On existence of solutions for some functional integral equations in Banach algebra by fixed point theorem

Document Type : Research Paper

Author

Department of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, Iran

Abstract

In this research, we analyze the existence of solution for some nonlinear functional integral equations using the techniques of measures of noncompactness and the Petryshyn's fixed point theorem in
Banach space. The results obtained in this paper cover many existence results obtained by numerous
authors under some weaker conditions. We also give an example satisfying the conditions of our main
theorem but not satisfying the conditions described by other authors.

Keywords

[1] R. P. Agarwal, N. Hussain, M.-A. Taoudi, Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations, Abstr. Appl. Anal., vol. 2012, Hindawi 2012.
[2] A. Aghajani, J. Bana´s, Y. Jalilian, Existence of solutions for a class of nonlinear Volterra singular integral equations, Comput. Math. Appl., 62(2011), no. 3, 1215-1227.
[3] I.K. Argyros, Quadratic equations and applications to Chandrasekhars and related equations, Bull. Austral. Math. Soc. 32 (1985) 275-292.
[4] J. Bana´s, Measures of noncompactness in the study of solutions of nonlinear differential and integral equations, Cent. Eur. J. Math., 10(2012), no. 6, 2003-2011.
[5] J. Bana´s , K. Goebel, Measures of noncompactness in Banach spaces, volume 60 of Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 1980.
[6] J. Banas , M. Lecko, Fixed points of the product of operators in Banach algebra, Panamer. Math. J., 12(2002) 101-109.
[7] J. Banas, B. Rzepka, On existence and asymptotic stability of solutions of a nonlinear integral equation, J. Math. Anal. Appl., 284 (2003) 165-173.
[8] J. Bana´s , K. Sadarangani, Solutions of some functional-integral equations in Banach algebra, Math. Comput. Modelling, 38(2003), no. 3-4, 245-250.[9] A. Ben Amar, A. Jeribi, M. Mnif, Some fixed point theorems and application to biological model, Numer. Funct. Anal. Optim., 29(2008), no. 1-2, 1-23.
[10] J. Caballero, A. B. Mingarelli, K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electron. J. Diff. Eq., 57(2006), 1-11.
[11] S. Chandrasekhar, Radiative Transfer, Oxford Univ. Press, London, 1950.
[12] C. Corduneanu, Integral Equations and Applications, Cambridge Univ. Press, New York, 1973.
[13] M. A. Darwish , S. K. Ntouyas, On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument, Nonlinear Anal., 74(2011), no 11, 3510-3517.
[14] A. Das, B. Hazarika, P. Kumam, Some new generalization of Darbo’s fixed point theorem and its applications on integral equations, Mathematics, 7(2019), no. 3, 214.
[15] A. Deep, Deepmala, J. R. Roshan, K. S. Nisar, T. Abdeljawad, An extension of Darbo’s fixed point theorem for a class of system of nonlinear integral equations, Advances in Difference Equations. 2020(2020), no. 1, 1–17.
[16] Deepmala, H.K. Pathak, Study on existence of solutions for some nonlinear functional-integral equations with application, Math. Commun. 18(2013), 97-107.
[17] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
[18] L. S. Goldenˇste˘ın , A. S. Markus. On the measure of non-compactness of bounded sets and of linear operators, Studies in Algebra and Math. Anal. (Russian), Izdat. “Karta Moldovenjaske”, Kishinev, (1965) 45–54 (Russian).
[19] S. Hu, M. Khavani , W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal. 34 (1989) 261-266.
[20] C. T. Kelley, Approximation of solutions of some quadratic integral equations in transport theory, J. Integral Eq. 4 (1982) 221-237.
[21] M. Kazemi, R. Ezzati, Existence of solution for some nonlinear two-dimensional volterra integral equations via measures of noncompactness, Appl. Math. Comput., 275 (2016) 165-171.
[22] K. Kuratowski. Sur les espaces completes Fund. Math., 15(1930) 301–335.
[23] K. Maleknejad, R. Mollapourasl, K. Nouri. Study on existence of solutions for some nonlinear functional-integral equations, Nonlinear Anal., 69(8) (2008) 2582-2588.
[24] K. Maleknejad, K. Nouri, , R. Mollapourasl. Existence of solutions for some nonlinear integral equations Commun. Nonlinear Sci. Numer. Simul., 14(2009), no. 6, 2559-2564.
[25] K. Maleknejad, K. Nouri, R. Mollapourasl. Invgatestiion on the existence of solutions for some nonlinear functional integral equations Nonlinear Anal., 71(2009), no. 12, 1575-1578.
[26] L. N. Mishra, R. P. Agarwal, On existence theorems for some nonlinear functional-integral equations. Dynamic systems and Applications, 25 (2016), no. 3 303-320.
[27] L. N. Mishra, M. Sen , R. N. Mohapatra, On existence theorems for some generalized nonlinear functional-integral equations with applications, Filomat, 31(2017), no. 7, 2081-2091.
[28] N. I. Muskhelishvili. Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending. P. Noordhoff, Ltd., Groningen, 1953. Translated by J. R. M. Radok.
[29] R. D. Nussbaum. The fixed-point index and fixed point theorem for k-set contractions. ProQuest LLC, Ann Arbor, MI, 1969, Thesis (Ph.D.)–The University of Chicago.
[30] D. O’Regan, Existence theory for nonlinear Volterra integrodifferential and integral equations, Nonlinear Anal. 31 (1998) 317-341.
[31] ˙I. Ozdemir, ¨ U. C¸ akan, B. ¨ ˙Ilhan. On the existence of the solutions for some nonlinear Volterra integral equations Abstr. Appl. Anal., vol. 2013, Hindawi, 2013.
[32] ˙I. Ozdemir, B. ¨ ˙Ilhan, U. C¸ akan, ¨ On the solutions of a class of nonlinear integral equations in Banach algebra of the continuous functions and some examples, An. Univ. Vest Timi Ser. Mat.-Inform., (2014) 121–140.
[33] ˙I. Ozdemir, ¨ U. C¸ akan, ¨ The solvability of some nonlinear functional integral equations, Studia Sci. Math. Hunger. 53(2016), 7-21.
[34] D. H. K. Pathak, A study on some problems on existence of solutions for nonlinear functional- integral equations, Acta Math. Scientia, 33(2013) 1305-1313.
[35] W. V. Petryshyn. Structure of the fixed points sets of k-set-contractions Arch. Rational Mech. Anal., 40(1971), no. 4, 312-328.
[36] M. Rabbani, R. Arab, B. Hazarika, Solvability of nonlinear quadratic integral equation by using simulation type condensing operator and measure of noncompactness, Appl. Math. Comput., 349(2019), 102-117.
[37] M. Rabbani, A. Das, B. Hazarika, R. Arab, Existence of solution for two dimensional non-linear fractional integral equation by measure of noncompactness and iterative algorithm to solve it, J. Comput. App. Math., 370(2020), 112654, 1-17.
[38] M. Rabbani, A. Deep,On some generalized non-linear functional integral equations of two variables via measuresof noncompactness and numerical method to solve it, Mathematical Sciences (2021) 1-8.
[39] A. G. Ramm. Dynamical systems method for solving operator equations, volume 208 of Mathematics in Science and Engineering. Elsevier B. V., Amsterdam, 2007.
[40] S. Singh, B. Watson, P. Srivastava. Fixed point theory and best approximation: the KKM-map principle, volume 424 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 1997.
Volume 13, Issue 1
March 2022
Pages 451-466
  • Receive Date: 09 June 2021
  • Revise Date: 04 July 2021
  • Accept Date: 21 August 2021