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

Abstract EN

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)+∫0tB(t-s)u(s)ds+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.