Existence and uniqueness of mild solutions for fractional semilinear differential equations

Abstract

In this article, we study the existence and uniqueness of a local mild solution for a class of semilinear differential equations involving the Caputo fractional time derivative of order $\alpha$ $(0<\alpha<1)$ and, in the linear part, a sectorial linear operator A. We put some conditions on a nonlinear term f and an initial data $u_0$ in terms of the fractional power of A. By applying Banach's Fixed Point Theorem, we obtain a unique local mild solution with smoothing effects, estimates, and a behavior at t close to 0. An example as an application of our results is also given.

Keywords

• Fractional semilinear differential equation
• sectorial operator
• Caputo fractional derivative
• fractional power
• mild solution