Weak Convergence Theorems on the Split Common Fixed Point Problem for Demicontractive Continuous Mappings

Abstract

We are concerned with the split common fixed point problem in Hilbert spaces. We propose a new method for solving this problem and establish a weak convergence theorem whenever the involved mappings are demicontractive and Lipschitz continuous. As an application, we also obtain a new method for solving the split equality problem in Hilbert spaces.

Iterative Algorithms for Split Common Fixed Point Problem Involved in Pseudo-Contractive Operators without Lipschitz Assumption

“…Wang obtained the weak convergence of algorithm (4). In this paper, we extend a previous author's results from the demicontractive operators [8,10,24], firmly-nonexpansive operators [25], quasi-nonexpansive operators [26], directed operators [1], nonexpansive operators [27], and strictly pseudo-contractive operators [28] to the more general pseudo-contractive operators. Subsequently, two algorithms are suggested based on (4) and (5) to solve the problem (1).…”

“…In this paper, we investigated the problem (1) involved in pseudo-contractive operators without Lipschitz assumption. By extending someone's results from [1,8,10,[24][25][26][27][28] to the more general pseudo-contractive operators, we constructed two algorithm for solving the problem (1). Weak and strong convergence theorems are obtained under some mild hypotheses.…”

“…• T is called strictly pseudo-contractive if there exists k < 1 such that 24]). Let T : H → H be a nonlinear operator with Fix(T) = ∅.…”

Multiple-Sets Split Common Fixed-Point Problems for Demicontractive Mappings

“…It is shown that if τ n is chosen in (0, 2/‖A‖ 2 ), then (7) converges weakly to a solution of (1). Subsequently, this result was extended to more general cases (see, e.g., [8][9][10][11][12][13][14][15][16][17]). Since the choice of the stepsize is related to ‖A‖, thus to implement (7), one has to compute (or at least estimate) the norm ‖A‖, which is generally not easy in practice.…”

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]…”