WCNA'2004

Asymptotic behavior of solutions of differential

and functional differential equations

session survey

(USA, Orlando, July, 2004)

Yu.V.Rogovchenko

Eastern Mediterranean University, Famagusta, North Cyprus

yuri.rogovchenko@emu.edu.tr

The Session "Asymptotic Behavior of Solutions of Differential and Functional Differential Equations" has been organized by Dr. Yuri V.Rogovchenko as a part of the scientific program at the Fourth World Congress of Nonlinear Analysts (WCNA-2004) held in Orlando, Florida, USA during June 30 - July 7, 2004. The topics planned for the discussion in the Session covered asymptotic properties of solutions of various classes of linear and nonlinear differential, difference, and functional differential equations including oscillatory and non-oscillatory solutions, solutions with prescribed asymptotics, positive, bounded, blowing-up, stable, periodic solutions, etc.

There were thirteen talks planned for this Session, including a one-hour invited address by Dr. Seenith Sivasundaram of the Embry-Riddle Aeronautical University. Unfortunately, three speakers could not attend the Congress and have been forced to cancel their talks in the Session.

The list of speakers who attended the Session and the titles of the talks follow.

-         Jaromír Baštinec, Josef Diblík (speaker). Asymptotic behavior of solutions of discrete equations (30 min);

-         Harry Gingold. Developments in asymptotic integration (45 min);

-         Armands Gritsans, Yurii Klokov, Felix Sadyrbaev (speaker). Asymptotic behavior of solutions to the Emden-Fowler type equations (30 min);

-         Nickolai Kosmatov. Fixed point methods and three-point boundary-value problems (45 min)

-         Bikkar S. Lalli. Oscillation of higher order difference equations (45 min);

-         Jan Melin. Some examples describing the need of modifying the Bendixson criterion for piecewise C¹-systems (45 min);

-         Octavian G.Mustafa, Yuri V.Rogovchenko (speaker). Global existence and asymptotic behavior of solutions of second-order equations (45 min);

-         Christine Nowak. Nonuniqueness points of ordinary differential equations (45 min);

-         Youssef N. Raffoul. Exponential stability in nonlinear functionaldifferential equations with applications to Volterra integro-differential equations (30 min);

-         Patricia J.Y.Wong. Constant-sign solutions for a system of third order generalized right focal problems (45 min).

In what follows, we briefly present the talks given in the Session using the materials kindly provided by the speakers.

In the talk entitled "Asymptotic behavior of solutions of discrete equations", Dr. Josef Diblík (Brno University of Technology, Czech Republic) presented a brief survey of results on the asymptotic behavior of solutions of discrete equations obtained recently by the author and his collaborators by using the retract idea. A key result is of a high degree of generality and provides conditions which guarantee that the graph of at least one solution of a given system of discrete equations remains indefinitely in the prescribed domain. As an application, sufficient conditions for the existence of δ-bounded solutions were provided. A version of the aforementioned general principle for discrete delayed equations was given, and its application to the problem of existence of positive solutions of delayed discrete equations has been considered. Finally, new sufficient conditions for positivity of solutions were presented.

The talk "Developments in asymptotic integration" given by Dr. Harry Gingold (West Virginia University, Morgantown, USA) was concerned with two groups of problems related naturally to the theory of singular perturbations. The first one deals with the reoccurrence of exponentially small terms in mathematical physics, whereas the second is related to differential equations with varying nonlinearities and is based on the joint work with Dr. E.Elias (Technion, Israel) and Dr. W.Fang (West Virginia University). There has been a renewed interest in the incorporation of exponentially small terms in asymptotic approximations which is essential for regaining information lost by asymptotic expansions in the sense of Poincaré. These terms occur, for example, in the theory of quantum mechanics, crystal needles and water waves. Various techniques which were successfully employed have been compared. In his talk, Dr. Gingold presented also a new approach to the study of solutions of differential equations with "varying nonlinearity" as functions of a parameter that measures the variation and growth of the non-linearity. It has been shown that these solutions exhibit properties that resemble those of solutions of singularly perturbed problems. Similarities and differences with singularly perturbed problems were discussed, including the natural link with the "Delta Method," a perturbation method that has been recently developed by physicists (Bender et al). In this method, a given equation of mathematical physics is embedded into a family of equations with varying non-linearity. The parameter of varying non-linearity is then utilized to obtain a non-traditional perturbation series.

Dr. Jan Melin (University of Kalmar, Sweden) presented a talk entitled "Some examples describing the need of modifying the Bendixson criterion for piecewise C¹-systems" in which several numerical examples were analyzed to clarify the use of distributions for the investigation of ordinary differential equations with discontinuous right-hand sides.

During the past decades, various important applications in control theory, electronics, navigation systems, optimal foraging theory in mathematical ecology called for the study of piecewise-linear systems. In his talk, Dr. Melin suggested a new approach based on the theory of distributions in the two-dimensional case. This approach requires calculation of the divergence of a given system in the distributional sense. The use of distributions has been efficiently explained by introducing a couple of interesting examples where the speed along the trajectories has been examined. First, the example due to Branicky has been revisited and then the system described in polar coordinates, where the phenomena can be explained in a simpler way, has been explored.

In the talk "Exponential stability in nonlinear functional differential equations with applications to Volterra integro-differential equations," Dr. Youssef N. Raffoul (University of Dayton, Dayton, USA) has made use of non-negative definite Lyapunov functions and obtained sufficient conditions that guarantee exponential stability of the zero solution for a certain system of functional differential equations. The results have also been applied to Volterra integro-differential equations.

Dr. Yuri Rogovchenko (Eastern Mediterranean University, Famagusta, North Cyprus) have reported recent results on asymptotic behavior of general nonlinear second-order differential equations obtained in collaboration with Dr. Octavian Mustafa (University of Craiova, Romania) in the talk entitled "Global existence and asymptotic behavior of solutions of second-order equations." In particular, existence of non-local asymptotically linear (AL) solutions has been addressed, and the following problems have been discussed in detail.

-         Existence of AL solutions under less restrictive conditions on the nonlinearity and coefficients in the equation.

-         Relationship between unbounded connected domains in the real plane with the smooth boundary where the initial values for the problems are taken and existence of AL solutions.

-         Extension of results on asymptotic integration to general n-th order nonlinear differential equations.

-         Application of the results on AL solutions to the study of radial solutions of elliptic differential equations.

Dr. Felix Sadyrbaev (Daugavpils University and University of Latvia, Riga, Latvia) has presented the talk entitled "Asymptotic behavior of solutions to the Emden-Fowler type equations" where the results of the joint work with Dr. Armands Gritsans (Daugavpils University) and Dr. Yuri Klokov (University of Latvia) concerning the qualitative investigation of solutions to Emden-Fowler equations have been discussed.

First, equations with integrally small coefficients were considered and conditions for the existence of asymptotically linear solutions with prescribed number of zeros were given. Then explicit formulas for solutions and coefficients for the expansions of solutions in Taylor series were provided. Finally, for a class of perturbed equations, sharp conditions (in terms of coefficients of the equation) for the equation to exhibit superlinear behavior have been obtained.

The talk "Constant-sign solutions for a system of third-order generalized right focal problems" given by Dr. Patricia J. Y. Wong (School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore) has been concerned with the system of third-order three-point generalized right focal boundary value problems. Using a nonlinear version of the Leray-Schauder alternative and Krasnosel'skii fixed point theorem, Dr. Wong established the existence of one or more constant-sign solutions for the given system.

Several new oscillation criteria for higher order difference equations obtained in collaboration with Dr. S.R. Grace were reported by Dr. Bikkar S. Lalli (University of British Columbia, Canada) in the talk entitled "Oscillation of higher order difference equations."

In his talk "Fixed point methods and three-point boundary-value problems" Dr. Nickolai Kosmatov (University of Arkansas at Little Rock, USA) presented very recent results on existence of solutions of three-point boundary-value obtained by using a refined fixed point technique.

Dr. Christine Nowak (University of Klagenfurt, Austria) gave a talk entitled "Nonuniqueness points of ordinary differential equations" presenting important results concerned with the cases where classical conditions that guarantee the uniqueness of the solution of nonlinear differential equations are violated. Theoretical results have been supported with carefully selected examples.

In summary, all the speakers in the Session as well as participants of the Congress who attended the talks have enjoyed creative and relaxed atmosphere during the talks and informal discussions that followed the talks. We all look forward towards the next Congress WCNA-2008 where we plan to meet again, hopefully in larger numbers.

 

 

Yuri V. Rogovchenko, Dr. received PhD degree in Differential Equations and Mathematical Physics from the Institute of Mathematics of the National Academy of Sciences of Ukraine in 1987 where he has been with the Department of Mathematical Physics and Nonlinear Oscillations from 1986 to 1997. In 1997, he has joined the Department of Applied Mathematics and Computer Science at the Eastern Mediterranean University in Famagusta, North Cyprus, where he is currently Professor of Mathematics. Research interests of Dr. Rogovchenko lie in the area of qualitative theory of nonlinear differential equations and impulsive differential equations.