# Strong Convergence of an ImplicitS-Iterative Process for Lipschitzian Hemicontractive Mappings

Authors: Shin Min Kang; Arif Rafiq; Sunhong Lee
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We establish the strong convergence for the implicitS-iterative process associated with Lipschitzian hemicontractive mappings in Hilbert spaces.


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## Body

## 1. Introduction

LetH be a Hilbert space and let T:H→H be a mapping.The mappingT is called Lipshitzian if there exists L>0 such that
(1.1)∥Tx-Ty∥≤L∥x-y∥,∀x,y∈H.IfL=1, then T is called nonexpansive and if 0≤L<1, then T is called contractive.The mappingT is said to be pseudocontractive ([1, 2]) if
(1.2)∥Tx-Ty∥2≤∥x-y∥2+∥(I-T)x-(I-T)y∥2,∀x,y∈H,
and the mapping T is said to be strongly pseudocontractive if there exists k∈(0,1) such that
(1.3)∥Tx-Ty∥2≤∥x-y∥2+k∥(I-T)x-(I-T)y∥2,∀x,y∈H.LetF(T):={x∈H:Tx=x} and the mapping T is called hemicontractive if F(T)≠∅ and
(1.4)∥Tx-x*∥2≤∥x-x*∥2+∥x-Tx∥2,∀x∈H,x*∈F(T).It is easy to see the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractive mappings. For the importance of fixed points of pseudocontractions the reader may consult [1].In 1974, Ishikawa [3] proved the following result.Theorem 1.1.
LetK be a compact convex subset of a Hilbert space H and let T:K→K be a Lipschitzian pseudocontractive mapping.
For arbitraryx1∈K, let {xn} be a sequence defined iteratively by(1.5)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n≥1,
where {αn} and {βn} are sequences satisfying the conditions:(i)
0≤αn≤βn≤1,(ii)
limn→∞βn=0,(iii)
∑n=1∞‍αnβn=∞.
Then the sequence{xn} converges strongly to a fixed point of T.Another iteration scheme which has been studied extensively in connection with fixed points of pseudocontractive mappings.In 2011, Sahu [4] and Sahu and Petruşel [5] introduced the S-iterative process as follows.LetK be a nonempty convex subset of a normed space X and let T:K→K be a mapping. Then, for arbitrary x1∈K, the S-iterative process is defined by
(1.6)xn+1=Tyn,yn=(1-βn)xn+βnTxn,n≥1,
where {βn} is a real sequence in [0,1].In this paper, we establish the strong convergence for the implicitS-iterative process associated with Lipschitzian hemicontractive mappings in Hilbert spaces.


## 2. Main Results

We need the follwing lemma.Lemma 2.1 (see [6]).
For allx,y∈H and λ∈[0,1], the following well-known identity holds
(2.1)∥(1-λ)x+λy∥2=(1-λ)∥x∥2+λ∥y∥2-λ(1-λ)∥x-y∥2.Now we prove our main results.Theorem 2.2.
LetK be a compact convex subset of a real Hilbert space H and let T:K→K be a Lipschitzian hemicontractive mapping satisfying
(C)∥x-Ty∥≤∥Tx-Ty∥,∀x,y∈K.
Let{βn} be a sequence in [0,1] satisfying(iv)
∑n=1∞‍βn=∞,(v)
∑n=1∞‍βn2<∞.
For arbitraryx0∈K, let {xn} be a sequence defined iteratively by
(2.2)xn=Tyn,yn=(1-βn)xn-1+βnTxn,n≥1.
Then the sequence{xn} converges strongly to the fixed point x* of T.Proof.
From Schauder’s fixed point theorem,F(T) is nonempty since K is a convex compact set and T is continuous, let x*∈F(T). Using the fact that T is hemicontractive we obtain
(2.3)∥Txn-x*∥2≤∥xn-x*∥2+∥xn-Txn∥2,(2.4)∥Tyn-x*∥2≤∥yn-x*∥2+∥yn-Tyn∥2.
Now by (v), there existsn0∈ℕ such that for all n≥n0,
(2.5)βn≤min{13,1L2},
which implies that
(2.6)2βn1-βn≤1.
With the help of (2.2), (2.3), and Lemma 2.1, we obtain the following estimates:
(2.7)∥yn-x*∥2=∥(1-βn)xn-1+βnTxn-x*∥2=∥(1-βn)(xn-1-x*)+βn(Txn-x*)∥2=(1-βn)∥xn-1-x*∥2+βn∥Txn-x*∥2-βn(1-βn)∥xn-1-Txn∥2≤(1-βn)∥xn-1-x*∥2+βn(∥xn-x*∥2+∥xn-Txn∥2)-βn(1-βn)∥xn-1-Txn∥2,∥yn-Tyn∥2=∥(1-βn)xn-1+βnTxn-Tyn∥2=∥(1-βn)(xn-1-Tyn)+βn(Txn-Tyn)∥2=(1-βn)∥xn-1-Tyn∥2+βn∥Txn-Tyn∥2-βn(1-βn)∥xn-1-Txn∥2.
Substituting (2.7) in (2.4) we obtain
(2.8)∥Tyn-x*∥2≤(1-βn)∥xn-1-x*∥2+βn(∥xn-x*∥2+∥xn-Txn∥2)+(1-βn)∥xn-1-Tyn∥2+βn∥Txn-Tyn∥2-2βn(1-βn)∥xn-1-Txn∥2.
Also with the help of condition(C) and (2.8), we have
(2.9)∥xn+1-x*∥2=∥Tyn-x*∥2≤(1-βn)∥xn-1-x*∥2+βn(∥xn-x*∥2+∥xn-Txn∥2)+(1-βn)∥xn-1-Tyn∥2+βn∥Txn-Tyn∥2-2βn(1-βn)∥xn-1-Txn∥2≤(1-βn)∥xn-1-x*∥2+βn∥xn-x*∥2+(1-βn)∥xn-1-Tyn∥2+2βn∥Txn-Tyn∥2-2βn(1-βn)∥xn-1-Txn∥2,
which implies that
(2.10)∥xn+1-x*∥2≤∥xn-1-x*∥2+∥xn-1-Tyn∥2+2βn1-βn∥Txn-Tyn∥2-2βn∥xn-1-Txn∥2≤∥xn-1-x*∥2+∥xn-1-Tyn∥2+∥Txn-Tyn∥2-2βn∥xn-1-Txn∥2,
where
(2.11)∥xn-1-Tyn∥2≤∥Txn-1-Tyn∥2≤L2∥xn-1-yn∥2=L2βn2∥xn-1-Txn∥2,(2.12)∥Txn-Tyn∥2≤L2∥xn-yn∥2≤L2(∥xn-xn-1∥+∥xn-1-yn∥)2≤L2(∥xn-xn-1∥+βn∥xn-1-Txn∥)2≤L2(∥xn-xn-1∥+βnM)2,∥xn-xn-1∥=∥xn-1-Tyn∥≤∥Txn-1-Tyn∥≤L∥xn-1-yn∥=Lβn∥xn-1-Txn∥≤LβnM
and consequently from (2.12), we obtain
(2.13)∥Txn-Tyn∥2≤L2(1+L)2M2βn2.
Hence by (2.5), (2.10), (2.11), and (2.13), we have
(2.14)∥xn-x*∥2≤∥xn-1-x*∥2+L2βn2∥xn-1-Txn∥2+L2(1+L)2M2βn2-2βn∥xn-1-Txn∥2=∥xn-1-x*∥2+L2(1+L)2M2βn2-βn(2-L2βn)∥xn-1-Txn∥2≤∥xn-1-x*∥2+L2(1+L)2M2βn2-βn∥xn-1-Txn∥2,
which implies that
(2.15)βn∥xn-1-Txn∥2≤∥xn-1-x*∥2-∥xn-x*∥2+L2(1+L)2M2βn2,
so that
(2.16)12∑j=Nn‍βj∥xj-1-Txj∥2≤∥xN-x*∥2-∥xn-x*∥2+L2(1+L)2M2∑j=Nn‍βj2.
Hence by conditions (iv) and (v), we get(2.17)∑j=0∞‍∥xj-1-Txj∥2<∞.
It implies that(2.18)limn→∞∥xn-1-Txn∥=0.
Consider(2.19)∥xn-Txn∥≤∥xn-xn-1∥+∥xn-1-Txn∥,
which implies that
(2.20)limn→∞∥xn-Txn∥=0.
The rest of the argument follows exactly as in the proof of Theorem of [3]. This completes the proof.Theorem 2.3.
LetK be a compact convex subset of a real Hilbert space H and let T:K→K be a Lipschitzian hemicontractive mapping satisfying the condition (C). Let {βn} be a sequence in [0,1] satisfying the conditions (iv) and (v).
Assume thatPK:H→K be the projection operator of H onto K. Let {xn} be a sequence defined iteratively by
(2.21)xn=PK(Tyn),yn=PK((1-βn)xn-1+βnTxn),n≥1.
Then the sequence{xn} converges strongly to a fixed point of T.Proof.
The operatorPK is nonexpansive (see, e.g., [2]). K is a Chebyshev subset of H so that, PK is a single-valued mapping. Hence, we have the following estimate:
(2.22)∥xn-x*∥2=∥PK(Tyn)-PKx*∥2≤∥Tyn-x*∥2≤∥xn-1-x*∥2+L2(1+L)2M2βn2-βn∥xn-1-Txn∥2.
The setK=K∪T(K) is compact and so the sequence {∥xn-Txn∥} is bounded. The rest of the argument follows exactly as in the proof of Theorem 2.2. This completes the proof.Remark 2.4.
In main results, the condition(C) is not new and it is due to Liu et al. [7].



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*Source: 804745-2012-11-29.xml*