JP Journal of Fixed Point Theory and Applications Volume 7, Number 2, 2012, Pages 113-127 Available online at http://pphmj.com/journals/jpfpta.htm Published by Pushpa Publishing House, Allahabad, INDIA WEAK CONVERGENCE THEOREMS FOR AN INFINITE FAMILY OF NONEXPANSIVE MAPPINGS AND EQUILIBRIUM PROBLEMS P N Anh*†, L B Long†, N V Quy and L Q Thuy †Department of Scientific Fundamentals Posts and Telecommunications Institute of Technology Hanoi, Vietnam e-mail: anhpn@ptit.edu.vn Department of Scientific Fundamentals Academy of Finance Hanoi, Vietnam Faculty of Applied Mathematics and Informatics Hanoi University of Technology Vietnam Abstract The purpose of this paper is to investigate a new iteration scheme for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings and the solution set of a pseudomonotone and Lipschitz-type continuous equilibrium problem The scheme is based on the extragradient-type methods and fixed point methods We show that the iterative sequences generated by this algorithm converge weakly to the common element in a real Hilbert space © 2012 Pushpa Publishing House 2010 Mathematics Subject Classification: 65K10, 65K15, 90C25, 90C33 Keywords and phrases: nonexpansive mapping, pseudomonotone, Lipschitz-type continuous, equilibrium problems, fixed point *Corresponding author Received December 5, 2011 P N Anh, L B Long, N V Quy and L Q Thuy 114 Introduction Let H be a real Hilbert space with inner product ⋅, ⋅ and norm ⋅ Let C be a closed convex subset of a real Hilbert space H and PrC be the projection of H onto C When {x n } is a sequence in H, then x n → x (resp x n x ) will denote strong (resp weak) convergence of the sequence {x n } to x as n → ∞ A mapping S : C → C is said to be nonexpansive if S (x) − S ( y) ≤ x − y , ∀x, y ∈ C Fix( S ) is denoted by the set of fixed points of S Let f : C × C → R be a bifunction such that f ( x, x ) = for all x ∈ C We consider the equilibrium problems in the sense of Blum and Oettli [8] which are presented as follows: Find x∗ ∈ C such that f ( x∗ , y ) ≥ for all y ∈ C EP( f , C ) The set of solutions of EP( f , C ) is denoted by Sol ( f , C ) The bifunction f is called strongly monotone on C with β > if f ( x, y ) + f ( y, x ) ≤ −β x − y , ∀x, y ∈ C ; monotone on C if f ( x, y ) + f ( y, x ) ≤ 0, ∀x, y ∈ C ; pseudomonotone on C if f ( x, y ) ≥ ⇒ f ( y, x ) ≤ 0, ∀x, y ∈ C ; Lipschitz-type continuous on C with constants c1 > and c2 > if f ( x, y ) + f ( y, z ) ≥ f ( x, z ) − c1 x − y − c2 y − z , ∀x, y, z ∈ C In this paper, we are interested in the problem of finding a common element of the solution set of the equilibrium problems EP( f , C ) and the set ∞ of fixed points ∩k =1 Fix( S k ) of an infinite family of nonexpansive mappings Weak Convergence Theorems for an Infinite Family … 115 {S k }, namely: Find x∗ ∈ ∞ ∩i =1 Fix(Si ) ∩ Sol ( f , C ) , where the bifunction f and the mappings S k following conditions: (1.1) (k = 1, 2, ) satisfy the A1 f is Lipschitz-type continuous on C, A2 f is pseudomonotone on C, A3 f is weakly continuous on C, A4 S k is nonexpansive on C for all k ≥ 1, ∞ ∩i =1 Fix(S1 ) ∩ Sol ( f , C ) ≠ ∅, ∞ A5 ∑ sup{ k =1 S k +1( x ) − S k ( x ) : x ∈ D} < ∞ for any bounded subset D of C An important special case of problem (1.1) is that f ( x, y ) = F ( x ), y − x , where F : C → H and this problem is reduced to finding a common element of the solution set of variational inequalities and the set of fixed points of an infinite family of nonexpansive mappings (see [6, 14, 16, 17, 22]) Motivated by the viscosity method in [16] and the approximation method in [7] via an improvement set of extragradient methods in [2-4], we introduce a new iteration algorithm for finding a common element of problem (1.1) At each main iteration n, we only solve two strongly convex programs with a pseudomonotone and Lipschitz-type continuous bifunction We show that all of the iterative sequences generated by this algorithm converge weakly to the common element in a real Hilbert space This paper is organized as follows: Section recalls some concepts in equilibrium problems and fixed point problems that will be used in the sequel P N Anh, L B Long, N V Quy and L Q Thuy 116 and an iterative algorithm for solving problem (1.1) Section investigates the convergence of the algorithms presented in Section as the main results of our paper Preliminaries In 1953, Mann [13] introduced a well-known classical iteration method to approximate a fixed point of a nonexpansive mapping S : C → C in a real Hilbert space H This iteration is defined as x k +1 = α k x k + (1 − α k ) S ( x k ) , ∀k ≥ 0, where C is nonempty closed convex subset of H and {α k } ⊂ [0, 1] Then {x k } converges weakly to x∗ ∈ Fix( S ) For finding a common fixed point of an infinite family of nonexpansive mappings {S k }, Aoyama et al [7] introduced an iterative sequence {x k } of C defined by x ∈ C and x k +1 = α k x k + (1 − α k ) S k ( x k ), ∀k ≥ 0, where C is a nonempty closed convex subset of a real Hilbert space, {α k } ⊂ [0, 1] and ∞ ∩i =1 Fix(Si ) ≠ ∅ The authors proved that the sequence {x k } ∞ converges strongly to x∗ ∈ ∩i =1 Fix( Si ) Recently, Yao et al [20] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1): ⎧x0 ∈ C, ⎪ ⎪ k k k k ⎨ f ( y , x ) + r x − y , y − x ≥ 0, ∀x ∈ C , k ⎪ ⎪ x k +1 = α k g ( x k ) + β k x k + γ kWk ( y k ) , ⎩ where g : C → C is contractive and Wk is W-mapping of {S k } Under mild Weak Convergence Theorems for an Infinite Family … 117 assumptions on parameters, the authors proved that the sequences {x k } and {y k } converge strongly to x∗ , where x ∗ = Pr ∩i∞=1 Fix ( Si )∩ Sol ( f , C ) ( g ( x∗ )) Methods for solving problem (1.1) have been well developed by many researchers (see [5, 9-11, 18-21]) These methods require solving approximation equilibrium problems with strongly monotone or monotone and Lipschitz-type continuous bifunctions In our scheme, the main steps are to solve two strongly convex problems ⎧ y k = arg ⎧λ f ( x k , y ) + y − x k : y ∈ C ⎫ , ⎨ k ⎬ ⎪⎪ ⎩ ⎭ ⎨ ⎪t k = arg ⎧⎨λ k f ( y k , y ) + y − x k : y ∈ C ⎫⎬ , ⎪⎩ ⎩ ⎭ (2.1) and compute the next iteration point by Mann-type fixed points x k +1 = α k x k + (1 − α k ) S k (t k ) (2.2) To investigate the convergence of this scheme, we recall the following technical lemmas which will be used in the sequel Lemma 2.1 (See [1]) Let C be a nonempty closed convex subset of a real Hilbert space H Let f : C × C → R be a pseudomonotone, Lipschitz- type continuous bifunction For each x ∈ C , let f ( x, ⋅) be convex and subdifferentiable on C Suppose that the sequences {x k }, { y k }, {t k } are generated by scheme (2.1) and x∗ ∈ Sol ( f , C ) Then t k − x∗ ≤ x k − x∗ − (1 − 2λ k c1 ) x k − y k − (1 − 2λ k c2 ) y k − t k , ∀k ≥ Lemma 2.2 [17] Let H be a real Hilbert space, {δ k } be a sequence of real numbers such that {δ k } ⊂ [α, β] ⊂ (0, 1) , c > and two sequences P N Anh, L B Long, N V Quy and L Q Thuy 118 {x k }, {y k } of H such that ⎧ sup x k ≤ c, ⎪lim ⎪⎪ k → ∞ k ≤ c, ⎨lim sup y ⎪ k →∞ ⎪lim sup δ x k + (1 − δ ) y k k k ⎪⎩ k → ∞ = c Then lim x k − y k k →∞ = Lemma 2.3 [7] Let C be a nonempty closed convex subset of a Banach space Let {S k } be a sequence of nonexpansive mappings of C into itself and S is a mapping of C into itself such that ⎧∞ ⎪⎪ sup{ S k +1 ( x ) − S k ( x ) : x ∈ C} < ∞, ⎨ i =1 ⎪ S k ( x ), ∀x ∈ C ⎪⎩S ( x ) = klim →∞ ∑ Then lim sup{ S k ( x ) − S ( x ) : x ∈ C} = k →∞ Lemma 2.4 (See [12]) Assume that S is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H If Fix( S ) ≠ ∅, then I − S is demiclosed; that is, whenever {x k } is a sequence in C weakly converging to some x ∈ C and the sequence {( I − S ) ( x k )} strongly converging to some y , it follows that ( I − S ) ( x ) = y Here I is the identity operator of H Lemma 2.5 (See [17]) Let C be a nonempty closed convex subset of a real Hilbert space H Suppose that, for all u ∈ C , the sequence {x k } satisfies x k +1 − u ≤ xk − u , ∀k ≥ Then the sequence {PrC ( x k )} converges strongly to some x ∈ C Weak Convergence Theorems for an Infinite Family … 119 Convergence Results Now, we prove the main convergence theorem Theorem 3.1 Suppose that Assumptions A1-A5 are satisfied, x ∈ C and two positive sequences {λ k }, {α k } satisfy the following restrictions: ⎧{α k } ⊂ [c, d ] ⊂ (0, 1) ⎪ ⎨ ⎛ 1⎞ ⎪⎩{λ k } ⊂ [a, b] for some a, b ∈ ⎜⎝ 0, L ⎟⎠, where L = max{2c1 , 2c2 } Then the sequences {x k }, {y k } and {t k } generated by (2.1) and (2.2) ∞ converge weakly to the same point x ∗ ∈ ∩i =1 Fix( Si ) ∩ Sol ( f , C ) , where x ∗ = lim Pr ∩i∞=1 Fix ( Si )∩ Sol ( f , C ) k →∞ ( x k ) The proof of this theorem is divided into several steps In Steps and 2, we will consider weak clusters of {x k } It follows from Lemma 2.1 that x k +1 − x∗ ≤ x k − x∗ , ∀x∗ ∈ ∞ ∩i =1 Fix(Si ) ∩ Sol ( f , C ) , k ≥ 0, and hence there exists c = lim x k − x∗ k →∞ (3.1) k The sequence {x k } is bounded and there exists a subsequence {x j } converges weakly to x as j → ∞ ∞ Step Claim that x ∈ ∩i =1 Fix( Si ) Proof of Step By Lemma 2.1 and (3.1), we have (1 − d ) (1 − 2bc1 ) x k − y k ≤ (1 − d ) (1 − 2λ k c1 ) x k − y k ≤ x k − x∗ − x k +1 − x∗ → as k → ∞ 120 P N Anh, L B Long, N V Quy and L Q Thuy Then lim x k − y k = yk − tk = k →∞ (3.2) By the similar way, also lim k →∞ xk − t k Combining this, (3.2) and the inequality ≤ xk − yk + y k − t k , we have lim x k − t k k →∞ = (3.3) ∞ Since x∗ ∈ ∩i =1 Fix( Si ) ∩ Sol ( f , C ) , Lemma 2.1 and (3.1), we have S k (t k ) − x∗ ≤ t k − x∗ ≤ x k − x∗ and hence lim sup S k (t k ) − x∗ k →∞ ≤ c Using (3.1) and x k +1 = α k x k + (1 − α k ) S k (t k ) , we have lim α k ( x k − x∗ ) + (1 − α k ) ( S k (t k ) − x ∗ ) = lim x k +1 − x ∗ k →∞ k →∞ = c By Lemma 2.2, we have lim S k (t k ) − x k k →∞ = (3.4) If follows from (3.1) and (3.3) that the sequence {t k } is bounded By Assumption A5, we have ∞ sup{ S k +1 ( x ) − S k ( x ) ∑ k =1 : x ∈ {t k }} < ∞ Weak Convergence Theorems for an Infinite Family … 121 Let S be a mapping of C into itself defined by S ( x ) = lim S k ( x ) (3.5) k →∞ for all x ∈ C and suppose that Fix( S ) = ∞ ∩i =1 Fix(Si ) Then, using Lemma 2.3, (3.3) and (3.4), we obtain S (xk ) − xk ≤ S ( x k ) − S (t k ) + S (t k ) − S k (t k ) + S k (t k ) − x k ≤ xk − t k + sup{ S ( x ) − S k ( x ) : x ∈ {t k }} + S k (t k ) − x k → as k → ∞ k Then, by Lemma 2.4 and the sequence {x j } converges weakly to x , we have x ∈ Fix( S ) , i.e., ∞ x∈ Step When x kj Fix( Si ) ∩ i =1 x as j → ∞, we show that x ∈ Sol ( f , C ) Proof of Step Since y k is the unique solution of the strongly convex problem y − x k + f ( x k , y ) : y ∈ C ⎫⎬ , ⎧⎨ ⎩2 ⎭ we have ∈ ∂ ⎛⎜ λ k f ( x k , y ) + y − x k ⎞⎟ ( y k ) + N C ( y k ) ⎝ ⎠ This follows that = λk w + y k − xk + w, P N Anh, L B Long, N V Quy and L Q Thuy 122 where w ∈ ∂ f ( x k , y k ) and w ∈ N C ( y k ) By the definition of the normal cone N C , we imply that y k − x k , y − y k ≥ λ k w, y k − y , ∀y ∈ C (3.6) On the other hand, since f ( x k , ⋅) is subdifferentiable on C, by the well known Moreau-Rockafellar theorem, there exists w ∈ ∂ f ( x k , y k ) such that f ( x k , y ) − f ( x k , y k ) ≥ w, y − y k , ∀y ∈ C Combining this with (3.6), we have λ k ( f ( x k , y ) − f ( x k , y k )) ≥ y k − x k , y k − y , ∀y ∈ C Hence k k λk j ( f (x j , y) − f (x j , y kj )) ≥ y kj k − x j, y k Then, using {λ k } ⊂ [a, b] ⊂ ⎛⎜ 0, ⎞⎟ , (3.2), x j ⎝ L⎠ kj −y , ∀y ∈ C x as j → ∞ and weak continuity of f, we have f ( x , y ) ≥ 0, ∀y ∈ C This means that x ∈ Sol ( f , C ) Step Claim that the sequences {x k }, {y k } and {t k } converge weakly to the same point x∗ , where x∗ = lim Pr k →∞ ∩i∞=1 Fix ( Si ) ∩ Sol ( f , C ) ( x k ) Proof of Step It follows from Steps and that for every weak cluster ∞ point x of the sequence {x k } satisfies x ∈ ∩i =1 Fix( Si ) ∩ Sol ( f , C ) We show that {x k } converges weakly to x Now, we assume that {x nk } is an Weak Convergence Theorems for an Infinite Family … another subsequence of {x k } such that x nk 123 xˆ as k → ∞ Then xˆ ∈ Sol ( f , C ) ∩ Fix( S ) , where S is defined by (3.5) We will show that xˆ = x If x ≠ xˆ , then from (3.1) and the Opial condition, it follows that c = lim x k − x k →∞ = lim inf x kj −x < lim inf x kj − xˆ j →∞ j →∞ = lim x k − xˆ k →∞ = lim inf x nk − xˆ k →∞ < lim inf x nk − x k →∞ = lim inf x k − x k →∞ = c This is a contraction Thus, we have x = xˆ It implies xk x ∈ Sol ( f , C ) ∩ Fix( S ) as k → ∞ It follows from (3.2) and (3.3) that yk x, t k x as k → ∞ Setting z k = PrSol ( f , C ) ∩ Fix( S ) ( x k ) 124 P N Anh, L B Long, N V Quy and L Q Thuy Then, from x ∈ Sol ( f , C ) ∩ Fix( S ) , it implies x − z k , z k − x k ≥ 0, ∀k ≥ By Lemma 2.5 and Step 1, the sequence {z k } converges strongly to z ∈ Sol ( f , C ) ∩ Fix( S ) Hence, we have x − z , z − x ≥ 0, so, we have x = z This shows that lim Pr k →∞ ∩i∞=1 Fix( Si ) ∩ Sol ( f , C ) ( x k ) = x ~ Applications Let C be a nonempty closed convex subset of a real Hilbert space H and F be a function from C into H In this section, we consider the variational inequality problem which is presented as follows: Find x∗ ∈ C such that F ( x∗ ), x − x∗ ≥ for all x ∈ C VI ( F , C ) Let f : C × C → R be defined by f ( x, y ) = F ( x ), y − x Then problem EP( f , C ) can be written in VI ( F , C ) The set of solutions of VI ( F , C ) is denoted by Sol ( F , C ) Recall that the function F is called strongly monotone on C with β > if F ( x ) − F ( y ), x − y ≥ β x − y , ∀x, y ∈ C ; monotone on C if F ( x ) − F ( y ), x − y ≥ 0, ∀x, y ∈ C ; pseudomonotone on C if F ( y ), x − y ≥ ⇒ F ( x ), x − y ≥ 0, ∀x, y ∈ C ; Lipschitz continuous on C with constants L > if F (x) − F ( y) ≤ L x − y , ∀x, y ∈ C Weak Convergence Theorems for an Infinite Family … 125 Since y k = arg ⎧⎨λ k f ( x k , y ) + y − x k : y ∈ C ⎫⎬ ⎩ ⎭ = arg ⎧⎨λ k F ( x k ), y − x k + y − x k : y ∈ C ⎫⎬ ⎭ ⎩ = PrC ( x k − λ k F ( x k )) , using (2.1), (2.2) and Theorem 3.1, we obtain the following convergence theorem for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings {Si } and the solution set of problem VI ( F , C ) Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let F be a function from C to H such that F is pseudomonotone and L-Lipschitz continuous on C For each i = 1, , Si : C → C is nonexpansive such that ∞ ∑k =1 sup{ ∞ ∩i =1 Fix(Si ) ∩ Sol ( F , C ) ≠ ∅ and S k +1 ( x ) − S k ( x ) : x ∈ D} < ∞ for any bounded subset D of C If positive sequences {α k } and {λ k } satisfy the following restrictions: ⎧{α k } ⊂ [c, d ] ⊂ (0, 1) , ⎪ ⎨ ⎛ 1⎞ ⎪⎩{λ k } ⊂ [a, b] for some a, b ∈ ⎜⎝ 0, L ⎟⎠ , then the sequences {x k }, { y k } and {t k } generated by ⎧ y k = PrC ( x k − λ k F ( x k )) , ⎪ ⎪k k k ⎨t = PrC ( x − λ k F ( y )) , ⎪ k +1 = α k x k + (1 − α k ) S k (t k ) , ⎪⎩ x ∞ converge weakly to the same point x ∗ ∈ ∩i =1 Fix( Si ) ∩ Sol ( F , C ) , where x ∗ = lim 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solution set of the equilibrium problems EP( f , C ) and the set ∞ of fixed points ∩k =1 Fix( S k ) of an infinite family of nonexpansive mappings Weak Convergence Theorems for an Infinite Family …... (2.1), (2.2) and Theorem 3.1, we obtain the following convergence theorem for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings {Si } and the solution... 2588-2595 [5] P N Anh, J K Kim and J M Nam, Strong convergence of an extragradient method for equilibrium problems and fixed point problems, J Korean Math Soc (2011), accepted [6] P N Anh and D X Son,