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 $u^{\prime }(t)=Au(t)+{\int }_0^{t}B(t-s)u(s)ds+f(t,u(t))$, $t\in [\mathrm{0,1}]$, $u(0)=g(u)$, where $A:D(A)\subseteq X\to X$, and for every $t\in [\mathrm{0,1}]$ the maps $B(t):D(B(t))\subseteq X\to X$ are linear closed operators defined in a Banach space $X$. We assume further that $D(A)\subseteq D(B(t))$ for every $t\in [\mathrm{0,1}]$, and the functions $f:[\mathrm{0,1}]\times X\to X$ and $g:C([\mathrm{0,1}];X)\to X$ are $X$-valued functions which satisfy appropriate conditions.