CONVEX BANACH SPACES

© 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 limsup_nsn<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=tnT

snT xn+ 1−sn

xn +

1−tn

xn, 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 convex 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 limsup_nsn<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},{ˆαn},{βˆn}, and{ˆγn}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) is called the Mann iteration sequence with errors.

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+bn

an+cn for alln≥1. (2.1)

If_∞

n=1bnand_∞

n=1cnconverges, thenlimn→∞anexists.

Proof. Forn,m≥1, we have an+m+1≤

1+bn+m

an+m+cn+m

≤

n+m

i=n

1+bi an+

n+m

i=n n+m

j=i+1

1+bj ci

≤ ··· ≤

n+m

i=n

1+bi an+

n+m

j=n

1+bj^n+m

i=n

ci.

(2.2)

It follows that

limsup

m→∞ am≤ ∞ i=n

1+bi an+

∞ j=n

1+bj^∞

i=n

ci. (2.3)

Hence, limsup_m→∞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+βn ˆ

αnxn−p+βˆnT xn−p+γˆnvn−p+γnun−p

≤xn−p+γnun−p+βnγˆnvn−p.

(2.5)

This completes the proof.

Lemma2.3[3]. LetCbe a closed convex subset of a uniformly convex Banach spaceX, and letT:C→Xa nonexpansive mapping. Then the mappingI−Tis demiclosed onC.

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 +βn

αnyn−xn+βnyn−T yn+γnyn−un+γnM

≤αnxn−T xn+βnxn−T yn+αnβnβˆnxn−T xn +β²_n

ˆ

αnxn−T yn+βˆnT xn−T yn+γnM+βnγnyn−un +αnγnxn−un+αnβnγˆnxn−vn+β²_nγˆnT yn−vn

≤αnxn−T xn+αnβnxn−T yn+αnβnβˆnxn−T xn +β²_nαˆnxn−T yn+β²_nβˆnxn−yn+2γnM+βnγˆnM

≤

αn+αnβnβˆn+β²nβˆ²nxn−T xn +

αnβn+β²_nαˆnxn−T xn+T xn−T yn +2γnM+βnγˆnM+β²_nβˆnγˆnM

≤

αn+αnβnβˆn+β²_nβˆ²_n+αnβn+β²_nαˆn+αnβnβˆn+β²_nαˆnβˆn

×xn−T xn+2γnM+βnγˆnM+

αnβn+β²_nαˆn+β²_nβˆn ˆ γnM

≤

1+2αnβnβˆnxn−T xn+2

γn+βnγˆn M.

(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{x_n^∗}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=αn

xn−x^∗n +βn

T yn−T yn^∗

≤

1−2αnβnδ xn−x_n^∗−

T yn−T y_n^∗

xn−xn^∗ xn−x^∗_n, (3.3) sinceT yn−T y_n^∗ ≤ xn−x_n^∗. Thus,

n i=0

2αiβiδ xi−x_i^∗−

T yi−T y_i^∗

xi−x_i^∗ xi−x_i^∗≤x0−x^∗₀−xn+1−x^∗_n+1. (3.4)

It follows that

∞ n=0

αnβnδ xn−x^∗_n−

T yn−T y_n^∗

xn−xn^∗ <∞. (3.5)

By condition_∞

n=0αnβˆn<∞,we have_∞

n=0αnβnβˆn<∞. Thus, ∞

n=0

αnβn

δ xn−x^∗_n−

T yn−T y_n^∗ xn−xn^∗

+βˆn

<∞. (3.6) It follows that

liminf

n→0 xn−x_n^∗−

T yn−T y_n^∗+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

k→∞limxnk−x^∗_nk−

T ynk−T y_n^∗_k=0, lim

k→∞βˆnk=0. (3.8) On the other hand, we have

xn_k−T xn_k−x^∗_nk−T x_n^∗_k

≤xn_k−T xn_k

−

x^∗n_k−T xn^∗_k

≤xnk−x_n^∗_k−

T ynk−T y_n^∗_k+T xnk−T ynk+T x_n^∗_k−T y_n^∗_k

≤xnk−x_n^∗_k−

T ynk−T y_n^∗_k+βˆnkxnk−T xnk+βˆnkx_n^∗_k−T x^∗_nk+2ˆγnM.

(3.9) Settingk→ ∞in (3.9), it follows from (3.8) that

k→∞limxnk−T xnk−x_n^∗_k−T x_n^∗_k=0. (3.10) Thus,

n→∞limxn−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.

The remainder of the proof is similar to that of [7, Theorem 3.1], so the details are omitted.

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

E-mail address:[email protected]