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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 262691, 12 pages doi:10.1155/2010/262691 Research Article A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces Urailuk Singthong1 and Suthep Suantai1, 2 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand Correspondence should be addressed to Suthep Suantai, scmti005@chiangmai.ac.th Received 10 February 2010; Revised 21 June 2010; Accepted 15 July 2010 Academic Editor: Massimo Furi Copyright q 2010 U Singthong and S Suantai 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 general iterative method by using the K-mapping for finding a common fixed point of a finite family of nonexpansive mappings in the framework of Hilbert spaces A strong convergence theorem of the purposed iterative method is established under some certain control conditions Our results improve and extend the results announced by many others Introduction Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H A mapping T of C into itself is called nonexpansive if T x − T y ≤ x − y for all x, y ∈ C A point x ∈ C is called a fixed point of T provided that T x x We denote by F T the set of fixed points of T i.e., F T {x ∈ H : T x x} Recall that a self-mapping f : C → C is a contraction on C, if there exists a constant α ∈ 0, such that fx − fy ≤ α x − y for all x, y ∈ C A bounded linear operator A on H is called strongly positive with coefficient γ if there is a constant γ > with the property Ax, x ≥ γ x , ∀x ∈ H 1.1 In 1953, Mann introduced a well-known classical iteration to approximate a fixed point of a nonexpansive mapping This iteration is defined as xn αn xn − αn T xn , n ≥ 0, 1.2 Fixed Point Theory and Applications where the initial guess x0 is taken in C arbitrarily, and the sequence {αn }∞ is in the interval n 0, But Mann’s iteration process has only weak convergence, even in a Hilbert space setting In general for example, Reich showed that if E is a uniformly convex Banach space ∞, and has a Frehet differentiable norm and if the sequence {αn } is such that Σ∞ αn − αn n then the sequence {xn } generated by process 1.2 converges weakly to a point in F T Therefore, many authors try to modify Mann’s iteration process to have strong convergence In 2005, Kim and Xu introduced the following iteration process: x ∈ C arbitrarily chosen, x0 βn xn yn xn αn u 1 − βn T xn , 1.3 − αn yn They proved in a uniformly smooth Banach space that the sequence {xn } defined by 1.3 converges strongly to a fixed point of T under some appropriate conditions on {αn } and {βn } In 2008, Yao et al alsomodified Mann’s iterative scheme 1.2 to get a strong convergence theorem N Let {Ti }N1 be a finite family of nonexpansive mappings with F : i n F Ti / ∅ There are many authors introduced iterative method for finding an element of F which is an optimal point for the minimization problem For n > N, Tn is understood as T n mod N with the mod function taking values in {1, 2, , N} Let u be a fixed element of H In 2003, Xu proved that the sequence {xn } generated by xn 1− nA Tn xn n 1u 1.4 converges strongly to the solution of the quadratic minimization problem x∈F under suitable hypotheses on F n Ax, x − x, u , 1.5 and under the additional hypothesis F T1 T2 · · · TN F TN T1 · · · TN−1 ··· F T2 T3 · · · T N T1 1.6 In 1999, Atsushiba and Takahashi defined the mapping Wn as follows: Un,0 I, Un,1 γn,1 T1 Un,2 γn,2 T2 Un,1 − γn,2 I, Un,3 γn,3 T3 Un,2 − γn,3 I, − γn,1 I, 1.7 Un,N−1 Wn γn,N−1 TN − 1Un,N−2 Un,N γn,N TN Un,N−1 − γn,N−1 I, − γn,N I, Fixed Point Theory and Applications where {γn,i }N ⊆ 0, This mapping is called the W-mapping generated by T1 , T2 , , TN and i γn,1 , γn,2 , , γn,N In 2000, Takahashi and Shimoji proved that if X is strictly convex Banach space, N 1, 2, , N then F Wn i F Ti , where < λn,i < 1, i In 2007,Shang et al introduced a composite iteration scheme as follows: x0 x ∈ C arbitrarily chosen, yn βn xn xn − βn Wn xn , αn γf xn 1.8 I − αn A yn , where f ∈ C is a contraction, and A is a linear bounded operator Note that the iterative scheme 1.8 is not well-defined, because xn n ≥ may not lie in C, so Wn xn is not defined However, if C H, the iterative scheme 1.8 is well-defined and Theorem 2.1 is obtained In the case C / H, we have to modify the iterative scheme 1.8 in order to make it well-defined In 2009, Kangtunyakarn and Suantai introduced a new mapping, called Kmapping, for finding a common fixed point of a finite family of nonexpansive mappings For a finite family of nonexpansive mappings {Ti }N1 and sequence {γn,i }N in 0, , the mapping i i Kn : C → C is defined as follows: Un,1 γn,1 T1 Un,2 γn,2 T2 Un,1 − γn,2 Un,1 , Un,3 γn,3 T3 Un,2 − γn,3 Un,2 , − γn,1 I, 1.9 Un,N−1 Kn γn,N−1 TN − 1Un,N−2 Un,N − γn,N−1 Un,N−2 , γn,N TN Un,N−1 − γn,N Un,N−1 The mapping Kn is called the K-mapping generated by T1 , , TN and γn,1 , γn,2 , , γn,N In this paper, motivated by Kim and Xu , Marino and Xu 10 , Xu , Yao et al , andShang et al , we introduce a composite iterative scheme as follows: x ∈ C arbitrarily chosen, x0 yn xn βn xn − βn Kn xn , PC αn γf xn 1.10 I − αn A yn , where f ∈ C is a contraction, and A is a bounded linear operator We prove, under certain appropriate conditions on the sequences {αn } and {βn } that {xn } defined by 1.10 converges strongly to a common fixed point of the finite family of nonexpansive mappings {Ti }N1 , which i solves a variational inequaility problem 4 Fixed Point Theory and Applications In order to prove our main results, we need the following lemmas Lemma 1.1 For all x, y ∈ H, there holds the inequality x y 2 ≤ x y, x y , x, y ∈ H 1.11 Lemma 1.2 see 11 Let {xn } and {zn } be bounded sequences in a Banach space X, and let {βn } be a sequence in 0, with < lim infn → ∞ βn ≤ lim supn → ∞ βn < Suppose that xn βn xn 1 − βn zn 1.12 for all integer n ≥ 0, and lim sup zn n→∞ Then limn → ∞ xn − zn − zn − xn − xn ≤ 1.13 Lemma 1.3 see Assume that {an } is a sequence of nonnegative real numbers such that an − γn an δn n ≥ 0, where {γn } ⊂ 0, and {δn } is a sequence in R such that i ∞ n γn ≤ ∞, ii lim supn → ∞ δn /γn ≤ or Then limn → ∞ an ∞ n |δn | < ∞ Lemma 1.4 see 10 Let A be a strongly positive linear bounded operator on a Hilbert space H with coefficient γ and < ρ ≤ A −1 Then I − ρA ≤ − ργ Lemma 1.5 see 10 Let H be a Hilbert space Let A be a strongly positive linear bounded operator with coefficient γ > Assume that < γ < γ/α Let T : C → C be a nonexpansive mapping with a − tA T x Then xt converges strongly as fixed point xt ∈ C of the contraction C x → tγf x t → to a fixed point x of T , which solves the variational inequality A − γf x, z − x ≥ 0, z∈F T 1.14 Lemma 1.6 see Demiclosedness principle Assume that T is nonexpansive self-mapping of closed convex subset C of a Hilbert space H If T has a fixed point, then I − T is demiclosed That is, whenever {xn } is a sequence in C weakly converging to some x ∈ C and the sequence { I − T xn } strongly converges to some y, it follows that I − T x y Here, I is identity mapping of H Lemma 1.7 see Let C be a nonempty closed convex subset of a strictly convex Banach space Let {Ti }N1 be a finite family of nonexpansive mappings of C into itself with N1 F Ti / ∅, and let i i λ1 , , λN be real numbers such that < λi < for every i 1, , N − and < λN ≤ Let K be N the K-mapping of C into itself generated by T1 , , TN and λ1 , , λN Then F K i F Ti Fixed Point Theory and Applications By using the same argument as in 9, Lemma 2.10 , we obtain the following lemma Lemma 1.8 Let C be a nonempty closed convex subset of Banach space Let {Ti }N1 be a finite family of i nonexpanxive mappings of C into itself and {λn,i }N1 sequences in 0, such that λn,i → λi , as n → i ∞, i 1, 2, , N Moreover, for every n ∈ N, let K and Kn be the K -mappings generated by T1 , T2 , , TN and λ1 , λ2 , , λN , and T1 , T2 , , TN and λn,1 , λn,2 , , λn,N , respectively Then, for every bounded sequence xn ∈ C, one has limn → ∞ Kn xn − Kxn Let H be real Hilbert space with inner product ·, · , C a nonempty closed convex subset of H Recall that the metric nearest point projection PC from a real Hilbert space H to a closed convex subset C of H is defined as follows Given that x ∈ H, PC x is the only inf{ x − y : y ∈ C} Below Lemma 1.9 can be found point in C with the property x − PC x in any standard functional analysis book Lemma 1.9 Let C be a closed convex subset of a real Hilbert space H Given that x ∈ H and y ∈ C then i y PC x if and only if the inequality x − y, y − z ≥ for all z ∈ C, ii PC is nonexpansive, iii x − y, PC x − PC y ≥ PC x − PC y for all x, y ∈ H, iv x − PC x, PC x − y ≥ for all x ∈ H and y ∈ C Main Result In this section, we prove strong convergence of the sequences {xn } defined by the iteration scheme 1.10 Theorem 2.1 Let H be a Hilbert space, C a closed convex nonempty subset of H Let A be a strongly positive linear bounded operator with coefficient γ > 0, and let f ∈ c· Let {Ti }N1 be a finite family of i nonexpansive mappings of C into itself, and let Kn be defined by 1.9 Assume that < γ < γ/α and ∞ ∞ N F i F Ti / ∅ Let x0 ∈ C, given that {αn }n and {βn }n are sequences in 0, , and suppose that the following conditions are satisfied: C1 αn → 0; C2 ∞ n αn ∞; C3 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1; C4 ∞ n |γn,i − γn−1,i | < ∞, for all i C5 ∞ n |αn − αn | < ∞; C6 ∞ n |βn 1, 2, , N and {γn,i }N1 ⊂ a, b , where < a ≤ b < 1; i − βn | < ∞ If {xn }∞ is the composite process defined by 1.10 , then {xn }∞ converges strongly to q ∈ F, which n n also solves the following variational inequality: γf q − Aq, p − q ≤ 0, p ∈ F 2.1 Fixed Point Theory and Applications Proof First, we observe that {xn }∞ is bounded Indeed, take a point u ∈ F, and notice that n yn − u ≤ βn xn − u − βn Kn xn − u ≤ xn − u 2.2 Since αn → 0, we may assume that αn ≤ A−1 for all n By Lemma 1.4, we have I − αn A ≤ − αn γ for all n It follows that xn PC αn γf xn −u I − αn A yn − PC u ≤ αn γf xn − Au I − αn A yn − u ≤ αn γf xn − Au − αn γ ≤ αn γf xn − γf u − γ − γα αn xn − u − γ − γα αn xn − u − αn γ yn − u xn − u 2.3 αn γf u − Au γf u − Au γ − γα γ − γα αn γf u − Au γ − γα xn − u , − αn γ αn γf u − Au αn γf u − Au ≤ αγαn xn − u ≤ max yn − u By simple inductions, we have xn − u ≤ max x0 − u , γf u − Au γ − γα , n ≥ 2.4 Therefore {xn } is bounded, so are {yn } and {f xn } Since Kn is nonexpansive and yn − βn Kn xn , we also have βn xn yn − yn ≤ βn xn βn xn − βn xn 1 − βn ≤ βn xn 1 xn 1 1 − xn 1 Kn xn βn − βn xn Kn xn − Kn xn βn − βn xn Kn xn − Kn xn βn − βn xn βn xn − βn xn Kn xn − Kn xn − xn − βn xn − xn − βn ≤ βn − βn − βn xn − βn Kn xn − βn − βn − βn − βn 1 − Kn xn Kn xn − Kn xn Kn xn 1 − βn βn − βn Kn xn Kn xn − − βn Kn xn βn − βn 1 xn − xn Kn xn Kn xn − Kn xn βn − βn Kn xn 2.5 Fixed Point Theory and Applications By using the inequalities 2.6 and 2.11 of 9, Lemma 2.11 , we can conclude that Kn xn−1 − Kn−1 xn−1 ≤ M N γn,j − γn−1,j , 2.6 j where M sup{ N Tj Un,j−1 xn j By 2.5 and 2.6 , we have xn − xn PC αn γf xn ≤ Un,j−1 xn I − αn A yn T1 xn xn } − PC αn−1 γf xn−1 I − αn−1 A yn−1 I − αn A yn − yn−1 − αn − αn−1 Ayn−1 γαn f xn − f xn−1 ≤ − αn γ yn − yn−1 |αn − αn−1 | Ayn−1 γ|αn − αn−1 | f xn−1 γααn xn − xn−1 ≤ − αn γ γ αn − αn−1 f xn−1 xn − xn−1 βn − βn−1 xn−1 − βn Kn xn−1 − Kn−1 xn−1 |αn − αn−1 | Ayn−1 ≤ − αn γ xn − xn−1 γααn xn − xn−1 − γ − γα αn N − βn M γ|αn − αn−1 | f xn−1 βn − βn−1 xn−1 − βn Kn xn−1 − Kn−1 xn−1 |αn − αn−1 | Ayn−1 βn−1 − βn Kn−1 xn−1 βn−1 − βn Kn−1 xn−1 γααn xn − xn−1 L βn−1 − βn xn − xn−1 γ|αn − αn−1 | f xn−1 M |αn − αn−1 | γn,j − γn−1,j , j 2.7 where L sup{ xn−1 Kn−1 xn−1 : n ∈ N}, M max{ Ayn−1 γ f xn−1 } Since ∞ |αn − n ∞ αn−1 | < ∞, n |βn−1 −βn | < ∞, and ∞ |γn,j −γn−1,j | < ∞, for all j 1, 2, , N, by Lemma 1.3, n we obtain xn − xn → It follows that xn − yn PC αn γf xn ≤ αn γf xn αn γf xn I − αn A yn − PC yn I − αn A yn − yn Ayn 2.8 Fixed Point Theory and Applications Since αn → and {f xn }, {Ayn } are bounded, we have xn xn − yn ≤ xn − xn xn 1 − yn → as n → ∞ Since − yn , 2.9 it implies that xn − yn → as n → ∞ On the other hand, we have yn − Kn xn Kn xn − xn ≤ xn − yn xn − yn βn xn − Kn xn , 2.10 which implies that − βn Kn xn − xn ≤ xn − yn From condition C3 and xn − yn → as n → ∞, we obtain Kn xn − xn → 2.11 1, 2, , N Let K be the K-mapping By C4 , we have limn → ∞ γn,i γi ∈ a, b for all i generated by T1 , , TN and γ1 , , γN Next, we show that lim sup γf q − Aq, xn − q ≤ 0, 2.12 n→∞ I − tA Kx where q limt → xt with xt being the fixed point of the contraction x → tγf x tγf xt I − tA Kxt By Lemma 1.5 and Thus, xt solves the fixed point equation xt Lemma 1.7, we have q ∈ F and γf q − Aq, p − q ≥ for all p ∈ F It follows by 2.11 I − tA Kxt − xn and Lemma 1.8 that Kxn − xn → Thus, we have xt − xn t γf xt − Axn It follows from Lemma 1.1 that for < t < A −1 , xt − xn I − tA Kxt − xn ≤ − γt ≤ − γt Kxt − xn t γf xt − Axn Kxt − Kxn 2t γf xt − Axn , xt − xn 2 Kxt − Kn xn 2t γf xt − Axt , xt − xn ≤ − 2γt γt 2 xt − xn Kxn − xn Kxn − xn 2.13 Axt − Axn , xt − xn fn t 2t γf xt − Axt , xt − xn 2t Axt − Axn , xt − xn , where fn t xt − xn xn − Kxn xn − Kxn −→ 0, as n → 2.14 Fixed Point Theory and Applications It follows that γt −2γt Axt − γf xt , xt − xn ≤ 2t γt −2 ≤ ≤ −1 γ xt − xn γt 2 xt − xn fn t 2t Axt − Axn , xt − xn Axt − Axn , xt − xn fn t 2t Axt − Axn , xt − xn γt Axt − Axn , xt − xn ≤ fn t 2t Axt − Axn , xt − xn fn t 2t 2.15 Letting n → ∞ in 2.15 and 2.14 , we get lim sup Axt − γf xt , xt − xn ≤ n→∞ t M0 , 2.16 where M0 > is a constant such that M0 ≥ γ Axt − Axn , xt − xn for all t ∈ 0, and n ≥ Taking t → in 2.16 , we have lim sup lim sup Axt − γf xt , xt − xn ≤ t→0 2.17 n→∞ On the other hand, one has γf q − Aq, xn − q γf q − Aq, xn − q − γf q − Aq, xn − xt γf q − Aq, xn − xt − γf q − Axt , xn − xt γf q − Axt , xn − xt − γf xt − Axt , xn − xt γf xt − Axt , xn − xt γf q − Aq, xt − q Axt − Aq, xn − xt γf q − γf xt , xn − xt ≤ γf q − Aq xt − q γf xt − Axt , xn − xt A xt − q A γα γα xt − q xn − xt γf xt − Axt , xn − xt γf q − Aq xt − q xt − q xn − xt γf xt − Axt , xn − xt 2.18 It follows that lim sup γf q − Aq, xn − q ≤ γf q − Aq n→∞ xt − q A γα xt − q lim sup xn − xt n→∞ lim sup γf xt − Axt , xn − xt n→∞ 2.19 10 Fixed Point Theory and Applications Therefore, from 2.17 and limt → xt − q 0, we have lim sup γf q − Aq, xn − q ≤ lim sup lim sup γf q − Aq, xn − q n→∞ n→∞ t→0 2.20 ≤ lim sup lim sup γf xt − Axt , xn − xt ≤ n→∞ t→0 Hence 2.12 holds Finally, we prove that xn → q By using 2.2 and together with the Schwarz inequality, we have xn −q I − αn A yn − PC q PC αn γf xn I − αn A yn − q ≤ αn γf xn − Aq I − αn A yn − q α2 γf xn − Aq n 2αn I − αn A yn − q , γf xn − Aq ≤ − αn γ yn − q α2 γf xn − Aq n 2αn yn − q, γf xn − Aq − 2α2 A yn − q , γf xn − Aq n ≤ − αn γ xn − q α2 γf xn − Aq n 2αn yn − q, γf xn − γf q 2αn yn − q, γf q − Aq − 2α2 A yn − q , γf xn − Aq n ≤ − αn γ xn − q α2 γf xn − Aq n γf xn − γf q 2αn yn − q 2αn yn − q, γf q − Aq − 2α2 A yn − q , γf xn − Aq n ≤ − αn γ xn − q α2 γf xn − Aq n xn − q 2γααn yn − q 2αn yn − q, γf q − Aq − 2α2 A yn − q , γf xn − Aq n ≤ − αn γ xn − q 2γααn xn − q 2 α2 γf xn − Aq n 2αn yn − q, γf q − Aq − 2α2 A yn − q , γf xn − Aq n ≤ − αn γ 2γααn α2 γf xn − Aq n − γ − γα αn xn − q 2αn yn − q, γf xn − Aq 2α2 A yn − q n γf xn − Aq xn − q αn yn − q, γf q − Aq αn γf xn − Aq 2 A yn − q γf xn − Aq γ xn − q 2.21 Fixed Point Theory and Applications 11 Since {xn }, {f xn }, and {yn } are bounded, we can take a constant η > such that η ≥ γf xn − Aq 2 A yn − q γf xn − Aq γ xn − q 2.22 for all n ≥ It then follows that xn −q ≤ − γ − γα αn xn − q αn βn , 2.23 where βn yn − q, γf q − Aq ηαn By lim supn → ∞ γf − A q, yn − q ≤ 0, we get lim supn → ∞ βn ≤ By applying Lemma 1.3 to 2.23 , we can conclude that xn → q This completes the proof If A I and γ in Theorem 2.1, we obtain the following result Corollary 2.2 Let H be a Hilbert space, C a closed convex nonempty subset of H, and let f ∈ c Let {Ti }N1 be a finite family of nonexpansive mappings of C into itself, and let Kn be defined by 1.9 i ∞ ∞ N Assume that F i F Ti / ∅ Let x0 ∈ C, given that {αn }n and {βn }n are sequences in 0, , and suppose that the following conditions are satisfied: C1 αn → 0; C2 ∞ n αn ∞; C3 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1; C4 C5 C6 ∞ n |γn,i − γn−1,i | < ∞, for all i b < 1; ∞ n ∞ n |αn − αn | < ∞; |βn 1, 2, , N and {γn,i }N1 ⊂ a, b , where < a ≤ i − βn | < ∞ If {xn }∞ is the composite process defined by n yn xn βn xn αn f xn − βn Kn xn , − αn yn , 2.24 then {xn }∞ converges strongly to q ∈ F, which also solves the following variational inequality: n f − I q, p − q ≤ 0, If N 1, A Kim and Xu I, γ p ∈ F 2.25 1, and f ≡ u ∈ C is a constant in Theorem 2.1, we get the results of Corollary 2.3 Let H be a Hilbert space, C a closed convex nonempty subset of H, and let f ∈ c Let T be a nonexpansive mapping of C into itself F T / ∅ Let x0 ∈ C, given that {αn }∞ and n {βn }∞ are sequences in 0, , and suppose that the following conditions are satisfied: n C1 αn → 0; C2 ∞ n αn ∞; 12 Fixed Point Theory and Applications C3 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1; C4 C5 ∞ n ∞ n |αn − αn | < ∞; |βn − βn | < ∞ If {xn }∞ is the composite process defined by n yn xn βn xn αn u − βn T xn , I − αn yn , 2.26 then {xn }∞ converges strongly to q ∈ F, which also solves the following variational inequality: n u − q, p − q ≤ 0, p ∈ F 2.27 Acknowledgments The authors would like to thank the referees for valuable suggestions on the paper and thank the Center of Excellence in Mathematics, the Thailand Research Fund, and the Graduate School of Chiang Mai University for financial support References W R Mann, “Mean 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mapping, called Kmapping, for finding a common fixed point of a finite family of nonexpansive mappings For a finite family of nonexpansive mappings {Ti }N1 and sequence... Proceedings of the American Mathematical Society, vol 4, pp 506–510, 1953 S Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces, ” Journal of Mathematical Analysis and Applications,... for a finite family of nonexpansive mappings and applications,” Indian Journal of Mathematics, vol 41, no 3, pp 435–453, 1999 W Takahashi and K Shimoji, “Convergence theorems for nonexpansive mappings