On a quasilinear functional integrodifferential equation in a Banach space (Q1323268)

The authors study the existence of a strong solution of the quasilinear abstract functional integro-differential equation of the type \[ x'(t) + A(t,x_ t) x(t) = f \left( t,x_ t, \int_ 0^ t k(t,s,x_ s) ds \right), \quad t \in [0,T), \] where \(x_ t(s) = x(t - s)\), \(-r \leq s \leq 0\) and \(x_ 0 = \varphi\) is given. For each \(t\) and for each piecewise continuous function \(\psi\) the operator \(v \mapsto A (t, \psi)v\) is assumed to be \(m\)-accretive in the Banach space \(X\). The mapping \((t,\psi) \mapsto A(t, \psi)v\) is assumed to satisfy a certain Lipschitz condition and similarly the functions \(f\) and \(k\) are locally of Lipschitz type. The proof uses the theory of accretive operators and the Banach fixed-point theorem.