Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 296759, ppt

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 296759, 16 pages doi:10.1155/2010/296759 Research Article Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert Spaces Somyot Plubtieng and Sukanya Chornphrom Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Correspondence should be addressed to Somyot Plubtieng, somyotp@nu.ac.th Received 30 June 2010; Revised 10 October 2010; Accepted 13 December 2010 Academic Editor: Brailey Sims Copyright q 2010 S Plubtieng and S Chornphrom This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonspreading mapping in a Hilbert space Then, we prove a strong convergence theorem which is connected with the work of S Takahashi and W Takahashi 2007 and Iemoto and Takahashi 2009 Introduction Let H be a real Hilbert space with inner product ·, · and norm · , respectively, and let C be a closed convex subset of H Let F : C × C → Ê be bifunction, where Ê is the set of real numbers The equilibrium problem for F : C × C → Ê is to find x ∈ C such that F x, y ≥ ∀y ∈ C 1.1 The set of solution of 1.1 is denoted by EP F Given a mapping A : C → H, let F x, y Ax, y − x for all x, y ∈ C Then, z ∈ EP F if and only if Az, y − z ≥ for all y ∈ C, that is, z is a solution of the variational inequality Numerous problems in physics, optimization, and economics reduce to find a solution of 1.1 ; see, for example, 1–9 and the references therein A mapping T of C into itself is said to be nonexpansive if Tx − Ty ≤ x − y for all x, y ∈ C, and a mapping F is said to be firmly nonexpansive if Fx − Fy ≤ x − y, Fx − Fy for all x, y ∈ C Let E be a smooth, strictly convex and reflexive Banach space, and let J be the Fixed Point Theory and Applications duality mapping of E and C a nonempty closed convex subset of E A mapping S : C → C is said to be nonspreading if φ Sy, Sx ≤ φ Sx, y φ Sx, Sy φ Sy, x 1.2 for all x, y ∈ C, where φ x, y x −2 x, Jy y for all x, y ∈ E; see, for instance, Kohsaka and Takahashi 10 In the case when E is a Hilbert space, we know that φ x, y x−y for all x, y ∈ E Then a nonspreading mapping S : C → C in a Hilbert space H is defined as follows: Sx − Sy ≤ Sx − y x − Sy 1.3 for all x, y ∈ C Let F Q be the set of fixed points of Q, and F Q nonempty; a mapping Q : C → C is said to be quasi-nonexpansive if Qx − y ≤ x − y for all x ∈ C and y ∈ F Q Remark 1.1 In a Hilbert space, we know that every firmly nonexpansive mapping is nonspreading and that if the set of fixed points of a nonspreading mapping is nonempty, the nonspreading mapping is quasi-nonexpansive; see 10, 11 In 1953, Mann 12 introduced the iteration as follows: a sequence {xn } defined by xn αn xn − αn Txn , 1.4 where the initial guess element x0 ∈ C is arbitrary and {αn } is a real sequence in 0, Mann iteration has been extensively investigated for nonexpansive mappings In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak convergence see 12, 13 Fourteen years later, Halpern 14 introduced the following iterative scheme for approximating a fixed point of T: xn αn x − αn Txn , 1.5 for all n ∈ Ỉ , where x1 x ∈ C and {αn } is a sequence of 0, Strong convergence of this type iterative sequence has been widely studied: Wittmann 15 discussed such a sequence in a Hilbert space On the other hand, Kohsaka and Takahashi 10 proved an existence theorem of fixed point for nonspreading mappings in a Banach space Recently, Lemoto and Takahashi 16 studied the approximation theorem of common fixed points for a nonexpansive mapping T of C into itself and a nonspreading mapping S of C into itself in a Hilbert space In particular, this result reduces to approximation fixed points of a nonspreading mapping S of C into itself in a Hilbert space by using iterative scheme xn αn xn − αn Sxn 1.6 Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping: see, for instance, 1, 2, 6, 7, 17–20 and the references Fixed Point Theory and Applications therein In 1997, Combettes and Hirstoaga introduced an iterative scheme of finding the best approximation to the initial data when EP F is nonempty and proved a strong convergence theorem Recently, S Takahashi and W Takahashi introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space Let S : C → H be a nonexpansive mapping In 2008, Plubtieng and Punpaeng introduced a new iterative sequence for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space which is the optimality condition for the minimization problem Very recently, S Takahashi and W Takahashi introduced an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain that the sequence converges strongly to a common element of two sets In this paper, motivated by S Takahashi and W Takahashi and Lemoto and Takahashi 16 , we introduce an iterative sequence and prove a strong convergence theorem for finding solution of equilibrium problems and the set of fixed points of a nonspreading mapping in Hilbert spaces Preliminaries Let H be a real Hilbert space When {xn } is a sequence in H, xn x implies that xn converges weakly to x and xn → x means the strong convergence Let C be a nonempty closed convex subset of H For every point x ∈ H, there exists a unique nearest point in C; denote by PC x, such that x − PC x ≤ x − y ∀y ∈ C 2.1 PC is called the metric projection of H onto C We know that PC is nonexpansive Further, for x ∈ H and z ∈ C, z PC x ⇐⇒ x − z, z − y ≥ ∀y ∈ C 2.2 Moreover, PC x is characterized by the following properties: PC x ∈ C and x − PC x, y − PC y ≤ 0, x−y ≥ x − PC x y − PC x 2.3 for all x ∈ H, y ∈ C We also know that H satisfies Opial’s condition 21 , that is, for any sequence {xn } ⊂ H with xn x, the inequality lim inf xn − x < lim inf xn − y n→∞ n→∞ holds for every y ∈ H with x / y; see 21, 22 for more details 2.4 Fixed Point Theory and Applications The following lemmas will be useful for proving the convergence result of this paper Lemma 2.1 see 23 Let E, ·, · 0, with α β γ 1, one has αx βy γz α x be an inner product space Then for all x, y, z ∈ E and α, β, γ ∈ β y γ z − αβ x − y − αγ x − z 2 − βγ y − z 2.5 Lemma 2.2 see 10 Let H be a Hilbert space, C a nonempty closed convex subset of H Let S be a nonspreading mapping of C into itself Then the following are equivalent There exists x ∈ C such that {Sn x} is bounded; F S is nonempty Lemma 2.3 see 10 Let H be a Hilbert space, C a nonempty closed convex subset of H Let S be a nonspreading mapping of C into itself Then F S is closed and convex Lemma 2.4 Let H be a real Hilbert space Then for all x, y ∈ H, x x y y ≤ x 2 y, x ≥ x 2 y, x y ; Lemma 2.5 see 24 Let {an }, {bn } ⊂ 0, ∞ , and let {cn } ⊂ 0, be sequences of real numbers such that ≤ ∞ n cn an − cn an ∞ and Then, limn → ∞ an bn , for all n ∈ Æ , ∞ n bn < ∞ Lemma 2.6 see 16 Let H be a Hilbert space, C a closed convex subset of H, and S : C → C a nonspreading mapping with F S / ∅ Then S is demiclosed, that is, xn u and xn − Sxn → imply u ∈ F S Lemma 2.7 see 16 Let H be a Hilbert space, C a nonempty closed convex subset of a real Hilbert space H, and let S be a nonspreading mapping of C into itself, and let A I − S Then Ax − Ay 2 ≤ x − y, Ax − Ay Ax Ay 2.6 Lemma 2.8 see 25 Assume {an } is a sequence of nonnegative real numbers such that an ≤ − αn an δn , n ≥ 0, where {αn } is a sequence in 0, and {δn } is a sequence in Ê such that ∞ n ∞; αn lim supn → ∞ δn /αn ≤ or Then limn → ∞ an ∞ n |δn | < ∞ 2.7 Fixed Point Theory and Applications For solving the equilibrium problems for a bifunction F : C × C → that F satisfies the following conditions: A1 F x, x Ê, let us assume for all x ∈ C; F y, x ≤ for all x, y ∈ C; A2 F is monotone, that is, F x, y − t x, y ≤ F x, y ; A3 for each x, y, z ∈ C, limt↓0 F tz A4 for each x ∈ C, y → F x, y is convex and lower semicontinuous The following lemma appears implicitly in 26 Lemma 2.9 see 26 Let C be a nonempty closed convex subset of H, and let F be a bifunction of C × C into Ê satisfying (A1)–(A4) Let r > and x ∈ H Then, there exists z ∈ C such that y − z, z − x ≥ ∀y ∈ C r F z, y 2.8 The following lemma was also given in Lemma 2.10 see Assume that F : C × C → define a mapping Tr : H → C as follows: Tr x z ∈ C : F z, y Ê satisfies (A1)–(A4) For r y − z, z − x ≥ 0, ∀y ∈ C r > and x ∈ H, 2.9 for all z ∈ H Then, the following hold: Tr is single-valued; Tr is firmly nonexpansive, that is, for any x, y ∈ H, Tr x − Tr y F Tr ≤ Tr x − Tr y, x − y ; EP F ; EP F is closed and convex Lemma 2.11 see 27 Let Γn be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence Γnj j≥0 of Γn which satisfies Γnj < Γnj for all j ≥ Also consider the sequence of integers τ n n≥n0 defined by τ n max{k ≤ n | Γk < Γk } Then τ n n≥n0 is a nondecreasing sequence verifying limn → ∞ τ n properties are satisfied for all n ≥ n0 : Γτ n ≤ Γτ n 1, Γn ≤ Γτ n 2.10 ∞, and the following 2.11 Main Result In this section, we prove a strong convergence theorem for finding a common element of the set of fixed points of a nonspreading mapping and the set of solutions of the equilibrium problems 6 Fixed Point Theory and Applications Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let F be a bifunctions from C × C → Ê satisfying (A1)–(A4), and let S be a nonspreading mapping of C into itself such that F S ∩ EP F / ∅ Let u ∈ C, and let {xn } and {un } be sequences generated by x1 ∈ C and y − un , un − xn ≥ 0, rn F un , y xn − βn S αn u βn xn ∀y ∈ C, 3.1 − αn un , for all n ∈ Ỉ , where {αn }, {βn } ∈ 0, and {rn } ∈ 0, ∞ satisfy limn → ∞ αn ∞ n ∞ n 0, ∞, < a ≤ βn ≤ b < 1, αn |αn − αn−1 | < ∞, lim infn → ∞ rn > 0, and ∞ n |βn − βn−1 | < ∞, ∞ n |rn − rn | < ∞ Then {xn } converges strongly to z ∈ F S ∩ EP F , where z Proof Let p ∈ F S ∩ EP F From un αn u S ∩EP F u Trn xn , we have un − p for all n ∈ Ỉ Put yn PF Trn xn − Trn p ≤ xn − p 3.2 − αn un We divide the proof into several steps Step We claim that the sequences {xn }, {un }, {yn }, and {Syn } are bounded First, we note that Syn − p ≤ yn − p αn u − αn un − p 3.3 ≤ αn u − p − αn un − p ≤ αn u − p − αn xn − p , and so xn −p βn xn − βn Syn − p ≤ βn xn − p − βn Syn − p ≤ βn xn − p − βn yn − p βn xn − p − βn αn u ≤ βn xn − p − βn αn u − p − αn un − p ≤ βn xn − p − βn αn u − p − αn xn − p − αn − βn xn − p − αn un − p αn − βn 3.4 u−p Fixed Point Theory and Applications Putting M max{ xn − p , u − p }, we note that xn − p ≤ M for all n ∈ Ỉ In fact, it is obvious that x1 − p ≤ M Assume that xk − p ≤ M for all k ∈ Æ Thus, we have xk − p ≤ − αk − βk ≤ − αk − βk αk − βk xk − p M u−p αk − βk M 3.5 M By induction, we obtain that xn − p ≤ M for all n ∈ Ỉ So, {xn } is bound Hence, {un }, {yn }, and {Syn } are also bounded Step Put tn xn βn yn − xn − βn Syn We claim that xn βn xn − βn Syn − βn−1 xn−1 βn xn − βn xn−1 βn xn−1 − βn−1 xn−1 − tn → as n → ∞ We note that − βn−1 Syn−1 − βn Syn − − βn Syn−1 − βn Syn−1 − − βn−1 Syn−1 ≤ βn xn − xn−1 βn − βn−1 xn−1 − βn − − βn−1 − βn Syn − Syn−1 yn − yn−1 Syn−1 ≤ βn xn − xn−1 βn − βn−1 xn−1 − βn βn xn − xn−1 βn − βn−1 xn−1 − βn × αn u − αn un − αn−1 u − − αn−1 un−1 ≤ βn xn − xn−1 × βn − βn−1 xn−1 αn u − αn−1 u βn xn − xn−1 − βn ≤ βn xn − xn−1 − αn un − − αn−1 un−1 βn − βn−1 xn−1 βn xn − xn−1 βn − βn−1 xn−1 βn xn − xn−1 βn − βn−1 Syn−1 − βn |αn − αn−1 | u − αn un−1 − − αn−1 un−1 − βn |αn − αn−1 | u − βn | − αn − − αn−1 | un−1 Syn−1 βn − βn−1 xn−1 − βn − αn un − un−1 βn − βn−1 Syn−1 Syn−1 − βn − αn un − un−1 βn − βn−1 βn − βn−1 Syn−1 − βn − αn un − − αn un−1 βn − βn−1 βn−1 − βn − βn |αn − αn−1 | u − βn |αn − αn−1 | un−1 Syn−1 βn − βn−1 K1 − βn − αn un − un−1 − βn |αn − αn−1 |K1 − βn |αn − αn−1 |K1 βn − βn−1 K1 , 3.6 Fixed Point Theory and Applications Syn where K1 sup{ xn un Trn xn , we have un−1 : n ∈ Ỉ } On the other hand, from un u y − un , un − xn ≥ 0, rn F un , y y − un , un rn F un , y for all y ∈ C Putting y un in 3.7 and y F un , un F un , un rn 3.7 ≥0 3.8 un in 3.8 , we have un rn − xn Trn xn and − un , un − xn ≥ 0, 3.9 un − un , un − xn ≥ So, from A2 , we note that un − un , un − xn un − xn − rn rn 1 ≥ 0, 3.10 and hence un − un , un − un un − xn − rn un rn 1 − xn ≥ 3.11 Without loss of generality, let us assume that there exists a real number d such that rn > d > for all n ∈ Ỉ Thus, we have un − un ≤ un ≤ un 1 − un , xn − un xn − xn 1− rn rn un rn 1− rn − xn − xn 3.12 un − xn , and hence un − un ≤ xn ≤ xn 1 − xn |rn rn − xn |rn d 1 − r n | un − rn |L, − xn 3.13 Fixed Point Theory and Applications where L xn sup{ un − xn : n ∈ Ỉ } So, from 3.6 , we note that − xn ≤ βn xn − xn−1 βn − βn−1 K1 − βn − αn − βn − αn − − βn αn − βn − αn |rn − rn−1 |L d xn − xn−1 − βn − αn βn − βn |αn − αn−1 |K1 xn − xn−1 βn − βn−1 K1 − βn |αn − αn−1 |K1 |rn − rn−1 |L d xn − xn−1 βn − βn−1 K1 − βn |αn − αn−1 |K1 L |rn − rn−1 | d 3.14 By Lemma 2.5, we have lim xn n→∞ for p ∈ F S ∪ EP F We note from un un − p 2 un − p 3.15 Trn xn that Trn xn − Trn p − xn ≤ Trn xn − Trn p, xn − p xn − p − xn − un un − p, xn − p 3.16 , and hence un − p Therefore, from the convexity of · xn −p βn xn ≤ xn − p − xn − un 3.17 , we have − βn Syn − p ≤ βn xn − p − βn Syn − p ≤ βn xn − p − βn yn − p βn xn − p − βn αn u − αn un − p ≤ βn xn − p αn − βn u−p − βn − αn ≤ βn xn − p αn − βn u−p − βn − αn − − βn αn xn − p 2 αn − βn u−p 2 un − p xn − p 2 − xn − un − βn − αn xn − un , 3.18 10 Fixed Point Theory and Applications and hence − βn − αn xn − un ≤ αn − βn u−p xn − p αn − βn − xn u−p xn − p − xn ≤ αn − βn u−p xn − xn lim xn − yn −p lim xn − αn u lim n→∞ αn xn − p xn xn xn − p −p 3.19 −p − αn un , it follows that − αn un − αn xn − αn u ≤ lim αn xn − u − αn un − αn xn − un n→∞ lim αn xn − u xn − p − αn − βn αn u n→∞ 2 − αn − βn xn − p So, we have xn − un → Indeed, since yn n→∞ −p xn − p − αn − βn 3.20 lim − αn xn − un n→∞ n→∞ Then, we note that xn − tn − βn Syn − βn yn βn xn − βn βn xn − yn − βn Syn Syn − Syn 3.21 βn xn − yn Since, < a ≤ βn ≤ b < and xn − yn → 0, it follows that lim xn n→∞ Step Put A tn − p I − S From Ap βn yn − tn 3.22 0, it follows by Lemma 2.7 that − βn Syn − p yn − p − − βn 2 yn − Syn yn − p − − βn Ayn yn − p 2 − − βn yn − p, Ayn − Ap − βn Ayn Fixed Point Theory and Applications ≤ yn − p − βn 11 − − βn − αn un − p − βn Ayn − 2 Ayn Ap 2 Ayn αn u − p Ayn − Ap − βn − − βn Ayn 2 Ayn ≤ αn u−p − αn un − p − βn − βn Ayn ≤ αn u−p − αn xn − p − βn − βn Ayn ≤ αn u − p xn − p − βn − βn Ayn 3.23 Since < a ≤ βn ≤ b < 1, we have βn − βn ≥ a − b : K2 Therefore, by 3.23 , we obtain K2 Ayn ≤ αn u − p K2 yn − Syn xn − p 2 − tn − p ≤ αn M2 xn − p − tn − p αn M2 xn − p − αn M2 xn − p − tn − xn ≤ αn M2 xn − p − xn tn − xn 1 −p xn 1 −p 2 − tn − xn , xn − p − xn − tn − xn , xn −p −p 3.24 Step Putting z PF S ∩EP F u, we claim that the sequence {xn } converges strongly to z PF S ∩EP F u Indeed, we discuss two possible cases Case Assume that there exists n0 such that the sequence { xn − p } is a nonincreasing sequence for all n ≥ n0 Then we have xn − p ≤ xn − p for n ≥ n0 , and hence limn → ∞ xn − p exists Therefore lim xn − p n→∞ lim xn n→∞ −p 3.25 By 3.22 , 3.24 , and 3.25 , we get yn − Syn −→ 3.26 Let {yni } be a subsequence of {yn } such that lim sup u − z, yn − z n→∞ lim u − z, yni − z n→∞ 3.27 12 Fixed Point Theory and Applications Since {yn } is bounded, there exists a subsequence {yni } of {yn } which converges weakly to w Since C is closed and convex, w Without loss of generality, we can assume that yni we note that C is weakly closed So, we have w ∈ C Since Syn − yn → 0, it follows by Lemma 2.6 that w ∈ F S From 3.27 and the property of metric projection, we have lim sup u − z, yn − z lim u − z, yni − z n→∞ n→∞ 3.28 u − z, w − z ≤ Finally, we prove that xn → z In fact, since yn − z xn −z − βn Syn − z βn xn αn u − z − αn un − z , it follows that 2 ≤ βn xn − z − βn Syn − z ≤ βn xn − z − βn yn − z ≤ βn xn − z − βn − αn ≤ βn xn − z − βn − αn xn − z 3.29 − αn − βn By 3.28 and ∞ n αn xn − z xn − z 2 2αn u − z, yn − z 2αn − βn u − z, yn − z 2αn − βn u − z, yn − z ∞, we immediately deduce by Lemma 2.8 that xn → z Case Assume that for all n ∈ Ỉ , there exits m ≥ n such that xm − p < xm − p Put am : xm − p for all m ∈ Ỉ Thus, it follows that there exists a subsequence ank k≥1 of an n≥1 such that ank < ank for all k ∈ Ỉ Let ϕ : Ỉ → Ỉ be a mapping defined by ϕ n max{k ≤ n : ak ≤ ak }, 3.30 where Ỉ {n ∈ Ỉ : n ≥ n1 } By Lemma 2.11, we note that ϕ n is a nondecreasing sequence such that ϕ n → ∞ as n → ∞ and that the following properties are satisfied by all numbers n ≥ n1 : aϕ n ≤ aϕ n 1, an ≤ aϕ n 3.31 From 3.24 , we have K2 yϕ n − Syϕ n ≤ αϕ n M2 xϕ n − p − tϕ n − xϕ n − xϕ n , xϕ n ≤ αϕ n M2 − tϕ n − xϕ n −p −p , xϕ n 3.32 −p Fixed Point Theory and Applications 13 This implies that −→ yϕ n − Syϕ n 3.33 Take a subsequence {yϕ n i } of {yϕ n } such that lim sup u − z, yϕ n − z lim u − z, yϕ n i − z 3.34 n→∞ n→∞ v Since C is closed and From the boundedness of {yϕ n i }, we can assume that yϕ n i convex, it follows that C is weakly closed So, we have v ∈ C Since Syϕ n − yϕ n → 0, it follows by Lemma 2.6 that v ∈ F S From 3.34 and the property of metric projection, we have lim sup u − z, yϕ n − z lim u − z, yϕ n i − z n→∞ n→∞ 3.35 u − z, v − z ≤ By the same argument as 3.29 in Case 1, we conclude immediately that, for all n ≥ 1, ≤ xϕ n −z 2 − xϕ n − z ≤ βϕ n xϕ n − z − βϕ n Syϕ n − z ≤ βϕ n xϕ n − z − βϕ n yϕ n − z ≤ βϕ n xϕ n − z − βϕ n × − αϕ n ≤ βϕ n xϕ n − z uϕ n − z 2αϕ n − βϕ n ≤ βϕ n xϕ n − z 2αϕ n − βϕ n αϕ n − βϕ n 2 − xϕ n − z − xϕ n − z 2αϕ n u − z, yϕ n − z − αϕ n − βϕ n uϕ n − z − αϕ n xϕ n − z ≤ u − z, yϕ n − z − xϕ n − z − xϕ n − z 2 2 u − z, yϕ n − z − xϕ n − z u − z, yϕ n − z − xϕ n − z 2 u − z, yϕ n − z − xϕ n − z − βϕ n 2 , 3.36 which implies that xϕ n − z ≤ u − z, yϕ n − z 3.37 14 Fixed Point Theory and Applications By 3.35 , we have lim xϕ n − z 0, n→∞ 3.38 and hence lim xϕ n n→∞ Since xn − z an ≤ aϕ n −z lim xϕ n − z n→∞ 3.39 xϕ n − z for all n ≥ n1 , we have lim xn − z n→∞ 3.40 This completes the proof As direct consequences of Theorem 3.1, we obtain corollaries Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H Let F be a bifunctions from C × C → Ê satisfying (A1)–(A4), and let S be a firmly nonexpansive mapping of C into itself such that F S ∩ EP F / ∅ Let u ∈ C, and let {xn } and {un } be sequences generated by x1 ∈ C and y − un , un − xn ≥ 0, rn F un , y xn βn xn 1 − βn S αn u ∀y ∈ C, 3.41 − αn un , for all n ∈ Æ , where {αn }, {βn } ∈ 0, and {rn } ∈ 0, ∞ satisfy limn → ∞ αn ∞ n 0, ∞ n αn |αn − αn−1 | < ∞, lim infn → ∞ rn > 0, and ∞, < a ≤ βn ≤ b ∞ n |βn − βn−1 | < ∞, ∞ n |rn − rn | < ∞ < 1, Then {xn } converges strongly to z ∈ F S ∩ EP F , where z PF S ∩EP F u Acknowledgments The authors would like to thank the referees for the insightful comments and suggestions Moreover, the authors gratefully acknowledge the Thailand Research Fund Master Research Grants TRF-MAG, MRG-WII515S029 for funding this paper References S.-S Chang, H W Joseph Lee, and C K Chan, “A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization,” Nonlinear Analysis Theory: Methods & Applications, vol 70, no 9, pp 3307–3319, 2009 Fixed Point Theory 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equilibrium problems, and fixed point problems in Hilbert spaces,” Fixed Point Theory and Applications, vol 2009, Article ID 567147, 20 pages, 2009 18 S Plubtieng and W Sriprad, “Hybrid methods for equilibrium... equilibrium problems and fixed points problems of a countable family of relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol 2010, Article ID 962628, 17 pages,... equilibrium problem and fixed point problem of nonexpansive mapping: see, for instance, 1, 2, 6, 7, 17–20 and the references Fixed Point Theory and Applications therein In 1997, Combettes and Hirstoaga