Iterative approximation of fixed point for Φ hemicontractive mapping without Lipschitz assumption

ITERATIVE APPROXIMATION OF FIXED POINT

FOR

Φ

-HEMICONTRACTIVE MAPPING

WITHOUT LIPSCHITZ ASSUMPTION

XUE ZHIQUN

Received 30 December 2004 and in revised form 7 July 2005

LetEbe an arbitrary real Banach space and letK be a nonempty closed convex subset of Esuch that K+K⊂K. Assume that T:K →K is a uniformly continuous and Φ -hemicontractive mapping. It is shown that the Ishikawa iterative sequence with errors converges strongly to the unique fixed point ofT.

1. Introduction

LetEbe a real Banach space and letE∗be the dual space onE. The normalized duality mappingJ:E→2E∗is defined by

Jx=f ∈E∗_{:x,f = x · f = f}2 _(1.1)

for allx∈E, where·,·denotes the generalized duality pairing. It is well known that ifE is a uniformly smooth Banach space, thenJis single valued and such thatJ(−x)= −J(x),

J(tx)=tJ(x) for allx∈E and t≥0; and J is uniformly continuous on any bounded subset ofE. In the sequel, we shall denote single-valued normalized duality mapping byj by means of the normalized duality mappingJ. In the following, we give some concepts.

Definition 1.1. A mappingT with domainD(T) and rangeR(T) is said to be strongly pseudocontractive if for anyx,y∈D(T), there existsj(x−y)∈J(x−y) such that

Tx−T y,j(x−y)≤kx−y2 _(1.2)

for some constantk∈(0, 1). The mappingT is calledΦ-strongly pseudocontractive if there exists a strictly increasing functionΦ: [0,∞)→[0,∞) withΦ(0)=0 such that the inequality

Tx−T y,j(x−y)≤ x−y2_{−Φx−yx−y (1.3)}

holds for allx,y∈D(T). Let F(T)= {x∈D(T) :Tx=x}. A mapping T is called Φ -hemicontractive if there exists a strictly increasing function Φ: [0,∞)→[0,∞) with

Copyright©2005 Hindawi Publishing Corporation

Φ(0)=0 such that the inequality

Tx−Tq,j(x−q)≤ x−q2_{−Φx−qx−q (1.4)}

holds for allx∈D(T) andq∈F(T).

It is shown in [5] that the class of strongly pseudocontractive mapping is a proper sub-class ofΦ-strongly pseudocontractive mapping. Furthermore, the example in [2] shows that the class ofΦ-strongly pseudocontractive mapping with the nonempty fixed point set is a proper subclass ofΦ-hemicontractive mapping. The classes of mappings intro-duced above have been studied by several authors. In [1], Chidume proved that ifE=Lp

(orlp),p≥2,Kis a nonempty closed convex and bounded subset ofE, andT:K→Kis a Lipschitz strongly pseudocontractive mapping, then Mann iteration process converges strongly to the unique fixed point of T. In [4], Deng extended the above result to the Ishikawa iteration process. After Tan and Xu [7] extended the results of both Chidume [1] and Deng [4] toq-uniformly smooth Banach spaces (1< q <2), Chidume and Osilike [3] extended to realq-uniformly smooth Banach spaces (1< q <∞). Recently, these results above have been extended from Lipschitz strongly pseudocontractive mapping to Lips-chitzΦ-strongly pseudocontractive mapping in realq-uniformly smooth Banach spaces (1< q <∞). More recently, Osilike [6] proved that ifKis a nonempty closed convex sub-set of arbitrary real Banach spaceEandT:K→K is a LipschitzianΦ-hemicontractive mapping, then Ishikawa iteration sequence{xn}∞n=1converges strongly to the unique fixed point ofT. It is our purpose in this paper to examine the strong convergence theorems of the Ishikawa iterative sequences with errors forΦ-hemicontractive mapping in arbitrary real Banach spaces.

Lemma 1.2. Let Ebe a real Banach space, then for all x,y∈E, there exists j(x+y)∈

J(x+y)such thatx+y2_{≤ x}2_{+ 2y,j(x+y).}

Proof. By definition of duality mapping, we may obtain directly the results of Lemma1.2.

2. Main results

Theorem2.1. LetEbe a real Banach space, and letKbe a nonempty closed convex subset of

Esuch thatK+K⊂K. Assume thatT:K→Kis a uniformly continuousΦ-hemicontractive mapping. Let{αn}∞n=0 and{βn}∞n=0 be two real sequences in [0,1] satisfying the following conditions: (i)αn,βn→0asn→ ∞; (ii)∞n=0αn= ∞. Suppose that{un}∞n=0and{vn}∞n=0are two sequences inKsatisfying that∞_n=0un<∞and

∞

n=0vn<∞. Define the Ishikawa iterative sequence{xn}∞n=0with errors inKby

(IS)

        

x0∈K,

yn=1−βnxn+βnTxn+vn, n≥0, xn+1=

1−αnxn+αnT yn+un, n≥0.

If {T yn}∞n=0and{Txn}∞n=0are bounded, then the sequence{xn}∞n=0converges strongly to the unique fixed point ofT.

Proof. We first observe that the iterative sequence{xn}defined by (2.1) is well defined,

sinceKis convex andTis a self-mapping fromKto itself withK+K⊂K. By the defini-tion of T, we known that ifF(T)= ∅, thenF(T) must be a singleton, letq∈Kdenote the unique fixed point. And we also obtain that for anyx∈K, there exists j(x−q)∈J(x−y) such that

Tx−Tq,j(x−q)≤x−q2

−Φx−qx−q. (2.2)

Now set

M=sup

n≥0

_{T yn−q+x0−q,}

D= ∞

n=0

_{un+M+ 1.} (2.3)

By using induction, we obtainxn−q ≤M+∞n=0un,n≥0, which implies thatxn−

q ≤D,n≥0. Using (2.1) and Lemma1.2, we have

_xn+1−q2

=1−αnxn−q+αnT yn−Tq+un2

≤1−αnxn−q+αnT yn−Tq2+ 2Dun.

(2.4)

Let An= T yn−T(xn+1−un). ThenAn→0 as n→ ∞. Indeed, sinceT is uniformly continuous, we observe that{xn}∞n=0,{Txn}∞n=0, and{T yn}∞n=0are all bounded andyn− (xn+1−un) →0 asn→ ∞, so thatAn→0 asn→ ∞. Using Lemma1.2, (2.1), and (2.2), we have

xn+1−un−q2

=1−αnxn−q+αnT yn−Tq2

≤1−αn2xn−q2+ 2αnT yn−Tq,jxn+1−un−q

≤1−αn2xn−q2+ 2αnT yn−Txn+1−un

,jxn+1−un−q

+ 2αnTxn+1−un−Tq,jxn+1−un−q

≤1−αn2xn−q2+ 2αnAnxn+1−un−q

+ 2αnxn+1−un−q 2

−2αnΦxn+1−un−qxn+1−un−q

+ 2αnxn+1−un−q2−2αnΦxn+1−un−qxn+1−un−q

≤1−αn2xn−q2+αnAn1−αn1 +xn−q2+ 2α2nAnD

+ 2αnxn+1−un−q 2

−2αnΦxn+1−un−qxn+1−un−q ≤1−αn2+αnAnxn−q2+αnAn1 + 2αnD

+ 2αnxn+1−un−q 2

−2αnΦxn+1−un−qxn+1−un−q,

(2.5)

which implies that

_{xn+1−un−q}2 ≤

1−αn2+αnAn

1−2αn

_xn−q2 +αnAn

1 + 2αnD

1−2αn

− 2αn 1−2αnΦ

_{xn+1−un−qxn+1−un−q}

≤xn−q2+ 2αn 1−2αn

D2_α

n+D2An+An+ 2αnAnD

2

−Φxn+1−un−qxn+1−un−q

.

(2.6)

Substituting (2.6) into (2.4) yields that

_xn+1−q2

≤ xn−q2+₁₋2α₂n_α n

D2_α

n+D2_A

n+An+ 2αnAnD

2

−Φxn+1−un−qxn+1−un−q

+ 2Dun

≤xn−q2+₁₋2α₂n_α n

Bn−Φxn+1−un−qxn+1−un−q+ 2Dun,

(2.7)

whereBn=D2αn+D2_A

n+An+ 2αnAnD/2. Now we consider the following two possible cases.

Case (i). limn→∞infxn+1−un−q =r >0. SinceBn→0,αn→0 asn→ ∞, then there exists a positive integerNsuch thatBn<1/2Φ(r)r,αn<1/2 for alln≥N. It follows from

(2.7) that

_xn+1−q2

≤xn−q2+₁₋α₂n_α

nΦ(r)r−

2αn

1−2αnΦ(r)r+ 2D

_un

≤xn−q2−₁₋αn_2α

nΦ(r)r+ 2D

_un (2.8)

which implies thatΦ(r)r∞_n=Nαn/1−2αn≤ xN−q2+ 2Dn∞=Nun<∞. This

Case (ii). limn→∞infxn+1−un−q =0. In this case, there exists a subsequence{xnj+1−

unj−q}such thatxnj+1−unj−q→0 as j→ ∞. Hence, for any 0< ε <1, there exists a positive integernj such thatxnj+1−unj−q< εandBn<Φ(ε)ε, 2D

_∞

k=nj+1uk< ε for alln≥nj for alln≥nj. Now we show thatxnj+m< εfor allm≥1. First, by (2.4), we havexnj+1−q2≤ε2+ 2Dunj. Again consider the following two possible cases. Case(ii-1). xnj+2−unj+1−q< ε. Using (2.4), we obtain

xnj+2−q2=1−αnj+1

xnj+1−q

+αnj+1

T ynj+1−Tq

+unj+1 2

≤xnj+2−unj+1−q 2

+ 2Dunj+1 ≤ε2_{+ 2Du}

nj+1.

(2.9)

Case(ii-2). xnj+2−unj+1−q ≥ε. Then using (2.7) yields that

_xn

j+2−q 2

≤ε2_{+ 2Du}

nj+unj+1. (2.10)

For all m≥1, using induction, we have xnj+m−q2≤ε2+ 2D

nj+m−1

k=nj uk<2ε. Thus we prove thatxn→qasn→ ∞. This completes the proof.

Remark 2.2. The assumptionK+K⊂K only is used to guarantee that the iterative se-quence{xn}∞n=0is well defined. We can drop this assumption in Theorem2.1by using a revised iterative scheme.

Corollary2.3. LetEbe a real Banach space, and letKbe a nonempty bounded and con-vex subset ofE. Assume thatT:K→Kis a uniformly continuousΦ-hemicontractive map-ping. Let{αn}∞n=0,{βn}∞n=0,{γn}∞n=0,{α}∞n=0,{β}∞n=0, and {γ}∞n=0 be six real sequences in [0,1] satisfying the following conditions: (i)βn→0,βn→0,γn→0asn→ ∞; (ii)∞n=0βn= ∞,∞_n=0γn<∞; (iii) αn+βn+γn=αn+βn+γn=1. Let {un}n∞=0 and {νn}∞n=0 be two bounded sequences inK. Define iteratively the Ishikawa sequence {xn}∞n=0 with errors in

Kas follows:

x0∈K,

yn=αnxn+βnTxn+γnvn, n≥0, xn+1=αnxn+βnT yn+γnun, n≥0.

(2.11)

Then the sequence{xn}∞n=0 defined by (2.11) converges strongly to the unique fixed point ofT.

Proof. We observe that (2.11) can be rewritten as follows:

x0∈K,

yn=1−βnxn+βnTxn+γn(vn−xn), n≥0, xn+1=

1−βnxn+βnT yn+γnun−xn, n≥0.

(2.12)

Theorem2.4. LetEbe a real Banach space, and letKbe a nonempty closed convex subset of

Esuch thatK+K⊂K. Assume thatT:K→Kis a uniformly continuousΦ-hemicontractive mapping. Let{αn}∞n=0and{βn}∞n=0 be two real sequences in [0,1] satisfying the following conditions: (i)αn,βn→0asn→ ∞; (ii)∞n=0αn= ∞. Suppose that{un}∞n=0and{vn}∞n=0 are two sequences inKsatisfyingun,vn →0asn→ ∞, whereun =o(αn). Define the

Ishikawa iterative sequence{xn}∞n=0with errors inKby

(IS)

        

x0∈K,

yn=1−βnxn+βnTxn+vn, n≥0,

xn+1=1−αnxn+αnT yn+un, n≥0.

(2.13)

If{T yn}∞n=0and{Txn}∞n=0are bounded, then the sequence{xn}∞n=0converges strongly to the unique fixed point ofT.

Proof. SinceK+K⊂KandKis convex, we see that the sequence{xn}∞n=0is well defined. By the definition ofT,T has a unique fixed point inK. Letq denote the unique fixed point. Now we shall show that{xn}∞n=0is bounded. In fact, we may setun =εnαn, where

εn→0 asn→ ∞. SetD=supn≥0{T yn−q+εn}+x0−q, by induction, we can show thatxn−q ≤Dfor alln≥0, so that{yn}is bounded. And we have

Tx−Tq,j(x−q)≤ x−q2_{−Φx−qx−q (2.14)}

for eachx∈K. By using Lemma1.2and (2.7), we have

_xn+1−q2

≤1−αnxn−q+αnT yn−Tq2+ 2Dun. (2.15)

After repeating the usage of the proof of Theorem2.1, we obtain

_1−αn

xn−q+αnT yn−Tq2

≤1−αn2+αnAnxn−q2+αnAn1 + 2αnD

+ 2αnxn+1−un−q 2

−2αnΦxn+1−un−qxn+1−un−q.

(2.16)

Thus, we have

xn+1−q2

≤ xn−q2+₁₋2α₂n_α n

D2_α

n+D2An+An+ 2αnAnD

2

−Φxn+1−un−qxn+1−un−q

+ 2Dun

≤ xn−q2+₁₋2α₂n_α n

Bn+Cn−Φ(xn+1−un−q)xn+1−un−q

+ 2Dun,

(2.17)

where Bn=D2αn+D2_A

positive integerN such thatBn+Cn<1/2Φ(r)r,αn<1/2 for alln≥N. It follows that xn+1−q2≤ xn−q2−αn/1−2αnΦ(r)r, which leads toΦ(r)r

_∞

n=Nαn≤ xN−q2<

∞, a contradiction. Hence, there exists a subsequence{xnj+ 1}such thatxnj+ 1→qas

j→ ∞. At this point, we can choose a positive integernj such thatxnj+1−q< εand

Bn+Cn<Φ(ε/2)ε/4,un< ε/2 for alln≥nj. We show thatxnj+2−q< ε. If not, we assume thatxnj+2−q ≥ε, thenxnj+2−unj+1−q ≥ xnj+2−q − unj+1≥ε/2 so that Φ(xnj+2−unj+1−q)≥Φ(ε/2). Thus, using (2.17), we have

_xn

j+2−q 2

≤xnj+1−q 2

− αnj+1 1−2αnj+1Φ

_ε

2

_ε

2< ε

2_{, (2.18)}

this is a contradiction and soxnj+2−q< ε. By induction,xnj+m−q< εfor allm≥

1.

Corollary 2.5. Let Ebe a real Banach space, and let K be a nonempty bounded and convex subset ofE. Assume thatT:K→K is a uniformly continuousΦ-hemicontractive mapping. Let{αn}∞n=0,{βn}∞n=0,{γn}∞n=0,{α}∞n=0,{β}∞n=0, and{γ}∞n=0be six real sequences in [0,1] satisfying the following conditions: (i)βn→0,βn→0,γn→0asn→ ∞; (ii)∞n=0βn= ∞,γn=o(βn); (iii)αn+βn+γn=αn+βn+γn=1,n≥0. Let{un}∞n=0and{vn}∞n=0be two bounded sequences inK. Define iteratively the Ishikawa sequence{xn}∞n=0with errors inK as follows:

x0∈K,

yn=αnxn+βnTxn+γnvn, n≥0, xn+1=αnxn+βnT yn+γnun, n≥0.

(2.19)

Then the sequence{xn}∞n=0 defined by (2.11) converges strongly to the unique fixed point ofT.

Proof. We observe that (2.11) can be rewritten as follows:

x0∈K,

yn=1−βnxn+βnTxn+γn(vn−xn), n≥0,

xn+1=

1−βnxn+βnT yn+γn(un−xn), n≥0.

(2.20)

It is easily to obtain the conclusion from Theorem2.4. This completes the proof.

3. Acknowledgment

The author is very grateful to the referees for careful reading of the original version of this paper and for some good comments.

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Xue Zhiqun: Department of Mathematics, Shijiazhuang Railway College, Shijiazhuang 050043, China