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# Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces

## Abstract

In this paper, we establish strong convergence for the iterative scheme introduced by Sahu and Petruşel associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

MSC:47H10, 47J25.

## 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

$∥Tx−Ty∥≤L∥x−y∥,∀x,y∈H.$

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

The mapping $T:H→H$ is said to be pseudocontractive (see, for example, [1, 2]) if

$∥ T x − T y ∥ 2 ≤ ∥ x − y ∥ 2 + ∥ ( I − T ) x − ( I − T ) y ∥ 2 ,∀x,y∈H$
(1.1)

and it is said to be strongly pseudocontractive if there exists $k∈(0,1)$ such that

$∥ T x − T y ∥ 2 ≤ ∥ x − y ∥ 2 +k ∥ ( I − T ) x − ( I − T ) y ∥ 2 ,∀x,y∈H.$
(1.2)

Let $F(T):={x∈H:Tx=x}$, and let K be a nonempty subset of H. A mapping $T:K→K$ is called hemicontractive if $F(T)≠∅$ and

$∥ T x − x ∗ ∥ 2 ≤ ∥ x − x ∗ ∥ 2 + ∥ x − T x ∥ 2 ,∀x∈H, x ∗ ∈F(T).$

It is easy to see that the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractions. For the importance of fixed points of pseudocontractions, the reader may consult .

In 1974, Ishikawa  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 the Ishikawa iterative scheme

${ x n + 1 = ( 1 − α n ) x n + α n T y n , y n = ( 1 − β n ) x n + β n T x n , n ≥ 1 ,$
(1.3)

where ${ α n }$ and ${ β n }$ are sequences satisfying the conditions

1. (i)
$0≤ α n ≤ β n ≤1$

;

2. (ii)
$lim n → ∞ β n =0$

;

3. (iii)
$∑ n = 1 ∞ α n β n =∞$

.

Then the sequence ${ x n }$ converges strongly to a fixed point of T.

Another iterative scheme which has been studied extensively in connection with fixed points of pseudocontractive mappings is the S-iterative scheme introduced by Sahu and Petruşel  in 2011.

In this paper, we establish strong convergence for the S-iterative scheme associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

## 2 Main results

We need the following lemma.

Lemma 2.1 

For all $x,y∈H$ and $λ∈[0,1]$, the following well-known identity holds:

$∥ ( 1 − λ ) x + λ y ∥ 2 =(1−λ) ∥ x ∥ 2 +λ ∥ y ∥ 2 −λ(1−λ) ∥ x − y ∥ 2 .$

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 Let ${ β n }$ be a sequence in $[0,1]$ satisfying

1. (iv)
$∑ n = 1 ∞ β n =∞$

;

2. (v)
$lim n → ∞ β n =0$

.

For arbitrary $x 1 ∈K$, let ${ x n }$ be a sequence defined iteratively by the S-iterative scheme

${ x n + 1 = T y n , y n = ( 1 − β n ) x n + β n T x n , n ≥ 1 .$
(2.1)

Then the sequence ${ x n }$ converges strongly to the fixed point of T.

Proof From Schauder’s fixed point theorem, $F(T)$ is nonempty since K is a compact convex set and T is continuous. Let $x ∗ ∈F(T)$. Using the fact that T is hemicontractive, we obtain

$∥ T x n − x ∗ ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 + ∥ x n − T x n ∥ 2$
(2.2)

and

$∥ T y n − x ∗ ∥ 2 ≤ ∥ y n − x ∗ ∥ 2 + ∥ y n − T y n ∥ 2 .$
(2.3)

With the help of (2.1), (2.2) and Lemma 2.1, we obtain the following estimates: (2.4) (2.5)

Substituting (2.4) and (2.5) in (2.3) we obtain

$∥ T y n − x ∗ ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 + ( 1 − β n ) ∥ x n − T y n ∥ 2 + β n ∥ T x n − T y n ∥ 2 − β n ( 1 − 2 β n ) ∥ x n − T x n ∥ 2 .$
(2.6)

Also, with the help of condition (C) and (2.6), we have

$∥ x n + 1 − x ∗ ∥ 2 = ∥ T y n − x ∗ ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 + ( 1 − β n ) ∥ x n − T y n ∥ 2 + β n ∥ T x n − T y n ∥ 2 − β n ( 1 − 2 β n ) ∥ x n − T x n ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 + ∥ T x n − T y n ∥ 2 − β n ( 1 − 2 β n ) ∥ x n − T x n ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 + L 2 ∥ x n − y n ∥ 2 − β n ( 1 − 2 β n ) ∥ x n − T x n ∥ 2 = ∥ x n − x ∗ ∥ 2 + L 2 β n 2 ∥ x n − T x n ∥ 2 − β n ( 1 − 2 β n ) ∥ x n − T x n ∥ 2 = ∥ x n − x ∗ ∥ 2 − β n ( 1 − ( 2 + L 2 ) β n ) ∥ x n − T x n ∥ 2 .$
(2.7)

Now, by $lim n → ∞ β n =0$, there exists $n 0 ∈N$ such that for all $n≥ n 0$,

$β n ≤ 1 2 ( 2 + L 2 ) ,$
(2.8)

and with the help of (2.8), (2.7) yields

$∥ x n + 1 − x ∗ ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 − 1 2 β n ∥ x n − T x n ∥ 2 ,$

which implies

$1 2 β n ∥ x n − T x n ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 − ∥ x n + 1 − x ∗ ∥ 2 ,$

so that

$1 2 ∑ j = N n β j ∥ x j − T x j ∥ 2 ≤ ∥ x N − x ∗ ∥ 2 − ∥ x n + 1 − x ∗ ∥ 2 .$

The rest of the argument follows exactly as in the proof of theorem of . 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 condition (C). Let ${ β n }$ be a sequence in $[0,1]$ satisfying conditions (iv) and (v).

Assume that $P K :H→K$ is the projection operator of H onto K. Let ${ x n }$ be a sequence defined iteratively by

${ x n + 1 = P K ( T y n ) , y n = P K ( ( 1 − β n ) x n + β n T x n ) , n ≥ 1 .$

Then the sequence ${ x n }$ converges strongly to a fixed point of T.

Proof The operator $P K$ is nonexpansive (see, e.g., ). K is a Chebyshev subset of H so that $P K$ is a single-valued mapping. Hence, we have the following estimate:

$∥ x n + 1 − x ∗ ∥ 2 = ∥ P K ( T y n ) − P K x ∗ ∥ 2 ≤ ∥ T y n − x ∗ ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 − β n ( 1 − ( 2 + L 2 ) β n ) ∥ x n − T x n ∥ 2 .$

The set $K=K∪T(K)$ is compact, and so the sequence ${∥ x n −T x n ∥}$ 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 Theorem 1.1, putting $α n =1$, $0≤ α n ≤ β n ≤1$ implies $β n =1$, which contradicts $lim n → ∞ β n =0$. Hence the S-iterative scheme is not the special case of Ishikawa iterative scheme.

Remark 2.5 In Theorems 2.2 and 2.3, condition (C) is not new; it is due to Liu et al. .

## References

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Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear Functional Analysis. Am. Math. Soc., Providence; 1976.

2. 2.

Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6

3. 3.

Ishikawa S: Fixed point by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5

4. 4.

Sahu DR, Petruşel A: Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces. Nonlinear Anal. 2011, 74: 6012–6023. 10.1016/j.na.2011.05.078

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Liu Z, Feng C, Ume JS, Kang SM: Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings. Taiwan. J. Math. 2007, 11: 27–42.

## Acknowledgements

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

## Author information

Correspondence to Sunhong Lee.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors read and approved the final manuscript.

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