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

1. Introduction

Let H be a Hilbert space and let T:H→H be a mapping.

The mapping T is called Lipshitzian if there exists L>0 such that
(1.1)∥Tx-Ty∥≤L∥x-y∥,∀x,y∈H.

If L=1, then T is called nonexpansive and if 0≤L<1, then T is called contractive.

The mapping T 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.

Let F(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.

Let K be a compact convex subset of a Hilbert space H and let T:K→K be a Lipschitzian pseudocontractive mapping.

For arbitrary x1∈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:

0≤αn≤βn≤1,

limn→∞βn=0,

∑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.

Let K 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 implicit S-iterative process associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

2. Main Results

We need the follwing lemma.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

For all x,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.

Let K 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

∑n=1∞βn=∞,

∑n=1∞βn2<∞.

For arbitrary x0∈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 exists n0∈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.

Let K 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 that PK: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 operator PK 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 set K=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].

Acknowledgment

The authors would like to thank the referees for thier useful comments and suggestions.

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