# Article

 Title: On some nonlinear alternatives of Leray-Schauder type and functional integral equations (English) Author: Dhage, Bapurao Chandra Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 42 Issue: 1 Year: 2006 Pages: 11-23 Summary lang: English . Category: math . Summary: In this paper, some new fixed point theorems concerning the nonlinear alternative of Leray-Schauder type are proved in a Banach algebra. Applications are given to nonlinear functional integral equations in Banach algebras for proving the existence results. Our results of this paper complement the results that appear in Granas et. al. (Granas, A., Guenther, R. B. and Lee, J. W., Some existence principles in the Caratherodony theory of nonlinear differential system, J. Math. Pures Appl. 70 (1991), 153–196.) and Dhage and Regan (Dhage, B. C. and O’Regan, D., A fixed point theorem in Banach algebras with applications to functional integral equations, Funct. Differ. Equ. 7(3-4)(2000), 259–267.). (English) Keyword: Banach algebra Keyword: fixed point theorem Keyword: integral equations MSC: 45G10 MSC: 47H10 MSC: 47N20 idZBL: Zbl 1164.47357 idMR: MR2227108 . Date available: 2008-06-06T22:47:00Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107977 . Reference: [1] Browder F. E.: Nonlinear operators and nonlinear equations of evolutions in Banach spaces.Proc. Symp. pure Math. Amer. Math. Soc. Providence, Rhode Island 1976. MR 0405188 Reference: [2] Chandrasekhar S.: Radiative Heat Transfer.Dover, New York, 1960. MR 0111583 Reference: [3] Deimling K.: Nonlinear Functional Analysis.Springer Verlag, 1985. Zbl 0559.47040, MR 0787404 Reference: [4] Dhage B. C.: On some variants of Schauder’s fixed point principle and applications to nonlinear integral equations.J. Math. Phys. Sci. 25 (1988), 603–611. Zbl 0673.47043, MR 0967242 Reference: [5] Dhage B. C.: A fixed point theorem and applications to nonlinear integral equations.Proc. Internat. Symp. Nonlinear Anal. Appl. Bio-Math., Waltair, India (1987), 53–59. Reference: [6] Dhage B. C.: On $\alpha$-condensing mappings in Banach algebras.The Math. Student 6 (1994), 146–152. Zbl 0882.47033, MR 1292381 Reference: [7] Dhage B. C.: On a fixed point theorem of Krasnoselskii type.Electron. J. Qual. Theory Differ. Equ. 2002, No. 6, 9 pp. (electronic). MR 1895277 Reference: [8] Dhage B. C., Ntouyas S. K.: Existence results for nonlinear functional integral equations via a fixed point theorem of Krasnoselskii-Schaefer type.Nonlinear Studies 9(3)(2002), 307–317. MR 1918909 Reference: [9] Dhage B. C., Jahagirdar P. G.: On nonlinear integral equations in Banach algebras.Applied Sciences Periodical II (2000), 131–133. MR 1840872 Reference: [10] Dhage B. C., O’Regan D.: A fixed point theorem in Banach algebras with applications to functional integral equations.Funct. Differ. Equ. 7(3-4)(2000), 259–267. Zbl 1040.45003, MR 1940503 Reference: [11] Dugundji J., Granas A.: Fixed point theory.Monographie Matematyczne, Warsaw, 1982. Zbl 0483.47038 Reference: [12] Granas A., Guenther R. B., Lee J. W.: Some existence principles in the Caratherodony theory of nonlinear differential system.J. Math. Pures Appl. 70 (1991), 153–196. MR 1103033 Reference: [13] Nashed M. Z., Wong J. S. W.: Some variants of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations.J. Math. Mech. 18 (1969), 767–777. MR 0238140 Reference: [14] Ntouyas S. K., Tsamatos P. G.: A fixed point theorem of Krasnoselskii-nonlinear alternative type with applications to functional integral equations.Differential Equations Dynam. Systems 7(2) (1999), 139–146. MR 1860784 Reference: [15] Subramanyam P. V., Sundarsanam S. K.: A note on functional integral equations.Differential Equations Dynam. Systems 4 (1996), 473–478. Reference: [16] Zeidler E.: Nonlinear Functional Analysis and Its Applications I.Springer Verlag, 1985. MR 0816732 .

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