International Journal of Nonlinear Analysis and Applications

Solvability and numerical method for non-linear Volterra integral equations by using Petryshyn’s fixed point theorem

Document Type : Research Paper

Authors

1 PDPM-IIITDM, Jabalpur 482005, India

2 Govt. PG college Bina 470113, India

3 IIT, Mandi 175005, India

4 Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran

Abstract

In this paper, utilizing the technique of Petryshyn’s fixed point theorem in Banach algebra, we analyze the existence of solution for functional integral equations, which includes as special cases many functional integral equations that arise in various branches of non-linear analysis and its application. Finally, we introduce the numerical method formed by modified homotopy perturbation approach to resolving the problem with acceptable accuracy.

Keywords

[1] M. A. Abdou, On the solution of linear and non-linear integral equation, Appl. Math. Comput. 146(2003), 857-871.
[2] G. Adomian, Solving Frontier Problem of Physics: The Decomposition Method; Kluwer Academic Press: Dordrecht the Netherlands, 1994.
[3] A. Aghajani, Y. Jalilian, Existence and global attractivity of solutions of non-linear functional integral equation, Commun. Non. Sci. Number. Simul. 15(2010), 3306-3312.
[4] I. K. Argyros, Quadratic equations and applications to Chandrasekhar and related equations, Bull. Aus. Math. Soc. 32(1985), 275-292.
[5] J. Banas, K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, 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. Banas, K. Sadarangani, Solutions of some functional integral equations in Banach algebra, Math. Comput. Modelling 38 (2003), 245-250.
[9] J. Biazar, H. Ghazvini, He homotopy perturbation method for solving systems of Volterra integral equations of the second kind, Cha. Solit. Frac. 39(2009), 770-777.
[10] J. Caballero, A. B. Mingarelli, K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Elect. J. Diff. Eqs. 57(2006), 1-11.
[11] S. Chandrasekhar, Radiative Transfer, Oxford Univ. Press, London, 1950.
[12] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, New York, 1990.
[13] A Das, M Rabbani, B Hazarika, R Arab, Solvability of infinite system of nonlinear singular integral equations in the C(I × I,c) space and modified semi-analytic method to find a closed-form of solution., Int. J. Non. Anal.Appl. 10(1)(2019), 63-76.
[14] 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, Adv. Diff. Eqs, 2020:483(2020).
[15] A. Deep, Deepmala, R. Ezzati, Application of Petryshyn’s fixed point theorem to solvability for functional integral equations, Appl. Math. Comput. 395(2021), 125878.
[16] Deepmala, H.K.Pathak, A study on some problems on existence of solutions for some nonlinear functional integral equations, Acta. Math. Sci.33(5)(2013),1305-1313.
[17] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.
[18] B. C. Dhage, On α-condensing mappings in Banach algebras, The Math. Student 63 (1994), 146-152.
[19] B. Hazarika, H.M.Srivastava, R. Arab, M. Rabbani, Application of simulation function and measure of noncompactness for solvability of nonlinear functional integral equations and introduction to an iteration algorithm to find solution, Appl. Math. Comput. 360(2019), 131-146.
[20] J. H. He, A new approach to nonlinear partial differential equations., Comm. Nonlinear Sci. Number. Simulation 2(4)(1997), 230-235.
[21] S. Hu, M. Khavanin, W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal. 34 (1989), 261-266.
[22] M. Kazemi, R. Ezzati, Existence of solutions for some nonlinear Volterra integral equations via Petryshyn’s fixed point theorem, Int. J. Anal. Appl. 9(2018).
[23] K. Maleknejad, K. Nouri, R. Mollapourasl, Existence of solutions for some nonlinear integral equations, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009), 2559-2564.
[24] S.A. Mohiuddine, H.M. Srivastava, A. Alotaibi, Applications of measures of noncompactness to the infinite system of second order differential equations in lp spaces, Adv. Differ. Equ. 2016(2016), 1-13.
[25] M. Mursaleen, S.A. Mohiuddine, Applications of noncompactness to the infinite system of differential equations in lp spaces, Nonlinear Anal. 4(2012), 2111-2115.
[26] 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.
[27] D. O’Regan, Existence results for nonlinear integral equations, J. Math. Anal. Appl. 192(1995), 705-726.
[28] I. Ozdemir, U. Cakan, B. Iihan, On the existence of the solution for some nonlinear Volterra integral equations, Abstr. Appl. Anal. 5(2013).
[29] W.V. Petryshyn, Structure of the fixed points sets of k-set-contractions, Arch. Rational Mech. Anal., 40(1970-1971), 312-328.
[30] M. Rabbani, An iterative algorithm to find a closed form of solution for Hammerstein nonlinear integral equation constructed by the concept of cosm-rs, Mathematical Sciences 13 (3), (2019)299-305.
[31] B. Hazarik, M. Rabbani, RP. Agarwal, A. Das, R. Arab, Existence of Solution for Infinite System of Nonlinear Singular Integral Equations and Semi-Analytic Method to Solve it., Iranian Journal of Science and Technology, Transactions A: Science 45(2021), 235-245.
[32] M. Rabbani, A. Gelayeri, Computation Adomian Method for Solving Non-linear Differential Equation in the Fluid Dynamic., Int. J. Mechat., Elect. Compu. Techn. 5(14)(2015), 2039-2043.
[33] J. R. Roshan, Existence of solutions for a class of system of functional integral equation via measure of noncompactness J. Comput. Appl. Math. 313(2017), 129-141.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 1-28
  • Receive Date: 08 March 2021
  • Accept Date: 31 August 2021