On a solution of a Delay Differential Equation, via a New Iteration Scheme

Manoj Kumar; Hemant Kumar Pathak

doi:https://doi.org/10.14445/22315373/IJMTT-V69I2P516

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 69 | Issue 2 | Year 2023 | Article Id. IJMTT-V69I2P516 | DOI : https://doi.org/10.14445/22315373/IJMTT-V69I2P516

On a solution of a Delay Differential Equation, via a New Iteration Scheme

Manoj Kumar, Hemant Kumar Pathak

Received	Revised	Accepted	Published
09 Jan 2023	11 Feb 2023	21 Feb 2023	28 Feb 2023

Abstract

In this paper, results on stability and data dependency for a new iteration scheme under contractive- like mappings are discussed. In the framework of uniformly convex Banach spaces, we also provide weak and strong convergence results for mappings satisfying the condition βγ,μ. To validate our proofs, numerical example are also provided which are supported by graphs and tables. Finally, we exhibit the applicability of our three-step iteration process in delay differential equations.

Keywords

Contractive-like mappings, Bγ,μ condition, stability, data dependence, fixed points, iterative algorithm, weak and strong convergence, delay differential equation.

References

[1] M. Abbas and T. Nazir A new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesn., 66(2), 223-234, (2014).
[2] T. Abdeljawad, K. Ullah and J. Ahmad, On Picard{Krasnoselskii Hybrid Iteration Process in Banach Spaces, Journal of Mathematics, Hindawi, 2020, 1-5, 2020.
[3] T. Abdeljawad, K. Ullah and J. Ahmad, Iterative Algorithm for Mappings Satisfying Bγ,μ Condition, Journal of Function Spaces Hindawi, (2020), 1-7, (2020).
[4] T. Abdeljawad, K. Ullah, J. Ahmad, M. D. L. Sen, and A. Ulhaq, Approximation of fixed points and best proximity points of relatively nonexpansive mappings, Journal of Mathematics, Hindawi, 2020, 1- 11, (2020).
[5] R.P. Agarwal, D. O'Regan and D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal., 8(1), 61-79, (2007).
[6] V. Berinde, On the stability of some fixed point procedures, Buletinul stiintific al Universitatii Baia Mare, Seria B, Fascicola matematica-Informatica, 18(1), 7-14, (2002).
[7] V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, (2007).
[8] O. Christopher, Imoru, O. Olatinwo, On the stability of Picard and Mann iteration processes, Carpathian Journal of Mathematics, 19, 155-160, (2003).
[9] G. H. Coman, G. Pavel, I. Rus and I. A. Rus, Introduction in the theory of operational equation, Ed. Dacia, Cluj-Napoca, (1976).
[10] J. Garcia-Falset, E. Llorens-Fuster and T. Suzuki, Fixed point theory for a class of generalized nonex- pansive mappings, J. Math. Anal. Appl., 375(1), 185-195, (2011).
[11] C. Garodia and I. Uddin, A new iterative method for solving split feasibility problem, Journal of applied amalysis and computation, 3, 986-1004, (2020).
[12] C. Garodia and I. Uddin, A new fixed point algorithm for finding the solution of a delay differential equation. AIMS Mathematics, 5(4), 3182{3200, (2020).
[13] F. Gursoy and V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, 16, (2014).
[14] A. M. Harder and T. L. Hicks, A stable iteration procedure for non-expansive mappings. Math Japon, 33, 687{692, (1988)
[15] N. Hussain, K. Ullah and M. Arshad, Fixed point approximation of Suzuki generalized non-expansive mappings via new faster iteration process, Journal of nonlinear and convex analysis, 19(8), 1383{1393, (2018).
[16] S. Ishikawa, Fixed points by a new iteration method. Proc. Am. Math. Soc., 44, 147-150, (1974).
[17] W.R. Mann, Mean value methods in iteration. Proc. Am. Math. Soc., 4, 506-510, (1953). 22
[18] M. Kumar and H. K. Pathak, Numerical reckoning of fixed points for generalized nonexpansive mappings in CAT(0) spaces with applications. U. P. B. Sci. Bull., Series A, 84(2), 117-128, (2022).
[19] M.A. Noor, New approximation schemes for general variational inequalities. J. Math. Anal. Appl., 251(1), 217-229, (2000).
[20] A. Ofem and U. E. Udofia, Iterative solutions for common fixed points of nonexpansive mappings and strongly pseudocontractive mappings with applications. Canad. J. Appl. Math., 3(1), 18-36, (2021).
[21] G. A. Okeke and M. Abbas, A solution of delay differential equations via Picard{Krasnoselskii hybrid iterative process. Arab. J. Math., 2017(6), 21-29, (2017).
[22] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc., 73, 591-597, (1967).
[23] M. O. Osilike, Stability results for fixed point iteration procedures. J Nigerian Math Soc., 14, 17{29, 1995.
[24] A. M. Ostrowski, The round-off stability of iterations. Z Angew Math Mech., 47, 77{81, (1967).
[25] R. Pant and R. Shukla, Approximating fixed point of generalized α nonexpansive mappings in Banach spaces. Numer. Funct. Anal. Optim., 38, 248-266, (2017).
[26] B. Patir, N. Goswami and V. N . Mishra, Some results on fixed point theory for a class of generalized nonexpansive mappings. Fixed Point Theory and Applications, 2018(1), 1-18, (2018).
[27] H. Piri, B. Daraby, S. Rahrovi and M. Ghasemi, Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces by new faster iteration process. Numer. Algorithm, 81, 1129- 1148 (2019).
[28] B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures. Indian J. Pure Applied Math., 21, 1-9, (1990).
[29] H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44, 375-380, (1974).
[30] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Appl., 340, 1088-1095, (2008).
[31] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Austral. Math. Soc., 43, 153-159, (1991). [32] S. M. Soltuz, Data dependence for Ishikawa iteration. Lecturas Matematicas, 25(2), 149{ 155, (2004).
[33] S.M. Soltuz and D. Otrocol, Classical results via Mann{Ishikawa iteration. Revue d'Analyse Numerique et de Theorie de l'Approximation. 36(2), 195{199, (2007).
[34] S. M. Soltuz, Grosan, T., Data dependence for Ishikawa iteration when dealing with contractive-like operators. Hindawi publishing corporation fixed point theory and applications, 2008, 1- 7, (2008).
[35] D. Thakur, B. S. Thakur and M. Postolache, New iteration scheme for numerical reckoning fixed points of nonexpansive mappings, Journal of Inequalities and Applications, (2014), 1-15, (2014).
[36] D. Thakur, B. S. Thakur and M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, Appl. Math. Comp., 275, 147-155, (2016).
[37] K. Ullah and M. Arshad, New iteration process and numerical reckoning fixed points in Banach spaces, U. P. B. Sci. Bull., Series A, 79(4), 113-122, (2017).
[38] K. Ullah and M. Arshad, Numerical reckoning fixed points for Suzuki's generalized non-expansive map- pings via new iteration process, Filomat, 32(1), 187-196, (2018).
[39] K. Ullah and M. Arshad, New three step iteration process and fixed point approximation in Banach spaces, Journal of Linear and Topological Algebra, 7(2), 87-100, (2018).
[40] H. Y. Zhou, Stable iteration procedures for of strong pseudocontractions and nonlinear equations. involving accretive operators without Lipschitz assumption. J Math Anal appl., 230, 1-10, (1999).

Citation :

Manoj Kumar, Hemant Kumar Pathak, "On a solution of a Delay Differential Equation, via a New Iteration Scheme," International Journal of Mathematics Trends and Technology (IJMTT), vol. 69, no. 2, pp. 124-146, 2023. Crossref, https://doi.org/10.14445/22315373/IJMTT-V69I2P516