Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 368137, 15 pages doi:10.1155/2011/368137 Research Article Convergence of Iterative Sequences for Fixed Point and Variational Inclusion Problems Li Yu and Ma Liang School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China Correspondence should be addressed to Li Yu, brucemath@139.com Received 14 November 2010; Accepted February 2011 Academic Editor: Yeol J Cho Copyright q 2011 L Yu and M Liang 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 An iterative process is considered for finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and an inverse strongly monotone mapping Strong convergence theorems of common elements are established in real Hilbert spaces Introduction and Preliminaries Throughout this paper, we always assume that H is a real Hilbert space with the inner product ·, · and the 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 κ-strictly pseudocontractive if there exists a constant κ ∈ 0, such that Sx − Sy ≤ x−y κ x − Sx − y − Sy , ∀x, y ∈ C 1.2 The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn in 1967 It is easy to see that every nonexpansive mapping is a 0-strictly pseudocontractive mapping 2 Fixed Point Theory and Applications Let A : C → H be a mapping Recall that A is said to be monotone if Ax − Ay, x − y ≥ 0, ∀x, y ∈ C 1.3 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 1.4 For such a case, A is also said to be α-inverse strongly monotone Let M : H → 2H be a set-valued mapping The set D M defined by D M {x ∈ H : Mx / ∅} is said to be the domain of M The set R M defined by R M x∈H Mx is said to be the range of M The set G M defined by G M { x, y ∈ H × H : x ∈ D M , y ∈ R M } is said to be the graph of M Recall that M is said to be monotone if x − y, f − g > 0, ∀ x, f , y, g ∈ G M 1.5 M is said to be maximal monotone if it is not properly contained in any other monotone operator Equivalently, M is maximal monotone if R I rM H for all r > For a maximal monotone operator M on H and r > 0, we may define the single-valued resolvent Jr F Jr I rM −1 : H → D M It is known that Jr is firmly nonexpansive and M−1 Recall that the classical variational inequality problem is to find x ∈ C such that Ax, y − x ≥ 0, ∀y ∈ C 1.6 Denote by VI C, A of the solution set of 1.6 It is known that x ∈ C is a solution to 1.6 if and only if x is a fixed point of the mapping PC I − λA , where λ > is a constant and I is the identity mapping Recently, many authors considered the convergence of iterative sequences for the variational inequality 1.6 and fixed point problems of nonlinear mappings see, for example, 1–32 In 2005, Iiduka and Takahashi proved the following theorem Theorem IT Let C be a closed convex subset of a real Hilbert space H Let A be an α-inversestrongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such that F S ∩ VI C, A / ∅ Suppose that x1 x ∈ C and {xn } is given by xn αn x − αn SPC xn − λn Axn , 1.7 for every n 1, 2, , where {αn } is a sequence in 0, and {λn } is a sequence in a, b If {αn } and {λn } are chosen so that {λn } ∈ a, b for some a, b with < a < b < 2α, ∞ lim αn n→∞ 0, αn ∞, n then {xn } converges strongly to PF ∞ n S ∩VI C,A x |αn − αn | < ∞, ∞ n |λn − λn | < ∞, 1.8 Fixed Point Theory and Applications In 2007, Y Yao and J.-C Yao 31 further obtained the following theorem Theorem YY Let C be a closed convex subset of a real Hilbert space H Let A be an α-inversestrongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such that F S ∩ Ω / ∅, where Ω denotes the set of solutions of a variational inequality for the α-inversestrongly monotone mapping Suppose that x1 u ∈ C and {xn }, {yn } are given by u ∈ C, x1 yn xn αn u βn xn PC xn − λn Axn , 1.9 γn SPC I − λn A yn , n ≥ 1, where {αn }, {βn }, and {γn } are three sequences in 0, and {λn } is a sequence in 0, 2a If {αn }, {βn }, {γn }, and {λn } are chosen so that λn ∈ a, b for some a, b with < a < b < 2a and a αn βn 1, for all n ≥ 1, γn b limn → ∞ αn 0, ∞ n αn ∞, c < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1, d limn → ∞ λn − λn 0, then {xn } converges strongly to PF S ∩Ω u In this work, motivated by the above results, we consider the problem of finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and a inverse strongly monotone mapping Strong convergence theorems of common elements are established in real Hilbert spaces The results presented in this paper improve and extend the corresponding results announced by Iiduka and Takahashi and Y Yao and J.-C Yao 31 In order to prove our main results, we also need the following lemmas Lemma 1.1 see 22 Let C be a nonempty closed convex subset of a Hilbert space H, A : C → H a mapping, and M : H → 2H a maximal monotone mapping Then, F Jr I − rA A M −1 , ∀r > 1.10 Lemma 1.2 see Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → C a κ-strict pseudocontraction with a fixed point Define S : C → C by Sa x ax − a Sx F S for each x ∈ C If a ∈ κ, , then Sa is nonexpansive with F Sa Lemma 1.3 see 25 Let C be a nonempty closed convex subset of a Hilbert space H and S : C → C a κ-strict pseudocontraction Then, a S is κ / − κ -Lipschitz, b I − S is demi-closed, this is, if {xn } is a sequence in C with xn then x ∈ F S x and xn − Sxn → 0, Fixed Point Theory and Applications Lemma 1.4 see 28 Let {xn } and {yn } be bounded sequences in a Hilbert space H, and let {βn } be a sequence in 0, with < lim inf βn ≤ lim sup βn < n→∞ Suppose that xn − βn yn βn xn for all integers n ≥ and lim sup yn n→∞ Then, limn → ∞ yn − xn 1.11 n→∞ − yn − xn − xn ≤ 1.12 Lemma 1.5 see 29 Assume that {αn } is a sequence of nonnegative real numbers such that αn ≤ − γn αn δn , 1.13 where {γn } is a sequence in 0, and {δn } is a sequence such that a ∞ n γn ∞, ∞ n b lim supn → ∞ δn /γn ≤ or Then, limn → ∞ αn |δn | < ∞ Lemma 1.6 see 24 Let H be a Hilbert space and M a maximal monotone operator on H Then, the following holds: Jr x − Js x where Jr I rM −1 and Js ≤ r−x Jr x − Js x, Jr x − x , r I sM −1 ∀s, t > 0, x ∈ H, 1.14 Main Results Theorem 2.1 Let H be a real Hilbert space H and C a nonempty close and convex subset of H Let M : H → 2H and W : H → 2H two maximal monotone operators such that D M ⊂ C and D W ⊂ C, respectively Let S : C → C be a κ-strict pseudocontraction, A : C → H an α-inverse strongly monotone mapping, and B : C → H a β-inverse strongly monotone mapping Assume that F : F S ∩ A M −1 ∩ B W −1 / ∅ Let {xn } be a sequence generated in the following manner: x1 ∈ C, yn xn αn u βn xn Jsn xn − sn Bxn , γn δn Jrn yn − rn Ayn − δn SJrn yn − rn Ayn 2.1 , ∀n ≥ 1, I rn M −1 and Jsn I sn W −1 , {rn } is a sequence where u ∈ C is a fixed element, Jrn in 0, 2α , {sn } is a sequence in 0, 2β and {αn }, {βn }, {γn }, and {δn } are sequences in 0, Fixed Point Theory and Applications Assume that the following restrictions are satisfied: a < a ≤ rn ≤ b < 2α, limn → ∞ rn − rn b < c ≤ sn ≤ d < 2βn , limn → ∞ sn − sn c ≤ κ ≤ δn < e < 1, limn → ∞ δn − δn d limn → ∞ αn 0, ∞ n 0, 0, 0, ∞, αn e < lim infn → ∞ βn ≤ lim infn → ∞ βn < Then, the sequence {xn } converges strongly to q PF u Proof The proof is split into five steps Step Show that {xn } is bounded Note that I − rn A and I − sn B are nonexpansive for each fixed n ≥ Indeed, we see from the restriction a that x−y − 2rn x − y, Ax − Ay ≤ x−y − rn 2α − rn ≤ x−y I − rn A x − I − rn A y 2 , Ax − Ay rn Ax − Ay 2 2.2 ∀x, y ∈ C This shows that I − rn A is nonexpansive for each fixed n ≥ 1, so is I − sn B Put Sn x δn x − δn Sx, ∀x ∈ C 2.3 In view of the restriction c , we obtain from Lemma 1.2 that Sn is a nonexpansive mapping F S for each fixed n ≥ Fixing p ∈ F and since Jrn and I − rn A are with F Sn nonexpansive, we see that βn xn − p γn Sn Jrn yn − rn Ayn − p βn xn − p γn Jrn yn − rn Ayn − p ≤ αn u − p − p ≤ αn u − p ≤ αn u − p xn βn xn − p γn yn − p ≤ αn u − p 2.4 − αn xn − p By mathematical inductions, we see that {xn } is bounded and so is {yn } This completes Step 6 Fixed Point Theory and Applications Step Show that xn − xn → as n → ∞ Notice from Lemma 1.6 that yn − yn ≤ xn Jsn ≤ xn − sn Bxn xn − sn Bxn − Jsn xn − sn Bxn − xn |sn |sn − sn | sn ≤ xn Jsn − xn − xn − sn Bxn 1 − sn | Bxn 2.5 xn − sn Bxn − xn − sn Bxn 2M1 |sn − sn |, where M1 is an appropriate constant such that M1 Jsn max sup{ Bxn }, sup n≥1 xn − sn Bxn − xn − sn Bxn sn 1 n≥1 2.6 Put Jrn yn − rn Ayn , zn ∀n ≥ 2.7 In a similar way, we can obtain from Lemma 1.6 that zn − zn ≤ yn Jrn ≤ yn − rn Ayn 1 |rn rn ≤ yn yn − rn Ayn − Jrn yn − rn Ayn − −yn − yn − rn Ayn − rn | |rn Jrn − yn − rn | Ayn 2.8 yn − rn Ayn − yn − rn Ayn 2M2 |rn − rn |, where M2 is an appropriate constant such that max sup Ayn , sup n≥1 M2 n≥1 Jrn yn − rn Axn − yn − rn Ayn rn 2.9 Substituting 2.5 into 2.8 yields that zn − zn ≤ xn − xn M3 |sn − sn | |rn − rn | , 2.10 Fixed Point Theory and Applications where M3 is an appropriate constant such that max{2M1 , 2M2 } M3 2.11 It follows from 2.10 that − Sn zn ≤ zn − zn ≤ xn Sn zn zn − Szn |δn − δn | − xn M4 |sn − sn | |rn − rn | |δn − δn | , 2.12 where M4 is an appropriate constant such that max sup{ zn − Szn }, M3 M4 2.13 n≥1 Put ln xn − βn xn , − βn ∀n ≥ 2.14 Note that ln − ln αn u γn Sn zn − βn αn 1 − βn γn 1 − βn αn 1 − βn − − − αn u − βn αn u γn Sn zn − βn γn 1 − βn Sn zn − Sn zn 2.15 γn − Sn zn − βn αn − βn γn 1 − βn u − Sn zn Sn zn − Sn zn It follows from 2.12 that − ln ≤ αn 1 − βn ≤ ln αn 1 − βn 1 M4 |sn − αn − βn − αn − βn u − Sn zn − sn | γn 1 − βn u − Sn zn |rn − rn | xn 1 Sn zn − xn − Sn zn 2.16 |δn − δn | This in turn implies from the restrictions a – e that lim sup ln n→∞ − ln − xn − xn ≤ 2.17 Fixed Point Theory and Applications From Lemma 1.4, we obtain that lim ln − xn n→∞ 2.18 Notice that − xn − βn ln − xn lim xn xn 2.19 It follows that n→∞ − xn 2.20 This completes Step Step Show that xn − Sxn → as n → ∞ Since Jrn and Jsn are nonexpansive, we see that zn − p ≤ xn − p − rn 2α − rn Ayn − Ap , 2.21 yn − p ≤ xn − p − sn 2β − sn Bxn − Bp 2.22 It follows from 2.21 that −p 2 βn xn − p γn Sn zn − p βn xn − p γn zn − p ≤ αn u − p ≤ αn u − p ≤ αn u − p xn xn − p 2 − γn rn 2α − rn 2.23 Ayn − Ap This in turn implies that γn rn 2α − rn Ayn − Ap ≤ αn u − p xn − xn xn − p xn −p 2.24 In view of 2.20 , we see from the restrictions a , d , and e that lim Ayn − Ap n→∞ 2.25 Fixed Point Theory and Applications It follows from 2.22 that βn xn − p γn Sn zn − p βn xn − p γn Jrn yn − rn Ayn − p βn xn − p γn yn − p ≤ αn u − p −p ≤ αn u − p ≤ αn u − p ≤ αn u − p xn xn − p 2 2 − γn sn 2β − sn 2.26 Bxn − Bp This in turn implies that Bxn − Bp γn sn 2β − sn ≤ αn u − p xn − xn xn − p xn −p 2.27 In view of 2.20 , we see from the restrictions a , d , and e that lim Bxn − Bp n→∞ 2.28 Since Jrn is firmly nonexpansive, we obtain that zn − p Jrn yn − rn Ayn − Jrn p − rn Ap ≤ zn − p, yn − rn Ayn − p − rn Ap zn − p − zn − p − zn − p 2 ≤ zn − p yn − rn Ayn − p − rn Ap yn − rn Ayn − p − rn Ap yn − p yn − p − zn − yn − zn − yn 2 rn Ayn − Ap 2.29 2 − rn Ayn − Ap −2rn zn − yn , Ayn − Ap ≤ zn − p yn − p − zn − yn 2rn zn − yn ≤ zn − p xn − p − zn − yn 2rn zn − yn Ayn − Ap Ayn − Ap This in turn implies that zn − p ≤ xn − p − zn − yn 2rn zn − yn Ayn − Ap 2.30 10 Fixed Point Theory and Applications In a similar way, we can obtain that yn − p ≤ xn − p − yn − xn 2sn yn − xn Bxn − Bp 2.31 In view of 2.30 , we see that −p 2 βn xn − p γn Sn zn − p βn xn − p γn zn − p ≤ αn u − p ≤ αn u − p ≤ αn u − p xn xn − p 2 2.32 − γn zn − yn 2rn zn − yn Ayn − Ap It follows that γn zn − yn ≤ αn u − p xn − xn 2rn zn − yn xn − p xn −p Ayn − Ap 2.33 In view of 2.25 , we obtain from the restrictions d and e that lim zn − yn n→∞ 2.34 Notice from 2.31 , we see that βn xn − p γn Sn zn − p βn xn − p γn zn − p 2 βn xn − p γn yn − p ≤ αn u − p −p ≤ αn u − p ≤ αn u − p ≤ αn u − p xn xn − p − γn yn − xn 2 2.35 2sn yn − xn Bxn − Bp It follows that γn yn − xn ≤ αn u − p 2sn yn − xn xn − xn xn − p Bxn − Bp xn −p 2.36 In view of 2.28 , we obtain from the restrictions d and e that lim yn − xn n→∞ 2.37 Fixed Point Theory and Applications 11 Combining 2.34 with 2.37 yields that lim zn − xn n→∞ 2.38 Note that xn − xn γn Sn zn − xn αn u − xn 2.39 In view of 2.20 , we see from the restriction d that lim Sn zn − xn n→∞ 2.40 Note that Szn − xn Sn zn − xn − δn δn xn − zn − δn 2.41 From 2.38 and 2.40 , we get from the restriction c that lim Szn − xn n→∞ 2.42 Notice that Sxn − xn ≤ Sxn − Szn Szn − xn 2.43 In view of 2.38 and 2.42 , we see from Lemma 1.3 that lim Sxn − xn n→∞ 2.44 This completes Step Step Show that lim supn → ∞ u − q, xn − q ≤ 0, where q PF u To show it, we may choose a subsequence {xni } of {xn } such that lim sup u − q, xn − q n→∞ lim sup u − q, xni − q i→∞ 2.45 Since {xni } is bounded, we can choose a subsequence {xnij } of {xni } converging weakly to x We may, without loss of generality, assume that xni x, where denotes the weak convergence Next, we prove that x ∈ F In view of 2.44 , we can conclude from Lemma 1.3 that x ∈ F S easily Notice that yn − rn Ayn ∈ zn rn Mzn 2.46 12 Fixed Point Theory and Applications Let μ ∈ Mν Since M is monotone, we have yn − zn − Ayn − μ, zn − ν rn ≥ 2.47 In view of the restriction a , we see from 2.34 that −Ax − μ, x − ν ≥ 2.48 This implies that −Ax ∈ Mx, that is, x ∈ A M −1 In similar way, we can obtain that x ∈ B W −1 This proves that x ∈ F It follows from 2.45 that lim sup u − q, xn − q ≤ 2.49 n→∞ This completes Step Step Show that xn → q as n → ∞ Notice that xn −q αn u − q, xn −q βn xn − q, xn γn Sn Jrn yn − rn Ayn − q, xn ≤ αn u − q, xn γn −q βn xn − q Sn Jrn yn − rn Ayn − q ≤ αn u − q, xn γn 1 yn − q ≤ αn u − q, xn −q βn xn −q 1 −q −q 2 xn − q −q xn xn 1 −q xn −q 2.50 −q − αn xn − q xn −q This in turn implies that xn −q ≤ − αn xn − q 2αn u − q, xn −q 2.51 In view of 2.49 , we conclude from Lemma 1.5 that lim xn − q n→∞ This completes Step This whole proof is completed 2.52 Fixed Point Theory and Applications 13 0, then Theorem 2.1 is reduced to the If S is a nonexpansive mapping and δn following Corollary 2.2 Let H be a real Hilbert space H and C a nonempty close and convex subset of H Let M : H → 2H and W : H → 2H be two maximal monotone operators such that D M ⊂ C and D W ⊂ C, respectively Let S : C → C be a nonexpansive mapping, A : C → H an α-inverse strongly monotone mapping and B : C → H a β-inverse strongly monotone mapping Assume that F : F S ∩ A M −1 ∩ B W −1 / ∅ Let {xn } be a sequence generated in the following manner: x1 ∈ C, yn xn αn u βn xn Jsn xn − sn Bxn , γn SJrn yn − rn Ayn , 2.53 ∀n ≥ 1, I rn M −1 and Jsn I sn W −1 , {rn } is a sequence in where u ∈ C is a fixed element, Jrn 0, 2α , {sn } is a sequence in 0, 2β and {αn }, {βn } and {γn } are sequences in 0, Assume that the following restrictions are satisfied: a < a ≤ rn ≤ b < 2α, limn → ∞ rn − rn b < c ≤ sn ≤ d < 2βn , limn → ∞ sn − sn c limn → ∞ αn 0, ∞ n αn 0, 0, ∞, d < lim infn → ∞ βn ≤ lim infn → ∞ βn < Then, the sequence {xn } converges strongly to q PF u Next, we consider the problem of finding common fixed points of three strict pseudocontractions Theorem 2.3 Let C be a nonempty closed convex subset of a real Hilbert space H and PC the metric projection from H onto C Let S : C → C be a κ-strict pseudocontraction, TA : C → H an α-strict pseudocontraction, and B : C → H a β-strict pseudocontraction Assume that F : F S ∩ F TA ∩ F TB / ∅ Let {xn } be a sequence generated in the following manner: x1 ∈ C, zn αn u sn TB xn , yn xn − sn xn − rn zn rn TA zn γn δn yn − δn Syn , βn xn 2.54 ∀n ≥ 1, where u ∈ C is a fixed element, {rn } is a sequence in 0, − α , {sn } is a sequence in 0, − β , and {αn }, {βn }, {γn }, and {δn } are sequences in 0, Assume that the following restrictions are satisfied a < a ≤ rn ≤ b < − α, limn → ∞ rn − rn 0, b < c ≤ sn ≤ d < − βn , limn → ∞ sn − sn c ≤ κ ≤ δn < e < 1, limn → ∞ δn − δn 0, 0, 14 Fixed Point Theory and Applications d limn → ∞ αn 0, ∞ n αn ∞, e < lim infn → ∞ βn ≤ lim infn → ∞ βn < Then, the sequence {xn } converges strongly to q PF u Proof Putting A I − TA , we see that A is − α /2 -inverse strongly monotone We also VI C, A and PC xn − rn Axn − rn xn rn T xn Putting B I − TB , we see that have F TA VI C, B and PC xn −sn Bxn B is 1−β /2-inverse strongly monotone 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pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and a inverse