Study On Existence Of Periodic Solutions And Controllability For Several Types Of Differential Inclusions

Differential inclusion is an important branch of nonlinear analysis, which has close relationships with other branches of mathematics such as differential equation, optimal control and optimization. The existence of periodic solutions and controllability are basic contents of differential inclusions. The main content of this paper is the periodic problems. Particularly, we give the existence theorems of the periodic solutions for several types of differential inclusions. Moreover we study the controllability several classes of differential inclusions.1. In finite dimensional spaces, we discuss the periodic problems for semi-linear evolution inclusions. WhenF (t , x) satisfies one-side Lipschitz condition, using techniques from multivalued analysis and fixed point theory, we establish the existence theorems for convex and nonconvex cases. In the nonconvex case, we obtain the sufficient conditions for the existence of periodic solutions by using single-valued Leray-Schauder alternative theorem. In the convex case, the desirable resut has been obtained by using set-valued Leray-Schauder alternative theorem. On the basis of Tolstonogov extremal continuous selection theorem, we prove the existence of extermal periodic solutions and the density of extermal periodic solutions (the strong relaxation theorem).2. In separable Banach Space, we study the periodic problems for a class of integrodifferential inclusions. By applying integral expression of solutions and fixed point theorems, we obtain the sufficient conditions for the existence of periodic solutions both convex and nonconvex case. For convex case, we use Kakutani fixed point theorem. For nonconvex case, the method of continuous selections and Tychonoff fixed point theorem are used.3. Periodic problems for a class of non-autonomous differential inclusions are debated. Utilizing Leray-Schauder alternative theorem we give the sufficient conditions for the existence of periodic solusions for both convex and nonconvex problems when orientor field F (t , x) satisfy one-side Lipschitz continuous. By the theorem of the continuous selection on extremal points, we give the sufficient conditions for existence of extremal solusions and prove the strong relaxation theorem.4. The controllability of a class of evolution inclusions has been debated. Our method is based on the transition from the problems discussed to the fixed point problems of set-valued integral operators. By applying a fixed point theorem for condesing maps, the sufficient conditions of controllability are obtained.5. The controllability of a class of integrodifferential inclusions is studied by fixed point theorem. For convex and nonconvex cases, we establish sufficient conditions for controllability separately. For convex case, our method is based on the transition from the problems discussed to the fixed point problems of set-valued integral operators. By using Kakutani fixed point theorem, we give the sufficient conditions of controllability. For nonconvex case, we convert the problem into fixed point problem of single-valued integral operators and obtain the desirable result by Schauder fixed point theorem.6. Using Galerkin approximation, we extend the finite dimensional results to infinite dimensional space. Under the framework of evolution triple of spaces, we proved the existence theorem of semi-linear evolution inclusion. Moreover we apply the results obtained to a class of partial differential inclusion, the sufficient conditions of existence for periodic solusions are given.