Grzegorz Gabor

Boundary value problems on infinite intervals

We present two methods, both based on topological ideas, to the solvability of boundary value problems for differential equations and inclusions on infinite intervals. In the first one, related to the rich family of asymptotic problems, we generalize and extend some statements due to the Florence group of mathematicians Anichini, Cecchi, Conti, Furi, Marini, Pera, and Zecca. Thus, their conclusions for differential systems are as well true for inclusions; all under weaker assumptions (for example, the convexity restrictions in the Schauder linearization device can be avoided). In the second, dealing with the existence of bounded solutions on the positive ray, we follow and develop the ideas of Andres, Górniewicz, and Lewicka, who considered periodic problems. A special case of these results was previously announced by Andres. Besides that, the structure of solution sets is investigated. The case of l.s.c. right hand sides of differential inclusions and the implicit differential equations are also considered. The large list of references also includes some where different techniques (like the Conley index approach) have been applied for the same goal, allowing us to envision the full range of recent attacks on the problem stated in the title.

Ważewski type theorem for non-autonomous systems of equations with a disconnected set of egress points

In this paper we study an asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form y Δ ( t ) = f ( t , y ( t ) ) , where f : T × R n ? R n , and T is a time scale. For a given set ? ? T × R n , we formulate conditions for function f which guarantee that at least one solution y of the above system stays in ?. Unlike previous papers the set ? is considered in more general form, i.e., the time section ?t is an arbitrary closed bounded set homeomorphic to the disk (for every t ? T ) and the boundary ? T ? does not contain only egress points. Thanks to this, we can investigate a substantially wider range of equations with various types of bounded solutions. A relevant example is considered.The results are new also for non-autonomous systems of difference equations and the systems of impulsive differential equations.

On attainability of a set by at least one solution to a differential inclusion

In many dynamical system problems, it is interesting not only to know that equilibria do exist but also to know if the equilibria can be reached by at least one trajectory (possibly asymptotically). It is worth pointing out that this question is different from those of stability, or attractiveness of the equilibria. In the present article, our purpose is to give sufficient conditions for answering positively to the above question. More generally we address the question of attainability of a closed set E by at least one trajectory starting from outside E. Our method is based on a kind of Lyapunov Analysis based on Viability Theory and topological methods.