The Hybrid Extragradient Method for the Split Feasibility and Fixed Point Problems Jitsupa Deepho, Wiyada Kumam and Poom Kumam, Member, IAENG, Abstract—In this paper, we suggest a hybrid extragradient method for finding a common element of the set of fixed point sets of an infinite family of nonexpansive mappings and the solution set of the split feasibility problem (SFP) in real Hilbert spaces. Index Terms—Fixed point problems, Split feasibility problem, CQ method, Projection, Strong convergence, Hybrid extragra- dient method I. I NTRODUCTION T HROUGHOUT this paper, let H be a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉 and ‖·‖, respectively. Let C and Q be a nonempty closed convex subset of infinite-dimensional real Hilbert spaces H 1 and H 2 , respectively. The split feasibility problem (SFP) is to find a point x ∗ with the property: x ∗ ∈ C and Ax ∗ ∈ Q, (1) where A ∈ B(H 1 ,H 2 ) and B(H 1 ,H 2 ) denotes the family of all bounded linear operators from H 1 to H 2 . We use Γ to denote the solution set of the (SFP), i.e., Γ= {x ∗ ∈ C : Ax ∗ ∈ Q}. In 1994, the SFP was introduced by Censor and Elfving [1], in finite dimensional Hilbert spaces, for modeling inverse problems which arise from phase retrievals and in medical image reconstruction and many researches; see, e.g., [2–5]. A special case of the SFP is the following convex con- strained linear inverse problem [6] of finding an element x such that x ∈ C such that Ax = b. (2) This problem, due to its applications in many applied disciplines, has extensively been investigated in the literature ever since Lanweber [7] introduced his iterative method in 1951. In 2002, Byrne [2] proposed his CQ algorithm to solve (1). The sequence {x n } is generated by the following iteration scheme: x n+1 = P C (I − γA ∗ (I − P Q )A)x n , n ∈ N, (3) Manuscript received January 28, 2014. This work was supported by Thai- land Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0033/2554) and the King Mongkut’s University of Technology Thonburi and the National Research Council of Thailand (NRCT). J. Deepho and K. Kumam are with the Department of Mathematics Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thrung Kru, Bangkok 10140, Thailand, e-mail: jitsupa.deepho@mail.kmutt.ac.th (J. Deepho) and poom.kum@kmutt.ac.th (P. Kumam). W. Kumam is with Department of Mathematics and Computer Sci- ence, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), 39, Moo 1, Rangsit-Nakhonnayok Rd., Klong 6, Thanyaburi, Pathumthani 12110, Thailand, e-mail: wiyada.kum@mail.rmutt.ac.th (W. Kumam) where γ ∈ (0, 2 λ ), with λ being the spectral radius of the operator A ∗ A. The variational inequality problem VI (C, A) is to find u ∈ C such that 〈Au, v − u〉≥ 0, ∀v ∈ C. (4) Variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral and equilibrium problems, which arise in several branches of pure and applied sciences in a unified and general framework. Several numerical methods have been developed for solving variational inequalities and related optimization problems, see, e.g., [8–10]. Let us start with Korpelevich’s extragradient method which was introduce by Korpelevich [10] in 1976 and which generates a sequence {x n } via the recursion;  y n = P C (x n − λAx n ), x n+1 = P C (x n − λAy n ), n ≥ 0, (5) where P C is the metric projection from R n onto C, A : C → H is a monotone operator and λ is a constant. Korpelevich [10] prove that the sequence {x n } converges strongly to a solution of VI (C, A). Note that the setting of the problems in the Euclidean space R n . We note that Nadezhkina and Takahashi [11] employed the monotonicity and Lipschitz-continuity of A to define a maximal monotone operator T as follows: Tv =  Av + N C v if v ∈ C, ∅ if v ∈ C. (6) where N C v = {w ∈ H : 〈v − u, w〉≥ 0, ∀u ∈ C} is the normal cone to C at v ∈ C (see, [12]). However, if the mapping A is a pseudomonotone Lipschitz-continuous, then T is not necessarily a maximal monotone operator. Yu, Yao and Liou [13] introduced a new iterative method as follows:            x 1 = x 0 ∈ C, y n = P C (x n − λ n Ax n ), z n = α n x n + (1 − α n )W n P C (x n − λ n Ay n ), C n+1 = {z ∈ C n : ‖z n − z ‖≤‖x n − z ‖}, x n+1 = P Cn+1 x 0 , n ≥ 1, (7) under their condition, they proved that the sequences {x n }, {y n } and {z n } converge strongly to the same point P ∩ ∞ n=1 F ix(Sn)∩Ω x 0 . Ceng, Ansari and Yao [14] introduce an extragradient method for solving split feasibility and fixed point problems. They propose the following method:    x 0 = x ∈ C, y n = P C (x n − λ n ∇f αn x n ), x n+1 = β n x n + (1 − β n )SP C (x n − λ n ∇f αn y n ). (8) They prove that the sequences {x n } and {y n } converge weakly to the same elemaent ˆ x ∈ F ix(S) ∩ Γ. Proceedings of the International MultiConference of Engineers and Computer Scientists 2014 Vol I, IMECS 2014, March 12 - 14, 2014, Hong Kong ISBN: 978-988-19252-5-1 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) IMECS 2014