On Some Problems For Fractional Evolution Systems

Evolution systems are the hot topics in dynamics research. This thesis studies the solvability and controllability of a class of stochastic evolution equations and a few types of fractional evolution differential equations. The systems considered in the thesis are in-fluenced by the impulses, delays and the nondensely defined operators. This Ph.D thesis contains six chapters.Chapter1is concerned with the historical background, current research situation and the main work of the thesis.In Chapter2, we consider the existence and controllability of impulsive stochastic evo-lution systems with infinite delay in Hilbert space. We firstly establish a stochastic inequality in an appropriate state space. Then by using skills of stochastic analysis, the theory of evo-lution systems and the Banach contraction theorem, sufficient conditions for existence and controllability are obtained. Some known results are generalized.Chapter3deals with the existence and uniqueness of integral solution for nondensely defined fractional evolution differential equations with nonlocal conditions. Our approach is based on the integrated semigroup theory, Krasnoselskii fixed point theorem and the Banach contraction mapping principle. we use the integrated semigroup and probability densities to define the integral solution. Some mistakes in recent papers are corrected.In Chapter4, we consider the controllability results for integral solution of nondensely defined fractional semilinear functional differential equations. The concept of controllability is established by the integrated semigroup theory. Our approach is based on some analysis skills and the Schauder fixed point theorem. An example is also given to illustrate our results.Chapter5studies the existence of mild solutions for fractional neutral evolution dif-ferential inclusions. Our approach is based on the fractional power of operators and a fixed point theorem for multivalued map due to Dhage. Moreover, we present a new general-ized Gronwall inequality with singularity, which is the key point in proving our main result. Some known results are generalized and improved.In Chapter6, we investigate the existence results for mild solutions of semilinear fractional evolution equations on an unbounded interval. We obtain some sufficient conditions of the existence results by applying the classical Tichonov fixed point theorem. The key point of the approach is choosing the appropriate inequalities and the measure of noncompactness in Frechet space.