Strong Convergence of an Implicit S-Iterative Process for Lipschitzian Hemicontractive Mappings

and Applied Analysis 3 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 ∥ ∥x − Ty∥∥ ≤ ∥∥Tx − Ty∥∥, ∀x, y ∈ K. C Let {βn} be a sequence in 0, 1 satisfying iv ∑∞ n 1 βn ∞, v ∑∞ n 1 β 2 n < ∞. For arbitrary x0 ∈ K, let {xn} be a sequence defined iteratively by xn Tyn, yn ( 1 − βn ) xn−1 βnTxn, n ≥ 1. 2.2 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 ‖Txn − x∗‖2 ≤ ‖xn − x∗‖2 ‖xn − Txn‖, 2.3 ∥ Tyn − x∗ ∥ ∥ 2 ≤ ∥∥yn − x∗ ∥ ∥ 2 ∥ yn − Tyn ∥ ∥ 2 . 2.4 Now by v , there exists n0 ∈ N such that for all n ≥ n0, βn ≤ min { 1 3 , 1 L2 } , 2.5


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 Tx − Ty ≤ L x − y , ∀x, y ∈ H.

1.1
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 and the mapping T is said to be strongly pseudocontractive if there exists k ∈ 0, 1 such that Let F T : {x ∈ H : Tx x} and the mapping T is called hemicontractive if F T / ∅ and 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 x 1 ∈ K, let {x n } be a sequence defined iteratively by where {α n } and {β n } are sequences satisfying the conditions: Then the sequence {x n } 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 Petrus ¸el 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 x 1 ∈ K, the S-iterative process is defined by where {β n } 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.

Main Results
We need the follwing lemma.Lemma 2.1 see 6 .For all x,y ∈ H and λ ∈ 0, 1 , the following well-known identity holds

2.1
Abstract and Applied Analysis 3 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 For arbitrary x 0 ∈ K, let {x n } be a sequence defined iteratively by

2.2
Then the sequence {x n } 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

2.4
Now by v , there exists n 0 ∈ N such that for all n ≥ n 0 , which implies that With the help of 2.2 , 2.3 , and Lemma 2.1, we obtain the following estimates:

2.7
Substituting 2.7 in 2.4 we obtain

2.8
Also with the help of condition C and 2.8 , we have 2.9 which implies that

2.11
2.12 and consequently from 2.12 , we obtain

2.16
Hence by conditions iv and v , we get ∞ j 0 x j−1 − Tx j 2 < ∞.

2.18
Consider x n − Tx n 0.

2.20
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 {β n } be a sequence in 0, 1 satisfying the conditions (iv) and (v).
Assume that P K : H → K be the projection operator of H onto K. Let {x n } be a sequence defined iteratively by

2.21
Then the sequence {x n } converges strongly to a fixed point of T .
Proof.The operator P K is nonexpansive see, e.g., 2 .K is a Chebyshev subset of H so that, P K is a single-valued mapping.Hence, we have the following estimate:

2.22
The set K K ∪ T K is compact and so the sequence { x n − Tx n } 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 .