Topological Structure Of Solution Set For Impulsive Evolution Inclusions

In recent years, impulsive differential equations and inclusions have being widely adopted in mathematical models of real processes, and they arise in phenomena since they are an appropriate model for describing phenomena studied in several fields of applied science as Biology, Engineering, Economics, Physics,Medical, where systems instantaneously change their state. In this thesis, we study topological structure of solution set for impulsive evolution inclusions. For details, the existence of mild solutions and topological structure of solutions set for impulsive evolution inclusions; topological structure of integral solutions set for impulsive evolution inclusions with Hille-Yosida operator; topological structure of C~0-solutions set for impulsive evolution inclusions with disspative operator.In Chapter 2, we introduce some basic facts on spaces of functions, weak topology, semigroup theory, multivalued analysis and some fundamental theorems.In Chapter 3, we investigate the topological characterizations of solution sets for impulsive evolution inclusions in the simple case that the evolution operator generated by linear part is compact. By using the weak topology method, we consider the existence of mild solutions, and also prove that the solutions set is nonempty and a compact R_δ-set. In particular, we concentrate on the case when the evolution operator generated by linear part is noncompact, we consider the existence of mild solutions and prove that the solution set is nonempty and a weakly compact R_δ-set. Since we assume the regularity of nonlinear terms with respect to the weak topology, we do not require compactness of the evolution operator and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. As samples of applications, we consider a partial impulsive differential inclusion at the end of this chapter.In Chapter 4, we consider the topological structure to solution set of impulsive evolution inclusion involving a nondensely defined closed linear operator satisfying the Hille-Yosida condition and source term of multivalued type in Banach space.Definition of integral solution for impulsive evolution inclusion is given in Section4.1. We state and prove the integral solution set for impulsive evolution inclusion is a compact R_δ-set on compact interval under the cases that the semigroup is compact as well as noncompact. Then, using the inverse limit method, we obtained the corresponding results on the noncompact interval. For detailed, Section 4.2 is devoted to studying the case that the semigroup is compact. Section 4.3 provides the main results on the case that the semigroup is noncompact by using the theory of measure of noncompactness.In Chapter 5, we investigate the topological structure to C~0-solutions set of impulsive evolution inclusion. Section 5.1 gives the concept of a C~0-solution for impulsive evolution inclusion. Section 5.2 is devoted to proving that the C~0-solution set for impulsive evolution differential inclusion in compact interval is a nonempty compact R_δ-set in the case that the semigroup is compact, then proceed to study the R_δ-set on noncompact interval by the inverse limit method. Section 5.2provides that the C~0-solution set for impulsive evolution inclusion in compact interval is a nonempty compact R_δ-set in the case that the semigroup is noncompact,then proceed to study the R_δ-structure of C~0-solution set of impulsive evolution inclusion on noncompact interval.