A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem q K.R. Kazmi ⇑ , S.H. Rizvi Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India article info Keywords: Mixed equilibrium problem Variational inequality problem System of variational inequality problems Strictly pseudocontractive mappings Fixed-point problem abstract In this paper, we give a hybrid extragradient iterative method for finding the approximate element of the common set of solutions of a generalized equilibrium problem, a system of variational inequality problems, a variational inequality problem and a fixed point problem for a strictly pseudocontractive mapping in a real Hilbert space. Further we establish a strong convergence theorem based on this method. The results presented in this paper improves and generalizes the results given in Yao et al. [36] and Ceng et al. [7], and some known corresponding results in the literature. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction Throughout the paper unless otherwise stated, let H be a real Hilbert space with inner product h,i and norm kk. Let C be a non-empty closed convex subset of H. Let {x n } be any sequence in H, then x n ? x (respectively, x n N x) will denote strong (respectively, weak) convergence of the sequence {x n }. Let T : C ? H be a nonlinear mapping. Then T is called (i) monotone, if hTx Ty; x yi P 0; 8x; y 2 H; (ii) a-strongly monotone, if there exists a constant a > 0 such that hTx Ty; x yi P akx yk 2 ; 8x; y 2 H; (iii) a-inverse strongly monotone, if there exists a constant a > 0 such that hTx Ty; x yi P akTx Tyk 2 ; 8x; y 2 H; (iv) k-Lipschitz continuous, if there exists a constant k > 0 such that kTx Tyk 6 kkx yk; 8x; y 2 H: 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.032 q This work has been done under a Major Research Project No. F.36-7/2008 (SR) sanctioned by the University Grants Commission, Government of India, New Delhi. ⇑ Corresponding author. E-mail addresses: krkazmi@gmail.com (K.R. Kazmi), shujarizvi07@gmail.com (S.H. Rizvi). Applied Mathematics and Computation 218 (2012) 5439–5452 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc