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Zbl 1154.45005
Banaś, Józef; Dhage, Bapurao C.
Global asymptotic stability of solutions of a functional integral equation.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 69, No. 7, A, 1945-1952 (2008). ISSN 0362-546X

The authors consider the solvability in the space $BC(\mathbb{R}_{+})$ of the following functional integral equation $$x(t)=f(t,x(\alpha(t)))+\int_0^{\beta(t)} g(t,s,x(\gamma(s))) \,ds,$$ where $\alpha,\beta,\gamma : \mathbb{R}_{+}\to\mathbb{R}_{+}$ are continuous and $\alpha(t)\to\infty$ as $t\to\infty$. Using the technique of measures of noncompactness, namely a fixed point theorem of Darbo type, they prove a theorem on the existence and global asymptotic stability of solutions of the above functional integral equation. A few realizations of the result obtained are indicated.
[Oleh Omel'chenko (Berlin)]
MSC 2000:
*45G10 Nonsingular nonlinear integral equations
45M10 Stability theory of integral equations
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
45M05 Asymptotic theory of integral equations

Keywords: functional integral equation; measure of noncompactness; fixed point theorem; global asymptotic stability

Cited in: Zbl 1197.45005

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