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

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## 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## 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 ).

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

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:

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

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 ).

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].

see [8].

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

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

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 .

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 .

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 .

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

Lemma 2.6 (see [11]).

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

(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 .

- (i)From (1.11) and (1.13), we obtain(2.11)Because Open image in new window , we have(2.12)
and, therefore, Open image in new window is contractive with constant Open image in new window .

- (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(2.13)

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.

Proof.

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

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

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

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

Corollary 3.2.

Proof.

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.

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.

Proof.

Therefore, applying Theorem 3.1, the result follows.

Corollary 4.2.

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.

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.

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.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.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.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.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.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.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.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.Zeidler E:
*Nonlinear Functional Analysis and Its Applications. III*. Springer, New York, NY, USA; 1985:xxii+662.CrossRefGoogle Scholar - 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.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.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.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.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.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.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.Takahashi W:
*Nonlinear Functional Analysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. Fixed Point Theory and Its ApplicationMATHGoogle Scholar - 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

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