Semilinear Fractional Evolution Inclusion Problem in the Frame of a Generalized Caputo Operator

Abstract

In this paper, we study the existence of solutions to initial value problems for a nonlinear generalized Caputo fractional differential inclusion with Lipschitz set-valued functions. The applied fractional operator is given by the kernel $k\left(\rho ,s\right)=\xi \left(\rho \right)-\xi \left(s\right)$ and the derivative operator $\left(1/{\xi }^{\prime }\left(\rho \right)\right)\left(d/d\rho \right)$ . The existence result is obtained via fixed point theorems due to Covitz and Nadler. Moreover, we also characterize the topological properties of the set of solutions for such inclusions. The obtained results generalize previous works in the literature, where the classical Caputo fractional derivative is considered. In the end, an example demonstrating the effectiveness of the theoretical results is presented.