RESEARCH ARTICLE

Implicit iterative algorithms of the split common fixed point problem for Bregman quasi-nonexpansive mapping in Banach spaces

  • Yuanheng WANG , 1 ,
  • Chanjuan PAN 2
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  • 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

Published date: 15 Oct 2022

Copyright

2022 Higher Education Press 2022

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.

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]. Frontiers of Mathematics in China, 2022 , 17(5) : 797 -811 . DOI: 10.1007/s11464-022-1027-9

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12171435) and the Natural Science Foundation of Zhejiang Province (Grant No. LY14A010011).
1
Bauschke H H, Borwein J M, Combettes P L. Bregman monotone optimization algorithms. SIAM J Control Optim 2003; 42(2): 596–636

2
Cai G. 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
Censor Y, Segal A. The split common fixed point problem for directed operators. J Convex Anal 2009; 16(2): 587–600

4
Chen J Z, Hu H Y, Ceng L C. 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
ChenJ WWan Z PYuanL Y, . Approximation of fixed points of weak Bregman relatively nonexpansive mappings in Banach spaces. Int J Math Math Sci, 2011, 2011: 420192 (23 pp)

6
Eskandani G Z, Raeisi M, Kim J K. 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
KeY FMa C F. The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl 2015, 2015, 190: (21 pp)

8
Liu Y. Strong convergence of iterative algorithms for generalized variational inequalities in Banach spaces. Adv Math China 2013; 42(6): 849–858

9
LuoPCaiG ShehuY. 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
Ma Z L, Wang L, Chang S-S. 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
Mainge P-E. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal 2008; 16(7/8): 899–912

12
MoudafiA. The split common fixed-point problem for demicontractive mappings. Inverse Problems, 2010, 26(5): 055007 (6 pp)

13
Padcharoen A, Kumam P, Cho Y J. Split common fixed point problems for demicontractive operators. Numer Algorithms 2019; 82(1): 297–320

14
PanC JWang Y H. Convergence theorems for modified inertial viscosity splitting methods in Banach spaces. Mathematics, 2019, 7(2): 156 (12 pp)

15
PanC JWang Y H. Generalized viscosity implicit iterative process for asymptotically non-expansive mappings in Banach spaces. Mathematics, 2019, 7(5): 379 (13 pp)

16
Pant R, Okeke C C, Izuchukwu C. Modified viscosity implicit rules for proximal split feasibility and fixed point problems. J Appl Math Comput 2020; 64(1/2): 355–378

17
Resmerita E. On total convexity, Bregman projections and stability in Banach spaces. J Convex Anal 2004; 11(1): 1–16

18
Shehu Y. Iterative methods for split feasibility problems in certain Banach spaces. J Nonlinear Convex Anal 2015; 16(12): 2351–2364

19
Shehu Y, Ogbuisi F U. Convergence analysis for proximal split feasibility problems and fixed point problems. J Appl Math Comput 2015; 48(1/2): 221–239

20
Suantai S, Witthayarat U, Shehu Y. . Iterative methods for the split feasibility problem and the fixed point problem in Banach spaces. Optimization 2019; 68(5): 955–980

21
TaiwoAJolaoso L OMewomoO T. 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
Thong D V, Hieu D V. An inertial method for solving split common fixed point problems. J Fixed Point Theory Appl 2017; 19(4): 3029–3051

23
Xu H-K. Inequalities in Banach spaces with applications. Nonlinear Anal 1991; 16(12): 1127–1138

24
XuH-KAlghamdi M AShahzadN. The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl, 2015, 2015: 41 (12 pp)

25
ZegeyeH. 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
Zhang S S, Wang L, Zhao Y H. . Strong convergence of multivalued Bregman totally quasi-asymptotically nonexpansive mappings. Acta Math Sin Chin Ser 2015; 58(2): 213–226

27
ZhouZTan BLiSX. 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
Zhou Z, Tan B, Li S X. An inertial shrinking projection algorithm for split common fixed point problems. J Appl Anal Comput 2020; 10(5): 2104–2120

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