Numerical algorithms for solutions of nonlinear problems in some distance spaces

Junaid Ahmad; Kifayat Ullah; Reny George; Junaid Ahmad; Kifayat Ullah; Reny George

doi:10.3934/math.2023426

AIMS Mathematics

2023, Volume 8, Issue 4: 8460-8477. doi: 10.3934/math.2023426

Previous Article Next Article

Research article Special Issues

Numerical algorithms for solutions of nonlinear problems in some distance spaces

1.
Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan
2.
Department of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat 28420, Khyber Pakhtunkhwa, Pakistan
3.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia

Received: 20 November 2022 Revised: 22 December 2022 Accepted: 02 January 2023 Published: 03 February 2023
MSC : 47H09, 47H10

This paper introduces some numerical algorithms for finding solutions of nonlinear problems like functional equations, split feasibility problems (SFPs) and variational inequality problems (VIPs) in the setting of Hilbert and Banach spaces. Our approach is based on the Thakur-Thakur-Postolache (TTP) iterative algorithm and the class of mean nonexpansive mappings. First we provide some convergence results (including weak and strong convergence) in the setting of Banach space. To support these results, we provide a numerical example and prove that our TTP algorithm in this case converges faster to fixed point compared to other iterative algorithms of the literature. After that, we consider two new TTP type projection iterative algorithms to solve SFPs and VIPs on the Hilbert space setting. Our result are new in analysis and suggest new type effective numerical algorithms for finding approximate solutions of some nonlinear problems.
- algorithm,
- fixed point,
- numerical solution,
- Hilbert space,
- Banach space
Citation: Junaid Ahmad, Kifayat Ullah, Reny George. Numerical algorithms for solutions of nonlinear problems in some distance spaces[J]. AIMS Mathematics, 2023, 8(4): 8460-8477. doi: 10.3934/math.2023426

Related Papers:

Abstract

This paper introduces some numerical algorithms for finding solutions of nonlinear problems like functional equations, split feasibility problems (SFPs) and variational inequality problems (VIPs) in the setting of Hilbert and Banach spaces. Our approach is based on the Thakur-Thakur-Postolache (TTP) iterative algorithm and the class of mean nonexpansive mappings. First we provide some convergence results (including weak and strong convergence) in the setting of Banach space. To support these results, we provide a numerical example and prove that our TTP algorithm in this case converges faster to fixed point compared to other iterative algorithms of the literature. After that, we consider two new TTP type projection iterative algorithms to solve SFPs and VIPs on the Hilbert space setting. Our result are new in analysis and suggest new type effective numerical algorithms for finding approximate solutions of some nonlinear problems.

References

[1]	H. A. Abass, C. Izuchukwu, O. T. Mewomo, On split equality generalized equilibrium problems and fixed point problems of Bregman relatively nonexpansive semigroups, J. Appl. Numer. Optim., 4 (2022), 357–380. 10.23952/jano.4.2022.3.04 doi: 10.23952/jano.4.2022.3.04
[2]	M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vestn., 66 (2014), 223–234.
[3]	R. P. Agarwal, D. O'Regan, D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications Series, New York: Springer, 6 (2009).
[4]	R. P. Agarwal, D. O'Regon, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61–79.
[5]	S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math., (1922), 133–181.
[6]	F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci., 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041 doi: 10.1073/pnas.54.4.1041
[7]	C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103–120. https://doi.org/10.1088/0266-5611/20/1/006 doi: 10.1088/0266-5611/20/1/006
[8]	Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221–239.
[9]	J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396–414.
[10]	D. Gohde, Zum Prinzip der Kontraktiven Abbildung, Math. Nachr., 30 (1965), 251–258. https://doi.org/10.1002/mana.19650300312 doi: 10.1002/mana.19650300312
[11]	S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. https://doi.org/10.1090/S0002-9939-1974-0336469-5 doi: 10.1090/S0002-9939-1974-0336469-5
[12]	L. Liu, B. Tan, A. Latif, Approximation of fixed points for a semigroup of Bregman quasi-nonexpansive mappings in Banach spaces, J. Nonlinear Var. Anal., 5 (2021), 9–22. https://doi.org/10.23952/jnva.5.2021.1.02 doi: 10.23952/jnva.5.2021.1.02
[13]	G. Lopez, V. Martín-Ma Themrquez, H. K. Xu, Halpern's iteration for nonexpansive mappings, In: Nonlinear Analysis and Optimization I: Nonlinear Analysis, 513 (2010), 211–231. http://doi.org/10.1090/conm/513/10085
[14]	W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. https://doi.org/10.2307/2032162 doi: 10.2307/2032162
[15]	G. Maniu, On a three-step iteration process for Suzuki mappings with qualitative study, Numer. Funct. Anal. Optim., 41 (2020), 929–949. https://doi.org/10.1080/01630563.2020.1719415 doi: 10.1080/01630563.2020.1719415
[16]	M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229. https://doi.org/10.1006jmaa.2000.70
[17]	Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597.
[18]	J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc., 43 (1991), 153–159. https://doi.org/10.1017/S0004972700028884 doi: 10.1017/S0004972700028884
[19]	H. F. Senter, W. G. Dotson, Approximating fixed points of non-expansive mappings, Proc. Amer. Math. Soc., 44 (1974), 375–380.
[20]	W. Takahashi, Nonlinear Functional Analysis, Yokohoma: Yokohoma Publishers, 2000.
[21]	B. S. Thakur, D. Thakur, M. Postolache, New iteration scheme for approximating fixed point of nonexpansive mappings, Filomat, 30 (2016), 2711–2720. https://doi.org/10.2298/FIL1610711T doi: 10.2298/FIL1610711T
[22]	T. Thianwan, A new iteration scheme for mixed type asymptotically nonexpansive mappings in hyperbolic spaces, J. Nonlinear Funct. Anal., 2021 (2021), 21. https://doi.org/10.23952/jnfa.202 doi: 10.23952/jnfa.202
[23]	C. X. Wu, L. J. Zhang, Fixed points for mean non-expansive mappings, Acta Math. Appl. Sin., 23 (2007), 489–494.
[24]	S. Zhang, About fixed point theory for mean nonexpansive mapping in Banach spaces. J. Sichuan Univ., 2 (1975), 67–78.
[25]	Z. F. Zuo, Fixed-point theorems for mean nonexpansive mappings in Banach spaces, Abstr. Appl. Anal. 2014 (2014), 746291. https://doi.org/10.1155/2014/746291 doi: 10.1155/2014/746291

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)