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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.

45G10Nonsingular nonlinear integral equations
45M10Stability theory of integral equations
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
45M05Asymptotic theory of integral equations
Full Text: DOI
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