Existence of Mild Solutions for a Semilinear Integrodifferential Equation with Nonlocal Initial Conditions

Using Hausdorff measure of noncompactness and a fixed-point argument we prove the existence of mild solutions for the semilinear integrodifferential equation subject to nonlocal initial conditions u'(t) =Au(t)+integral(t)(0) B(t - s)u(s)ds + f(t, u(t)), t epsilon [0, 1], u(0) = g(u), where A : D(A) subset of X -> X, and for every t is an element of [0, 1] the maps B(t) : D(B(t)) subset of X -> X are linear closed operators defined in a Banach space X. We assume further that D(A) subset of D(B(t)) for every t is an element of [0, 1], and the functions f : [0, 1] x X -> X and g : C([0, 1]; X) -> X are X-valued functions which satisfy appropriate conditions.