) ≤ −L4w(t).

Integrating the above inequality from t4 to ∞, we get∫ ∞

t4

q(t)G(R(t− α), T )dt <∞,

a contradiction to (H9). Hence c = 0. Consequently, (H6) implies thatG(L3w(t)) ≥ γL3w(t) for t ≥ t3. Therefore, (14) yields

L4w(t) + γG(1− a)q(t)G(R(t− α, T ))L3w(t− α) ≤ 0,

for t ≥ t3 + α. From Lemma 4, it follows that

u′(t) + γG(1− a)q(t)G(R(t− α, T ))u(t− α) ≤ 0

admits a positive solution L3w(t), which is a contradiction due to (H7).If L1z(t) < 0, then limt→∞ z(t) exists. Using the same type of rea-

soning as in Theorem 1, it is easy to verify that lim inft→∞ y(t) = 0 andlimt→∞ z(t) = 0. Consequently, limt→∞ y(t) = 0.

The rest of the proof follows from the proof of the Theorem 1. Hencethe theorem is proved.

Theorem 4. Let 0 ≤ p(t) ≤ a < 1. Assume that (H0), (H1), (H3), (H6) −(H8) hold. Then every unbounded solution of (7) oscillates.

The proof of the theorem follows from the Theorem 3 and hence thedetails are omitted.Theorem 5. Let 0 ≤ p(t) ≤ a < 1 or 1 < p(t) ≤ a < ∞, τ ≤ α and(H0)− (H3) hold. Assume that

(H10)G(x1)xσ1

≥ G(x2)xσ2

for x1 ≥ x2 > 0 and σ ≥ 1

and (H5) hold. Then every solution of (7) either oscillates or tends to zeroas t→ ∞.

Proof. Proceeding as in the proof of Theorem 1, we obtain

L4w(t) +G(a)L4w(t− τ) + λQ(t)G(z(t− α)) ≤ 0,(15)

for t ≥ t2. Using the fact that w(t) is nondecreasing, there exists k > 0 andt3 > 0 such that w(t) > k for t ≥ t3. Hence use of (H10) along with Lemma

2 we obtain for t ≥ T + α > t3 + α,

G(w(t− α)) = (G(w(t− α))/wσ(t− α))wσ(t− α)

≥ (G(k)/kσ)wσ(t− α)

> (G(k)/kσ)Rσ(t− α, T )(L3w(t− α))σ.

Thus (15) yields

λ(G(k)/kσ)Rσ(t− α, T )Q(t)(L3w(t− α))σ < λQ(t)G(w(t− α))

≤ λQ(t)G(z(t− α))

≤ −L4w(t)−G(a)L4w(t− τ),

that is,

λ(G(k)/kσ)Rσ(t− α, T )Q(t) < −(L3w(t− α))−σ[L4w(t)

+ G(a)L4w(t− τ)]

< −(L3w(t))−σL4w(t)

− G(a)(L3w(t− τ))−σL4w(t− τ).

Since limt→∞(L3w(t) exists and R(t, T ) is nondecreasing, then proceeding asin the proof of Theorem 1, we obtain∫ ∞

T+σ

Rσ(t− α, T )Q(t)dt <∞,

a contradiction due to (H5). The rest of the proof follows from Theorem 1.Thus the proof theorem is complete.

Theorem 6. Let 0 ≤ p(t) ≤ a < ∞ and τ ≤ α hold. If (H0) - (H3), (H5)and (H10) hold, then every unbounded solution of (7) oscillates.

The proof of the theorem follows from the Theorem 5. Hence thedetails are omitted.

Theorem 7. Let 0 ≤ p(t) ≤ a < 1 or 1 < p(t) ≤ a < ∞,τ ≤ α. Supposethat (H0), (H1), (H2), (H5) and(H11) G(u)G(v) ≥ G(uv) for u > 0, v > 0,(H12) G(−u) = −G(u), u ∈ Rhold. Then every solution of (7) either oscillates or converges to zero ast→ ∞.

Proof. Proceeding as in the proof of the Theorem 5 we have (15) for t ≥ t2.Since w(t) is nondecreasing, then there exist k > 0 and t3 > 0 such thatw(t) > k for t ≥ t3, that is, z(t) ≥ w(t) > k for t ≥ t3. Consequently, (15)yields

λG(k)

∫ ∞

t3

Q(t)dt <∞,

a contradiction to (H5). The rest of the proof follows from Theorem 1. Thiscompletes the proof of the theorem.

Remark 2. In Theorems 1 - 4, G is sublinear only, whereas in Theorems 5and 6, G is superlinear. But in Theorem 7, G could be linear, sublinear orsuperlinear.Theorem 8. Let 0 ≤ p(t) ≤ a < ∞. If (H0), (H1), (H2), (H5), (H11) and(H12) hold, then every unbounded solution of (7) is oscillatory.Theorem 9. Let −1 < b ≤ p(t) ≤ 0. If (H0), (H1), (H3), (H4) and (H8)hold, then every solution of (7) either oscillates or tends to zero as t→ ∞.

Proof. Let y(t) be a nonoscillatory solution of (7). Because of (H3), withoutloss of generality we may suppose that y(t) > 0 for t ≥ t0 > 0. Setting as in(9), (10) and (11) we obtain (12) for t ≥ t0 + ρ. Hence w(t) is monotonic on[t1,∞], t1 ≥ t0 + ρ. If w(t) > 0 for t ≥ t1, then any one of the cases (a) or(b) of Lemma 1 holds. Consequently, w(t) ≥ R(t, T )L3w(t) for t ≥ t2 > t1due to Lemma 2. Moreover, w(t) ≤ y(t) implies that y(t) ≥ R(t, T )L3w(t)for t ≥ t2 and hence (12) becomes∫ ∞

t2+α

q(t)G(R(t− α, T ))dt <∞,

a contradiction to (H8). Hence w(t) < 0 for t ≥ t1. Then any one of thecases (b), (c), (d) or (e) of Lemma 1 holds. We claim that y(t) is bounded.If not, let there be an increasing sequence ηn∞n=1 such that ηn → ∞ andy(ηn) → ∞ as n→ ∞ and y(ηn) = maxy(t) : t1 ≤ t ≤ ηn. We may choosen large enough such that ηn − τ > t1. Hence

w(ηn) ≥ y(ηn) + by(ηn − τ)−K(ηn)

≥ (1 + b)y(ηn)−K(ηn).

Since K(ηn) is bounded and (1 + b) > 0, then w(ηn) > 0 for large n which isa contradiction. Thus our claim holds and the cases (c), (d) and (e) are easyto verity following to Theorem 1. Using same type of reasoning as in the case

(b) of Theorem 1, we obtain lim inft→∞ y(t) = 0 and hence limt→∞ z(t) = 0due to Lemma 4. Consequently,

0 = lim supt→∞

z(t) ≥ lim supt→∞

(y(t) + by(t− τ))

≥ lim supt→∞

y(t) + lim inft→∞

(b(y(t− τ))

= lim supt→∞

y(t) + b lim supt→∞

y(t− τ)

= (1 + b) lim supt→∞

y(t)

implies that lim supt→∞ y(t) = 0. Hence limt→∞ y(t) = 0. This completesthe proof of the theorem.

Theorem 10. Let −1 < b ≤ p(t) ≤ 0. If (H0), (H1), (H3), (H4) and (H8)hold, then every unbounded solution of (7) is oscillatory.Theorem 11. Let −∞ < p(t) < −1. Assume that (H0), (H1), (H3), (H4)and (H8) hold. Then the following statements are hold:(i) if −∞ < b ≤ p(t) ≤ b1 < −1, then every bounded solution of (7) eitheroscillates or tends to zero as t→ ∞.(ii) if −∞ < b ≤ p(t) ≤ −1, then unbounded solutions of (7) oscillate.

The proof of the theorem follows from the proofs of the Theorems 9 and10. Hence the details are omitted.

3. Oscillation Properties of Non-homogeneous Equations.This section is devoted to study the oscillatory and asymptotic behavior ofsolutions of forced equations (8) with suitable forcing function. Our attentionis restricted to the forcing functions which are changing sign eventually. Wehave the following hypotheses regarding f(t):(H13) There exists F , a real-valued continuously differentiable function on[0,∞) such that F (t) changes sign, r1F

′, r2(r1F′), r3(r2(r1F

′)′) are all real-valued continuously differentiable function on [0,∞) consecutively and(r3(r2(r1F

′)′)′)′ = f and−∞ < lim inft→∞ F (t) < 0 < lim supt→∞ F (t) <∞.(H14) All conditions are same as (H13), only difference is lim inft→∞ F (t) =−∞, lim supt→∞ F (t) = ∞.Theorem 12. Let 0 ≤ p(t) ≤ a <∞. Assume that (H0), (H1), (H2), (H11),(H12) and (H14) hold. If

(H15)

∫ ∞

α

Q(t)G(F+(t− α))dt = ∞ =

∫ ∞

α

Q(t)G(F−(t− α))dt,

where F+(t) = max0, F (t) and F−(t) = max−F (t), 0, then (8) is oscil-latory.

Proof. Let y(t) be a non oscillatory solution of (8) such that y(t) > 0 fort ≥ t0 > 0. Setting as in (9), (10) and (11), let

V (t) = w(t)− F (t) = z(t)−K(t)− F (t).(16)

Hence for t ≥ t0 + ρ, (2) becomes

L4V (t) = −q(t)G(y(t− α)) ≤ 0, ≡ 0.(17)

Thus V (t) is monotonic on [t1,∞], t1 > t0 + ρ. Since F (t) changes sign,then V (t) > 0 implies that z(t) −K(t) > F (t) in which z(t) −K(t) < 0 isnot possible due to (H14). Hence z(t)−K(t) > 0 and z(t)−K(t) > F+(t).Ultimately, that is,

z(t) > K(t) + F+(t) > F+(t),

for t ≥ t1. For t ≥ t2 > t1, we have

0 = L4V (t) + q(t)G(y(t− α)) +G(a)L4V (t− τ)

+ G(a)q(t− τ)G(y(t− α− τ))

≥ L4V (t) +G(a)L4V (t− τ) + λQ(t)G(y(t− α) + ay(t− α− τ))

≥ L4V (t) +G(a)L4V (t− τ) + λQ(t)G(z(t− α))

≥ L4V (t) +G(a)L4V (t− τ) + λQ(t)G(F+(t− α)).(18)

Integrating the inequality (18) from t2 + α to ∞, we get∫ ∞

t2+α

Q(t)G(F+(t− α))dt <∞,

which is a contradiction to (H15). Consequently, V (t) < 0 for t ≥ t1. Thusany one of the cases (b), (c), (d) or (e) of Lemma 1 holds. Since limt→∞ V (t)exists, then for each case z(t) = V (t) +K(t) + F (t)implies that

lim inft→∞

z(t) = lim inft→∞

[K(t) + V (t) + F (t)]

≤ lim supt→∞

K(t) + lim inft→∞

[V (t) + F (t)]

≤ lim supt→∞

K(t) + lim supt→∞

V (t) + lim inft→∞

F (t)

= limt→∞



F (t) → −∞

that is, z(t) < 0 for large t, a contradiction.If y(t) < 0 for t ≥ t0, we set x(t) = −y(t) to obtain x(t) > 0 for t ≥ t0

and

L4(x(t) + p(t)x(t− τ)) + q(t)G(x(t− α))− h(t)H(x(t− β)) = f(t),

where f(t) = −f(t). If F (t) = −F (t), then F (t) changes sign, F+(t) = F−(t)and (r3(r2(r1F

′)′)′)′ = f(t). Proceeding as above we obtain a contradiction.This completes the proof of the theorem.

Theorem 13. Let −1 < p(t) ≤ 0. Suppose that (H0), (H1), (H12) and (H14)hold. If

(H16)

∫ ∞

α

q(t)G(F+(t− α))dt = ∞ =

∫ ∞

α

q(t)G(F−(t− α))dt,

then every solution of (8) oscillates.

Proof. Proceeding as in the proof of the Theorem 12, we obtain V (t) > 0 or< 0 when y(t) > 0 for t ≥ t1 > t0 + ρ. When V (t) > 0, then any one of thecases (a) and (b) of Lemma 1 holds for t ≥ t1. Since V (t) is monotonic, thenlimt→∞ V (t) exists and hence z(t)−K(t) > F (t) implies that z(t)−K(t) > 0due to (H14). Further, K(t) > 0 yields that z(t) > 0 for t ≥ t1. Hence fort2 > t1,

y(t) > z(t) > z(t)−K(t) > F+(t), t ≤ t2.

Consequently, (17) becomes

q(t)G(F+(t− α)) ≤ −L4V (t), t ≥ t2 + α,

that is, ∫ ∞

t2+α

q(t)G(F+(t− α)) <∞,

a contradiction to (H16). Ultimately, V (t) < 0 for t ≥ t1. As a result,z(t)−K(t) < F (t) yields that z(t)−K(t) < 0 due to (H14). Thus we havetwo cases z(t) > 0 or < 0 for t > t1. Assume that z(t) > 0 and because V (t)is a monotonic function, then limt→∞ V (t) exists and

lim inft→∞

z(t) ≤ lim inft→∞

[K(t) + F (t)]

≤ lim supt→∞

K(t) + lim inft→∞

F (t)

implies that lim inft→∞ z(t) < 0, a contradiction. Hence, z(t) < 0 for t ≥ t1,that is, y(t) < y(t − τ) for t ≥ t1 + ρ. Consequently, y(t) is bounded fort ≥ t2 > t1+ρ. On the other hand, lim inft→∞ z(t) = −∞ (as above) impliesthat z(t) is unbounded, a contradiction.

Using the same type of reasoning as in Theorem 12, for the case y(t) <0 for t ≥ t0, we obtain the desired contradiction. Hence the theorem isproved.

Theorem 14. Let −∞ < p(t) ≤ −1. If (H0), (H1), (H12), (H14) and (H16)hold, then every bounded solution of (8) is oscillatory.

The proof of the theorem follows from the proof of the Theorem 13.Hence, the details are omitted.Theorem 15. Let 0 ≤ p(t) ≤ a <∞. Assume that (H0), (H1), (H2), (H11)- (H13) and (H15) hold. Then every unbounded solution of (8) oscillates.

Proof. Suppose on the contrary that y(t) is an unbounded nonoscillatorysolution of (8) such that y(t) > 0 for t ≥ t0. Setting as in (9)-(11) and (16)we obtain (17) for t ≥ t0+ρ. Hence V (t) is monotonic on [t1,∞), t1 > t0+ρ.Proceeding as in the proof of the Theorem 12, we obtain contradiction whenV (t) > 0 for t ≥ t1. Hence V (t) < 0 for t ≥ t1. From Lemma 1, it followsthat any one of the cases (b), (c), (d) or (e) holds. In case (b), limt→∞ V (t)exists and hence

z(t) = V (t) +K(t) + F (t),

implies that

y(t) ≤ V (t) +K(t) + F (t),(19)

that is, y(t) is bounded, which is absurd. For each of the cases (c), (d) and(e), V (t) is a nonincreasing on [t1,∞). Let limt→∞ V (t) = C, C ∈ [−∞, 0).If C = −∞, then (19) yields

lim inft→∞

y(t) ≤ lim supt→∞


(K(t) + F (t))

≤ lim supt→∞



F (t)

= limt→∞

V (t) + limt→∞


F (t),

that is, lim inft→∞ y(t) = −∞, which is absurd. The contradiction is obviouswhen −∞ < C < 0 as 0 < y(t) < z(t) = V (t) + K(t) + F (t) implies thaty(t) is bounded (because V (t), K(t), F (t) all are bounded) which is absurd.

The case y(t) < 0 for t ≥ t0 is similar. This completes the proof of thetheorem.

Theorem 16. Let −1 0 on [t0,∞). Then proceeding as in the proof of the Theorem 13, wehave contradiction when V (t) > 0 for t ≥ t1.

Next, we suppose that V (t) < 0 for t ≥ t1. In what follows, z(t)−K(t) <

0 due to (H13). So, we have two cases viz. z(t) < 0 or z(t) > 0 for t ≥ t1. Ifz(t) < 0, that is, y(t) < y(t − τ) for t ≥ t1, then y(t) is bounded and henceit contradicts to the fact that y(t) is unbounded. Ultimately, z(t) > 0 fort ≥ t1. Since y(t) is unbounded, then there exists ηn∞n=1 such that ηn → ∞,y(ηn) → ∞ as n→ ∞ and

y(ηn) = maxy(t) : t1 ≤ t ≤ ηn.

We may choose n large enough such that ηn − τ > t1. Hence

z(ηn) ≥ (1 + b)y(ηn).

By the Lemma 1, any one of the cases (b), (c), (d), or (e) holds. In case (b),limt→∞ |V (t)| <∞ and z(t) = V (t) +K(t) + F (t) implies that

∞ = (1 + b) y(ηn) ≤ V (ηn) +K(ηn) + F (ηn)(20)

≤ |V (ηn)|+K(ηn) + |F (ηn)| <∞,

which is absurd. In each of the cases (c), (d) and (e), V (t) is nonincreasing.Let limt→∞ V (t) = c, c ∈ [−∞, 0). If −∞ < c < 0, then (20) gives the desiredcondition. Again for c = −∞, it follows from (20) that ∞ ≤ −∞, which isabsurd. Hence the Theorem is proved.

Example 1. Consider

L4(y(t) + e−5ty(t− π)) + 4eπy(t− π)− reθ−πe−5t(e2(t−θ) sin2(t− θ)(21)

+e4(t−θ) sin4(t− θ))y(t− θ)

y2(t− θ) + y4(t− θ)= −8et sin t,

where, r1(t) = r2(t) = r3(t) = 1, p(t) = e−5t, q(t) = 4eπ, h(t) = reθ−πe−5t(e2(t−θ)

sin2(t−θ)+e4(t−θ) sin4(t−θ)), G(u) = u,H(u) = uu2+u4 , r

2 = (161)2+(240)2,

tan θ = 240161

and f(t) = −8et sin t. Indeed, if we choose F (t) = 2et sin t, then(r3(r2(r1F

′)′)′)′ = f .Since

F+(t− π) =

0, t ∈ [2nπ, (2n+ 1)π]−2et−π sin t, t ∈ [(2n+ 1)π, (2n+ 2)π],

and

F−(t− π) =

2et−π sin t, t ∈ [2nπ, (2n+ 1)π]0, t ∈ [(2n+ 1)π, (2n+ 2)π]


for n = 0, 1, 2 . . . .Then∫ ∞

π

Q(t)F+(t− π)dt = −8∞∑n=0

∫ (2n+2)π

(2n+1)π

et sin tdt

> 4∞∑n=0

e(2n+2)π = ∞,

and ∫ ∞

π

Q(t)F−(t− π)dt > 4∞∑n=1

e(2n+1)π2 = ∞.

Clearly, (H0), (H1), (H2), (H11), (H12) and (H14) are satisfied. Hence The-orem 12 can be applied to (21), that is, every solution of (21) oscillates.Indeed, y(t) = et sin t is such an oscillatory solution.

REFERENCES

[1] I. Gyori, G. Ladas, Oscillation Theory of Delay Differential Equation with Applica-tion, Claredon Press, Oxford, 1991.

[2] I. T. Kiguradze, T. A. Chanturia, Asymptotic properties of solutions of nonau-tonomous ordinary differential equations, Kluwer Academic, Dordrecht, 1993.

[3] T. Kusano, M. Naito, Nonlinear oscillation of fourth order differential equations,Canad. J. Math. 4 (1976), 840–852.

[4] T. Kusano, M. Naito, On fourth order nonlinear oscillations, J. London Math. Soc.(2)14 (1976), 91–105.

[5] N. Parhi, A. K. Tripathy, On oscillatory fourth order nonlinear neutral differentialequations - II, Math. Slovaca, 55 (2005), 183–202.

[6] A. K. Tripathy, S. Panigrahi, R. Basu, Oscillation results for fourth order nonlinearmixed neutral differential equation - I (Communicated).

[7] A. K. Tripathy, Oscillation theorems for fourth order nonlinear neutral equationswith quasi-derivatives - I, Int. J. Math. Sci. Appl. 1 (2011), 377–393.


VOLUME 19

2012, NO 1–2

PP. 213–230

CORRECT SOLVABILITY OF INTEGRO-DIFFERENTIALEQUATIONS ARISING IN THE THEORY OF HEAT

TRANSFER AND ACOUSTICS ∗

V. VLASOV † AND N. RAUTIAN‡

Abstract. We study the correct solvability of an abstract functional differential equa-tions in Hilbert space including integro-differential equations arising in the theory of heattransfer and acoustics.

Dedicated to Professor A. D. Myshkis

Key Words. Functional differential equations, Integrodifferential equations, Sobolevspace, Gurtin-Pipkin heat equation.

AMS(MOS) subject classification. 34D05, 34C23

1. Introduction. This paper studies integro-differential equationswith unbounded operator coefficients in a Hilbert space. Most of the equa-tions under consideration are abstract hyperbolic equations perturbed byterms containing Volterra integral operators. These equations are abstractforms of the Gurtin-Pipkin integro-differential equations (see [19], [5] formore details), which describes heat propagation in media; it also arises inhomogenization problems in porous media (Darcy law).

It is shown that the initial boundary value problems for these equationsare well-solvable in Sobolev spaces on the positive half-axis.

∗ The research of V.Vlasov is financially supported by the Russian Foundation for BasicResearch (grant N 11-01-00790), and also Leading Scientific School 7322.2010.1.

† Department of Mechanics and Mathematics, Moscow Lomonosov State University,Vorobievi Gori, Moscow, 117234, Russia, ≪[email protected]≫,

‡ Department of Economics and Mathematics, Plekhanov Russian University of Eco-nomics, Stremyanny per. 36, Moscow, 117997, Russia, ≪[email protected]≫

213

214 V. VLASOV AND N. RAUTIAN

The main attention is paid to the asymptotic behavior of solutions to evo-lution equations. In this connection, it is natural and convenient to considerintegro-differential equations with unbounded operator coefficients (abstractintegro-differential equations), which can be realized as integro-differentialpartial differential equations with respect to spatial variables when necessary.For the self-adjoin positive operator A considered in what follows we can take,in particular, the operator A2y = −y′′, where x ∈ (0, π), y(0) = y(π) = 0,or the operator A2y = −∆y satisfying the Dirichlet conditions on a boundeddomain with sufficiently smooth boundary. At present, there is an extensiveliterature on abstract integro-differential equations (see, e.g., [1]-[27] and thereferences therein).

2. Statement of the problem. Let H be a separate Hilbert space,and let A be a self-adjoint positive operator with bounded inverse actingonH. We turn the domain Dom(Aβ) of the operator Aβ, β > 0 into aHilbert space Hβ by endowing Dom(Aβ) with the norm ∥ · ∥β = ∥Aβ · ∥,which is equivalent to the graph norm of the operator Aβ.

By W n2,γ (R+, A

n) we denote the Sobolev space of vector functions on thehalf-axis R+ = (0,∞) taking values in H endowed with the norm

∥u∥Wn2,γ(R+,An) ≡

(∫ ∞

0

e−2γt(∥∥u(n)(t)∥∥2

H+ ∥Anu(t)∥2H

)dt

)1/2

, γ ≥ 0.

For more information about the spaces W n2,γ (R+, A

n), see monograph[11], Chapter 1. For n = 0 we set W 0

2,γ (R+, A0) ≡ L2,γ (R+, H), and for

γ = 0, we write W n2,0 =W n

2 .Consider the following problem for a first-order integro-differential equa-

tion on R+ = (0,∞):

(1)dv(t)

dt+

∫ t

0

K(t− s)A2v(s)ds = q(t), t ∈ R+,

(2) v(+0) = ψ0.

It is assumed that the scalar function K(t) admits the representation

(3) K(t) =∞∑j=1

cjγje−γjt,

where cj > 0, γj+1 > γj > 0, j ∈ N, γj → +∞ (j → +∞) and

(4) K(0) =∞∑j=1

cjγj<∞.

Simultaneously, we consider the following problem for a second-orderintegro-differential equation:

(5)d2u(t)

dt2+K(0)A2u(t) +

∫ t

0

K ′(t− s)A2u(s)ds = f(t), t ∈ R+,

(6) u(+0) = φ0, u(1)(+0) = φ1,

Remark 1. If f(t) = q′(t), φ0 = ψ0 and φ1 = q(0), then problem (5), (6) isobtained from (1), (2) by differentiation with respect to t.

In the case under consideration, condition (4) means that K ′(t) ∈L1(R+). If, in addition to (4), condition

(7)∞∑j=1

cj < +∞,

holds, then the kernel K ′(t) belongs to the space W 11 (R+).

Definition 1. We say that a vector function v is a strong solution of theproblem (1), (2), if it belongs to the space W 1

2,γ(R+, A2) for some γ > 0,

satisfies the equation (1) almost everywhere on the half-axis R+, and satisfiesthe initial condition (2).Definition 2. We say that a vector function u is a strong solution of theproblem (5), (6), if it belongs to the space W 2

2,γ(R+, A2) for some γ > 0,

satisfies (5) almost everywhere on the half-axis R+, and satisfies the initialconditions (6).

The following theorems give conditions for problems (1), (2) and (5), (6)to be well-solvable.Theorem 1. If Aq′(t) ∈ L2,γ1 (R+, H) for some γ1 ≥ 0, q(0) = 0 andcondition (4), then the following assertion are valid.(i) If condition (7) holds and ψ0 ∈ H2, then, for any γ > γ1, problem (1),(2) is uniquely solvable in the space W 2

2,γ (R+, A2), and its solution satisfies

the inequality

(8) ∥v∥W 22,γ(R+,A2) ≤ d

(∥Aq′(t)∥L2,γ(R+,H) +

∥∥A2ψ0

∥∥H

)where the constant d does not depend on the vector function q and the vectorψ0;(ii) If condition (7) is violated and ψ0 ∈ H3, then, for any γ > γ1, prob-lem (1), (2) is uniquely solvable in the space W 1

2,γ (R+, A2) and its solution

satisfies the inequality

(9) ∥v∥W 22,γ(R+,A2) ≤ d

(∥Aq′(t)∥L2,γ(R+,H) +

∥∥A3ψ0

∥∥H

)

where the constant d does not depend on the vector function q and the vectorψ0.Theorem 2. If condition (4) holds and f ′(t) ∈ L2,γ2(R+, H) and f(0) = 0or Af(t) ∈ L2,γ2 (R+, H) for some γ2 ≥ 0, then the following assertions arevalid.(i) If condition (7) holds and φ0 ∈ H2, φ1 ∈ H1, then, for any γ > γ2, prob-lem (5), (6) is uniquely solvable in the space W 2

2,γ (R+, A2), and its solution

satisfies the inequality

(10) ∥u∥W 22,γ(R+,A2) ≤ d

(∥f ′(t)∥L2,γ(R+,H) +

∥∥A2φ0

∥∥H+ ∥Aφ1∥H

)where the constant d does not depend on the vector function f and the vectorsφ0 and φ1;(ii) If condition (7) is violated and φ0 ∈ H3, φ1 ∈ H2, then, for any γ >γ2, problem (5), (6) is uniquely solvable in the space W 2

2,γ (R+, A2), and its

solution satisfies the inequality

(11) ∥u∥W 22,γ(R+,A2) ≤ d

(∥f ′(t)∥L2,γ(R+,H) +

∥∥A3φ0

∥∥H+∥∥A2φ1

∥∥H

)where the constant d does not depend on the vector function f and the vectorsφ0 and φ1.

If Af(t) ∈ L2,γ2(R+, H) for some γ2 ≥ 0, then estimates (10) and (11)remain valid for ∥f ′(t)∥L2,γ(R+,H) replaced by ∥Af(t)∥L2,γ(R+,H). The proof ofthis fact is given in [32].

3. Proofs. The proof of the theorem 1 is given in [32]. We shallbegin the proof of the theorem 2 in the case of homogeneous (zero) initialconditions (φ0 = φ1 = 0). We use Laplace transformation in oder to provethe correct solvability of the problem (5) and (6). Now we are going toremind the base assertions that will be used later.Definition 3. We denote by H2(ℜλ > γ,H) the Hardy space of vector-functions f(λ) taking values in the space H, holomorphic (analytic) in thesemiplane λ ∈ C : ℜλ > γ > 0 endowed with the norm

(12) ∥f∥H2(ℜλ>γ,H) = supx>γ

(∫ +∞

−∞

∥∥∥f(x+ iy)∥∥∥2Hdy

)1/2

<∞, (λ = x+ iy).

We formulate well-known Paley-Wiener theorem for Hardy spaceH2(ℜλ > γ,H).Theorem 3. (Paley-Wiener). 1) The space H2(ℜλ > γ,H) coincides withthe set of vector-functions (Laplace transformations) representing in the form

(13) f(λ) =1√2π

∫ ∞

0

e−λtf(t)dt,


with vector-function f(t) ∈ L2,γ (R+, H), λ ∈ C, ℜλ > γ > 0.2) There exists unique vector-function f(t) ∈ L2,γ (R+, H) for arbitrary

vector-function f(λ) ∈ H2(ℜλ > γ,H) and the following inversion formulatake place

(14) f(t) =1√2π

∫ +∞

−∞f(γ + iy)e(γ+iy)tdy, t ∈ R+, γ > 0

3) The following equality take place for vector-function f(λ) ∈ H2(ℜλ >γ,H) and f(t) ∈ L2,γ (R+, H), connected by relation (13):

(15) ∥f∥2H2(ℜλ>γ,H) ≡ supx>γ

∫ +∞

−∞

∥∥∥f(x+ iy)∥∥∥2Hdy =

=

∫ +∞

0

e−2γt ∥f(t)∥2H dt ≡ ∥f∥2L2,γ(R+,H)

The theorem formulated above is well-known for the scalar functions. How-ever it is easily generalized for the vector-functions taking values in the sep-arable Hilbert space.(see [29]).

Proof of the theorem 2. Let us consider the case f ′(t) ∈ L2,γ2 (R+, H),f(0) = 0. The Laplace transformation of the strong solution of the problem(5) and (6) with initial conditions u(+0) = 0, u(1)(+0) = 0 has the followingrepresentation

(16) u(λ) = L−1(λ)f(λ).

Here the operator-function L(λ) is the symbol of the equation (5) and it hasthe following form

(17) L(λ) = λ2I + λA2

∞∑j=1

cjγj(λ+ γj)

,

We suppose there exists γ∗ ≥ 0, so that u ∈ W 22,γ∗ (R+, A

2). We needthis supposition in oder to use the Laplace transformation to the equation(5).

It is sufficient (for proof of the theorem 2) to prove that vector-functionsλ2u(λ) and A2u(λ) belong to Hardy space H2(ℜλ > γ,H) for some γ >γ2 ≥ 0. Then we shall obtain by the Paley-Wiener theorem that vector-

functions d2udt2

and A2u(t) belong to the space L2,γ (R+, H) and hence u(t) ∈W 2

2,γ (R+, A2). Thus we shall prove the solvability of the problem (5) and (6)

with homogeneous initial conditions in the space W 22,γ (R+, A

2).

Let denote by ej∞j=1 the orthonormal basis composed of the eigen-vectors of the operator A corresponding to eigenvalues aj, i.e., such thatAej = ajej for j ∈ N. The eigenvalues aj are numbered in increasing or-der with multiplicity taken into account: 0 < a1 6 a2 6 ...; an → +∞ asn→ +∞.

The restriction un(λ) of vector-function u(λ) to the one-dimensional sub-space spanned by the vector en can be presented as follows:

un(λ) = l−1n (λ)fn(λ),

where fn(λ) =(f(λ), en

), ln(λ) = (L(λ)en, en) that is

ln(λ) = λ2 + a2nλ∞∑j=1

cjγj(λ+ γj)

= λ2 + a2n

(∞∑j=1

cjγj

−∞∑j=1

cj(λ+ γj)

),

Let us prove that vector-function λ2u(λ) belongs to the spaceH2 (ℜλ > γ,H). So we consider the restriction of the vector-function onλ2u(λ) to the one-dimensional subspace spanned by the vector en:

(18)(λ2u(λ)en, en

)=λhn(λ)

ln(λ).

Here we denote by hn(λ) the restriction of the vector-function h(λ) = λf(λ)to the one-dimensional subspace spanned by the vector en. Due to the con-ditions of the theorem the vector-function h(t) = f ′(t) belongs to the spaceL2,γ2 (R+, H) and f(0) = 0. Hence the Laplace transformation h(λ) = λf(λ)belongs to Hardy space H2 (ℜλ > γ2, H).

Thus, it is sufficient to prove the following uniform estimate:

(19) supReλ>γ, n∈N

∣∣∣∣ λ

ln(λ)

∣∣∣∣ ≤ const.

We consider the following function

Nn(λ) :=ln(λ)

λ= λ+ a2n

∞∑k=1

ckγk(λ+ γk)

Let us obtain low estimate of the function |Nn(λ)|. The real part of thefunction Nn(λ) has the following form

ReNn(λ) = x+ a2n

∞∑j=1

cj(x+ γj)

γj((x+ γj)2 + y2), λ = x+ iy,

Let us estimate |Nn(λ)| for x > γ > γ1 ≥ 0:

(20) |ReNn(λ)| =

∣∣∣∣∣x+ a2n

∞∑j=1

cj(x+ γj)

γj((x+ γj)2 + y2)

∣∣∣∣∣ > γ

Using the estimate (20), for x > γ > γ1 ≥ 0 we obtain∣∣∣∣ λ

ln(λ)

∣∣∣∣ < 1

Re N(λ)<

1

γ.

Hence, we have 1

(22) supℜλ>γ,n∈N

∣∣∣∣ λ

ln(λ)

∣∣∣∣ < 1

γ.

So, we obtain from the inequality (21) and Paley-Wiener theorem thefollowing assertions

(23)

∥u(2)(t)∥2L2,γ(R+,H) = ∥λ2u(λ)∥2H2(ℜλ>γ,H) =∥∥∥λL−1(λ)λf(λ)

∥∥∥2H2(ℜλ>γ,H)

=

= supµ>γ

∫ +∞

−∞

[∞∑j=1

∣∣∣∣ (µ+ iν)

lj(µ+ iν)· (µ+ iν)fj(µ+ iν)

∣∣∣∣2]dν 6

6 supℜλ>γj∈N

∣∣∣∣ λ

lj(λ)

∣∣∣∣2 · supµ>γ

∫ +∞

−∞

[∞∑j=1

∣∣∣(µ+ iν)fj(µ+ iν)∣∣∣2]dν =

= supℜλ>γj∈N

∣∣∣∣ λ

lj(λ)

∣∣∣∣2 ∥λf(λ)∥2H2(ℜλ>γ,H) 6 d21∥f ′(t)∥2L2,γ(R+,H),

where d1 = supℜλ>γj∈N

∣∣∣∣ λlj(λ)

∣∣∣∣, λ = µ + iν. Thus, we conclude that the vector-

function λu(λ) belongs to the Hardy space H2 (ℜλ > γ,H) and the followingestimate is valid

(24)

∥∥∥∥d2udt2∥∥∥∥2L2,γ(R+,H)

6 d1∥f ′∥2L2,γ(R+,H).

1 The estimate (22) implies the inequality

(21) supℜλ>γ

∥λL−1(λ)∥ ≤ const.

Now we shall prove that the vector-function A2u(λ) also belongs to the

space H2 (ℜλ > γ,H). Let us denote by Ψ(λ) :=∞∑j=1

cjγj(λ+γj)

. Evidently that

the following representation is valid

(25) I = λ2L−1(λ) + λΨ(λ)A2L−1(λ).

for ℜλ > γ. Hence, we have

f(λ) = λ2L−1(λ)f(λ) + λΨ(λ)A2L−1(λ)f(λ)

for ℜλ > γ. So, due to (16) we have

(26) f(λ) = λ2u(λ) + λΨ(λ)A2u(λ).

According to the conditions of the theorem 2, f ′(t) ∈ L2,γ2(R+, H) and

f(0) = 0. Hence the Laplace transformation λf(λ) belongs to Hardy spaceH2 (ℜλ > γ2, H), that is

∥λf(λ)∥2H2(ℜλ>γ2,H) = supx>γ2

∫ +∞

−∞

∥∥∥(x+ iy)f(x+ iy)∥∥∥2Hdy <∞, λ = x+ iy.

So we have the following chain of the inequalities

∥λf(λ)∥2H2(ℜλ>γ2,H) = supx>γ2

∫ +∞

−∞

[∞∑n=1

∣∣∣(x+ iy)fn(x+ iy)∣∣∣2]dy >

> supx>γ2

γ22

∫ +∞

−∞

[∞∑n=1

∣∣∣fn(x+ iy)∣∣∣2]dy = γ22 sup

x>γ2

∫ +∞

−∞

∥∥∥f(x+ iy)∥∥∥2Hdy =

= γ22∥f(λ)∥2H2(ℜλ>γ2,H).

Hence, f(λ) ∈ H2 (ℜλ > γ2, H) and f(t) ∈ L2,γ2(R+, H).Owing to suppositions on the sequences cj∞j=1 and γj∞j=1, the func-

tion λΨ(λ) is analytic and bounded in the half-plane λ : ℜλ > γ. Really,according to the condition (4)

(27) |λΨ(λ)| =

∣∣∣∣∣∞∑j=1

cjγj

−∞∑j=1

cjλ+ γj

∣∣∣∣∣ 6∞∑j=1

cjγj

+∞∑j=1

cj|λ+ γj|

=

=∞∑j=1

cjγj

+∞∑j=1

cj√(x+ γj)2 + y2

< 2∞∑j=1

cjγj< +∞.

Moreover, we have for x ≥ γ

(28)∞∑j=1

cj(x+ γj)

(x+ γj)2 + y2<

∞∑j=1

cjx+ γj

<

∞∑j=1

cjγ + γj

<

∞∑j=1

cjγj< +∞.

Owing to the estimate (28) we have

|λΨ(λ)| =

∣∣∣∣∣∞∑j=1

cjγj

−∞∑j=1

cjλ+ γj

∣∣∣∣∣ >∞∑j=1

cjγj

−∞∑j=1

cj(x+ γj)

(x+ γj)2 + y2>

>∞∑j=1

cjγj

−∞∑j=1

cjγ + γj

> 0.

Keeping in mind the inequality (26) and estimates (23) we obtain

(29) ∥A2u(λ)∥H2(ℜλ>γ,H) 6

6 |λΨ(λ)|−1(∥λf(λ)∥H2(ℜλ>γ,H) + ∥λ2u(λ)∥H2(ℜλ>γ,H)

)6

6 d2∥λf(λ)∥H2(ℜλ>γ,H),

where the constant d2 > 0 does not depend on the vector-function f . Thus,we have

(30) ∥A2u(λ)∥L2,γ(R+,H) 6 d2∥f ′(t)∥L2,γ(R+,H).

Hence, uniting the estimates (24) and (30), we establish the inequality

(31) ∥u∥W 22,γ(R+,A2) 6 d ∥f ′(t)∥L2,γ(R+,H)

where the constant d2 > 0 does not depend on the vector-function f .So we proved that the equation (5) with homogeneous initial conditions

has a unique solution u(t) in the space W 22,γ (R+, A

2) and the estimate (31)is valid.

Now we prove that the obtained solution u(t) satisfy the initial condi-tions u(+0) = 0, u(1)(+0) = 0.

2

2 If the vector-function φ(λ) ∈ H2 (ℜλ > γ,H), then it is possible to find such sequanceRk for arbitrary R > γ that lim

k→∞Rk = +∞ and

limk→∞

∫ R

γ

∥φ(x± iRk)∥Hdx = 0.

Really, for arbitrary R > γ and Rk > 0∫ Rk

−Rk

(∫ R

γ

∥φ(x± iy)∥2H dx)dy 6

∫ R

γ

(∫ ∞

−∞∥φ(x± iy)∥2H dy

)dx <∞.

Hence, there exists the sequance Rk for any R > 0 such that limk→∞

Rk = +∞

and limk→∞

∫ R

γ∥φ(x± iRk)∥2Hdx = 0. Then we apply the Cauchy-Bunyakovskii

inequality.Let us suppose that p = 0, 1. Owing to the Paley-Wiener theorem, we

obtain that

u(p)(0) =1√2π

limRk→∞

∫ Rk

−Rk

(γ + iy)pu(γ + iy)dy =

=1√2πi

limRk→∞

∫ γ+iRk

γ−iRk

λpu(λ)dλ,

where u(0)(t) ≡ u(t).The vector-function λpu(λ) (p = 0, 1) is analytic in the half-plane ℜλ > γ

so, owing to the Cauchy theorem∫ γ+iRk

γ−iRk

λpu(λ)dλ =

(∫ R−iRk

γ−iRk

−∫ R+iRk

γ+iRk

+

∫ R+iRk

R−iRk

)λpu(λ)dλ =

=

∫ R

γ

(x− iRk)pu(x− iRk)dx−

∫ R

γ

(x+ iRk)pu(x+ iRk)dx+

+ i

∫ Rk

−Rk

(R + iy)pu(R + iy)dy.

Due to the remark 2,

limk→∞

∫ R

γ

|(x± iRk)pu(x± iRk)|dx = 0,

because λpu(λ) ∈ H2 (ℜλ > γ,H). Hence,∣∣u(p)(0)∣∣ ≤ 1√2π

limRk→∞

∫ Rk

−Rk

∥(R + iy)pu(R + iy)∥ dy =

=1√2π

∫ +∞

−∞

∥∥∥∥(R + iy)2u(R + iy)

(R + iy)2−p

∥∥∥∥ dy ≤

≤ 1√2π

(∫ +∞

−∞

∥∥(R + iy)2u(R + iy)∥∥2 dy) 1

2(∫ +∞

−∞

dy

(R2 + y2)2−p

) 12

.

. 1

R(3−2p)/2.


Thus tending R → +∞ we obtain u(p)(0) = 0, p = 0, 1.Let us prove finally that vector-function u(t) satisfies the equation (5).

Owing to the Paley-Wiener theorem

u(t) = limR→+∞

1√2π

∫ +R

−R

L−1(γ + iy)f(γ + iy)e(γ+iy)tdy =

= limR→+∞

1√2πi

∫ γ+iR

γ−iR

L−1(λ)f(λ)eλtdλ.

Thus, we have

(32)d2u(t)

dt2= lim

R→+∞

1√2πi

∫ γ+iR

γ−iR

λ2L−1(λ)f(λ)eλtdλ,

(33) A2u(t) = limR→+∞

1√2πi

∫ γ+iR

γ−iR

A2L−1(λ)f(λ)eλtdλ,

(34)

∫ t

0

K(t− s)A2u(s)ds =

= limR→+∞

1√2πi

∫ t

0

∞∑j=1

cje−γj(t−s)

(∫ γ+iR

γ−iR

A2L−1(λ)f(λ)eλtdλ

)ds =

= limR→+∞

1√2πi

∫ γ+iR

γ−iR

A2L−1(λ)f(λ)

(∞∑j=1

cj

∫ t

0

e−γj(t−s)eλsds

)dλ =

= limR→+∞

1√2πi

∫ γ+iR

γ−iR

(A2

∞∑j=1

cjλ+ γj

)L−1(λ)f(λ)eλtdλ.

The assertions (32)-(34) have as a consequence that vector-function u(t) sat-isfies the equation (5).

Now let us prove theorem 2 with nonhomogeneous initial conditions. Letus set in the problem (5), (6)

u(t) = cos(√SAt)φ0 + (

√SA)−1 sin(

√SAt)φ1 + ω(t),

where S := K(0). Then we have the following problem for the vector-functionω(t):

(35)d2ω(t)

dt2+K(0)A2ω(t) +

∫ t

0

K ′(t− s)A2ω(s)ds = f1(t), t ∈ R+,


(36) ω(+0) = ω(1)(+0) = 0.

where f1(t) = f(t)− g(t),

(37) g(t) =

∫ t

0

K ′(t− s)A2(cos(

√SAs)φ0 + (

√SA)−1 sin(

√SAs)φ1

)ds.

(38) g′(t) = K ′(t)A2φ0+

+

∫ t

0

K ′(t− s)(−√SA3sin(

√SAs)φ0 + A2 cos(

√SAs)φ1

)ds.

Let us prove that vector-function f1(t) satisfies the conditions of the theorem2 with homogeneous initial conditions. Really, we have

(39) ∥f ′1(t)∥L2,γ(R+,H) 6 ∥f ′(t)∥L2,γ(R+,H) + ∥g′(t)∥L2,γ(R+,H) .

Let us estimate the norm ∥g′(t)∥L2,γ(R+,H).Integrating by parts we have

(40)

∫ t

0

e−γj(t−s) cos(√SAs)ds =

=(SA2 + γ2j I

)−1(γj

(cos(

√SAt)− e−γjt

)+√SA sin(

√SAt)

)

(41)

∫ t

0

e−γj(t−s) sin(√SAs)ds =

=(SA2 + γ2j I

)−1(√

SA(e−γjt − cos(

√SAt)

)+ γj sin(

√SAt)

)

Then we need the following remark3.

3 The inequality

(42)∥∥∥(SA2 + γ2j I

)−1∥∥∥2H

. γ−2j

∥∥A−1∥∥2H.

is valid.


Really for arbitrary vector x ∈ H such that ∥x∥H = 1 the following chainof inequalities is true∥∥∥(SA2 + γ2j I

)−1x∥∥∥2H=

∞∑n=1

(Sa2n + γ2j

)−2 |xn|2 .

.∞∑n=1

(γjan)−2 |xn|2 =

∥∥γ−1j A−1x

∥∥2H,

where xn = (x, en), n ∈ N. Hence, so as A is a self-adjoint operator and γjis positive, we obtain (42).

Let us suppose that condition (7) is satisfied.Using (38), (40), (41) and ∥cos (At)∥ 6 1, ∥sin (At)∥ 6 1 and remark 3,

we have the following chain of the inequalities

∥g′(t)∥L2,γ(R+,H) 6∥∥∥∥∥

∞∑j=1

cje−γjtA2φ0

∥∥∥∥∥L2,γ(R+,H)

+

+

∥∥∥∥∥∥t∫

0

∞∑j=1

cje−γj(t−s)

(−√SA3 sin

(√SAs

)φ0+

+A2 cos(√

SAs)φ1

)ds∥∥∥L2,γ(R+,H)

.

.(

∞∑j=1

cj

)∥∥A2φ0

∥∥H+

+

∥∥∥∥∥∥∞∑j=1

cj

−√SA3

t∫0

e−γj(t−s) sin(√

SAs)ds

φ0+

+A2

t∫0

e−γj(t−s) cos(√

SAs)ds

φ1

∥∥∥∥∥∥L2,γ(R+,H)

.

.(

∞∑j=1

cj

)∥∥A2φ0

∥∥H+

+

∥∥∥∥∥∞∑j=1

cj(SA2 + γjI

)−1(−√SA3

)(√SA(e−γjt − cos

(√SAt

))+


+γj sin(√

SAt))

φ0

∥∥∥L2,γ(R+,H)

+

+

∥∥∥∥∥∞∑j=1

cj(SA2 + γjI

)−1A2(γj

(cos(√

SAt)− e−γjt

)+

+√SA sin

(√SAt

)φ1

∥∥∥L2,γ(R+,H)

=

=

(∞∑j=1

cj

)∥∥A2φ0

∥∥H+

∥∥∥∥∥−∞∑j=1

cje−γjt

(SA2 + γjI

)−1SA4φ0−

− cos(√

SAt) ∞∑

j=1

cj(SA2 + γjI

)−1SA4φ0+

+sin(√

SAt) ∞∑

j=1

cjγj(SA2 + γjI

)−1 √SA3φ0

∥∥∥∥∥L2,γ(R+,H)

+

+

∥∥∥∥∥cos(√SAt)∞∑j=1

cjγj(SA2 + γjI

)−1A2φ1+

+∞∑j=1

cjγje−γjt

(SA2 + γjI

)−1A2φ1 +

+sin(√

SAt) ∞∑

j=1

cj(SA2 + γjI

)−1 √SA3φ1

∥∥∥∥∥L2,γ(R+,H).

.

.(

∞∑j=1

cj

)∥∥A2φ0

∥∥H+

∥∥∥∥∥∞∑j=1

cj

(√S)−1

A−2SA4φ0

∥∥∥∥∥H

+

+

∥∥∥∥∥∞∑j=1

cjγj

(√Sγj

)−1

A−1√SA3φ0

∥∥∥∥∥H

+

+

∥∥∥∥∥∞∑j=1

cjγj

(√Sγj

)−1

A−1A2φ1

∥∥∥∥∥H

+

+

∥∥∥∥∥∞∑j=1

cjS−1A−2

√SA3φ1

∥∥∥∥∥H

.


.(

∞∑j=1

cj

)(∥∥A2φ0

∥∥H+ ∥Aφ1∥H

)Hence, we obtain from (31), (39) and the last estimate that

∥ω∥W 22,γ(R+,A2) ≤ d ∥f ′

1(t)∥L2,γ(R+,H) ≤

≤ d(∥f ′(t)∥L2,γ(R+,H) +

∥∥A2φ0

∥∥H+ ∥Aφ1∥H

).

In turn from this we establish the estimate (10).

b) Let us suppose that condition (7) is not satisfied. Then, from (38)and condition (4) we have

∥g′(t)∥L2,γ(R+,H) 6∥∥∥∥∥

∞∑j=1

cje−γjtA2φ0

∥∥∥∥∥L2,γ(R+,H)

+

+

∥∥∥∥∥∥t∫

0

∞∑j=1

cje−γj(t−s)

(e −

√SA3 sin

(√SAs

)φ0+

+√SA2 cos

(√SAs

)φ1

)ds∥∥∥ .

.

∞∫0

(∞∑j=1

cje−γjt

)2 ∥∥A2φ0

∥∥2He−2γtdt

1/2

+

∞∫0

∥∥∥∥∥∥t∫

0

∞∑j=1

cje−γj(t−s)

(−√SA3 sin

(√SAs

)φ0+

+√SA2 cos

(√SAs

)φ1

)ds∥∥∥2He−2γtdt

)1/2

.

.(

∞∑j=1

cjγj

)(∥∥A3φ0

∥∥H+∥∥A2φ1

∥∥H

).

Thus we obtain the estimates (10), (11) for the solution u(t) of the problem(5), (6). Theorem 2 is proved.

4. Remarks. Analysis of the proof of the theorem 2 shows thatsupposition of compactness of the operator A−1 can be substitute to thesupposition that operator A has a bounded universe. Really due to well-known theorem about representation of self-adjoint operator A =

∫R adEa,

where Ea (−∞ ≤ a ≤ +∞) -spectral resolution of the identity of operatorA the following inequalities

supa∈σ(A)

∣∣∣∣ λl(λ)∣∣∣∣ < +∞, sup

a∈σ(A)

∣∣∣∣ a

l(λ)

∣∣∣∣ < +∞,

are valid, where l(λ) := λ2 + λa2K(λ), and σ(A) is spectrum of operator A.Thus, it is possible to obtain the inequalities

∥λL−1(λ)∥ < +∞, ∥AL−1(λ)∥ < +∞, Reλ > γ,

that are essentially used while proving the correct solvability. Let us notethat this substitution implies the possibility to consider operator A withcontinious spectrum and as follows to study Cauchy problems (1), (2) or (5),(6) when A2y = −∆y for wide class of unbounded domains.

In conclusion, it is pertinent to mention that integro-differential equa-tions considered in this paper were derived in [5] and after that, apparently,acquired the name Gurtin-Pipkin equations.

In [1]–[3], [13], [21]–[23], [26] (see also the references therein), integro-differential equations with principal part being an abstract parabolic equa-tion were studied. Equations with principal part being an abstract hyperbolicequation are significantly less studied. The works most closely related to thisquestions are [24], [25]–[27], [4]–[8].

The main difference between the solvability results obtained in this pa-per and those already known (see, e.g., [4], [17], [19]) is that we allow thepresence of (integrable) singularities in the Volterra kernela of integral opera-tors. In works on hyperbolic-type equations mentioned above, requirementson the kernels of integral operators are more severe. The situation withparabolic-type equations is fundamentally different because in this case, theconditions on the kernels of integral operators are substantially weaker (see[1]–[3], [13], [21]–[23], [26]).

We emphasize that our method of the proof of the theorem on the correctsolvability of the initial boundary value problem for an abstract Gurtin-Pipkin equation differs substantially from the approach used by Pandolfiin [19]. Moreover, Pandolfi studied solvability in function space on a finitetime interval (0, T ), whereas we study solvability in the weighted Sobolevspaces W 2

2,γ(R+, A2) on the half-axis R+.


Our proof of the solvability theorems 1, 2 essentially uses the Hilbertstructure of the spaceW 2

2,γ(R+, A2), L2,γ(R+, H) and Paley-Wiener theorem,

while in [19], considerations are performed in Banach function spaces on afinite time interval (0, T ).

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