Functional differential equations with nonlocal initial conditions (Q1366405)

The object of the paper is the study of the nonlocal Cauchy problem \[ u'(t)+ Au(t)+ G(u)(t) \ni 0\;(0<t<T),\;u(0) =g(u). \tag{1} \] Here \(A\) is an \(m\)-accretive multivalued operator in a Banach space \(X\), \(G:C([0,T],X) \to L^1((0,T),X)\) and \(g:C ([0,T],X) \to X\). The functions \(G\) and \(g\) are assumed to be Lipschitz continuous. Both mild and strong solutions of the problem (1) are considered. Under some extra technical assumptions a few results are proved on the existence, uniqueness, regularity, dependence upon data and asymptotic properties \((T\to \infty)\) of solutions of (1). The basic tools used in the considerations are methods of the theory of \(m\)-accretive operators and the Banach fixed point principle. An example illustrating the obtained results is also provided.