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IMPLICIT PREDICTOR-CORRECTOR ITERATION PROCESS FOR FINITELY MANY ASYMPTOTICALLY (QUASI-)NONEXPANSIVE MAPPINGS L. C. CENG, N. C. WONG, AND J. C. YAO Received 13 February 2006; Rev ised 3 June 2006; Accepted 5 June 2006 We study an implicit predictor-corrector iteration process for finitely many asymptoti- cally quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space E. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case E is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial’s condition is satisfied (resp., one of these mappings is semicom- pact). Our results improve and extend earlier and recent ones in the literature. Copyright © 2006 L. C. Ceng 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. 1. Introduction and preliminaries Let E be a real Banach space equipped with norm ·,letC be a nonempty subset of E, and let T : C → C. The set F(T) ={x ∈ C : Tx = x} consists of all fixed points of T. Definit ion 1.1. T is said to be (1) nonexpansive if Tx− Ty≤x − y, ∀x, y ∈ C; (1.1) (2) asymptotically nonexpansive [3] if there exists a sequence {k n } ∞ n=1 ⊂ [1,∞)with lim n→∞ k n = 1suchthat T n x − T n y ≤ k n x − y, ∀x, y ∈ C, n ≥ 1; (1.2) (3) asymptotically quasi-nonexpansive if F(T) =∅, and there exists a sequence {k n } ∞ n=1 ⊂ [1,∞) with lim n→∞ k n = 1suchthat T n x − p ≤ k n x − p, ∀x ∈ C, p ∈ F(T), n ≥ 1; (1.3) Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 65983, Pages 1–11 DOI 10.1155/JIA/2006/65983 2 Implicit predictor-corrector iteration process (4) s emicompact [9]ifforanyboundedsequence {x n }⊂C with lim n→∞ x n − Tx n = 0, there exists a strongly convergent subsequence of {x n }. The class of asymptotically nonexpansive mappings, as a natural extension of that of nonexpansive mappings, was introduced by Goebel and Kirk [3]. They proved that if C is a nonempty bounded closed convex subset of a uniformly convex Banach space E,then every asymptotically nonexpansive self-mapping T on C has a fixed point. Furthermore, the study of iterative construction for fixed points of asymptotically nonexpansive map- pings began in 1978. Bose [1] first proved that if the uniformly convex B anach space E satisfies Opial’s condition [5], then {T n x} convergesweaklytoafixedpointofT,pro- vided T is asymptotically regular at x, that is, lim n→∞ T n x − T n+1 x=0. A Banach space E is said to satisfy Opial’s condition [5]ifwhenever {x n } is a sequence in E which con- verges weakly to x,onehas liminf n→∞ x n − x < liminf n→∞ x n − y , ∀y ∈ E, y = x. (1.4) It is well known that every Hilbert space satisfies Opial’s condition (see, e.g., [5]). Xu and Ori [8] first introduced an implicit iteration process for N nonexpansive map- pings in a Hilbert space and proved the following weak convergence theorem. Theorem 1.2 (see [8]). Let H be a Hilbert space and let C be a nonempty closed convex subset of H.Let {T i } N i =1 be N nonexpansive self-mappings on C such that F = N i =1 F(T i ) = ∅ .Letx 0 ∈ C and let {α n } ∞ n=1 beasequencein(0,1) such that lim n→∞ α n = 0.Thenthe sequence {x n } defined implicity by x n = α n x n−1 + 1 − α n T n(modN ) x n , n ≥ 1, (1.5) converges weakly to a common fixed point of mappings {T j } N j =1 . Later, Sun [7] introduced and studied another implicit iteration process x n = α n x n−1 + 1 − α n T l n +1 n(modN ) x n , n ≥ 1, (1.6) for N asymptotically quasi-nonexpansive self-mappings {T j } N j =1 on a nonempty bounded closed convex subset C ofaBanachspaceE,where {α n } is a sequence in (0,1), x 0 is an initial point in C,andn = l n N + n (modN). Moreover, he proved that the sequence {x n } defined by his iteration process converges strongly to a common fi xed point of {T j } N j =1 under suitable conditions. Atthesametime,in[10], Zhou and Chang introduced and studied the following im- plicit iteration process: x n = α n x n−1 + β n T n n(modN ) x n + γ n u n , n ≥ 1, (1.7) for N asymptotically nonexpansive self-mappings {T j } N j =1 on a nonempty closed convex subset C of a Banach space E,where {α n }, {β n }, {γ n } are three sequences in [0,1], x 0 is an initial point in C,and {u n } is a bounded sequence in C. Moreover, they proved that the sequence {x n } defined by their iteration process converges weakly to a common fixed point of {T j } N j =1 under suitable conditions. L. C. Ceng et al. 3 As indicated in [10], if T 1 ,T 2 , ,T N : C → C are N asymptotically nonexpansive map- pings, then there exists a sequence, called common Lipschitz constants, {k n }⊂[1,∞)with lim n→∞ k n = 1suchthatforeachi = 1,2, ,N, T n i x − T n i y ≤ k n x − y, ∀x, y ∈ C, n ≥ 1. (1.8) A similar situation occurs when T 1 ,T 2 , ,T N are asymptotically quasi-nonexpansive. By convention, we write T n := T n(modN ) ,forintegern ≥ 1, with the mod function taking values in the set {1,2, ,N}. In other words, if n = l n N + q for some unique integers l n ≥ 0and1≤ q ≤ N,thenwesetT n = T q . In this paper, we introduce the following implicit predictor-corrector iteration process with an auxiliary finite family of asymptotically quasi-nonexpansive self-mappings on C. Definit ion 1.3 (basic setup). Let C be a nonempty closed convex subset of a Banach space E,andlet {T 1 ,T 2 , ,T N } and { T 1 , T 2 , , T N } be two families of asymptotically quasi- nonexpansive mappings from C into C with common Lipschitz constants {k n } and { k n } such that ∞ n=1 (k n − 1) < +∞ and ∞ n=1 ( k n − 1) < +∞, respectively. Let {x n } be an itera- tive sequence in C generated from an arbitrar y x 0 ∈ C by the following three steps. Auxiliary step. With x n−1 (n ≥ 1) established, y n is computed implicitly by y n = α n x n−1 + β n T l n n y n + γ n u n . (1.9a) Predictor step. With y n obtained in the auxiliary step, z n is computed implicitly by z n = α n y n + β n T l n n z n + γ n u n . (1.9b) Corrector step. With z n obtained in the predictor step, x n is computed explicitly by x n = α n y n + β n T l n n z n + γ n u n . (1.9c) Here, T n := T n(modN ) and T n := T n(mod N) for n = 1,2, On the other hand, {u n } ∞ n=1 , {u n } ∞ n=1 , {u n } ∞ n=1 are three bounded sequences in C;and{α n } ∞ n=1 , {α n } ∞ n=1 , {α n } ∞ n=1 , {β n } ∞ n=1 , { β n } ∞ n=1 , {β n } ∞ n=1 , {γ n } ∞ n=1 , {γ n } ∞ n=1 , {γ n } ∞ n=1 are nine real sequences in [0,1] such that α n + β n + γ n = 1(∀n ≥ 1), ∞ n=1 γ n < +∞, α n + β n + γ n = 1(∀n ≥ 1), ∞ n=1 γ n < +∞, α n + β n + γ n = 1(∀n ≥ 1), ∞ n=1 γ n < +∞, 0 < β n ,β n ≤ c<K −1 (∀n ≥ 1), K = max sup n≥1 k n ,sup n≥1 k n ≥ 1. (1.10) 4 Implicit predictor-corrector iteration process Remark 1.4. Since 0< β n ,β n ≤c<K −1 , it is clear that the mappings y → α n x n−1 + β n T l n n y + γ n u n and z → α n y n + β n T l n n z + γ n u n are two contractions from the nonempty closed con- vex set C into itself. Thus, by the Banach contraction principle, there exist the unique points y n ,z n ∈ C such that (1.9a)and(1.9b) hold, respectively. Therefore, the sequence {x n } is well defined. Our aim is to consider and study the strong and weak convergences of the above im- plicit predictor-corrector iteration process. To this end, we need the following lemmas. Lemma 1.5. Let {b n }, {b n }, { b n } be three nonnegative real sequences with finite sums. Then ∞ n=1 λ n < +∞,whereλ n = (1 + b n )(1 + b n )(1 + b n ) − 1 for each ≥ 1. Lemma 1.6 (see [10]). Let {a n }, {λ n }, {μ n } be three nonnegative real sequences such that ∞ n=1 λ n < +∞, ∞ n=1 μ n < +∞,and a n+1 ≤ 1+λ n a n + μ n , ∀n ≥ 1. (1.11) Then lim n→∞ a n exists. Lemma 1.7 (see [6]). Let E be a uniformly convex Banach space, {t n }⊂[b,c] ⊂ (0,1), and {x n },{y n }⊂E.Iflim n→∞ t n x n +(1− t n )y n =d<+∞, lim sup n→∞ x n ≤d,and limsup n→∞ y n ≤d, then lim n→∞ x n − y n =0. Lemma 1.8 (demiclosed principle [2]). Let E be a uniformly convex Banach space, let C be a nonempt y closed convex subset of E,andletT : C → C be an asymptotically nonexpansive mapping with F(T) =∅. Then I − T is demiclosed at zero, that is, for any sequence {x n }⊂ C, x n −→ q ∈ C weakly (I − T)x n −→ 0 strongly =⇒ (I − T)q = 0. (1.12) 2. Main results Lemma 2.1. Let C be a nonempty clos ed convex subs et of a B anach space E,andlet {T i } N i =1 and { T j } N j =1 be two finite families of asymptotically quasi-nonexpansive self-mappings on C such that N i =1 F(T i ) ∩ N j =1 F( T j ) =∅.If{x n }, {y n },and{z n } are the iterative sequences defined by (1.9a), (1.9b), and (1.9c), then for each p ∈ N i =1 F(T i ) ∩ N j =1 F( T j ), there hold lim n→∞ x n − p = d,limsup n→∞ y n − p ≤ d,limsup n→∞ z n − p ≤ d. (2.1) Proof. Since {u n } ∞ n=1 , {u n } ∞ n=1 , {u n } ∞ n=1 are three bounded sequences in C,foranygiven p ∈ N i =1 F(T i ) ∩ N j =1 F( T j ), we have M : = max sup n≥1 u n − p ,sup n≥1 u n − p ,sup n≥1 u n − p < +∞. (2.2) L. C. Ceng et al. 5 Note that 1 − β n k l n ≥ 1 − cK > 0and1− β n k l n ≥ 1 − cK > 0. Put L = 1 1 − cK , b n = β n k l n − 1 , b n = 1 − β n 1 − β n k l n − 1, b n = 1 − β n 1 − β n k l n − 1. (2.3) Then we have 0 ≤ b n = β n k l n − 1 ≤ k l n − 1, 1 + b n ≤ K, 0 ≤ b n = β n k l n − 1 1 − β n k l n ≤ L k l n − 1 ,1+b n ≤ L, 0 ≤ b n = β n k l n − 1 1 − β n k l n ≤ L k l n − 1 ,1+ b n ≤ L. (2.4) Observe that y n − p = α n x n−1 − p + β n T l n n y n − p + γ n u n − p ≤ α n x n−1 − p + β n k l n y n − p + γ n u n − p . (2.5) It follows y n − p ≤ α n 1 − β n k l n x n−1 − p + γ n 1 − β n k l n u n − p ≤ 1 − β n 1 − β n k l n x n−1 − p + LMγ n = 1+ b n x n−1 − p + LMγ n . (2.6) Similarly, z n − p = α n y n − p + β n T l n n z n − p + γ n u n − p ≤ α n y n − p + β n k l n z n − p + γ n u n − p (2.7) Consequently, z n − p ≤ α n 1 − β n k l n y n − p + γ n 1 − β n k l n u n − p ≤ 1 − β n 1 − β n k l n y n − p + LMγ n = 1+b n y n − p + LMγ n . (2.8) 6 Implicit predictor-corrector iteration process Therefore, x n − p = α n y n − p + β n T l n n z n − p + γ n u n − p ≤ α n y n − p + β n k l n z n − p + γ n u n − p ≤ 1 − β n y n − p + β n k l n 1+b n y n − p + LMγ n + γ n M ≤ 1+β n k l n − 1 1+b n y n − p + M KLγ n + γ n ≤ 1+b n 1+b n y n − p + KLM γ n + γ n ≤ 1+b n 1+b n 1+ b n x n−1 − p + LMγ n + KLM γ n + γ n ≤ 1+b n 1+b n 1+ b n x n−1 − p + KL 2 Mγ n + KLM γ n + γ n ≤ 1+b n 1+b n 1+ b n x n−1 − p + KL 2 M γ n + γ n + γ n = 1+λ n x n−1 − p + μ n , (2.9) where λ n = (1 + b n )(1 + b n )(1 + b n ) − 1, and μ n = KL 2 M[γ n + γ n + γ n ]. Since ∞ n=1 (k l n − 1) < +∞ and ∞ n=1 ( k l n − 1) < +∞,itfollowsfrom(2.4)that ∞ n=1 b n < + ∞, ∞ n=1 b n < +∞,and ∞ n=1 b n < +∞. Hence, we derive ∞ n=1 λ n < +∞ by Lemma 1.5. Note that ∞ n=1 γ n < +∞, ∞ n=1 γ n < +∞,and ∞ n=1 γ n < +∞. This provides ∞ n=1 μ n < +∞. By Lemma 1.6,lim n→∞ x n − p exists. Let lim n→∞ x n − p=d. Since lim n→∞ b n = lim n→∞ γ n = 0, from (2.6), we obtain limsup n→∞ y n − p ≤ limsup n→∞ 1+ b n x n−1 − p + LM limsup n→∞ γ n ≤ d. (2.10) Further, since lim n→∞ b n = lim n→∞ γ n = 0, from (2.8), we obtain limsup n→∞ z n − p ≤ limsup n→∞ 1+b n y n − p + LM limsup n→∞ γ n ≤ d. (2.11) Theorem 2.2. Let C be a nonempt y closed convex subs et of a Banach space E.Let{T i } N i =1 and { T j } N j =1 be two finite families of asymptotically quasi-nonexpansive self-mappings on C such that F : = N i =1 F(T i ) ∩ N j =1 F( T j ) =∅.Let{x n } be the iterative sequence defined by (1.9a), (1.9b), and (1.9c). Then {x n } converges strongly to an element of F if and only if liminf n→∞ d x n ,F = 0. (2.12) Proof. The necessity is obvious. For the sufficiency, we assume liminf n→∞ d(x n ,F) = 0. Let p be any given element in F.Thenfrom(2.9), we obtain x n − p ≤ 1+λ n x n−1 − p + μ n , (2.13) L. C. Ceng et al. 7 where ∞ n=1 λ n < +∞ and ∞ n=1 μ n < +∞. Taking the infimum over all p ∈ F,weget d x n ,F ≤ 1+λ n d x n−1 ,F + μ n . (2.14) Hence, lim n→∞ d(x n ,F) exists. Furthermore, we have lim n→∞ d(x n ,F) = 0. By Lemma 2.1, we know that lim n→∞ x n − p exists. Hence {x n } is bounded. Put δ n = λ n x n−1 − p + μ n .Then ∞ n=1 δ n < +∞,and(2.13)canberewrittenas x n − p ≤ x n−1 − p + δ n . (2.15) For arbitrary ε>0, choose N 0 such that d(x N 0 ,F) <ε/4and ∞ j=N 0 δ j <ε/4. Conse- quently, for all n,m ≥ N 0 ,wehave x n − x m ≤ x n − p + x m − p ≤ x N 0 − p + n j=N 0 +1 δ j + x N 0 − p + m j=N 0 +1 δ j ≤ 2 x N 0 − p +2 ∞ j=N 0 δ j . (2.16) Taking the infimum over all p ∈ F,weobtain x n − x m ≤ 2d x N 0 ,F +2 ∞ j=N 0 δ j ≤ 2ε 4 + 2ε 4 = ε. (2.17) This shows that {x n } ∞ n=1 is Cauchy. Let lim n→∞ x n = u.ItiseasytoverifythatF is closed. Since lim n→∞ d(x n ,F) = 0, we must have that u ∈ F. As a consequence of Lemma 2.1, the iterated sequence {x n } is bounded. If the under- lying space E is reflexive, then we can expect that its weak cluster points provide common fixed points of T 1 ,T 2 , ,T N . This leads to the following theorem. Theorem 2.3. Let E be a uniformly convex Banach space, let C be a nonempty closed convex subset of E,andlet {T i } N i =1 (resp., { T j } N j =1 ) be a finite family of asymptotically nonexpan- sive (resp., asymptotically quasi-nonexpansive) self-mappings on C such that N j =1 F( T j ) ∩ N i =1 F(T i ) =∅.Supposelim n→∞ β n = 0 and {β n } ∞ n=1 ⊂ [b,c] ⊂ (0,K −1 ),whereK is as in (1.10). Then ever y weak cluster point of the bounded iterative sequence {x n } defined by (1.9a), (1.9b), and (1.9c)belongsto N i =1 F(T i ). Proof. Let p ∈ N j =1 F( T j ) ∩ N i =1 F(T i ). By Lemma 2.1,wehave lim n→∞ x n − p = d,limsup n→∞ y n − p ≤ d,limsup n→∞ z n − p ≤ d. (2.18) Obviously, {x n }, {y n },and{z n } are bounded sequences in C. 8 Implicit predictor-corrector iteration process Observe that x n − p = 1 − β n y n − p + γ n u n − y n + β n T l n n z n − p + γ n u n − y n −→ d, (2.19) as n →∞.Sincelim n→∞ γ n = 0and{u n } is bounded, we have limsup n→∞ y n − p + γ n u n − y n ≤ limsup n→∞ y n − p + γ n u n − y n ≤ d, limsup n→∞ T l n n z n − p + γ n u n − y n ≤ limsup n→∞ k l n z n − p + γ n u n − y n ≤ d. (2.20) It follows from Lemma 1.7 that lim n→∞ T l n n z n − y n = 0. (2.21) Thus, lim n→∞ z n − y n = lim n→∞ α n y n + β n T l n n z n + γ n u n − y n = lim n→∞ β n T l n n z n − y n + γ n u n − y n = 0. (2.22) Similarly, lim n→∞ x n − y n = lim n→∞ α n y n + β n T l n n z n + γ n u n − y n = lim n→∞ β n T l n n z n − y n + γ n u n − y n = 0. (2.23) Moreover, y n − x n−1 = α n x n−1 + β n T l n n y n + γ n u n − x n−1 = β n T l n n y n − x n−1 + γ n u n − x n−1 ≤ β n T l n n y n − x n−1 + γ n u n − x n−1 −→ 0, as n −→ ∞ , (2.24) since lim n→∞ β n = lim n→∞ γ n = 0. As a result, we have x n − x n−1 ≤ x n − y n + y n − x n−1 −→ 0, as n −→ ∞ . (2.25) It forces lim n→∞ x n − x n+i = 0, for each i = 1,2, ,N. (2.26) On the other hand, we have x n − T l n n x n ≤ x n − y n + y n − T l n n z n + T l n n z n − T l n n x n ≤ x n − y n + y n − T l n n z n + k l n z n − x n −→ 0, as n −→ ∞ . (2.27) L. C. Ceng et al. 9 As n = l n N + n (modN)forn>N,weget n − N = l n − 1 N + n (modN), (2.28) and hence l n−N = l n − 1. Thus, we have T l n −1 n = T l n−N n−N . (2.29) Consequently, we derive x n − T n x n ≤ x n − T l n n x n + T l n n x n − T n x n ≤ x n − T l n n x n + K T l n −1 n x n − x n = x n − T l n n x n + K T l n−N n−N x n − x n ≤ x n − T l n n x n + K T l n−N n−N x n − T l n−N n−N x n−N + T l n−N n−N x n−N − x n−N + x n−N − x n ≤ x n − T l n n x n + K (1 + K) x n−N − x n + T l n−N n−N x n−N − x n−N −→ 0, as n −→ ∞ . (2.30) This implies that for each j = 1,2, ,N, x n − T n+ j x n ≤ x n − x n+ j + x n+ j − T n+ j x n+ j + T n+ j x n+ j − T n+ j x n ≤ (1 + K) x n − x n+ j + x n+ j − T n+ j x n+ j −→ 0, as n −→ ∞ . (2.31) Note that the closedness and convexity of C imply the weak closedness of C.Let x ∈ C be any weak cluster point of the bounded sequence {x n }.Let{x n i } be a subsequence of {x n } such that x n i → x weakly (see, e.g., [4, page 313]). Since the pool of mappings {T i :1≤ i ≤ N} is finite, we may further assume (passing to a further subsequence if necessary) that for some integer l ∈{1,2, ,N}, T n i = T l for all i ≥ 1. Then it follows from (2.31)thatforeach j = 1,2, ,N, x n i − T l+ j x n i −→ 0, as i −→ ∞ , (2.32) that is, for each j = 1,2, ,N, x n i − T j x n i −→ 0, as i −→ ∞ . (2.33) By Lemma 1.8, we can conclude that x ∈ N j =1 F(T j ). Theorem 2.4. In addition to the conditions in Theorem 2.3, assume further that ∅ = N i =1 F(T i ) ⊆ N j =1 F( T j ). (a) If E satisfies Opial’s condition, then {x n } converges weakly to an element of N i =1 F(T i ). (b) If one of {T i } N i =1 is semicompact, then {x n } converges strongly to an element of N i =1 F(T i ). 10 Implicit predictor-corrector iteration process Proof. We continue the argument in the proof of Theorem 2.3. For (a), we claim that {x n } is weakly convergent. Were this false, there existed another subsequence {x n j } of {x n } such that x n j → x ∈ C weakly and x = x. Utilizing the same argument as in Theorem 2.3,wecanprovethat x ∈ N j =1 F(T j ). Note that by Lemma 2.1, both lim n→∞ x n − x and lim n→∞ x n − x exist. It follows from the Opial condition of E that lim n→∞ x n − x = liminf i→∞ x n i − x < liminf i→∞ x n i − x = lim n→∞ x n − x = liminf j→∞ x n j − x < liminf j→∞ x n j − x = lim n→∞ x n − x . (2.34) This contradiction indicates that x = x,andso{x n } converges weakly to x. For (b), by (2.33), we can assume that a subsequence {x n i } of {x n } exists such that x n i → x ∈ N i =1 F(T i )innorm.ItthenfollowsfromLemma 2.1 that lim n→∞ x n − x = lim i→∞ x n i − x = 0. (2.35) This completes the proof. Acknowledgments The first author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai. 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C Ceng et al 11 [9] S S Zhang, On the iterative approximation problem of fixed points for asymptotically nonexpansive type mappings in Banach spaces, Applied Mathematics and Mechanics (English Edition) 22 (2001), no 1, 25–34 [10] Y Zhou and S.-S Chang, Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numerical Functional Analysis . IMPLICIT PREDICTOR-CORRECTOR ITERATION PROCESS FOR FINITELY MANY ASYMPTOTICALLY (QUASI-)NONEXPANSIVE MAPPINGS L. C. CENG, N. C. WONG,. 2006; Rev ised 3 June 2006; Accepted 5 June 2006 We study an implicit predictor-corrector iteration process for finitely many asymptoti- cally quasi-nonexpansive self-mappings on a nonempty closed. sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case E is a uniformly convex Banach space and the mappings are asymptotically
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