An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings

Abstract

a b s t r a c tIn this paper, we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces. We establish some weak and strong convergence theorems of the sequences generated by our proposed scheme. Our results combine the ideas of Marino and Xu's result [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336-346], and Takahashi and Takahashi's result [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506-515]. In particular, necessary and sufficient conditions for strong convergence of our iterative scheme are obtained.

Retracted Article: Strong convergence theorem by a new iterative method for equilibrium problems and symmetric generalized hybrid mappings

“…In the recent years, many authors studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces, respectively; see, for instance, [6][7][8][9][10][11][12][13][14] and the references therein.…”

Strong and weak convergence theorems for split equilibrium problems and fixed point problems in Banach spaces

“…In order to relax the restriction, Wang et al [31] introduced a new iterative algorithm to solve the split equilibrium problem as follows:      u i,n = T F r n (I − γA * i (I − T F i r n )A i )x n , i = 1, · · · , N 1 , 5) for each n 1, where F : C × C → R, F 1 , · · · , F N 1 : Q × Q → R are bifunctions, A 1 , · · · , A N 1 : H 1 → H 2 are linear bounded operators, B 1 , · · · , B N 2 : C → H 1 are inverse strongly monotone mappings, for each i 1, S i : C → C is nonexpansive mapping. Under some suitable conditions on the control sequences {r n }, {α n }, {λ n }, they proved that the sequence {x n } generated by (1.5) converges strongly to an element…”

Modified hybrid iterative methods for generalized mixed equilibrium, variational inequality and fixed point problems

“…If B = 0, then the problem (1.1) reduces the following mixed equilibrium problem (for short, MEP) of finding x ∈ C such that Θ(x, y) + ϕ(y) − ϕ(x) 0, ∀y ∈ C, (1. 3) which was studied by Ceng and Yao [5] (see also [39]). The set of solutions of the problem (1.3) is denoted by MEP(Θ, ϕ).…”

“…In 2008, Moudafi [21] proposed an iterative method for the GEP (1.2) related to an α-inverse-strongly monotone mapping B and nonexpansive mapping S and showed weak convergence to a point w ∈ GEP(Θ, B) ∩ Fix(S). In 2009, Ceng et al [3] provided an iterative method for the EP (1.4) and k-strictly pseudocontractive mapping T and proved weak convergence to a point w ∈ EP(Θ) ∩ Fix(T ). In 2015, Lv [19] also studied an iterative method for the GEP (1.2) and k-strictly pseudocontractive mapping T and proved weak convergence to a point w ∈ GEP(Θ) ∩ Fix(T ).…”