ARTICLE IN PRESS Nonlinear Analysis: Hybrid Systems ( ) – Contents lists available at ScienceDirect Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces ✩ Narin Petrot a,c , Kriengsak Wattanawitoon b , Poom Kumam b,c,∗ a Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand b Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thungkru, Bangkok 10140, Thailand c Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand article info Article history: Received 26 February 2009 Accepted 22 March 2010 Keywords: Hybrid projection iterative scheme Strong convergence Generalized mixed equilibrium problem Quasi-φ-nonexpansive mapping Relatively nonexpansive mapping abstract We introduce a hybrid projection iterative scheme for approximating a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of two quasi-φ-nonexpansive mappings in a real uniformly convex and uniformly smooth Banach space. Then, we establish strong convergence theorems for this algorithm which are connected with results by Takahashi and Zembayashi [W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Analysis 70 (2009) 45–57], Qin et al. [X. Qin, Y.J. Cho, S.M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, Journal of Computational and Applied Mathematics 225 (2009) 20–30], Wattanawitoon and Kumam [K. Wattanawitoon, P. Kumam, Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Analysis: Hybrid Systems 3 (2009) 11–20], and many others. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction The equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization, and it has been extended and generalized in many directions; see [1,2]. In particular, equilibrium problems are related to the problem of finding fixed points of nonexpansive mappings. Therefore it is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of the set of the solutions of the equilibrium problems and the set of the fixed points, (see also [3–25] and the references therein). In this paper, we suggest and analyze a hybrid iterative method for finding a common element of the set of the solutions of the equilibrium problem and the set of fixed points of two quasi-φ-nonexpansive mappings in the framework of Banach spaces. Let E be a real Banach space, and E ∗ the dual space of E . Let C be a nonempty closed convex subset of E . Let Θ : C ×C −→ R be a bifunction, ϕ : C −→ R be a real-valued function, and A : C −→ E ∗ be a nonlinear mapping. The generalized mixed equilibrium problem, is to find x ∈ C such that Θ(x, y) +〈Ax, y − x〉+ ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C . (1.1) ✩ The project was supported by the ‘‘Centre of Excellence in Mathematics’’ under the Commission on Higher Education, Ministry of Education, Thailand. ∗ Corresponding author at: Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thungkru, Bangkok 10140, Thailand. Tel.: +66 2 4708822; fax: +66 2 4284025. E-mail addresses: narinp@nu.ac.th (N. Petrot), s9510105@st.kmutt.ac.th (K. Wattanawitoon), poom.kum@kmutt.ac.th (P. Kumam). 1751-570X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.03.008 Please cite this article in press as: N. Petrot, et al., A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces, Nonlinear Analysis: Hybrid Systems (2010), doi:10.1016/j.nahs.2010.03.008