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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 165098, 18 pages doi:10.1155/2010/165098 Research Article A New Iterative Method for Solving Equilibrium Problems and Fixed Point Problems for Infinite Family of Nonexpansive Mappings Shenghua Wang,1 Yeol Je Cho,2 and Xiaolong Qin3 School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China Correspondence should be addressed to Yeol Je Cho, yjcho@gnu.ac.kr Received January 2010; Revised 21 May 2010; Accepted 11 July 2010 Academic Editor: Simeon Reich Copyright q 2010 Shenghua Wang et al 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 a new iterative scheme for finding a common element of the solutions sets of a finite family of equilibrium problems and fixed points sets of an infinite family of nonexpansive mappings in a Hilbert space As an application, we solve a multiobjective optimization problem using the result of this paper Introduction Let H be a Hilbert space and C be a nonempty, closed, and convex subset of H Let Φ be a bifunction of C × C into R, where R is the set of real numbers The equilibrium problem for the bifunction Φ : C × C → R is to find x ∈ C such that Φ x, y ≥ 0, ∀y ∈ C 1.1 The set of solutions of the above inequality is denoted by EP Φ Many problems arising from physics, optimization, and economics can reduce to finding a solution of an equilibrium problem In 2007, S Takahashi and W Takahashi first introduced an iterative scheme by the viscosity approximation method for finding a common element of the solutions set of equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space Fixed Point Theory and Applications H and proved a strong convergence theorem which is based on Combettes and Hirstoaga’s result and Wittmann’s result More precisely, they obtained the following theorem Theorem 1.1 see Let C be a nonempty closed and convex subset of H Let Φ : C × C → R be a bifunction which satisfies the following conditions: A1 Φ x, x for all x ∈ C; A2 Φ is monotone, that is, Φ x, y Φ y, x ≤ for all x, y ∈ C; A3 For all x, y, z ∈ C, limΦ tz − t x, y ≤ Φ x, y ; t↓0 1.2 A4 For each x ∈ C, y → Φ x, y is convex and lower semicontinuous Let S : C → H be a nonexpansive mapping with Fix S ∩ EP Φ / ∅, where Fix S denotes the set of fixed points of the mapping S, and let f : H → H be a contraction, if there exists a constant λ ∈ 0, such that fx − fy ≤ λ x − y for all x, y ∈ H Let {xn } and {un } be the sequences generated by x1 ∈ H and y − un , un − xn ≥ 0, rn Φ un , y xn αn f xn 1 − αn Sun , ∀y ∈ C, 1.3 ∀n ≥ 1, where {αn } ⊂ 0, and {rn } ⊂ 0, ∞ satisfy the following conditions: lim αn n→∞ ∞ αn 0, ∞ ∞, n lim inf rn > 0, n→∞ |αn − αn | < ∞, n ∞ |rn 1.4 − rn | < ∞ n Then the sequences {xn } and {un } converge strongly to a point z ∈ Fix S ∩ EP Φ , where z PFix S ∩EP Φ f z 1.5 P is the metric projection of H onto C and PFix S ∩EP Φ f z denotes nearest point in Fix S ∩ EP Φ from f z Recently, many results on equilibrium problems and fixed points problems in the context of the Hilbert space and Banach space are introduced see, e.g., 4–8 Fixed Point Theory and Applications Let F : H → H be a nonlinear mapping The variational inequality problem corresponding to the mapping F is to find a point x∗ ∈ C such that F x∗ , x − x∗ ≥ 0, ∀x ∈ C 1.6 The variational inequality problem is denoted by VI F, C The mapping F is called κ-Lipschitzian and η-strongly monotone if there exist constants κ, η > such that Fx − Fy ≤ κ x − y , ∀x, y ∈ H, Fx − Fy, x − y ≥ η x − y 1.7 , 1.8 ∀x, y ∈ H, respectively It is well known that if F is strongly monotone and Lipschitzian on C, then VI F, C has a unique solution An important problem is how to find a solution of VI F, C Recently, there are many results to solve the VI F, C see, e.g., 10–14 Let C be a nonempty closed and convex subset of a Hilbert space H, {Tn }∞ : H → H n be a countable family of nonexpansive mappings, and {Φi }m : C × C → R be m bifunctions i ∞ satisfying conditions A1 – A4 such that Ω n Fix Tn ∩ EP Φ1 ∩ · · · ∩ EP Φm / ∅ Let r1 , , rm ∈ 0, ∞ For each i 1, , m, define the mapping Tri : H → C by Tri x z ∈ C : Φi z, y y − z, z − x ≥ 0, ∀y ∈ C , ri ∀x ∈ H 1.9 Lemma 2.5 see below shows that, for each ≤ i ≤ m, Tri is firmly nonexpansive and EP Φi Suppose that F : H → H is a κ-Lipschitzian and hence nonexpansive and Fix Tri η-strong monotone operator and let μ ∈ 0, 2η/κ2 Assume that VI Φi F, Ω / ∅ In this paper, motivated and inspired by the above research results, we introduce the following iterative process for finding an element in Ω: for an arbitrary initial point x1 ∈ H, zn xn αn xn n γ1 Tr1 xn γ2 Tr2 xn αi−1 − αi σn Ti xn ··· γm Trm xn , − αn − σn T λn zn , ∀n ≥ 1, 1.10 i where T λn zn zn − λn μF zn , α0 1, {αn }∞ is a strictly decreasing sequence in 0, α with n < α < 1, {λn }∞ ⊂ 0, , {γi }m ⊂ 0, with m γi 1, and {σn }∞ ⊂ a, b with < a, b < n n i i Then we prove that the iterative process {xn } defined by 1.10 strongly converge to an element x∗ ∈ Ω, which is the unique solution of the variational inequality F x∗ , x − x∗ ≥ 0, ∀x ∈ Ω As an application of our main result, we solve a multiobjective optimization problem 1.11 Fixed Point Theory and Applications Preliminaries Let H be a Hilbert space and T a nonexpansive mapping of H into itself such that Fix T / ∅ For all x ∈ Fix T and x ∈ H, we have x−x ≥ Tx − Tx Tx − x 2 Tx − x x−x Tx − x x−x T x − x, x − x 2.1 and hence Tx − x ≤ x − T x, x − x , ∀x ∈ Fix T , x ∈ H 2.2 It is well known that, for all x, y ∈ H and t ∈ 0, , tx 1−t y ≤t x 1−t y , 2.3 which implies that n ti xi i ≤ n ti xi 2.4 i for all {xi }n ⊂ H and {ti }n ⊂ 0, with n ti i i i Let C be a nonempty closed and convex subset of H and, for any x ∈ H, there exists unique nearest point in C, denoted by PC x, such that PC x − x ≤ y − x , ∀y ∈ C 2.5 Moreover, we have the following: z PC x ⇐⇒ x − z, z − y ≥ 0, ∀y ∈ C 2.6 Let I denote the identity operator of H and let {xn } be a sequence in a Hilbert space H and x ∈ H Throughout this paper, xn → x denotes that {xn } strongly converges to x and x denotes that {xn } weakly converges to x xn We need the following lemmas for our main results Lemma 2.1 see 15 Let C be a nonempty closed and convex subset of a Hilbert space H and T a nonexpansive mapping from C into itself Then I − T is demiclosed at zero, that is, xn x, xn − T xn −→ implies x T x 2.7 Fixed Point Theory and Applications Lemma 2.2 see 10, Lemma 3.1 b Let H be a Hilbert space and T : H → H be a nonexpansive mapping Let F : H → H be a mapping which is κ-Lipschitzian and η-strong monotone on T H Assume that λ ∈ 0, and μ ∈ 0, 2η/κ2 Define a mapping T λ : H → H by T λx T x − λμF T x , ∀x ∈ H Then T λ x − T λ y ≤ − λτ x − y for all x, y ∈ H, where τ If T 1− 2.8 − μ 2η − μκ2 ∈ 0, I, Lemma 2.2 still holds Lemma 2.3 see 16 Let {sn }, {cn } be the sequences of nonnegative real numbers and {an } ⊂ 0, Suppose that {bn } is a real number sequence such that sn ≤ − an sn bn cn , ∀n ≥ 2.9 Assume that ∞ cn < ∞ Then the following results hold n (1) If bn ≤ βan for all n ≥ 0, where β ≥ 0, then {sn } is a bounded sequence (2) If ∞ an ∞, n then limn → ∞ sn lim sup n→∞ bn ≤ 0, an 2.10 Lemma 2.4 see 17 Let C be a nonempty closed and convex subset of a Hilbert space H and Φ : C × C → R be a bifunction which satisfies the conditions (A1)–(A4) Let r > and x ∈ H Then there exists z ∈ C such that Φ z, y y − z, z − x ≥ 0, r ∀y ∈ C 2.11 Lemma 2.5 see Let H be a Hilbert space and C be a nonempty closed and convex subset of H Assume that Φ : C × C → R satisfies the conditions (A1)–(A4) For all r > and x ∈ H, define a mapping Tr : H → C as follows: Tr x z ∈ C : Φ z, y y − z, z − x ≥ 0, ∀y ∈ C , r ∀x ∈ H 2.12 Then the following holds: Tr is single-valued; Tr is firmly nonexpansive, that is, for any x, y ∈ H, Tr x − Tr y Fix Tr EP Φ ; EP Φ is closed and convex ≤ Tr x − Tr y, x − y ; 2.13 Fixed Point Theory and Applications The following lemma is an immediate consequence of an inner product Lemma 2.6 Let H be a real Hilbert space Then the following identity holds: x y ≤ x 2 y, x y , ∀x, y ∈ H 2.14 Main Results First, we prove some lemmas as follows Lemma 3.1 The sequence {xn } generated by 1.10 is bounded Proof Let uin Tri xn for each i 1, 2, , m Lemma 2.5 shows that each Tri is firmlynonexpansive and hence nonexpansive Hence, for each ≤ i ≤ m and p ∈ Ω, we have Tri xn − Tri p ≤ xn − p , uin − p m zn − p ≤ ∀n ≥ 1, 3.1 ∀n ≥ 3.2 ∀x, y ∈ H, 3.3 γi uin − p ≤ xn − p , i By Lemma 2.2, we have T λ n x − T λ n y ≤ − λn τ where τ 1− − μ 2η − μκ2 ∈ 0, Therefore, by 3.2 and 3.3 , we obtain note that {αn } is strictly decreasing and T λn p − p xn −p x−y , αn xn − p n −λn μF p αi−1 − αi σn Ti xn − p − αn − σn T λn zn − p αi−1 − αi σn Ti xn − p − αn − σn T λn zn − p i ≤ αn xn − p n i ≤ αn xn − p n αi−1 − αi σn xn − p i 1 − αn − σn ≤ αn xn − p n T λn zn − T λn p T λn p − p αi−1 − αi σn xn − p i 1 − αn − σn − λn τ zn − p λn μ F p Fixed Point Theory and Applications n ≤ αn xn − p αi−1 − αi σn xn − p i 1 − αn − σn xn − p − λn τ − − αn − σn λn τ xn − p λn μ F p − αn − σn λn μ F p 3.4 By induction, we obtain xn are {zn } and {uin } for each i F zn ≤ max{ x1 − p , μ/τ F p } Hence {xn } is bounded and so 1, 2, , m Since F is κ-Lipschitzian, we have ≤ F zn − F p F p F p ≤ κ zn − p ≤ κ zn F p κ p , 3.5 which shows that {F zn } is bounded This completes the proof Lemma 3.2 If the following conditions hold: ∞ λn ∞, n then limn → ∞ xn Proof For each i ∞ ∞ |λn − λn | < ∞, n − xn |σn − σn | < ∞, 3.6 n 1, 2, , m, since each Tri is nonexpansive, we have uin−1 − uin Tri xn−1 − Tri xn ≤ xn−1 − xn , ∀n ≥ 3.7 By 3.7 , we have γ1 u1n − u1n−1 zn − zn−1 ≤ m γ2 u2n − u2n−1 γi uin − uin−1 ≤ i m ··· γm umn − umn−1 γi xn − xn−1 3.8 i xn−1 − xn , ∀n ≥ By the definition of the iterative sequence 1.10 , we have xn − xn αn xn − xn−1 αn xn−1 n αi−1 − αi σn Ti xn − Ti xn−1 i n αi−1 − αi σn Ti xn−1 − αn − σn T λn zn − T λn zn−1 i 1 − αn − σn T λn zn−1 − αn−1 xn−1 − n−1 i − − αn−1 − σn−1 T λn−1 zn−1 αi−1 − αi σn−1 Ti xn−1 Fixed Point Theory and Applications αn xn − xn−1 αn − αn−1 xn−1 n αi−1 − αi σn Ti xn − Ti xn−1 i n T λn zn − T λn zn−1 − αn − σn αi−1 − αi σn Ti xn−1 i − n−1 αi−1 − αi σn−1 Ti xn−1 − αn − σn T λn zn−1 i − − αn−1 − σn−1 T λn−1 zn−1 αn xn − xn−1 αn − αn−1 xn−1 n αi−1 − αi σn Ti xn − Ti xn−1 i n−1 T λn zn − T λn zn−1 − αn − σn αi−1 − αi σn − σn−1 Ti xn−1 i αn−1 − αn σn Tn xn−1 αn−1 − αn − σn σn−1 − σn − αn−1 zn−1 { − αn−1 − σn−1 λn−1 − λn − αn−1 − αn − σn σn−1 − σn − αn−1 λn }μF zn−1 , 3.9 and hence xn − xn ≤ αn xn − xn−1 n αn−1 − αn xn−1 αi−1 − αi σn xn − xn−1 i n−1 − αn − σn − λn τ zn − zn−1 αi−1 − αi |σn − σn−1 | Ti xn−1 i αn−1 − αn Tn xn−1 |λn−1 − λn | αn xn − xn−1 αn−1 − αn αn−1 − αn n |σn−1 − σn | zn−1 |σn−1 − σn | μ F zn−1 αi−1 − αi σn xn − xn−1 i 1 − αn − σn − λn τ zn − zn−1 αn−1 − αn n−1 xn−1 Tn xn−1 αi−1 − αi |σn − σn−1 | Ti xn−1 zn−1 μ F zn−1 |σn−1 − σn | zn−1 μ F zn−1 i |λn−1 − λn |μ F zn−1 3.10 Fixed Point Theory and Applications It follows from 3.8 and 3.10 that xn n − xn ≤ αn xn − xn−1 αi−1 − αi σn xn − xn−1 i 1 − αn − σn − λn τ xn−1 − xn αn−1 − αn n−1 Tn xn−1 xn−1 zn−1 αi−1 − αi |σn − σn−1 | Ti xn−1 μ F zn−1 |σn−1 − σn | zn−1 μ F zn−1 i 3.11 |λn−1 − λn |μ F zn−1 ≤ − − αn − σn λn τ xn − xn−1 |σn − σn−1 | μ M |λn−1 − λn |μM μ M ≤ − − α − b λn τ xn − xn−1 |σn − σn−1 | αn−1 − αn αn−1 − αn μ M |λn−1 − λn |μM, μ M where M max{supn≥1 xn , supn≥1 zn , supi≥1,n≥1 Ti xn , supn≥1 F zn } Since {αn } is α1 < ∞ Further, from the assumptions, it strictly decreasing, we have ∞ αn−1 − αn n follows that ∞ αn−1 − αn |σn − σn−1 | μ M μ M |λn−1 − λn |μM < ∞ 3.12 n Therefore, by Lemma 2.3, we have limn → ∞ xn − xn This completes the proof Lemma 3.3 If the following conditions hold: lim λn n→∞ ∞ 0, λn ∞ ∞, n then lim xn − uin n→∞ n for each i Proof For any p ∈ Ω and i uin − p |λn − λn | < ∞, ∞ |σn − σn | < ∞, 3.13 n 1, 2, , m 1, 2, , m, it follows from Lemma 2.5 that Tri xn − Tri p uin − p 2 ≤ Tri xn − Tri p, xn − p xn − p − uin − xn uin − p, xn − p 3.14 , 10 Fixed Point Theory and Applications and hence uin − p ≤ xn − p − uin − xn Further, we have m zn − p γi uin − p m ≤ i ≤ m γi uin − p i xn − p γi − uin − xn 3.15 i xn − p m − γi uin − xn , ∀n ≥ i Therefore, from 2.4 and 3.3 , we have xn 1−p n αn xn − p − αn − σn T λn zn − p αi−1 − αi σn Ti xn − p i ≤ αn xn − p n αi−1 − αi σn Ti xn − p − αn − σn T λn zn − p i ≤ αn xn − p − αn σn xn − p − αn − σn ≤ αn xn − p ≤ αn xn − p × − λn τ 2 − m 2 2 λn μ F p − αn − σn 2λn − λn τ μ zn − p − αn σn xn − p xn − p T λn p − p zn − p − λn τ − αn σn xn − p zn − p − λn τ ≤ αn xn − p × T λn zn − T λn p − αn σn xn − p − αn − σn F p λn μ F p − αn − σn γi uin − xn i 2λn − λn τ μ zn − p − − αn − σn λn τ λn μ F p F p xn − p 2 − − αn − σn − λn τ m γi uin − xn i 2λn μ − αn − σn − λn τ zn − p F p − αn − σn λn μ2 F p Fixed Point Theory and Applications ≤ xn − p 11 m − − αn − σn − λn τ γi uin − xn i − αn − σn λn μ2 F p 2λn μ − αn − σn − λn τ zn − p F p 3.16 It follows that γi − αn − σn − λn τ uin − xn xn − p ≤ xn −p xn − xn λn μ F p 2μ zn − p F p 3.17 for each i 1, 2, , m Note that < γi < for i 1, 2, , m From the assumptions, Lemma 3.2, and the previous inequality, we conclude that uin − xn → as n → ∞ for each i 1, 2, , m Further, we have zn − xn ≤ m γi uin − xn −→ n −→ ∞ 3.18 i This completes the proof Lemma 3.4 If the following conditions hold: lim λn n→∞ ∞ 0, λn n then limn → ∞ xn − Ti xn ∞ ∞, ∞ |λn − λn | < ∞, |σn − σn | < ∞, 3.19 − αn − σn T λn zn , n 3.20 n for all i ≥ Proof By the definition of the iterative sequence 1.10 , we have xn n αi−1 − αi σn xn − Ti xn − − αn σn xn αn xn i that is, n αi−1 − αi σn xn − Ti xn xn − xn − xn αn xn − αn σn xn xn − xn 1 − αn σn − xn xn − xn 1 − αn − σn − αn − σn T λn zn i 1 − αn − σn T λn zn T λn zn − xn 3.21 12 Fixed Point Theory and Applications Hence, for any p ∈ Ω, we get n − αn − σn T λn zn − xn , xn − p αi−1 − αi σn xn − Ti xn , xn − p xn − xn , xn − p i 3.22 Since each Ti is nonexpansive, by 2.2 , we have Ti xn − xn ≤ xn − Ti xn , xn − p 3.23 Hence, combining this inequality with 3.22 , we get n αi−1 − αi σn Ti xn − xn 2i ≤ − αn − σn T λn zn − xn , xn − p xn − xn , xn − p , 3.24 which implies that note that {αn } is a strictly decreasing sequence ≤ − αn − σn αi−1 − αi σn T λn zn − xn , xn − p ≤ Ti xn − xn − αn − σn αi−1 − αi σn T λn zn − xn xn − xn , xn − p αi−1 − αi σn xn − xn αi−1 − αi σn xn − p xn − p 3.25 From Lemma 3.3, limn → ∞ λn 0, and the inequality T λn zn − xn ≤ zn − xn λn μ F zn , 3.26 we obtain lim T λn zn − xn n→∞ 3.27 Therefore, from Lemma 3.2, 3.25 , and 3.27 , it follows that lim Ti xn − xn n→∞ This completes the proof 0, ∀i ≥ 3.28 Fixed Point Theory and Applications 13 Next we prove the main results of this paper Theorem 3.5 Assume that the following conditions hold: lim λn n→∞ ∞ λn 0, ∞ ∞, n ∞ γn − γn |λn − λn | < ∞, n ∞ < ∞, n 3.29 |σn − σn | < ∞ n Then the sequence {xn } generated by 1.10 converges strongly to an element in Ω, which is the unique solution of the variational inequality VI F, Ω Proof Since VI F, Ω / ∅, we can select an element x∗ ∈ VI F, Ω , which implies that F x∗ , x∗ − x ≥ 0, ∀x ∈ Ω 3.30 − x∗ ≤ 3.31 First, we prove that lim sup −F x∗ , xn n→∞ Since {xn } is bounded, there exists a subsequence {xnj } of {xn } such that lim sup −F x∗ , xn − x∗ n→∞ lim −F x∗ , xnj − x∗ j →∞ 3.32 x for some x ∈ H From Without loss of generality, we may further assume that xnj Lemmas 3.4 and 2.1, we get x ∈ Fix Tn for all n ≥ Hence we have x ∈ ∞ Fix Tn n It follows from Lemma 2.5 that each Tri is firmly nonexpansive and hence nonexpansive Lemma 3.3 shows that Tri xn − xn → as n → ∞ Therefore, from Lemma 2.1, it follows that x ∈ Fix Tri for each i 1, , m, which shows that x ∈ m Fix Tri Lemma 2.5 shows i EP Φi for each i 1, , m Hence x ∈ m EP Φi By using the above that Fix Tri i argument, we conclude that x∈Ω ∞ Fix Tn ∩ EP Φ1 ∩ · · · ∩ EP Φm 3.33 n Noting that x∗ is a solution of the VI F, Ω , we obtain lim sup −F x∗ , xn − x∗ n→∞ −F x∗ , x − x∗ ≤ 3.34 14 Fixed Point Theory and Applications It follows from Lemma 2.6 that xn − x∗ n αn xn − x∗ αi−1 − αn σn Ti xn − x∗ − αn − σn T λn zn − T λn x∗ i − αn − σn n ≤ αn xn − x∗ T λn x ∗ − x ∗ αi−1 − αn σn Ti xn − x∗ − αn − σn T λn zn − T λn x∗ i − αn − σn T λn x∗ − x∗ , xn ≤ αn xn − x∗ n − x∗ αi−1 − αn σn Ti xn − x∗ − αn − σn T λn zn − T λn x∗ i 3.35 ∗ − αn − σn λn μ −F x , xn ≤ αn xn − x∗ n −x αi−1 − αn σn xn − x∗ ∗ − αn − σn − λn τ zn − x∗ i − αn − σn λn μ −F x∗ , xn αn xn − x∗ − αn σn xn − x∗ − x∗ − αn − σn − λn τ zn − x∗ 2 − αn − σn λn μ −F x∗ , xn ≤ αn xn − x∗ 1 − αn σn xn − x∗ − x∗ − αn − σn − λn τ xn − x∗ 2 − αn − σn λn μ −F x∗ , xn 1 − − αn − σn λn τ xn − x∗ − αn − σn λn μ −F x∗ , xn ∞ n − x∗ Let an − αn − σn λn τ and bn − αn − σn λn μ −F x∗ , xn Then, from the assumptions and 3.31 , we have < an < 1, an ∞, lim sup n→∞ bn an 1 − x∗ − x∗ for all n ≥ 3.36 Therefore, by applying Lemma 2.3 to 3.35 , we conclude that the sequence {xn } strongly converges to a point x∗ In order to prove the uniqueness of solution of the VI F, Ω , we assume that u∗ is another solution of VI F, Ω Similarly, we can conclude that {xn } converges strongly to a u∗ , that is, x∗ is the unique solution of VI F, Ω This completes the point u∗ Hence x∗ proof Fixed Point Theory and Applications 15 As direct consequences of Theorem 3.5, we obtain the following corollaries Corollary 3.6 Let C be a nonempty closed and convex subset of a Hilbert space H For each i 1, 2, , m let Φi : C × C → R be m bifunctions which satisfy conditions (A1)–(A4) such that ∞ m i EP Φi / ∅ Let μ ∈ 0, , and let {αn }n ⊂ 0, α be a strictly decreasing sequence with < α < ∞ m 1, {λn }n ⊂ 0, , {γi }i ⊂ 0, with m γi 1, r1 , r2 , , rm ∈ 0, ∞ , and {σn }n ⊂ a, b n i with < a, b < For an arbitrary initial x1 ∈ H, define the iterative sequence {xn } by zn xn αn γ1 Tr1 xn γ2 Tr2 xn − αn σn xn ··· γm Trm xn , − αn − σn − λn μ zn , ∀n ≥ 3.37 If the following conditions hold: lim λn n→∞ ∞ 0, λn ∞, n ∞ ∞ |λn − λn | < ∞, n |σn − σn | < ∞, 3.38 n then the sequence {xn } converges strongly to an element x∗ ∈ m i EP Φi Proof Put F I and Ti I for each i ≥ in Theorem 3.5 Then we know that F is 11−αn xn and T λn zn 1−λn μ zn Lipschitzian and 1-strongly monotone, n αi−1 −αi Ti xn i Therefore, by Theorem 3.5, we conclude the desired result Corollary 3.7 Let C be a nonempty closed and convex subset of a Hilbert space H Let {Ti }∞1 be i ∞ a countable family of nonexpansive mappings of H such that C i Fix Ti and F : H → H an operator which is κ-Lipschitzian and η-strong monotone on H Let μ ∈ 0, 2η/κ2 Assume that VI F, C / ∅ Let {αn }∞ ⊂ 0, α with < α < be a strictly decreasing sequence, {λn }∞ ⊂ 0, n n and {σn }∞ ⊂ a, b with < a, b < For an arbitrary initial x1 ∈ H, define the iterative sequence n {xn } by xn αn xn n αi−1 − αi σn Ti xn − αn − σn PC xn − λn μF PC xn , ∀n ≥ 1, 3.39 i where α0 If the following conditions hold: lim λn n→∞ ∞ 0, n λn ∞, ∞ |λn − λn | < ∞, n ∞ |σn − σn | < ∞, 3.40 n then the sequence {xn } strongly converges to an element x∗ ∈ C, which is the unique solution of the variational inequality F x∗ , x − x∗ ≥ 0, ∀x ∈ C Proof Put Φi x, y for each i 1, 2, , m and x, y ∈ C Set r1 Tr2 xn · · · Trm xn in Theorem 3.5 Then, by 2.6 , we have Tr1 xn Theorem 3.5, we conclude the desired result 3.41 r2 ··· rm PC xn Therefore, by 16 Fixed Point Theory and Applications Remark 3.8 Recently, many authors have studied the iteration sequences for infinite family of nonexpansive mappings But our iterative sequence 1.10 is very different from others because we not use W-mapping generated by the infinite family of nonexpansive mappings and we have no any restriction with the infinite family of nonlinear mappings We not use Suzuki’s lemma 18 for obtaining the result that limn → ∞ xn − However, many authors have used Suzuki’s lemma 18 for obtaining the result that xn in the process of studying the similar algorithms For example, see limn → ∞ xn − xn 5, 19, 20 and so on Application In this section, we study a kind of multiobjective optimization problem based on the result of this paper That is, we give an iterative sequence which solves the following multiobjective optimization problem with nonempty set of solutions: h1 x , h2 x , 4.1 x ∈ C, where h1 x and h2 x are both convex and lower semicontinuous functions defined on a nonempty closed and convex subset of C of a Hilbert space H We denote by A the set of solutions of 4.1 and assume that A / ∅ We denote the sets of solutions of the following two optimization problems by A1 and A2 , respectively, h1 x x ∈ C, 4.2 h2 x x ∈ C Obviously, if we find a solution x ∈ A1 ∩ A2 , then one must have x ∈ A Now, let Φ1 and Φ2 be two bifunctions from C × C to R defined by Φ1 x, y h2 y − h2 x , respectively It is easy to see that EP Φ1 A1 h1 y − h1 x and Φ2 x, y and EP Φ2 A2 , where EP Φi denotes the set of solutions of the equilibrium problem: Φi x, y ≥ 0, ∀y ∈ C, i 1, 2, 4.3 respectively In addition, it is easy to see that Φ1 and Φ2 satisfy the conditions A1 – A4 Therefore, by setting m in Corollary 3.6, we know that, for any initial guess x1 ∈ H, h1 y − h1 u1n y − u1n , u1n − xn ≥ 0, r1n ∀y ∈ C, h2 y − h2 u2n y − u2n , u2n − xn ≥ 0, r2n ∀y ∈ C, zn xn αn − αn σn xn γ1 u1n − γ1 u2n , − αn − σn − λn μ zn , ∀n ≥ 4.4 Fixed Point Theory and Applications 17 By Corollary 3.6, we know that the sequence {xn } converges strongly to a solution x∗ ∈ A1 ∩ A2 , which is a solution of the multiobjective optimization problem EP Φ1 ∩ EP Φ2 4.1 Acknowledgment This work was supported by the Korea Research Foundation Grant funded by the Korean Government KRF-2008-313-C00050 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