# On the Convergence of Multistep Iteration for Uniformly ContinuousΦ-Hemicontractive Mappings

Authors: Zhiqun Xue; Arif Rafiq; Haiyun Zhou
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

It is shown that the convergence of the multistep iterative process with errors is obtained for uniformly continuousΦ-hemicontractive mappings in real Banach spaces. We also revise the problems of C. E. Chidume and C. O. Chidume (2005).


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## Body

## 1. Introduction

LetE be a real Banach space with norm ∥·∥ and let E* be its dual space. The normalized duality mapping J:E→2E* is defined by
(1.1)J(x)={f∈E*:〈x,f〉=∥x∥2=∥f∥2},∀x∈E,
where 〈·,·〉 denotes the generalized duality pairing. The single-valued-normalized duality mapping is denoted by j.A mappingT with domain D(T) and range R(T) in E is said to be strongly pseudocontractive if there is a constant k∈(0,1), and for all x,y∈D(T),∃j(x-y)∈J(x-y) such that
(1.2)〈Tx-Ty,j(x-y)〉≤k∥x-y∥2.
The mapping T is called Φ-pseudocontractive if there exists a strictly increasing continuous function Φ:[0,+∞)→[0,+∞) with Φ(0)=0 such that
(1.3)〈Tx-Ty,j(x-y)〉≤∥x-y∥2-Φ(∥x-y∥)
holds for all x,y∈D(T). It is well known that the strongly pseudocontractive mapping must be the Φ-pseudocontractive mapping in the special case in which Φ(t)=(1-k)t2, but the converse is not true in general. That is, the class of strongly pseudocontractive mappings is a proper subclass of the class of Φ-pseudocontractive mappings. Let F(T)={x∈D(T):Tx=x}≠∅. If the inequalities (1.2) and (1.3) hold for any x∈D(T) and y∈F(T), then the corresponding mapping T is called strongly hemicontractive and Φ-hemicontractive, respectively.LetN(T)={x∈E:Tx=0}≠∅. An operator T:D(T)⊆E→E is called strongly quasiaccretive, Φ-quasiaccretive if and only if I-T is strongly hemicontractive, Φ-hemicontractive, respectively, where I denotes the identity mapping on E. That is, if T is Φ-quasi-accretive, then N(T)≠∅ and there exists a strictly increasing continuous function Φ:[0,+∞)→[0,+∞) with Φ(0)=0 such that
(1.4)〈Tx-Ty,j(x-y)〉≥Φ(∥x-y∥)
holds for all x∈D(T) and y∈N(T). Many authors have studied extensively the strongly convergence problems of the iterative algorithms for the class of operators.In 2004, Rhoades and Soltuz [1] introduced the multistep iteration as follows.LetD be a nonempty closed convex subset of real Banach space E and let T:D→D be a mapping. The multistep iteration {xn} is defined by
(1.5)x0∈D,ynp-1=(1-bnp-1)xn+bnp-1Txn,n≥0,p≥2,ynk=(1-bnk)xn+bnkTynk+1,k=p-2,p-3,…,2,1,xn+1=(1-an)xn+anTyn1,n≥0,
where {an},{bnk}(k=1,2,…,p-1) in [0,1] satisfy certain conditions. Obviously, the iteration defined above is generalization of Mann, Ishikawa, and Noor iterations.Inspired and motivated by the work of Xu [2] and the iteration above, we discuss the following multistep iteration with errors:
(1.6)x0∈D,ynp-1=(1-bnp-1-dnp-1)xn+bnp-1Txn+dnp-1wnp-1,n≥0,p≥2,ynk=(1-bnk-dnk)xn+bnkTynk+1+dnkwnk,k=p-2,p-3,…,2,1,xn+1=(1-an-cn)xn+anTyn1+cnun,n≥0,
where {an},{cn},{bnk},{dnk}(k=1,2,…,p-1) in [0,1] with an+cn≤1,bnk+dnk≤1, {un},{wnk}(k=1,2,…,p-1) are the bounded sequences of D.In 2005, C. E. Chidume and C. O. Chidume [3] proved the convergence theorems for fixed points of uniformly continuous generalized Φ-hemicontractive mappings and published in [3]. However, there exists a gap in the proof course of their theorems.The aim of this paper is to show the convergence of the multistep iteration with errors for fixed points of uniformly continuousΦ-hemicontractive mappings and revise the results of C. E. Chidume and C. O. Chidume [3]. For this, we need the following Lemmas.Lemma 1.1 (see [4]).
LetE be a real Banach space and let J:E→2E* be a normalized duality mapping. Then
(1.7)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,
for all x,y∈E and for all j(x+y)∈J(x+y).Lemma 1.2 (see [5]).
Let{δn}n=0∞, {λn}n=0∞ and {γn}n=0∞ be three nonnegative real sequences and let Φ:[0,+∞)→[0,+∞) be a strictly increasing and continuous function with Φ(0)=0 satisfying the following inequality:
(1.8)δn+12≤δn2-λnΦ(δn+1)+γn,n≥0,
where λn∈[0,1] with ∑n=0∞‍λn=∞, γn=o(λn). Then δn→0 as n→∞.


## 2. Main Results

Theorem 2.1.
LetE be an arbitrary real Banach space, D a nonempty closed convex subset of E, and T:D→D a uniformly continuous Φ-hemicontractive mapping with q∈F(T)≠∅. Let {an},{bnk},{cn},{dnk} be real sequences in [0,1] and satisfy the conditions:(i)
an+cn≤1,bnk+dnk≤1,k=1,2,…,p-1;(ii)
an,cn,bnk,dnk→0 as n→∞,k=1,2,…,p-1;(iii)
cn=o(an), ∑n=0∞‍an=∞.For somex0∈D, let {un},{wn1},{wn2},…,{wnp-1} be any bounded sequences of D, and let {xn} be the multistep iterative sequence with errors defined by (1.6). Then (1.6) converges strongly to the fixed point q of T.Proof.
SinceT:D→D is Φ-hemicontractive mapping, then there exists a strictly increasing continuous function Φ:[0,+∞)→[0,+∞) with Φ(0)=0 such that
(2.1)〈Tx-Tq,j(x-q)〉≤∥x-q∥2-Φ(∥x-q∥),
for x∈D,q∈F(T), that is
(@)〈Tx-x,j(x-q)〉≤-Φ(∥x-q∥).
Choose some x0∈D and x0≠Tx0 such that ∥x0-Tx0∥·∥x0-q∥∈R(Φ) and denote that r0=∥x0-Tx0∥·∥x0-q∥, R(Φ) is the range of Φ. Indeed, if Φ(r)→+∞ as r→+∞, then r0∈R(Φ); if sup{Φ(r):r∈[0,+∞)}=r1<+∞ with r1<r0 (here, we only give a example. If r0=2,Φ(t)=t/(1+t), then sup{Φ(r):r∈[0,+∞)}=r1=1<2=r0), then for q∈D, there exists a sequence {wn} in D such that wn→q as n→∞ with wn≠q. Furthermore, we obtain that Twn→Tq as n→∞. So {wn-Twn} is the bounded sequence. Hence, there exists natural number n0 such that ∥wn-Twn∥·∥wn-q∥<r1/2 for n≥n0, then we redefine x0=wn0 and ∥x0-Tx0∥·∥x0-q∥∈R(Φ). This is to ensure that Φ-1(r0) is defined well.
Step  1. We show that {xn} is a bounded sequence.
SetR=Φ-1(r0), then from above formula (@), we obtain that ∥x0-q∥≤R. Denote
(2.2)B1={x∈D:∥x-q∥≤R},B2={x∈D:∥x-q∥≤2R}.
Since T is the uniformly continuous, so T is a bounded mapping. We let
(2.3)M=supx∈B2{∥Tx-q∥+1}+max{supn{∥wn1-q∥},supn{∥wn2-q∥},…,supn{∥wnp-1-q∥},supn{∥un-q∥}}.
Next, we want to prove that xn∈B1. If n=0, then x0∈B1. Now, assume that it holds for some n, that is, xn∈B1. We prove that xn+1∈B1. Suppose that it is not the case, then ∥xn+1-q∥>R>R/2. Since T is uniformly continuous, then for ϵ0=Φ(R/2)/8R, there exists δ>0 such that ∥Tx-Ty∥<ϵ0 when ∥x-y∥<δ. Denote
(2.4)τ0=min{1,RM,Φ(R/2)8R(M+2R),δ2M+4R}.
Since an,bnk,cn,dnk→0 as n→∞ for k=1,2,…,p-1. Without loss of generality, we assume that 0≤an,bnk,cn,dnk≤τ0 for any n≥0. Since cn=o(an), let cn<anτ0. Now, estimate ∥ynk-q∥ for k=1,2,…,p-1. By using (1.6), we have
(2.5)∥ynp-1-q∥≤(1-bnp-1-dnp-1)∥xn-q∥+bnp-1∥Txn-q∥+dnp-1∥wnp-1-q∥≤R+τ0M≤2R,
then ynp-1∈B2. Similarly, we have
(2.6)∥ynp-2-q∥≤(1-bnp-2-dnp-2)∥xn-q∥+bnp-2∥Tynp-1-q∥+dnp-2∥wnp-2-q∥≤R+τ0M≤2R,
then ynp-2∈B2,…. We have
(2.7)∥yn1-q∥≤(1-bn1-dn1)∥xn-q∥+bn1∥Tyn2-q∥+dn1∥wn1-q∥≤R+τ0M≤2R,
then yn1∈B2. Therefore, we get
(2.8)∥xn+1-q∥≤(1-an-cn)∥xn-q∥+an∥Tyn1-q∥+cn∥un-q∥≤R+τ0M≤2R.
And we have
(2.9)∥xn+1-xn∥≤an∥Tyn1-xn∥+cn∥un-xn∥≤an(∥Tyn1-q∥+∥xn-q∥)+cn(∥un-q∥+∥xn-q∥)≤τ0[(∥Tyn1-q∥+∥un-q∥)+2∥xn-q∥]≤τ0(M+2R)≤Φ(R/2)8R,∥xn+1-yn1∥≤an∥Tyn1-xn∥+cn∥un-xn∥+bn1∥Tyn2-xn∥+dn1∥wn1-xn∥≤an(∥Tyn1-q∥+∥xn-q∥)+cn(∥un-q∥+∥xn-q∥)+bn1(∥Tyn2-q∥+∥xn-q∥)+dn1(∥wn1-q∥+∥xn-q∥)≤τ0[(∥Tyn1-q∥+∥un-q∥+2∥xn-q∥)≤τ0+(∥Tyn2-q∥+∥wn1-q∥+2∥xn-q∥)]≤τ0(2M+4R)≤δ.
Further, by using uniform continuity of T, we have
(2.10)∥Txn+1-Tyn1∥<Φ(R/2)8R.
In view of Lemma 1.1 and the above formulas, we obtain
(2.11)∥xn+1-q∥2=∥(xn-q)+an(Tyn1-xn)+cn(un-xn)∥2≤∥xn-q∥2+2an〈Tyn1-xn,j(xn+1-q)〉+2cn〈un-xn,j(xn+1-q)〉≤∥xn-q∥2+2an〈Txn+1-xn+1+xn+1-xn-Txn+1+Tyn1,j(xn+1-q)〉+2cn∥un-xn∥·∥xn+1-q∥≤∥xn-q∥2-2anΦ(∥xn+1-q∥)+2an∥xn+1-xn∥·∥xn+1-q∥+2an∥Txn+1-Tyn1∥·∥xn+1-q∥+2cn(∥un-q∥+∥xn-q∥)∥xn+1-q∥≤∥xn-q∥2-2anΦ(R2)+2anΦ(R/2)8R·2R+2anΦ(R/2)8R·2R+2anτ0(R+M)2R≤∥xn-q∥2-anΦ(R2)+2anΦ(R/2)8R(M+2R)(R+M)2R≤∥xn-q∥2-an2Φ(R2)≤R2,
which is a contradiction. Hence, xn+1∈B1, that is, {xn} is a bounded sequence; it leads to that {yn1},{yn2},…,{ynp-1} are all bounded sequences as well.
Step
2. We want to prove ∥xn-q∥→0 as n→∞.
Sincean,bnk,cn,dnk→0 as n→∞ and {xn},{yn1} are bounded. From (2.9), we obtain
(2.12)limn→∞∥xn+1-xn∥=0,limn→∞∥xn+1-yn1∥=0,limn→∞∥Txn+1-Tyn1∥=0.
By (2.11), we have
(2.13)∥xn+1-q∥2=∥(xn-q)+an(Tyn1-xn)+cn(un-xn)∥2≤∥xn-q∥2+2an〈Tyn1-xn,j(xn+1-q)〉+2cn〈un-xn,j(xn+1-q)〉≤∥xn-q∥2+2an〈Txn+1-xn+1+xn+1-xn-Txn+1+Tyn1,j(xn+1-q)〉+2cn∥un-xn∥·∥xn+1-q∥≤∥xn-q∥2-2anΦ(∥xn+1-q∥)+2an∥xn+1-xn∥·∥xn+1-q∥+2an∥Txn+1-Tyn1∥·∥xn+1-q∥+2cn∥un-xn∥·∥xn+1-q∥=∥xn-q∥2-2anΦ(∥xn+1-q∥)+o(an),
where 2an∥xn+1-xn∥·∥xn+1-q∥+2an∥Txn+1-Tyn1∥·∥xn+1-q∥+2cn∥un-xn∥·∥xn+1-q∥=o(an). By Lemma 1.2, we obtain that limn→∞∥xn-q∥=0.Theorem 2.2.
LetE be an arbitrary real Banach space and let T:E→E be a uniformly continuous Φ-quasi-accretive operator with q∈N(T)≠∅. Let {an},{bnk},{cn},{dnk} be real sequences in [0,1] and satisfy the conditions:(i)
an+cn≤1,bnk+dnk≤1,k=1,2,…,p-1;(ii)
an,cn,bnk,dnk→0 as n→∞,k=1,2,…,p-1;(iii)
cn=o(an), ∑n=0∞‍an=∞.For some x0∈E, let {un},{wn1},{wn2},…,{wnp-1} be any bounded sequences of E, and let {xn} be the multistep iterative sequence with errors defined by
(2.14)x0∈D,ynp-1=(1-bnp-1-dnp-1)xn+bnp-1Sxn+dnp-1wnp-1,n≥0,p≥2,ynk=(1-bnk-dnk)xn+bnkSynk+1+dnkwnk,k=p-2,p-3,…,2,1,xn+1=(1-an-cn)xn+anSyn1+cnun,n≥0,
where S:E→E is defined by Sx=x-Tx for all x∈E. Then (2.14) converges strongly to the fixed point q of S.Proof.
We find easily thatS is a uniformly continuous Φ-hemicontractive. Then the conclusion of Theorem 2.2 is obtained directly by Theorem 2.1.Remark 2.3.
In Theorems2.1 and 2.2, if bnk=dnk=0(k=p-1,p-2,…,2,1), then, the conclusions are as follows.Corollary 2.4.
LetE be an arbitrary real Banach space, D a nonempty closed convex subset of E, and T:D→D a uniformly continuous Φ-hemicontractive mapping with q∈F(T)≠∅. Let {an},{cn} be real sequences in [0,1] and satisfy the conditions (i)  an+cn≤1; (ii)  an,cn→0 as n→∞; (iii)  cn=o(an) and ∑n=0∞‍an=∞. For some x0∈D, let {un} be any bounded sequence of D, and let {xn} be Mann iterative sequence with errors defined by xn+1=(1-an-cn)xn+anTxn+cnun,n≥0. Then {xn} converges strongly to the fixed point q of T.Corollary 2.5.
LetE be an arbitrary real Banach space and let T:E→E be a uniformly continuous Φ-quasi-accretive operator with q∈N(T)≠∅. Let {an},{cn} be real sequences in [0,1] and satisfy the conditions (i)an+cn≤1; (ii)an,cn→0 as n→∞; (iii)cn=o(an) and ∑n=0∞‍an=∞. For some x0∈E, let {un} be any bounded sequence of E, and let {xn} be Mann iterative sequence with errors defined by xn+1=(1-an-cn)xn+anSxn+cnun,n≥0. where S:E→E is defined by Sx=x-Tx for all x∈E. Then {xn} converges strongly to the fixed point q of S.Remark 2.6.
It is mentioned to notice that there exists a serious shortcoming in the proof process of Theorem 2.3 of [3]. That is, M1cn≤(Φ(ϵ)/4)αn does not hold in line 15 of Claim 2 of page 552. The reason is that the conditions ∑n=0∞cn<+∞ and ∑n=0∞bn=+∞, bn→0 as n→∞ can not obtain cn=o(bn).
Counterexample, let the iteration parameters bean=1-bn-cn,bn,cn in the following:
(2.15){bn}:b0=b1=0,bn=1n,n≥2,{cn}:0,11,122,132,14,152,162,172182,19,1102,1112,1122,1132,1142,1152,116,1172,1182,…,1232,1242,125,1262,…,1352,136,1372,…
Then, ∑n=0∞bn=+∞, ∑n=0∞cn<2∑n=1∞(1/n2)<+∞, but cn≠o(bn).Application 1.
LetE=R be a real number space with the usual norm and D=[0,+∞). Define T:D→D by
(2.16)Tx=x31+x2
for all x∈D. Then T is uniformly continuous with F(T)={0}. Define Φ:[0,+∞)→[0,+∞) by
(2.17)Φ(t)=t21+t2,
then Φ is a strictly increasing function with Φ(0)=0. For all x∈D,q∈F(T), we obtain that
(2.18)〈Tx-Tq,j(x-q)〉=〈x31+x2-0,j(x-0)〉=〈x31+x2,x〉=x41+x2=|x-q|2-|x-q|21+|x-q|2=|x-q|2-Φ(|x-q|).
Therefore, T is a Φ-hemicontractive mapping. Set
(2.19)an=1n+2,cn=1(n+2)2,bnk=dnk=1n+2,k=1,2,…,p-1
for all n≥0.



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