Dhage, Bapurao C. Local asymptotic attractivity for nonlinear quadratic functional integral equations. (English) Zbl 1173.47056 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 5, 1912-1922 (2009). The paper deals with the quadratic functional-integral equation of mixed type \[ x(t)=[f(t,x(\alpha(t)))]\big(q(t)+\int_0^{\beta(t)}g(t,s,x(\gamma(s)))\,ds\big). \tag{1} \] Using a result about operator equations in Banach algebras, the author proves the existence of a solution of (1) in the space \(BC(\mathbb{R}_+,\mathbb{R})\). It is also proved that the solutions of (1) are uniformly locally asymptotically attractive on \(\mathbb{R}_+\). Reviewer: Mirosława Zima (Rzeszow) Cited in 9 Documents MSC: 47N20 Applications of operator theory to differential and integral equations 47H10 Fixed-point theorems 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 45G10 Other nonlinear integral equations Keywords:quadratic functional integral equation; fixed point theorem; uniformly locally asymptotically attractive solution PDF BibTeX XML Cite \textit{B. C. Dhage}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 5, 1912--1922 (2009; Zbl 1173.47056) Full Text: DOI OpenURL References: [1] Banas, J., Measures of noncompactness in the space of continuous tempered functions, Demonstratio math., 14, 127-133, (1981) · Zbl 0462.47035 [2] Banas, J.; Dhage, B.C., Global asymptotic stability of solutions of a functional integral equation, Nonlinear anal., (2007) [3] Banas, J.; Rzepka, B., An application of measures of noncompactness in the study of asymptotic stability, Appl. math. lett., 16, 1-6, (2003) · Zbl 1015.47034 [4] Burton, T.A., Volterra integral and differential equations, (1983), Academic Press New York · Zbl 0515.45001 [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] Burton, T.A.; Furumochi, T., A note on stability by schauder’s theorem, Funkc. ekvac., 44, 73-82, (2001) · Zbl 1158.34329 [7] Dhage, B.C., A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook math. J., 44, 145-155, (2004) · Zbl 1057.47062 [8] Dhage, B.C., Nonlinear functional boundary value problems involving Carathéodory, Kyungpook math. journal, 46, 427-441, (2006) [9] Dhage, B.C.; O’Regan, A fixed point theorem in Banach algebras with application to functional integral equations, Funct. differential equations, 7, 259-267, (2000) · Zbl 1040.45003 [10] Dugundji, J.; Granas, A., () [11] Lusternik, L.A.; Sobolev, V.J., Elements of functional analysis, (1985), Hindustan Publishing Corpn India · Zbl 0293.46001 [12] Krasnoselskii, M.A., Topological methods in the theory of nonlinear integral equations, (1964), Pergamon Press New York [13] 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 [14] Väth, M., Volterra and integral equations of vector functions, (2000), PAM, Marcel Dekker New York · Zbl 0940.45002 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.