# Strong Convergence of Non-Implicit Iteration Process with Errors in Banach Spaces

Authors: Yan Hao; Xiaoshuang Wang; Aihua Tong
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The purpose of this paper is to study the strong convergence of a non-implicit iteration process with errors for asymptoticallyI-nonexpansive mappings in the intermediate sense in the framework of Banach spaces. The results presented in this paper extend and improve the corresponding results recently announced.


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## Body

## 1. Introduction and Preliminaries

LetK be a nonempty, closed, and convex subset of a real Banach space X and let T:K→K be a mapping. In this paper, we use F(T) to stand for the set of fixed points of T, that is F(T)={x∈K:Tx=x}.Recall thatT is said to be nonexpansive if
(1.1)∥Tx-Ty∥≤∥x-y∥,∀x,y∈K.Tis said to be asymptotically nonexpansive if there exists a sequence {hn} with hn⊂[1,+∞) with limn→∞hn=1 such that
(1.2)∥Tnx-Tny∥≤hn∥x-y∥,∀x,y∈K,n≥1.T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:
(1.3)limsupn→∞supx,y∈k(∥Tnx-Tny∥-∥x-y∥)≤0.Observe that if we definean=supx,y∈k(∥Tnx-Tny∥-∥x-y∥),σn=max{0,an}, then σn→0 as n→∞ and (1.3) reduces to
(1.4)∥Tnx-Tny∥≤∥x-y∥+σn,∀x,y∈K,n≥1.It is easy to see that every nonexpansive mapping is asymptotically nonexpansive. And every asymptotically nonexpansive mapping is asymptotically nonexpansive in the intermediated sense. In [1], Goebel and Kirk proved that, if K is a nonempty closed convex bounded subset of a real uniformly convex Banach space X, and T is an asymptotically nonexpansive self-mapping on K, then T has a fixed point in K. The class of mappings which are asymptotically nonexpansive in the intermediat sense was investigated by Bruck et al. [2] and Kirk [3]. Since then, many authors have investigated the fixed point problem of these mappings based on implicit iterative methods or non-implicit iterative methods; see, for example, [4–21].LetI:K→K be a mapping. Recall that T is said to be asymptotically I-nonexpansive if there exists a sequence {hn} with {hn}⊂[1,+∞) with limn→∞hn=1 such that
(1.5)∥Tnx-Tny∥≤hn∥Inx-Iny∥,∀x,y∈K,n≥1.
Recently, weak and strong convergence theorems for fixed points of I-nonexpansive mappings, and asymptotically I-nonexpansive mappings have been established by many scholar, see, for example, [22–25].In this paper, we consider a new mapping based on asymptotically nonexpansive mappings in the intermediate sense and asymptoticallyI-nonexpansive mappings.LetT:K→K,I:K→K be two mappings. T is said to be asymptotically I-nonexpansive in the intermediate sense if it is continuous and the following inequality holds:
(1.6)limsupn→∞supx,y∈k(∥Tnx-Tny∥-∥Inx-Iny∥)≤0.
Observe that if we define an=supx,y∈k(∥Tnx-Tny∥-∥Inx-Iny∥), σn=max{0,an}, then σn→0 as n→∞ and (1.6) reduces to
(1.7)∥Tnx-Tny∥≤∥Inx-Iny∥+σn,∀x,y∈K,n≥1.
Note that if I=Id, where Id is the identity mapping, then (1.7) reduces to (1.4).In this paper, we investigate asymptoticallyI-nonexpansive mappings in the intermediate sense based on a non-implicit iterative algorithm. Strong convergence of the implicit iterative algorithm is obtained in the framework of Banach spaces.In order to prove our main results, we need the following lemmas.Lemma 1.1 (see [21]).
letX be a uniformly convex Banach space. Let b and c be two constants with 0<b<c<1. Suppose that {tn} is a sequence in [b,c]. Let {xn} and {yn} be two sequences in X such that
(1.8)limsupn→∞∥xn∥≤d,limsupn→∞∥yn∥≤d,limn→∞∥tnxn+(1-tn)yn∥=d
hold for some d≥0, then limn→∞∥xn-yn∥=0.Lemma 1.2 (see [26]).
Let{an}, {bn}, and {cn} be three nonnegative sequences satisfying the following condition:
(1.9)an+1≤(1+bn)an+cn,∀n≥n0,
where n0 is some nonnegative integer, ∑n=1∞‍bn<∞ and ∑n=1∞‍cn<∞. Then the limit limn→∞an exists.


## 2.   Main Results

Lemma 2.1.
LetX be a real Banach space and K a nonempty closed and convex subset of X. Let T:K→K be a asymptotically I-nonexpansive in the intermediate sense and I:K→K a asymptotically nonexpansive in the intermediate sense. Assume that F:=F(T)∩F(I)≠∅. Let σn=max{0,supx,y∈k(∥Tnx-Tny∥-∥Inx-Iny∥)} and σ-n=max{0,supx,y∈k(∥Tnx-Tny∥-∥x-y∥)}. Let {αn}, {βn}, {γn}, {α^n}, {β^n}, {γ^n} be six real number sequences in (0,1). Let {xn} be a sequence generated in the following iterative process:
(2.1)x1∈C,yn=α^nxn+β^nInxn+γ^nvn,xn+1=αnxn+βnTnyn+γnun,n≥1,
where {un} and {vn} be two bounded sequences in K. Assume that the following restrictions are satisfied:  (a)
αn+βn+γn=α^n+β^n+γ^n=1;(b)
∑n=1∞‍σn<∞,∑n=1∞‍σ-n<∞;(c)
∑n=1∞‍γn<∞,∑n=1∞‍γ^n<∞. Then limn→∞∥xn-p∥ exists for all p∈F.Proof.
Lettingp∈F, we see that
(2.2)∥yn-p∥=∥(1-β^n-γ^n)xn+β^nInxn+γ^nvn-p∥≤(1-β^n-γ^n)∥xn-p∥+β^n∥Inxn-p∥+γ^n∥vn-p∥≤(1-β^n-γ^n)∥xn-p∥+β^n(∥xn-p∥+σ-n)+γ^n∥vn-p∥=(1-γ^n)∥xn-p∥+γ^n∥vn-p∥+β^nσ-n≤∥xn-p∥+γ^n∥vn-p∥+β^nσ-n,(2.3)∥xn+1-p∥=∥(1-βn-γn)xn+βnTnyn+γnun-p∥≤(1-βn-γn)∥xn-p∥+βn∥Tnyn-p∥+γn∥un-p∥=(1-βn-γn)∥xn-p∥+βn∥Tnyn-Tnp∥+γn∥un-p∥≤(1-βn-γn)∥xn-p∥+βn(∥Inyn-Inp∥+σn)+γn∥un-p∥=(1-βn-γn)∥xn-p∥+βn∥Inyn-Inp∥+βnσn+γn∥un-p∥≤(1-βn-γn)∥xn-p∥+βn(∥yn-p∥+σ-n)+βnσn+γn∥un-p∥=(1-βn-γn)∥xn-p∥+βn∥yn-p∥+γn∥un-p∥+βn(σn+σ-n)≤(1-βn)∥xn-p∥+βn∥yn-p∥+γn∥un-p∥+βn(σn+σ-n).
Substituting (2.2) into (2.3),we obtain that
(2.4)∥xn+1-p∥≤(1-βn)∥xn-p∥+βn(∥xn-p∥+γ^n∥vn-p∥+β^nσ-n)+γn∥un-p∥+βn(σn+σ-n)=∥xn-p∥+[βnγ^n∥vn-p∥+γn∥un-p∥+βnσn+βnσ-n(1+β^n)].
Let an=∥xn-p∥, bn=0, and
(2.5)cn=βnγ^n∥vn-p∥+γn∥un-p∥+βnσn+βnσ-n(1+β^n).
It follows from (2.4) that
(2.6)an+1≤an+cn.
In view of the restrictions (b) and (c), we see that ∑n=1∞‍cn<∞. We can easily conclude the desired conclusion with the aid of Lemma 1.2. This completes the proof of Lemma 2.1.Theorem 2.2.
LetX be a real Banach space and K a nonempty closed and convex subset of X. Let T:K→K be a asymptotically I-nonexpansive in the intermediate sense and I:K→K a asymptotically nonexpansive in the intermediate sense. Assume that F:=F(T)∩F(I)≠∅. Let σn=max{0,supx,y∈k(∥Tnx-Tny∥-∥Inx-Iny∥)} and σ-n=max{0,supx,y∈k(∥Tnx-Tny∥-∥x-y∥)}. Let {αn},{βn},{γn},{α^n},{β^n},{γ^n} be six real number sequences in (0,1). Let {xn} be a sequence generated in the following iterative process:
(2.7)x1∈C,yn=α^nxn+β^nInxn+γ^nvn,xn+1=αnxn+βnTnyn+γnun,n≥1,
where {un} and {vn} be two bounded sequences in K. Assume that the following restrictions are satisfied:  (a)
αn+βn+γn=α^n+β^n+γ^n=1;(b)
∑n=1∞‍σn<∞,∑n=1∞‍σ-n<∞;(c)
∑n=1∞‍γn<∞,∑n=1∞‍γ^n<∞. If both T and I are continuous, then the sequence {xn} strongly converges to a common fixed point of T and I if and only if
(2.8)liminfn→∞d(xn,F)=0.Proof.
The necessity is obvious. Next, we prove the sufficiency part of the theorem. Note that continuity ofT and I implies that the set F(T) and F(I) are closed. It follows from (2.6) that
(2.9)∥xn+1-p∥≤∥xn-p∥+cn.
This implies in turn that
(2.10)d(xn+1,F)≤d(xn,F)+cn.
Now applying Lemma 1.2 to (2.10), we obtain the existence of the limit limn→∞d(xn,F). By condition (2.8), we have
(2.11)limn→∞d(xn,F)=liminfn→∞d(xn,F)=0.
Next we prove that the sequence{xn} is a Cauchy sequence in K. For any positive integers n, m, from (2.9) it follows that
(2.12)∥xn+m-p∥≤∥xn+m-1-p∥+cn+m-1≤(∥xn+m-2-p∥+cn+m-2)+cn+m-1≤⋯≤∥xn-p∥+∑i=nn+m-1ci‍≤∥xn-p∥+∑i=n∞‍ci.
Since limn→∞d(xn,F)=0, and ∑n=1∞‍cn<∞, for any given ϵ>0, there exists a positive integer n0 such that
(2.13)d(xn,F)<ϵ8,∑i=n∞‍ci<ϵ2,∀n≥n0.
Therefore there exists p1∈F such that d(xn,p1)<(ϵ/4), ∀n≥n0. Consequently, for any n≥n0 and for all m≥1, we have
(2.14)∥xn+m-xn∥≤∥xn+m-p1∥+∥xn-p1∥≤2∥xn-p1∥+∑i=n∞‍ci≤ϵ4·2+ϵ2=ϵ.
This implies that {xn} is a Cauchy sequence in K. Let xn→x*∈K. Since F is closed, this implies that x*∈F. This shows that {xn} strongly converges to a common fixed of T and I. This completes the proof of Theorem 2.2.Lemma 2.3.
LetX be a real Banach space and K a nonempty closed and convex subset of X. Let T:K→K be a asymptotically I-nonexpansive in the intermediate sense and I:K→K a asymptotically nonexpansive in the intermediate sense. Assume that F:=F(T)∩F(I)≠∅. Let σn=max{0,supx,y∈k(∥Tnx-Tny∥-∥Inx-Iny∥)} and σ-n=max{0,supx,y∈k(∥Tnx-Tny∥-∥x-y∥)}. Let {αn}, {βn}, {γn}, {α^n}, {β^n}, {γ^n} be six real number sequences in (0,1). Let {xn} be a sequence generated in the following iterative process:
(2.15)x1∈C,yn=α^nxn+β^nInxn+γ^nvn,xn+1=αnxn+βnTnyn+γnun,n≥1,
where {un} and {vn} be two bounded sequences in K. Assume that the following restrictions are satisfied:  (a)
αn+βn+γn=α^n+β^n+γ^n=1, ∀n≥1;(b)
∑n=1∞‍σn<∞,  ∑n=1∞‍σ-n<∞;(c)
there exist constantsτ1,τ2∈(0,1) such that τ1≤βn, β^n≤τ2, ∀n≥1;(d)
∑n=1∞‍γn<∞, ∑n=1∞‍γ^n<∞.Then
(2.16)limn→∞∥xn-Tnxn∥=0,(2.17)limn→∞∥xn-Inxn∥=0.Proof.
According to Lemma2.1, for any p∈F, we have limn→∞∥xn-p∥ exists. Without loss of generality, we may assume that
(2.18)limn→∞∥xn-p∥=d,
where d>0 is some constant. It follows that
(2.19)limn→∞∥xn+1-p∥=limn→∞∥(1-βn)[xn-p+γn(un-xn)]+βn[Tnyn-p+γn(un-xn)]∥=d.
Notice that
(2.20)∥xn-p+γn(un-xn)∥≤∥xn-p∥+γn∥un-xn∥.
It follows from the restriction (d) and (2.18) that
(2.21)limsupn→∞∥xn-p+γn(un-xn)∥≤d.
Notice that
(2.22)limsupn→∞∥Tnyn-p+γn(un-xn)∥≤limsupn→∞∥Tnyn-p∥+limsupn→∞γn∥un-xn∥=limsupn→∞∥Tnyn-p∥≤limsupn→∞(∥Inyn-Inp∥+σn)≤limsupn→∞(∥yn-p∥+σ-n+σn)=limsupn→∞∥yn-p∥≤limsupn→∞(∥xn-p∥+γ^n∥vn-p∥+β^nσ-n)=d.
In view of (2.19), (2.21) and (2.22), we obtain from Lemma 1.1 that
(2.23)limn→∞∥xn-Tnyn∥=0.
Notice that
(2.24)∥xn+1-xn∥=∥βn(Tnyn-xn)+γn(un-xn)∥≤βn∥Tnyn-xn∥+γn∥un-xn∥.
It follows from (2.23) and the restriction (d) that
(2.25)limn→∞∥xn+1-xn∥=0.
Notice that
(2.26)∥Inxn-p∥≤∥xn-p∥+σ-n.
It follows that
(2.27)limsupn→∞∥Inxn-p∥≤d.
On the other hand, we have
(2.28)limn→∞∥yn-p∥=limn→∞∥(1-β^n)[xn-p+γ^n(vn-xn)]+β^n[Inxn-p+γ^n(vn-xn)]∥=d.
Notice that
(2.29)∥xn-p+γ^n(vn-xn)∥≤∥xn-p∥+γ^n∥vn-xn∥.
It follows that
(2.30)limsupn→∞∥xn-p+γ^n(vn-xn)∥≤d.
Notice that
(2.31)∥Inxn-p+γ^n(vn-xn)∥≤∥Inxn-p∥+γ^n∥vn-xn∥.
It follows from (2.27) that
(2.32)limsupn→∞∥Inxn-p+γ^n(vn-xn)∥≤d.
In view of (2.28), (2.30), and (2.32), we obtain from Lemma 1.1 that
(2.33)limn→∞∥xn-Inxn∥=0.
On the other hand, we have
(2.34)∥xn-Tnxn∥≤∥xn-Tnyn∥+∥Tnyn-Tnxn∥≤∥xn-Tnyn∥+∥yn-xn∥+σn+σ-n≤∥xn-Tnyn∥+β^n∥Inxn-xn∥+γ^n∥vn-xn∥+σn+σ-n.
In view of (2.23) and (2.33), we have limn→∞∥xn-Tnxn∥=0. This completes the proof of Lemma 2.3.Theorem 2.4.
LetX be a real Banach space and K a nonempty closed and convex subset of X. Let T:K→K be a asymptotically I-nonexpansive in the intermediate sense and I:K→K a asymptotically nonexpansive in the intermediate sense. Assume that F:=F(T)∩F(I)≠∅. Let σn=max{0,supx,y∈k(∥Tnx-Tny∥-∥Inx-Iny∥)} and σ-n=max{0,supx,y∈k(∥Tnx-Tny∥-∥x-y∥)}. Let {αn}, {βn}, {γn}, {α^n}, {β^n}, {γ^n} be six real number sequences in (0,1). Assume that both T and I are Lipschitz continuous. Let {xn} are a sequence generated in the following iterative process:
(2.35)x1∈C,yn=α^nxn+β^nInxn+γ^nvn,xn+1=αnxn+βnTnyn+γnun,n≥1,
where {un} and {vn} be two bounded sequences in K. Assume that the following restrictions are satisfied:  (a)
αn+βn+γn=α^n+β^n+γ^n=1,∀n≥1;(b)
∑n=1∞‍σn<∞,∑n=1∞‍σ-n<∞;(c)
there exist constantsτ1,τ2∈(0,1) such that τ1≤βn,β^n≤τ2,∀n≥1;(d)
∑n=1∞‍γn<∞,∑n=1∞‍γ^n<∞. If at least one of the mappings T and I is compact, then the sequence convergence strongly to a common fixed point of  T and I.Proof.
Without loss of generality, we may assume thatT is compact; this means that there exists a subsequence {Tnkxnk} of {Tnxn} such that {Tnkxnk} converges strongly to x*∈K, then (2.16) implies that {xnk} converges strongly to x*. Since T is continuous, then {Tnk+1xnk} converges strongly to Tx*. On the other hand, according to (2.17) and the continuity of I, we obtain that {Inkxnk}, {Ink+1xnk} converge strongly to x*, Ix*, respectively. Since limk→∞∥xnk+1-xnk∥=0, then
(2.36)∥Ink+1xnk+1-Ink+1xnk∥≤∥xnk+1-xnk∥+σ-nk→0,ask→∞,∥Tnk+1xnk+1-Tnk+1xnk∥≤∥Ink+1xnk+1-Ink+1xnk∥+σnk≤∥xnk+1-xnk∥+σ-nk+σnk→0,ask→∞.
Observe that
(2.37)∥x*-Tx*∥≤∥x*-xnk+1∥+∥xnk+1-Tnk+1xnk+1∥+∥Tnk+1xnk+1-Tnk+1xnk∥+∥Tnk+1xnk-Tx*∥,∥x*-Ix*∥≤∥x*-xnk+1∥+∥xnk+1-Ink+1xnk+1∥+∥Ink+1xnk+1-Ink+1xnk∥+∥Ink+1xnk-Ix*∥.
Taking limit as k→∞ in the above inequality, we find x*=Tx*,x*=Ix*, which means x*∈F. However, due to Lemma 2.1, the limit limn→∞∥xn-x*∥ exists, therefore
(2.38)limn→∞∥xn-x*∥=limk→∞∥xnk-x*∥=0,
which means that {xn} converges strongly to x*∈F. This completes the proof of Theorem 2.4.



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