Iterative algorithm for solving monotone inclusion and fixed point problem of a finite family of demimetric mappings

Anjali; Seema Mehra; Renu Chugh; Salma Haque; Nabil Mlaiki; Anjali; Seema Mehra; Renu Chugh; Salma Haque; Nabil Mlaiki

doi:10.3934/math.2023986

AIMS Mathematics

2023, Volume 8, Issue 8: 19334-19352. doi: 10.3934/math.2023986

Previous Article Next Article

Research article Special Issues

Iterative algorithm for solving monotone inclusion and fixed point problem of a finite family of demimetric mappings

1.
Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, India
2.
Department of Mathematics, Gurugram University, Gurugram 122413, India
3.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia

Received: 20 April 2023 Revised: 25 May 2023 Accepted: 01 June 2023 Published: 08 June 2023
MSC : 47H05, 47H10, 47J25, 65K15

The goal of this study is to develop a novel iterative algorithm for approximating the solutions of the monotone inclusion problem and fixed point problem of a finite family of demimetric mappings in the context of a real Hilbert space. The proposed algorithm is based on the inertial extrapolation step strategy and combines forward-backward and Tseng's methods. We introduce a demimetric operator with respect to $ M $-norm, where $ M $ is a linear, self-adjoint, positive and bounded operator. The algorithm also includes a new step for solving the fixed point problem of demimetric operators with respect to the $ M $-norm. We study the strong convergence behavior of our algorithm. Furthermore, we demonstrate the numerical efficiency of our algorithm with the help of an example. The result given in this paper extends and generalizes various existing results in the literature.
- fixed point problem,
- monotone inclusion problem,
- strong convergence,
- forward-backward splitting algorithm
Citation: Anjali, Seema Mehra, Renu Chugh, Salma Haque, Nabil Mlaiki. Iterative algorithm for solving monotone inclusion and fixed point problem of a finite family of demimetric mappings[J]. AIMS Mathematics, 2023, 8(8): 19334-19352. doi: 10.3934/math.2023986

Related Papers:

Abstract

The goal of this study is to develop a novel iterative algorithm for approximating the solutions of the monotone inclusion problem and fixed point problem of a finite family of demimetric mappings in the context of a real Hilbert space. The proposed algorithm is based on the inertial extrapolation step strategy and combines forward-backward and Tseng's methods. We introduce a demimetric operator with respect to $ M $-norm, where $ M $ is a linear, self-adjoint, positive and bounded operator. The algorithm also includes a new step for solving the fixed point problem of demimetric operators with respect to the $ M $-norm. We study the strong convergence behavior of our algorithm. Furthermore, we demonstrate the numerical efficiency of our algorithm with the help of an example. The result given in this paper extends and generalizes various existing results in the literature.

References

[1]	A. B. Abubakar, K. Muangchoo, A. H. Ibrahim, A. B. Muhammad, L. O. Jolaoso, K. O. Aremu, A new three-term Hestenes-Stiefel type method for nonlinear monotone operator equations and image restoration, IEEE Access, 9 (2021), 18262–18277. https://doi.org/10.1109/ACCESS.2021.3053141 doi: 10.1109/ACCESS.2021.3053141
[2]	P. L. Combettes, V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Sim., 4 (2005), 1168–1200. https://doi.org/10.1137/050626090 doi: 10.1137/050626090
[3]	V. Dadashi, M. Postolache, Forward–backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators, Arab. J. Math., 9 (2020), 89–99. https://doi.org/10.1007/s40065-018-0236-2 doi: 10.1007/s40065-018-0236-2
[4]	A. Dixit, D. R. Sahu, P. Gautam, T. Som, J. C. Yao, An accelerated forward-backward splitting algorithm for solving inclusion problems with applications to regression and link prediction problems, J. Nonlinear Var. Anal., 5 (2021), 79–101. https://doi.org/10.23952/jnva.5.2021.1.06 doi: 10.23952/jnva.5.2021.1.06
[5]	A. Hanjing, S. Suantai, A fast image restoration algorithm based on a fixed point and optimization method, Mathematics, 8 (2020), 378. https://doi.org/10.3390/math8030378 doi: 10.3390/math8030378
[6]	K. Janngam, S. Suantai, An accelerated forward-backward algorithm with applications to image restoration problems, Thai J. Math., 19 (2021), 325–339.
[7]	D. Kitkuan, K. Muangchoo, Inertial relaxed CQ algorithm with an application to signal processing, Thai J. Math., 18 (2020), 1091–1103.
[8]	D. Kitkuan, P. Kumam, J. Martínez-Moreno, Generalized Halpern-type forward–backward splitting methods for convex minimization problems with application to image restoration problems, Optimization, 69 (2020), 1557–1581. 10.1080/02331934.2019.1646742 doi: 10.1080/02331934.2019.1646742
[9]	G. López, V. Martín-Márquez, F. Wang, H. K. Xu, Forward-backward splitting methods for accretive operators in Banach spaces, Abstract Appl. Anal., 2012 (2012), 109236. https://doi.org/10.1155/2012/109236 doi: 10.1155/2012/109236
[10]	D. A. Lorenz, T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vis., 51 (2015), 311–325. https://doi.org/10.1007/s10851-014-0523-2 doi: 10.1007/s10851-014-0523-2
[11]	D. R. Sahu, A. Pitea, M. Verma, A new iteration technique for nonlinear operators as concerns convex programming and feasibility problems, Numer. Algor., 83 (2020), 421–449. https://doi.org/10.1007/s11075-019-00688-9 doi: 10.1007/s11075-019-00688-9
[12]	K. Sitthithakerngkiet, J. Deepho, P. Kumam, A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems, Appl. Math. Comput., 250 (2015), 986–1001. https://doi.org/10.1016/j.amc.2014.10.130 doi: 10.1016/j.amc.2014.10.130
[13]	S. Sra, S. Nowozin, S. J. Wright, Optimization for machine learning, 2012.
[14]	P. Sunthrayuth, P. Cholamjiak, Iterative methods for solving quasi-variational inclusion and fixed point problem in q-uniformly smooth Banach spaces, Numer. Algor., 78 (2018), 1019–1044. https://doi.org/10.1007/s11075-017-0411-0 doi: 10.1007/s11075-017-0411-0
[15]	P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431–446. https://doi.org/10.1137/S0363012998338806 doi: 10.1137/S0363012998338806
[16]	G. I. Usurelu, M. Postolache, Algorithm for generalized hybrid operators with numerical analysis and applications, J. Nonlinear Var. Anal., 6 (2022), 255–277.
[17]	Y. Dong, X. Zhu, An inertial splitting method for monotone inclusions of three operators, Int. J. Math. Stat. Oper. Res., 2 (2022), 43–60. https://DOI:10.47509/IJMSOR.2022.v02i01.04 doi: 10.47509/IJMSOR.2022.v02i01.04
[18]	P. L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964–979. https://doi.org/10.1137/0716071 doi: 10.1137/0716071
[19]	A. Gibali, D. V.Thong, Tseng type method for solving inclusion problems and its applications, Calcolo, 55 (2018), 49. https://doi.org/10.1007/s10092-018-0292-1 doi: 10.1007/s10092-018-0292-1
[20]	A. Moudafi, M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 155 (2003), 447–454. https://doi.org/10.1016/S0377-0427(02)00906-8 doi: 10.1016/S0377-0427(02)00906-8
[21]	E. Altiparmak, I. Karahan, A new preconditioning algorithm for finding a zero of the sum of two monotone operators and its application to image restoration problems, Int. J. Comput. Math., 99 (2022), 2482–2498. https://doi.org/10.1080/00207160.2022.2068146 doi: 10.1080/00207160.2022.2068146
[22]	E. Altiparmak, I. Karahan, A Modified Preconditioning Algorithm for solving monotone inclusion problem and application to image restoration problem, U. Politeh. Buch. Ser. A, 84 (2022), 81–92.
[23]	W. Shatanawi, T. A. Shatnawi, New fixed point results in controlled metric type spaces based on new contractive conditions, AIMS Math., 8 (2023), 9314–9330. https://doi.org/10.3934/math.2023468 doi: 10.3934/math.2023468
[24]	A. Z. Rezazgui, A. A. Tallafha, W. Shatanawi, Common fixed point results via $A\nu$-$\alpha$-contractions with a pair and two pairs of self-mappings in the frame of an extended quasi b-metric space, AIMS Math., 8 (2023), 7225–7241. https://doi.org/10.3934/math.2023363 doi: 10.3934/math.2023363
[25]	M. Joshi, A. Tomar, T. Abdeljawad, On fixed points, their geometry and application to satellite web coupling problem in $S$-metric spaces, AIMS Math., 8 (2023), 4407–4441. https://doi.org/10.3934/math.2023220 doi: 10.3934/math.2023220
[26]	P. Debnath, Z. D. Mitrović, S. Y. Cho, Common fixed points of Kannan, Chatterjea and Reich type pairs of self-maps in a complete metric space, São Paulo J. Math. Sci., 15 (2021), 383–391. https://doi.org/10.1007/s40863-020-00196-y doi: 10.1007/s40863-020-00196-y
[27]	P. Debnath, Banach, Kannan, Chatterjea and Reich‐type contractive inequalities for multivalued mappings and their common fixed points, Math. Method. Appl. Sci., 45 (2022), 1587–1596. https://doi.org/10.1002/mma.7875 doi: 10.1002/mma.7875
[28]	P. Debnath, New common fixed point theorems for Gornicki-type mappings and enriched contractions, São Paulo J. Math. Sci., 16 (2022), 1401–1408. https://doi.org/10.1007/s40863-022-00283-2 doi: 10.1007/s40863-022-00283-2
[29]	B. T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Comp. Math. Math. Phys., 4 (1964), 1–17. https://doi.org/10.1016/0041-5553(64)90137-5 doi: 10.1016/0041-5553(64)90137-5
[30]	H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, New York: Springer, 2011.
[31]	A. Cegielski, Iterative methods for fixed point problems in Hilbert spaces, Springer, 2012.
[32]	B. V. Limaye, Functional Analysis, New Age International, 1996.
[33]	K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, 1990. https://doi.org/10.1017/CBO9780511526152
[34]	S. Takahashi, W. Takahashi, The split common fixed point problem and the shrinking projection method in Banach spaces, Optimization, 65 (2016), 281–287. https://doi.org/10.1080/02331934.2015.1020943 doi: 10.1080/02331934.2015.1020943
[35]	H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256. https://doi.org/10.1112/S0024610702003332 doi: 10.1112/S0024610702003332
[36]	P. E. Maingé, A hybrid extra gradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499–1515. https://doi.org/10.1137/060675319 doi: 10.1137/060675319
[37]	W. Takahashi, C. F. Wen, J. C. Yao, The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space, Fixed Point Theor., 19 (2018), 407-419. 10.24193/fpt-ro.2018.1.32 doi: 10.24193/fpt-ro.2018.1.32

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)