Chapter 2 - Fractional Evolution Equations

Abstract

In this chapter, we first study the existence of Cauchy problems for fractional evolution equations. The suitable mild solutions of fractional Cauchy problems with Riemann-Liouville derivative and Caputo derivative are introduced, respectively. By using fixed point theorems and Hausdorff measure of noncompactness, we give existence results of mild solutions in the cases that the almost sectorial operator is compact and noncompact, respectively. In Section 2.2, we discuss the existence and uniqueness of the bounded solutions on real axis for fractional evolution equations with Liouville fractional derivative of order q∈01 with the lower limit –∞. Some sufficient conditions are established for the existence and uniqueness of periodic solutions, S-asymptotically periodic solutions, and other types of bounded solutions.