ERRORS FOR MAPPINGS WITH UNBOUNDED
DOMAIN IN BANACH SPACES
HAFIZ FUKHAR-UD-DIN AND ABDUL RAHIM KHAN
Received 13 March 2005
We prove that an implicit iterative process with errors converges weakly and strongly to a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings on unbounded sets in a uniformly convex Banach space. Our results generalize and improve upon, among others, the corresponding recent results of Sun (2003) in the following two different directions: (i) domain of the mappings is unbounded, (ii) the iterative sequence contains an error term.
1. Introduction
In 1967, Browder [1] studied the iterative construction for fixed points of nonexpansive mappings on closed and convex subsets of a Hilbert space (see also [2]). The Ishikawa iteration process in the context of nonexpansive mappings on bounded closed convex subsets of a Banach space has been considered by a number of authors (see, e.g., Tan and Xu [16] and the references therein).
Ghosh and Debnath [4], in 1997, established a necessary and sufficient condition (in
Theorem 3.1) for convergence of the Ishikawa iterates of a quasi-nonexpansive mapping on a closed convex subsetCof a Banach space. Qihou [11] has extended this result (in Theorem 1) for Ishikawa iterates with errors, in the sense of Liu [10], of an asymptotically quasi-nonexpansive mapping onC.
Fixed point results for asymptotically nonexpansive mappings have been obtained by Kirk and Ray [9] on unbounded sets in a Banach space. Recently, Hussain and Khan [7] have constructed (inTheorem 3.8) approximating sequences to fixed points of a class of mappings, containing nonexpansive mappings as a subclass, on closed convex un-bounded subsets of a Hilbert space (see [13,14] as well).
Sun [15] has recently extended an implicit iteration process for a finite family of nonexpansive mappings due to Xu and Ori [20] to the case of asymptotically quasi-nonexpansive mappings.
The aim of this paper is to prove the weak convergence and strong convergence of an implicit iterative process with errors, in the sense of Xu [19], for a finite family of asymptotically quasi-nonexpansive mappings on a closed convex unbounded set in a real
Copyright©2005 Hindawi Publishing Corporation
uniformly convex Banach space. Our results unify, improve, and generalize the corre-sponding results of Sun [15], Wittmann [17], and Xu and Ori [20].
2. Preliminaries and lemmas
In 1967, Diaz and Metcalf [2] introduced the notion of a quasi-nonexpansive mapping and Goebel and Kirk [5] introduced, in 1972, the concept of an asymptotically nonex-pansive mapping.
LetC be a nonempty subset of a normed spaceEand T:C→C a given mapping. We assume that the set of fixed points ofT,F(T)= {x∈C:T(x)=x} is nonempty. The mappingT is said to be (1) nonexpansive providedTx−T y ≤ x−yfor all x,y∈C; (2) quasi-nonexpansive if Tx−p ≤ x−p for all x∈C, p∈F(T); (3) asymptotically nonexpansive if there is a sequence{un} ⊂[0,∞) with limn→∞un=0 such thatTnx−Tny ≤(1 +un)x−yfor allx,y∈Cand for alln≥1; (4) asymptotically quasi-nonexpansive if there is a sequence{un} ⊂[0,∞) with limn→∞un=0 such that
Tnx−p ≤(1 +un)x−pfor allx∈C,p∈F(T) and for alln≥1, and (5) uniformly L-Lipschitzian if for someL >0,Tnx−Tny ≤Lx−yholds for allx,y∈Cand for alln≥1.
Denote the indexing set{1, 2, 3,. . . N}byI. We say that a finite family{Ti:i∈I}ofN self-mappings onCis
(i) asymptotically nonexpansive if it satisfies Tnx
i−Tnyi≤
1 +uinx−y ∀x,y∈C,∀n≥1, (2.1)
(ii) asymptotically quasi-nonexpansive provided Tnx
i−qi≤
1 +uinx−qi ∀x∈C,qi∈F
Ti
,∀n≥1, (2.2)
where{uin}is the sequence of reals as in (3) andi∈I.
It is obvious, in view of these definitions, that a nonexpansive mapping is asymptot-ically nonexpansive, nonexpansive mapping with the nonempty fixed point set is quasi-nonexpansive, and an asymptotically nonexpansive mapping is uniformlyL−Lipschitzian withL=sup{1 +un:n≥1}. However, the converses of these claims are not true in gen-eral.
The Mann and Ishikawa iteration processes have been used by a number of authors to approximate the fixed points of nonexpansive mappings, asymptotically nonexpansive mappings, and quasi-nonexpansive mappings on Banach spaces (see, e.g., [4,8,10,12,
14,19]).
For a nonempty subsetCof a normed spaceEandT:C→E, Liu [10] introduced in 1995 the concept of Ishikawa iteration process with errors by the iterative sequence{xn} defined as follows:
x1=x∈C,
xn+1=1−αnxn+αnTnyn+un, yn=
1−βn
xn+βnTnxn+vn, n≥1,
where {αn}, {βn} are real sequences in [0, 1] satisfying appropriate conditions and ∞
n=1un<∞, ∞
n=1vn<∞. Ifβn=0,vn=0 for alln≥1, then this process becomes the Mann iteration process with errors.
The above definitions of Liu depend on the convergence of the error termsunandvn. The occurrence of errors is random and so the conditions imposed on the error terms are unreasonable. Moreover, there is no assurance that the iterates defined by Liu will fall within the domain under consideration.
In 1998, Xu [19] gave the following new definitions in place of these noncompatible ones.
For a nonempty convex subsetCof a normed spaceEandT:C→C, the Ishikawa iteration process with errors is the iterative sequence{xn}defined by
x1=x∈C,
xn+1=αnxn+βnTnyn+γnun, yn=anxn+βnTnxn+γnvn, n≥1,
(2.4)
with{un},{vn}bounded sequences inCand{αn},{βn},{γn},{αn},{βn},{γn}are se-quences in [0, 1] such thatαn+βn+γn=1=αn+βn+γnfor alln≥1.
It reduces to the Mann iteration process with errors whenβn=0=γnfor alln≥1. Clearly, the normal Ishikawa and Mann iteration processes are special cases of the Ishikawa iteration process with errors.
Huang [6] has computed fixed points of asymptotically nonexpansive mappings while Fukhar and Khan [3] have approximated common fixed points of two asymptotically nonexpansive mappings, using iteration process with errors in the sense of Liu [10] (see also [12]).
Let{Ti:i∈I}be a finite family of asymptotically quasi-nonexpansive self-mappings on a convex subset Cof a normed spaceE. The implicit iterative process of Sun [15] with an error term, in the sense of Xu [19], and with an initial valuex0∈C, is defined as follows:
x1=α1x0+β1T1x1+γ1u1, x2=α2x1+β2T2x2+γ2u2, xN=αNxN−1+βNTNxN+γNuN, xN+1=αN+1xN+βN+1T12xN+1+γN+1uN+1,
x2N=α2Nx2N−1+β2NTN2x2N+γ2Nu2N, x2N+1=α2N+1x2N+β2N+1T13x2N+1+γ2N+1u2N+1,
(2.5)
where{un}is a bounded sequence inCand{αn},{βn},{γn}are sequences in [0, 1] such thatαn+βn+γn=1.
The above table in compact form is
xn=αnxn−1+βnTikxn+γnun (2.6)
In the sequel, we assume that the sequence{xn}, defined by the implicit iteration pro-cess with errors (2.6), exists and the setF=N
i=1F(Ti),i∈I, is nonempty.
The distance between a pointxand a setCand closed ball with centre zero and radius rinEare, respectively, defined by
d(x,C)=inf
y∈Cx−y, Br(0)=
x∈E:x ≤r. (2.7)
Definition 2.1(see [15]). LetCbe a closed subset of a normed spaceEand letT:C→Cbe a mapping. ThenTis said to be semicompact if for any bounded sequence{xn}inCwith
xn−Txn →0 asn→ ∞, there is a subsequence{xni}of{xn}such thatxni→x∗∈Cas ni→ ∞.
Definition 2.2(see [8]). A normed spaceEis said to satisfy the Opial condition if for any sequence{xn}in E,xnconverging weakly tox implies that lim supn→∞xn−x< lim supn→∞xn−yfor ally∈Ewithy=x.
Lemma 2.3 (see [3]). Let {rn},{sn},{tn}be three nonnegative sequences satisfying the following condition:
rn+1≤
1 +sn
rn+tn ∀n≥1. (2.8)
If∞n=1sn<∞,n∞=1tn<∞, thenlimn→∞rnexists.
Lemma2.4 (see [18]). Letp >1andr >0be two fixed real numbers. Then a Banach space Eis uniformly convex if and only if there is a continuous strictly increasing convex function g: [0,∞)→[0,∞)withg(0)=0and
λx+ (1−λ)yp
≤λxp+ (1−λ)yp−wp(λ)gx−y (2.9)
for allx,y∈Br(0)where0≤λ≤1andwp(λ)=λp(1−λ) +λ(1−λ)p.
Lemma2.5 (see [8]). LetEbe a uniformly convex Banach space satisfying the Opial condi-tion and letCbe a nonempty closed convex subset ofE. LetTbe an asymptotically nonex-pansive mapping ofCinto itself. ThenI−Tis demiclosed at0(i.e., for any sequence{xn}in C, the conditionsxnconverge weakly tox0andxn−Txn→0implyx0−Tx0=0).
3. An implicit iterative process with errors
We begin with a necessary and sufficient condition for convergence of{xn}generated by the implicit iteration process with errors in (2.6) to a point ofF; for this we follow the arguments of Sun (see [15, Theorem 3.1]) and useLemma 2.3to prove the following general result.
Theorem3.1. Let Cbe a nonempty closed convex subset of a Banach spaceE. Let {Ti: i∈I}beN asymptotically quasi-nonexpansive mappings ofC, with∞n=1uin<∞for all i∈I. Suppose thatx0∈C,{un}is a bounded sequence inC, and{αn},{βn},{γn}are real sequences in[0, 1]such thatαn+βn+γn=1,{αn} ⊂(s, 1−s)for somes∈(0, 1),∞n=1γn<
Proof. The necessity is obvious. We prove the sufficiency of the conditions. Let p∈F. Usingxn=αnxn−1+βnTikxn+γnun,wheren=(k−1)N+i, it follows that
xn−p=αnxn−1+βnTk
ixn+γnun−p
=αn
xn−1−p
+βn
Tikxn−p
+γn
un−p
≤αnxn−1−p+βn1 +uikxn−p+γnun−p
≤αnxn−1−p+
1−αn
1 +uikxn−p+γnun−p
≤αnxn−1−p+
1−αn+uikxn−p+γnun−p.
(3.1)
This implies that
αnxn−p≤αnxn−1−p+uikxn−p+γnun−p. (3.2)
Now using the information that 0< s < αn<1−s <1, we have that
s−u ik
s xn−p≤xn−1−p+ γn
s un−p. (3.3)
That is,
xn−p≤1 + uik s−uik
xn−1−p+ γn s−uik
un−p. (3.4)
Since∞k=1uik<∞for alli∈I, therefore limk→∞uik=0 for alli∈I. This gives that there exists a natural numbern0(ask >(n0/N) + 1) such thats−uik>0 anduik< s/2 for alln > n0.
Now the above inequality, withM=supn≥1un−p, becomes
xn−p≤1 +2
suik xn−1−p+ 2M
s γn. (3.5)
This further implies that
dxn,F≤1 +2 suik d
xn−1,F
+2M
s γn. (3.6)
Applying Lemma 2.3 to the inequalities (3.5) and (3.6), we conclude that both limn→∞xn−p and limn→∞d(xn,F) exist. Since lim infn→∞d(xn,F)=0, therefore
x >0, 1 +x≤ex, and hence by (3.4), we have xn+m−p
≤exp N
i=1
∞
k=1 u
ik s−uik
xn−p+2 sMexp
N
i=1
∞
k=1 u
ik s−uik
∞
n=1 γn
< Qxn−p+R ∀m≥1,n≥1,
(3.7)
where
Q=exp N
i=1
∞
k=1
uik s−uik
+ 1,
R=2 sMexp
N
i=1
∞
k=1
uik s−uik
∞
n=1 γn.
(3.8)
Let>0. As limn→∞xn−pexists, therefore for> R, there exists a natural number n1 such thatxn−p<(−R)/(1 +Q) for alln≥n1. Now it follows from the above inequality that
xn+m−xn≤xn+m−p+xn−p
< Qxn−p+R+xn−p
=(1 +Q)xn−p+R < ∀m≥1,n≥n1.
(3.9)
This proves that {xn} is a Cauchy sequence in E and so it must converge. Let limn→∞xn=q(say). As limn→∞d(xn,F)=0, therefored(q,F)=0. This implies that there
existsp∈Fsuch thatq−p =0. That is,q=p. Hence,qis a common fixed point ofTi
for alli∈I. This completes the proof of the theorem.
The above theorem can be restated as follows.
Corollary 3.2. Suppose that all the conditions of Theorem 3.1hold. Then the implicit iterative sequence{xn}with errors converges to a point p∈F if and only if{xn}has an infinite subsequence{xnj}with limitp.
We prove a lemma which plays an important role in establishing weak and strong convergence of the implicit iteration process with errors in a uniformly convex Banach space.
Lemma3.3. LetCbe a closed convex subset of a real uniformly convex Banach spaceE. Let {Ti:i∈I}beNuniformlyL−Lipschitzian asymptotically quasi-nonexpansive mappings of Cwith∞n=1uin<∞for alli∈I. Suppose thatx0∈C,{un}is a sequence inC, and{αn},
{βn},{γn}are real sequences in[0, 1]such thatαn+βn+γn=1,{αn} ⊂(s, 1−s)for some s∈(0, 1),∞n=1γn<∞. Thenlimn→∞xn−Tlxn =0for alll∈I.
can obtain a closed ballBr(0)⊃ {xn−q,Tikxn−q}for somer >0. ByLemma 2.4and the scheme (2.6), we get
xn−q2
=αnxn−1−q
+1−αnTikxn−q+γnun−Tikxn2
≤αnxn−1−q+1−αnTikxn−q2+γnM for someM >0
≤αnxn−1−q2+
1−αnTikxn−q2
−W2αn
gσn
+γnM
≤αnxn−1−q 2
+1−αn
1 +uik 2
xn−q 2
−W2αn
gσn
+γnM
≤αnxn−1−q 2
+1−αn+vikxn−q 2
−W2αngσn+γnM,
(3.10)
wherevik=2uik+u2ik. Hence, ∞
k=1vik<∞for alli∈I. Thus from the above inequality ands < αn<1−s, we have that
xn−q2
≤xn−1−q2+vik
s xn−q 2
−1−αngσn+γn
s M. (3.11)
Therefore, as inTheorem 3.1, it can be shown that limn→∞xn−q2=dexists. From (3.11), it follows that
1−αngσn≤xn−1−q 2
−xn−q2+vik
s xn−q 2
+γn s M ≤xn−1−q
2
−xn−q 2
+vikQ + γn
s M for someQ >0.
(3.12)
From (1−αn)≥s, we have
sgσn≤xn−1−q2−xn−q2+vikQ +γn
s M. (3.13)
Letmbe a positive integer such thatm≥n. Then
m
n=1 gσn
≤ 1
s2x0−q 2
+Q s2
m
k=1 vik+Ms3
m
k=1
γn. (3.14)
Whenm→ ∞in (3.14), we have that limn→∞g(σn)=0. Sinceg is strictly increasing
and continuous withg(0)=0, it follows that limn→∞σn=0. Hence, xn−xn−1≤βnTk
nxn−xn−1+γnun−xn−1
≤1−αnTnkxn−xn−1+γnR for someR >0
≤(1−s)Tnkxn−xn−1+γnR for someR >0,
which proves that limn→∞xn−xn−1 =0. That is, limn→∞xn−xn+l =0 for alll <2N. Forn > N, we have
xn−1−Tnxn≤xn−1−Tk
nxn+Tnkxn−Txn≤σn+LTnk−1xn−xn
≤σn+L
Tnk−1xn−Tnk−−1Nxn−N+Tnk−−N1xn−N−x(n−N)−1 +Lx(n−N)−1−xn.
(3.16)
Byn≡(n−N)(mod,N), we getTn=Tn−N. Now the above inequality becomes
xn−1−Txn≤σn+L2x
n−xn−N+Lσn−N+Lx(n−N)−1−xn, (3.17)
which yields that limn→∞xn−1−Txn =0. Since
xn−Txn≤xn−xn−1+xn−1−Tnxn, (3.18)
so we have that
lim
n→∞xn−Txn=0. (3.19)
Hence, for alll∈I,
xn−Tn+lxn≤xn−xn+l+xn+l−Tn+lxn+l+Tn+lxn+l−Tn+lxn
≤(1 +L)xn−xn+l+xn+l−Tn+lxn+l,
(3.20)
which implies that
lim
n→∞xn−Tn+lxn=0 ∀l∈I. (3.21)
Thus
lim
n→∞xn−Tlxn=0 ∀l∈I. (3.22)
Now we are in a position to prove our convergence theorems.
Theorem3.4. LetCbe a closed convex subset of a real uniformly convex Banach spaceE satisfying the Opial condition. Let{Ti:i∈I}beNuniformlyL−Lipschitzian asymptotically quasi-nonexpansive mappings ofCwith∞n=1uin<∞for alli∈I. Suppose thatx0∈C,
{un}is a sequence inC, and{αn},{βn},{γn}are real sequences in[0, 1]such thatαn+ βn+γn=1,{αn} ⊂(s, 1−s)for somes∈(0, 1),∞n=1γn<∞. If for every memberT in
{Ti:i∈I},I−T is demiclosed at 0, then the iterative sequence {xn}generated by (2.6) converges weakly to a point inF.
limn→∞xn−Tlxn =0 for alll∈I. Also,I−Tl is demiclosed at 0 for alll∈I. There-fore, we obtain thatTlu=ufor alll∈I. Similarly, we can prove thatTlv=vfor alll∈I. That is,u,v∈F. For the uniqueness, assume thatuandvare distinct. Then by the Opial condition,
lim
n→∞xn−u=nlimi→∞ xn
i−u
< lim ni→∞
xn i−v
=lim
n→∞xn−v
< lim nj→∞
xn j−u
=lim
n→∞xn−u,
(3.23)
a contradiction. Hence, the proof.
With the help ofLemma 2.5, the following corollary is an immediate consequence of the above theorem; this result includes, as a special case, Theorem 2 of Xu and Ori [20]. Corollary3.5. LetCbe a closed convex subset of a real uniformly convex Banach spaceE satisfying the Opial condition. Let{Ti:i∈I}beNuniformly asymptotically nonexpansive mappings ofCwith∞n=1uin<∞for alli∈I. Suppose thatx0∈C,{un}is a sequence inC and{αn},{βn},{γn}are real sequences in[0, 1]such thatαn+βn+γn=1,{αn} ⊂(s, 1− s)for some s∈(0, 1),∞n=1γn<∞. Then the iterative sequence{xn} generated by (2.6) converges weakly to a point inF.
Next, we prove strong convergence theorems.
Theorem3.6. LetCbe a closed convex subset of a real uniformly convex Banach spaceE. Let {Ti:i∈I}beNuniformlyL−Lipschitzian asymptotically quasi-nonexpansive mappings of Cwith∞n=1uin<∞for alli∈I. Suppose thatx0∈C,{un}is a sequence inC, and{αn},
{βn},{γn}are real sequences in[0, 1]such thatαn+βn+γn=1,{αn} ⊂(s, 1−s)for some s∈(0, 1),∞n=1γn<∞, If at least one memberT in{Ti:i∈I}is semicompact, then the implicitly defined iterative sequence{xn}in (2.6) converges strongly to a point inF.
Proof. ByLemma 3.3, it follows that
lim
n→∞xn−Tlxn=0 ∀l∈I. (3.24)
Without any loss of generality, assume thatT1is semicompact. Therefore, by (3.24), it follows that limn→∞xn−T1xn =0. SinceT1is semicompact, therefore there exists a subsequence{xnj}of{xn}such thatxnj→x∗∈C. Now consider
x∗−T
lx∗=nlim j→∞
xn
j−Tlxnj=0 ∀l∈I. (3.25)
This proves thatx∗∈F. As limn→∞xn−qexists for allq∈F, thereforexnconverges
Remark 3.7. If we takeγn=0, for alln≥1, then the above theorem becomes Theorem 3.3 due to Sun [15] without the boundedness ofCwhich in turn generalizes Theorem 2 by Wittmann [17] from Hilbert spaces to uniformly convex Banach spaces.
An asymptotically nonexpansive mapping is both uniformly L-Lipschitzian and asymptotically quasi-nonexpansive. Hence, the following generalization of [15, Theorem 3.4] is an immediate consequence ofTheorem 3.6.
Theorem3.8. LetCbe a closed convex subset of a real uniformly convex Banach spaceE. Let{Ti:i∈I}beNasymptotically nonexpansive mappings ofCwith
∞
n=1uin<∞for all i∈I. Suppose thatx0∈C,{un}is a sequence inC, and{αn},{βn},{γn}are real sequences in[0, 1] such thatαn+βn+γn=1,{αn} ⊂(s, 1−s)for some s∈(0, 1),∞n=1γn<∞. If at least one memberT in{Ti:i∈I}is semicompact, then the implicitly defined iterative sequence{xn}generated by (2.6) strongly converges to a point inF.
Definition 3.9(condition (∗)). The family{Ti:i∈I}ofN-self-mappings on a subsetC of a normed spaceEsatisfies condition (∗) if there exists a nondecreasing function f : [0,∞)→[0,∞) with f(0)=0, f(r)>0 for allr∈(0,∞) such that 1/nNi=1x−Tix ≥
f(d(x,F)) for allx∈Cwhered(x,F)=inf{x−p:p∈F}.
Note that condition (∗) defined above reduces to the [16, condition (A)] if we choose Ti=T(say) for alli∈I.
Finally, an application of the convergence criteria established inTheorem 3.1is given below to obtain yet another strong convergence result in our setting.
Theorem3.10. LetCbe a closed convex subset of a real uniformly convex Banach spaceE. Let{Ti:i∈I}beNuniformlyL−Lipschitzian asymptotically quasi-nonexpansive mappings ofCwith∞n=1uin<∞for alli∈Iand satisfy the condition (∗). Suppose thatx0∈C,{un} is a sequence inCand{αn},{βn},{γn}are real sequences in[0, 1]such thatαn+βn+γn=1,
{αn} ⊂(s, 1−s)for somes∈(0, 1), ∞
n=1γn<∞. Then the iterative sequence{xn}generated by (2.6) strongly converges to a point inF.
Proof. As in the proof of Theorems3.6, (3.24) holds. Taking lim inf on both sides of condition (∗) and using (3.24), we have that lim infn→∞f(d(xn,F))=0. Since f is a nondecreasing function with f(0)=0 and f(r)>0 for allr∈(0,∞), it follows that lim infn→∞d(xn,F)=0. Now byTheorem 3.1,xn→p∈F.
Acknowledgment
The second author gratefully acknowledges the support provided by King Fahd University of Petroleum and Minerals during this research.
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Hafiz Fukhar-Ud-Din: Department of Mathematics, Islamia University, Bahawalpur-63100, Pak-istan
Current address: Department of Mathematical and Computing Science, Tokyo Institute of Tech-nology, Tokyo 152-8552, Japan
E-mail address:[email protected]
Abdul Rahim Khan: Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia