Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 392741, 24 pages doi:10.1155/2011/392741 Research Article Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings in Hilbert Spaces Shenghua Wang1, and Baohua Guo1, National Engineering Laboratory for Biomass Power Generation Equipment, North China Electric Power University, Baoding 071003, China Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China Correspondence should be addressed to Shenghua Wang, sheng-huawang@hotmail.com Received 15 October 2010; Accepted 18 November 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2011 S Wang and B Guo 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 algorithm for finding a common element of the set of solutions of an infinite family of equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space We prove some strong convergence theorems for the proposed iterative scheme to a fixed point of the family of nonexpansive mappings, which is the unique solution of a variational inequality As an application, we use the result of this paper to solve a multiobjective optimization problem Our result extends and improves the ones of Colao et al 2008 and some others Introduction Let H be a real Hilbert space and T be a mapping of H into itself T is said to be nonexpansive if Tx − Ty ≤ x − y , ∀x, y ∈ H 1.1 If there exists a point u ∈ H such that T u u, then the point u is called a fixed point of T The set of fixed points of T is denoted by F T It is well known that F T is closed convex and also nonempty if T has a bounded trajectory see Fixed Point Theory and Applications Let f : H → H be a mapping If there exists a constant ≤ κ < such that ∀x, y ∈ H, fx − fy ≤ κ x − y , 1.2 then f is called a contraction with the constant κ Recall that an operator A : H → H is called to be strongly positive with coefficient γ > if Ax, x ≥ γ x , ∀x ∈ H 1.3 Let u ∈ H be a fixed point, A be a strongly positive linear bounded operator on H and {T }N be a finite family of nonexpansive mappings of H into itself such that F n N n F Tn / ∅ In 2003, Xu introduced the following iterative scheme: xn I− n 1A Tn xn n u, ∀n ≥ 1, 1.4 where I is the identical mapping on H and Tn Tn mod N , and proved some strong convergence theorems for the iterative scheme to the solution of the quadratic minimization problem x∈F under suitable hypotheses on F n F T1 T2 · · · TN Ax, x − x, u 1.5 and the additional hypothesis: F TN T1 · · · TN−1 ··· F T2 T3 · · · TN T1 1.6 Recently, Marino and Xu introduced a new iterative scheme from an arbitrary point x0 ∈ H by the viscosity approximation method as follows: xn n γf xn I− nA T xn , ∀n ≥ 1, 1.7 and prove that the scheme strongly converges to the unique solution x∗ of the variational inequality: A − γf x∗ , x − x∗ ≥ 0, ∀x ∈ F T , 1.8 which is the optimality condition for the minimization problem: x∈F Ax, x − h x , where h is a potential function for γf i.e., h x γf x for all x ∈ H 1.9 Fixed Point Theory and Applications Let {Tn }N be a finite family of nonexpansive mappings of H into itself In 2007, Yao n defined the mappings − λn,1 I, Un,1 λn,1 T1 Un,2 λn,2 T2 Un,1 − λn,2 I, 1.10 λn,N−1 TN−1 Un,N−2 Un,N−1 Wn ≡ Un,N − λn,N−1 I, − λn,N I λn,N TN Un,N−1 and, by extending 1.10 , proposed the iterative scheme: xn n γf xn βxn 1−β I − nA Wn xn , ∀n ≥ 1.11 Then he proved that the iterative scheme 1.10 strongly converges to the unique solution x∗ of the variational inequality: A − γf x∗ , x − x∗ ≥ 0, where F N n ∀x ∈ F, 1.12 F Tn , which is the optimality condition for the minimization problem: x∈F Ax, x − h x , 1.13 where h is a potential function for γf However, Colao et al pointed out in that there is a gap in Yao’s proof Let C be a nonempty closed convex subset of H and G : C × C → R be a bifunction The equilibrium problem for the function G is to determine the equilibrium points, that is, the set EP G x ∈ C : G x, y ≥ 0, ∀y ∈ C 1.14 Let A : C → H be a nonlinear mapping Let EP G, A denote the set of all solutions to the following equilibrium problem: EP cG, A x ∈ C : G x, y Az, y − z ≥ 0, ∀y ∈ C 1.15 In the case of A ≡ 0, EP G, A is deduced to EP In the case of G ≡ 0, EP G, A is also denoted by VI C, A Fixed Point Theory and Applications In 2007, S Takahashi and W Takahashi introduced a viscosity approximation method for finding a common element of EP G and F T from an arbitrary initial element x1 ∈ H y − un , un − xn ≥ 0, rn G un , y xn nf xn 1− n T un , ∀y ∈ C, 1.16 ∀n ≥ 1, and proved that, under certain appropriate conditions over n and rn , the sequences {xn } and {un } both converge strongly to z PF T ∩EP G f z By combing the schemes 1.7 and 1.16 , Plubtieg and Punpaeng proposed the following algorithm: y − un , un − xn ≥ 0, rn G un , y xn n γf xn I− nA ∀y ∈ C, 1.17 ∀n ≥ 1, T un , and proved that the iterative schemes {xn } and {un } converge strongly to the unique solution z of the variational inequality: A − γf z, x − z ≥ 0, ∀x ∈ F T ∩ EP G , 1.18 which is the optimality condition for the minimization problem: x∈F T ∩EP G Ax, x − h x , 1.19 where h is a potential function for γf Very recently, for finding a common element of the set of a finite family of nonexpansive mappings and the set of solutions of an equilibrium problem, by combining the schemes 1.11 and 1.17 , Colao et al proposed the following explicit scheme: G un , y xn n γf xn y − un , un − xn ≥ 0, rn βxn 1−β I − nA ∀y ∈ C, W n un , 1.20 ∀n ≥ 1, and proved under some certain hypotheses that both sequences {xn } and {un } converge strongly to a point x∗ ∈ F which is an equilibrium point for G and is the unique solution of the variational inequality: A − γf x∗ , x − x∗ ≥ 0, where F N n F Tn ∀x ∈ F ∩ EP G , 1.21 Fixed Point Theory and Applications The equilibrium problems have been considered by many authors; see, for example, 6, 8–19 and the reference therein But, in these references, the authors only considered at most finite family of equilibrium problems and few of authors investigate the infinite family of equilibrium problems in a Hilbert space or Banach space In this paper, we consider a new iterative scheme for obtaining a common element in the solution set of an infinite family of generalized equilibrium problems and in the common fixed-point set of a finite family of nonexpansive mappings in a Hilbert space Let {Tn }N N ≥ be a finite family of n nonexpansive mappings of H into itself, be {Gn } : C × C → R be an infinite family of bifunctions, and be {An } : C → H be an infinite family of kn -inverse-strongly monotone mappings Let {rn } be a sequence such that rn ⊂ r, 2kn with r > for each n ≥ Define the mapping Tri : H → C by Tri x y − z, z − x ≥ 0, ∀y ∈ C , ri z ∈ C : Gi z, y x ∈ H, i ≥ 1.22 N ∞ Assume that Ω i EP Gi , Ai / ∅ For an arbitrary initial point x1 ∈ H, we i F Ti define the iterative scheme {xn } by n zn αn xn αi−1 − αi Tri I − ri Ai xn , 1.23 i xn n γf xn δn Bxn I − δn B − nA Wn zn , ∀n ≥ 1, where α0 1, {αn }, { n } and {δn } are three sequences in 0,1 , A and B are both strongly positive linear bounded operators on H, Wn is defined by 1.10 , and prove that, under some certain appropriate hypotheses on the control sequences, the sequence {xn } strongly converges to a point x∗ ∈ Ω, which is the unique solution of the variational inequality: A − γf x∗ , x − x∗ ≥ 0, ∀x ∈ Ω 1.24 If Ai ≡ A0 , Gi ≡ G and ri ≡ r, then 1.23 is reduced to the iterative scheme: zn xn n γf xn αn xn δn Bxn − αn Tr I − rA0 xn , I − δn B − nA Wn zn , ∀n ≥ 1.25 The proof method of the main result of this paper is different with ones of others in the literatures and our result extends and improves the ones of Colao et al and some others 6 Fixed Point Theory and Applications Preliminaries Let C be a closed convex subset of a Hilbert space H For any point x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that x − PC x ≤ x − y , ∀y ∈ C 2.1 Then PC is called the metric projection of H onto C It is well known that PC is a nonexpansive mapping of H onto C and satisfies the following: x − y, PC x − PC y ≥ PC x − PC y , ∀x, y ∈ H 2.2 Let A be a mapping from C into H, then A is called monotone if x − y, Ax − Ay ≥ 2.3 for all x, y ∈ C However, A is called an α-inverse-strongly monotone mapping if there exists a positive real number α such that x − y, Ax − Ay ≥ α Ax − Ay 2.4 for all x, y ∈ C Let I denote the identity mapping of H, then for all x, y ∈ C and λ > 0, one has 20 I − λA x − I − λA y ≤ x−y λ λ − 2α Ax − Ay 2.5 Hence, if λ ∈ 0, 2α , then I − λA is a nonexpansive mapping of C into H If there exists u ∈ C such that v − u, Au ≥ 2.6 for all v ∈ C, then u is called the solution of this variational inequality The set of all solutions of the variational inequality is denoted by VI C, A In this paper, we need the following lemmas Fixed Point Theory and Applications Lemma 2.1 see 21 Given x ∈ H and y ∈ C Then PC x inequality x − y, y − z ≥ 0, y if and only if there holds the ∀z ∈ C 2.7 Lemma 2.2 see 22 Let {sn } be a sequence of nonnegative real numbers satisfying sn ≤ − η n sn ηn τn ξn , ∀n ≥ 0, 2.8 where {ηn }, {τn }, and {ξn } satisfy the conditions: ∞ n 1 {ηn } ⊂ 0, , ∞ or, equivalently, ηn ∞ n − ηn 0; lim supn → ∞ τn ≤ 0; ξn ≥ n ≥ , Then limn → ∞ sn ∞ n ξn < ∞ Let H be a Hilbert space For all x, y ∈ H, the following equality holds: x y x 2 y, x y − y 2.9 Therefore, the following lemma naturally holds Lemma 2.3 Let H be a real Hilbert space The following identity holds: x y ≤ x 2 y, x y , ∀x, y ∈ H 2.10 Lemma 2.4 see Assume that A is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ > and < ρ ≤ A −1 Then I − ρA ≤ − ργ Lemma 2.5 see Assume that {an } is a sequence of nonnegative numbers such that an ≤ − γn an δn , ∀n ≥ 0, where {γn } is a sequence in 0, and δn is a sequence in R such that ∞ n γn ∞; lim supn → ∞ δn /γn ≤ or Then limn → ∞ an ∞ n |δn | < ∞ 2.11 Fixed Point Theory and Applications Lemma 2.6 see 23 Let C be a nonempty closed convex subset of a Hilbert space H and let G : C × C → R be a bifunction which satisfies the following: for all x ∈ C; A1 G x, x A2 G is monotone, that is, G x, y G y, x ≤ for all x, y ∈ C; A3 For each x, y, z ∈ C, − t x, y ≤ G x, y ; lim G tz t↓0 2.12 A4 For each x ∈ C, y → G x, y is convex and lower semicontinuous For x ∈ H and r > 0, define a mapping Tr : H → C by Tr x y − z, z − x ≥ 0, ∀y ∈ C r z ∈ C : G z, y 2.13 Then Tr is well defined and 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 ; 2.14 EP G ; EP G is closed and convex It is easy to see that if there exists some point v ∈ C such that v Tr I − rA v, where A : C → H is an α-inverse strongly monotone mapping, then v ∈ EP G, A In fact, since v Tr I − rA v, one has G v, y y − v, v − I − rA v ≥ 0, r ∀y ∈ C, 2.15 that is, G v, y y − v, Av ≥ 0, ∀y ∈ C 2.16 Hence, v ∈ EP G, A Let C be a nonempty convex subset of a Banach space Let {Ti }N1 be a finite family of i nonexpansive mappings of C into itself and λ1 , λ2 , , λN be real numbers such that ≤ λi ≤ Fixed Point Theory and Applications for each i 1, 2, , N Define a mapping W of C into itself as follows: − λ1 I, U1 λ1 T U2 λ2 T2 U1 − λ2 I, UN−1 W 2.17 − λN−1 I, λN−1 TN−1 UN−2 λN TN UN−1 UN − λN I Such a mapping W is called the W-mapping generated by T1 , T2 , , TN and λ1 , λ2 , , λN see 5, 24, 25 Lemma 2.7 see 26 Let C be a nonempty closed convex subset of a Banach space Let T1 , T2 , , TN be nonexpansive mappings of C into itself such that N1 F Ti / ∅ and let i λ1 , λ2 , , λN be real numbers such that < λi < for each i 1, 2, , N − and < λN ≤ Let N W be the W-mapping of C generated by T1 , T2 , , TN and λ1 , λ2 , , λN Then F W i F Ti Lemma 2.8 see Let C be a nonempty convex subset of a Banach space Let {Ti }N1 be a i finite family of nonexpansive mappings of C into itself and let {λn,i }N1 be sequences in 0, such i 1, 2, , N Moreover, for each n ∈ N, let W and Wn be the Wthat λn,i → λi for each i mappings generated by T1 , T2 , , TN and λ1 , λ2 , , λN and T1 , T2 , , TN and λn,1 , λn,2 , , λn,N , respectively Then, for all x ∈ C, it follows that lim Wn x − Wx n→∞ 2.18 Main Results Now, we give our main results in this paper Theorem 3.1 Let H be a Hilbert space and C be a nonempty closed convex subset of H Let f : H → H be a contraction with coefficient < κ < 1, A, B : H → H be strongly positive linear bounded B , respectively, {Tn }N : H → self-adjoint operators with coefficients γ > and β > B n H N ≥ be a finite family of nonexpansive mappings, {Gn } : C × C → R be an infinite family of bifunctions satisfying A1 – A4 , and {An } : C → H be an infinite family of inverse-strongly N ∩ ∞1 EP Gi , Ai / ∅ Let monotone mappings with constants {kn } such that Ω i i F Ti N { n } and {δn } be two sequences in 0, , {λn,i }i be asequence in a, b with < a ≤ b < 1, {rn } be a sequence in r, 2kn with r > and {αn } be a strictly decreasing sequence 0, Set α0 Take a fixed number γ > with < γ − γκ < Assume that E1 limn → ∞ and n E2 limn → ∞ |λn 1,i E3 ≤ δn ∞; ≤ for all n ≥ 1; n − λn,i | ∞ n n for each i E4 {δn } ⊂ 0, min{c, 1/β, B c < 1; E5 ∞ n | n − n| < ∞, ∞ n |αn 1, 2, , N; B −β β− B − αn | < ∞, ∞ n −2 B |δn 8β B /4β B } with − δn | < ∞ 10 Fixed Point Theory and Applications Then the sequence {xn } defined by 1.23 converges strongly to x∗ ∈ Ω, which is the unique solution of the variational inequality: 1.24 , that is, x∗ x∗ PΩ I − A − γf 3.1 Proof Since n → as n → ∞ by the condition E1 , we may assume, without loss of generality, that n < − δn B A −1 for all n ≥ Noting that A and B are both the linear bounded self-adjoint operators, one has A sup{| Ax, x | : x ∈ H, x 1}, B sup{| Bx, x | : x ∈ H, x 1} 3.2 Observing that I − δn B − nA − δn Bx, x − x, x ≥ − δn B − n n Ax, x A 3.3 ≥ 0, we obtain that I − δn B − I − δn B − nA is positive for all n ≥ It follows that sup{ I − δn B − nA x, x : x ∈ H, x 1} sup{1 − δn B nA x, x : x ∈ H, x 1} ≤ − δn β − n γ nA 3.4 For each n ≥ 1, define a quadratic function f δn in δn as follows: 2β B δn f δn β− B −2 B δn 3.5 Note that f ⎛ ⎜2 B f⎜ ⎝ B f −β B −β B 0, 2β B β− B −2 B 4β B ⎞ 8β B ⎟ ⎟ ⎠ 3.6 3.7 Hence, for each δn satisfying the condition E4 , one has < 2β B δn β− B −2 B δn < 3.8 Fixed Point Theory and Applications 11 Moreover, it follows from 3.7 , f 1/ B > and E4 that , B δn < ∀n ≥ 3.9 Next, we proceed the proof with following steps Step {xn } is bounded Let p ∈ Ω Lemma 2.6 shows that every Tri is firmly nonexpansive and hence nonexpansive Since r < ri < 2ki , I − ri Ai is nonexpansive for each i ≥ Therefore, Tri I − ri Ai is nonexpansive for each i ≥ Noting that {αn } is strictly decreasing, α0 1, we have zn − p n αn xn − p αi−1 − αi Tri I − ri Ai xn − Tri I − ri Ai p i n ≤ αn xn − p αi−1 − αi Tri I − ri Ai xn − Tri I − ri Ai p αi−1 − αi xn − p 3.10 i n ≤ αn xn − p i xn − p and hence Wn zn − Wn p ≤ zn − p ≤ xn − p Wn zn − p 3.11 Then, from 3.4 and 3.11 , it follows that noting that B is linear and β > B B xn B ⇒β> −p δn Bxn − Bp γf xn − Ap n I − δn B − I − δn B − nA Wn zn − p ≤ n ≤ n γκ xn − p n γf p − Ap δn B xn − p − δn β − nγ Wn zn − p ≤ n γκ xn − p n γf p − Ap δn B xn − p − δn β − nγ xn − p δn B xn − p γf xn − Ap ≤ 1− n γ − γκ xn − p n nA Wn zn − p γf p − Ap 3.12 12 Fixed Point Theory and Applications It follows from n ∈ 0, and < γ − γκ < that < induction, we have xn − p ≤ max γ − γκ < Therefore, by the simple n γf p − Ap γ − γκ x1 − p , , ∀n ≥ 1, 3.13 which shows that {xn } is bounded, so is {zn } Step xn − xn → as n → ∞ First, we prove lim Wn zn − Wn zn n→∞ 3.14 Let i ∈ {0, 1, , N − 2} and set N M1 sup T1 zn zn Ti Un,i−1 zn n < ∞ 3.15 i It follows from the definition of Wn that Un − Un,N−i zn 1,N−i zn λn ≤ λn − λn 1,N−i TN−i Un 1,N−i−1 zn 1,N−i |λn TN−i Un 1,N−i 1,N−i−1 zn 1,N−i zn − λn,N−i TN−i Un,N−i−1 zn − − λn,N−i zn − TN−i Un,N−i−1 zn − λn,N−i | TN−i Un,N−i−1 zn ≤ Un 1,N−i−1 zn − Un,N−i−1 zn zn ≤ Un 1,N−i−1 zn − Un,N−i−1 zn M1 |λn |λn 1,N−i − λn,N−i | zn TN−i Un,N−i−1 zn |λn 1,N−i 1,N−i − λn,N−i | − λn,N−i | 3.16 for each i ∈ {0, 1, , N − 2} Thus, using the above recursive inequalities repeatedly, we have Wn zn − Wn zn Un ≤ M1 1,N zn N − Un,N zn |λn 1,i − λn,i | |λn 1,i − λn,i | i ≤ M1 N i |λn 1,1 − λn,1 | zn T1 zn 3.17 Fixed Point Theory and Applications 13 Also, we have n zn − zn−1 αn xn αi−1 − αi Tri I − ri Ai xn − αn−1 xn−1 i n − αi−1 − αi Tri I − ri Ai xn−1 αn−1 − αn Trn I − rn An xn−1 i ≤ αn xn − xn−1 n |αn − αn−1 | xn−1 αi−1 − αi xn − xn−1 3.18 |αn−1 − αn | Trn I − rn An xn−1 i xn − xn−1 ≤ xn − xn−1 |αn − αn−1 | xn−1 |αn−1 − αn | Trn I − rn An xn−1 |αn − αn−1 |L, where L sup{ xn−1 Trn I − rn An xn−1 } Observe noting that B is linear that Next, we prove limn → ∞ xn − xn xn − xn nγ f xn − f xn−1 I − δn B − − − n−1 γf xn−1 f xn − f xn−1 n − n−1 γf xn−1 δn−1 − δn BWn−1 zn−1 n−1 I − δn B − δn B xn − xn−1 δn − δn−1 Bxn−1 δn−1 BWn−1 zn−1 − δn BWn zn−1 nγ δn B xn − xn−1 Wn zn − Wn zn−1 f xn − f xn−1 n xn−1 xn−1 − δn−1 Bxn−1 − I − δn−1 B − n−1 γf nγ nA n γf − n AWn−1 zn−1 n−1 A nA δn − δn−1 Bxn−1 Wn zn−1 Wn−1 zn−1 I − δn B − nA Wn zn − Wn zn−1 Wn zn−1 − Wn−1 zn−1 n−1 AWn−1 zn−1 δn B xn − xn−1 δn Bxn−1 − I − δn B − n AWn zn−1 nA Wn zn − Wn zn−1 Wn zn−1 − Wn−1 zn−1 δn B Wn−1 zn−1 − Wn zn−1 nA Wn−1 zn−1 − Wn zn−1 3.19 14 Fixed Point Theory and Applications Hence, by 3.4 and 3.18 , we get xn − xn ≤ n γκ | n xn − xn−1 − n−1 |γ δn B xn − xn−1 |δn − δn−1 | Bxn−1 f xn−1 |δn−1 − δn | BWn−1 zn−1 | ≤ n−1 n γκ − n| − δn β − | n − n−1 |γ n−1 − n| zn − zn−1 Wn zn−1 − Wn−1 zn−1 A Wn−1 zn−1 − Wn zn−1 n δn B xn − xn−1 xn − xn−1 nγ n 2|δn−1 − δn |M2 Wn zn−1 − Wn−1 zn−1 δn B Wn−1 zn−1 − Wn zn−1 AWn−1 zn−1 ≤ − δn β − B |αn − αn−1 |L |δn − δn−1 | Bxn−1 f xn−1 |δn−1 − δn | BWn−1 zn−1 | nγ δn B Wn−1 zn−1 − Wn zn−1 AWn−1 zn−1 xn − xn−1 − δn β − A Wn−1 zn−1 − Wn zn−1 n γ − γκ xn − xn−1 δn B n L|αn − αn−1 | 2| n−1 − n |M2 Wn−1 zn−1 − Wn zn−1 , A 3.20 where M2 supn {γ f xn−1 Bxn−1 BWn−1 zn−1 AWn−1 zn−1 } Set M3 min{β − B , γ − γκ} It follows from ≤ γ − γκ < and β > B β>2 B B that ≤ M2 < Thus we have xn − xn ≤ − δn × n M3 xn − xn−1 n M3 δn δn δn B δn n M3 × Wn−1 zn−1 − Wn zn−1 M3 n n δn L|αn − αn−1 | due to A n 2| 3.21 M3 n−1 − n |M2 2|δn−1 − δn |M2 Set ηn τn δn n δn B n ξn δn M3 δn L|αn − αn−1 | n M3 , n M3 2| n−1 δn − A n n |M2 M3 Wn−1 zn−1 − Wn zn−1 , 2|δn−1 − δn |M2 3.22 Fixed Point Theory and Applications 15 Then it follows from 3.21 that xn − xn ≤ − ηn xn − xn−1 ηn τn ξn 3.23 It follows from the assumption condition E1 , E3 , E5 , and 3.14 that ∞ ηn ⊂ 0, , ηn ∞ ∞, lim τn 0, n→∞ n By applying Lemma 2.2 to 3.23 , we obtain xn ξn < ∞ 3.24 n − xn → as n → ∞ Step xn − Wn zn → as n → ∞ For all n ≥ 1, we have xn − Wn zn ≤ xn − xn xn − xn xn − Wn zn n γf xn I − δn B − δn Bxn nA Wn zn − Wn zn 3.25 ≤ xn − xn ≤ xn − xn n γf xn − AWn zn δn Bxn − BWn zn n γf xn − AWn zn δn B xn − Wn zn and hence noting 3.9 xn − Wn zn ≤ 1 − δn B xn − xn n − δn γ f xn AWn zn 3.26 It follows from the assumption conditions E1 , E2 , and Step that xn − Wn zn −→ Step xn − zn → as n → ∞ n −→ ∞ 3.27 16 Fixed Point Theory and Applications Notice that, for any x ∈ Ω, zn − x n αn xn − x αi−1 − αi Tri I − ri Ai xn − Tri I − ri Ai x i ≤ αn xn − x n αi−1 − αi I − ri Ai xn − I − ri Ai x i ≤ αn xn − x 3.28 n αi−1 − αi xn − x ri ri − 2ki Ai xn − Ai x i n xn − x αi−1 − αi ri ri − 2ki Ai xn − Ai x i Let yn γf xn −AWn zn and λ sup{ γf xn −AWn zn : n ≥ 1} By using 3.8 , 3.9 , 3.28 , Lemmas 2.3, and 2.4, we have noting that δn < 1/β xn −x I − δn B Wn zn − x ≤ δn Bxn − Bx I − δn B Wn zn − x δn Bxn − Bx zn − x ≤ − δn β xn − x δn B n 2 n yn , xn n δn B xn − x I − δn B Wn zn − Wn x ≤ − δn β γf xn − AWn zn n 2 2δn − δn β xn − x yn , xn −x −x B xn − x αi−1 − αi ri ri − 2ki Ai xn − Ai x 2λ2 n i δn B xn − x − δn β − δn B n − δn β 2δn − δn β − 2δn − δn β B xn − x xn − x B αi−1 − αi ri ri − 2ki Ai xn − Ai x 2λ2 n 2 2λ2 n i ≤ xn − x n − δn β αi−1 − αi ri ri − 2ki Ai xn − Ai x 2λ2 n i 3.29 This shows that − δn β n αi−1 − αi ri 2ki − ri Ai xn − Ai x i ≤ xn − x ≤ xn − x − xn −x xn −x 2λ2 3.30 n xn − xn 2λ2 n Fixed Point Theory and Applications 17 and hence, for each i ≥ 1, − δn β ≤ xn − x ≤ Since δn → 0, xn − xn αi−1 − αi ri 2ki − ri Ai xn − Ai x − xn −x xn −x xn − x 2λ2 3.31 n xn − xn 2λ2 n → and αi−1 − αi > 0, we have lim Ai xn − Ai x 0, n→∞ i ≥ 3.32 Now, for x ∈ Ω, we have, from Lemma 2.2, Tri I − ri Ai xn − x Tri I − ri Ai xn − Tri I − ri Ai x ≤ Tri I − ri Ai xn − Tri I − ri Ai x, I − ri Ai xn − I − ri Ai x Tri I − ri Ai xn − x, xn − x Tri I − ri Ai xn − x ri Tri I − ri Ai xn − x, Ai x − Ai xn 2 xn − x − xn − Tri I − ri Ai xn 3.33 ri Tri I − ri Ai xn − x, Ai x − Ai xn and hence Tri I − ri Ai xn − x ≤ xn − x − xn − Tri I − ri Ai xn 3.34 2ri Tri I − ri Ai xn − x, Ai x − Ai xn Therefore, zn − x ≤ αn xn − x n αi−1 − αi Tri I − ri Ai xn − x i ≤ αn xn − x n αi−1 − αi xn − x − xn − Tri I − ri Ai xn i 2ri Tri I − ri Ai xn − x, Ai x − Ai xn xn − x − n αi−1 − αi xn − Tri I − ri Ai xn i n i αi−1 − αi ri Tri I − ri Ai xn − x, Ai x − Ai xn 3.35 18 Fixed Point Theory and Applications By using 3.8 , 3.9 , 3.35 , Lemmas 2.3 and 2.4, we have noting that δn < 1/β xn −x I − δn B Wn zn − x I − δn B Wn zn − x ≤ δn Bxn − Bx δn Bxn − Bx zn − x ≤ − δn β xn − x δn B n − 2 γf xn − AWn zn n yn , xn n δn B xn − x I − δn B Wn zn − Wn x ≤ − δn β n 2 −x 2δn − δn β xn − x 2 yn , xn −x B xn − x 2λ2 n αi−1 − αi xn − Tri I − ri Ai xn i n αi−1 − αi ri Tri I − ri Ai xn − x, Ai x − Ai xn i δn B 2 xn − x − δ n β − δn B − − δn β n 2δn − δn β B xn − x − 2δn − δn β B xn − x αi−1 − αi xn − Tri I − ri Ai xn 2λ2 n 2 i − δn β n αi−1 − αi ri Tri I − ri Ai xn − x, Ai x − Ai xn 2λ2 n 2λ2 n i ≤ xn − x − − δn β n αi−1 − αi xn − Tri I − ri Ai xn i − δn β n αi−1 − αi ri Tri I − ri Ai xn − x, Ai x − Ai xn i 3.36 and hence − δn β n αi−1 − αi xn − Tri I − ri Ai xn i ≤ xn − x × n − xn −p 2 − δn β αi−1 − αi ri Tri I − ri Ai xn − x, Ai x − Ai xn 2λ2 3.37 n i ≤ xn − x − δn β xn n i 1 −x xn − xn αi−1 − αi ri Tri I − ri Ai xn − x, Ai x − Ai xn 2λ2 n Fixed Point Theory and Applications 19 This shows that for, each i ≥ 1, − δn β ≤ αi−1 − αi xn − Tri I − ri Ai xn xn − x xn − δn β n −x xn − xn 3.38 αi−1 − αi ri Tri I − ri Ai xn − x, Ai x − Ai xn 2λ2 n i Since {αn } is strictly decreasing, δn → 0, have, for each i ≥ 1, n → 0, Ai xn − Ai x → and xn − xn xn − Tri I − ri Ai xn −→ 0, n i Now, from zn − xn n −→ ∞ → 0, we 3.39 αi−1 − αi Tri xn − xn we get zn − xn ≤ n αi−1 − αi Tri xn − xn 3.40 i Since xn − Tri xn → and < αi−1 − αi for each i ≥ 1, one has zn − xn −→ 0, as n −→ ∞ 3.41 Step lim supn → ∞ γf − A x∗ , xn − x∗ ≤ To prove this, we pick a subsequence {xnj } of {xn } such that lim sup γf − A x∗ , xn − x∗ n→∞ lim j →∞ γf − A x∗ , xnj − x∗ 3.42 Without loss of generality, we may further assume that xnj x Obviously, to prove Step 5, we only need to prove that x ∈ Ω Indeed, for each i ≥ 1, since xn − Tri I − ri Ai xn → 0, xnj → x and Tri I − ri Ai is nonexpansive, by demiclosed principle of nonexpansive mapping we have x ∈ F Tri I − ri Ai EP Gi , Ai , i ≥ 3.43 Assume that λnm ,k → λk ∈ 0, for each k 1, 2, , N Let W be the W-mapping generated by T1 , , TN and λ1 , , λN Then, by Lemma 2.8, we have Wnm x −→ Wx, ∀x ∈ H 3.44 20 Fixed Point Theory and Applications F W Assume that x / F W ∈ Moreover, it follows from Lemma 2.7 that N F Ti n Then x / W x Since x ∈ F Tri I − ri Ai for each i ≥ 1, by Step 3, 3.44 and Opial’s property of the Hilbert space H, we have lim inf xnm − x n→∞ < lim inf xnm − W x n→∞ ≤ lim inf xnm − Wnm znm Wnm znm − Wnm x ≤ lim inf xnm − Wnm znm znm − x ≤ lim inf xnm − Wnm znm znm − xnm n→∞ n→∞ n→∞ Wnm x − W x Wnm x − W x 3.45 xnm − Tri I − ri Ai xnm Wnm x − W x x − Tri I − ri Ai xnm ≤ lim inf xnm − Wnm znm znm − xnm n→∞ Tri I − ri Ai x − Tri I − ri Ai xnm xnm − Tri I − ri Ai xnm Wnm x − W x ≤ lim inf xnm − x , n→∞ N i which is a contradiction Therefore, x ∈ F W Hence, x ∈ Ω F Ti ∩ ∞ i EP Gi , Ai Step The sequence {xn } strongly converges to some point x∗ ∈ H By using Lemmas 2.3 and 2.4, we have xn − x∗ I − δn B − ≤ nA Wn zn − x∗ δn Bxn − Bx∗ I − δn B − nA Wn zn − x∗ δn Bxn − Bx∗ δn B xn − x∗ δn B xn − x∗ 2 ≤ γf xn − Ax∗ , xn − δn B − ≤ n n n 1− ≤ 1− ≤ 1− xn − x∗ nγ xn − x∗ 2 xn − x∗ 3.46 − x∗ n γf xn − Ax∗ , xn n γκ xn − x∗ γf x∗ − Ax∗ , xn nγ n 2 γf x∗ − Ax∗ , xn xn 1 − x∗ − x∗ − x∗ n γκ − x∗ xn − x∗ γf xn − Ax∗ , xn nγ n nγ γf xn − Ax∗ − x∗ zn − x∗ γf xn − Ax∗ , xn − δn B − nγ n xn − x∗ − x∗ , xn − x∗ Fixed Point Theory and Applications 21 which implies that xn − x∗ ≤ 1− nγ 1− n γκ n γκ − nγ n γκ − n γκ ≤ 1− 2 γ − κγ − n γκ n γ − κγ − n γκ xn − x∗ xn − x∗ n γf x∗ − Ax∗ , xn − n γκ xn − x∗ 2 nγ 1− n γκ − x∗ n γf x∗ − Ax∗ , xn − n γκ − x∗ γf x∗ − Ax∗ , xn γ − κγ n xn − x∗ 1 nγ − x∗ 2 γ − κγ M , 3.47 supn≥1 { xn − x∗ } Put where M is an appropriate constant such that M sn tn γ − κγ , − n κγ n γf x∗ − Ax∗ , xn γ − κγ −x nγ ∗ 3.48 2 γ − κγ M Then we have xn − x∗ ≤ − sn xn − x∗ sn t n 3.49 It follows from the assumption condition E1 and 3.42 that ∞ lim sn n→∞ 0, sn n ∞, lim sup tn ≤ n→∞ 3.50 Thus, applying Lemma 2.5 to 3.49 , it follows that xn → x∗ as n → ∞ This completes the proof By Theorem 3.1, we have the following direct corollaries Corollary 3.2 Let H be a Hilbert space and C be a nonempty closed convex subset of H Let f : H → H be a contraction with coefficient < κ < 1, A : H → H be strongly positive linear bounded self-adjoint operator with coefficient γ > 0, {Tn }N : H → H N ≥ be a finite family of n nonexpansive mappings, G : C × C → R be a bifunction satisfying (A1)–(A4), and A0 : C → H N be an α-inverse strongly monotone mapping such that Ω i F Ti ∩ EP G, A / ∅ Let {εn } and N {δn } be two sequences in 0, , {λn,i }i be a sequence in a, b with < a ≤ b < 1, r be a number in 0, 2α , and {αn } be a sequence 0, Take a fixed number γ > with < γ − γκ < Assume that 22 Fixed Point Theory and Applications E1 limn → ∞ and n ∞; E2 limn → ∞ |λn 1,i E3 ≤ δn ≤ for all n ≥ 1; n − λn,i | ∞ n εn for each i E4 {δn } ⊂ 0, min{c, 1/β, B c < 1; E5 ∞ n | n − n| < ∞, ∞ n 1, 2, , N; B −β |αn β− B − αn | < ∞, ∞ n |δn −2 B 8β B /4β B } with − δn | < ∞ Then the sequence {xn } defined by 1.25 converges strongly to x∗ ∈ Ω, which is the unique solution of the variational inequality: x∗ PΩ I − A − γf x∗ 3.51 Remark 3.3 In the proof process of Theorem 3.1, we not use Suzuki’s Lemma see 27 , which was used by many others for obtaining xn − xn → as n → ∞ see 4, 5, 28 The proof method of x ∈ ∞1 EP Gi , Ai is simple and different with ones of others i Applications for Multiobjective Optimization Problem In this section, we study a kind of multiobjective optimization problem by using the result of this paper That is, we will give an iterative algorithm of solution for the following multiobjective optimization problem with the nonempty set of solutions: h1 x , h2 x , x ∈ C, 4.1 where h1 x and h2 x are both the convex and lower semicontinuous functions defined on a closed convex subset of C of a Hilbert space H We denote by A the set of solutions of the problem 4.1 and assume that A / ∅ Also, 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 4.3 and Note that, if we find a solution x ∈ A1 ∩ A2 , then one must have x ∈ A obviously Fixed Point Theory and Applications 23 Now, let G1 and G2 be two bifunctions from C × C to R defined by G1 x, y h1 y − h1 x , h2 y − h2 x , G2 x, y A1 and EP G2 respectively It is easy to see that EP G1 set of solutions of the equilibrium problem: Gi x, y ≥ 0, ∀y ∈ C, i ∀ x, y ∈ C × C, 4.4 A2 , where EP Gi denotes the 1, 2, 4.5 respectively In addition, it is easy to see that G1 and G2 satisfy the 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