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.


45G10 Other nonlinear integral equations
45M10 Stability theory for integral equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
45M05 Asymptotics of solutions to integral equations
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