Banaś, Józef; Dhage, Bapurao C. Global asymptotic stability of solutions of a functional integral equation. (English) Zbl 1154.45005 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) Cited in 1 ReviewCited in 34 Documents 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 PDF BibTeX XML Cite \textit{J. Banaś} and \textit{B. C. Dhage}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 7, 1945--1952 (2008; Zbl 1154.45005) Full Text: DOI OpenURL References: [1] Banaś, J.; Goebel, K., () [2] Banaś, J., Measures of noncompactness in the space of continuous tempered functions, Demonstratio math., 14, 127-133, (1981) · Zbl 0462.47035 [3] Banaś, J.; Rzepka, B., An application of a measure of noncompactness in the study of asymptotic stability, Appl. math. lett., 16, 1-6, (2003) · Zbl 1015.47034 [4] J. Banaś, D. O’Regan, On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order, Preprint [5] Burton, T.A.; Zhang, B., Fixed points and stability of an integral equation: nonuniqueness, Appl. math. lett., 17, 839-846, (2004) · Zbl 1066.45002 [6] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040 [7] Dhage, B.C.; Ntouyas, S., Existence results for nonlinear functional integral equations via a fixed point theorem of krasnosel’skii – schaefer type, Nonlinear stud., 9, 307-317, (2002) · Zbl 1009.47054 [8] Hu, X.; Yan, J., The global attractivity and asymptotic stability of solution of a nonlinear integral equation, J. math. anal. appl., 321, 147-156, (2006) · Zbl 1108.45006 [9] O’Regan, D.; Meehan, M., Existence theory for nonlinear integral and integrodifferential equations, (1998), Kluwer Academic Dordrecht · Zbl 0932.45010 [10] Väth, M., () This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.