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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 312602, 18 pages doi:10.1155/2010/312602 Research Article Hybrid Projection Algorithms for Generalized Equilibrium Problems and Strictly Pseudocontractive Mappings Jong Kyu Kim,1 Sun Young Cho,2 and Xiaolong Qin3 Department of Mathematics Education, Kyungnam University, Masan 631-701, Republic of Korea Department of Mathematics, Gyeongsang National University, Chinju 660-701, Republic of Korea Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China Correspondence should be addressed to Jong Kyu Kim, jongkyuk@kyungnam.ac.kr Received 12 October 2009; Accepted 19 July 2010 Academic Editor: Andr´ s Ronto a ´ Copyright q 2010 Jong Kyu Kim 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 The purpose of this paper is to consider the problem of finding a common element in the solution set of equilibrium problems and in the fixed point set of a strictly pseudocontractive mapping Strong convergence of the purposed hybrid projection algorithm is obtained in Hilbert spaces Introduction and Preliminaries Let H be a real Hilbert space with inner product ·, · and norm · Let C be a nonempty closed convex subset of H and S : C → C a nonlinear mapping In this paper, we use F S to denote the fixed point set of S Recall that the mapping S is said to be nonexpansive if Sx − Sy ≤ x − y , ∀x, y ∈ C 1.1 S is said to be k-strictly pseudocontractive if there exists a constant k ∈ 0, such that Sx − Sy ≤ x−y k x − Sx − y − Sy , ∀x, y ∈ C 1.2 ∀x, y ∈ C 1.3 S is said to be pseudocontractive if Sx − Sy ≤ x−y x − Sx − y − Sy , Journal of Inequalities and Applications The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn in 1967 It is easy to see that the class of strictly pseudocontractive mappings falls into the class of nonexpansive mappings and the class of pseudocontractions Let A : C → H be a mapping Recall that A is said to be monotone if Ax − Ay, x − y ≥ 0, ∀x, y ∈ C 1.4 A is said to be inverse-strongly monotone if there exists a constant α > such that Ax − Ay, x − y ≥ α Ax − Ay , ∀x, y ∈ C 1.5 Let F be a bifunction of C × C into R, where R denotes the set of real numbers and A : C → H an inverse-strongly monotone mapping In this paper, we consider the following generalized equilibrium problem Find x ∈ C such that F x, y Ax, y − x ≥ 0, ∀y ∈ C 1.6 In this paper, the set of such an x ∈ C is denoted by EP F, A , that is, EP F, A x ∈ C : F x, y Ax, y − x ≥ 0, ∀y ∈ C 1.7 To study the generalized equilibrium problems 1.6 , we may assume that F satisfies the following conditions: A1 F x, x for all x ∈ C; A2 F is monotone, that is, F x, y F y, x ≤ for all x, y ∈ C; A3 for each x, y, z ∈ C, lim sup F tz − t x, y ≤ F x, y ; 1.8 t↓0 A4 for each x ∈ C, y → F x, y is convex and weakly lower semicontinuous Next, we give two special cases of the problem 1.6 I If A ≡ 0, then the generalized equilibrium problem 1.6 is reduced to the following equilibrium problem: Find x ∈ C such that F x, y ≥ 0, ∀y ∈ C 1.9 In this paper, the set of such an x ∈ C is denoted by EP F , that is, EP F x ∈ C : F x, y ≥ 0, ∀y ∈ C 1.10 Journal of Inequalities and Applications II If F ≡ 0, then the problem 1.6 is reduced to the following classical variational inequality Find x ∈ C such that Ax, y − x ≥ 0, ∀y ∈ C 1.11 It is known that x ∈ C is a solution to 1.11 if and only if x is a fixed point of the mapping PC I − ρA , where ρ > is a constant and I is the identity mapping Recently, many authors studied the problems 1.6 and 1.9 based on iterative methods; see, for example, 2–18 In 2007, Tada and Takahashi 17 considered the problem 1.9 and proved the following result Theorem TT Let C be a nonempty closed convex subset of H Let F be a bifunction from C × C to R satisfying A1 – A4 and let S be a nonexpansive mapping of C into H such that F S ∩ EP F / ∅ Let {xn } and {un } be sequences generated by x1 x ∈ H and let F un , y y − un , un − xn ≥ 0, rn wn − αn xn ∀y ∈ C, αn Sun , Cn {z ∈ H : wn − z ≤ xn − z }, Dn 1.12 {z ∈ H : xn − z, x − xn ≥ 0}, xn PCn ∩Dn x, for every n ≥ 1, where {αn } ⊂ a, for some a ∈ 0, and {rn } ⊂ 0, ∞ satisfies lim infn → ∞ rn > Then, {xn } converges strongly to PF S ∩EP F x In this paper, we consider the generalized equilibrium problem 1.6 and a strictly pseudocontractive mapping based on the shrinking projection algorithm which was first introduced by Takahashi et al 18 A strong convergence of common elements of the fixed point sets of the strictly pseudocontractive mapping and of the solution sets of the generalized equilibrium problem is established in the framework of Hilbert spaces The results presented in this paper improve and extend the corresponding results announced by Tada and Takahashi 17 In order to prove our main results, we also need the following definitions and lemmas Lemma 1.1 see 19 Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C a k-strict pseudocontraction Then T is k / − k -Lipschitz and I − T is demiclosed, this is, if x and xn − T xn → 0, then x ∈ F T {xn } is a sequence in C with xn The following lemma can be found in 2, Journal of Inequalities and Applications Lemma 1.2 Let C be a nonempty closed convex subset of H and let F : C × C → R be a bifunction satisfying A1 – A4 Then, for any r > and x ∈ H, there exists z ∈ C such that F z, y y − z, z − x ≥ 0, r ∀y ∈ C 1.13 Further, define Tr x y − z, z − x ≥ 0, ∀y ∈ C r z ∈ C : F z, y 1.14 for all r > and x ∈ H Then, the following hold: a Tr is single-valued; b Tr is firmly nonexpansive, that is, for any x, y ∈ H, Tr x − Tr y c F Tr ≤ Tr x − Tr y, x − y ; 1.15 EP F ; d EP F is closed and convex Lemma 1.3 see Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → C a k-strict pseudocontraction with a fixed point Define S : C → C by Sa x ax − a Sx F S for each x ∈ C If a ∈ k, , then Sa is nonexpansive with F Sa Main Results Theorem 2.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let F1 and F2 be two bifunctions from C × C to R which satisfies A1 – A4 Let A : C → H be an α-inversestrongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping, and S : C → C Journal of Inequalities and Applications a k-strict pseudocontraction Let {rn } and {sn } be two positive real sequences Assume that F : EP F1 , A ∩FP F2 , B ∩F S is not empty Let {xn } be a sequence generated in the following manner: x1 ∈ C, C1 C, F1 un , u Axn , u − un u − un , un − xn ≥ 0, rn ∀u ∈ C, F2 , v Bxn , v − v − , − xn ≥ 0, sn ∀v ∈ C, γn un zn Cn − γn , − αn βn zn αn xn yn − βn Szn , w ∈ Cn : yn − w ≤ xn − w xn Υ PCn x1 , , n ≥ 1, where {αn }, {βn }, and {γn } are sequences in 0, Assume that {αn }, {βn }, {γn }, {rn }, and {sn } satisfy the following restrictions: a ≤ αn ≤ a < 1; b ≤ k ≤ βn < b < 1; c ≤ c ≤ γn ≤ d < 1; d < e ≤ rn ≤ f < 2α and < e ≤ sn ≤ f < 2β Then the sequence {xn } generated in Υ converges strongly to some point x, where x PF x1 Proof Note that un can be rewritten as un Trn xn − rn Axn , ∀n ≥ 2.1 Tsn xn − sn Bxn , ∀n ≥ 2.2 and can be rewritten as Fix p ∈ F It follows that p Sp Trn p − rn Ap Tsn p − sn Bp , ∀n ≥ 2.3 Journal of Inequalities and Applications Note that I − rn A is nonexpansive for each n ≥ Indeed, for any x, y ∈ C, we see from the restriction d that I − rn A x − I − rn A y x − y − rn Ax − Ay x−y − 2rn x − y, Ax − Ay ≤ x−y − rn 2α − rn ≤ x−y Ax − Ay rn Ax − Ay 2 2.4 This shows that I − rn A is nonexpansive for each n ≥ In a similar way, we can obtain that I − sn B is nonexpansive for each n ≥ It follows that un − p ≤ xn − p , un − p ≤ xn − p 2.5 This implies that zn − p ≤ γn un − p − γn − p ≤ xn − p 2.6 Now, we are in a position to show that Cn is closed and convex for each n ≥ From the assumption, we see that C1 C is closed and convex Suppose that Cm is closed and convex for some m ≥ We show that Cm is closed and convex for the same m Indeed, for any w ∈ Cm , we see that ym − w ≤ xm − w 2.7 is equivalent to ym − xm − w, ym − xm ≥ 2.8 Thus Cm is closed and convex This shows that Cn is closed and convex for each n ≥ Next, we show that F ⊂ Cn for each n ≥ From the assumption, we see that F ⊂ C C1 Suppose that F ⊂ Cm for some m ≥ Putting Sn βn I − βn S, ∀n ≥ 1, 2.9 we see from Lemma 1.3 that Sn is a nonexpansive mapping for each n ≥ For any w ∈ F ⊂ Cm , we see from 2.6 that ym − w αm xm − αm Sm zm − w ≤ αm xm − w ≤ xm − w − αm zm − w 2.10 Journal of Inequalities and Applications This shows that w ∈ Cm This proves that F ⊂ Cn for each n ≥ Note xn w ∈ F ⊂ Cn , we have PCn x1 For each x1 − xn ≤ x1 − w 2.11 x1 − xn ≤ x1 − PF x1 2.12 In particular, we have This implies that {xn } is bounded Since xn PCn x1 and xn ≤ x1 − xn , xn − xn ⊂ Cn , we have x1 − xn , xn − x1 ≤ − x1 − xn PCn x1 ∈ Cn x1 − xn x1 − xn 2.13 x1 − xn It follows that xn − x1 ≤ xn − x1 2.14 This proves that limn → ∞ xn − x1 exists Notice that xn − xn xn − x1 x1 − xn xn − x1 2 xn − x1 , x1 − xn xn − x1 2 xn − x1 , x1 − xn xn − x1 ≤ x1 − xn − xn − x1 2 x1 − xn xn − xn x1 − xn xn − x1 , xn − xn 1 x1 − xn 2.15 − xn − x1 It follows that lim xn − xn n→∞ Since xn 1 2.16 PCn x1 ∈ Cn , we see that yn − xn ≤ xn − xn 2.17 This implies that yn − xn ≤ yn − xn xn − xn ≤ xn − xn 2.18 Journal of Inequalities and Applications From 2.16 , we obtain that lim xn − yn n→∞ 2.19 On the other hand, we have xn − yn xn − αn xn − − αn Sn zn − αn xn − Sn zn 2.20 From the assumption ≤ αn ≤ a < and 2.19 , we have lim xn − Sn zn n→∞ 2.21 For any p ∈ F, we have un − p Trn I − rn A xn − Trn I − rn A p 2 xn − p − rn Axn − Ap xn − p − 2rn xn − p, Axn − Ap ≤ xn − p − rn 2α − rn rn Axn − Ap Axn − Ap 2.22 In a similar way, we also have − p ≤ xn − p − sn 2β − sn Bxn − Bp 2.23 Note that yn − p − αn Sn zn − p αn xn ≤ αn xn − p − αn Sn zn − p ≤ αn xn − p − αn zn − p ≤ αn xn − p − αn γn un − p 2 2.24 − αn − γn − p Substituting 2.22 and 2.23 into 2.24 , we arrive at yn − p ≤ xn − p − − αn γn rn 2α − rn − − αn − γn sn 2β − sn Axn − Ap Bxn − Bp 2.25 Journal of Inequalities and Applications It follows that − αn γn rn 2α − rn Axn − Ap ≤ xn − p ≤ − yn − p xn − p yn − p xn − yn 2.26 In view of the restrictions a – d and 2.19 , we obtain that lim Axn − Ap n→∞ 2.27 It also follows from 2.25 that − αn − γn sn 2β − sn Bxn − Bp ≤ xn − p ≤ xn − p − yn − p yn − p xn − yn 2.28 By virtue of the restrictions a – d and 2.19 , we get that lim Bxn − Bp n→∞ 2.29 On the other hand, we have from Lemma 1.1 that un − p 2 Trn I − rn A xn − Trn I − rn A p ≤ I − rn A xn − I − rn A p, un − p I −rn A xn − I −rn A p xn − p ≤ xn −p 2 un − p un −p 2 un −p − − xn − un − rn Axn − Ap xn −un −2rn xn −un , Axn −Ap − I −rn A xn − I −rn A p− un −p rn Axn −Ap 2.30 This implies that un − p ≤ xn − p − xn − un 2rn xn − un Axn − Ap 2.31 2sn xn − Bxn − Bp 2.32 In a similar way, we can also obtain that − p ≤ xn − p − xn − 10 Journal of Inequalities and Applications Substituting 2.31 and 2.32 into 2.24 , we obtain that yn − p ≤ xn − p − − αn − γn ≤ xn − p 2 − − αn γn xn − un xn − 2sn − αn − γn − − αn γn xn − un − − αn − γn xn − 2rn − αn γn xn − un 2rn xn − un 2sn xn − Axn − Ap xn − Bxn − Bp Axn − Ap 2.33 Bxn − Bp It follows that − αn γn xn − un ≤ xn − p − yn − p 2sn xn − ≤ xn − p 2rn xn − un Axn − Ap Bxn − Bp yn − p 2sn xn − xn − yn 2rn xn − un Axn − Ap 2.34 Bxn − Bp In view of the restrictions a and c , we obtain from 2.27 and 2.29 that lim xn − un n→∞ 2.35 It also follows from 2.33 that − αn − γn xn − ≤ xn − p − yn − p 2sn xn − ≤ xn − p 2rn xn − un Axn − Ap Bxn − Bp yn − p 2sn xn − xn − yn 2rn xn − un Axn − Ap Bxn − Bp 2.36 Thanks to the restrictions a and c , we obtain from 2.27 and 2.29 that lim xn − n→∞ 2.37 Note that zn − xn ≤ γn un − xn − γn − xn 2.38 From 2.35 and 2.37 , we see that lim xn − zn n→∞ 2.39 Journal of Inequalities and Applications 11 On the other hand, we see from 2.21 that βn zn − xn − βn Szn − xn −→ 2.40 as n → ∞ In view of 2.39 and the restriction b , we obtain that lim xn − Szn n→∞ 2.41 Note that Sxn − xn ≤ Sxn − Szn Szn − xn ≤ k xn − zn 1−k Szn − xn 2.42 It follows from 2.39 and 2.41 that lim xn − Sxn n→∞ 2.43 Since {xn } is bounded, we assume that a subsequence {xni } of {xn } converges weakly to ξ Next, we show that ξ ∈ F S ∩ EP F1 , A ∩ EP F2 , B First, we prove that ξ ∈ EP F1 , A Since un Trn xn − rn Axn for any u ∈ C, we have F1 un , u u − un , un − xn ≥ rn Axn , u − un 2.44 From the condition A2 , we see that Axn , u − un u − un , un − xn ≥ F1 u, un rn 2.45 Replacing n by ni , we arrive at Axni , u − uni For any t with < t ≤ and u ∈ C, let ut It follows from 2.46 that u − uni , tu uni − xni rni ≥ F1 u, uni − t ξ Since u ∈ C and ξ ∈ C, we have ut ∈ C ut − uni , Aut ≥ ut − uni , Aut − Axni , ut − uni − ut − uni , ut −uni , Aut −Auni 2.46 uni − xni rni ut −uni , Auni −Axni − ut −uni , F1 ut , uni uni −xni rni F1 ut , uni 2.47 12 Journal of Inequalities and Applications Since A is Lipschitz continuous, we obtain from 2.35 that Auni − Axni → as i → ∞ On the other hand, we get from the monotonicity of A that ut − uni , Aut − Auni ≥ 2.48 It follows from A4 and 2.47 that ut − ξ, Aut ≥ F1 ut , ξ 2.49 From A1 , A4 , and 2.49 , we see that F1 ut , ut ≤ tF1 ut , u ≤ tF1 ut , u − t F1 ut , ξ − t ut − ξ, Aut tF1 ut , u − t t u − ξ, Aut , F1 ut , u − t u − ξ, Aut ≥ 2.50 which yields that 2.51 Letting t → in the above inequality, we arrive at u − ξ, Aξ ≥ F1 ξ, u 2.52 This shows that ξ ∈ EP F1 , A In a similar way, we can obtain that ξ ∈ EP F2 , B Next, we show that ξ ∈ F S We can conclude from Lemma 1.1 the desired conclusion easily This proves that ξ ∈ F Put x PF x1 Since x PF x1 ⊂ Cn and xn PCn x1 , we have x1 − xn ≤ x1 − x 2.53 On the other hand, we have x1 − x ≤ x1 − ξ ≤ lim inf x1 − xni i→∞ ≤ lim sup x1 − xni i→∞ ≤ x1 − x 2.54 Journal of Inequalities and Applications 13 We, therefore, obtain that x1 − ξ lim x1 − xni x1 − x i→∞ 2.55 x Since {xni } is an arbitrary subsequence of {xn }, we obtain that This implies xni → ξ xn → x as n → ∞ This completes the proof If S is nonexpansive, then we have from Theorem 2.1 the following result immediately Corollary 2.2 Let C be a nonempty closed convex subset of a real Hilbert space H Let F1 and F2 be two bifunctions from C × C to R which satisfies A1 – A4 Let A : C → H be an αinverse-strongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping, and S : C → C a nonexpansive mapping Let {rn } and {sn } be two positive real sequences Assume that F : EP F1 , A ∩ FP F2 , B ∩ F S is not empty Let {xn } be a sequence generated in the following manner: x1 ∈ C, C1 C, F1 un , u Axn , u − un u − un , un − xn ≥ 0, rn ∀u ∈ C, F2 , v Bxn , v − v − , − xn ≥ 0, sn ∀v ∈ C, αn xn yn Cn 1 − αn S γn un − γn , w ∈ Cn : yn − w ≤ xn − w xn PCn x1 , 2.56 , n ≥ 1, where {αn } and {γn } are sequences in 0, Assume that {αn }, {γn }, {rn }, and {sn } satisfy the following restrictions: a ≤ αn ≤ a < 1; b ≤ c ≤ γn ≤ d < 1; c < e ≤ rn ≤ f < 2α and < e ≤ sn ≤ f < 2β Then the sequence {xn } converges strongly to some point x, where x PF x1 As applications of Theorem 2.1, we consider the problems 1.9 and 1.11 14 Journal of Inequalities and Applications Theorem 2.3 Let C be a nonempty closed convex subset of a real Hilbert space H Let A : C → H be an α-inverse-strongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping, and S : C → C a k-strict pseudocontraction Let {rn } and {sn } be two positive real sequences Assume that F : V I C, A ∩ V I C, B ∩ F S is not empty Let {xn } be a sequence generated in the following manner: x1 ∈ C, C1 γn PC I − rn A xn zn αn xn yn Cn C, − γn PC I − sn B xn , − αn βn zn − βn Szn , w ∈ Cn : yn − w ≤ xn − w xn PCn x1 , 2.57 , n ≥ 1, where {αn }, {βn }, and {γn } are sequences in 0, Assume that {αn }, {βn }, {γn }, {rn }, and {sn } satisfy the following restrictions: a ≤ αn ≤ a < 1; b ≤ k ≤ βn < b < 1; c ≤ c ≤ γn ≤ d < 1; d < e ≤ rn ≤ f < 2α and < e ≤ sn ≤ f < 2β Then the sequence {xn } converges strongly to some point x, where x Proof Putting F1 PF x1 F2 ≡ 0, we see that Axn , u − un u − un , un − xn ≥ 0, rn ∀u ∈ C, 2.58 is equivalent to xn − rn Axn − un , un − u ≥ 0, This implies that un PC xn − rn Axn We also have Theorem 2.1 the desired results immediately ∀u ∈ C 2.59 PC xn − sn Bxn We can obtain from Corollary 2.4 Let C be a nonempty closed convex subset of a real Hilbert space H Let A : C → H be an α-inverse-strongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping, and S : C → C a nonexpansive mapping Let {rn } and {sn } be two positive real sequences Assume Journal of Inequalities and Applications 15 that F : V I C, A ∩ V I C, B ∩ F S is not empty Let {xn } be a sequence generated in the following manner: x1 ∈ C, C1 γn PC I − rn A xn zn αn xn yn Cn C, − γn PC I − sn B xn , − αn Szn , w ∈ Cn : yn − w ≤ xn − w xn PCn x1 , 2.60 , n ≥ 1, where {αn } and {γn } are sequences in 0, Assume that {αn }, {γn }, {rn }, and {sn } satisfy the following restrictions: a ≤ αn ≤ a < 1; b ≤ c ≤ γn ≤ d < 1; c < e ≤ rn ≤ f < 2α and < e ≤ sn ≤ f < 2β Then the sequence {xn } converges strongly to some point x, where x PF x1 Theorem 2.5 Let C be a nonempty closed convex subset of a real Hilbert space H Let F1 and F2 be two bifunctions from C × C to R which satisfies A1 – A4 Let S : C → C be a kstrict pseudocontraction Let {rn } and {sn } be two positive real sequences Assume that F : EP F1 ∩ FP F2 ∩ F S is not empty Let {xn } be a sequence generated in the following manner: x1 ∈ C, C1 C, F1 un , u u − un , un − xn ≥ 0, rn ∀u ∈ C, F2 , v v − , − xn ≥ 0, sn ∀v ∈ C, γn un zn αn xn yn Cn 1 − γn , − αn βn zn − βn Szn , w ∈ Cn : yn − w ≤ xn − w xn 2.61 PCn x1 , , n ≥ 1, where {αn }, {βn }, and {γn } are sequences in 0, Assume that {αn }, {βn }, {γn }, {rn }, and {sn } satisfy the following restrictions: a ≤ αn ≤ a < 1; b ≤ k ≤ βn < b < 1; 16 Journal of Inequalities and Applications c ≤ c ≤ γn ≤ d < 1; d < e ≤ rn ≤ f < ∞ and < e ≤ sn ≤ f < ∞ Then the sequence {xn } converges strongly to some point x, where x Proof Putting A immediately B PF x1 0, we can obtain from Theorem 2.1 the desired conclusion Remark 2.6 Theorem 2.5 is generalization of Theorem TT To be more precise, we consider a pair of bifunctions and a strictly pseudocontractive mapping Let T : C → C be a k-strict pseudocontraction It is known that I − T is a − k /2inverse-strongly monotone mapping The following results are not hard to derive Theorem 2.7 Let C be a nonempty closed convex subset of a real Hilbert space H Let F1 and F2 be two bifunctions from C × C to R which satisfies A1 – A4 Let TA : C → C be a kα -strict pseudocontraction, B : C → C a kβ -strict pseudocontraction, and S : C → C a k-strict pseudocontraction Let {rn } and {sn } be two positive real sequences Assume that F : EP F1 , I − TA ∩ FP F2 , I − TB ∩ F S is not empty Let {xn } be a sequence generated in the following manner: x1 ∈ C, C1 C, F1 un , u I − TA xn , u − un u − un , un − xn ≥ 0, rn ∀u ∈ C, F2 , v I − TB xn , v − v − , − xn ≥ 0, sn ∀v ∈ C, γn un zn αn xn yn Cn 1 − γn , − αn βn zn − βn Szn , w ∈ Cn : yn − w ≤ xn − w xn 2.62 PCn x1 , , n ≥ 1, where {αn }, {βn }, and {γn } are sequences in 0, Assume that {αn }, {βn }, {γn }, {rn }, and {sn } satisfy the following restrictions: a ≤ αn ≤ a < 1; b ≤ k ≤ βn < b < 1; c ≤ c ≤ γn ≤ d < 1; d < e ≤ rn ≤ f < − kα and < e ≤ sn ≤ f < − kβ Then the sequence {xn } converges strongly to some point x, where x PF x1 Journal of Inequalities and Applications 17 Acknowledgment This work was supported by a National Research Foundation of Korea Grant funded by the Korean Government 2009-0076898 References F E Browder and W V Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol 20, pp 197–228, 1967 E Blum and W Oettli, “From optimization and variational 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