On the Solutions of a Class of Nonlinear Integral Equations in the Banach Algebra of the Continuous Functions and Some Examples

Özdemir, Ismet; Ilhan, Bekir; Çakan, Ümit

doi:10.2478/awutm-2014-0008

Abstract

In this paper, we study the existence of the solutions of a class of functional integral equations which contain a lot of classical nonlinear integral equations as special cases. We consider the solvability of the equations in the Banach algebra of continuous functions on a closed and bounded interval. The main tools here are the measure of noncompactness and the suitable fixed point theorem for the product of two operators in the Banach algebra.

References

[1] R. P. Agarval, D. O’Regan, andWong P. J. Y., Positive solutions of differential, difference and integral equations, Kluwer Academic, Dordrecht, 1999.10.1007/978-94-015-9171-3_22
Search in Google Scholar
[2] I. K. Argyros, Quadratic equations and applications to Chandrasekhar’s and related equations, Bull. Austral. Math. Soc., 32, (1985), 275-292.10.1017/S0004972700009953
Search in Google Scholar
[3] J. Banas and K. Goebel, Measures of noncompactness in Banach space, Lecture Notes in Pure and Applied Mathematics, New York: Dekker, 60, 1980.
Search in Google Scholar
[4] J. Banas, Integrable solutions of Hammerstein and Urysohn integral equations, J. Austral. Math. Soc., 46, (1989), 61-6810.1017/S1446788700030378
Search in Google Scholar
[5] J. Banas, Applications of measure of weak compactness and some classes of operators in the theory of functional equations in the Lebesque space, Nonlinear Anal., 30, (1997), 3283-3293.10.1016/S0362-546X(96)00157-5
Search in Google Scholar
[6] J. Banas, M. Lecko, and W. G. El-Sayed, Existence theorems for some quadratic integral equations, J. Math. Anal. Appl., 222, (1998), 276-285.10.1006/jmaa.1998.5941
Search in Google Scholar
[7] J. Banas and M. Lecko, Fixed points of the product of operators in Banach algebra, Panamer. Math. J., 12, (2002), 101-109.
Search in Google Scholar
[8] J. Banas and L. Olszowy, On a class of mesures of noncompactness in Banach algebras and their application to nonlinear integral equations, J. for Analy. and its Appl., 28, (2009), 1-24.10.4171/ZAA/1394
Search in Google Scholar
[9] J. Caballero, A. B. Mingarelli, and K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electron. J. Differential Equations, 57, (2006), 1-11.
Search in Google Scholar
[10] S. Chandrasekhar, Radiative Transfer, Oxford University Press, London, 1950.
Search in Google Scholar
[11] K. Deimling, Nonlinear Functional Analysis, Springer- Verlag, Berlin, 1985.10.1007/978-3-662-00547-7
Search in Google Scholar
[12] S. Hu, M. Khavanin, and W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Analysis, 34, (1989), 261-266.10.1080/00036818908839899
Search in Google Scholar
[13] A. Karoui, On the existence of continuous solutions of nonlinear integral equations, Appl. Math. Lett., 18, (2005), 299-305.10.1016/j.aml.2004.09.007
Search in Google Scholar
[14] P. P. Zabrejko, A. I. Koshelev, M. A. Krasnosel’skii, S. G. Mikhlin, Rakovshchik L. S., and Stecenko V. J., Integral equations, Noordhoff, Leyden, 1975. 10.1007/978-94-010-1909-5_10
Search in Google Scholar

On the Solutions of a Class of Nonlinear Integral Equations in the Banach Algebra of the Continuous Functions and Some Examples

Abstract

Paradigm

My account