editorial qualitative analysis of differential...

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2013, Article ID 598956, 2 pageshttp://dx.doi.org/10.1155/2013/598956

EditorialQualitative Analysis of Differential Equations

Ondlej Došlý,1 Jaroslav Jaroš,2 Mervan PašiT,3 and Norio Yoshida4

1 Department of Mathematics and Statistics, Masaryk University, 611 37 Brno, Czech Republic2 Department of Mathematical Analysis and Numerical Mathematics, Comenius University, 842 48 Bratislava, Slovakia3 Department of Mathematics, University of Zagreb, 10000 Zagreb, Croatia4Department of Mathematics, University of Toyama, Toyama 930-8555, Japan

Correspondence should be addressed to Norio Yoshida; [email protected]

Received 1 August 2013; Accepted 1 August 2013

Copyright © 2013 Ondřej Došlý et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It is our pleasure to provide this special issue on qualitativeanalysis of differential equations in International Journal ofDifferential Equations.

Qualitative analysis has proved to be an important anduseful tool to investigate the properties of solutions of dif-ferential equations, because it enables to analyze differentialequations without solving analytically and numerically. Thestudy of qualitative properties of differential equations hasa long history, and qualitative theories have been developedfor various equations such as ordinary differential equa-tions, functional differential equations, abstract differentialequations, dynamical systems, integral equations, differenceequations, partial differential equations, functional partialdifferential equations and so forth. Since the qualitativeanalysis of differential equations is related to both pure andapplied mathematics, its applications to various fields suchas science, engineering, and ecology have been extensivelydeveloped.

This special issue contains papers which treat a numberof important and attractive problems related to existenceof positive periodic solutions for periodic neutral Lotka-Volterra systemwith distributed delays and impulses, dynam-ics of a Gross-Pitaevskii equation with phenomenologicaldamping, characterization for rectifiable and nonrectifiableattractivity of nonautonomous systems of linear differentialequations, positive periodic solutions of cooperative systemswith delays and feedback controls, oscillations of a classof forced second-order differential equations with possiblediscontinuous coefficients, analysis of mixed elliptic andparabolic boundary layers with corners, unboundedness of

solutions of Timoshenko beam equations with damping andforcing terms, and fractal oscillations of chirp functions andapplications to second-order linear differential equations.

The paper by Z. Luo and L. Luo discusses the existenceof positive periodic solutions for a class of impulsive neutralLotka-Volterra system with distributed delays by using afixed point theorem of strict-set contraction. Some criteriathat guarantee the existence of at least one positive periodicsolution of the system are established.

The research article of R. Colucci et al. deals withthe dynamical behavior of solutions of an 𝑛-dimensionalnonlinear Schrödinger equation with potential and linearderivative terms under the presence of phenomenologicaldamping. The authors of that paper obtain conditions for theexistence of a compact global attractor and find bounds forits Hausdorff and fractal dimensions.

In the paper by Y. Naito and M. Pašić, attractivity of allsolutions near zero as time variable goes to zero of a classof nonautonomous linear differential system is studied by akind of singular behaviour of eigenvalues of a correspondingmatrix system.Moreover, a necessary and sufficient conditionfor a new notion, the so-called nonrectifiable attractivesolutions (solution’s curves of infinite length), is derived byan additional singular condition on eigenvalues of the matrixsystem. These results are based on some new lemmas on therectification of planar curves near a point.

T. Niyaz and A. Muhammadhaji study a class of periodic𝑛 species cooperative Lotka-Volterra systemswith continuoustime delays and feedback controls. Based on the continuationtheorem of the coincidence degree theory developed by

2 International Journal of Differential Equations

Gaines and Mawhin, some new sufficient conditions on theexistence of positive periodic solutions are established.

The paper authored by S. Miličić et al. enlarges somerecently published results on the classic oscillation of second-order forced differential equations to some related equationsthat allow discontinues coefficients. In such a class of differ-ential equations, instead of the classic 𝐶2-solutions, the firstderivatives of solutions are absolutely continuous functions.It causes some difficulties in using the Riccati transformation.It is resolved by a new pointwise comparison principlefor ordinary differential inequalities with locally Lipschitznonlinear term and bounded measurable coefficients.

The work of G.-M. Gie et al. addresses the asymptoticbehavior at small diffusivity of solutions to a convection-diffusion equation in a rectangular domain. The validity oftheir proposed asymptotic expansions in suitable Sobolevspaces is proved via the energy estimates rather than themaximum principle.

In the paper by K. Kobayashi and N. Yoshida, unbound-edness results for Timoshenko beam equations with dampingand forcing terms are presented. Three kinds of end condi-tions are considered, and it is shown that themagnitude of thedisplacement of the beam grows up to infinity as time passesunder some assumptions on the forcing term.

The paper of M. Pašić and S. Tanaka deals with a newlemma for calculating the fractal (box-counting) dimensionof graph of any real continuous function. It is illustrated to ageneral form of chirp functions which oscillates near a finitepoint. As a consequence, fractal oscillations near a point arederived for some classes of second-order linear differentialequations generated by chirp functions.

Acknowledgment

The guest editors of this special issue would like to expresstheir sincere gratitude to all authors for sending their newpapers for publication. Thanks are also due to the reviewers,whose professional comments and valuable suggestions guar-anteed the high quality of these selected papers.We hope thatthis special issue will be useful and fruitful for researchersworking in qualitative analysis of differential equations andrelated areas and will help stimulate further progress in thisimportant branch of differential equations.

Ondřej DošlýJaroslav JarošMervan PašićNorio Yoshida

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