Some Problems Of Fractional Differential Equations

In recent years, many new papers appeared about fractional integral equa-tions and fractional evolution equations, but, compared with the di?erential equa-tions of integer order, fractional order is far from perfect in the theory, many areasare not involved, we need to do further study.In this paper, we discuss the existence of solutions and the extremal solutionsfor integral equations and a class of fractional evolution equations with nonlocalCauchy conditions. In Chapter 2, firstly, by the use of Krasnoselskii's fixed pointtheorem we prove the existence of positive solutions for certain Volterra integralequations; then by the use of the hybrid fixed point theorem we prove the ex-istence of extremal solutions for certain Volterra integral equations, and finally,the results are applied to a variety of fractional di?erential equations. In Chapter3, by considering probability density and semigroup, we give definitions of mildsolutions for fractional evolution equations with nonlocal conditions; by using thefunctional analysis concerning to the semigroup of operators and some fixed pointtheorems e?ectively, we give the criteria on existence of mild solutions.