The Cauchy Problem Of Fractional Evolution Equations

In this thesis, we study the Cauchy problems for three classes of fractional evolution systems, i.e., the existence of mild solutions for evolution equation with Hilfer fractional derivative; the existence of integral solutions for non-dense de-?ned evolution equation with Caputo fractional derivative; topological structure of solution set for non-dense de?ned evolution inclusion with Caputo fractional derivative.In Chapter 2, we introduce some preliminary knowledge about the fractional order calculus, noncompact measure, semigroup theory, the nonlinear multivalued analysis and some fundamental theorems.Hilfer fractional derivative, which includes Riemann-Liuoville fractional derivative and Caputo fractional derivative, has an important role in practical application. However, there is no results about evolution equation with Hilfer fractional derivative. In Chapter 3, the de?nition of mild solution for Hilfer fractional evolution equation is given ?rstly, then by using noncompact measure method and Ascoli–Arzela theorem, we obtain some su?cient conditions to ensure the existence of mild solution. Here, we don’t require the strongly continuous semigroup be compact, and the results we obtain are new and more general to known results.In Chapter 4, we deal with the existence of integral solutions for two classes of fractional order evolution equations with non-dense domain. First, we investigate the nonhomogeneous fractional order evolution equation and obtain its integral solution by Laplace transform and probability density function. Subsequently, based on the form of integral solution of nonhomogeneous fractional order evolution equation, we study the existence of integral solution for nonlinear fractional order evolution equation by noncompact measure method. Here, we don’t require the strongly continuous semigroup be compact, which is di?erent from [100]. Finally, the controllability for the fractional control system with nonlocal condition is investigated.In Chapter 5, based on the de?nition of integral solution for fractional order evolution equations with non-dense domain in Chapter 4, we ?rstly give the concept of a integral solution for fractional di?erential inclusion with non-dense domain. In Section 5.2, we use weak topology approach to obtain the existence of integral solutions, avoiding hypotheses of compactness on the semigroup generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of noncompact measures. Section 5.3 is devoted to prove that the solution set for fractional evolution inclusion is a nonempty compact Rδ-set in two cases that the semigroup is compact and noncompact, respectively. In Section 5.4, the controllability for the fractional evolution inclusion is investigated. Finally, an example is given to illustrate the obtained theory.