Nonlocal Conformable-Fractional Differential Equations with a Measure of Noncompactness in Banach Spaces

Source abstract

This paper deals with the existence of mild solutions for the following Cauchy problem: d α x t / d t α = A x t + f t , x t , x 0 = x 0 + g x , t ∈ 0 , τ , where d α . / d t α is the so-called conformable fractional derivative. The linear part A is the infinitesimal generator of a uniformly continuous semigroup T t t ≥ 0 on a Banach space X , f and g are given functions. The main result is proved by using the Darbo–Sadovskii fixed point theorem without assuming the compactness of the family T t t > 0 and the Lipshitz condition on the nonlocal part g .