Nonlocal Problems for Fractional Differential Equations via Resolvent Operators

We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory. 1. Introduction In this paper, we are concerned with the existence of mild solutions for a fractional differential equation with nonlocal conditions of the form where is the Caputo fractional derivative of order with , is the infinitesimal generator of a resolvent , , is a real Banach space endowed with the norm , and and are appropriate continuous functions to be specified later. The theory of fractional differential equations has received much attention over the past twenty years, since they are important in describing the natural models such as diffusion processes, stochastic processes, finance, and hydrology. Many notions associated with resolvent are developed such as integral resolvent, solution operators, -resolvent operator functions, -regularized resolvent, and -order fractional semigroups. All of these notions play a central role in the study of Volterra equations, especially the fractional differential equations. Concerning the literature, we refer the reader to the books [1, 2], the recent papers [3–20], and the references therein. On the other hand, abstract differential equations with nonlocal conditions have also been studied extensively in the literature, since it is demonstrated that the nonlocal problems have better effects in applications than the classical ones. It was Byszewski and Lakshmikantham [21] who first studied the existence and uniqueness of mild solutions for nonlocal differential equations. And the main difficulty in dealing with the nonlocal problem is how to get the compactness of solution operator at zero, especially when the nonlocal item is only assumed to be Lipschitz continuous or continuous. Many authors developed different techniques and methods to solve this problem. For more details on this topic, we refer to [10, 11, 22–33] and references therein. In this paper, we combine the above two directions and study the nonlocal fractional differential equation (1) governed by operator generating an analytic resolvent. A standard approach in deriving the mild solution of (1) is to define the solution operator . Then, conditions are given such that some fixed point theorems such as