© Hindawi Publishing Corp.
ISHIKAWA ITERATION PROCESS WITH ERRORS FOR
NONEXPANSIVE MAPPINGS IN UNIFORMLY
CONVEX BANACH SPACES
DENG LEI and LI SHENGHONG
(Received 6 May 1999)
Abstract.We shall consider the behaviour of Ishikawa iteration with errors in a uniformly convex Banach space. Then we generalize the two theorems of Tan and Xu without the restrictions thatCis bounded and limsupnsn<1.
Keywords and phrases. Uniformly convex Banach space, nonexpansive mapping, Ishikawa iteration process with errors.
2000 Mathematics Subject Classification. Primary 47H10; Secondary 40A05.
1. Introduction. LetCbe a closed convex subset of a Banach spaceXandT:C→C be nonexpansive (that is,T x−T y ≤ x−yfor allx,yinC). In 1974, Ishikawa [1] introduced a new iteration process as
xn+1=tnTsnT xn+1−snxn+1−tnxn, n=0,1,2,..., (1.1)
where{tn}and{sn}are sequences in[0,1]satisfying certain restrictions. The Mann iteration process is a special case of Ishikawa wheresn=0 for alln≥0 [4].
In 1993, Tan and Xu [7] obtained following result: letC be a bounded closed con-vex subset of a uniformly concon-vex Banach spaceX,T :C →C a nonexpansive map-ping. If for any initial guessx0inC,{xn}defined by (1.1), with the restrictions that ∞
n=0tn(1−tn)= ∞,∞n=0sn(1−tn) <∞,and limsupnsn<1, then limn→∞xn−T xn
=0.
LetCbe a closed convex subset of a Banach spaceXandT:C→Cbe nonexpansive. For any givenx0∈Cthe sequence{xn}defined by
xn+1=αnxn+βnT yn+γnun, yn=αˆnxn+βˆnT xn+γˆnvn, n≥0. (1.2)
is called the Ishikawa iteration sequence with errors. Here {un}and {vn}are two bounded sequences inC, and{αn},{βn},{γn},{αˆn},{βˆn}, and{γˆn}are six sequences in [0, 1] satisfying the conditions
αn+βn+γn=αˆn+βˆn+γˆn=1 for alln≥0. (1.3)
In particular, if ˆβn=γˆn=0 for alln≥0, the{xn}defined by
x0∈C, xn+1=αnxn+βnT xn+γnun, n≥0, (1.4)
Remark1.1. Note the Ishikawa and Mann iterative processes are all special cases of the Ishikawa and Mann iterative processes with errors.
It has been shown that ifC is a nonempty bounded closed convex subset of a uni-formly convex Banach spaceX, then every nonexpansive mappingT :C→C has a fixed point (see [2]). In this paper, we first extend [7, Lemma 2.3] to the Ishikawa it-eration sequence with errors (1.2), without the restrictions thatC is bounded and limsupnsn<1. Then we generalize [7, Theorems 3.1, 3.2, and 3.4].
2. Lemmas
Lemma2.1. Suppose that{an},{bn}, and{cn}are three sequences of nonnegative numbers such that
an+1≤1+bnan+cn for alln≥1. (2.1) If∞n=1bnandn∞=1cnconverges, thenlimn→∞anexists.
Proof. Forn,m≥1, we have
an+m+1≤1+bn+man+m+cn+m
≤ n+m
i=n
1+bian+ n+m
i=n n+m
j=i+1
1+bjci
≤ ··· ≤ n+m
i=n
1+bian+ n+m
j=n
1+bj n+m
i=n
ci.
(2.2)
It follows that
limsup
m→∞ am≤ ∞
i=n
1+bian+
∞
j=n
1+bj
∞
i=n
ci. (2.3)
Hence, limsupm→∞am≤liminfn→∞an. This completes the proof.
Lemma2.2. LetCbe a closed convex subset of a Banach spaceX, and letT:C→X
a nonexpansive mapping. Then for any initial guessx0inC,{xn}defined by (1.2), xn+1−p≤xn−p+γnun−p+βnγˆnvn−p, (2.4)
for alln≥1and for allp∈F(T ), whereF(T ), denotes the set of fixed points ofT.
Proof. For allp∈F(T ), we have
xn+1−p
≤αnxn−p+βnT yn−p+γnun−p
≤αnxn−p+βnαˆnxn−p+βˆnT xn−p+γˆnvn−p+γnun−p ≤xn−p+γnun−p+βnγˆnvn−p.
(2.5)
This completes the proof.
3. Main Results
Theorem3.1. LetCbe a closed convex subset of a uniformly convex Banach space
X,T :C →C a nonexpansive mapping with a fixed point. If for any initial guessx0 inC, {xn}defined by (1.2), with the restrictions that∞n=0αnβn= ∞, ∞n=0αnβˆn< ∞,∞n=0γn<∞and∞n=0γˆn<∞, thenlimn→∞xn−T xn =0.
Proof. By Lemma 2.2 andT with a fixed point, we set
M=sup
n≥0
T xn−un,xn−un,T yn−vn,yn−un,xn−vn (3.1)
It follows from (1.2) that
xn+1−T xn+1≤αnxn−T xn+1+βnT yn−T xn+1+γnT xn+1−un ≤αnxn−T xn+T xn−T xn+1
+βnαnyn−xn+βnyn−T yn+γnyn−un+γnM ≤αnxn−T xn+βnxn−T yn+αnβnβˆnxn−T xn
+β2
nαˆnxn−T yn+βˆnT xn−T yn+γnM+βnγnyn−un +αnγnxn−un+αnβnγˆnxn−vn+β2nγˆnT yn−vn ≤αnxn−T xn+αnβnxn−T yn+αnβnβˆnxn−T xn
+β2
nαˆnxn−T yn+β2nβˆnxn−yn+2γnM+βnγˆnM ≤αn+αnβnβˆn+β2nβˆ2nxn−T xn
+αnβn+β2nαˆnxn−T xn+T xn−T yn +2γnM+βnγˆnM+β2nβˆnγˆnM
≤αn+αnβnβˆn+β2nβˆ2n+αnβn+β2nαˆn+αnβnβˆn+β2nαˆnβˆn ×xn−T xn+2γnM+βnγˆnM+αnβn+β2nαˆn+β2nβˆnγˆnM ≤1+2αnβnβˆnxn−T xn+2γn+βnγˆnM.
(3.2)
Setting an =T xn−xn, bn = 2αnβnβˆn,andcn =2(γn+βnγˆn)M, it follows from
Lemma 2.1 that limn→∞anexists.
Letr (x0)=limn→∞xn−T xn. To reach the desired conclusion, it suffices to show
thatr (x0)is independent of the initial valuex0. We let{xn∗}denote iteration (1.2)
com-mencing atx∗
0. Sincexn+1−xn∗+1 ≤ xn−xn∗, we may assume that limn→∞xn−
x∗
n =d >0. Then, we obtain xn+1−x∗
n+1=αnxn−x∗n+βnT yn−T yn∗ ≤
1−2αnβnδ xn−x
∗
n−T yn−T yn∗
xn−xn∗
xn−x∗
n,
(3.3)
sinceT yn−T yn∗ ≤ xn−xn∗. Thus, n
i=0
2αiβiδ xi−x
∗
i −
T yi−T yi∗ xi−x∗
i
xi−x∗
It follows that
∞
n=0
αnβnδ xn−x
∗
n−T yn−T yn∗ xn−x∗
n <∞. (3.5)
By condition∞n=0αnβˆn<∞,we have∞n=0αnβnβˆn<∞. Thus,
∞
n=0
αnβn
δ xn−x∗n−
T yn−T yn∗ xn−x∗
n
+βˆn
<∞. (3.6)
It follows that
liminf
n→0 xn−x
∗
n−
T yn−T yn∗+βˆn=0 (3.7)
since∞n=0αnβn= ∞andδis the modulus of convexity of uniformly convex Banach
spaceX. Hence, there is a sequence{nk} ⊂ {n}such that
lim
k→∞xnk−x ∗
nk−
T ynk−T yn∗k=0, klim→∞βˆnk=0. (3.8)
On the other hand, we have
xn
k−T xnk−x∗nk−T xn∗k ≤xnk−T xnk
−x∗
nk−T xn∗k ≤xnk−xn∗k−
T ynk−T yn∗k+T xnk−T ynk+T xn∗k−T yn∗k ≤xnk−xn∗k−
T ynk−T yn∗k+βˆnkxnk−T xnk+βˆnkxn∗k−T x
∗
nk+2ˆγnM. (3.9)
Settingk→ ∞in (3.9), it follows from (3.8) that
lim
k→∞xnk−T xnk−x ∗
nk−T xn∗k=0. (3.10)
Thus,
lim
n→∞xn−T xn−x ∗
n−T xn∗=0, (3.11)
that is,r (x0)=r (x∗0).This completes the proof.
Recall that a Banach spaceXis said to satisfy Opial’s condition [5] if the condition xn→x0weakly implies
limsup
n→∞
xn−x0<limsup n→∞
xn−y for ally≠x0. (3.12)
Theorem3.2. LetCbe a closed convex subset of a uniformly convex Banach space
X which satisfies Opial’s condition,T :C →C a nonexpansive mapping with a fixed point, and{xn}as in Theorem 3.1. Then{xn}converges weakly to a fixed point ofT.
Proof. Let ωw(xn) be the weak limit ω-set of {xn}. By Lemma 2.3 and
Theorem 3.1,ωw(xn)is contained inF(T ), the fixed point set ofT.
Remark3.3. Theorem 3.2 generalizes [7, Theorem 3.1].
Recall that a mappingT:C→Cwith a nonempty fixed points setF(T )inCwill be said to satisfy conditionA[6] if there is a nondecreasing functionf:[0,∞)→[0,∞) with f (0)=0, f (r ) >0 forr ∈(0,∞), such that x−T x ≥f (d(x,F(T )))for all x∈C, whered(x,F(T ))=inf{x−z:z∈F(T )}.
The following two theorems generalize Theorem 3.2 and [7, Theorem 3.4] respec-tively. Since a similar proof is in [7], we omit their proof here.
Theorem3.4. LetX,C,T, and{xn}be as in Theorem 3.1. If the range ofC under
T is contained in a compact subset ofX, then{xn}converges strongly to a fixed point ofT.
Theorem3.5. LetX,C,T and{xn}be as in Theorem 3.1. IfT with a nonempty fixed points setF(T )satisfies conditionA, then{xn}converges strongly to a fixed point ofT.
Acknowledgement. Research partially supported by NNSF(79790130) and ZJPNSF(198013).
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Deng Lei: Department of Mathematics, Southwest China Normal University, Beibei, Chongqing400715, China
Li Shenghong: Department of Mathematics, Zhejiang University Hangzhou,310027, China
