Iterative schemes for numerical reckoning of fixed points of new nonexpansive mappings with an application

Kifayat Ullah; Junaid Ahmad; Hasanen A. Hammad; Reny George; Kifayat Ullah; Junaid Ahmad; Hasanen A. Hammad; Reny George

doi:10.3934/math.2023543

AIMS Mathematics

2023, Volume 8, Issue 5: 10711-10727. doi: 10.3934/math.2023543

Previous Article Next Article

Research article

Iterative schemes for numerical reckoning of fixed points of new nonexpansive mappings with an application

1.
Department of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat 28420, Khyber Pakhtunkhwa, Pakistan
2.
Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad-44000, Pakistan
3.
Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Buraydah 52571, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
5.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia

Received: 13 December 2022 Revised: 12 February 2023 Accepted: 16 February 2023 Published: 03 March 2023
MSC : 47H09, 47H10

The goal of this manuscript is to introduce a new class of generalized nonexpansive operators, called $ (\alpha, \beta, \gamma) $-nonexpansive mappings. Furthermore, some related properties of these mappings are investigated in a general Banach space. Moreover, the proposed operators utilized in the $ K $-iterative technique estimate the fixed point and examine its behavior. Also, two examples are provided to support our main results. The numerical results clearly show that the $ K $-iterative approach converges more quickly when used with this new class of operators. Ultimately, we used the $ K $-type iterative method to solve a variational inequality problem on a Hilbert space.
- generalized operator,
- demiclosed principle,
- fixed point,
- convergence result,
- Banach space
Citation: Kifayat Ullah, Junaid Ahmad, Hasanen A. Hammad, Reny George. Iterative schemes for numerical reckoning of fixed points of new nonexpansive mappings with an application[J]. AIMS Mathematics, 2023, 8(5): 10711-10727. doi: 10.3934/math.2023543

Related Papers:

Abstract

The goal of this manuscript is to introduce a new class of generalized nonexpansive operators, called $ (\alpha, \beta, \gamma) $-nonexpansive mappings. Furthermore, some related properties of these mappings are investigated in a general Banach space. Moreover, the proposed operators utilized in the $ K $-iterative technique estimate the fixed point and examine its behavior. Also, two examples are provided to support our main results. The numerical results clearly show that the $ K $-iterative approach converges more quickly when used with this new class of operators. Ultimately, we used the $ K $-type iterative method to solve a variational inequality problem on a Hilbert space.

References

[1]	S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math., 3 (1922), 133–181. https://cir.nii.ac.jp/crid/1570572700413342720
[2]	D. R. Sahu, D. O'Regan, R. P. Agarwal, Fixed point theory for Lipschitzian-type mappings with applications series, New York: Springer, 2009. https://doi.org/10.1007/978-0-387-75818-3
[3]	W. Takahashi, Nonlinear functional analysis, Yokohoma: Yokohoma Publishers, 2000.
[4]	H. A. Hammad, M. De la Sen, Tripled fixed point techniques for solving system of tripled-fractional differential equations, Aims Math., 6 (2021), 2330–2343. https://doi:10.3934/math.2021141 doi: 10.3934/math.2021141
[5]	H. A. Hammad, M. Zayed, Solving a system of differential equations with infinite delay by using tripled fixed point techniques on graphs, Symmetry, 14 (2022), 1388. https://doi.org/10.3390/sym14071388 doi: 10.3390/sym14071388
[6]	K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry and nonexpansive mappings, New York: Marcel Dekkae, 1984.
[7]	F. E. Browder, Nonexpansive nonlinear operators in a Banach space, P. Natl. A. Sci., 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041 doi: 10.1073/pnas.54.4.1041
[8]	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
[9]	W. A. Kirk, A fixed point theorem for mappings which do not increase distance, Am. Math. Mon., 72 (1965), 1004–1006. https://doi.org/10.2307/2313345 doi: 10.2307/2313345
[10]	J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396–414. https://doi.org/10.2307/1989630 doi: 10.2307/1989630
[11]	Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597. https: file: ///C: /Users/Hp/Downloads/1183528964.pdf
[12]	H. F. Senter, W. G. Dotson, Approximating fixed points of non-expansive mappings, Proc. Amer. Math. Soc., 44 (1974), 375–380. https://doi.org/10.2307/2040440 doi: 10.2307/2040440
[13]	J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, B. Aust. Math. Soc., 43 (1991), 153–159. https://doi:10.1017/S0004972700028884 doi: 10.1017/S0004972700028884
[14]	X. Zhang, L. Dai, Image enhancement based on rough set and fractional order differentiator, Fractal Fract., 6 (2022), 214. https://doi.org/10.3390/fractalfract6040214 doi: 10.3390/fractalfract6040214
[15]	J. X. Zhang, G. H. Yang, Fault-tolerant output-constrained control of unknown Euler-Lagrange systems with prescribed tracking accuracy, Automatica, 111 (2020), 108606. https://doi.org/10.1016/j.automatica.2019.108606 doi: 10.1016/j.automatica.2019.108606
[16]	W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510.
[17]	S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. https://doi.org/10.2307/2039245 doi: 10.2307/2039245
[18]	M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229. https://doi.org/10.1006/jmaa.2000.7042 doi: 10.1006/jmaa.2000.7042
[19]	R. P. Agarwal, D. O'Regon, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex A., 8 (2007), 61–79.
[20]	M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vestn., 66 (2014), 223–234.
[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]	N. Hussain, K. Ullah, M. Arshad, Fixed point approximation of Suzuki generalized non-expansive mappings via new faster iteration process, J. Nonlinear Convex A., 19 (2018), 1383–1393. https://doi.org/10.48550/arXiv.1802.09888 doi: 10.48550/arXiv.1802.09888
[23]	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
[24]	F. Facchinei, J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, New York: Springer, 2003.
[25]	I. Konnov, Combined relaxation methods for variational inequalities, Berlin: Springer-Verlag, 2001.
[26]	G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), 747–756.
[27]	C. Martinez-Yanes, H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. Theor., 64 (2006), 2400–2411. https://doi.org/10.1016/j.na.2005.08.018 doi: 10.1016/j.na.2005.08.018
[28]	H. A. Hammad, H. U. Rahman, M. De la Sen, Shrinking projection methods for accelerating relaxed inertial Tseng-type algorithm with applications, Math. Probl. Eng., 2020 (2020), 7487383. https://doi.org/10.1155/2020/7487383 doi: 10.1155/2020/7487383
[29]	T. M. Tuyen, H. A. Hammad, Effect of shrinking projection and $CQ$-methods on two inertial forward–backward algorithms for solving variational inclusion problems, Rend. Circ. Mat. Palerm. Ser. Ⅱ, 70 (2021), 1669–1683. https://doi.org/10.1007/s12215-020-00581-8 doi: 10.1007/s12215-020-00581-8
[30]	H. A. Hammad, W. Cholamjiak, D. Yambangwai, H. Dutta, A modified shrinking projection methods for numerical reckoning fixed points of $G$-nonexpansive mappings in Hilbert spaces with graph, Miskolc Math. Notes, 20 (2019), 941–956. https://doi.org/10.18514/MMN.2019.2954 doi: 10.18514/MMN.2019.2954
[31]	H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, New York: Springer, 2011. https://doi.org/10.1007/978-1-4419-9467-7
[32]	H. H. Bauschke, J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1096), 367–426. https://doi.org/10.1137/S0036144593251710 doi: 10.1137/S0036144593251710
[33]	P. Chen, J. Huang, X. Zhang, A primal-dual fixed point algorithm for convex separable minimization with applications to image restoration, Inverse Probl., 29 (2013), 025011. https://doi.org/10.1088/0266-5611/29/2/025011 doi: 10.1088/0266-5611/29/2/025011
[34]	Y. Dang, J. Sun, H. Xu, Inertial accelerated algorithms for solving a split feasibility problem, J. Ind. Manag. Optim., 13 (2017), 1383-1394. https://doi.org/10.3934/jimo.2016078 doi: 10.3934/jimo.2016078

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)