Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces

Abstract

We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo-Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.