Weak Convergence Theorems for Strictly Pseudocontractive Mappings and Generalized Mixed Equilibrium Problems

Abstract

We introduce a new iterative method for finding a common element of the set of fixed points of a strictly pseudocontractive mapping, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of a variational inequality problem for an inverse-strongly-monotone mapping in Hilbert spaces and then show that the sequence generated by the proposed iterative scheme converges weakly to a common element of the above three sets under suitable control conditions. The results in this paper substantially improve, develop, and complement the previous well-known results in this area.

Weak Convergence Theorems for Generalized Mixed Equilibrium Problems, Monotone Mappings and Pseudocontractive Mappings

“…1) For finding a common element of GM EP (Θ, ϕ, B) ∩ V I(C, F ) ∩ F ix(T ), where B is a continuous monotone mapping, F is a continuous monotone mapping, and T is a continuous pseudocontractive mapping, Theorem 3.1 is a new ones different from previous those introduced by several authors. Consequently, in the sense that our convergence is for the more general class of continuous monotone and continuous pseudocontractive mappings, our results improve, develop and complement the corresponding results, which were obtained recently by several authors in references; for example, see [5,18,25,29,31] and references therein.…”

“…Recently, many authors have introduced some iterative algorithms for finding a common element of the set of the solutions of the GMEP, the GEP, the MEP, the EP, and the VIP and the set of fixed points of a countable family of nonexpansive mappings, and have proved strong convergence of the sequences generated by the proposed iterative algorithms; see [6,13,15,17,25,26,27,29,30,31,32] and the references therein. Also we refer to [4,5,7,18,21] for the GMEP, the GEP, the EP, and the VIP combined with the fixed point problem for nonexpansive semigroups and strictly pseudocontractrive mappings.…”

“…In 2009, Ceng et al [5] presented an iterative algorithm for finding a common element of the set of solutions of the EP and the set of fixed points of a k-strictly pseudocontractive mapping, and showed weak convergence of the sequence generated by the proposed iterative algorithm. In 2012, Jung [18] considered an iterative algorithm for finding a common element of the set of solutions of the GMEP related to α-inverse-strongly monotone mapping B, the set of solutions of the VIP for β-inverse-strongly monotone mapping F and the set of fixed points of a k-strictly pseudocontractive mapping, and established weak convergence of the sequence generated by the proposed iterative algorithm.…”

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

“…In 2012, Jung [16] considered an iterative method for GMEP (1.1) related to a β-inverse-strongly monotone mapping B, the VIP (1.5) for an α-inverse-strongly monotone mapping and k-strictly pseudocontractive mapping T and proved weak convergence to a point w ∈ GMEP(Θ, ϕ, B) ∩ VI(C, F) ∩ Fix(T ). In 2015 Jung [17] also proposed an iterative method for GMEP (1.1) related to a continuous monotone mapping B, the VIP (1.5) for a continuous monotone mapping F and a continuous pseudocontractive mapping T and proved weak convergence to a point w ∈ GMEP(Θ, ϕ, B) ∩ VI(C, F) ∩ Fix(T ).…”