Qualitative Properties Of Solutions To Some Evolution Equations And Inclusions

The thesis is mainly concerned with two classes of problems:the existence of asymptotically anti-periodic mild solutions to a Cauchy problem of nonlinear evolution equation with nonlocal initial condition; the Rδ-structure of C0-solution set for nonlinear delay differential inclusion of evolution type on non-compact intervals and its application to nonlocal inclusion.In Chapter1, we introduce some recent developments and significant works in the study of anti-periodic problem and Rδ-structure of solution set for evolution equations and inclusions.In Chapter2, we first introduce an innovative concept of semi-Lipschitz continuity.Then, new criterions, ensuring the existence of asymptotically anti-periodic mild solutions to a Cauchy problem of nonlinear evolution equation with nonlocal initial condition, are established. It is worth mentioning that one of our main results is proved in the absence of the compactness and Lipschitz continuity of nonlocal item and of the Lipschitz continuity of nonlinearity.Chapter3is devoted to a nonlinear delay differential inclusion of evolution type involving m-dissipative operator and source term of multi-valued type in a Banach space. Under rather mild conditions,Rδ-structure of C0-solution set on compact intervals is studied, which is then used to study the Rδ-structure on non-compact intervals. At last, this result is further employed to show the existence of C0-solution for the inclusion (mentioned above) subject to nonlocal condition.In Chapter4, we first present partial differential equations with homogeneous Dirichlet boundary condition and no local condition to illustrate the abstract result established in Chapter2. Then, a system of partial differential inclusion with time delay is presented. Using the abstract result derived in Chapter3, we first discuss the toplogical characteration of C0-solution set for the system. Then, for the system subject to a nonlocal condition, we established the existence of C0-solution.