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

Abstract

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 , 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.