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Strong convergence of modified Halpern’s iterations for a k-strictly pseudocontractive mapping

  • Suhong Li
  • Lihua Li
  • Lingmin Zhang
  • Xiujuan He
Open Access
Research

Abstract

In this paper, we discuss three modified Halpern iterations as follows:

and obtained the strong convergence results of the iterations (I)-(III) for a k-strictly pseudocontractive mapping, where { α n } Open image in new window satisfies the conditions: (C1) lim n α n = 0 Open image in new window and (C2) n = 1 α n = + Open image in new window, respectively. The results presented in this work improve the corresponding ones announced by many other authors.

Keywords

Hilbert Space Nonexpansive Mapping Strong Convergence Real Hilbert Space Nonempty Closed Convex Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Let H be a real Hilbert space with the inner product , Open image in new window and the norm Open image in new window and let C be a nonempty closed convex subset of H.

Recall that a mapping T with domain D ( T ) Open image in new window and range R ( T ) Open image in new window in the Hilbert space H is called strongly pseudo-contractive if, for all x , y D ( T ) Open image in new window, there exists k ( 0 , 1 ) Open image in new window such that
T x T y , x y k x y 2 , Open image in new window
(1.1)
while T is said to be pseudo-contractive if (1.1) holds for k = 1 Open image in new window. A mapping T is said to be Lipschitzian if, for all x , y D ( T ) Open image in new window, there exists L > 0 Open image in new window such that
T x T y L x y . Open image in new window
(1.2)
A mapping T is called nonexpansive if (1.2) holds for L = 1 Open image in new window and, further, T is said to be contractive if L < 1 Open image in new window. T is said to be firmly nonexpansive if for all x , y D ( T ) Open image in new window,
T x T y 2 x y , T x T y . Open image in new window
Firmly nonexpansive mappings could be looked upon as an important subclass of nonexpansive mappings. A mapping T is called k-strictly pseudocontractive, if for all x , y D ( T ) Open image in new window, there exists λ > 0 Open image in new window such that
T x T y , x y x y 2 λ x y ( T x T y ) 2 . Open image in new window
(1.3)
Without loss of generality, we may assume that λ ( 0 , 1 ) Open image in new window. In Hilbert spaces H, (1.3) is equivalent to the inequality
T x T y 2 x y 2 + k ( I T ) x ( I T ) y 2 , k = ( 1 2 λ ) < 1 , Open image in new window

and we can assume also that k 0 Open image in new window so that k [ 0 , 1 ) Open image in new window.

It is obvious that a k-strictly pseudocontractive mapping is Lipschitzian with L = k + 1 k Open image in new window. The class of nonexpansive mappings is a subclass of strictly pseudocontractive mappings in a Hilbert space, but the converse implication may be false. We remark that the class of strongly pseudo-contractive mappings is independent from the class of k-strict pseudo-contractions.

In 1967, Halpern [1] was the first who introduced the following iteration scheme for a nonexpansive mapping T which was referred to as Halpern iteration: For any initialization x 0 C Open image in new window and any anchor u C Open image in new window, α n [ 0 , 1 ] Open image in new window,
x n + 1 = α n u + ( 1 α n ) T x n , n 0 . Open image in new window
(1.4)
He proved that the sequence (1.4) converges weakly to a fixed point of T, where α n = n a Open image in new window, a ( 0 , 1 ) Open image in new window. In 1977, Lions [2] further proved that the sequence (1.4) converges strongly to a fixed point of T in a Hilbert space, where { α n } Open image in new window satisfies the following conditions:
But, in [1, 2], the real sequence { α n } Open image in new window excluded the canonical choice α n = 1 n + 1 Open image in new window. In 1992, Wittmann [3] proved, still in Hilbert spaces, the strong convergence of the sequence (1.4) to a fixed point of T, where { α n } Open image in new window satisfies the following conditions:

The strong convergence of Halpern’s iteration to a fixed point of T has also been proved in Banach spaces; see, e.g., [4, 5, 6, 7, 8, 9, 10]. Reich [4, 5] has showed the strong convergence of the sequence (1.4), where { α n } Open image in new window satisfies the conditions (C1), (C2) and (C5), { α n } Open image in new window is decreasing (noting that the condition (C5) is a special case of condition(C4)). In 1997, Shioji and Takahashi [6] extended Wittmann’s result to Banach spaces. In 2002, Xu [9] obtained a strong convergence theorem, where { α n } Open image in new window satisfies the following conditions: (C1), (C2) and (C6) lim n | α n + 1 α n | α n + 1 = 0 Open image in new window. In particular, the canonical choice of α n = 1 n + 1 Open image in new window satisfies the conditions (C1), (C2) and (C6).

However, is a real sequence { α n } Open image in new window satisfying the conditions (C1) and (C2) sufficient to guarantee the strong convergence of Halpern’s iteration (1.4) for nonexpansive mappings? It remains an open question, see [1].

Some mathematicians considered the open question. In [11], Song proved that for a firmly nonexpansive mapping T, an important subclass of nonexpansive mappings, the answer of the Halpern open problem is affirmative. A partial answer to this question was given independently by Chidume and Chidume [12] and Suzuki [7]. They defined the sequence { x n } Open image in new window by
x n + 1 = α n u + ( 1 α n ) ( ( 1 δ ) x n + δ T x n ) , Open image in new window
(1.5)
where δ [ 0 , 1 ] Open image in new window, I is the identity, and obtained the strong convergence of the iteration (1.5), where { α n } Open image in new window satisfies the conditions (C1) and (C2). Recently, Xu [10] gave another partial answer to this question. He obtained the strong convergence of the iterative sequence
x n + 1 = α n ( ( 1 δ ) u + δ x n ) + ( 1 α n ) T x n , Open image in new window
(1.6)

where δ [ 0 , 1 ] Open image in new window and { α n } Open image in new window satisfies the conditions (C1) and (C2).

In [13], Liang-Gen Hu introduced the modified Halpern’s iteration: For any u , x 0 C Open image in new window, the sequence { x n } Open image in new window is defined by
x n + 1 = α n u + β n x n + γ n T x n , n 0 , Open image in new window
(1.7)

where { α n } Open image in new window, { β n } Open image in new window, { γ n } Open image in new window are three real sequences in [ 0 , 1 ] Open image in new window, satisfying α n + β n + γ n = 1 Open image in new window. He showed that the sequence { α n } Open image in new window satisfying the conditions (C1) and (C2) is sufficient to guarantee the strong convergence of the modified Halpern’s iterative sequence (1.7) for nonexpansive mappings.

The purpose of this paper is to present a significant answer to the above open question. We will show that the sequence { α n } Open image in new window satisfying the conditions (C1) and (C2) is sufficient to guarantee the strong convergence of the modified Halpern’s iterative sequences (1.5)-(1.7) for k-strictly pseudocontractive mappings, respectively. The results present in this paper improve and develop the corresponding results of [7, 10, 12, 13].

2 Preliminaries

In what follows we will need the following:

Lemma 2.1 [14]

Let H be a real Hilbert space, then the following well-known results hold:
  1. (i)

    t x + ( 1 t ) y 2 = t x 2 + ( 1 t ) y 2 t ( 1 t ) x y 2 Open image in new window for all x , y H Open image in new window and for all t [ 0 , 1 ] Open image in new window.

     
  2. (ii)

    x + y 2 x 2 + 2 y , x + y Open image in new window, x , y H Open image in new window.

     
Let C be a nonempty closed convex subset of a real Hilbert space H. The nearest point projection P C : H C Open image in new window defined from H onto C is the function which assigns to each x H Open image in new window its nearest point denoted by P C x Open image in new window in C. Thus P C x Open image in new window is the unique point in C such that
x P C x x y , y C . Open image in new window
It is known that for each x H Open image in new window,
x P C x , y P C x 0 , y C . Open image in new window
(2.1)

Lemma 2.2 [15]

Let C be a nonempty closed convex subset of a real Hilbert space H, T : C C Open image in new window be a k-strictly pseudocontractive mapping. Then ( I T ) Open image in new window is demiclosed at zero.

Lemma 2.3 [9]

Let { a n } Open image in new window be a sequence of nonnegative real numbers such that a n + 1 ( 1 δ n ) a n + δ n ξ n Open image in new window, n 0 Open image in new window, where { δ n } Open image in new window is a sequence in [ 0 , 1 ] Open image in new window and { ξ n } Open image in new window is a sequence in R satisfying the following conditions:
  1. (i)

    n = 1 δ n = + Open image in new window;

     
  2. (ii)

    lim sup n ξ n 0 Open image in new window or n = 1 δ n | ξ n | < + Open image in new window.

     

Then lim n a n = 0 Open image in new window.

3 Main results

In this section, proving the following theorems, we show that the conjunction of (C1) and (C2) is a sufficient condition on our iteration (I)-(III), respectively.

Theorem 3.1 Let C be a closed and convex subset of a real Hilbert space H, T : C C Open image in new window be a k-strictly pseudocontractive mapping such that F ( T ) Open image in new window. For an arbitrary initial value x 0 C Open image in new window and fixed anchor u C Open image in new window, define iteratively a sequence { x n } Open image in new window as follows:

where { α n } Open image in new window, { β n } Open image in new window, { γ n } Open image in new window are three real sequences in ( 0 , 1 ) Open image in new window, satisfying α n + β n + γ n = 1 Open image in new window and 0 < k < β n β n + γ n Open image in new window. Suppose that { α n } Open image in new window satisfies the conditions: (C1) lim n α n = 0 Open image in new window; (C2) n = 1 α n = + Open image in new window. Then as n Open image in new window, { x n } Open image in new window converges strongly to some fixed point x Open image in new window of T and x = P F ( T ) u Open image in new window, where P F ( T ) Open image in new window is the metric projection from H onto F ( T ) Open image in new window.

Proof Firstly, we show that { x n } Open image in new window is bounded. Rewrite the iterative process (I) as follows:
x n + 1 = α n u + β n x n + γ n T x n = α n u + ( 1 α n ) β n x n + γ n T x n 1 α n = α n u + ( 1 α n ) y n , Open image in new window
(3.1)
where y n = β n x n + γ n T x n 1 α n Open image in new window. Take any p F ( T ) Open image in new window, then, from Lemma 2.1 and (3.1), we estimate as follows:
By induction,
x n + 1 p 2 max { u p 2 , x 0 p 2 } . Open image in new window

This proves the boundedness of the sequence { x n } Open image in new window, which leads to the boundedness of { T x n } Open image in new window.

Next, we claim that
lim n x n T x n = 0 . Open image in new window
In fact, we have from (3.2) (for some appropriate constant M > 0 Open image in new window) that
which implies that
( β n 1 α n k ) γ n x n T x n 2 α n M x n p 2 x n + 1 p 2 . Open image in new window
(3.3)
If ( β n 1 α n k ) γ n x n T x n 2 α n M 0 Open image in new window, then

and hence the desired result is obtained by the condition (C1) and 0 < k < β n 1 α n Open image in new window.

If ( β n 1 α n k ) γ n x n T x n 2 α n M > 0 Open image in new window, then following (3.3), we have
n = 0 m [ ( β n 1 α n k ) γ n x n T x n 2 α n M ] x 0 p 2 x m p 2 x 0 p 2 . Open image in new window
Then
n = 0 [ ( β n 1 α n k ) γ n x n T x n 2 α n M ] < + . Open image in new window
Thus
lim n [ ( β n 1 α n k ) γ n x n T x n 2 α n M ] = 0 , Open image in new window
and hence
lim n x n T x n = 0 . Open image in new window
(3.4)
In order to prove x n x = P F ( T ) u Open image in new window, we next show that
lim sup n P F ( T ) u u , P F ( T ) u x n + 1 0 . Open image in new window
Indeed, we can take a subsequence { x n k } Open image in new window of { x n } Open image in new window such that
lim sup n P F ( T ) u u , P F ( T ) u x n + 1 = lim k P F ( T ) u u , P F ( T ) u x n k + 1 . Open image in new window

We may assume that x n k z Open image in new window since H is reflexive and { x n } Open image in new window is bounded. From (3.4), it follows from Lemma 2.2 that z F ( T ) Open image in new window.

From (2.1), we conclude
lim sup n P F ( T ) u u , P F ( T ) u x n + 1 = lim k P F ( T ) u u , P F ( T ) u x n k + 1 = P F ( T ) u u , P F ( T ) u z 0 . Open image in new window
(3.5)
Finally, we show that x n P F ( T ) u Open image in new window. As a matter of fact, from Lemma 2.1 and (3.1), we obtain
It follows from the conditions (C1), (C2) and (3.5), using Lemma 2.3, that
lim n x n P F ( T ) u = 0 . Open image in new window

This completes the proof of Theorem 3.1. □

Theorem 3.2 Let C be a closed and convex subset of a real Hilbert space H, T : C C Open image in new window be a k-strictly pseudocontractive mapping such that F ( T ) Open image in new window. For an arbitrary initial value x 0 C Open image in new window and fixed anchor u C Open image in new window, define iteratively a sequence { x n } Open image in new window as follows:

where { α n } ( 0 , 1 ) Open image in new window, 0 < k < α n δ Open image in new window. Suppose that { α n } Open image in new window satisfies the conditions: (C1) lim n α n = 0 Open image in new window; (C2) n = 1 α n = Open image in new window. Then as n Open image in new window, { x n } Open image in new window converges strongly to some fixed point x Open image in new window of T and x = P F ( T ) u Open image in new window, where P F ( T ) Open image in new window is the metric projection from H onto F ( T ) Open image in new window.

Proof Firstly, we show that { x n } Open image in new window is bounded. Rewrite the iterative process (II) as follows:
y n = ( 1 δ ) u + δ x n , x n + 1 = α n y n + ( 1 α n ) T x n . Open image in new window
(3.6)
Take any p F ( T ) Open image in new window, then, from Lemma 2.1 and (3.6), we estimate as follows:
By induction,
x n + 1 p 2 max { u p 2 , x 0 p 2 } . Open image in new window

This proves the boundedness of the sequence { x n } Open image in new window, which implies that the sequence { T x n } Open image in new window is bounded also.

Using the same technique as in Theorem 3.1, we can prove
lim n x n T x n = 0 Open image in new window
and
lim sup n P F ( T ) u u , P F ( T ) u x n + 1 0 . Open image in new window
Finally, we show that x n P F ( T ) u Open image in new window. Writing z n = α n δ x n + ( 1 α n ) T x n 1 ( 1 δ ) α n Open image in new window, then
x n + 1 = ( 1 δ ) α n u + [ 1 ( 1 δ ) α n ] z n , Open image in new window
from Lemma 2.1 and (3.1), we obtain
It follows from the conditions (C1), (C2) and (3.5), using Lemma 2.3, that
lim n x n P F ( T ) u = 0 . Open image in new window

This completes the proof of Theorem 3.2. □

Theorem 3.3 Let C be a closed and convex subset of a real Hilbert space H, T : C C Open image in new window be a k-strictly pseudocontractive mapping such that F ( T ) Open image in new window. For an arbitrary initial value x 0 C Open image in new window and fixed anchor u C Open image in new window, define iteratively a sequence { x n } Open image in new window as follows:

where T β = β I + ( 1 β ) T Open image in new window, { α n } [ 0 , 1 ] Open image in new window, β ( k , 1 ) Open image in new window. Suppose that { α n } Open image in new window satisfies the conditions: (C1) lim n α n = 0 Open image in new window; (C2) n = 1 α n = Open image in new window. Then as n Open image in new window, { x n } Open image in new window converges strongly to some fixed point x Open image in new window of T, and x = P F ( T ) u Open image in new window, where P F ( T ) Open image in new window is the metric projection from H onto F ( T ) Open image in new window.

Proof It is easy to see that F ( T β ) = F ( T ) Open image in new window. For any x , y C Open image in new window, we have
Thus, for all x C Open image in new window and for all p F ( T β ) = F ( T ) Open image in new window, we have
T β x p 2 x p 2 ( β k ) x T β x 2 . Open image in new window

This implies that T β Open image in new window is a quasi-firmly type nonexpansive mapping (see, for example, [11]). T β Open image in new window is also a strongly quasi-nonexpansive mapping (see, for example, [16]). Hence it follows from [11, 16] (see Theorem 3.1 and Remark 1 of [11] or Corollary 8 of [16]) that { x n } Open image in new window converges strongly to a point x F ( T β ) = F ( T ) Open image in new window.

Finally, we show x = P F ( T ) u Open image in new window. From Lemma 2.1 and the iterative process (III), we estimate as follows:
It follows from the conditions (C1), (C2) and
lim n u P F ( T ) u , x n + 1 P F ( T ) u = u P F ( T ) u , x P F ( T ) u 0 , Open image in new window
using Lemma 2.3, that
lim n x n P F ( T ) u = 0 . Open image in new window

This completes the proof of Theorem 3.3. □

Remark 3.1 Theorems 3.1-3.3 improve the main results of [7, 10, 12, 13] from a nonexpansive mapping to a k-strictly pseudocontractive mapping, respectively. Theorems 3.1-3.3 show that the real sequence { α n } Open image in new window satisfying the two conditions (C1) and (C2) is sufficient for the strong convergence of the iterative sequences (I)-(III) for k-strictly pseudocontractive mappings, respectively. Therefore, our results give a significant partial answer to the open question.

Notes

Acknowledgements

The authors are very grateful to the referees for their careful reading, comments and suggestions, which improved the presentation of this article. The first author was supported by the Natural Science Foundational Committee of Hebei Province (Z2011113) and Hebei Normal University of Science and Technology (ZDJS 2009 and CXTD2010-05).

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Copyright information

© Li 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.

Authors and Affiliations

  • Suhong Li
    • 1
    • 2
  • Lihua Li
    • 1
  • Lingmin Zhang
    • 1
  • Xiujuan He
    • 3
  1. 1.College of Mathematics and Information TechnologyHebei Normal University of Science and TechnologyQinhuangdaoChina
  2. 2.Institute of Mathematics and Systems ScienceHebei Normal University of Science and TechnologyQinhuangdaoChina
  3. 3.Hebei Business and Trade SchoolShijiazhuangChina

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