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

This paper deals with the existence of mild solutions for the following Cauchy problem: d(alpha)x(t)/dt(alpha). Ax(t) + f(t, x(t)), x(0) = x(0) + g(x), t subset of [0, t], where d(alpha)(.)/dt(alpha) 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.