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Asymptotically almost periodic and almost periodic solutions for partial neutral integrodifferential equations. (English) Zbl 1139.34051

The authors consider the existence of almost periodic and asymptotically almost periodic solutions for a class of partial neutral integro-differential equations with unbounded delay of the form:
\[ {d\over dt} D(t, u_t)= AD(t, u_t)+ \int^t_0 B(t- s)D(s, u_s)\,ds+ g(t, u_t),\tag{1} \]
where \(A\) and \(B(t)\) are linear closed densely defined operators on a Banach space \(X\), \(x_t(\varphi)= x(t+\varphi)\), \(D(t,\varphi)= \varphi(0)+ f(t,\varphi)\) and \(f(.)\), \(g(.)\) are appropriate functions. Throughout it is assumed that the associated abstract integro-differential Cauchy problem has the resolvent family of bounded linear operators.
By the contraction mapping theorem the existence of a unique mild solution of equation (1) is obtained. Several cases are considered.
An example of a partial functional differential equation with unbounded delay is given.

MSC:

34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
45N05 Abstract integral equations, integral equations in abstract spaces
35R10 Partial functional-differential equations
34K40 Neutral functional-differential equations
45D05 Volterra integral equations
34K30 Functional-differential equations in abstract spaces
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