An Existence Result for Some Fractional Evolution Equation with Nonlocal Conditions and Compact Resolvent Operator

Abstract

We are concerned with the existence of mild solutions to the fractional differential equation $D^{\alpha }_{t}u(t)=Au(t)+J_{t}^{1-\alpha }f(t,u(t)),\;\;0<t\le T,$ with nonlocal conditions $u_0=u(0)+g(u)$, where $0<\alpha <1$, $D^{\alpha }_{t}$ is the Caputo derivative, $J_{t}^{1-\alpha }h(t):=\frac{1}{\Gamma (1-\alpha )}\int _{0}^{t}\frac{h(s)}{(t-s)^{\alpha }}ds$ is the fractional integral of order $\alpha$ of the function h, and $A:D(A)\subset X\rightarrow X$ is a linear operator which generates a compact analytic resolvent family $(R_{\alpha }(t))_{t\ge 0}$, X being a Banach space. We obtain our results using the Krasnoleskii’s fixed point theorem.