Recent Progress in Differential and Difference...
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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2011, Article ID 971371, 3 pages doi:10.1155/2011/971371 Editorial Recent Progress in Differential and Difference Equations J. Dibl´ ık, 1, 2 E. Braverman, 3 Yu. Rogovchenko, 4 and M. R ˚ uˇ ziˇ ckov ´ a 5 1 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Veveˇ r´ ı 331/95, 602 00 Brno, Czech Republic 2 Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick´ a 10, 616 00 Brno, Czech Republic 3 Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N1N4 4 Department of Mathematics and Mathematical Statistics, Ume˚ a University, SE-901 87 Ume˚ a, Sweden 5 Department of Mathematics, ˇ Zilina University, Universitn´ a 8215/1, 01026 ˇ Zilina, Slovakia Correspondence should be addressed to J. Dibl´ ık, [email protected] Received 4 September 2011; Accepted 4 September 2011 Copyright q 2011 J. Dibl´ ık 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. This special issue on Recent Progress in Differential and Difference Equations contains forty- four research articles. Most papers originate from the talks at the Conference on Differential and Difference Equations and Applications where all Guest Editors served as organizers held in Rajeck´ e Teplice, Slovak Republic, during June 21–25, 2010 http://fpv.uniza.sk/ cddea2010/page.php?id19&actionshow. At the conference, more than 50 contributed papers and posters were presented along with eighteen invited lectures delivered by leading researchers such as Professors A. A. Boichuk Slovak Republic, T. A. Burton USA, V. Covachev Oman, O. Doˇ sl´ y Czech Republic, J. Dˇ zurina Slovak Republic, J. Jaroˇ s Slovak Republic, D. Khusainov Ukraine, W. Kratz Germany, N. Partsvania Georgia, I. Rach ˚ unkov´ a Czech Republic, V. Rˇ asvan Romania, M. Ront ´ o Hungary, S. Stanˇ ek Czech Republic, M. Tvrd´ y Czech Republic, and F. Sadyrbaev Latvia. In addition to the papers discussed at the conference, many articles have been written for publication in this issue. As a result, seventy-seven authors from sixteen countries contributed to the success of this thematic collection of papers. The issue covers a wide variety of problems for different classes of ordinary, functional, impulsive, stochastic, fractional, partial differential equations, as well as difference and integrodifferential equations, inclusions, and dynamic equations on time scales. The topics
Transcript of Recent Progress in Differential and Difference...
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2011, Article ID 971371, 3 pagesdoi:10.1155/2011/971371
EditorialRecent Progress in Differentialand Difference Equations
J. Diblık,1, 2 E. Braverman,3 Yu. Rogovchenko,4 and M. Ruzickova5
1 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering,Brno University of Technology, Veverı 331/95, 602 00 Brno, Czech Republic
2 Department of Mathematics, Faculty of Electrical Engineering and Communication,Brno University of Technology, Technicka 10, 616 00 Brno, Czech Republic
3 Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W.,Calgary, Alberta, Canada T2N1N4
4 Department of Mathematics and Mathematical Statistics, Umea University,SE-901 87 Umea, Sweden
5 Department of Mathematics, Zilina University, Universitna 8215/1,01026 Zilina, Slovakia
Correspondence should be addressed to J. Diblık, [email protected]
Received 4 September 2011; Accepted 4 September 2011
Copyright q 2011 J. Diblık et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
This special issue on Recent Progress in Differential and Difference Equations contains forty-four research articles. Most papers originate from the talks at the Conference on Differentialand Difference Equations and Applications (where all Guest Editors served as organizers)held in Rajecke Teplice, Slovak Republic, during June 21–25, 2010 (http://fpv.uniza.sk/cddea2010/page.php?id=19&action=show). At the conference, more than 50 contributedpapers and posters were presented along with eighteen invited lectures delivered by leadingresearchers such as Professors A. A. Boichuk (Slovak Republic), T. A. Burton (USA),V. Covachev (Oman), O. Dosly (Czech Republic), J. Dzurina (Slovak Republic), J. Jaros(Slovak Republic), D. Khusainov (Ukraine), W. Kratz (Germany), N. Partsvania (Georgia), I.Rachunkova (Czech Republic), V. Rasvan (Romania), M. Ronto (Hungary), S. Stanek (CzechRepublic), M. Tvrdy (Czech Republic), and F. Sadyrbaev (Latvia). In addition to the papersdiscussed at the conference, many articles have been written for publication in this issue.As a result, seventy-seven authors from sixteen countries contributed to the success of thisthematic collection of papers.
The issue covers awide variety of problems for different classes of ordinary, functional,impulsive, stochastic, fractional, partial differential equations, as well as difference andintegrodifferential equations, inclusions, and dynamic equations on time scales. The topics
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discussed in the contributed papers are traditional for qualitative theory of differential,functional differential, difference, and other classes of equations. The issue contains paperson asymptotic behavior of positive solutions of functional differential equations of delayedtype, solutions to third-order trinomial delay differential equations, half-linear q-differenceequations, solutions to discrete equations with two delays in the critical case, solutionsto delay difference and integrodifferential equations, as well as research on weightedasymptotically periodic solutions to linear Volterra difference equations. Furthermore,asymptotic properties of third-order nonlinear functional differential equations with mixedarguments and variational equations are investigated.
Traditionally, stability problems receive a great deal of attention at various conferences.Papers included in this issue address stability of linear differential equations with severaldelays, stability of linear delay differential equations under Perron’s condition, exponentialstability of solutions to stochastic control systems with delay, and instability of the trivialsolution of autonomous differential systems with quadratic right-hand sides in a cone.
As usual, many papers deal with oscillation and nonoscillation of various classesof equations. In particular, a number of papers are concerned with oscillation of second-order neutral delay dynamic equations of Emden-Fowler type, second-order neutralfunctional differential equations with mixed nonlinearities and of mixed type, second-ordersuperlinear neutral differential equations, singular nonlinear differential equations, second-order sublinear impulsive differential equations, and half-linear differential equations. Inaddition, nonoscillation of advanced differential equations with several terms, second-orderdynamic equations with several delays, and first-order neutral differential equations arestudied.
Several authors deal with different aspects of the theory of boundary value problemsfor nonlinear fractional differential equations, q-difference inclusions, and weakly nonlineardelay differential systems. Interesting results are obtained for a class of fourth-orderboundary value problems, singular boundary value problems for nonlinear fractional dif-ferential equations, nonseparated three-point boundary value problems for linear functionaldifferential equations, and periodic problems for difference equations.
Papers collected in this special issue are also concerned with a maximal numberof period annuli, Lie groups in infinite dimension, Weyl-Titchmarsh theory for time-scalesymplectic systems on a half line, compatible and incompatible nonuniqueness conditionsfor the classical Cauchy problem, optimization of solutions to dynamic systems with randomstructure, application of discrete Mittag-Leffler functions in linear fractional differenceequations, conjugacy of self-adjoint even order difference equations, H∞ estimates forLipschitz nonlinear discrete-time systems with delay, and reducibility of quasiperiodicHamiltonian systems with a small perturbation. Existence of invariant sets for impulsivedifferential equations with particularities in ω-limit set and existence of pseudosymmetricsolutions to p-Laplacian differential equations involving derivative are explored.
Finally, some applied problems are also considered—a two-species cooperativeLotka-Volterra system of degenerate parabolic equations, equations of Emden-Fowler type,and oscillatory periodic solutions for two differential-difference equations that model phase-locked loop control of high-frequency generators and nonlinear growth of a fluctuatingpopulation.
Although it is not possible to adequately represent in this special issue all directions ofcurrent research in ordinary, functional, partial, impulsive, dynamic, stochastic differentialequations, difference, and integrodifferential equations, we believe that it reflects many
Abstract and Applied Analysis 3
important recent trends in research, indicates current challenging problems, and outlines newideas for future studies in the field.
J. DiblıkE. Braverman
Yu. RogovchenkoM. Ruzickova
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