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Existence results for partial neutral functional integrodifferential equations with unbounded delay. (English) Zbl 1056.45012

Assume that A is a (possibly unbounded) infinitesimal generator of an analytic semigroup of bounded linear operators (T(t)) t0 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 dt(x(t)+G(t,x t ))=Ax(t)+Ft , x t , 0 t h (t,x,x s ) d s(fortI=[δ,T]),x(δ)=φ·(1)

Here the history x t :(-,0]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.

45N05Abstract integral equations, integral equations in abstract spaces
45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations