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Fixed Point Theory and Applications

, 2010:907275 | Cite as

Strong Convergence of a Generalized Iterative Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces

  • Husain Piri
  • Hamid Vaezi
Open Access
Research Article

Abstract

Using Open image in new window -strongly accretive and Open image in new window -strictly pseudocontractive mapping, we introduce a general iterative method for finding a common fixed point of a semigroup of non-expansive mappings in a Hilbert space, with respect to a sequence of left regular means defined on an appropriate space of bounded real-valued functions of the semigroup. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality.

Keywords

Hilbert Space Banach Space Variational Inequality Iterative Algorithm Nonexpansive Mapping 
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 Open image in new window be a real Hilbert space. A mapping Open image in new window of Open image in new window into itself is called non-expansive if Open image in new window , for all Open image in new window . By Open image in new window , we denote the set of fixed points of Open image in new window (i.e., Open image in new window ).

Mann [1] introduced an iteration procedure for approximation of fixed points of a non-expansive mapping Open image in new window on a Hilbert space as follows. Let Open image in new window and

where Open image in new window is a sequence in Open image in new window . See also [2].

On the other hand, Moudafi [3] introduced the viscosity approximation method for fixed point of non-expansive mappings (see [4] for further developments in both Hilbert and Banach spaces). Let Open image in new window be a contraction on a Hilbert space Open image in new window (i.e., Open image in new window for all Open image in new window and Open image in new window ). Starting with an arbitrary initial Open image in new window , define a sequence Open image in new window recursively by
where Open image in new window is sequence in Open image in new window . It is proved in [3, 4] that, under appropriate condition imposed on Open image in new window , the sequence Open image in new window generated by (1.2) converges strongly to the unique solution Open image in new window in Open image in new window of the variational inequality:
Assume that Open image in new window is strongly positive, that is, there is a constant Open image in new window with the property
In [4] (see also [5]), it is proved that the sequence Open image in new window defined by the iterative method below, with the initial guess Open image in new window chosen arbitrarily,
converges strongly to the unique solution of the minimization problem
provided that the sequence Open image in new window satisfies certain conditions. Marino and Xu [6] combined the iterative (1.5) with the viscosity approximation method (1.2) and considered the following general iterative methods:

where Open image in new window . They proved that if Open image in new window is a sequence in Open image in new window satisfying the following conditions:

Open image in new window

Open image in new window

either Open image in new window or Open image in new window ,

then, the sequence Open image in new window generated by (1.7) converges strongly, as Open image in new window , to the unique solution of the variational inequality:
which is the optimality condition for minimization problem

where Open image in new window is a potential function for Open image in new window (i.e., Open image in new window , for all Open image in new window ).

Let Open image in new window be the topological dual of a Banach space Open image in new window . The value of Open image in new window at Open image in new window will be denoted by Open image in new window or Open image in new window . With each Open image in new window , we associate the set

Using the Hahn-Banach theorem, it is immediately clear that Open image in new window for each Open image in new window . The multivalued mapping Open image in new window from Open image in new window into Open image in new window is said to be the (normalized) duality mapping. A Banach space Open image in new window is said to be smooth if the duality mapping Open image in new window is single valued. As it is well known, the duality mapping is the identity when Open image in new window is a Hilbert space; see [7].

Let Open image in new window and Open image in new window be two positive real numbers such that Open image in new window . Recall that a mapping Open image in new window with domain Open image in new window and range Open image in new window in Open image in new window is called Open image in new window -strongly accretive if, for each Open image in new window , there exists Open image in new window such that
Recall also that a mapping Open image in new window is called Open image in new window -strictly pseudo-contractive if, for each Open image in new window , there exists Open image in new window such that
It is easy to see that (1.12) can be rewritten as

see [8].

In this paper, motivated and inspired by Atsushiba and Takahashi [9], Lau et al. [10], Marino and Xu [6] and Xu [4, 11], we introduce the iterative below, with the initial guess Open image in new window chosen arbitrarily,
where Open image in new window is Open image in new window -strongly accretive and Open image in new window -strictly pseudo-contractive with Open image in new window , Open image in new window is a contraction on a Hilbert space Open image in new window with coefficient Open image in new window , Open image in new window is a positive real number such that Open image in new window , and Open image in new window is a non-expansive semigroup on Open image in new window such that the set Open image in new window of common fixed point of Open image in new window is nonempty, Open image in new window is a subspace of Open image in new window such that Open image in new window and the mapping Open image in new window is an element of Open image in new window for each Open image in new window , and Open image in new window is a sequence of means on Open image in new window . Our purpose in this paper is to introduce this general iterative algorithm for approximating a common fixed points of semigroups of non-expansive mappings which solves some variational inequality. We will prove that if Open image in new window is left regular and Open image in new window is a sequence in Open image in new window satisfying the conditions Open image in new window and Open image in new window , then Open image in new window converges strongly to Open image in new window , which solves the variational inequality:

Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of non-expansive mappings are also presented. It is worth mentioning that we obtain our result without assuming condition Open image in new window .

2. Preliminaries

Let Open image in new window be a semigroup and let Open image in new window be the space of all bounded real-valued functions defined on Open image in new window with supremum norm. For Open image in new window and Open image in new window , we define elements Open image in new window and Open image in new window in Open image in new window by
Let Open image in new window be a subspace of Open image in new window containing Open image in new window , and let Open image in new window be its dual. An element Open image in new window in Open image in new window is said to be a mean on Open image in new window if Open image in new window . We often write Open image in new window instead of Open image in new window for Open image in new window and Open image in new window . Let Open image in new window be left invariant (resp., right invariant), that is, Open image in new window (resp., Open image in new window ) for each Open image in new window . A mean Open image in new window on Open image in new window is said to be left invariant (right invariant) if Open image in new window (resp. Open image in new window ) for each Open image in new window and Open image in new window . Open image in new window is said to be left (resp., right) amenable if Open image in new window has a left (resp., right) invariant mean. Open image in new window is amenable if Open image in new window is both left and right amenable. As it is well known, Open image in new window is amenable when Open image in new window is a commutative semigroup; see [12]. A net Open image in new window of means on Open image in new window is said to be left regular if

for each Open image in new window , where Open image in new window is the adjoint operator of Open image in new window .

Let Open image in new window be a nonempty closed and convex subset of a reflexive Banach space Open image in new window . A family Open image in new window of mapping from Open image in new window into itself is said to be a non-expansive semigroup on Open image in new window if Open image in new window is non-expansive and Open image in new window for each Open image in new window . We denote by Open image in new window the set of common fixed points of Open image in new window , that is,

The open ball of radius Open image in new window centered at Open image in new window is denoted by Open image in new window . For subset Open image in new window of Open image in new window , by Open image in new window , we denote the closed convex hull of Open image in new window . Weak convergence is denoted by Open image in new window , and strong convergence is denoted by Open image in new window .

Lemma 2.1 (see [12, 13]).

Let f be a function of semigroup Open image in new window into a reflexive Banach space Open image in new window such that the weak closure of Open image in new window is weakly compact, and let Open image in new window be a subspace of Open image in new window containing all functions Open image in new window with Open image in new window . Then, for any Open image in new window , there exists a unique element Open image in new window in Open image in new window such that

One can write Open image in new window by Open image in new window

Lemma 2.2 (see [13]).

Let Open image in new window be a closed convex subset of a Hilbert space Open image in new window , Open image in new window a semigroup from Open image in new window into Open image in new window such that Open image in new window , the mapping Open image in new window an element of Open image in new window for each Open image in new window and Open image in new window , and Open image in new window a mean on Open image in new window . If one writes Open image in new window instead of Open image in new window , then the following holds.

(i) Open image in new window is non-expansive mapping from Open image in new window into Open image in new window .

(ii) Open image in new window for each Open image in new window .

(iii) Open image in new window for each Open image in new window .

(iv)If Open image in new window is left invariant, then Open image in new window is a non-expansive retraction from Open image in new window onto Open image in new window .

Let Open image in new window be a nonempty subset of a normed space Open image in new window , and let Open image in new window . An element Open image in new window is said to be the best approximation to Open image in new window if
where Open image in new window . The number Open image in new window is called the distance from Open image in new window to Open image in new window or the error in approximating Open image in new window by Open image in new window . The (possibly empty) set of all best approximation from Open image in new window to Open image in new window is denoted by

This defines a mapping Open image in new window from Open image in new window into Open image in new window and is called metric (the nearest point) projection onto Open image in new window .

Lemma 2.3 (see [7]).

Let Open image in new window be a nonempty convex subset of a smooth Banach space Open image in new window and let Open image in new window and Open image in new window . Then, the following is equivalent.

(i) Open image in new window is the best approximation to Open image in new window .

(ii) Open image in new window is a solution of the variational inequality
Let Open image in new window be a nonempty subset of a Banach space Open image in new window and Open image in new window a mapping. Then Open image in new window is said to be demiclosed at Open image in new window if, for any sequence Open image in new window in Open image in new window , the following implication holds:

Lemma 2.4 (see [14]).

Let Open image in new window be a nonempty closed convex subset of a Hilbert space Open image in new window and suppose that Open image in new window is non-expansive. Then, the mapping Open image in new window is demiclosed at zero.

The following lemma is well known.

Lemma 2.5.

Let Open image in new window be a real Hilbert space. Then, for all Open image in new window

(i) Open image in new window

(ii) Open image in new window

Lemma 2.6 (see [11]).

Let Open image in new window be a sequence of nonnegative real numbers such that

where Open image in new window and Open image in new window are sequences of real numbers satisfying the following conditions:

(i) Open image in new window

(ii)either Open image in new window or Open image in new window

Then, Open image in new window

The following lemma will be frequently used throughout the paper. For the sake of completeness, we include its proof.

Lemma 2.7.

Let Open image in new window be a real smooth Banach space and Open image in new window a mapping.

(i)If Open image in new window is Open image in new window -strongly accretive and Open image in new window -strictly pseudo-contractive with Open image in new window , then, Open image in new window is contractive with constant Open image in new window .

(ii)If Open image in new window is Open image in new window -strongly accretive and Open image in new window -strictly pseudo-contractive with Open image in new window , then, for any fixed number Open image in new window , Open image in new window is contractive with constant Open image in new window .

Proof.
  1. (i)
    From (1.11) and (1.13), we obtain

    and, therefore, Open image in new window is contractive with constant Open image in new window .

     
  2. (ii)
    Because Open image in new window is contractive with constant Open image in new window , for each fixed number Open image in new window , we have
     

This shows that Open image in new window is contractive with constant Open image in new window .

Throughout this paper, Open image in new window will denote a Open image in new window -strongly accretive and Open image in new window -strictly pseudo-contractive mapping with Open image in new window , and Open image in new window is a contraction with coefficient Open image in new window on a Hilbert space Open image in new window . We will also always use Open image in new window to mean a number in Open image in new window .

3. Strong Convergence Theorem

The following is our main result.

Theorem 3.1.

Let Open image in new window be a non-expansive semigroup on a real Hilbert space Open image in new window such that Open image in new window . Let Open image in new window be a left invariant subspace of Open image in new window such that Open image in new window , and the function Open image in new window is an element of Open image in new window for each Open image in new window . Let Open image in new window be a left regular sequence of means on Open image in new window , and let Open image in new window be a sequence in Open image in new window such that Open image in new window and Open image in new window . Let Open image in new window and Open image in new window be generated by the iteration algorithm (1.14). Then, Open image in new window converges strongly, as Open image in new window , to Open image in new window , which is a unique solution of the variational inequality (1.15). Equivalently, one has

Proof.

First, we claim that Open image in new window is bounded. Let Open image in new window ; by Lemmas 2.2 and 2.7 we have
By induction,

Therefore, Open image in new window is bounded and so is Open image in new window .

Set Open image in new window . We remark that Open image in new window is Open image in new window -invariant bounded closed convex set and Open image in new window . Now we claim that
Also by [15, Corollary Open image in new window ], there exists a natural number Open image in new window such that
By Lemma 2.2 we have
It follows from (3.5), (3.6), (3.7), and (3.8) that
Since Open image in new window is arbitrary, we get (3.4). In this stage, we will show that
Let Open image in new window and Open image in new window . Then, there exists Open image in new window , which satisfies (3.5). Take
for all Open image in new window . Therefore, we have
for all Open image in new window . This shows that

Since Open image in new window is arbitrary, we get (3.11).

Let Open image in new window . Then Open image in new window is a contraction of Open image in new window into itself. In fact, we see that

and hence Open image in new window is a contraction due to Open image in new window

Therefore, by Banach contraction principal, Open image in new window has a unique fixed point Open image in new window . Then using Lemma 2.3, Open image in new window is the unique solution of the variational inequality
We show that
Indeed, we can choose a subsequence Open image in new window of Open image in new window such that
Because Open image in new window is bounded, we may assume that Open image in new window . In terms of Lemma 2.4 and (3.11), we conclude that Open image in new window . Therefore,
Finally, we prove that Open image in new window as Open image in new window . By Lemmas 2.5 and 2.7 we have
On the other hand
Since Open image in new window and Open image in new window are bounded, we can take a constant Open image in new window such that
So from the above, we reach the following:
Substituting (3.24) in (3.21), we obtain
It follows that
Since Open image in new window is bounded and Open image in new window , by (3.18), we get

Consequently, applying Lemma 2.6, to (3.26), we conclude that Open image in new window .

Corollary 3.2.

Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be as in Theorem 3.1. Suppose that Open image in new window a strongly positive bounded linear operator on Open image in new window with coefficient Open image in new window and Open image in new window . Let Open image in new window be defined by the iterative algorithm
Then, Open image in new window converges strongly, as Open image in new window , to Open image in new window , which is a unique solution of the variational inequality

Proof.

Because Open image in new window is strongly positive bounded linear operator on Open image in new window with coefficient Open image in new window , we have
Therefore, Open image in new window is Open image in new window -strongly accretive. On the other hand,
Since Open image in new window is strongly positive if and only if Open image in new window is strongly positive, we may assume, with no loss of generality, that Open image in new window , so that

This shows that Open image in new window is Open image in new window -strictly pseudo-contractive. Now apply Theorem 3.1 to conclude the result.

Corollary 3.3.

Then, Open image in new window converges strongly, as Open image in new window , to a Open image in new window , which is a unique solution of the variational inequality

Proof.

It is sufficient to take Open image in new window and Open image in new window in Theorem 3.1.

4. Some Application

Corollary 4.1.

Then, Open image in new window converges strongly, as Open image in new window , to Open image in new window which solves the variational inequality:

Proof.

Let Open image in new window for each Open image in new window . Then Open image in new window is a semigroup of non-expansive mappings on Open image in new window . Now, for each Open image in new window and Open image in new window , we define Open image in new window Then, Open image in new window is regular sequence of means [16]. Next, for each Open image in new window and Open image in new window , we have

Therefore, applying Theorem 3.1, the result follows.

Corollary 4.2.

Let Open image in new window be a strongly continuous semigroup of non-expansive mappings on a Hilbert space Open image in new window such that Open image in new window . Let Open image in new window be a sequence in Open image in new window satisfying conditions Open image in new window and Open image in new window . Let Open image in new window and Open image in new window . Let Open image in new window be a sequence defined by the iterative algorithm:
where Open image in new window is an increasing sequence in Open image in new window such that Open image in new window and Open image in new window . Then, Open image in new window converges strongly, as Open image in new window , to Open image in new window , which solves the variational inequality

Proof.

For Open image in new window , we define Open image in new window for each Open image in new window , where Open image in new window denotes the space of all real-valued bounded continuous functions on Open image in new window with supremum norm. Then, Open image in new window is regular sequence of means [16]. Furthermore, for each Open image in new window , we have Open image in new window . Now, apply Theorem 3.1 to conclude the result.

Corollary 4.3.

Let Open image in new window be a strongly continuous semigroup of non-expansive mappings on a Hilbert space Open image in new window such that Open image in new window . Let Open image in new window be a sequence in Open image in new window satisfying conditions Open image in new window and Open image in new window . Let Open image in new window and Open image in new window . Let Open image in new window be a sequence defined by the iterative algorithm
where Open image in new window is an decreasing sequence in Open image in new window such that Open image in new window . Then Open image in new window converges strongly, as Open image in new window , to Open image in new window , which solves the variational inequality

Proof.

For Open image in new window , we define Open image in new window for each Open image in new window . Then Open image in new window is regular sequence of means [16]. Furthermore, for each Open image in new window , we have Open image in new window . Now, apply Theorem 3.1 to conclude the result.

Corollary 4.4.

Let Open image in new window be a non-expansive mapping on a Hilbert space Open image in new window such that Open image in new window . Let Open image in new window be a sequence in Open image in new window satisfying conditions Open image in new window and Open image in new window and let Open image in new window be a strongly regular matrix. Let Open image in new window and Open image in new window . Let Open image in new window be a sequence defined by the iterative algorithm
Then, Open image in new window converges strongly, as Open image in new window , to Open image in new window which solves the variational inequality

Proof.

for each Open image in new window . Since Open image in new window is a strongly regular matrix, for each Open image in new window , we have Open image in new window , as Open image in new window ; see [17]. Then, it is easy to see that Open image in new window is regular sequence of means. Furthermore, for each Open image in new window , we have Open image in new window Now, apply Theorem 3.1 to conclude the result.

Notes

Acknowledgments

The authors thank the referee(s) for the helpful comments, which improved the presentation of this paper. This paper is dedicated to Professor Anthony To Ming Lau. This paper is based on final report of the research project of the Ph.D. thesis which is done with financial support of research office of the University of Tabriz.

References

  1. 1.
    Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Xu HK: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Yamada I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Studies in Computational Mathematics. Volume 8. North-Holland, Amsterdam, The Netherlands; 2001:473–504.Google Scholar
  6. 6.
    Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006,318(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Agarwal RP, O'Regan D, Sahu DR: Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Topological Fixed Point Theory and Its Applications. Springer, New York, NY, USA; 2009:x+368.MATHGoogle Scholar
  8. 8.
    Zeidler E: Nonlinear Functional Analysis and Its Applications. III. Springer, New York, NY, USA; 1985:xxii+662.CrossRefGoogle Scholar
  9. 9.
    Atsushiba S, Takahashi W: Approximating common fixed points of nonexpansive semigroups by the Mann iteration process. Annales Universitatis Mariae Curie-Skłodowska A 1997,51(2):1–16.MathSciNetMATHGoogle Scholar
  10. 10.
    Lau AT, Miyake H, Takahashi W: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces. Nonlinear Analysis. Theory, Methods & Applications 2007,67(4):1211–1225. 10.1016/j.na.2006.07.008MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Xu HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003,116(3):659–678. 10.1023/A:1023073621589MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lau AT, Shioji N, Takahashi W: Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces. Journal of Functional Analysis 1999,161(1):62–75. 10.1006/jfan.1998.3352MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Takahashi W: A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space. Proceedings of the American Mathematical Society 1981,81(2):253–256. 10.1090/S0002-9939-1981-0593468-XMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jung JS: Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 2005,302(2):509–520. 10.1016/j.jmaa.2004.08.022MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bruck RE: On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces. Israel Journal of Mathematics 1981,38(4):304–314. 10.1007/BF02762776MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. Fixed Point Theory and Its ApplicationMATHGoogle Scholar
  17. 17.
    Hirano N, Kido K, Takahashi W: Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces. Nonlinear Analysis. Theory, Methods & Applications 1988,12(11):1269–1281. 10.1016/0362-546X(88)90059-4MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© H. Piri and H. Vaezi. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesUniversity of TabrizTabrizIran

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