Abstract
In this paper, motivated and inspired by Ceng and Yao (J. Comput. Appl. Math. 214(1):186-201, 2008), Iiduka and Takahashi (Nonlinear Anal. 61(3):341-350, 2005), Jaiboon and Kumam (Nonlinear Anal. 73(5):1180-1202, 2010), Kim (Nonlinear Anal. 73:3413-3419, 2010), Marino and Xu (J. Math. Anal. Appl. 318:43-52, 2006) and Saeidi (Nonlinear Anal. 70:4195-4208, 2009), we introduce a new iterative scheme for finding a common element of the set of solutions of a mixed equilibrium problem for an equilibrium bifunction, the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of some variational inequality problem, and the set of fixed points of a left amenable semigroup of nonexpansive mappings with respect to W-mappings and a left regular sequence of means defined on an appropriate space of bounded real-valued functions of the semigroup S. Furthermore, we prove that the iterative scheme converges strongly to a common element of the above four sets. Our results extend and improve the corresponding results of many others. MSC:43A65, 47H05, 47H09, 47H10, 47J20, 47J25, 74G40.
Highlights
Let H be a real Hilbert space, let C be a nonempty closed convex subset of H, and let PC be the metric projection of H onto C
In this paper, motivated and inspired by Ceng and Yao [ ], Iiduka and Takahashi [ ], Jaiboon and Kumam [ ], Kim [ ], Marino and Xu [ ] and Saeidi [ ], we introduce a new iterative scheme:
( – γn)xn + γnTμn WnPC(I – δnB)zn, = αnγ f (Wnxn) + βnxn + (( – βn)I – αn for all x ∈ C, n ≥, for finding a common element of the set of solutions of a mixed equilibrium problem for an equilibrium bifunction, the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of some variational inequality problem and the set of fixed points of a left amenable semigroup {Tt : t ∈ S} of nonexpansive mappings with respect to W -mappings and a left regular sequence {μn} of means defined on an appropriate space of bounded real-valued functions of the semigroup S
Summary
Let H be a real Hilbert space, let C be a nonempty closed convex subset of H, and let PC be the metric projection of H onto C. We consider the mixed equilibrium problem (for short, MEP) is to find x* ∈ C such that. Denote the set of solutions of MEP by. Recall that a mapping f : C → C is called contractive if there exists a constant α ∈ ( , ) such that f (x) – f (y) ≤ α x – y , for all x, y ∈ C. In , Moudafi [ ] introduced the viscosity approximation method for nonexpansive mappings (see [ ] for further developments in both Hilbert and Banach spaces). Let A be a strongly positive bounded linear operator on H, that is, there exists a constant γ > such that.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
