Please wait a minute...
Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

Postal Subscription Code 80-964

2018 Impact Factor: 0.565

Front. Math. China    2022, Vol. 17 Issue (5) : 797-811    https://doi.org/10.1007/s11464-022-1027-9
RESEARCH ARTICLE
Implicit iterative algorithms of the split common fixed point problem for Bregman quasi-nonexpansive mapping in Banach spaces
Yuanheng WANG1(), Chanjuan PAN2
1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China
2. Department of Basic Teaching, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018, China
 Download: PDF(555 KB)   HTML
 Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks
Abstract

In this paper, we study a modified implicit rule for finding a solution of split common fixed point problem of a Bregman quasi-nonexpansive mapping in Banach spaces. We propose a new iterative algorithm and prove the strong convergence theorem under appropriate conditions. As an application, the results are applied to solving the zero problem and the equilibrium problem.

Keywords Split common fixed point      implicit rule      Bregman quasi-nonexpansive mapping      strong convergence      Banach space     
Corresponding Author(s): Yuanheng WANG   
Online First Date: 22 December 2022    Issue Date: 28 December 2022
 Cite this article:   
Yuanheng WANG,Chanjuan PAN. Implicit iterative algorithms of the split common fixed point problem for Bregman quasi-nonexpansive mapping in Banach spaces[J]. Front. Math. China, 2022, 17(5): 797-811.
 URL:  
https://academic.hep.com.cn/fmc/EN/10.1007/s11464-022-1027-9
https://academic.hep.com.cn/fmc/EN/Y2022/V17/I5/797
1 H H Bauschke, J M Borwein, P L Combettes. Bregman monotone optimization algorithms. SIAM J Control Optim 2003; 42(2): 596–636
2 G Cai. Viscosity iterative algorithms for a new variational inequality problem and fixed point problem in Hilbert spaces. Acta Math Sin Chin Ser 2019; 62(5): 765–776
3 Y Censor, A Segal. The split common fixed point problem for directed operators. J Convex Anal 2009; 16(2): 587–600
4 J Z Chen, H Y Hu, L C Ceng. Strong convergence of hybrid Bergman projection algorithm for split feasibility and fixed point problems in Banach spaces. J Nonlinear Sci Appl 2017; 10(1): 192–204
5 J W ChenZ P WanL Y Yuan, et al.. Approximation of fixed points of weak Bregman relatively nonexpansive mappings in Banach spaces. Int J Math Math Sci, 2011, 2011: 420192 (23 pp)
6 G Z Eskandani, M Raeisi, J K Kim. A strong convergence theorem for Bregman quasi-noexpansive mappings with applications. Rev R Acad Cienc Exactas Fis Nat Ser A Mat RACSAM 2019; 113(2): 353–366
7 Y F KeC F Ma. The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl 2015, 2015, 190: (21 pp)
8 Y Liu. Strong convergence of iterative algorithms for generalized variational inequalities in Banach spaces. Adv Math China 2013; 42(6): 849–858
9 P LuoG Cai Y Shehu. The viscosity iterative algorithms for the implicit midpoint rule of nonexpansive mappings in uniformly smooth Banach spaces. J Inequal Appl, 2017, 2017: 154 (12 pp)
10 Z L Ma, L Wang, S-S Chang. On the split feasibility problem and fixed point problem of quasi-ϕ-nonexpansive mapping in Banach spaces. Numer Algorithms 2019; 80(4): 1203–1218
11 P-E Mainge. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal 2008; 16(7/8): 899–912
12 A Moudafi. The split common fixed-point problem for demicontractive mappings. Inverse Problems, 2010, 26(5): 055007 (6 pp)
13 A Padcharoen, P Kumam, Y J Cho. Split common fixed point problems for demicontractive operators. Numer Algorithms 2019; 82(1): 297–320
14 C J PanY H Wang. Convergence theorems for modified inertial viscosity splitting methods in Banach spaces. Mathematics, 2019, 7(2): 156 (12 pp)
15 C J PanY H Wang. Generalized viscosity implicit iterative process for asymptotically non-expansive mappings in Banach spaces. Mathematics, 2019, 7(5): 379 (13 pp)
16 R Pant, C C Okeke, C Izuchukwu. Modified viscosity implicit rules for proximal split feasibility and fixed point problems. J Appl Math Comput 2020; 64(1/2): 355–378
17 E Resmerita. On total convexity, Bregman projections and stability in Banach spaces. J Convex Anal 2004; 11(1): 1–16
18 Y Shehu. Iterative methods for split feasibility problems in certain Banach spaces. J Nonlinear Convex Anal 2015; 16(12): 2351–2364
19 Y Shehu, F U Ogbuisi. Convergence analysis for proximal split feasibility problems and fixed point problems. J Appl Math Comput 2015; 48(1/2): 221–239
20 S Suantai, U Witthayarat, Y Shehu. et al.. Iterative methods for the split feasibility problem and the fixed point problem in Banach spaces. Optimization 2019; 68(5): 955–980
21 A TaiwoL O JolaosoO T Mewomo. A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces. Comput Appl Math, 2019, 38(190): 77 (28 pp)
22 D V Thong, D V Hieu. An inertial method for solving split common fixed point problems. J Fixed Point Theory Appl 2017; 19(4): 3029–3051
23 H-K Xu. Inequalities in Banach spaces with applications. Nonlinear Anal 1991; 16(12): 1127–1138
24 H-K XuM A AlghamdiN Shahzad. The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl, 2015, 2015: 41 (12 pp)
25 H Zegeye. The general split equality problem for Bregman quasi-nonexpansive mappings in Banach spaces. J Fixed Point Theory Appl, 2018, 20(1): 6 (17 pp)
26 S S Zhang, L Wang, Y H Zhao. et al.. Strong convergence of multivalued Bregman totally quasi-asymptotically nonexpansive mappings. Acta Math Sin Chin Ser 2015; 58(2): 213–226
27 Z ZhouB TanSX Li. A new accelerated self-adaptive stepsize algorithm with excellent stability for split common fixed point problems. Comput Appl Math, 2020, 39(3): 220 (17 pp)
28 Z Zhou, B Tan, S X Li. An inertial shrinking projection algorithm for split common fixed point problems. J Appl Anal Comput 2020; 10(5): 2104–2120
[1] Jipu MA. A geometry characteristic of Banach spaces with c1-norm[J]. Front. Math. China, 2014, 9(5): 1089-1103.
[2] Qiaofen JIANG, Huaijie ZHONG. Components of generalized Kato resolvent set and single-valued extension property[J]. Front Math Chin, 2012, 7(4): 695-702.
[3] Yunnan ZHANG, Huaijie ZHONG. Strongly irreducible operators and Cowen-Douglas operators on c0, lp (1≤p<∞)[J]. Front Math Chin, 2011, 6(5): 987-1001.
[4] MA Ji-pu. A Rank Theorem of Operators between Banach Spaces[J]. Front. Math. China, 2006, 1(1): 138-143.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed