Viscosity approximation methods for split common fixed point problems without prior knowledge of the operator norm

Abstract

In this work, we study the split common fixed point problem which was first introduced by Censor and Segal [14]. We introduce an algorithm based on the viscosity approximation method without prior knowledge of the operator norm by selecting the stepsizes in the same adaptive way as L?opez et al. [22] for solving the problem for two attracting quasi-nonexpansive operators in real Hilbert spaces. A strong convergence result of the proposed algorithm is established under some suitable conditions. We also modify our algorithm to extend to the class of demicontractive operators and the class of hemicontractive operators, and obtain strong convergence results. Moreover, we apply our main result to other split problems, that is, the split feasibility problem and the split variational inequality problem. Finally, a numerical result is also given to illustrate the convergence behavior of our algorithm.

Single-Valued Demicontractive Mappings: Half a Century of Developments and Future Prospects

“…For other important recent results regarding the study of the SCFP problem with its particular instances (FPP, SFP, CFP etc. ), we refer the readers to Boikanyo [43], Chen [44], Chen et al [45], Chidume et al [46] (which deals with the split equality fixed-point problem, a generalization of the split feasibility problem), Cui [47], Cui et al [48], Cui and Wang [49,50], Dang et al [51], Eslamian [52,53], Eslamian et al [54], Fan et al [55], Gupta et al [56], Hanjing and Suantai [57][58][59][60], He et al [61][62][63][64], Jailoka and Suantai [65,66], Jirakitpuwapat et al [67], Kitkuan et al [68], Mouktonglang and Suparatulatorn [69], Padcharoen et al [70], Qin and Wang [71], Shehu [72], Shehu and Cholamjiak [73], Shehu and Mewomo [74], Suparatulatorn et al [75,76], Su and Zhao [77], Tang et al [20,78], Wang [79], Wang and Zhao [80], Wang and Cui [29], Wang and Fang [81,82], Wang et al [83]…”

Algorithms for Approximating Solutions of Split Variational Inclusion and Fixed-Point Problems

“…There are various ways to solve the split problems, see [2][3][4][5][6][7][8][9][10][11][12][13]. To solve (3), a remarkable channel brought up by Censor and Segal [14] is of the manner:…”