Mild Solutions For Some Nonlinear Fractional Evolution Equations

In this dissertation, we investigate the properties of mild solutions for some non-linear fractional evolution equations in Banach spaces (involving existence and unique-ness).Chapter 1 provides the research background and the main results of this disserta-tion.Chapter 2 contains some preliminaries including properties of fractional deriva-tive, Bochner integral of vector-valued functions, analytic semigroups of linear opera-tors, phase spaces, and mild solutions.Chapter 3 is devoted to discussing existence and uniqueness of mild solutions tofractional integro-differential equations. Based on the semigroup theory, two differentresults are obtained. The idea of the first result is by means of Banach's contractionmapping principle. The second result is derived from the Krasnoselkii's fixed pointtheorem. Finally, we give an example to illustrate our main result.Chapter 4 is assigned to study existence and uniqueness of the mild solution of thenonlocal Cauchy problem for nonautonomous fractional evolution equations. We ob-tain the existence by using the measure of noncompactness and Sadovskii's fixed pointtheorem. Furthermore, we get the existence and uniqueness by means of Banach'scontraction mapping principle. At last, an example is given to illustrate our result.Chapter 5 studies the neutral fractional evolution equations with infinite delay.Making use of the properties of phase spaces, we obtain the existence of the mildsolution by means of a fixed point theorem in the theory of measure of noncompactness.In addition, an example in form of a partial differential equation is given to illustrateour result.