Existence results for partial neutral functional integrodifferential equations with unbounded delay
Assume that \(A\) is a (possibly unbounded) infinitesimal generator of an analytic semigroup of bounded linear operators \((T(t))_{t\geq 0}\) on a Banach space \(X\). The author studies the existence of mild solutions of the partial neutral functional integrodifferential equation with unbounded delay having the form \[ {d\over dt}(x(t)+ G(t, x_t))= Ax(t)+ F\Biggl(t, x_t,\int^t_0 h(t,x,x_s)\,ds\Biggr)\,(\text{for }t\in I= [\delta, T]),\, x(\delta)= \varphi.\tag{1} \] Here the history \(x_t:(-\infty, 0]\to X\), \(x_t(s)= x(t+ s)\) is assumed to belong to some abstract phase space \(B\) defined axiomatically. There are imposed some other regularity assumptions on the components of the problem (1). The existence result concerning (1) is obtained with help of the Leray-Schauder alternative. Some applications are also given.
partial neutral functional integrodifferential equations
analytic semigroup of bounded linear operators
scientific article; zbMATH DE number 2081790