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On existence and asymptotic behaviour of solutions of a functional integral equation. (English) Zbl 1128.45004
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.

45G10Nonsingular nonlinear integral equations
45M05Asymptotic theory of integral equations
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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