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On global existence and attractivity results for nonlinear functional integral equations. (English) Zbl 1197.45005
The authors study the existence and global attractivity of the solutions for the following nonlinear functional integral equation (FIE) $$ x\left( t\right) =F\left( t,f\left( t,\alpha \left( t\right) \right) ,\int\limits_{0}^{\beta \left( t\right) }g\left( t,s,x\left( \gamma \left( s\right) \right) \right) ds\right) $$ for all $t\in \mathbb{R}_{+}$, where $f:\mathbb{R}_{+}\times \mathbb{R} \rightarrow \mathbb{R}$, $g:\mathbb{R}_{+}\times \mathbb{R}_{+}\times \mathbb{R}\rightarrow \mathbb{R}$, $F:\mathbb{R}_{+}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ and $\alpha ,\beta ,\gamma:\mathbb{R} _{+}\rightarrow \mathbb{R}_{+}$. They give two existence and global attractivity results for the solution of the FIE via a measure theoretic fixed point theorem of {\it B. Dhage} [Commun. Appl. Nonlinear Anal. 15, No. 2, 89--101 (2008; Zbl 1160.47041)]. These results improve and generalize the attractivity results of {\it J. Banas} and {\it B. Rzepka} [Appl. Math. Lett. 16, No. 1, 1--6 (2003; Zbl 1015.47034)], {\it J. Banas} and {\it B. Dhage} [Nonlinear Anal., Theory Methods Appl. 69, No. 7 (A), 1945--1952 (2008; Zbl 1154.45005)], {\it B. Dhage} [Nonlinear Anal., Theory Methods Appl. 70, No. 7 (A), 2485--2493 (2009; Zbl 1163.45005)] and {\it X. Hu} and {\it J. Yan} [J. Math. Anal. Appl. 321, No. 1, 147--156 (2006; Zbl 1108.45006)] under some weaker Lipschitz conditions.

45G10Nonsingular nonlinear integral equations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
45M05Asymptotic theory of integral equations
45M10Stability theory of integral equations
47H08Measures of noncompactness and condensing mappings, $K$-set contractions, etc.
Full Text: DOI
[1] Banas, J.; Dhage, B. C.: Global asymptotic stability of solutions of a functional integral equations, Nonlinear anal. 69, 1945-1952 (2008) · Zbl 1154.45005 · doi:10.1016/j.na.2007.07.038
[2] Dhage, B. C.; O’regan, D.: A fixed point theorem in Banach algebras with applications to nonlinear integral equations, Functional diff. Equations 7, No. 3--4, 259-267 (2000) · Zbl 1040.45003
[3] 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 · doi:10.1016/j.jmaa.2005.08.010
[4] Burton, T. A.; Zhang, B.: Fixed points and stability of an integral equation: nonuniqueness, Appl. math. Lett. 17, 839-846 (2004) · Zbl 1066.45002 · doi:10.1016/j.aml.2004.06.015
[5] Burton, T. A.; Furumochi, T.: A note on stability by Schauder’s theorem, Funkcialaj ekvacioj 44, 73-82 (2001) · Zbl 1158.34329
[6] Dhage, B. C.: Asymptotic stability of nonlinear functional integral equations via measures of noncompactness, Comm. appl. Nonlinear anal. 15, No. 2, 89-101 (2008) · Zbl 1160.47041
[7] Banas, 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 · doi:10.1016/S0893-9659(02)00136-2
[8] Akhmerov, R. R.; Kamenskii, M. I.; Potapov, A. S.; Rodhina, A. E.; Sadovskii, B. N.: Measures of noncompactness and condensing operators, (1992) · Zbl 0748.47045
[9] Appell, J.: Measures of noncompactness, condensing operators and fixed points: an application-oriented survey, Fixed point theory 6, 157-229 (2005) · Zbl 1102.47041
[10] Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040
[11] Väth, M.: Volterra and integral equations of vector functions, (2000) · Zbl 0940.45002
[12] B.C. Dhage, Attractivity and positivity results for nonlinear functional integral equations via measures of noncompactness, Differ. Equ. Appl. (in press) · Zbl 1201.45007 · http://files.ele-math.com/abstracts/dea-02-20-abs.pdf
[13] Dhage, B. C.: Local asymptotic attractivity for nonlinear quadratic functional integral equations, Nonlinear anal. 70, No. 5, 1912-1922 (2009) · Zbl 1173.47056 · doi:10.1016/j.na.2008.02.109
[14] M. Kuczma, Functional equations in single variable, in: Monografie Math., vol. 46, Warszawa, 1968 · Zbl 0196.16403
[15] Dhage, B. C.: Global attractivity results for the nonlinear functional integral equations via a Krasnoselskii type fixed point theorem, Nonlinear anal. 70, 2485-2493 (2009) · Zbl 1163.45005 · doi:10.1016/j.na.2008.03.033