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) = A u (t) + ? 0 t B (t - s) u (s) d s + f (t, u (t)), t [ 0,1 ], u (0) = g (u), where A: D (A) X ? X, and for every t [ 0,1 ] the maps B (t): D (B (t)) X ? X are linear closed operators defined in a Banach space X. We assume further that D (A) D (B (t)) for every t [ 0,1 ], and the functions f: [ 0,1 ] × X ? X and g: C ([ 0,1 ]; X) ? X are X -valued functions which satisfy appropriate conditions. © 2012 Carlos Lizama and Juan C. Pozo.