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

Banaś, Józef; Dhage, Bapurao C.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 7, 1945-1952 (2008).

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.

Reviewer: Oleh Omel’chenko (Berlin)

MSC:

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

Keywords:

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

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