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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 $(T(t))_{t\ge 0}$ 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 $${d\over dt}(x(t)+ G(t, x_t))= Ax(t)+ F\Biggl(t, x_t,\int^t_0 h(t,x,x_s)\,ds\Biggr)\,(\text{for }t\in I= [\delta, T]),\, x(\delta)= \varphi.\tag1$$ 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.

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations 45G10 Nonsingular nonlinear integral equations
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##### References:
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