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

Journal of Inequalities and Applications20132013:132

https://doi.org/10.1186/1029-242X-2013-132

  • Received: 24 October 2012
  • Accepted: 9 March 2013
  • Published:

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.

Keywords

  • iterative scheme
  • Lipschitzian mappings
  • hemicontractive mappings
  • 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
T x T y 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 : T x = 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 [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 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 [4] 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 [5]

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 [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 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., [2]). 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. [6].

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea
(2)
School of CS and Mathematics, Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, 54660, Pakistan

References

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Copyright

© Kang et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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