PROCESS FOR FINITELY MANY ASYMPTOTICALLY (QUASI-)NONEXPANSIVE MAPPINGS

PROCESS FOR FINITELY MANY ASYMPTOTICALLY (QUASI-)NONEXPANSIVE MAPPINGS

L. C. CENG, N. C. WONG, AND J. C. YAO

Received 13 February 2006; Revised 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 spaceE. 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 caseEis 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

LetEbe a real Banach space equipped with norm · , letCbe a nonempty subset ofE, and letT:C→C. The setF(T)= {x∈C:Tx=x}consists of all fixed points ofT.

Definition 1.1. Tis said to be (1)nonexpansiveif

Tx−T y ≤ x−y, ∀x,y∈C; (1.1) (2)asymptotically nonexpansive [3] if there exists a sequence{kn}^∞_n=1⊂[1,∞) with

lim_n→∞k_n=1 such that

Tⁿx−Tⁿy≤knx−y, ∀x,y∈C,n≥1; (1.2) (3)asymptotically quasi-nonexpansiveifF(T)=∅, and there exists a sequence{kn}^∞_n=1⊂

[1,∞) with lim_n→∞kn=1 such that

Tⁿx−p≤k_nx−p, ∀x∈C, p∈F(T), n≥1; (1.3)

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 65983, Pages1–11 DOI 10.1155/JIA/2006/65983

(4)semicompact[9] if for any bounded sequence{xn} ⊂Cwith lim_n→∞xn−Txn = 0, there exists a strongly convergent subsequence of{xn}.

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 ifCis a nonempty bounded closed convex subset of a uniformly convex Banach spaceE, then every asymptotically nonexpansive self-mappingTonChas 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 Banach spaceE satisfies Opial’s condition [5], then{Tⁿx}converges weakly to a fixed point ofT, pro- videdTis asymptotically regular atx, that is, lim_n→∞Tⁿx−Tⁿ⁺¹x =0. A Banach space Eis said to satisfyOpial’s condition[5] if whenever{xn}is a sequence inEwhich con- verges weakly tox, one has

lim inf

n→∞ x_n−x<lim inf

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 forNnonexpansive map- pings in a Hilbert space and proved the following weak convergence theorem.

Theorem1.2 (see [8]). LetH be a Hilbert space and letCbe a nonempty closed convex subset ofH. Let{T_i}^N_i=1beNnonexpansive self-mappings onCsuch thatF=_N

i=1F(T_i)=

∅. Letx0∈Cand let{αn}^∞_n=1be a sequence in(0, 1)such thatlim_n→∞αn=0. Then the sequence{xn}defined implicity by

x_n=α_nx_n−1+1−α_nT_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_nx_n−1+1−α_nT_n^ln_(mod⁺¹ _N)x_n, n≥1, (1.6) forNasymptotically quasi-nonexpansive self-mappings{T_j}^N_j=1on a nonempty bounded closed convex subsetCof a Banach spaceE, where{αn}is a sequence in (0, 1),x0is an initial point inC, andn=l_nN+n(modN). Moreover, he proved that the sequence{x_n} defined by his iteration process converges strongly to a common fixed point of{T_j}^N_j=1

under suitable conditions.

At the same time, in [10], Zhou and Chang introduced and studied the following im- plicit iteration process:

xn=αnxn−1+βnT_nⁿ(modN)xn+γnun, n≥1, (1.7) forNasymptotically nonexpansive self-mappings{Tj}^N_j=1on a nonempty closed convex subsetCof a Banach spaceE, where{α_n},{β_n},{γ_n}are three sequences in [0, 1],x0 is an initial point inC, and{un}is a bounded sequence inC. Moreover, they proved that the sequence{xn}defined by their iteration process converges weakly to a common fixed point of{T_j}^N_j=1under suitable conditions.

As indicated in [10], ifT1,T2,...,TN:C→CareNasymptotically nonexpansive map- pings, then there exists a sequence, calledcommon Lipschitz constants,{kn} ⊂[1,∞) with lim_n→∞k_n=1 such that for eachi=1, 2,...,N,

T_iⁿx−T_iⁿy≤knx−y, ∀x,y∈C,n≥1. (1.8) A similar situation occurs whenT1,T2,...,T_Nare asymptotically quasi-nonexpansive. By convention, we write Tn:=Tn(modN), for integern≥1, with the mod function taking values in the set{1, 2,...,N}. In other words, if n=l_nN+q for some unique integers l_n≥0 and 1≤q≤N, then we setT_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 onC.

Definition 1.3(basic setup). LetCbe a nonempty closed convex subset of a Banach space E, and let{T1,T2,...,TN}and{T1,T2,...,TN}be two families of asymptotically quasi- nonexpansive mappings fromCintoCwith common Lipschitz constants{k_n}and{k_n} such that^∞_n=1(kn−1)<+∞and^∞_n=1(kn−1)<+∞, respectively. Let{xn}be an itera- tive sequence inCgenerated from an arbitraryx0∈Cby the following three steps.

Auxiliary step. Withx_n−1(n≥1) established,y_nis computed implicitly by

yn=αnxn−1+βnT_n^lnyn+γnun. (1.9a) Predictor step. Withy_nobtained in the auxiliary step,z_nis computed implicitly by

z_n=α_ny_n+β_nT_n^lnz_n+γ_nu_n. (1.9b) Corrector step. Withznobtained in the predictor step,xnis computed explicitly by

x_n=α_ny_n+β_nT_n^lnz_n+γ_nu_n. (1.9c) Here, T_n:=T_n(modN) andT_n:=T_n(modN)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, {βn}^∞_n=1,{βn}^∞_n=1,{β_n}^∞_n=1,{γn}^∞_n=1,{γn}^∞_n=1,{γ_n}^∞_n=1are nine real sequences in [0, 1] such that

αn+βn+γn=1 (∀n≥1), ∞ n=1

γn<+∞,

αn+βn+γn=1 (∀n≥1), ∞ n=1

γn<+∞, αn+β_n+γ_n=1 (∀n≥1),

∞ n=1

γ_n<+∞, 0<βn,β_n≤c < K⁻¹ (∀n≥1), K=max sup

n≥1kn, sup

n≥1

kn

≥1.

(1.10)

Remark 1.4. Since 0<βn,β_n≤c <K⁻¹, it is clear that the mappingsy→αnxn−1+βnTn^lny+

γ_nu_nandz→α_ny_n+β_nTn^lnz+γ_nu_nare two contractions from the nonempty closed con- vex setCinto itself. Thus, by the Banach contraction principle, there exist the unique pointsyn,zn∈Csuch 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.

Lemma1.5. Let{bn},{bn},{bn}be three nonnegative real sequences with finite sums. Then _∞

n=1λ_n<+∞, whereλ_n=(1 +b_n)(1 +b_n)(1 +b_n)−1for each≥1.

Lemma1.6 (see [10]). Let{a_n},{λ_n},{μ_n}be three nonnegative real sequences such that _∞

n=1λn<+∞,^∞_n=1μn<+∞, and an+1≤

1 +λn

an+μn, ∀n≥1. (1.11)

Thenlim_n→∞anexists.

Lemma 1.7 (see [6]). Let E be a uniformly convex Banach space,{tn} ⊂[b,c]⊂(0, 1), and {x_n},{y_n} ⊂E. If lim_n→∞t_nx_n+ (1−t_n)y_n =d <+∞,lim sup_n→∞x_n ≤d, and lim sup_n→∞yn ≤d, thenlim_n→∞xn−yn =0.

Lemma1.8 (demiclosed principle [2]). LetEbe a uniformly convex Banach space, letCbe a nonempty closed convex subset ofE, and letT:C→Cbe an asymptotically nonexpansive mapping withF(T)= ∅. ThenI−Tis demiclosed at zero, that is, for any sequence{x_n} ⊂ C,

x_n−→q∈Cweakly

(I−T)xn−→0strongly ^=⇒ (I−T)q=0. (1.12) 2. Main results

Lemma2.1. LetCbe a nonempty closed convex subset of a Banach spaceE, and let{Ti}^N_i=1

and{T_j}^N_j=1be two finite families of asymptotically quasi-nonexpansive self-mappings onC such that^N_i=1F(T_i)∩N

j=1F(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 eachp∈_N

i=1F(T_i)∩N

j=1F(T_j), there hold

nlim→∞x_n−p=d, lim sup

n→∞

y_n−p≤d, lim sup

n→∞

z_n−p≤d. (2.1)

Proof. Since{un}^∞_n=1,{un}^∞_n=1,{un}^∞_n=1are three bounded sequences inC, for any given p∈_N

i=1F(T_i)∩N

j=1F(T_j), we have M:=max

sup

n≥1

un−p, sup

n≥1un−p, sup

n≥1

un−p

<+∞. (2.2)

Note that 1−β_nk_ln≥1−cK >0 and 1−β_nk_ln≥1−cK >0. Put L= 1

1−cK, bn=βn

kln−1, bn= 1−β_n 1−β_nk_ln

−1, bn= 1−βn

1−βnkln

−1.

(2.3) Then we have

0≤bn=βn

kln−1≤kln−1, 1 +bn≤K, 0≤bn=β_nk_ln−1

1−β_nk_ln

≤Lkln−1, 1 +bn≤L,

0≤bn=β_nk_ln−1 1−βnkln

≤Lkln−1, 1 +bn≤L.

(2.4)

Observe that

yn−p=αn

xn−1−p+βn T_n^lnyn−p+γn

un−p

≤α_nx_n−1−p+β_nk_lny_n−p+γ_nu_n−p.

(2.5) It follows

y_n−p≤ α_n 1−βnkln

x_n−1−p+ γn

1−βnkln

u_n−p

≤ 1−βn

1−βnkln

xn−1−p+LMγn

=

1 +bnxn−1−p+LMγn.

(2.6)

Similarly,

zn−p=αn

yn−p+β_nT_n^lnzn−p+γ_nun−p

≤α_ny_n−p+β_nk_lnz_n−p+γ_nu_n−p (2.7) Consequently,

z_n−p≤ αn

1−β_nkln

y_n−p+ γ_n 1−β_nkln

u_n−p

≤ 1−β_n 1−β_nkln

yn−p+LMγ_n

=

1 +b_ny_n−p+LMγ_n.

(2.8)

Therefore,

xn−p=αn

yn−p+βn

T_n^lnzn−p+γn

un−p

≤αnyn−p+βnklnzn−p+γnun−p

≤

1−β_ny_n−p+β_nk_ln

1 +b_ny_n−p+LMγ_n+γ_nM

≤ 1 +βn

kln−11 +bnyn−p+MKLγ_n+γn

≤

1 +b_n1 +b_ny_n−p+KLMγ_n+γ_n

≤

1 +b_n1 +b_n1 +b_nx_n−1−p+LMγ_n+KLMγ_n+γ_n

≤

1 +b_n1 +b_n1 +b_nx_n−1−p+KL²Mγ_n+KLMγ_n+γ_n

≤

1 +b_n1 +b_n1 +b_nx_n−1−p+KL²Mγ_n+γ_n+γ_n

=

1 +λ_nx_n−1−p+μ_n,

(2.9)

whereλn=(1 +bn)(1 +bn)(1 +bn)−1, andμn=KL²M[γn+γ_n+γn].

Since^∞_n=1(kln−1)<+∞and^∞_n=1(kln−1)<+∞, it follows from (2.4) that^∞_n=1bn<

+∞,^∞_n=1bn<+∞, and^∞_n=1bn<+∞. Hence, we derive^∞_n=1λn<+∞byLemma 1.5.

Note that^∞_n=1γn<+∞,^∞_n=1γ_n<+∞, and^∞_n=1γn<+∞. This provides^∞_n=1μn<+∞. ByLemma 1.6, lim_n→∞x_n−pexists. Let lim_n→∞x_n−p =d.

Since lim_n→∞bn=lim_n→∞γn=0, from (2.6), we obtain lim sup

n→∞

yn−p≤lim sup

n→∞

1 +bnxn−1−p+LMlim sup

n→∞ γn≤d. (2.10) Further, since lim_n→∞bn=lim_n→∞γ_n=0, from (2.8), we obtain

lim sup

n→∞

z_n−p≤lim sup

n→∞

1 +b_ny_n−p+LMlim sup

n→∞ γ_n≤d. (2.11) Theorem2.2. LetCbe a nonempty closed convex subset of a Banach spaceE. Let{Ti}^N_i=1

and{Tj}^N_j=1be two finite families of asymptotically quasi-nonexpansive self-mappings onC such thatF:=_N

i=1F(Ti)∩N

j=1F(Tj)= ∅. Let{xn}be the iterative sequence defined by (1.9a), (1.9b), and (1.9c). Then{xn}converges strongly to an element ofFif and only if

lim inf

n→∞ dx_n,F=0. (2.12)

Proof. The necessity is obvious. For the sufficiency, we assume lim infn→∞d(xn,F)=0.

Letpbe any given element inF. Then from (2.9), we obtain x_n−p≤

1 +λ_nx_n−1−p+μ_n, (2.13)

where^∞_n=1λn<+∞and^∞_n=1μn<+∞. Taking the infimum over allp∈F, we get dxn,F≤

1 +λn

dxn−1,F+μn. (2.14) Hence, lim_n→∞d(x_n,F) exists. Furthermore, we have lim_n→∞d(x_n,F)=0.

ByLemma 2.1, we know that lim_n→∞xn−pexists. Hence{xn}is bounded. Putδn= λnxn−1−p+μn. Then^∞_n=1δn<+∞, and (2.13) can be rewritten as

x_n−p≤x_n−1−p+δ_n. (2.15)

For arbitraryε >0, chooseN0 such thatd(x_N0,F)< ε/4 and^∞_j=N0δ_j< ε/4. Conse- quently, for alln,m≥N0, we have

xn−xm≤xn−p+xm−p

≤xN0−p+ n j=N0+1

δj+xN0−p+ m j=N0+1

δj

≤2xN0−p+ 2 ∞ j=N0

δj.

(2.16)

Taking the infimum over allp∈F, we obtain x_n−x_m≤2dx_N0,F+ 2

∞ j=N0

δ_j≤2ε 4 +2ε

4 ⁼ε. (2.17)

This shows that{xn}^∞_n=1is Cauchy. Let lim_n→∞xn=u. It is easy to verify thatFis closed.

Since lim_n→∞d(x_n,F)=0, we must have thatu∈F.

As a consequence ofLemma 2.1, the iterated sequence{x_n}is bounded. If the under- lying spaceEis reflexive, then we can expect that its weak cluster points provide common fixed points ofT1,T2,...,TN. This leads to the following theorem.

Theorem2.3. LetEbe a uniformly convex Banach space, letCbe a nonempty closed convex subset ofE, and let{Ti}^N_i=1(resp.,{Tj}^N_j=1) be a finite family of asymptotically nonexpan- sive (resp., asymptotically quasi-nonexpansive) self-mappings onCsuch that^N_j=1F(Tj)∩ _N

i=1F(Ti)= ∅. Supposelim_n→∞βn=0and{βn}^∞_n=1⊂[b,c]⊂(0,K⁻¹), whereK is as in (1.10). Then every weak cluster point of the bounded iterative sequence {xn} defined by (1.9a), (1.9b), and (1.9c) belongs to^N_i=1F(Ti).

Proof. Letp∈N

j=1F(Tj)∩_N

i=1F(Ti). ByLemma 2.1, we have

nlim→∞xn−p=d, lim sup

n→∞

yn−p≤d, lim sup

n→∞

zn−p≤d. (2.18)

Obviously,{xn},{yn}, and{zn}are bounded sequences inC.

Observe that

xn−p=1−βn

yn−p+γn

un−yn +βn

T_n^lnzn−p+γn

un−yn−→d, (2.19) asn→ ∞. Since lim_n→∞γ_n=0 and{u_n}is bounded, we have

lim sup

n→∞

yn−p+γn

un−yn≤lim sup

n→∞

yn−p+γnun−yn≤d, lim sup

n→∞

T_n^lnzn−p+γn

un−yn≤lim sup

n→∞

klnzn−p+γnun−yn≤d. (2.20)

It follows fromLemma 1.7that

nlim→∞T_n^lnz_n−y_n=0. (2.21) Thus,

nlim→∞z_n−y_n=lim

n→∞α_ny_n+β_nT_n^lnz_n+γ_nu_n−y_n

=lim

n→∞β_nT_n^lnzn−yn

+γ_nun−yn=0. (2.22) Similarly,

nlim→∞x_n−y_n=lim

n→∞α_ny_n+β_nT_n^lnz_n+γ_nu_n−y_n

=lim

n→∞βn

T_n^lnzn−yn +γn

un−yn=0. (2.23) Moreover,

yn−xn−1=αnxn−1+βnT_n^lnyn+γnun−xn−1

=βn T_n^lnyn−xn−1

+γn

un−xn−1

≤βn T_n^lnyn−xn−1+γnun−xn−1−→0, asn−→ ∞,

(2.24)

since lim_n→∞βn=lim_n→∞γn=0. As a result, we have

xn−xn−1≤xn−yn+yn−xn−1−→0, asn−→ ∞. (2.25) It forces

nlim→∞xn−xn+i=0, for eachi=1, 2,...,N. (2.26) On the other hand, we have

x_n−T_n^lnx_n≤x_n−y_n+y_n−T_n^lnz_n+T_n^lnz_n−T_n^lnx_n

≤x_n−y_n+y_n−T_n^lnz_n+k_lnz_n−x_n−→0, asn−→ ∞. (2.27)