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Zbl 1003.34068
Adimy, Mostafa; Bouzahir, Hassane; Ezzinbi, Khalil
Existence for a class of partial functional differential equations with infinite delay. (English)
[J] Nonlinear Anal., Theory Methods Appl. 46, No. 1, A, 91-112 (2001). ISSN 0362-546X

The authors study the following class of functional-differential equations with infinite delay in a Banach space $E$ $$ x'(t) = Ax(t) + F(t, x_t),\quad t \ge 0,\qquad x_0 = \varphi,$$ where $A$ is a closed linear operator. Existence, uniqueness and regularity results for the same class of equations were previously furnished by {\it 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 $E$. \par 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.
[Cristina Marcelli (Ancona)]

MSC 2000:: *34K30 Functional-differential equations in abstract spaces
35R10 Difference-partial differential equations
35R20 Partial operator-differential equations
47D62 Integrated semigroups

Keywords: functional-differential equations; integrated semigroup; semigroup of linear operators

Citations: Zbl 0814.35141; Zbl 0853.34072; Zbl 0864.35112

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