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 = f2
(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−y
x−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
International Journal of Mathematics and Mathematical Sciences 2005:17 (2005) 2711–2718 DOI:10.1155/IJMMS.2005.2711
Φ(0)=0 such that the inequality
Tx−Tq,j(x−q)≤ x−q2−Φx−q
x−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≤ x2+ 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−βn
xn+βnTxn+vn, n≥0, xn+1=
1−αn
xn+αnT yn+un, n≥0.
(2.1)
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−αn
xn−q+αn
T yn−Tq+un2
≤1−αn
xn−q+αn
T 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−αn
xn−q+αn
T yn−Tq2
≤
1−αn2xn−q2+ 2αn
T yn−Tq,jxn+1−un−q
≤
1−αn2xn−q2+ 2αn
T yn−Txn+1−un
,jxn+1−un−q + 2αn
Txn+1−un
−Tq,jxn+1−un−q
≤
1−αn2xn−q2+ 2αnAnxn+1−un−q
+ 2αnxn+1−un−q2−2αnΦxn+1−un−qxn+1−un−q
≤
1−αn2xn−q2+ 2αnAn
1−αnxn−q+ 2α2nAnD
+ 2αnxn+1−un−q2−2αnΦxn+1−un−qxn+1−un−q
≤
1−αn2xn−q2+αnAn 1−αn
1 +xn−q2+ 2α2nAnD + 2αnxn+1−un−q2−2αnΦxn+1−un−qxn+1−un−q
≤ 1−αn2
+αnAnxn−q2+αnAn
1 + 2αnD
+ 2αnxn+1−un−q2−2αnΦxn+1−un−qxn+1−un−q,
(2.5) which implies that
xn+1−un−q2≤
1−αn2
+αnAn
1−2αn
xn−q2+αnAn
1 + 2αnD 1−2αn
− 2αn
1−2αnΦxn+1−un−qxn+1−un−q
≤xn−q2+ 2αn
1−2αn
D2αn+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+ 2αn
1−2αn
D2αn+D2An+An+ 2αnAnD 2
−Φxn+1−un−qxn+1−un−q
+ 2Dun
≤xn−q2+ 2αn
1−2αn
Bn−Φxn+1−un−qxn+1−un−q+ 2Dun, (2.7) whereBn=D2αn+D2An+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+ αn
1−2αnΦ(r)r− 2αn
1−2αnΦ(r)r+ 2Dun
≤xn−q2− αn
1−2αnΦ(r)r+ 2Dun (2.8) which implies thatΦ(r)r∞n=Nαn/1−2αn≤ xN−q2+ 2D∞n=Nun<∞. This con- tradicts the assumption that∞n=0αn= ∞and so the case (i) is impossible.
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+12
≤xnj+2−unj+1−q2+ 2Dunj+1
≤ε2+ 2Dunj+1.
(2.9)
Case(ii-2). xnj+2−unj+1−q ≥ε. Then using (2.7) yields that
xnj+2−q2≤ε2+ 2Dunj+unj+1. (2.10) For all m≥1, using induction, we have xnj+m−q2≤ε2+ 2Dnk=j+nmj−1uk<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−βn
xn+βnTxn+γn(vn−xn), n≥0, xn+1=
1−βn
xn+βnT yn+γn un−xn
, n≥0.
(2.12)
It is easily seen that under the assumptions of Corollary 2.3, the sequence {xn}∞n=0 is bounded. Now the conclusion follows from Theorem2.1. This completes the proof.
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−βn
xn+βnTxn+vn, n≥0, xn+1=
1−αn
xn+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−q
x−q (2.14)
for eachx∈K. By using Lemma1.2and (2.7), we have xn+1−q2≤1−αn
xn−q+αn
T yn−Tq2+ 2Dun. (2.15) After repeating the usage of the proof of Theorem2.1, we obtain
1−αn
xn−q+αn
T yn−Tq2
≤ 1−αn2
+αnAnxn−q2+αnAn
1 + 2αnD
+ 2αnxn+1−un−q2−2αnΦxn+1−un−qxn+1−un−q.
(2.16)
Thus, we have xn+1−q2
≤ xn−q2+ 2αn
1−2αn
D2αn+D2An+An+ 2αnAnD 2
−Φxn+1−un−qxn+1−un−q
+ 2Dun
≤ xn−q2+ 2αn
1−2αn
Bn+Cn−Φ(xn+1−un−q)xn+1−un−q
+ 2Dun, (2.17) where Bn=D2αn+D2An+An+ 2αnAnD/2→0,Cn=1−2αn/αnDun →0 as n→ ∞. Then limn→∞infxn+1−un−q =0. If it is not the case, then there existδ >0 and
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
xnj+2−q2≤xnj+1−q2− αnj+1
1−2αnj+1Φε 2
ε
2< ε2, (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−βn
xn+βnTxn+γn(vn−xn), n≥0, xn+1=
1−βn
xn+βnT yn+γn(un−xn), n≥0.
(2.20)
It is easily to obtain the conclusion from Theorem2.4. This completes the proof.
Remark 2.6. Theorems2.1and2.4extend the results of [5] from realq-uniformly smooth Banach spaces to arbitrary real Banach spaces. It is also easy to see that our results are significant extensions of the results of [1,2,3,4,7] to arbitrary real Banach spaces and to the more general classes of mapping (Φ-hemicontractive mapping) considered here.
Moreover, our iteration schemes extend from the usual iterative sequences to the iterative sequences with errors.
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
E-mail address:[email protected]
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