Fractional Retarded Evolution Equations with Measure of Noncompactness Subjected to Mixed Nonlocal Plus Local Initial Conditions

We consider the fractional retarded evolution equations D-C(t)q u(t) + Au(t) = f(t, u(t), integral(t)(0) w(t, s, us) ds), t is an element of[0, a], where D-C(t)q, q is an element of(0, 1], is the fractional derivative in the Caputo sense, - A is the infinitesimal generator of a C-0-semigroup of uniformly bounded linear operators T(t) (t >= 0) on a Banach space X and the nonlinear operators f and w are given functions satisfying some assumptions, subjected to a general mixed nonlocal plus local initial condition of the form u(t) = g(u)(t) + phi(t), t. [-h, 0]. Undermore general conditions, the existence of mild solutions and positive mild solutions are obtained by means of fractional calculus and fixed point theory for condensing maps. Moreover, we present an example to illustrate the application of abstract results.