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Zbl 1128.45004
Banaś, Józef; Cabrera, Ignacio J.
On existence and asymptotic behaviour of solutions of a functional integral equation.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 66, No. 10, A, 2246-2254 (2007). ISSN 0362-546X

The authors deal with the following functional integral equation: $$x(t)=f(t,\;\int_0^t x(s)\,ds,\;\int_0^t x(h(s,x(s)))\,ds),\quad t\geq 0. \tag1$$ Under reasonable hypotheses on $f$ and $h$ the authors show that problem (1) has at least one solution and specify its asymptotic behaviour. To achieve their goal they use the classical Schauder fixed point principle and the concept of measure of noncompactness. Concluding, they provide two examples to illustrate the applicability of their result.
[Nikolaos G. Yannakakis (Athens)]
MSC 2000:
*45G10 Nonsingular nonlinear integral equations
45M05 Asymptotic theory of integral equations
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: functional integral equations; asymptotic behaviour; Schauder's fixed point theorem; measure of noncompactness; tempered functions

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