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Existence for a class of partial functional differential equations with infinite delay. (English) Zbl 1003.34068

The authors study the following class of functional-differential equations with infinite delay in a Banach space E

x ' (t)=Ax(t)+F(t,x t ),t0,x 0 =ϕ,

where A is a closed linear operator. Existence, uniqueness and regularity results for the same class of equations were previously furnished by H. R. Henríquez [Funkc. Ekvacioj, Ser. Int. 37, No. 2, 329–343 (1994; Zbl 0814.35141); Indian J. Pure Appl. Math. 27, No. 4, 357–386 (1996; Zbl 0853.34072) and Nonlinear Anal., Theory Methods Appl. 28, No. 3, 513–531 (1997; Zbl 0864.35112)] in the case where A is an infinitesimal generator of a C 0 -semigroup on E.

The aim of the present paper is to extend such existence results to the case where A is a Hille-Yosida operator with domain D(A) not necessarily dense in E. The theory of integrated semigroups is the main tool used. The authors apply the result to a partial integrodifferential equation.

34K30Functional-differential equations in abstract spaces
35R10Partial functional-differential equations
35R20Partial operator-differential equations
47D62Integrated semigroups