Existence of solutions of quasilinear integrodifferential equations with nonlocal condition (Q1583812)

The authors prove the existence and uniqueness of mild and classical solutions of the quasilinear abstract differential equation \[ du(t)/dt + Au(t)=f (t, u(t))\quad (t\in (0, a]) \] with the nonlocal condition \[ u (0) + g(t_1, t_2,\dots , t_p, u(t_1),\dots , u(t_p))=u_0 \] where \(-A\) is the infinitesimal generator of a \(C_0\)-semigroup in a Banach space \(X\), \(u_0\in X\) and \(f : [0, a] \times X\rightarrow X, f : [0, a]^p \times X^p\rightarrow X\) are given functions, and \[ 0\leq t_1\leq t_2\leq \dots \leq t_p\leq a. \] The main results of the paper are obtained by using \(C_0\)-semigroups and the Banach fixed point theorem.