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Global asymptotic stability of solutions of a functional integral equation. (English) Zbl 1154.45005
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.

##### MSC:
 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 topological linear spaces 45M05 Asymptotic theory of integral equations
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##### References:
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