Research ArticleOn Some Classes of Linear Volterra Integral Equations

Anatoly S Apartsyn

Melentiev Energy Systems Institute SB RAS Irkutsk Russia

Correspondence should be addressed to Anatoly S Apartsyn apartsynisemseiirkru

Received 19 December 2013 Revised 29 May 2014 Accepted 29 May 2014 Published 6 July 2014

Academic Editor Hossein Jafari

Copyright copy 2014 Anatoly S ApartsynThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The sufficient conditions are obtained for the existence and uniqueness of continuous solution to the linear nonclassical Volterraequation that appears in the integral models of developing systems The Volterra integral equations of the first kind with piecewisesmooth kernels are considered Illustrative examples are presented

1 Introduction

Volterra integral equations of the first kind with variableupper and lower limits of integrationwere studied byVolterrahimself [1] The publications on this topic in the first half ofthe 20th century were reviewed in [2] and later studies werediscussed in [3ndash5]

A noticeable impetus to the development of this areais related to the research [6] which suggested a macroe-conomic two-sector integral model The Glushkovrsquos modelsof developing systems were further extended in [7 8] andused in many applications (see [9] and references therein)In particular a one-sector version of the Glushkovrsquos modelapplied to the power engineering problems was considered in[10ndash12] In the recent years the researchers have got attractedby the equation (see [13] and references therein) that in ageneral case has the following form

119899

sum119894=1

int119886119894minus1(119905)

119886119894(119905)

119870119894(119905 119904) 119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879] (1)

where

0 le 119886119899(119905) lt 119886

119899minus1(119905) lt sdot sdot sdot lt 119886

0(119905) equiv 119905

119886119894(0) = 0 119894 = 0 119899

(2)

kernels 119870119894and right-hand side 119910(119905) are given and 119909(119905) is an

unknown desired solutionAt 119899 = 1 the problems of the existence and uniqueness

of solution to (1) in the space 119862[0119879]

as well as the numerical

methods are studied in detail in [5] In this paper we will beinterested in the same problems for (1) at 119899 gt 1 Further forsimplicity we will consider only the case 119899 = 2 since manyresults are easily generalized for the case 119899 gt 2

2 Sufficient Conditions for the Correctness of

(1) at 119899=2 in Pair (119862[0119879]

∘

119862(1)

[0119879])

For convenience present (1) with 119899 = 2 in operator form

1198811119909 + 1198812119909 ≜ int

119905

1198861(119905)

1198701(119905 119904) 119909 (119904) 119889119904

+ int1198861(119905)

0

1198702(119905 119904) 119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879]

(3)

(in (3) 1198862(119905) = 0 is assumed with no loss of generality)

Let kernels 1198701and 119870

2be continuous in arguments and

continuously differentiable with respect to 119905 in regions Δ1=

(119905 119904) 0 le 1198861(119905) le 119904 le 119905 le 119879 and Δ

2= (119905 119904) 0 le 119904 le

1198861(119905) respectively so that Δ

1cup Δ2= Δ Δ = (119905 119904) 0 le 119904 le

119905 le 119879 Δ1cap Δ2= 119897 119897 = (119905 119904) 119904 = 119886

1(119905) We will assume

that

11988610158401(119905) isin 119862

+[0119879] 11988610158401(0) lt 1 (4)

In particular (4) holds true for 1198861(119905) = 120572119905 120572 isin (0 1)

∘

119862(1)

[0119879]is further taken to mean the space of continu-

ously differentiable functions 119910(119905) on [0 119879] with the norm

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 532409 6 pageshttpdxdoiorg1011552014532409

2 Abstract and Applied Analysis

119910(119905)∘

119862

(1)

[0119879]

= max0le119905le119879

|119910(119905)| + |1199101015840(119905)| and additional condit-

ion 119910(0) = 0 If

min119905isin[0119879]

10038161003816100381610038161198701 (119905 119905)1003816100381610038161003816 = 119896 gt 0 (5)

then as established in [5 page 106] the following estimate istrue

10038171003817100381710038171003817119881minus1

1

10038171003817100381710038171003817 ∘119862

(1)

[0119879]rarr119862[0119879]

le 119878119896minus1119890119896minus11198711119879 (6)

where

1198711= max(119905119904)isinΔ 1

100381610038161003816100381610038161198701015840

1119905

(119905 119904)10038161003816100381610038161003816

119878 =infin

sum119895=0

119895

prod119894=1

120574119894ge 1

120574119894= 120573119894+ (119911119894minus 119911119894+1) 1198711119896minus1

119911119894= 1198861198941(119879) = 119886

1(1198861(sdot sdot sdot 1198861(119879))) 1198860

1(119879) = 119879

120573119894= max119905isin[119911119894 119911119894minus1]

11988610158401(119905) 10038161003816100381610038161198701 (119905 119886 (119905))

100381610038161003816100381610038161003816100381610038161198701 (119905 119905)1003816100381610038161003816

(7)

Estimating (6) makes it possible to obtain the sufficientcondition for the existence uniqueness and stability of the

solution to (3) in pair (119862[0119879]

∘

119862(1)

[0119879])

Theorem 1 Let the following inequality hold true

1198861(119879) (119872

2+ 1198712) + 11986011198722lt 119896119878minus1119890minus119896

minus11198711119879 (8)

where

1198601= max119905isin[0119879]

11988610158401(119905)

1198722= max(119905119904)isinΔ 2

10038161003816100381610038161198702 (119905 119904)1003816100381610038161003816

1198712= max(119905119904)isinΔ 2

100381610038161003816100381610038161198701015840

2119905

(119905 119904)10038161003816100381610038161003816

(9)

Then (3) is correct in the sense of Hadamard in pair

(119862[0119879]

∘

119862(1)

[0119879])

Proof By virtue of a well-known theorem of functionalanalysis (see eg [14 page 212]) if

100381710038171003817100381711988121003817100381710038171003817119862[0119879]rarr

∘

119862

(1)

[0119879]

lt 11003817100381710038171003817119881minus1

1

1003817100381710038171003817 ∘119862

(1)

[0119879]rarr119862[0119879]

(10)

then the operator 119881 = 1198811+ 1198812has a bounded inverse and

consequently (3) is correct in the sense of Hadamard in pair

(119862[0119879]

∘

119862(1)

[0119879]) We show that under (8)-(9) inequality (10)

holds true

As

100381710038171003817100381711988121199091003817100381710038171003817 ∘119862

(1)

[0119879]

= max0le119905le119879

100381610038161003816100381610038161003816100381610038161003816int1198861(119905)

0

1198702(119905 119904) 119909 (119904) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816100381610038161003816100381611988610158401(119905) 1198702(119905 1198861(119905))

+int1198861(119905)

0

11987010158402119905

(119905 119904) 119909 (119904) 119889119904100381610038161003816100381610038161003816100381610038161003816

le 1198861(119879) (119872

2+ 1198712) + 11986011198722 119909(119905)

119862[0119879]

(11)

then10038171003817100381710038171198812

1003817100381710038171003817119862[0119879]rarr

∘

119862

(1)

[0119879]

le 1198861(119879) (119872

2+ 1198712) + 11986011198722 (12)

and (10) follows from (6) and (12)

Condition (8) was obtained in the assumption that kernel1198701is defined on Δ

1 If it is possible to expand the domain of

definition 1198701to Δ so that Δ

1cap Δ2= Δ cap Δ

2= Δ2 then the

sufficient condition for the correctness of (3) is modified inthe following way Represent the first term in (3) in the form

int119905

1198861(119905)

1198701(119905 119904) 119909 (119904) 119889119904 = int

119905

0

1198701(119905 119904) 119909 (119904) 119889119904

minus int1198861(119905)

0

1198701(119905 119904) 119909 (119904) 119889119904

(13)

Then (3) can be represented as

1119909 + 2119909 ≜ int

119905

0

1198701(119905 119904) 119909 (119904) 119889119904

+ int1198861(119905)

0

(1198702(119905 119904) minus 119870

1(119905 119904)) 119909 (119904) 119889119904 = 119910 (119905)

119905 isin [0 119879] (14)

Since (see [5 page 12])10038171003817100381710038171003817minus1

1

10038171003817100381710038171003817 ∘119862

(1)

[0119879]rarr119862[0119879]

le 119896minus1119890119896minus11119879 (15)

where1= max(119905119904)isinΔ

100381610038161003816100381610038161198701015840

1119905

(119905 119904)10038161003816100381610038161003816 (16)

then sufficient conditions for the correctness of (14) give thefollowing theorem

Theorem 2 Let inequality

1198861(119879) (

2+ 2) + 119860

12lt 119896119890minus1119890minus119896

minus11119879 (17)

where2= max(119905119904)isinΔ 2

10038161003816100381610038161198702 (119905 119904) minus 1198701 (119905 119904)1003816100381610038161003816 (18)

2= max(119905119904)isinΔ 2

100381610038161003816100381610038161198701015840

2119905

(119905 119904) minus 11987010158401119905

(119905 119904)10038161003816100381610038161003816 (19)

hold trueThen (14) is correct in the sense of Hadamard in pair

(119862[0119879]

∘

119862(1)

[0119879])

Abstract and Applied Analysis 3

Proof With obvious changes repeat the proof of Theorem 1

Let us illustrate the obtained results with the followingexample

Consider the equation

int119905

120572119905

119909 (119904) 119889119904 + 120598int120572119905

0

119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879] (20)

Here by (5)ndash(7) 119896 = 1 1198722= |120598| 119872

2= |1 minus 120598| 119871

1= 1198711=

1198712= 0 119886

1(119879) = 120572119879 119860

1= 120572 120574

119894= 120573119894= 120572 and 119878 = 1(1 minus 120572)

therefore based on (8) inequality

120572119879 |120598| + 120572 |120598| lt 1 minus 120572 (21)

and based on (17) inequality

120572119879 |1 minus 120598| + 120572 |1 minus 120598| lt 1 (22)

give the following estimates 120598 which guarantee the existenceuniqueness and stability of solution to (20) in the space119862

[0119879]

|120598| lt 1 minus 120572120572 (1 + 119879)

|1 minus 120598| lt 1120572 (1 + 119879)

(23)

It is useful to compare (23) with the estimate obtained byshifting from (20) to the equivalent functional equationDifferentiation of (20) gives

119909 (119905) = 120572 (1 minus 120598) 119909 (120572119905) + 1199101015840 (119905) (24)

whence

119909 (119905) = lim119899rarrinfin

[

[120572119899(1 minus 120598)119899119909 (120572119899119905) +

119899minus1

sum119895=0

120572119895(1 minus 120598)1198951199101015840 (120572119895119905)]

](25)

and condition

|1 minus 120598| lt 1120572

(26)

provides convergence of series (25) to continuous function119909(119905) on [0 119879]

If in (20)

120598 = 1 minus 1120572 (27)

then condition (26) is violated Then it is easy to see that thehomogeneous equation

int119905

120572119905

119909 (119904) 119889119904 + (1 minus 1120572)int120572119905

0

119909 (119904) 119889119904 = 0 (28)

has a nontrivial solution 119909(119905) = const and if for example119910(119905) = 119905 the solution to the nonhomogeneous equation

int119905

120572119905

119909 (119904) 119889119904 + (1 minus 1120572)int120572119905

0

119909 (119904) 119889119904 = 119905 119905 isin [0 119879] (29)

is a one-parameter family

119909 (119905) = minus ln 119905ln120572

+ 119909 (1) (30)

Let now

120598 = 1 + 1120572 (31)

Then according to (24)

119909 (119905) = minus119909 (120572119905) + 1199101015840 (119905) (32)

whence

119909 (119905) = lim119899rarrinfin

[

[(minus1)119899119909 (120572119899119905) +

119899minus1

sum119895=0

(minus1)1198951199101015840 (120572119895119905)]

](33)

so that for the right-hand side of (20) 119910(119905) = 119910(119905) = 119905119896119896119896 = 1 2 3 from (33) we obtain

119909 (119905) = 119905119896minus1

1 + 120572119896minus1 119896 = 1 2 (34)

In conclusion of this section it should be noted thatinequalities (8) and (17) can be interpreted as constraints onthe value 119879 which guarantee at given 119870

1(119905 119904) 119870

2(119905 119904) and

1198861(119905) the correct solvability of (3) in 119862

[0119879] Since all param-

eters in the left-hand side of (8) and (17) are nondecreasingfunctions of119879 and the right-hand side of (8) and (17) at 119871

1= 0

(1

= 0) on the contrary monotonously decreases then thereal positive root of corresponding nonlinear equation thatgives a guaranteed lower-bound estimate of 119879 exists and isunique if 1198861015840(0) is sufficiently small In some special casesthis root can be found analytically in terms of the Lambertfunction119882 [15 16]

In [17ndash22] the authors studied the characteristic of con-tinuous solution locality and the role of the Lambert functionas applied to the polynomial (multilinear) Volterra equationsof the first kind The calculations of the test examples showthat the locality feature of the solution to the linear equation(3) is not the result of the inaccuracy of estimates (8) and (17)and reflects the specifics of the considered class of problemsIn this paper we do not dwell on the problem of numericallysolving (3) It is of independent interest and deserves specialconsideration

3 The Volterra Integral Equations of the FirstKind with Discontinuous Kernels

Equation (2) can be written in the form of Volterra integralequation of the first kind

int119905

0

119870 (119905 119904) 119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879] (35)

4 Abstract and Applied Analysis

with discontinuous kernel

119870 (119905 119904)

=

1198701(119905 119904) 119886

1(119905) lt 119904 le 119905

119870119894(119905 119904) 119886

119894(119905) lt 119904 lt 119886

119894minus1(119905)

119894 = 2 119899 minus 1(119870119894(119905 119904) + 119870

119894+1(119905 119904))

2 119904 = 119886

119894(119905) 119894 = 1 119899 minus 1

119870119899(119905 119904) 0 le 119904 lt 119886

119899minus1(119905)

(36)

To illustrate the fundamental difference between (35)(36) and classical Volterra equation of the first kind withsmooth kernel we confine ourselves to (20) that has the formof (35) at

119870 (119905 119904) =

1 120572119905 lt 119904 le 1199051 + 1205982

119904 = 120572119905120598 0 le 119904 lt 120572119905

(37)

where 120598 = 0 1 and 120572 isin (0 1) In particular at 120572 = 12 120598 = minus1

119870 (119905 119904) = sign(119904 minus 1199052) =

1 119904 gt 1199052

0 119904 = 1199052

minus1 119904 lt 1199052

(38)

For this case the solution to (35) with 119910(119905) = 119905 given in[23] is

119909 (119905) = ln 119905ln 2

+ 119909 (119904) (39)

For kernel (38)

119870 (0 0) = 0 119870 (119905 119905) = 0 119905 gt 0 (40)

If119870(119905 119904) is continuous in arguments and continuously differ-entiable with respect to 119905 inΔ then condition (40)means that(35) is Volterra integral equation of the third kind

The theory (whose foundation was laid by Volterra (see[24 pages 104ndash106])) of such equations is developed in theresearch done by Magnitsky [25ndash28]

In particular the author of [25ndash28] studies the structureof one- or many-parameter family of solutions to (35)

If 119870(119905 119904) is discontinuous then the solution to (35) maybe nonunique even if 119870(119905 119905) = 0 forall119905 ge 0

For example if120572 = 12 and 120598 = 1minus(1120572) = minus1 the solutionto equation

int119905

120572119905

119909 (119904) 119889119904 + 120598int120572119905

0

119909 (119904) 119889119904 = 119905 119905 isin [0 119879] (41)

is a one-parameter family

119909 (119905) = minus ln 119905ln120572

+ 119909 (1) (42)

but by (37) 119870(0 0) = (1 + 120598)2 = 0 119870(119905 119905) = 1 119905 gt 0

Now we show that there can be a nonunique solution to(35) and (36) even in the case 119870(119905 119905) equiv 1 Let

119870 (119905 119904) = 1 119904 ge 120572119905120598 119904 lt 120572119905

(43)

so that condition119870(119905 119905) equiv 1 is trueWe prove that solutions to (35) (37) and (35) (43)

coincide It suffices to show that the equivalent functionalequations for (35) (37) and (35) (43) coincide Recall that for(35) (37) the equivalent functional equation is (24)

Theorem 3 The equivalent functional equations for (35) (37)and (35) (43) coincide

Proof Let us represent (43) by

119870 (119905 119904) equiv 1 + (120598 minus 1) 119890 (120572119905 minus 119904) (44)

where 119890(sdot)ndash is a Heaviside function

119890 (]) = 1 ] ge 00 ] lt 0

(45)

Substitution of (44) in (35) gives

int119905

0

119909 (119904) 119889119904 + (120598 minus 1) int119905

0

119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905)

119905 isin [0 119879] (46)

Transform the second integral Let ] = 120572119905 minus 119904 Then

int119905

0

119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = int120572119905

(120572minus1)119905

119890 (]) 119909 (120572119905 minus ]) 119889]

= int120572119905

0

119909 (120572119905 minus ]) 119889]

(47)

By virtue of (47) differentiation of (46) results in

119909 (119905) + (120598 minus 1) 120572119909 (0) + (120598 minus 1) int120572119905

0

1199091015840119905(120572119905 minus ]) 119889] = 1199101015840 (119905)

(48)

But

1199091015840119905(120572119905 minus ]) = minus1205721199091015840] (120572119905 minus ]) (49)

By virtue of (49) we have

119909 (119905) + (120598 minus 1) 120572119909 (0) minus (120598 minus 1) 120572 [119909 (120572119905 minus ])|1205721199050] = 1199101015840 (119905)

(50)

from (48) whence finally

119909 (119905) + 120572 (120598 minus 1) 119909 (120572119905) = 1199101015840 (119905) (51)

and (51) coincides with (24)

Abstract and Applied Analysis 5

The solution to (35) (43) in the class of piecewisecontinuous functions with a jump on line 119904 = 120572119905 is interestingfrom the application perspective

It is easy to see that this solution is

119909 (119905 119904) =

1199101015840 (119904) 119904 ge 12057211990511205981199101015840 (119904) 119904 lt 120572119905

(52)

At last consider the concept of 120572-convolution Volterraintegral equations of convolution type

119870 (119905) lowast 119909 (119905) Δ= int119905

0

119870 (119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879]

(53)

are important for applicationExamples (38) and (44) show the usefulness of the 120572-

convolution concept

119870 (119905) 120572lowast 119909 (119905) Δ= int119905

0

119870 (120572119905 minus 119904) 119909 (119904) 119889119904

= 119910 (119905) 120572 isin (0 1] 119905 isin [0 119879] (54)

Give some inversion formulas of the integral equation

119870 (119905) 120572lowast 119909 (119905) = 119910 (119905) 119905 isin [0 119879] (55)

(1) If 119870(119905) = 120575(119905) 119910(119905) isin 119862[0119879]

and 120572 isin (0 1] then

119909 (120572119905) = 119910 (119905) (56)

(2) If 119870(119905) = 119890(119905) 119910(119905) isin∘

119862(1)

[0119879] and 120572 isin (0 1] then

119909 (120572119905) = 11205721199101015840 (119905) (57)

(3) If 119870(119905) = sign 119905 119910(119905) = 119905 and 120572 isin (0 1) then

119909 (119905) = ln 119905ln120572

+ 119909 (1) (58)

At 119870(119905) = 119905119899 119899 ge 1 (55) is Volterra integral equationof the third kind

(4) If 119870(119905) = 119905 119910(119905) = 11990522 and 120572 = 12 then

119909 (119905) = minus2 ln 119905 + 119909 (1) (59)

(5) If 119870(119905) = 119905 119910(119905) = 11990522 and 120572 isin (0 1) 120572 = 12 then

119909 (119905) = 119909 (1)119905(2120572minus1)(120572minus1)

(60)

4 Conclusion

As is mentioned in the introduction the main results of thisstudy can be easily applied to the case 119899 gt 2 in (1) Theequations of type (1) not only are of theoretical interest butalso play an important role in the mathematical modeling ofdeveloping dynamic systemsMoreover by119910(119905) we canmeansome criterion that characterizes the level of developmentof the system as a whole and the 119894th term in (1) representsa contribution of the system components 119909(119904) of the 119894thage group whose operation is reflected by the efficiencycoefficient 119870

119894(119905 minus 119904) As a rule 119870

1ge sdot sdot sdot ge 119870

119899ge 0

Such an approach is implemented for instance in [29 30]in the problem of the analysis of strategies for the long-termexpansion of the Russian electric power system with theconsideration of aging of the power plants equipment

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author wishes to thank the reviewers for their helpfulnotes The study is supported by the Russian Foundation forBasic Research Grant no 12-01-00722a

References

[1] V Volterra ldquoSopra alcune questioni di inversione di integralidefinitirdquo Annali di Matematica Pura ed Applicata Series 2 vol25 no 1 pp 139ndash178 1897

[2] H Brunner ldquo1896ndash1996 One hundred years of Volterra integralequations of the first kindrdquoAppliedNumericalMathematics vol24 no 2-3 pp 83ndash93 1997

[3] H Brunner and P J van der HouwenTheNumerical Solution ofVolterra Equations vol 3 of CWI Monographs North-HollandAmsterdam The Netherlands 1986

[4] H Brunner Collocation Methods for Volterra Integral andRelated Functional Differential Equations vol 15 of CambridgeMonographs on Applied and Computational Mathematics Cam-bridge University Press Cambridge Mass USA 2004

[5] A S Apartsyn Nonclassical Linear Volterra Equations of theFirst Kind VSP Utrecht The Netherlands 2003

[6] V M Glushkov ldquoOn one class of dynamic macroeconomicmodelsrdquo Upravlyayushchiye Sistemy I Mashiny no 2 pp 3ndash61977 (Russian)

[7] V M Glushkov V V Ivanov and V M Yanenko Modeling ofDeveloping Systems Nauka Moscow Russia 1983 (Russian)

[8] Y P Yatsenko Integral Models of Systems with ControlledMemory Naukova Dumka Kiev Ukraine 1991 (Russian)

[9] N Hritonenko and Y Yatsenko Applied Mathematical Mod-elling of Engineering Problems vol 81 of Applied OptimizationKluwer Academic Publishers Dordrecht The Netherlands2003

[10] A S Apartsyn E V Markova and V V Trufanov IntegralModels of Electric Power System Development Energy SystemsInstitute SB RAS Irkutsk Russia 2002 (Russian)

6 Abstract and Applied Analysis

[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004

[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)

[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008

[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)

[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993

[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004

[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007

[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007

[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008

[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008

[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)

[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013

[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)

[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)

[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)

[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)

[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)

[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012

[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

119910(119905)∘

119862

(1)

[0119879]

= max0le119905le119879

|119910(119905)| + |1199101015840(119905)| and additional condit-

ion 119910(0) = 0 If

min119905isin[0119879]

10038161003816100381610038161198701 (119905 119905)1003816100381610038161003816 = 119896 gt 0 (5)

then as established in [5 page 106] the following estimate istrue

10038171003817100381710038171003817119881minus1

1

10038171003817100381710038171003817 ∘119862

(1)

[0119879]rarr119862[0119879]

le 119878119896minus1119890119896minus11198711119879 (6)

where

1198711= max(119905119904)isinΔ 1

100381610038161003816100381610038161198701015840

1119905

(119905 119904)10038161003816100381610038161003816

119878 =infin

sum119895=0

119895

prod119894=1

120574119894ge 1

120574119894= 120573119894+ (119911119894minus 119911119894+1) 1198711119896minus1

119911119894= 1198861198941(119879) = 119886

1(1198861(sdot sdot sdot 1198861(119879))) 1198860

1(119879) = 119879

120573119894= max119905isin[119911119894 119911119894minus1]

11988610158401(119905) 10038161003816100381610038161198701 (119905 119886 (119905))

100381610038161003816100381610038161003816100381610038161198701 (119905 119905)1003816100381610038161003816

(7)

Estimating (6) makes it possible to obtain the sufficientcondition for the existence uniqueness and stability of the

solution to (3) in pair (119862[0119879]

∘

119862(1)

[0119879])

Theorem 1 Let the following inequality hold true

1198861(119879) (119872

2+ 1198712) + 11986011198722lt 119896119878minus1119890minus119896

minus11198711119879 (8)

where

1198601= max119905isin[0119879]

11988610158401(119905)

1198722= max(119905119904)isinΔ 2

10038161003816100381610038161198702 (119905 119904)1003816100381610038161003816

1198712= max(119905119904)isinΔ 2

100381610038161003816100381610038161198701015840

2119905

(119905 119904)10038161003816100381610038161003816

(9)

Then (3) is correct in the sense of Hadamard in pair

(119862[0119879]

∘

119862(1)

[0119879])

Proof By virtue of a well-known theorem of functionalanalysis (see eg [14 page 212]) if

100381710038171003817100381711988121003817100381710038171003817119862[0119879]rarr

∘

119862

(1)

[0119879]

lt 11003817100381710038171003817119881minus1

1

1003817100381710038171003817 ∘119862

(1)

[0119879]rarr119862[0119879]

(10)

then the operator 119881 = 1198811+ 1198812has a bounded inverse and

consequently (3) is correct in the sense of Hadamard in pair

(119862[0119879]

∘

119862(1)

[0119879]) We show that under (8)-(9) inequality (10)

holds true

As

100381710038171003817100381711988121199091003817100381710038171003817 ∘119862

(1)

[0119879]

= max0le119905le119879

100381610038161003816100381610038161003816100381610038161003816int1198861(119905)

0

1198702(119905 119904) 119909 (119904) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816100381610038161003816100381611988610158401(119905) 1198702(119905 1198861(119905))

+int1198861(119905)

0

11987010158402119905

(119905 119904) 119909 (119904) 119889119904100381610038161003816100381610038161003816100381610038161003816

le 1198861(119879) (119872

2+ 1198712) + 11986011198722 119909(119905)

119862[0119879]

(11)

then10038171003817100381710038171198812

1003817100381710038171003817119862[0119879]rarr

∘

119862

(1)

[0119879]

le 1198861(119879) (119872

2+ 1198712) + 11986011198722 (12)

and (10) follows from (6) and (12)

Condition (8) was obtained in the assumption that kernel1198701is defined on Δ

1 If it is possible to expand the domain of

definition 1198701to Δ so that Δ

1cap Δ2= Δ cap Δ

2= Δ2 then the

sufficient condition for the correctness of (3) is modified inthe following way Represent the first term in (3) in the form

int119905

1198861(119905)

1198701(119905 119904) 119909 (119904) 119889119904 = int

119905

0

1198701(119905 119904) 119909 (119904) 119889119904

minus int1198861(119905)

0

1198701(119905 119904) 119909 (119904) 119889119904

(13)

Then (3) can be represented as

1119909 + 2119909 ≜ int

119905

0

1198701(119905 119904) 119909 (119904) 119889119904

+ int1198861(119905)

0

(1198702(119905 119904) minus 119870

1(119905 119904)) 119909 (119904) 119889119904 = 119910 (119905)

119905 isin [0 119879] (14)

Since (see [5 page 12])10038171003817100381710038171003817minus1

1

10038171003817100381710038171003817 ∘119862

(1)

[0119879]rarr119862[0119879]

le 119896minus1119890119896minus11119879 (15)

where1= max(119905119904)isinΔ

100381610038161003816100381610038161198701015840

1119905

(119905 119904)10038161003816100381610038161003816 (16)

then sufficient conditions for the correctness of (14) give thefollowing theorem

Theorem 2 Let inequality

1198861(119879) (

2+ 2) + 119860

12lt 119896119890minus1119890minus119896

minus11119879 (17)

where2= max(119905119904)isinΔ 2

10038161003816100381610038161198702 (119905 119904) minus 1198701 (119905 119904)1003816100381610038161003816 (18)

2= max(119905119904)isinΔ 2

100381610038161003816100381610038161198701015840

2119905

(119905 119904) minus 11987010158401119905

(119905 119904)10038161003816100381610038161003816 (19)

hold trueThen (14) is correct in the sense of Hadamard in pair

(119862[0119879]

∘

119862(1)

[0119879])

Abstract and Applied Analysis 3

Proof With obvious changes repeat the proof of Theorem 1

Let us illustrate the obtained results with the followingexample

Consider the equation

int119905

120572119905

119909 (119904) 119889119904 + 120598int120572119905

0

119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879] (20)

Here by (5)ndash(7) 119896 = 1 1198722= |120598| 119872

2= |1 minus 120598| 119871

1= 1198711=

1198712= 0 119886

1(119879) = 120572119879 119860

1= 120572 120574

119894= 120573119894= 120572 and 119878 = 1(1 minus 120572)

therefore based on (8) inequality

120572119879 |120598| + 120572 |120598| lt 1 minus 120572 (21)

and based on (17) inequality

120572119879 |1 minus 120598| + 120572 |1 minus 120598| lt 1 (22)

give the following estimates 120598 which guarantee the existenceuniqueness and stability of solution to (20) in the space119862

[0119879]

|120598| lt 1 minus 120572120572 (1 + 119879)

|1 minus 120598| lt 1120572 (1 + 119879)

(23)

It is useful to compare (23) with the estimate obtained byshifting from (20) to the equivalent functional equationDifferentiation of (20) gives

119909 (119905) = 120572 (1 minus 120598) 119909 (120572119905) + 1199101015840 (119905) (24)

whence

119909 (119905) = lim119899rarrinfin

[

[120572119899(1 minus 120598)119899119909 (120572119899119905) +

119899minus1

sum119895=0

120572119895(1 minus 120598)1198951199101015840 (120572119895119905)]

](25)

and condition

|1 minus 120598| lt 1120572

(26)

provides convergence of series (25) to continuous function119909(119905) on [0 119879]

If in (20)

120598 = 1 minus 1120572 (27)

then condition (26) is violated Then it is easy to see that thehomogeneous equation

int119905

120572119905

119909 (119904) 119889119904 + (1 minus 1120572)int120572119905

0

119909 (119904) 119889119904 = 0 (28)

has a nontrivial solution 119909(119905) = const and if for example119910(119905) = 119905 the solution to the nonhomogeneous equation

int119905

120572119905

119909 (119904) 119889119904 + (1 minus 1120572)int120572119905

0

119909 (119904) 119889119904 = 119905 119905 isin [0 119879] (29)

is a one-parameter family

119909 (119905) = minus ln 119905ln120572

+ 119909 (1) (30)

Let now

120598 = 1 + 1120572 (31)

Then according to (24)

119909 (119905) = minus119909 (120572119905) + 1199101015840 (119905) (32)

whence

119909 (119905) = lim119899rarrinfin

[

[(minus1)119899119909 (120572119899119905) +

119899minus1

sum119895=0

(minus1)1198951199101015840 (120572119895119905)]

](33)

so that for the right-hand side of (20) 119910(119905) = 119910(119905) = 119905119896119896119896 = 1 2 3 from (33) we obtain

119909 (119905) = 119905119896minus1

1 + 120572119896minus1 119896 = 1 2 (34)

In conclusion of this section it should be noted thatinequalities (8) and (17) can be interpreted as constraints onthe value 119879 which guarantee at given 119870

1(119905 119904) 119870

2(119905 119904) and

1198861(119905) the correct solvability of (3) in 119862

[0119879] Since all param-

eters in the left-hand side of (8) and (17) are nondecreasingfunctions of119879 and the right-hand side of (8) and (17) at 119871

1= 0

(1

= 0) on the contrary monotonously decreases then thereal positive root of corresponding nonlinear equation thatgives a guaranteed lower-bound estimate of 119879 exists and isunique if 1198861015840(0) is sufficiently small In some special casesthis root can be found analytically in terms of the Lambertfunction119882 [15 16]

In [17ndash22] the authors studied the characteristic of con-tinuous solution locality and the role of the Lambert functionas applied to the polynomial (multilinear) Volterra equationsof the first kind The calculations of the test examples showthat the locality feature of the solution to the linear equation(3) is not the result of the inaccuracy of estimates (8) and (17)and reflects the specifics of the considered class of problemsIn this paper we do not dwell on the problem of numericallysolving (3) It is of independent interest and deserves specialconsideration

3 The Volterra Integral Equations of the FirstKind with Discontinuous Kernels

Equation (2) can be written in the form of Volterra integralequation of the first kind

int119905

0

119870 (119905 119904) 119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879] (35)

4 Abstract and Applied Analysis

with discontinuous kernel

119870 (119905 119904)

=

1198701(119905 119904) 119886

1(119905) lt 119904 le 119905

119870119894(119905 119904) 119886

119894(119905) lt 119904 lt 119886

119894minus1(119905)

119894 = 2 119899 minus 1(119870119894(119905 119904) + 119870

119894+1(119905 119904))

2 119904 = 119886

119894(119905) 119894 = 1 119899 minus 1

119870119899(119905 119904) 0 le 119904 lt 119886

119899minus1(119905)

(36)

To illustrate the fundamental difference between (35)(36) and classical Volterra equation of the first kind withsmooth kernel we confine ourselves to (20) that has the formof (35) at

119870 (119905 119904) =

1 120572119905 lt 119904 le 1199051 + 1205982

119904 = 120572119905120598 0 le 119904 lt 120572119905

(37)

where 120598 = 0 1 and 120572 isin (0 1) In particular at 120572 = 12 120598 = minus1

119870 (119905 119904) = sign(119904 minus 1199052) =

1 119904 gt 1199052

0 119904 = 1199052

minus1 119904 lt 1199052

(38)

For this case the solution to (35) with 119910(119905) = 119905 given in[23] is

119909 (119905) = ln 119905ln 2

+ 119909 (119904) (39)

For kernel (38)

119870 (0 0) = 0 119870 (119905 119905) = 0 119905 gt 0 (40)

If119870(119905 119904) is continuous in arguments and continuously differ-entiable with respect to 119905 inΔ then condition (40)means that(35) is Volterra integral equation of the third kind

The theory (whose foundation was laid by Volterra (see[24 pages 104ndash106])) of such equations is developed in theresearch done by Magnitsky [25ndash28]

In particular the author of [25ndash28] studies the structureof one- or many-parameter family of solutions to (35)

If 119870(119905 119904) is discontinuous then the solution to (35) maybe nonunique even if 119870(119905 119905) = 0 forall119905 ge 0

For example if120572 = 12 and 120598 = 1minus(1120572) = minus1 the solutionto equation

int119905

120572119905

119909 (119904) 119889119904 + 120598int120572119905

0

119909 (119904) 119889119904 = 119905 119905 isin [0 119879] (41)

is a one-parameter family

119909 (119905) = minus ln 119905ln120572

+ 119909 (1) (42)

but by (37) 119870(0 0) = (1 + 120598)2 = 0 119870(119905 119905) = 1 119905 gt 0

Now we show that there can be a nonunique solution to(35) and (36) even in the case 119870(119905 119905) equiv 1 Let

119870 (119905 119904) = 1 119904 ge 120572119905120598 119904 lt 120572119905

(43)

so that condition119870(119905 119905) equiv 1 is trueWe prove that solutions to (35) (37) and (35) (43)

coincide It suffices to show that the equivalent functionalequations for (35) (37) and (35) (43) coincide Recall that for(35) (37) the equivalent functional equation is (24)

Theorem 3 The equivalent functional equations for (35) (37)and (35) (43) coincide

Proof Let us represent (43) by

119870 (119905 119904) equiv 1 + (120598 minus 1) 119890 (120572119905 minus 119904) (44)

where 119890(sdot)ndash is a Heaviside function

119890 (]) = 1 ] ge 00 ] lt 0

(45)

Substitution of (44) in (35) gives

int119905

0

119909 (119904) 119889119904 + (120598 minus 1) int119905

0

119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905)

119905 isin [0 119879] (46)

Transform the second integral Let ] = 120572119905 minus 119904 Then

int119905

0

119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = int120572119905

(120572minus1)119905

119890 (]) 119909 (120572119905 minus ]) 119889]

= int120572119905

0

119909 (120572119905 minus ]) 119889]

(47)

By virtue of (47) differentiation of (46) results in

119909 (119905) + (120598 minus 1) 120572119909 (0) + (120598 minus 1) int120572119905

0

1199091015840119905(120572119905 minus ]) 119889] = 1199101015840 (119905)

(48)

But

1199091015840119905(120572119905 minus ]) = minus1205721199091015840] (120572119905 minus ]) (49)

By virtue of (49) we have

119909 (119905) + (120598 minus 1) 120572119909 (0) minus (120598 minus 1) 120572 [119909 (120572119905 minus ])|1205721199050] = 1199101015840 (119905)

(50)

from (48) whence finally

119909 (119905) + 120572 (120598 minus 1) 119909 (120572119905) = 1199101015840 (119905) (51)

and (51) coincides with (24)

Abstract and Applied Analysis 5

The solution to (35) (43) in the class of piecewisecontinuous functions with a jump on line 119904 = 120572119905 is interestingfrom the application perspective

It is easy to see that this solution is

119909 (119905 119904) =

1199101015840 (119904) 119904 ge 12057211990511205981199101015840 (119904) 119904 lt 120572119905

(52)

At last consider the concept of 120572-convolution Volterraintegral equations of convolution type

119870 (119905) lowast 119909 (119905) Δ= int119905

0

119870 (119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879]

(53)

are important for applicationExamples (38) and (44) show the usefulness of the 120572-

convolution concept

119870 (119905) 120572lowast 119909 (119905) Δ= int119905

0

119870 (120572119905 minus 119904) 119909 (119904) 119889119904

= 119910 (119905) 120572 isin (0 1] 119905 isin [0 119879] (54)

Give some inversion formulas of the integral equation

119870 (119905) 120572lowast 119909 (119905) = 119910 (119905) 119905 isin [0 119879] (55)

(1) If 119870(119905) = 120575(119905) 119910(119905) isin 119862[0119879]

and 120572 isin (0 1] then

119909 (120572119905) = 119910 (119905) (56)

(2) If 119870(119905) = 119890(119905) 119910(119905) isin∘

119862(1)

[0119879] and 120572 isin (0 1] then

119909 (120572119905) = 11205721199101015840 (119905) (57)

(3) If 119870(119905) = sign 119905 119910(119905) = 119905 and 120572 isin (0 1) then

119909 (119905) = ln 119905ln120572

+ 119909 (1) (58)

At 119870(119905) = 119905119899 119899 ge 1 (55) is Volterra integral equationof the third kind

(4) If 119870(119905) = 119905 119910(119905) = 11990522 and 120572 = 12 then

119909 (119905) = minus2 ln 119905 + 119909 (1) (59)

(5) If 119870(119905) = 119905 119910(119905) = 11990522 and 120572 isin (0 1) 120572 = 12 then

119909 (119905) = 119909 (1)119905(2120572minus1)(120572minus1)

(60)

4 Conclusion

As is mentioned in the introduction the main results of thisstudy can be easily applied to the case 119899 gt 2 in (1) Theequations of type (1) not only are of theoretical interest butalso play an important role in the mathematical modeling ofdeveloping dynamic systemsMoreover by119910(119905) we canmeansome criterion that characterizes the level of developmentof the system as a whole and the 119894th term in (1) representsa contribution of the system components 119909(119904) of the 119894thage group whose operation is reflected by the efficiencycoefficient 119870

119894(119905 minus 119904) As a rule 119870

1ge sdot sdot sdot ge 119870

119899ge 0

Such an approach is implemented for instance in [29 30]in the problem of the analysis of strategies for the long-termexpansion of the Russian electric power system with theconsideration of aging of the power plants equipment

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author wishes to thank the reviewers for their helpfulnotes The study is supported by the Russian Foundation forBasic Research Grant no 12-01-00722a

References

[1] V Volterra ldquoSopra alcune questioni di inversione di integralidefinitirdquo Annali di Matematica Pura ed Applicata Series 2 vol25 no 1 pp 139ndash178 1897

[2] H Brunner ldquo1896ndash1996 One hundred years of Volterra integralequations of the first kindrdquoAppliedNumericalMathematics vol24 no 2-3 pp 83ndash93 1997

[3] H Brunner and P J van der HouwenTheNumerical Solution ofVolterra Equations vol 3 of CWI Monographs North-HollandAmsterdam The Netherlands 1986

[4] H Brunner Collocation Methods for Volterra Integral andRelated Functional Differential Equations vol 15 of CambridgeMonographs on Applied and Computational Mathematics Cam-bridge University Press Cambridge Mass USA 2004

[5] A S Apartsyn Nonclassical Linear Volterra Equations of theFirst Kind VSP Utrecht The Netherlands 2003

[6] V M Glushkov ldquoOn one class of dynamic macroeconomicmodelsrdquo Upravlyayushchiye Sistemy I Mashiny no 2 pp 3ndash61977 (Russian)

[7] V M Glushkov V V Ivanov and V M Yanenko Modeling ofDeveloping Systems Nauka Moscow Russia 1983 (Russian)

[8] Y P Yatsenko Integral Models of Systems with ControlledMemory Naukova Dumka Kiev Ukraine 1991 (Russian)

[9] N Hritonenko and Y Yatsenko Applied Mathematical Mod-elling of Engineering Problems vol 81 of Applied OptimizationKluwer Academic Publishers Dordrecht The Netherlands2003

[10] A S Apartsyn E V Markova and V V Trufanov IntegralModels of Electric Power System Development Energy SystemsInstitute SB RAS Irkutsk Russia 2002 (Russian)

6 Abstract and Applied Analysis

[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004

[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)

[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008

[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)

[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993

[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004

[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007

[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007

[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008

[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008

[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)

[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013

[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)

[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)

[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)

[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)

[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)

[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012

[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Proof With obvious changes repeat the proof of Theorem 1

Let us illustrate the obtained results with the followingexample

Consider the equation

int119905

120572119905

119909 (119904) 119889119904 + 120598int120572119905

0

119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879] (20)

Here by (5)ndash(7) 119896 = 1 1198722= |120598| 119872

2= |1 minus 120598| 119871

1= 1198711=

1198712= 0 119886

1(119879) = 120572119879 119860

1= 120572 120574

119894= 120573119894= 120572 and 119878 = 1(1 minus 120572)

therefore based on (8) inequality

120572119879 |120598| + 120572 |120598| lt 1 minus 120572 (21)

and based on (17) inequality

120572119879 |1 minus 120598| + 120572 |1 minus 120598| lt 1 (22)

give the following estimates 120598 which guarantee the existenceuniqueness and stability of solution to (20) in the space119862

[0119879]

|120598| lt 1 minus 120572120572 (1 + 119879)

|1 minus 120598| lt 1120572 (1 + 119879)

(23)

It is useful to compare (23) with the estimate obtained byshifting from (20) to the equivalent functional equationDifferentiation of (20) gives

119909 (119905) = 120572 (1 minus 120598) 119909 (120572119905) + 1199101015840 (119905) (24)

whence

119909 (119905) = lim119899rarrinfin

[

[120572119899(1 minus 120598)119899119909 (120572119899119905) +

119899minus1

sum119895=0

120572119895(1 minus 120598)1198951199101015840 (120572119895119905)]

](25)

and condition

|1 minus 120598| lt 1120572

(26)

provides convergence of series (25) to continuous function119909(119905) on [0 119879]

If in (20)

120598 = 1 minus 1120572 (27)

then condition (26) is violated Then it is easy to see that thehomogeneous equation

int119905

120572119905

119909 (119904) 119889119904 + (1 minus 1120572)int120572119905

0

119909 (119904) 119889119904 = 0 (28)

has a nontrivial solution 119909(119905) = const and if for example119910(119905) = 119905 the solution to the nonhomogeneous equation

int119905

120572119905

119909 (119904) 119889119904 + (1 minus 1120572)int120572119905

0

119909 (119904) 119889119904 = 119905 119905 isin [0 119879] (29)

is a one-parameter family

119909 (119905) = minus ln 119905ln120572

+ 119909 (1) (30)

Let now

120598 = 1 + 1120572 (31)

Then according to (24)

119909 (119905) = minus119909 (120572119905) + 1199101015840 (119905) (32)

whence

119909 (119905) = lim119899rarrinfin

[

[(minus1)119899119909 (120572119899119905) +

119899minus1

sum119895=0

(minus1)1198951199101015840 (120572119895119905)]

](33)

so that for the right-hand side of (20) 119910(119905) = 119910(119905) = 119905119896119896119896 = 1 2 3 from (33) we obtain

119909 (119905) = 119905119896minus1

1 + 120572119896minus1 119896 = 1 2 (34)

In conclusion of this section it should be noted thatinequalities (8) and (17) can be interpreted as constraints onthe value 119879 which guarantee at given 119870

1(119905 119904) 119870

2(119905 119904) and

1198861(119905) the correct solvability of (3) in 119862

[0119879] Since all param-

eters in the left-hand side of (8) and (17) are nondecreasingfunctions of119879 and the right-hand side of (8) and (17) at 119871

1= 0

(1

= 0) on the contrary monotonously decreases then thereal positive root of corresponding nonlinear equation thatgives a guaranteed lower-bound estimate of 119879 exists and isunique if 1198861015840(0) is sufficiently small In some special casesthis root can be found analytically in terms of the Lambertfunction119882 [15 16]

In [17ndash22] the authors studied the characteristic of con-tinuous solution locality and the role of the Lambert functionas applied to the polynomial (multilinear) Volterra equationsof the first kind The calculations of the test examples showthat the locality feature of the solution to the linear equation(3) is not the result of the inaccuracy of estimates (8) and (17)and reflects the specifics of the considered class of problemsIn this paper we do not dwell on the problem of numericallysolving (3) It is of independent interest and deserves specialconsideration

3 The Volterra Integral Equations of the FirstKind with Discontinuous Kernels

Equation (2) can be written in the form of Volterra integralequation of the first kind

int119905

0

119870 (119905 119904) 119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879] (35)

4 Abstract and Applied Analysis

with discontinuous kernel

119870 (119905 119904)

=

1198701(119905 119904) 119886

1(119905) lt 119904 le 119905

119870119894(119905 119904) 119886

119894(119905) lt 119904 lt 119886

119894minus1(119905)

119894 = 2 119899 minus 1(119870119894(119905 119904) + 119870

119894+1(119905 119904))

2 119904 = 119886

119894(119905) 119894 = 1 119899 minus 1

119870119899(119905 119904) 0 le 119904 lt 119886

119899minus1(119905)

(36)

To illustrate the fundamental difference between (35)(36) and classical Volterra equation of the first kind withsmooth kernel we confine ourselves to (20) that has the formof (35) at

119870 (119905 119904) =

1 120572119905 lt 119904 le 1199051 + 1205982

119904 = 120572119905120598 0 le 119904 lt 120572119905

(37)

where 120598 = 0 1 and 120572 isin (0 1) In particular at 120572 = 12 120598 = minus1

119870 (119905 119904) = sign(119904 minus 1199052) =

1 119904 gt 1199052

0 119904 = 1199052

minus1 119904 lt 1199052

(38)

For this case the solution to (35) with 119910(119905) = 119905 given in[23] is

119909 (119905) = ln 119905ln 2

+ 119909 (119904) (39)

For kernel (38)

119870 (0 0) = 0 119870 (119905 119905) = 0 119905 gt 0 (40)

If119870(119905 119904) is continuous in arguments and continuously differ-entiable with respect to 119905 inΔ then condition (40)means that(35) is Volterra integral equation of the third kind

The theory (whose foundation was laid by Volterra (see[24 pages 104ndash106])) of such equations is developed in theresearch done by Magnitsky [25ndash28]

In particular the author of [25ndash28] studies the structureof one- or many-parameter family of solutions to (35)

If 119870(119905 119904) is discontinuous then the solution to (35) maybe nonunique even if 119870(119905 119905) = 0 forall119905 ge 0

For example if120572 = 12 and 120598 = 1minus(1120572) = minus1 the solutionto equation

int119905

120572119905

119909 (119904) 119889119904 + 120598int120572119905

0

119909 (119904) 119889119904 = 119905 119905 isin [0 119879] (41)

is a one-parameter family

119909 (119905) = minus ln 119905ln120572

+ 119909 (1) (42)

but by (37) 119870(0 0) = (1 + 120598)2 = 0 119870(119905 119905) = 1 119905 gt 0

Now we show that there can be a nonunique solution to(35) and (36) even in the case 119870(119905 119905) equiv 1 Let

119870 (119905 119904) = 1 119904 ge 120572119905120598 119904 lt 120572119905

(43)

so that condition119870(119905 119905) equiv 1 is trueWe prove that solutions to (35) (37) and (35) (43)

coincide It suffices to show that the equivalent functionalequations for (35) (37) and (35) (43) coincide Recall that for(35) (37) the equivalent functional equation is (24)

Theorem 3 The equivalent functional equations for (35) (37)and (35) (43) coincide

Proof Let us represent (43) by

119870 (119905 119904) equiv 1 + (120598 minus 1) 119890 (120572119905 minus 119904) (44)

where 119890(sdot)ndash is a Heaviside function

119890 (]) = 1 ] ge 00 ] lt 0

(45)

Substitution of (44) in (35) gives

int119905

0

119909 (119904) 119889119904 + (120598 minus 1) int119905

0

119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905)

119905 isin [0 119879] (46)

Transform the second integral Let ] = 120572119905 minus 119904 Then

int119905

0

119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = int120572119905

(120572minus1)119905

119890 (]) 119909 (120572119905 minus ]) 119889]

= int120572119905

0

119909 (120572119905 minus ]) 119889]

(47)

By virtue of (47) differentiation of (46) results in

119909 (119905) + (120598 minus 1) 120572119909 (0) + (120598 minus 1) int120572119905

0

1199091015840119905(120572119905 minus ]) 119889] = 1199101015840 (119905)

(48)

But

1199091015840119905(120572119905 minus ]) = minus1205721199091015840] (120572119905 minus ]) (49)

By virtue of (49) we have

119909 (119905) + (120598 minus 1) 120572119909 (0) minus (120598 minus 1) 120572 [119909 (120572119905 minus ])|1205721199050] = 1199101015840 (119905)

(50)

from (48) whence finally

119909 (119905) + 120572 (120598 minus 1) 119909 (120572119905) = 1199101015840 (119905) (51)

and (51) coincides with (24)

Abstract and Applied Analysis 5

The solution to (35) (43) in the class of piecewisecontinuous functions with a jump on line 119904 = 120572119905 is interestingfrom the application perspective

It is easy to see that this solution is

119909 (119905 119904) =

1199101015840 (119904) 119904 ge 12057211990511205981199101015840 (119904) 119904 lt 120572119905

(52)

At last consider the concept of 120572-convolution Volterraintegral equations of convolution type

119870 (119905) lowast 119909 (119905) Δ= int119905

0

119870 (119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879]

(53)

are important for applicationExamples (38) and (44) show the usefulness of the 120572-

convolution concept

119870 (119905) 120572lowast 119909 (119905) Δ= int119905

0

119870 (120572119905 minus 119904) 119909 (119904) 119889119904

= 119910 (119905) 120572 isin (0 1] 119905 isin [0 119879] (54)

Give some inversion formulas of the integral equation

119870 (119905) 120572lowast 119909 (119905) = 119910 (119905) 119905 isin [0 119879] (55)

(1) If 119870(119905) = 120575(119905) 119910(119905) isin 119862[0119879]

and 120572 isin (0 1] then

119909 (120572119905) = 119910 (119905) (56)

(2) If 119870(119905) = 119890(119905) 119910(119905) isin∘

119862(1)

[0119879] and 120572 isin (0 1] then

119909 (120572119905) = 11205721199101015840 (119905) (57)

(3) If 119870(119905) = sign 119905 119910(119905) = 119905 and 120572 isin (0 1) then

119909 (119905) = ln 119905ln120572

+ 119909 (1) (58)

At 119870(119905) = 119905119899 119899 ge 1 (55) is Volterra integral equationof the third kind

(4) If 119870(119905) = 119905 119910(119905) = 11990522 and 120572 = 12 then

119909 (119905) = minus2 ln 119905 + 119909 (1) (59)

(5) If 119870(119905) = 119905 119910(119905) = 11990522 and 120572 isin (0 1) 120572 = 12 then

119909 (119905) = 119909 (1)119905(2120572minus1)(120572minus1)

(60)

4 Conclusion

As is mentioned in the introduction the main results of thisstudy can be easily applied to the case 119899 gt 2 in (1) Theequations of type (1) not only are of theoretical interest butalso play an important role in the mathematical modeling ofdeveloping dynamic systemsMoreover by119910(119905) we canmeansome criterion that characterizes the level of developmentof the system as a whole and the 119894th term in (1) representsa contribution of the system components 119909(119904) of the 119894thage group whose operation is reflected by the efficiencycoefficient 119870

119894(119905 minus 119904) As a rule 119870

1ge sdot sdot sdot ge 119870

119899ge 0

Such an approach is implemented for instance in [29 30]in the problem of the analysis of strategies for the long-termexpansion of the Russian electric power system with theconsideration of aging of the power plants equipment

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author wishes to thank the reviewers for their helpfulnotes The study is supported by the Russian Foundation forBasic Research Grant no 12-01-00722a

References

[1] V Volterra ldquoSopra alcune questioni di inversione di integralidefinitirdquo Annali di Matematica Pura ed Applicata Series 2 vol25 no 1 pp 139ndash178 1897

[2] H Brunner ldquo1896ndash1996 One hundred years of Volterra integralequations of the first kindrdquoAppliedNumericalMathematics vol24 no 2-3 pp 83ndash93 1997

[3] H Brunner and P J van der HouwenTheNumerical Solution ofVolterra Equations vol 3 of CWI Monographs North-HollandAmsterdam The Netherlands 1986

[4] H Brunner Collocation Methods for Volterra Integral andRelated Functional Differential Equations vol 15 of CambridgeMonographs on Applied and Computational Mathematics Cam-bridge University Press Cambridge Mass USA 2004

[5] A S Apartsyn Nonclassical Linear Volterra Equations of theFirst Kind VSP Utrecht The Netherlands 2003

[6] V M Glushkov ldquoOn one class of dynamic macroeconomicmodelsrdquo Upravlyayushchiye Sistemy I Mashiny no 2 pp 3ndash61977 (Russian)

[7] V M Glushkov V V Ivanov and V M Yanenko Modeling ofDeveloping Systems Nauka Moscow Russia 1983 (Russian)

[8] Y P Yatsenko Integral Models of Systems with ControlledMemory Naukova Dumka Kiev Ukraine 1991 (Russian)

[9] N Hritonenko and Y Yatsenko Applied Mathematical Mod-elling of Engineering Problems vol 81 of Applied OptimizationKluwer Academic Publishers Dordrecht The Netherlands2003

[10] A S Apartsyn E V Markova and V V Trufanov IntegralModels of Electric Power System Development Energy SystemsInstitute SB RAS Irkutsk Russia 2002 (Russian)

6 Abstract and Applied Analysis

[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004

[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)

[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008

[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)

[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993

[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004

[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007

[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007

[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008

[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008

[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)

[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013

[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)

[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)

[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)

[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)

[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)

[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012

[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

with discontinuous kernel

119870 (119905 119904)

=

1198701(119905 119904) 119886

1(119905) lt 119904 le 119905

119870119894(119905 119904) 119886

119894(119905) lt 119904 lt 119886

119894minus1(119905)

119894 = 2 119899 minus 1(119870119894(119905 119904) + 119870

119894+1(119905 119904))

2 119904 = 119886

119894(119905) 119894 = 1 119899 minus 1

119870119899(119905 119904) 0 le 119904 lt 119886

119899minus1(119905)

(36)

To illustrate the fundamental difference between (35)(36) and classical Volterra equation of the first kind withsmooth kernel we confine ourselves to (20) that has the formof (35) at

119870 (119905 119904) =

1 120572119905 lt 119904 le 1199051 + 1205982

119904 = 120572119905120598 0 le 119904 lt 120572119905

(37)

where 120598 = 0 1 and 120572 isin (0 1) In particular at 120572 = 12 120598 = minus1

119870 (119905 119904) = sign(119904 minus 1199052) =

1 119904 gt 1199052

0 119904 = 1199052

minus1 119904 lt 1199052

(38)

For this case the solution to (35) with 119910(119905) = 119905 given in[23] is

119909 (119905) = ln 119905ln 2

+ 119909 (119904) (39)

For kernel (38)

119870 (0 0) = 0 119870 (119905 119905) = 0 119905 gt 0 (40)

If119870(119905 119904) is continuous in arguments and continuously differ-entiable with respect to 119905 inΔ then condition (40)means that(35) is Volterra integral equation of the third kind

The theory (whose foundation was laid by Volterra (see[24 pages 104ndash106])) of such equations is developed in theresearch done by Magnitsky [25ndash28]

In particular the author of [25ndash28] studies the structureof one- or many-parameter family of solutions to (35)

If 119870(119905 119904) is discontinuous then the solution to (35) maybe nonunique even if 119870(119905 119905) = 0 forall119905 ge 0

For example if120572 = 12 and 120598 = 1minus(1120572) = minus1 the solutionto equation

int119905

120572119905

119909 (119904) 119889119904 + 120598int120572119905

0

119909 (119904) 119889119904 = 119905 119905 isin [0 119879] (41)

is a one-parameter family

119909 (119905) = minus ln 119905ln120572

+ 119909 (1) (42)

but by (37) 119870(0 0) = (1 + 120598)2 = 0 119870(119905 119905) = 1 119905 gt 0

Now we show that there can be a nonunique solution to(35) and (36) even in the case 119870(119905 119905) equiv 1 Let

119870 (119905 119904) = 1 119904 ge 120572119905120598 119904 lt 120572119905

(43)

so that condition119870(119905 119905) equiv 1 is trueWe prove that solutions to (35) (37) and (35) (43)

coincide It suffices to show that the equivalent functionalequations for (35) (37) and (35) (43) coincide Recall that for(35) (37) the equivalent functional equation is (24)

Theorem 3 The equivalent functional equations for (35) (37)and (35) (43) coincide

Proof Let us represent (43) by

119870 (119905 119904) equiv 1 + (120598 minus 1) 119890 (120572119905 minus 119904) (44)

where 119890(sdot)ndash is a Heaviside function

119890 (]) = 1 ] ge 00 ] lt 0

(45)

Substitution of (44) in (35) gives

int119905

0

119909 (119904) 119889119904 + (120598 minus 1) int119905

0

119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905)

119905 isin [0 119879] (46)

Transform the second integral Let ] = 120572119905 minus 119904 Then

int119905

0

119890 (120572119905 minus 119904) 119909 (119904) 119889119904 = int120572119905

(120572minus1)119905

119890 (]) 119909 (120572119905 minus ]) 119889]

= int120572119905

0

119909 (120572119905 minus ]) 119889]

(47)

By virtue of (47) differentiation of (46) results in

119909 (119905) + (120598 minus 1) 120572119909 (0) + (120598 minus 1) int120572119905

0

1199091015840119905(120572119905 minus ]) 119889] = 1199101015840 (119905)

(48)

But

1199091015840119905(120572119905 minus ]) = minus1205721199091015840] (120572119905 minus ]) (49)

By virtue of (49) we have

119909 (119905) + (120598 minus 1) 120572119909 (0) minus (120598 minus 1) 120572 [119909 (120572119905 minus ])|1205721199050] = 1199101015840 (119905)

(50)

from (48) whence finally

119909 (119905) + 120572 (120598 minus 1) 119909 (120572119905) = 1199101015840 (119905) (51)

and (51) coincides with (24)

Abstract and Applied Analysis 5

The solution to (35) (43) in the class of piecewisecontinuous functions with a jump on line 119904 = 120572119905 is interestingfrom the application perspective

It is easy to see that this solution is

119909 (119905 119904) =

1199101015840 (119904) 119904 ge 12057211990511205981199101015840 (119904) 119904 lt 120572119905

(52)

At last consider the concept of 120572-convolution Volterraintegral equations of convolution type

119870 (119905) lowast 119909 (119905) Δ= int119905

0

119870 (119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879]

(53)

are important for applicationExamples (38) and (44) show the usefulness of the 120572-

convolution concept

119870 (119905) 120572lowast 119909 (119905) Δ= int119905

0

119870 (120572119905 minus 119904) 119909 (119904) 119889119904

= 119910 (119905) 120572 isin (0 1] 119905 isin [0 119879] (54)

Give some inversion formulas of the integral equation

119870 (119905) 120572lowast 119909 (119905) = 119910 (119905) 119905 isin [0 119879] (55)

(1) If 119870(119905) = 120575(119905) 119910(119905) isin 119862[0119879]

and 120572 isin (0 1] then

119909 (120572119905) = 119910 (119905) (56)

(2) If 119870(119905) = 119890(119905) 119910(119905) isin∘

119862(1)

[0119879] and 120572 isin (0 1] then

119909 (120572119905) = 11205721199101015840 (119905) (57)

(3) If 119870(119905) = sign 119905 119910(119905) = 119905 and 120572 isin (0 1) then

119909 (119905) = ln 119905ln120572

+ 119909 (1) (58)

At 119870(119905) = 119905119899 119899 ge 1 (55) is Volterra integral equationof the third kind

(4) If 119870(119905) = 119905 119910(119905) = 11990522 and 120572 = 12 then

119909 (119905) = minus2 ln 119905 + 119909 (1) (59)

(5) If 119870(119905) = 119905 119910(119905) = 11990522 and 120572 isin (0 1) 120572 = 12 then

119909 (119905) = 119909 (1)119905(2120572minus1)(120572minus1)

(60)

4 Conclusion

As is mentioned in the introduction the main results of thisstudy can be easily applied to the case 119899 gt 2 in (1) Theequations of type (1) not only are of theoretical interest butalso play an important role in the mathematical modeling ofdeveloping dynamic systemsMoreover by119910(119905) we canmeansome criterion that characterizes the level of developmentof the system as a whole and the 119894th term in (1) representsa contribution of the system components 119909(119904) of the 119894thage group whose operation is reflected by the efficiencycoefficient 119870

119894(119905 minus 119904) As a rule 119870

1ge sdot sdot sdot ge 119870

119899ge 0

Such an approach is implemented for instance in [29 30]in the problem of the analysis of strategies for the long-termexpansion of the Russian electric power system with theconsideration of aging of the power plants equipment

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author wishes to thank the reviewers for their helpfulnotes The study is supported by the Russian Foundation forBasic Research Grant no 12-01-00722a

References

[1] V Volterra ldquoSopra alcune questioni di inversione di integralidefinitirdquo Annali di Matematica Pura ed Applicata Series 2 vol25 no 1 pp 139ndash178 1897

[2] H Brunner ldquo1896ndash1996 One hundred years of Volterra integralequations of the first kindrdquoAppliedNumericalMathematics vol24 no 2-3 pp 83ndash93 1997

[3] H Brunner and P J van der HouwenTheNumerical Solution ofVolterra Equations vol 3 of CWI Monographs North-HollandAmsterdam The Netherlands 1986

[4] H Brunner Collocation Methods for Volterra Integral andRelated Functional Differential Equations vol 15 of CambridgeMonographs on Applied and Computational Mathematics Cam-bridge University Press Cambridge Mass USA 2004

[5] A S Apartsyn Nonclassical Linear Volterra Equations of theFirst Kind VSP Utrecht The Netherlands 2003

[6] V M Glushkov ldquoOn one class of dynamic macroeconomicmodelsrdquo Upravlyayushchiye Sistemy I Mashiny no 2 pp 3ndash61977 (Russian)

[7] V M Glushkov V V Ivanov and V M Yanenko Modeling ofDeveloping Systems Nauka Moscow Russia 1983 (Russian)

[8] Y P Yatsenko Integral Models of Systems with ControlledMemory Naukova Dumka Kiev Ukraine 1991 (Russian)

[9] N Hritonenko and Y Yatsenko Applied Mathematical Mod-elling of Engineering Problems vol 81 of Applied OptimizationKluwer Academic Publishers Dordrecht The Netherlands2003

[10] A S Apartsyn E V Markova and V V Trufanov IntegralModels of Electric Power System Development Energy SystemsInstitute SB RAS Irkutsk Russia 2002 (Russian)

6 Abstract and Applied Analysis

[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004

[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)

[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008

[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)

[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993

[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004

[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007

[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007

[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008

[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008

[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)

[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013

[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)

[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)

[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)

[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)

[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)

[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012

[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

The solution to (35) (43) in the class of piecewisecontinuous functions with a jump on line 119904 = 120572119905 is interestingfrom the application perspective

It is easy to see that this solution is

119909 (119905 119904) =

1199101015840 (119904) 119904 ge 12057211990511205981199101015840 (119904) 119904 lt 120572119905

(52)

At last consider the concept of 120572-convolution Volterraintegral equations of convolution type

119870 (119905) lowast 119909 (119905) Δ= int119905

0

119870 (119905 minus 119904) 119909 (119904) 119889119904 = 119910 (119905) 119905 isin [0 119879]

(53)

are important for applicationExamples (38) and (44) show the usefulness of the 120572-

convolution concept

119870 (119905) 120572lowast 119909 (119905) Δ= int119905

0

119870 (120572119905 minus 119904) 119909 (119904) 119889119904

= 119910 (119905) 120572 isin (0 1] 119905 isin [0 119879] (54)

Give some inversion formulas of the integral equation

119870 (119905) 120572lowast 119909 (119905) = 119910 (119905) 119905 isin [0 119879] (55)

(1) If 119870(119905) = 120575(119905) 119910(119905) isin 119862[0119879]

and 120572 isin (0 1] then

119909 (120572119905) = 119910 (119905) (56)

(2) If 119870(119905) = 119890(119905) 119910(119905) isin∘

119862(1)

[0119879] and 120572 isin (0 1] then

119909 (120572119905) = 11205721199101015840 (119905) (57)

(3) If 119870(119905) = sign 119905 119910(119905) = 119905 and 120572 isin (0 1) then

119909 (119905) = ln 119905ln120572

+ 119909 (1) (58)

At 119870(119905) = 119905119899 119899 ge 1 (55) is Volterra integral equationof the third kind

(4) If 119870(119905) = 119905 119910(119905) = 11990522 and 120572 = 12 then

119909 (119905) = minus2 ln 119905 + 119909 (1) (59)

(5) If 119870(119905) = 119905 119910(119905) = 11990522 and 120572 isin (0 1) 120572 = 12 then

119909 (119905) = 119909 (1)119905(2120572minus1)(120572minus1)

(60)

4 Conclusion

As is mentioned in the introduction the main results of thisstudy can be easily applied to the case 119899 gt 2 in (1) Theequations of type (1) not only are of theoretical interest butalso play an important role in the mathematical modeling ofdeveloping dynamic systemsMoreover by119910(119905) we canmeansome criterion that characterizes the level of developmentof the system as a whole and the 119894th term in (1) representsa contribution of the system components 119909(119904) of the 119894thage group whose operation is reflected by the efficiencycoefficient 119870

119894(119905 minus 119904) As a rule 119870

1ge sdot sdot sdot ge 119870

119899ge 0

Such an approach is implemented for instance in [29 30]in the problem of the analysis of strategies for the long-termexpansion of the Russian electric power system with theconsideration of aging of the power plants equipment

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author wishes to thank the reviewers for their helpfulnotes The study is supported by the Russian Foundation forBasic Research Grant no 12-01-00722a

References

[1] V Volterra ldquoSopra alcune questioni di inversione di integralidefinitirdquo Annali di Matematica Pura ed Applicata Series 2 vol25 no 1 pp 139ndash178 1897

[2] H Brunner ldquo1896ndash1996 One hundred years of Volterra integralequations of the first kindrdquoAppliedNumericalMathematics vol24 no 2-3 pp 83ndash93 1997

[3] H Brunner and P J van der HouwenTheNumerical Solution ofVolterra Equations vol 3 of CWI Monographs North-HollandAmsterdam The Netherlands 1986

[4] H Brunner Collocation Methods for Volterra Integral andRelated Functional Differential Equations vol 15 of CambridgeMonographs on Applied and Computational Mathematics Cam-bridge University Press Cambridge Mass USA 2004

[5] A S Apartsyn Nonclassical Linear Volterra Equations of theFirst Kind VSP Utrecht The Netherlands 2003

[6] V M Glushkov ldquoOn one class of dynamic macroeconomicmodelsrdquo Upravlyayushchiye Sistemy I Mashiny no 2 pp 3ndash61977 (Russian)

[7] V M Glushkov V V Ivanov and V M Yanenko Modeling ofDeveloping Systems Nauka Moscow Russia 1983 (Russian)

[8] Y P Yatsenko Integral Models of Systems with ControlledMemory Naukova Dumka Kiev Ukraine 1991 (Russian)

[9] N Hritonenko and Y Yatsenko Applied Mathematical Mod-elling of Engineering Problems vol 81 of Applied OptimizationKluwer Academic Publishers Dordrecht The Netherlands2003

[10] A S Apartsyn E V Markova and V V Trufanov IntegralModels of Electric Power System Development Energy SystemsInstitute SB RAS Irkutsk Russia 2002 (Russian)

6 Abstract and Applied Analysis

[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004

[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)

[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008

[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)

[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993

[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004

[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007

[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007

[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008

[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008

[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)

[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013

[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)

[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)

[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)

[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)

[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)

[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012

[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

[11] D V Ivanov V Karaulova E VMarkova V V Trufanov andOV Khamisov ldquoControl of power grid development numericalsolutionsrdquo Automation and Remote Control vol 65 no 3 pp472ndash482 2004

[12] A S Apartsyn I V Karaulova E V Markova and V VTrufanov ldquoApplication of the Volterra integral equations for themodeling of strategies of technical re-equipment in the electricpower industryrdquo Electrical Technology Russia no 10 pp 64ndash752005 (Russian)

[13] E Messina E Russo and A Vecchio ldquoA stable numericalmethod for Volterra integral equations with discontinuouskernelrdquo Journal of Mathematical Analysis and Applications vol337 no 2 pp 1383ndash1393 2008

[14] L V Kantorovich and G P Akilov Functional Analysis NaukaMoscow Russia 1977 (Russian)

[15] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert 119882 functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

[16] R M Corless G H Gonnet D E G Hare and D JJeffrey ldquoLambertrsquos 119882 function in Maplerdquo The Maple TechnicalNewsletter no 9 pp 12ndash22 1993

[17] A S Apartsyn ldquoMultilinear Volterra equations of the first kindrdquoAutomation and Remote Control vol 65 no 2 pp 263ndash2692004

[18] A S Apartsyn ldquoPolilinear integral Volterra equations of thefirst kind the elements of the theory and numeric meth-odsrdquo Izvestiya Irkutskogo Gosudarstvennogo Universiteta SeriesMathematics no 1 pp 13ndash41 2007

[19] A S Apartsin ldquoOn the convergence of numerical methodsfor solving a Volterra bilinear equations of the first kindrdquoComputational Mathematics and Mathematical Physics vol 47no 8 pp 1323ndash1331 2007

[20] A S Apartsin ldquoMultilinear Volterra equations of the firstkind and some problems of controlrdquo Automation and RemoteControl vol 69 no 4 pp 545ndash558 2008

[21] A S Apartsyn ldquoUnimprovable estimates of solutions for someclasses of integral inequalitiesrdquo Journal of Inverse and Ill-PosedProblems vol 16 no 7 pp 651ndash680 2008

[22] A S Apartsyn ldquoPolynomial Volterra integral equations of thefirst kind and the Lambert functionrdquo Proceedings of the Instituteof Mathematics and Mechanics Ural Branch of RAS vol 18 no1 pp 69ndash81 2012 (Russian)

[23] D N Sidorov ldquoOn parametric families of solutions of Volterraintegral equations of the first kind with piecewise smoothkernelrdquo Differential Equations vol 49 no 2 pp 210ndash216 2013

[24] V Volterra Theory of Functionals and of Integral and Integro-Differential Equations Nauka Moscow Russia 1982 (Russian)

[25] N A Magnitsky ldquoThe existence of multiparameter familiesof solutions of a Volterra integral equation of the first kindrdquoReports of theUSSRAcademy of Sciences vol 235 no 4 pp 772ndash774 1977 (Russian)

[26] N A Magnitsky ldquoLinear Volterra integral equations of the firstand third kindsrdquo Computational Mathematics and Mathemati-cal Physics vol 19 no 4 pp 970ndash988 1979 (Russian)

[27] N A Magnitsky ldquoThe asymptotics of solutions to the Volterraintegral equation of the first kindrdquoReports of the USSRAcademyof Sciences vol 269 no 1 pp 29ndash32 1983 (Russian)

[28] N A Magnitsky Asymptotic Methods for Analysis of Non-Stationary Controlled Systems Nauka Moscow Russia 1992(Russian)

[29] A S Apartsyn ldquoOn one approach to modeling of developingsystemsrdquo in Proceedings of the 6th International WorkshopldquoGeneralized Statments and Solutions of Control Problemsrdquo pp32ndash35 Divnomorskoe Russia 2012

[30] A S Apartsin and I V Sidler ldquoUsing the nonclassical Volterraequations of the first kind to model the developing systemsrdquoAutomation and Remote Control vol 74 no 6 pp 899ndash9102013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of