Solvability of infinite system of general order differential equations via generalized Meir-Keeler condensing operator and semi-analytic method

Document Type : Research Paper

Authors

1 Department of Mathematics, Cotton University, Panbazar, Guwahati-781001, Assam, India

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

3 Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India

4 Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Abstract

In this article, a new fixed point theory and generalized condensing operator have been established to prove the existence of solutions for an infinite system of differential equations of $n^{th}$ order. Also, some interesting examples are employed to support the findings. To validate our discussion the solutions of the examples are approximated by an iterative algorithm with high accuracy. The algorithm is convergent and constructed based on the modified homotopy perturbation method.

Keywords

[1] A. Aghajani, M. Mursaleen and A.S. Haghighi, Fixed point theorems for Meir-Keeler condensing operators via MNC, Acta. Math. Sci. 35 (2015), no. 3, 552–566.

[2] A. Alotaibi, M. Mursaleen, Badriah A.S. Alamri, Solvability of second order linear differential equations in the sequence space n(ϕ),, Adv. Differ. Equ. 2018 (2018), no. 1, 1–8.

[3] A. Alotaibi, M. Mursaleen and S.A. Mohiuddine, Application of measure of noncompactness to infinite system of linear equations in sequence spaces, Bull. Iran. Math. Soc. 41 (2015), 519–527.

[4] J. Bana´s and K. Goebel, Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60, Marcel Dekker, New York, 1980.

[5] J. Bana´s and M. Lecko, Solvability of infinite systems of differential equations in Banach sequence spaces, J. Comput. Appl. Math. 137 (2001), 363–375.

[6] A. Das, M. Rabbani, B. Hazarika and R. Arab, Solvability of infinite systems of nonlinear singular integral equations in the C(I × I, c) space and using semi-analytic method to find a closed-form of solution, Int. J. Nonlinear Anal. Appl. 10 (2019), no. 1, 63–76.

[7] D.G. Duffy, Green’s function with applications, Chapman and Hall/CRC, London, (2001).

[8] L.S. Goldenˇstein and A.S. Markus, On a measure of noncompactness of bounded sets and linear operators, Studies in Algebra and Mathematical Analysis, Kishinev, 1965.

[9] Bipan Hazarika, H.M. Srivastava, Reza Arab and M. Rabbani, Existence of solution for an infinite system of nonlinear integral equations via measure of noncompactness and homotopy perturbation method to solve it, J. Comput. Appl. Math. 343 (2018), 341—352.

[10] B. Hazarika, H.M. Srivastava, R. Arab and 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.

[11] E. Malkowsky and M. Mursaleen, Matrix transformations between FK-spaces and the sequence spaces m(ϕ) and n(ϕ), J. Math. Anal. Appl. 196 (1995), no. 2, 659–665.

[12] E. Malkowsky and M. Mursaleen, Compact matrix operators between the spaces m(ϕ) and n(ϕ), Bull. Korean Math. Soc. 48 (2011), no. 5, 1093–1103.

[13] M. Mursaleen, S.M. H. Rizvi and B. Samet, Solvability of a class of boundary value problems in the space of convergent sequences, Appl. Anal. 97 (2018), no. 11, 1829–1845.

[14] M. Mursaleen, E. Pourhadi and R. Saadati, Solvability of infinite systems of second-order differential equations with boundary conditions in ℓp, Q. Math. 43 (2020), no. 9, 1311–1330.

[15] S.A. Mohiuddine, H.M. Srivastava and A. Alotaibi, Application of measures of noncompactness to the infinite system of second-order differential equations in ℓp spaces, Adv. Difference Equ. 2016 (2016), Article 317.

[16] M. Mursaleen and S.A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in ℓp spaces, Nonlinear Anal. 75 (2012), 2111–2115.

[17] M. Mursaleen and S.M.H. Rizvi, Solvability of infinite systems of second order differential equations in c0 and ℓ1 by Meir-Keeler condensing operators, Proc. Amer. Math. Soc. 144 (2016), no. 10, 4279–4289.

[18] M. Rabbani, Modified homotopy method to solve non-linear integral equations, Int. J. Nonlinear Anal. Appl. 6 (2015), no. 2, 133–136.

[19] M. Rabbani and R. Arab, Extension of some theorems to find solution of nonlinear integral equation and homotopy perturbation method to solve it, Math. Sci. 11 (2017), no. 2, 87—94.

[20] W.L.C. Sargent, On compact matrix transformations between sectionally bounded BK-spaces, J. London Math. Soc. 41 (1966), 79–87.
Volume 14, Issue 2
February 2023
Pages 233-246
  • Receive Date: 14 April 2022
  • Accept Date: 19 July 2022