Research Article | Open Access

Kamonrat Nammanee, Suthep Suantai, Prasit Cholamjiak, "A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions", *Journal of Applied Mathematics*, vol. 2012, Article ID 506976, 14 pages, 2012. https://doi.org/10.1155/2012/506976

# A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions

**Academic Editor:**Giuseppe Marino

#### Abstract

We study strong convergence of the sequence generated by implicit and explicit general iterative methods for a one-parameter nonexpansive semigroup in a reflexive Banach space which admits the duality mapping , where is a gauge function on . Our results improve and extend those announced by G. Marino and H.-K. Xu (2006) and many authors.

#### 1. Introduction

Let be a real Banach space and the dual space of . Let be a nonempty, closed, and convex subset of . A (one-parameter) nonexpansive semigroup is a family of self-mappings of such that(i) for all ,(ii) for all and ,(iii)for each , the mapping is continuous,(iv)for each , is nonexpansive, that is, We denote by the common fixed points set of , that is, .

In 1967, Halpern [1] introduced the following classical iteration for a nonexpansive mapping in a real Hilbert space: where and .

In 1977, Lions [2] obtained a strong convergence provide the real sequence satisfies the following conditions:

C1: ; C2: ; C3: .

Reich [3] also extended the result of Halpern from Hilbert spaces to uniformly smooth Banach spaces. However, both Halpern’s and Lion’s conditions imposed on the real sequence excluded the canonical choice .

In 1992, Wittmann [4] proved that the sequence converges strongly to a fixed point of if satisfies the following conditions:

C1: ; C2: ; C3: .

Shioji and Takahashi [5] extended Wittmann’s result to real Banach spaces with uniformly Gâteaux differentiable norms and in which each nonempty closed convex and bounded subset has the fixed point property for nonexpansive mappings. The concept of the Halpern iterative scheme has been widely used to approximate the fixed points for nonexpansive mappings (see, e.g., [6–12] and the reference cited therein).

Let be a contraction. In 2000, Moudafi [13] introduced the explicit viscosity approximation method for a nonexpansive mapping as follows: where . Xu [14] also studied the iteration process (1.3) in uniformly smooth Banach spaces.

Let be a strongly positive bounded linear operator on a real Hilbert space , that is, there is a constant such that

A typical problem is to minimize a quadratic function over the fixed points set of a nonexpansive mapping on a Hilbert space : where is the fixed points set of a nonexpansive mapping on and is a given point in .

In 2006, Marino and Xu [15] introduced the following general iterative method for a nonexpansive mapping in a Hilbert space : where , is a contraction on , and is a strongly positive bounded linear operator on . They proved that the sequence generated by (1.6) converges strongly to a fixed point which also solves the variational inequality which is the optimality condition for the minimization problem: , where is a potential function for (i.e., for ).

Suzuki [16] first introduced the following implicit viscosity method for a nonexpansive semigroup in a Hilbert space: where and . He proved strong convergence of iteration (1.8) under suitable conditions. Subsequently, Xu [17] extended Suzuki’s [16] result from a Hilbert space to a uniformly convex Banach space which admits a weakly sequentially continuous normalized duality mapping.

Motivated by Chen and Song [18], in 2007, Chen and He [19] investigated the implicit and explicit viscosity methods for a nonexpansive semigroup without integral in a reflexive Banach space which admits a weakly sequentially continuous normalized duality mapping: where .

In 2008, Song and Xu [20] also studied the iterations (1.9) and (1.10) in a reflexive and strictly convex Banach space with a Gâteaux differentiable norm. Subsequently, Cholamjiak and Suantai [21] extended Song and Xu’s results to a Banach space which admits duality mapping with a gauge function. Wangkeeree and Kamraksa [22] and Wangkeeree et al. [23] obtained the convergence results concerning the duality mapping with a gauge function in Banach spaces. The convergence of iterations for a nonexpansive semigroup and nonlinear mappings has been studied by many authors (see, e.g., [24–38]).

Let be a real reflexive Banach space which admits the duality mapping with a gauge . Let be a nonexpansive semigroup on . Recall that an operator is said to be *strongly positive* if there exists a constant such that
where and .

Motivated by Chen and Song [18], Chen and He [19], Marino and Xu [15], Colao et al. [39], and Wangkeeree et al. [23], we study strong convergence of the following general iterative methods: where , is a contraction on and is a positive bounded linear operator on .

#### 2. Preliminaries

A Banach space is called *strictly convex* if for all with and . A Banach space is called *uniformly convex* if for each there is a such that for with and holds. The *modulus of convexity* of is defined by
for all . is uniformly convex if , and for all . It is known that every uniformly convex Banach space is strictly convex and reflexive. Let . Then the norm of is said to be *Gâteaux differentiable* if
exists for each . In this case is called *smooth*. The norm of is said to be *Fréchet differentiable* if for each , the limit is attained uniformly for . The norm of is called *uniformly Fréchet differentiable*, if the limit is attained uniformly for . It is well known that (uniformly) *Fréchet differentiability* of the norm of implies (uniformly) *Gâteaux differentiability* of the norm of .

Let be the *modulus of smoothness* of defined by

A Banach space is called *uniformly smooth* if as . See [40–42] for more details.

We need the following definitions and results which can be found in [40, 41, 43].

*Definition 2.1. *A continuous strictly increasing function is said to be gauge function if and .

*Definition 2.2. *Let be a normed space and a gauge function. Then the mapping defined by
is called the duality mapping with gauge function .

In the particular case , the duality mapping is called the normalized duality mapping.

In the case , the duality mapping is called the generalized duality mapping. It follows from the definition that and .

*Remark 2.3. *For the gauge function , the function defined by
is a continuous convex and strictly increasing function on . Therefore, has a continuous inverse function .

It is noted that if , then . Further

*Remark 2.4. *For each in a Banach space , , where denotes the sub-differential.

We also know the following facts:(i) is a nonempty, closed, and convex set in for each ,(ii) is a function when is strictly convex,(iii)If is single-valued, then Following Browder [43], we say that a Banach space has a weakly continuous duality mapping if there exists a gauge for which the duality mapping is single-valued and continuous from the weak topology to the weak* topology, that is, for any with , the sequence converges weakly* to . It is known that the space has a weakly continuous duality mapping with a gauge function for all . Moreover, is invariant on .

Lemma 2.5 (See [44]). *Assume that a Banach space has a weakly continuous duality mapping with gauge .*(i)*For all , the following inequality holds:
In particular, for all ,
*(ii)*Assume that a sequence in converges weakly to a point . Then the following holds:
for all . *

Lemma 2.6 (See [23]). *Assume that a Banach space has a weakly continuous duality mapping with gauge . Let be a strongly positive bounded linear operator on with coefficient and . Then .*

Lemma 2.7 (See [12]). * Assume that is a sequence of nonnegative real numbers such that
**
where is a sequence in and is a sequence in such that**(a) ; (b) or .** Then .*

#### 3. Implicit Iteration Scheme

In this section, we prove a strong convergence theorem of an implicit iterative method (1.12).

Theorem 3.1. *Let be a reflexive which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a nonexpansive semigroup on such that . Let be a contraction on with the coefficient and a strongly positive bounded linear operator with coefficient and . Let and be real sequences satisfying , and . Then defined by (1.12) converges strongly to which solves the following variational inequality:
*

* Proof. *First, we prove the uniqueness of the solution to the variational inequality (3.1) in . Suppose that satisfy (3.1), so we have
Adding the above inequalities, we get
This shows that
which implies by the strong positivity of
Since is invariant on ,
It follows that
Therefore since .

We next prove that is bounded. For each , by Lemma 2.6, we have
which yields
Hence is bounded. So are and .

We next prove that is relatively sequentially compact. By the reflexivity of and the boundedness of , there exists a subsequence of and a point in such that as . Now we show that . Put , and for , fix . We see that
So we have
On the other hand, by Lemma 2.5 (ii), we have
Combining (3.11) and (3.12), we have
This implies that . Further, we see that
So we have
By the definition of , it is easily seen that
Hence
Therefore as since is weakly continuous; consequently, as by the continuity of . Hence is relatively sequentially compact.

Finally, we prove that is a solution in to the variational inequality (3.1). For any , we see that
On the other hand, we have
which implies
Observe
as . Replacing by and letting in (3.20), we obtain
So is a solution of variational inequality (3.1); and hence by the uniqueness. In a summary, we have proved that is relatively sequentially compact and each cluster point of (as ) equals . Therefore as . This completes the proof.

#### 4. Explicit Iteration Scheme

In this section, utilizing the implicit version in Theorem 3.1, we consider the explicit one in a reflexive Banach space which admits the duality mapping .

Theorem 4.1. *Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on [0,1]. Let be a nonexpansive semigroup on such that . Let be a contraction on with the coefficient and a strongly positive bounded linear operator with coefficient and . Let and be real sequences satisfying , , and . Then defined by (1.13) converges strongly to which also solves the variational inequality (3.1).*

* Proof. *Since , we may assume that and for all . First we prove that is bounded. For each , by Lemma 2.6, we have
It follows from induction that
Thus is bounded, and hence so are and . From Theorem 3.1, there is a unique solution to the following variational inequality:
Next we prove that
Indeed, we can choose a subsequence of such that
Further, we can assume that by the reflexivity of and the boundedness of . Now we show that . Put and for , fix . We obtain
It follows that . From Lemma 2.5 (ii) we have
So we have and hence . Since the duality mapping is weakly sequentially continuous,
Finally, we show that . From Lemma 2.5 (i), we have
Note that and . Using Lemma 2.7, we have as by the continuity of . This completes the proof.

*Remark 4.2. *Theorems 3.1 and 4.1 improve and extend the main results proved in [15] in the following senses:(i)from a nonexpansive mapping to a nonexpansive semigroup,(ii)from a real Hilbert space to a reflexive Banach space which admits a weakly continuous duality mapping with gauge functions.

#### Acknowledgments

The authors wish to thank the editor and the referee for valuable suggestions. K. Nammanee was supported by the Thailand Research Fund, the Commission on Higher Education, and the University of Phayao under Grant MRG5380202. S. Suantai and P. Cholamjiak wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, Thailand.

#### References

- B. Halpern, “Fixed points of nonexpanding maps,”
*Bulletin of the American Mathematical Society*, vol. 73, pp. 957–961, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P.-L. Lions, “Approximation de points fixes de contractions,”
*Comptes Rendus de l'Académie des Sciences*, vol. 284, no. 21, pp. A1357–A1359, 1977. View at: Google Scholar | Zentralblatt MATH - S. Reich, “Approximating fixed points of nonexpansive mappings,”
*Panamerican Mathematical Journal*, vol. 4, no. 2, pp. 23–28, 1994. View at: Google Scholar | Zentralblatt MATH - R. Wittmann, “Approximation of fixed points of nonexpansive mappings,”
*Archiv der Mathematik*, vol. 58, no. 5, pp. 486–491, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH - N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 125, no. 12, pp. 3641–3645, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 67, no. 8, pp. 2350–2360, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. E. Chidume and C. O. Chidume, “Iterative approximation of fixed points of nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 318, no. 1, pp. 288–295, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. J. Cho, S. M. Kang, and H. Zhou, “Some control conditions on iterative methods,”
*Communications on Applied Nonlinear Analysis*, vol. 12, no. 2, pp. 27–34, 2005. View at: Google Scholar | Zentralblatt MATH - T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 61, no. 1-2, pp. 51–60, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 75, no. 1, pp. 287–292, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H.-K. Xu, “Another control condition in an iterative method for nonexpansive mappings,”
*Bulletin of the Australian Mathematical Society*, vol. 65, no. 1, pp. 109–113, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H.-K. Xu, “Iterative algorithms for nonlinear operators,”
*Journal of the London Mathematical Society*, vol. 66, no. 1, pp. 240–256, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. Moudafi, “Viscosity approximation methods for fixed-points problems,”
*Journal of Mathematical Analysis and Applications*, vol. 241, no. 1, pp. 46–55, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 298, no. 1, pp. 279–291, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 318, no. 1, pp. 43–52, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T. Suzuki, “On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces,”
*Proceedings of the American Mathematical Society*, vol. 131, no. 7, pp. 2133–2136, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H.-K. Xu, “A strong convergence theorem for contraction semigroups in Banach spaces,”
*Bulletin of the Australian Mathematical Society*, vol. 72, no. 3, pp. 371–379, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Chen and Y. Song, “Convergence to common fixed point of nonexpansive semigroups,”
*Journal of Computational and Applied Mathematics*, vol. 200, no. 2, pp. 566–575, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Chen and H. He, “Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space,”
*Applied Mathematics Letters*, vol. 20, no. 7, pp. 751–757, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. Song and S. Xu, “Strong convergence theorems for nonexpansive semigroup in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 338, no. 1, pp. 152–161, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. Cholamjiak and S. Suantai, “Viscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions,”
*Journal of Global Optimization*. In press. View at: Publisher Site | Google Scholar - R. Wangkeeree and U. Kamraksa, “Strong convergence theorems of viscosity iterative methods for a countable family of strict pseudo-contractions in Banach spaces,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 579725, 21 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Wangkeeree, N. Petrot, and R. Wangkeeree, “The general iterative methods for nonexpansive mappings in Banach spaces,”
*Journal of Global Optimization*, vol. 51, no. 1, pp. 27–46, 2011. View at: Publisher Site | Google Scholar - I. K. Argyros, Y. J. Cho, and X. Qin, “On the implicit iterative process for strictly pseudo-contractive mappings in Banach spaces,”
*Journal of Computational and Applied Mathematics*, vol. 233, no. 2, pp. 208–216, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S.-S. Chang, Y. J. Cho, H. W. J. Lee, and C. K. Chan, “Strong convergence theorems for Lipschitzian demicontraction semigroups in Banach spaces,”
*Fixed Point Theory and Applications*, vol. 11, Article ID 583423, 10 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. J. Cho, L. Ćirić, and S.-H. Wang, “Convergence theorems for nonexpansive semigroups in CAT(0) spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 74, no. 17, pp. 6050–6059, 2011. View at: Publisher Site | Google Scholar - Y. J. Cho, S. M. Kang, and X. Qin, “Some results on
*k*-strictly pseudo-contractive mappings in Hilbert spaces,”*Nonlinear Analysis. Theory, Methods & Applications*, vol. 70, no. 5, pp. 1956–1964, 2009. View at: Publisher Site | Google Scholar - Y. J. Cho, S. M. Kang, and X. Qin, “Strong convergence of an implicit iterative process for an infinite family of strict pseudocontractions,”
*Bulletin of the Korean Mathematical Society*, vol. 47, no. 6, pp. 1259–1268, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. J. Cho, X. Qin, and S. M. Kang, “Strong convergence of the modified Halpern-type iterative algorithms in Banach spaces,”
*Analele Stiintifice ale Universitatii Ovidius Constanta*, vol. 17, no. 1, pp. 51–67, 2009. View at: Google Scholar - W. Guo and Y. J. Cho, “On the strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings,”
*Applied Mathematics Letters*, vol. 21, no. 10, pp. 1046–1052, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. He, S. Liu, and Y. J. Cho, “An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,”
*Journal of Computational and Applied Mathematics*, vol. 235, no. 14, pp. 4128–4139, 2011. View at: Publisher Site | Google Scholar - J. Kang, Y. Su, and X. Zhang, “General iterative algorithm for nonexpansive semigroups and variational inequalities in Hilbert spaces,”
*journal of Inequalities and Applications*, vol. 2010, Article ID 264052, 10 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. N. Li and J. S. Gu, “Strong convergence of modified Ishikawa iteration for a nonexpansive semigroup in Banach spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 73, no. 4, pp. 1085–1092, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Li, L. Li, and Y. Su, “General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 70, no. 9, pp. 3065–3071, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Q. Lin, “Viscosity approximation to common fixed points of a nonexpansive semigroup with a generalized contraction mapping,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 11, pp. 5451–5457, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Plubtieng and R. Punpaeng, “Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces,”
*Mathematical and Computer Modelling*, vol. 48, no. 1-2, pp. 279–286, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Qin, Y. J. Cho, S. M. Kang, and H. Zhou, “Convergence of a modified Halpern-type iteration algorithm for quasi-$\varphi $-nonexpansive mappings,”
*Applied Mathematics Letters*, vol. 22, no. 7, pp. 1051–1055, 2009. View at: Publisher Site | Google Scholar - T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,”
*Journal of Mathematical Analysis and Applications*, vol. 305, no. 1, pp. 227–239, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - V. Colao, G. Marino, and H.-K. Xu, “An iterative method for finding common solutions of equilibrium and fixed point problems,”
*Journal of Mathematical Analysis and Applications*, vol. 344, no. 1, pp. 340–352, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. P. Agarwal, D. O'Regan, and D. R. Sahu,
*Fixed Point Theory for Lipschitzian-Type Mappings with Applications*, vol. 6, Springer, New York, NY, USA, 2009. View at: Zentralblatt MATH - C. Chidume,
*Geometric Properties of Banach Spaces and Nonlinear Iterations*, vol. 1965 of*Lecture Notes in Mathematics*, Springer, London, UK, 2009. - W. Takahashi,
*Nonlinear Functional Analysis*, Yokohama Publishers, Yokohama, Japan, 2000. - F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,”
*Mathematische Zeitschrift*, vol. 100, pp. 201–225, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T.-C. Lim and H. K. Xu, “Fixed point theorems for asymptotically nonexpansive mappings,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 22, no. 11, pp. 1345–1355, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2012 Kamonrat Nammanee et al. 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.