Table of Contents
ISRN Applied Mathematics
Volumeย 2011, Article IDย 484061, 17 pages
http://dx.doi.org/10.5402/2011/484061
Research Article

A Viscosity Approximation Method with a Weakly Contractive Mapping of General Iterative Processes for Nonexpansive Semigroups in Banach Spaces

1Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
3Department of Mathematics, Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng Saen Campus, Nakhonpathom 73140, Thailand

Received 21 April 2011; Accepted 9 June 2011

Academic Editor: R.ย Couturier

Copyright ยฉ 2011 Pongsakorn Sunthrayuth 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.

Abstract

We introduce a new viscosity approximation method with a weakly contractive mapping of general iterative processes for finding common fixed point of nonexpansive semigroups {๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+} in the framework of Banach spaces. We proved that under some mild conditions these iterative processes converge strongly to the common fixed point of {๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+}, which is the unique solution of some variational inequality. The results obtained in this paper extend and improve on the recent results of Li et al. (2009), Chen and He (2007), and many others as special cases.

1. Introduction

Let ๐‘‹ be a real Banach space, and let ๐ถ be a nonempty closed convex subset of ๐‘‹. A mapping ๐‘‡ of ๐ถ into itself is said to be nonexpansive if โ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€– for each ๐‘ฅ,๐‘ฆโˆˆ๐ถ. We denote ๐น(๐‘‡) as the set of fixed points of ๐‘‡. We know that ๐น(๐‘‡) is nonempty if ๐ถ is bounded; for more detail see [1]. Throughout this paper we denote by โ„• and โ„+ the set of all positive integers and all positive real numbers, respectively. A one-parameter family ๐’ฎ={๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+} from ๐ถ of ๐‘‹ into itself is said to be a nonexpansive semigroup on ๐ถ if it satisfies the following conditions: (i)๐‘‡(0)๐‘ฅ=๐‘ฅforall๐‘ฅโˆˆ๐ถ,(ii)๐‘‡(๐‘ +๐‘ก)=๐‘‡(๐‘ )โˆ˜๐‘‡(๐‘ก) for all ๐‘ ,๐‘กโˆˆโ„+,(iii)for each ๐‘ฅโˆˆ๐ถ the mapping ๐‘กโ†ฆ๐‘‡(๐‘ก)๐‘ฅ is continuous,(iv)โ€–๐‘‡(๐‘ก)๐‘ฅโˆ’๐‘‡(๐‘ก)๐‘ฆโ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€– for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐‘กโˆˆโ„+.

We denote by ๐น(๐’ฎ) the set of all common fixed points of ๐’ฎ, that is โ‹‚๐น(๐’ฎ)โˆถ=๐‘กโˆˆโ„+๐น(๐‘‡(๐‘ก))={๐‘ฅโˆˆ๐ถโˆถ๐‘‡(๐‘ก)๐‘ฅ=๐‘ฅ}. We know that ๐น(๐’ฎ) is nonempty if ๐ถ is bounded; see [2]. Recall that a self-mapping ๐‘“โˆถ๐ถโ†’๐ถ is a contraction if there exists a constant ๐›ผโˆˆ(0,1) such that โ€–๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)โ€–โ‰ค๐›ผโ€–๐‘ฅโˆ’๐‘ฆโ€– for each ๐‘ฅ,๐‘ฆโˆˆ๐ถ.

Definition 1.1. A mapping ๐œ™โˆถ๐ถโ†’๐ถ is said to be weakly contractive if there exists a continuous and strictly increasing function ๐œ“โˆถโ„+โ†’โ„+,๐œ“(0)=0 such that โ€–๐œ™(๐‘ฅ)โˆ’๐œ™(๐‘ฆ)โ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’๐œ“(โ€–๐‘ฅโˆ’๐‘ฆโ€–),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(1.1)

Remark 1.2. As a special case, if we consider ๐œ“(๐‘ก)โˆถ=(1โˆ’๐›ผ)๐‘ก, for each ๐‘กโˆˆ[0,โˆž) and ๐›ผโˆˆ(0,1), then we get the contraction mapping with the coefficient ๐›ผ.

In the last ten years or so, the iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [3โ€“5]. Let ๐ป be a real Hilbert space, whose inner product and norm are denoted by โŸจโ‹…,โ‹…โŸฉ and โ€–โ‹…โ€–, respectively. Let ๐ด be a strongly positive bounded linear operator on ๐ป: that is, there is a constant ๐›พ>0 with property โŸจ๐ด๐‘ฅ,๐‘ฅโŸฉโ‰ฅ๐›พโ€–๐‘ฅโ€–2โˆ€๐‘ฅโˆˆ๐ป.(1.2) A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space ๐ป: min๐‘ฅโˆˆ๐น12โŸจ๐ด๐‘ฅ,๐‘ฅโŸฉโˆ’โŸจ๐‘ฅ,๐‘โŸฉ,(1.3) where ๐ถ is the fixed point set of a nonexpansive mapping ๐‘‡ on ๐ป and ๐‘ is a given point in ๐ป.

In 2003, Xu [5] proved that the sequence {๐‘ฅ๐‘›} defined by the iterative method below, with the initial guess ๐‘ฅ0โˆˆ๐ป chosen arbitrarily: ๐‘ฅ๐‘›+1=๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฅ๐‘›+๐›ผ๐‘›๐‘ข,โˆ€๐‘›โ‰ฅ0,(1.4) converges strongly to the unique solution of the minimization problem (1.3) provided the sequence {๐›ผ๐‘›} satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the following iterative process for nonexpansive mappings (see [7, 8] for further developments in both Hilbert and Banach spaces). Let ๐‘“ be a contraction on ๐ป. Starting with an arbitrary initial ๐‘ฅ0โˆˆ๐ป, we defined the sequence {๐‘ฅ๐‘›} recursively by ๐‘ฅ๐‘›+1=๐œŽ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐œŽ๐‘›๎€ธ๐‘‡๐‘ฅ๐‘›,โˆ€๐‘›โ‰ฅ0,(1.5) where {๐œŽ๐‘›} is a sequence in (0,1). It is proved in [6, 8] that under certain appropriate conditions imposed on {๐œŽ๐‘›}, the sequence {๐‘ฅ๐‘›} generated by (1.5) strongly converges to a unique solution ๐‘ฅโˆ— of the variational inequality โŸจ(๐‘“โˆ’๐ผ)๐‘ฅโˆ—,๐œ”โˆ’๐‘ฅโˆ—โŸฉโ‰ค0,โˆ€๐œ”โˆˆ๐น(๐‘‡).(1.6) Recently, Marino and Xu [9] combined the iterative method (1.4) with the viscosity approximation method (1.5) considering the following general iterative process: ๐‘ฅ๐‘›+1=๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฅ๐‘›,โˆ€๐‘›โ‰ฅ0,(1.7) where 0<๐›พ<(๐›พ/๐›ผ). They proved that the sequence {๐‘ฅ๐‘›} generated by (1.7) converges strongly to a unique solution ๐‘ฅโˆ— of the variational inequality โŸจ(๐›พ๐‘“โˆ’๐ด)๐‘ฅโˆ—,๐œ”โˆ’๐‘ฅโˆ—โŸฉโ‰ค0,โˆ€๐œ”โˆˆ๐น(๐‘‡),(1.8) which is the optimality condition for the minimization problem: min๐‘ฅโˆˆ๐ถ(1/2)โŸจ๐ด๐‘ฅ,๐‘ฅโŸฉโˆ’โ„Ž(๐‘ฅ), where โ„Ž is a potential function for ๐›พ๐‘“ (i.e., โ„Ž๎…ž(๐‘ฅ)=๐›พ๐‘“(๐‘ฅ) for ๐‘ฅโˆˆ๐ป).

On the other hand, Shioji and Takahashi [10] introduced in a Hilbert space the implicit iteration as follows: ๐‘ฅ๐‘›=๐›ผ๐‘›๎€ท๐‘ข+1โˆ’๐›ผ๐‘›๎€ธ1๐‘ก๐‘›๎€œ๐‘ก๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,โˆ€๐‘›โˆˆโ„•,(1.9) where {๐›ผ๐‘›} is a sequence in (0,1)and{๐‘ก๐‘›} is a positive real divergent sequence and for ๐‘ขโˆˆ๐ถ. They proved, under certain restrictions on the sequence {๐›ผ๐‘›}, that the sequence {๐‘ฅ๐‘›} defined by (1.9) converges strongly to a member of ๐น(๐’ฎ) (see also [11]).

Chen and Song [12] studied the strong convergence of the following sequence (1.10) for a nonexpansive semigroup ๐’ฎ={๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+} with ๐น(๐’ฎ)โ‰ โˆ… in a uniformly convex Banach space: ๐‘ฅ๐‘›=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ1๐‘ก๐‘›๎€œ๐‘ก๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,โˆ€๐‘›โˆˆโ„•.(1.10) Suzuki [13] was the first to introduced again in a Hilbert space the following implicit iteration process: ๐‘ฅ๐‘›=๐›ผ๐‘›๎€ท๐‘ข+1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๎€ท๐‘ก๐‘›๎€ธ๐‘ฅ๐‘›,โˆ€๐‘›โˆˆโ„•,(1.11) for a nonexpansive semigroup case. Xu [14] established a Banach space version of the sequence (1.11) of Suzuki [13].

In 2007, Chen and He [15] extended the result of Suzuki [13] and Xu [14] and studied the strong convergence theorem of viscosity implicit iteration process for a nonexpansive semigroup ๐’ฎ={๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+} with ๐น(๐’ฎ)โ‰ โˆ…, in Banach spaces as follows: ๐‘ฅ๐‘›=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๎€ท๐‘ก๐‘›๎€ธ๐‘ฅ๐‘›,โˆ€๐‘›โˆˆโ„•.(1.12) Very recently, S. Li et al. [16] considered a general iterative process for a nonexpansive semigroup ๐’ฎ={๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+} in a Hilbert space as follows: ๐‘ฅ๐‘›=๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ1๐‘ก๐‘›๎€œ๐‘ก๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,โˆ€๐‘›โˆˆโ„•,(1.13) where {๐›ผ๐‘›}โŠ‚(0,1] and {๐‘ก๐‘›} are two sequences satisfying certain conditions. They proved that the sequence {๐‘ฅ๐‘›} defined by (1.13) converges strongly to ๐‘ฅโˆ—โˆˆ๐น(๐’ฎ), which solves the following variational inequality (1.8).

Question 1. Can Theorem of S. Li et al. [16] be extended from Hilbert space to a general Banach space?

Question 2. Can we extend the iterative method of algorithms (1.10) and (1.12) to general algorithms?

Question 3. We know that the weakly contractive mapping is more general than the contractive mapping. What happens if the contractive mapping is replaced by the weakly contractive mapping?

The purpose of this paper is to give affirmative answer to these questions mentioned above. Motivated by the iterative process (1.10), (1.12), and (1.13), we introduce a new viscosity approximation method with a weakly contractive mapping of general iterative processes for nonexpansive semigroups, which is a unique solution of some variational inequality. We prove the strong convergence theorems of these iterative processes in a Banach space which admits a weakly sequentially continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Chen and He [15] and S. Li et al. [16] and many others as special cases.

2. Preliminaries

Throughout this paper, let ๐‘‹ be a real Banach space and ๐ถ a closed convex subset of ๐‘‹. Let ๐ฝโˆถ๐‘‹โ†’2๐‘‹โˆ— be a normalized duality mapping by ๐ฝ(๐‘ฅ)={๐‘“โˆ—โˆˆ๐‘‹โˆ—โˆถโŸจ๐‘ฅ,๐‘“โˆ—โŸฉ=โ€–๐‘ฅโ€–2=โ€–๐‘“โˆ—โ€–2}, where ๐‘‹โˆ— denotes the dual space of ๐‘‹ and โŸจโ‹…,โ‹…โŸฉ denotes the generalized duality paring. In the following, the notations โ‡€ and โ†’ denote the weak and strong convergence, respectively. Also, a mapping ๐ผโˆถ๐ถโ†’๐ถ denotes the identity mapping.

The norm of a Banach space ๐‘‹ is said to be Gรขteaux differentiable if the limit lim๐‘กโ†’0โ€–๐‘ฅ+๐‘ก๐‘ฆโ€–โˆ’โ€–๐‘ฅโ€–๐‘ก(2.1) exists for each ๐‘ฅ,๐‘ฆโˆˆ๐ถ on the unit sphere ๐‘†(๐‘‹) of ๐‘‹. In this case ๐‘‹ is smooth. Recall that the Banach space ๐‘‹ is said to be smooth if duality mapping ๐ฝ is single valued. In a smooth Banach space, we always assume that ๐ด is strongly positive (see [17]), that is, a constant ๐›พ>0 with the property โŸจ๐ด๐‘ฅ,๐ฝ(๐‘ฅ)โŸฉโ‰ฅ๐›พโ€–๐‘ฅโ€–2,โ€–๐‘Ž๐ผโˆ’๐‘๐ดโ€–=supโ€–๐‘ฅโ€–โ‰ค1||โŸจ||[][](๐‘Ž๐ผโˆ’๐‘๐ด)๐‘ฅ,๐ฝ(๐‘ฅ)โŸฉ๐‘Žโˆˆ0,1,๐‘โˆˆโˆ’1,1.(2.2)

Moreover, if for each ๐‘ฆ in ๐‘†(๐‘‹) the limit (2.1) is uniformly attained for ๐‘ฅโˆˆ๐‘†(๐‘‹), we say that the norm ๐‘‹ is uniformly Gรขteaux differentiable. The norm of ๐‘‹ is said to be Frรชchet differentiable if, for each ๐‘ฅโˆˆ๐‘†(๐‘‹), the limit (2.1) is attained uniformly for ๐‘ฆโˆˆ๐‘†(๐‘‹). The norm of ๐‘‹ is said to be uniformly Frรชchet differentiable (or ๐‘‹ is said to be uniformly smooth) if the limit (2.1) is attained uniformly for (๐‘ฅ,๐‘ฆ)โˆˆ๐‘†(๐‘‹)ร—๐‘†(๐‘‹). A Banach space ๐‘‹ is said to be strictly convex if โ€–๐‘ฅโ€–=โ€–๐‘ฆโ€–=1,๐‘ฅโ‰ ๐‘ฆ, implies โ€–๐‘ฅ+๐‘ฆโ€–/2<1โ€‰and uniformly convex if ๐›ฟ๐‘‹(๐œ–)>0 for all ๐œ–>0, where ๐›ฟ๐‘‹(๐œ–) is modulus of convexity of ๐‘‹ defined by ๐›ฟ๐‘‹(๐œ–)=inf{1โˆ’(โ€–๐‘ฅ+๐‘ฆโ€–/2)โˆถโ€–๐‘ฅโ€–โ‰ค1,โ€–๐‘ฆโ€–โ‰ค1,โ€–๐‘ฅ+๐‘ฆโ€–โ‰ฅ๐œ–}, for all ๐œ–โˆˆ[0,2]. A uniformly convex Banach space ๐‘‹ is reflexive and strictly convex (see Theorems 4.1.6, and 4.1.2 of [18]) and every uniformly smooth Banach space ๐‘‹ is a reflexive Banach with uniformly Gรขteaux differentiable norm (see Theorems 4.3.7, and 4.1.6 of [18] and also [19]).

In the sequel we will use the following lemmas, which will be used in the proofs for the main results in the next section.

Lemma 2.1 (see [17]). Assume that A is a strongly positive linear bounded operator on a smooth Banach space ๐‘‹ with coefficient ๐›พ>0 and 0<๐œŒโ‰คโ€–๐ดโ€–โˆ’1. Then โ€–๐ผโˆ’๐œŒ๐ดโ€–โ‰ค1โˆ’๐œŒ๐›พ.

Lemma 2.2 (see [20]). Let (๐‘‹,๐‘‘) be a complete metric space and ๐‘‡โˆถ๐‘‹โ†’๐‘‹ a weakly contractive mapping. Then, ๐‘‡ has a unique fixed point in ๐‘‹.

If a Banach space ๐‘‹ admits a sequentially continuous duality mapping ๐ฝ from weak topology to weak star topology, then, by Lemma 1 of [21], we have that duality mapping ๐ฝ is a single value. In this case, the duality mapping ๐ฝ is said to be a weakly sequentially continuous duality mapping, that is, for each {๐‘ฅ๐‘›}โŠ‚๐‘‹ with ๐‘ฅ๐‘›โ‡€๐‘ฅ, we have ๐ฝ(๐‘ฅ๐‘›)โ‡€โˆ—๐ฝ(๐‘ฅ) (see [21โ€“23] for more details).

A Banach space ๐‘‹ is said to be satisfying Opialโ€™s condition if for any sequence ๐‘ฅ๐‘›โ‡€๐‘ฅ for all ๐‘ฅโˆˆ๐‘‹ implies limsup๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ฅ<limsup๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ฆโˆ€๐‘ฆโˆˆ๐‘‹,with๐‘ฅโ‰ ๐‘ฆ.(2.3)

By Theorem 1 in [21], it is well known that, if ๐‘‹ admits a weakly sequentially continuous duality mapping, then ๐‘‹ satisfies Opialโ€™s condition and ๐‘‹ is smooth.

Lemma 2.3 ([22] (Demiclosed Principle)). Let C be a nonempty closed convex subset of a reflexive Banach space ๐‘‹ which satisfies Opial's condition, and that suppose ๐‘‡โˆถ๐ถโ†’๐‘‹ is nonexpansive. Then the mapping ๐ผโˆ’๐‘‡ is demiclosed at zero, that is, ๐‘ฅ๐‘›โ‡€๐‘ฅ and ๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ†’0 imply that ๐‘ฅ=๐‘‡๐‘ฅ.

Lemma 2.4 (see [15]). Let ๐ถ be a closed convex subset of a uniformly convex Banach space ๐‘‹ and ๐’ฎ={๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+} a nonexpansive semigroup on ๐ถ such that ๐น(๐’ฎ)โ‰ โˆ…. Then, for each ๐‘Ÿ>0 and โ„Žโ‰ฅ0, limtโ†’โˆžsup๐‘ฅโˆˆ๐ถโˆฉ๐ต๐‘Ÿโ€–โ€–โ€–1๐‘ก๎€œ๐‘ก0๎‚ต1๐‘‡(๐‘ )๐‘ฅ๐‘‘๐‘ โˆ’๐‘‡(โ„Ž)๐‘ก๎€œ๐‘ก0๎‚ถโ€–โ€–โ€–๐‘‡(๐‘ )๐‘ฅ๐‘‘๐‘ =0.(2.4)

3. Main Results

In this section, we prove our main results.

Theorem 3.1. Let ๐‘‹ be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping ๐ฝ from ๐‘‹ into ๐‘‹โˆ— and ๐ถ a nonempty closed convex subset of ๐‘‹ such that ๐ถยฑ๐ถโŠ‚๐ถ. Let ๐’ฎ={๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+} be a nonexpansive semigroup from ๐ถ into itself such that ๐น(๐’ฎ)โ‰ โˆ…. Let ๐œ™ be a weakly contractive mapping and ๐ด a strongly positive linear bounded operator with a coefficient ๐›พ>0 such that 0<๐›พ<๐›พ. Let {๐‘ฅ๐‘›} be a sequence defined by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ1๐‘ก๐‘›๎€œ๐‘ก๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘ฅ๐‘‘๐‘ ,๐‘›=๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘ฆ๐‘›,โˆ€๐‘›โ‰ฅ0,(3.1) where {๐›ผ๐‘›},{๐›ฝ๐‘›} are two sequences in (0,1) and {๐‘ก๐‘›} is a positive real divergent sequence satisfying the following conditions: (๐ถ1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (๐ถ2)0<liminf๐‘›โ†’โˆž๐›ฝ๐‘›โ‰คlimsup๐‘›โ†’โˆž๐›ฝ๐‘›<1. Then the sequence {๐‘ฅ๐‘›} defined by (3.1) converges strongly to the common fixed point ๐‘ฅโˆ—โˆˆ๐น(๐’ฎ), where ๐‘ฅโˆ— is the unique solution of the variational inequality ๎ซ๎€ท๐‘ฅ๐›พ๐œ™โˆ—๎€ธโˆ’๐ด๐‘ฅโˆ—๎€ท,๐ฝ๐œ”โˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ค0,โˆ€๐œ”โˆˆ๐น(๐’ฎ).(3.2)

Proof. Firstly, we show that {๐‘ฅ๐‘›} defined by (3.1) is well define. Since lim๐‘›โ†’โˆž๐›ผ๐‘›=0, we may assume, with no loss of generality, that ๐›ผ๐‘›<โ€–๐ดโ€–โˆ’1 for each ๐‘›โ‰ฅ0. Define the mapping ๐‘‡๐œ™๐‘›โˆถ๐ถโ†’๐ถ by ๐‘‡๐œ™๐‘›โˆถ=๐›ผ๐‘›๎€ท๐›พ๐œ™+๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๎‚ธ๐›ฝ๐‘›๎€ท๐ผ+1โˆ’๐›ฝ๐‘›๎€ธ1๐‘ก๐‘›๎€œ๐‘ก๐‘›0๎‚น๐‘‡(๐‘ )๐‘‘๐‘ .(3.3) Indeed, from Lemma 2.1, we have, for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ, โ€–โ€–๐‘‡๐œ™๐‘›๐‘ฅโˆ’๐‘‡๐œ™๐‘›๐‘ฆโ€–โ€–=๐›ผ๐‘›๐›พ(๐œ™(๐‘ฅ)โˆ’๐œ™(๐‘ฆ))+๐›ฝ๐‘›๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ+๎€ท(๐‘ฅโˆ’๐‘ฆ)1โˆ’๐›ฝ๐‘›๎€ธ๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ1๐‘ก๐‘›๎€œ๐‘ก๐‘›0(๐‘‡(๐‘ )๐‘ฅโˆ’๐‘‡(๐‘ )๐‘ฆ)๐‘‘๐‘ โ‰ค๐›ผ๐‘›๐›พโ€–๐œ™(๐‘ฅ)โˆ’๐œ™(๐‘ฆ)โ€–+๐›ฝ๐‘›โ€–โ€–๐ผโˆ’๐›ผ๐‘›๐ดโ€–โ€–+๎€ทโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐ผโˆ’๐›ผ๐‘›๐ดโ€–โ€–1๐‘ก๐‘›๎€œ๐‘ก๐‘›0โ€–๐‘‡(๐‘ )๐‘ฅโˆ’๐‘‡(๐‘ )๐‘ฆโ€–๐‘‘๐‘ โ‰ค๐›ผ๐‘›๐›พ[]โ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’๐œ“(โ€–๐‘ฅโˆ’๐‘ฆโ€–)+๐›ฝ๐‘›๎€ท1โˆ’๐›ผ๐‘›๐›พ๎€ธ+๎€ทโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐›ฝ๐‘›๎€ธ๎€ท1โˆ’๐›ผ๐‘›๐›พ๎€ธโ€–๐‘ฅโˆ’๐‘ฆโ€–=๐›ผ๐‘›๐›พ[]+๎€ทโ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’๐œ“(โ€–๐‘ฅโˆ’๐‘ฆโ€–)1โˆ’๐›ผ๐‘›๐›พ๎€ธ=๎€บโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐›ผ๐‘›๎€ท๐›พโˆ’๐›พ๎€ธ๎€ปโ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’๐›ผ๐‘›๐›พ๐œ“(โ€–๐‘ฅโˆ’๐‘ฆโ€–)โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’๐›ผ๐‘›๐›พ๐œ“(โ€–๐‘ฅโˆ’๐‘ฆโ€–).(3.4) This shows that ๐‘‡๐œ™๐‘› is weakly contractive. It follows from Lemma 2.2 that ๐‘‡๐œ™๐‘› has a unique fixed point ๐‘ฅ๐‘›โˆˆ๐ถ, that is, {๐‘ฅ๐‘›} defined by (3.1) is well defined.
Next, we show that {๐‘ฅ๐‘›} is bounded. Letting ๐‘โˆˆ๐น(๐’ฎ), we getโ€–โ€–๐‘ฆ๐‘›โ€–โ€–=โ€–โ€–โ€–๐›ฝโˆ’๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ+๎€ทโˆ’๐‘1โˆ’๐›ฝ๐‘›๎€ธ1๐‘ก๐‘›๎€œ๐‘ก๐‘›0๎€ท๐‘‡(๐‘ )๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โˆ’๐‘๐‘‘๐‘ โ‰ค๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–+๎€ทโˆ’๐‘1โˆ’๐›ฝ๐‘›๎€ธ1๐‘ก๐‘›๎€œ๐‘ก๐‘›0โ€–โ€–๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘๐‘‘๐‘ โ‰ค๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–+๎€ทโˆ’๐‘1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–=โ€–โ€–๐‘ฅโˆ’๐‘๐‘›โ€–โ€–,โ€–โ€–๐‘ฅโˆ’๐‘๐‘›โ€–โ€–โˆ’๐‘2=๎ซ๐›ผ๐‘›๎€ท๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธ๎€ธ+๎€ทโˆ’๐ด๐‘๐ผโˆ’๐›ผ๐‘›๐ด๐‘ฆ๎€ธ๎€ท๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘,๐ฝ๐‘›โˆ’๐‘๎€ธ๎ฌ=๐›ผ๐‘›๎€ท๐‘ฅ๎ซ๎€ท๐›พ๐œ™๐‘›๎€ธ๎€ธ๎€ท๐‘ฅโˆ’๐›พ๐œ™(๐‘)+(๐›พ๐œ™(๐‘)โˆ’๐ด๐‘),๐ฝ๐‘›+โˆ’๐‘๎€ธ๎ฌ๎ซ๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๐‘ฆ๎€ธ๎€ท๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘,๐ฝ๐‘›โˆ’๐‘๎€ธ๎ฌโ‰ค๐›ผ๐‘›๐›พโ€–โ€–๐œ™๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โ€–๐ฝ๎€ท๐‘ฅโˆ’๐œ™(๐‘)๐‘›๎€ธโ€–โ€–โˆ’๐‘+๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐›พ๐œ™(๐‘)โˆ’๐ด๐‘,๐ฝ๐‘›+โ€–โ€–โˆ’๐‘๎€ธ๎ฌ๐ผโˆ’๐›ผ๐‘›๐ดโ€–โ€–โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๐ฝ๎€ท๐‘ฅโˆ’๐‘๐‘›๎€ธโ€–โ€–โˆ’๐‘โ‰ค๐›ผ๐‘›๎€บ๐›พโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ทโ€–โ€–๐‘ฅโˆ’๐‘โˆ’๐›พ๐œ“๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘๎€ธ๎€ป๐‘›โ€–โ€–โˆ’๐‘+๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐›พ๐œ™(๐‘)โˆ’๐ด๐‘,๐ฝ๐‘›+๎€ทโˆ’๐‘๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๐›พ๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2=๎€บ1โˆ’๐›ผ๐‘›๎€ทโ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๎€ธ๎€ป๐‘›โ€–โ€–โˆ’๐‘2โˆ’๐›ผ๐‘›๎€ทโ€–โ€–๐‘ฅ๐›พ๐œ“๐‘›โ€–โ€–๎€ธโ€–โ€–๐‘ฅโˆ’๐‘๐‘›โ€–โ€–โˆ’๐‘+๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐›พ๐œ™(๐‘)โˆ’๐ด๐‘,๐ฝ๐‘›,โˆ’๐‘๎€ธ๎ฌ(3.5) and so ()โ€–โ€–๐‘ฅ๐›พ๐œ“โ€–๐‘ฅโˆ’๐‘โ€–๐‘›โ€–โ€–+๎€ทโˆ’๐‘๎€ธโ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐‘›โ€–โ€–โ‰ค๎ซ๎€ท๐‘ฅโˆ’๐‘๐›พ๐œ™(๐‘)โˆ’๐ด๐‘,๐ฝ๐‘›โˆ’๐‘๎€ธ๎ฌ.(3.6) Thus, ()โ€–โ€–๐‘ฅ๐›พ๐œ“โ€–๐‘ฅโˆ’๐‘โ€–๐‘›โ€–โ€–โ‰ค๎ซ๎€ท๐‘ฅโˆ’๐‘๐›พ๐œ™(๐‘)โˆ’๐ด๐‘,๐ฝ๐‘›โ€–โ€–๐ฝ๎€ท๐‘ฅโˆ’๐‘๎€ธ๎ฌโ‰คโ€–๐›พ๐œ™(๐‘)โˆ’๐ด๐‘โ€–๐‘›๎€ธโ€–โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘=โ€–๐›พ๐œ™(๐‘)โˆ’๐ด๐‘๐‘›โ€–โ€–.โˆ’๐‘(3.7) It follows that ๐›พ๐œ“(โ€–๐‘ฅ๐‘›โˆ’๐‘โ€–)โ‰คโ€–๐›พ๐œ™(๐‘)โˆ’๐ด๐‘โ€–. Hence, โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘โ‰ค๐œ“โˆ’1๎‚ต(โ€–๐›พ๐œ™๐‘)โˆ’๐ด๐‘โ€–๐›พ๎‚ถ,(3.8) which implies that {๐‘ฅ๐‘›} is bounded. Since ๐œ™ is weakly contractive, we have โ€–โ€–๐œ™๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐œ™(๐‘)๐‘›โ€–โ€–๎€ทโ€–โ€–๐‘ฅโˆ’๐‘โˆ’๐œ“๐‘›โ€–โ€–๎€ธโ‰คโ€–โ€–๐‘ฅโˆ’๐‘๐‘›โ€–โ€–โˆ’๐‘.(3.9) Then, {๐œ™(๐‘ฅ๐‘›)} is bounded. From (3.1), we have ๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘ฆ๐‘›=๐‘ฅ๐‘›โˆ’๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โ‰ค11โˆ’๐›ผ๐‘›๐›พโ€–โ€–๐‘ฅ๐‘›โ€–โ€–+๐›ผ๐‘›๐›พ1โˆ’๐›ผ๐‘›๐›พโ€–โ€–๐œ™๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–.(3.10) Thus, {๐‘ฆ๐‘›} is also bounded. We denoted ๐‘ง๐‘›โˆถ=(1/๐‘ก๐‘›)โˆซ๐‘ก๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ . And since ๐‘ง๐‘›=11โˆ’๐›ฝ๐‘›๐‘ฆ๐‘›โˆ’๐›ฝ๐‘›1โˆ’๐›ฝ๐‘›๐‘ฅ๐‘›,โ€–โ€–๐‘ง๐‘›โ€–โ€–โ‰ค11โˆ’๐›ฝ๐‘›โ€–โ€–๐‘ฆ๐‘›โ€–โ€–+๐›ฝ๐‘›1โˆ’๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–,(3.11){๐‘ง๐‘›} is also bounded by the boundedness of {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›}.
Next, we show that โ€–๐‘ฅ๐‘›โˆ’๐‘‡(โ„Ž)๐‘ฅ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž, for all โ„Žโ‰ฅ0. We note thatโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ง๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–+โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ง๐‘›โ€–โ€–=๐›ผ๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธโˆ’๐ด๐‘ฆ๐‘›โ€–โ€–+๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ง๐‘›โ€–โ€–.(3.12) It follows that ๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ง๐‘›โ€–โ€–โ‰ค๐›ผ๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธโˆ’๐ด๐‘ฆ๐‘›โ€–โ€–.(3.13) By conditions (๐ถ1) and (๐ถ2), we obtain lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ง๐‘›โ€–โ€–=0.(3.14) Moreover, we note that, for all โ„Žโ‰ฅ0, โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡(โ„Ž)๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ง๐‘›โ€–โ€–+โ€–โ€–๐‘ง๐‘›โˆ’๐‘‡(โ„Ž)๐‘ง๐‘›โ€–โ€–+โ€–โ€–๐‘‡(โ„Ž)๐‘ง๐‘›โˆ’๐‘‡(โ„Ž)๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅโ‰ค2๐‘›โˆ’๐‘ง๐‘›โ€–โ€–+โ€–โ€–๐‘ง๐‘›โˆ’๐‘‡(โ„Ž)๐‘ง๐‘›โ€–โ€–.(3.15) Define the set ๐พ={๐‘งโˆˆ๐ถโˆถโ€–๐‘งโˆ’๐‘โ€–โ‰ค๐œ“โˆ’1(โ€–๐›พ๐œ™(๐‘)โˆ’๐ด๐‘โ€–/๐›พ)}; then ๐พ is a nonempty bounded closed convex subset of ๐ถ, which is๐‘‡(โ„Ž)-invariant for each โ„Žโ‰ฅ0 (i.e., ๐‘‡(โ„Ž)๐ถโŠ‚๐ถ). Since {๐‘ฅ๐‘›}โŠ‚๐ถ and ๐ถ is bounded, there exists ๐‘Ÿ>0 such that ๐พโŠ‚๐ต๐‘Ÿ, and it follows from Lemma 2.4 that lim๐‘›โ†’โˆžโ€–โ€–๐‘ง๐‘›โˆ’๐‘‡(โ„Ž)๐‘ง๐‘›โ€–โ€–=0,โˆ€โ„Žโ‰ฅ0.(3.16) Noting (3.14) and (3.16), then, from (3.15), we obtain lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡(โ„Ž)๐‘ฅ๐‘›โ€–โ€–=0,โˆ€โ„Žโ‰ฅ0.(3.17)
Next, we show that {๐‘ฅ๐‘›} contains a subsequence converging strongly to ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ). Since {๐‘ฅ๐‘›} is bounded and Banach space ๐‘‹ is a uniformly convex, it is reflexive and there exists a subsequence {๐‘ฅ๐‘›๐‘—}โŠ‚{๐‘ฅ๐‘›}, which converges weakly to some ฬƒ๐‘ฅโˆˆ๐ถ as ๐‘—โ†’โˆž. Again since Banach space ๐‘‹ has a weakly sequentially continuous duality mapping satisfying Opial's condition, noting (3.17) and by Lemma 2.3, we have ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ). From (3.7), replace ๐‘ by ฬƒ๐‘ฅ to obtainโ€–โ€–๐‘ฅ๐›พ๐œ“(โ€–๐‘ฅโˆ’ฬƒ๐‘ฅโ€–)๐‘›๐‘—โ€–โ€–โ‰ค๎‚ฌ๎‚€๐‘ฅโˆ’ฬƒ๐‘ฅ๐›พ๐œ™(ฬƒ๐‘ฅ)โˆ’๐ดฬƒ๐‘ฅ,๐ฝ๐‘›๐‘—โˆ’ฬƒ๐‘ฅ๎‚๎‚ญ.(3.18) Since ๐ฝ is single valued and weakly sequentially continuous from ๐‘‹ to ๐‘‹โˆ—, we get that lim๐‘—โ†’โˆž๎‚€โ€–โ€–๐‘ฅ๐›พ๐œ“๐‘›๐‘—โ€–โ€–๎‚โ€–โ€–๐‘ฅโˆ’ฬƒ๐‘ฅ๐‘›๐‘—โ€–โ€–โˆ’ฬƒ๐‘ฅโ‰คlim๐‘—โ†’โˆž๎‚ฌ๎‚€๐‘ฅ๐›พ๐œ™(ฬƒ๐‘ฅ)โˆ’๐ดฬƒ๐‘ฅ,๐ฝ๐‘›๐‘—โˆ’ฬƒ๐‘ฅ๎‚๎‚ญ=0.(3.19) Thus, ๐‘ฅ๐‘›๐‘—โ†’ฬƒ๐‘ฅ as ๐‘—โ†’โˆž.
Next, we show that ฬƒ๐‘ฅ is a solution of the variational inequality (3.2). Firstly, sinceโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–=๐›ผ๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธโˆ’๐ด๐‘ฆ๐‘›โ€–โ€–,(3.20) by condition (๐ถ1), we obtain lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–=0. Since ๐‘ฅ๐‘›๐‘—โ†’ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ), then ๐‘ฆ๐‘›๐‘—โ†’ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ).
From (3.1), we derive that ๐›พ๐œ™(๐‘ฅ๐‘›)โˆ’๐ด๐‘ฆ๐‘›=(1/๐›ผ๐‘›)(๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›). Then, for each ๐œ”โˆˆ๐น(๐’ฎ),๎ซ๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธโˆ’๐ด๐‘ฆ๐‘›๎€ท,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›=1๎€ธ๎ฌ๐›ผ๐‘›๎ซ๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›๎€ท,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›=1๎€ธ๎ฌ๐›ผ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๎ƒก๐‘ฅ๐‘›โˆ’1๐‘ก๐‘›๎€œ๐‘ก๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎€ท๐‘‘๐‘ ,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›๎€ธ๎ƒข=1๐›ผ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ1๎‚ธ๎ƒก๐œ”โˆ’๐‘ก๐‘›๎€œ๐‘ก๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎€ท๐‘‘๐‘ ,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›๎€ธ๎ƒขโˆ’โ€–โ€–๐œ”โˆ’๐‘ฅ๐‘›โ€–โ€–2๎‚นโ‰ค1๐›ผ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๎‚ƒโ€–โ€–๐œ”โˆ’๐‘ฅ๐‘›โ€–โ€–2โˆ’โ€–โ€–๐œ”โˆ’๐‘ฅ๐‘›โ€–โ€–2๎‚„=0.(3.21) Therefore, ๎€ท๐‘ฅโŸจ๐›พ๐œ™๐‘›๎€ธโˆ’๐ด๐‘ฆ๐‘›๎€ท,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›๎€ธโ‰ค0.(3.22) Since the duality mapping ๐ฝ is single-valued and weakly sequentially continuous duality mapping from ๐‘‹ to ๐‘‹โˆ—, for each ๐œ”โˆˆ๐น(๐’ฎ),๐‘ฅ๐‘›๐‘—โ†’ฬƒ๐‘ฅ and ๐‘ฆ๐‘›๐‘—โ†’ฬƒ๐‘ฅ, then, from (3.22), we obtain lim๐‘—โ†’โˆž๎‚ฌ๎‚€๐‘ฅ๐›พ๐œ™๐‘›๐‘—๎‚โˆ’๐ด๐‘ฆ๐‘›๐‘—๎‚€,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›๐‘—๎‚๎‚ญ=โŸจ๐›พ๐œ™(ฬƒ๐‘ฅ)โˆ’๐ดฬƒ๐‘ฅ,๐ฝ(๐œ”โˆ’ฬƒ๐‘ฅ)โŸฉโ‰ค0.(3.23) That is, ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ) is a solution of the variational inequality (3.2).
Next, we show the uniqueness of the solution of the variational inequality (3.2). Suppose that ฬƒ๐‘ฅ,๐‘ฅโˆ—โˆˆ๐น(๐’ฎ) satisfy (3.2). Then,๎ซ๎€ท๐‘ฅ๐›พ๐œ™(ฬƒ๐‘ฅ)โˆ’๐ดฬƒ๐‘ฅ,๐ฝโˆ—๎ซ๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ค0,๐›พ๐œ™โˆ—๎€ธโˆ’๐ด๐‘ฅโˆ—๎€ท,๐ฝฬƒ๐‘ฅโˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ค0.(3.24) Adding up (3.24), we get ๎ซ0โ‰ฅ(๐›พ๐œ™โˆ’๐ด)ฬƒ๐‘ฅโˆ’(๐›พ๐œ™โˆ’๐ด)๐‘ฅโˆ—๎€ท๐‘ฅ,๐ฝโˆ—=๎ซ๐ด๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโˆ—๎€ธ๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅ,๐ฝโˆ—๎ซ๐œ™๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโˆ’๐›พโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐œ™(ฬƒ๐‘ฅ),๐ฝโˆ—โ‰ฅโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ๐›พโ€–๐‘ฅโˆ—โˆ’ฬƒ๐‘ฅโ€–2โ€–โ€–๐œ™๎€ท๐‘ฅโˆ’๐›พโˆ—๎€ธโ€–โ€–โ€–โ€–๐ฝ๎€ท๐‘ฅโˆ’๐œ™(ฬƒ๐‘ฅ)โˆ—๎€ธโ€–โ€–โ‰ฅโˆ’ฬƒ๐‘ฅ๐›พโ€–๐‘ฅโˆ—โˆ’ฬƒ๐‘ฅโ€–2โˆ’๐›พโ€–๐‘ฅโˆ—โˆ’ฬƒ๐‘ฅโ€–2๎€ท+๐›พ๐œ“โ€–๐‘ฅโˆ—๎€ธโˆ’ฬƒ๐‘ฅโ€–โ€–๐‘ฅโˆ—โˆ’ฬƒ๐‘ฅโ€–.(3.25) Thus ๐›พ๐œ“(โ€–๐‘ฅโˆ—โˆ’ฬƒ๐‘ฅโ€–)โ‰ค(๐›พโˆ’๐›พ)โ€–๐‘ฅโˆ—โˆ’ฬƒ๐‘ฅโ€–. By the property of ๐œ“, we must have ฬƒ๐‘ฅ=๐‘ฅโˆ— and the uniqueness is proved.
Finally, we show that {๐‘ฅ๐‘›} converges strongly to ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ). Suppose that there exists another subsequence ๐‘ฅ๐‘›๐‘–โ†’ฬ‚๐‘ฅ as ๐‘–โ†’โˆž. We note that ฬ‚๐‘ฅโˆˆ๐น(๐’ฎ) is the solution of the variational inequality (3.2). Hence, ฬƒ๐‘ฅ=ฬ‚๐‘ฅ=๐‘ฅโˆ— by uniqueness. In summary, we have shown that {๐‘ฅ๐‘›} is sequentially compact and each cluster point of the sequence {๐‘ฅ๐‘›} is equal to ๐‘ฅโˆ—. Therefore, we conclude that ๐‘ฅ๐‘›โ†’๐‘ฅโˆ— as ๐‘›โ†’โˆž. This proof is complete.

Remark 3.2. (1) Theorem 3.1 improves and generalizes Theorem 3.1 of S. Li et al. [16] from a contractive mapping to a weakly contractive mapping and from Hilbert spaces to Banach spaces.
(2) Theorem 3.1 also improves and generalizes Theorem 3.2 of Marino and Xu [9] from a nonexpansive mapping to a nonexpansive semigroup, from a contractive mapping to a weakly contractive mapping and from Hilbert spaces to Banach spaces.

A strong mean convergence theorem for nonexpansive mapping was first established by Baillon [24], and it was generalized to that for nonlinear semigroups by Reich [25โ€“27]. It is clear that Theorem 3.1 is valid for nonexpansive mappings. Thus, we have the following mean ergodic theorem of viscosity iteration process for nonexpansive mappings in Hilbert spaces.

Corollary 3.3. Let ๐ป be a real Hilbert space and ๐ถ a nonempty closed convex subset of ๐ป such that ๐ถยฑ๐ถโŠ‚๐ถ. Let T be a nonexpansive mapping from C into itself such that ๐น(๐‘‡)โ‰ โˆ…. Let ๐œ™ be a weakly contractive mapping and ๐ด a strongly positive linear bounded operator with a coefficient ๐›พ>0 such that 0<๐›พ<๐›พ. Let {๐‘ฅ๐‘›} be a sequence defined by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ1๐‘›+1๐‘›๎“๐‘—=0๐‘‡๐‘—๐‘ฅ๐‘›,๐‘ฅ๐‘›=๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘ฆ๐‘›,โˆ€๐‘›โ‰ฅ0,(3.26) where {๐›ผ๐‘›},{๐›ฝ๐‘›} are two sequences in (0,1) satisfying the following conditions: (๐ถ1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (๐ถ2)0<liminf๐‘›โ†’โˆž๐›ฝ๐‘›โ‰คlimsup๐‘›โ†’โˆž๐›ฝ๐‘›<1. Then the sequence {๐‘ฅ๐‘›} defined by (3.26) converges strongly to the common fixed point ๐‘ฅโˆ—โˆˆ๐น(๐’ฎ), where ๐‘ฅโˆ— is the unique solution of the variational inequality ๎ซ๎€ท๐‘ฅ๐›พ๐œ™โˆ—๎€ธโˆ’๐ด๐‘ฅโˆ—๎€ท,๐ฝ๐œ”โˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ค0,โˆ€๐œ”โˆˆ๐น(๐’ฎ).(3.27)

Taking ๐ด=๐ผ and ๐›พ=1 in Theorem 3.1, we get the following corollary.

Corollary 3.4. Let ๐‘‹ be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping ๐ฝ from ๐‘‹ into ๐‘‹โˆ—,โ€‰and ๐ถ a nonempty closed convex subset of ๐‘‹ such that ๐ถยฑ๐ถโŠ‚๐ถ. Let ๐’ฎ={๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+} be a nonexpansive semigroup from ๐ถ into itself such that ๐น(๐’ฎ)โ‰ โˆ…. Let ๐œ™ be a weakly contractive mapping. Let {๐‘ฅ๐‘›} be a sequence defined by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ1๐‘ก๐‘›๎€œ๐‘ก๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘ฅ๐‘‘๐‘ ,๐‘›=๐›ผ๐‘›๐œ™๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฆ๐‘›,โˆ€๐‘›โ‰ฅ0,(3.28) where {๐›ผ๐‘›}, {๐›ฝ๐‘›} are two sequences in (0,1) and {๐‘ก๐‘›} is a positive real divergent sequence satisfying the following conditions: (๐ถ1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (๐ถ2)0<liminf๐‘›โ†’โˆž๐›ฝ๐‘›โ‰คlimsup๐‘›โ†’โˆž๐›ฝ๐‘›<1. Then the sequence {๐‘ฅ๐‘›} defined by (3.28) converges strongly to the common fixed point ๐‘ฅโˆ—โˆˆ๐น(๐’ฎ), where ๐‘ฅโˆ— is the unique solution of the variational inequality ๎ซ๐œ™๎€ท๐‘ฅโˆ—๎€ธโˆ’๐‘ฅโˆ—๎€ท,๐ฝ๐œ”โˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ค0,โˆ€๐œ”โˆˆ๐น(๐’ฎ).(3.29)

Next, we prove a strong convergence theorem under different conditions.

Theorem 3.5. Let ๐‘‹ be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping ๐ฝ from ๐‘‹ into ๐‘‹โˆ— and ๐ถ a nonempty closed convex subset of ๐‘‹ such that ๐ถยฑ๐ถโŠ‚๐ถ. Let ๐’ฎ={๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+} be a nonexpansive semigroup from ๐ถ into itself such that ๐น(๐’ฎ)โ‰ โˆ…. Let ๐œ™ be a weakly contractive mapping and ๐ด a strongly positive linear bounded operator with a coefficient ๐›พ>0 such that 0<๐›พ<๐›พ. Let {๐‘ฅ๐‘›} be a sequence defined by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‡๎€ท๐‘ก๐‘›๎€ธ๐‘ฅ๐‘›,๐‘ฅ๐‘›=๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘ฆ๐‘›,โˆ€๐‘›โ‰ฅ0,(3.30) where {๐›ผ๐‘›}, {๐›ฝ๐‘›} are two sequences in (0,1) and {๐‘ก๐‘›} is a positive real sequence satisfying the following conditions: (๐ถ1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0,(๐ถ2)lim๐‘›โ†’โˆž๐‘ก๐‘›=lim๐‘›โ†’โˆž(๐›ผ๐‘›/๐‘ก๐‘›)=0. Then, the sequence {๐‘ฅ๐‘›} defined by (3.30) converges strongly to the common fixed point ๐‘ฅโˆ—โˆˆ๐น(๐’ฎ), where ๐‘ฅโˆ— is the unique solution of the variational inequality ๎ซ๎€ท๐‘ฅ๐›พ๐œ™โˆ—๎€ธโˆ’๐ด๐‘ฅโˆ—๎€ท,๐ฝ๐œ”โˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ค0,โˆ€๐œ”โˆˆ๐น(๐’ฎ).(3.31)

Proof. Firstly, we show that {๐‘ฅ๐‘›} defined by (3.30) is well defined. Since lim๐‘›โ†’โˆž๐›ผ๐‘›=0, we may assume, with no loss of generality, that ๐›ผ๐‘›<โ€–๐ดโ€–โˆ’1 for each ๐‘›โ‰ฅ0. Define the mapping ๐‘‡๐œ™๐‘›โˆถ๐ถโ†’๐ถ by ๐‘‡๐œ™๐‘›โˆถ=๐›ผ๐‘›๎€ท๐›พ๐œ™+๐ผโˆ’๐›ผ๐‘›๐ด๐›ฝ๎€ธ๎€บ๐‘›๎€ท๐ผ+1โˆ’๐›ฝ๐‘›๎€ธ๐‘‡๎€ท๐‘ก๐‘›๎€ธ๎€ป.(3.32) From Lemma 2.1, we have for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ, โ€–โ€–๐‘‡๐œ™๐‘›๐‘ฅโˆ’๐‘‡๐œ™๐‘›๐‘ฆโ€–โ€–=๐›ผ๐‘›๐›พ(๐œ™(๐‘ฅ)โˆ’๐œ™(๐‘ฆ))+๐›ฝ๐‘›๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ+๎€ท(๐‘ฅโˆ’๐‘ฆ)1โˆ’๐›ฝ๐‘›๎€ธ๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๐‘‡๎€ท๐‘ก๎€ธ๎€ท๐‘›๎€ธ๎€ท๐‘ก๐‘ฅโˆ’๐‘‡๐‘›๎€ธ๐‘ฆ๎€ธโ‰ค๐›ผ๐‘›๐›พโ€–๐œ™(๐‘ฅ)โˆ’๐œ™(๐‘ฆ)โ€–+๐›ฝ๐‘›โ€–โ€–๐ผโˆ’๐›ผ๐‘›๐ดโ€–โ€–+๎€ทโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐ผโˆ’๐›ผ๐‘›๐ดโ€–โ€–โ€–โ€–๐‘‡๎€ท๐‘ก๐‘›๎€ธ๎€ท๐‘ก๐‘ฅโˆ’๐‘‡๐‘›๎€ธ๐‘ฆโ€–โ€–โ‰ค๐›ผ๐‘›๐›พ[]โ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’๐œ“(โ€–๐‘ฅโˆ’๐‘ฆโ€–)+๐›ฝ๐‘›๎€ท1โˆ’๐›ผ๐‘›๐›พ๎€ธ+๎€ทโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐›ฝ๐‘›๎€ธ๎€ท1โˆ’๐›ผ๐‘›๐›พ๎€ธโ€–๐‘ฅโˆ’๐‘ฆโ€–=๐›ผ๐‘›๐›พ[]+๎€ทโ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’๐œ“(โ€–๐‘ฅโˆ’๐‘ฆโ€–)1โˆ’๐›ผ๐‘›๐›พ๎€ธ=๎€บโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐›ผ๐‘›๎€ท๐›พโˆ’๐›พ๎€ธ๎€ปโ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’๐›ผ๐‘›๐›พ๐œ“(โ€–๐‘ฅโˆ’๐‘ฆโ€–)โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’๐›ผ๐‘›๐›พ๐œ“(โ€–๐‘ฅโˆ’๐‘ฆโ€–).(3.33) This show that ๐‘‡๐œ™๐‘› is weakly contractive. It follows from Lemma 2.2 that ๐‘‡๐œ™๐‘› has a unique fixed point ๐‘ฅ๐‘›โˆˆ๐ถ, that is, {๐‘ฅ๐‘›} defined by (3.30) is well defined.
Next, we show that {๐‘ฅ๐‘›} is bounded. Letting ๐‘โˆˆ๐น(๐’ฎ), we getโ€–โ€–๐‘ฆ๐‘›โ€–โ€–=โ€–โ€–๐›ฝโˆ’๐‘๐‘›๎€ท๐‘ฅ๐‘›๎€ธ+๎€ทโˆ’๐‘1โˆ’๐›ฝ๐‘›๐‘‡๎€ท๐‘ก๎€ธ๎€ท๐‘›๎€ธ๐‘ฅ๐‘›๎€ธโ€–โ€–โˆ’๐‘โ‰ค๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–+๎€ทโˆ’๐‘1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘‡๎€ท๐‘ก๐‘›๎€ธ๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘โ‰ค๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–+๎€ทโˆ’๐‘1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–=โ€–โ€–๐‘ฅโˆ’๐‘๐‘›โ€–โ€–,โ€–โ€–๐‘ฅโˆ’๐‘๐‘›โ€–โ€–โˆ’๐‘2=๎ซ๐›ผ๐‘›๎€ท๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธ๎€ธ+๎€ทโˆ’๐ด๐‘๐ผโˆ’๐›ผ๐‘›๐ด๐‘ฆ๎€ธ๎€ท๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘,๐ฝ๐‘›โˆ’๐‘๎€ธ๎ฌ=๐›ผ๐‘›๎€ท๐‘ฅ๎ซ๎€ท๐›พ๐œ™๐‘›๎€ธ๎€ธ๎€ท๐‘ฅโˆ’๐›พ๐œ™(๐‘)+(๐›พ๐œ™(๐‘)โˆ’๐ด๐‘),๐ฝ๐‘›+โˆ’๐‘๎€ธ๎ฌ๎ซ๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๐‘ฆ๎€ธ๎€ท๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘,๐ฝ๐‘›โˆ’๐‘๎€ธ๎ฌโ‰ค๐›ผ๐‘›๐›พโ€–โ€–๐œ™๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โ€–๐ฝ๎€ท๐‘ฅโˆ’๐œ™(๐‘)๐‘›๎€ธโ€–โ€–โˆ’๐‘+๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐›พ๐œ™(๐‘)โˆ’๐ด๐‘,๐ฝ๐‘›+โ€–โ€–โˆ’๐‘๎€ธ๎ฌ๐ผโˆ’๐›ผ๐‘›๐ดโ€–โ€–โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๐ฝ๎€ท๐‘ฅโˆ’๐‘๐‘›๎€ธโ€–โ€–โˆ’๐‘โ‰ค๐›ผ๐‘›๎€บ๐›พโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ทโ€–โ€–๐‘ฅโˆ’๐‘โˆ’๐›พ๐œ“๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘๎€ธ๎€ป๐‘›โ€–โ€–โˆ’๐‘+๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐›พ๐œ™(๐‘)โˆ’๐ด๐‘,๐ฝ๐‘›+๎€ทโˆ’๐‘๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๐›พ๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2=๎€บ1โˆ’๐›ผ๐‘›๎€ทโ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๎€ธ๎€ป๐‘›โ€–โ€–โˆ’๐‘2โˆ’๐›ผ๐‘›๎€ทโ€–โ€–๐‘ฅ๐›พ๐œ“๐‘›โ€–โ€–๎€ธโ€–โ€–๐‘ฅโˆ’๐‘๐‘›โ€–โ€–โˆ’๐‘+๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐›พ๐œ™(๐‘)โˆ’๐ด๐‘,๐ฝ๐‘›,โˆ’๐‘๎€ธ๎ฌ(3.34) and so ()โ€–โ€–๐‘ฅ๐›พ๐œ“โ€–๐‘ฅโˆ’๐‘โ€–๐‘›โ€–โ€–+๎€ทโˆ’๐‘๎€ธโ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐‘›โ€–โ€–โ‰ค๎ซ๎€ท๐‘ฅโˆ’๐‘๐›พ๐œ™(๐‘)โˆ’๐ด๐‘,๐ฝ๐‘›โˆ’๐‘๎€ธ๎ฌ.(3.35) Thus, ()โ€–โ€–๐‘ฅ๐›พ๐œ“โ€–๐‘ฅโˆ’๐‘โ€–๐‘›โ€–โ€–โ‰ค๎ซ๎€ท๐‘ฅโˆ’๐‘๐›พ๐œ™(๐‘)โˆ’๐ด๐‘,๐ฝ๐‘›โ€–โ€–๐ฝ๎€ท๐‘ฅโˆ’๐‘๎€ธ๎ฌโ‰คโ€–๐›พ๐œ™(๐‘)โˆ’๐ด๐‘โ€–๐‘›๎€ธโ€–โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘=โ€–๐›พ๐œ™(๐‘)โˆ’๐ด๐‘๐‘›โ€–โ€–.โˆ’๐‘(3.36) It follows that ๐›พ๐œ“(โ€–๐‘ฅ๐‘›โˆ’๐‘โ€–)โ‰คโ€–๐›พ๐œ™(๐‘)โˆ’๐ด๐‘โ€–. Hence, โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘โ‰ค๐œ“โˆ’1๎‚ต(โ€–๐›พ๐œ™๐‘)โˆ’๐ด๐‘โ€–๐›พ๎‚ถ.(3.37) This implies that {๐‘ฅ๐‘›} is bounded, so are {๐œ™(๐‘ฅ๐‘›)}, {๐ด๐‘ฆ๐‘›} and {๐‘‡(๐‘ก๐‘›)๐‘ฅ๐‘›}.
Next, we show that {๐‘ฅ๐‘›} contains a subsequence converging strongly to ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ). By reflexivity of ๐‘‹ and boundedness of the sequence {๐‘ฅ๐‘›}, there exists subsequence {๐‘ฅ๐‘›๐‘—}โŠ†{๐‘ฅ๐‘›} such that ๐‘ฅ๐‘›๐‘—โ‡€ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ) as ๐‘—โ†’โˆž. Now, we show that ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ). Put ๐‘ฅ๐‘—โˆถ=๐‘ฅ๐‘›๐‘—,โ€‰๐‘ฆ๐‘—โˆถ=๐‘ฆ๐‘›๐‘—, ๐›ผ๐‘—โˆถ=๐›ผ๐‘›๐‘—, ๐›ฝ๐‘—โˆถ=๐›ฝ๐‘›๐‘—, and ๐‘ก๐‘—โˆถ=๐‘ก๐‘›๐‘— for ๐‘—โˆˆโ„•, and fix ๐‘ก>0. We note thatโ€–โ€–๐‘ฅ๐‘—โ€–โ€–โ‰คโˆ’๐‘‡(๐‘ก)ฬƒ๐‘ฅ[๐‘ก/๐‘ก๐‘—]โˆ’1๎“๐‘˜=0โ€–โ€–๐‘‡๎€ท(๐‘˜+1)๐‘ก๐‘—๎€ธ๐‘ฅ๐‘—๎€ทโˆ’๐‘‡๐‘˜๐‘ก๐‘—๎€ธ๐‘ฅ๐‘—โ€–โ€–+โ€–โ€–โ€–๐‘‡๐‘ก๎‚ต๎‚ธ๐‘ก๐‘—๎‚น๐‘ก๐‘—๎‚ถ๐‘ฅ๐‘—๐‘กโˆ’๐‘‡๎‚ต๎‚ธ๐‘ก๐‘—๎‚น๐‘ก๐‘—๎‚ถโ€–โ€–โ€–+โ€–โ€–โ€–๐‘‡๐‘กฬƒ๐‘ฅ๎‚ต๎‚ธ๐‘ก๐‘—๎‚น๐‘ก๐‘—๎‚ถโ€–โ€–โ€–โ‰ค๎‚ธ๐‘กฬƒ๐‘ฅโˆ’๐‘‡(๐‘ก)ฬƒ๐‘ฅ๐‘ก๐‘—๎‚นโ€–โ€–๐‘‡๎€ท๐‘ก๐‘—๎€ธ๐‘ฅ๐‘—โˆ’๐‘ฅ๐‘—โ€–โ€–+โ€–โ€–๐‘ฅ๐‘—โ€–โ€–+โ€–โ€–โ€–๐‘‡๎‚ต๎‚ธ๐‘กโˆ’ฬƒ๐‘ฅ๐‘กโˆ’๐‘ก๐‘—๎‚น๐‘ก๐‘—๎‚ถโ€–โ€–โ€–=๎‚ธ๐‘กฬƒ๐‘ฅโˆ’ฬƒ๐‘ฅ๐‘ก๐‘—๎‚น๐›ผ๐‘—1โˆ’๐›ฝ๐‘—โ€–โ€–๎€ท๐‘ฅ๐›พ๐œ™๐‘—๎€ธโˆ’๐ด๐‘ฆ๐‘—โ€–โ€–+โ€–โ€–๐‘ฅ๐‘—โ€–โ€–+โ€–โ€–โ€–๐‘‡๎‚ต๎‚ธ๐‘กโˆ’ฬƒ๐‘ฅ๐‘กโˆ’๐‘ก๐‘—๎‚น๐‘ก๐‘—๎‚ถโ€–โ€–โ€–โ‰ค๐‘กฬƒ๐‘ฅโˆ’ฬƒ๐‘ฅ1โˆ’๐›ฝ๐‘—๐›ผ๐‘—๐‘ก๐‘—โ€–โ€–๎€ท๐‘ฅ๐›พ๐œ™๐‘—๎€ธโˆ’๐ด๐‘ฆ๐‘—โ€–โ€–+โ€–โ€–๐‘ฅ๐‘—โ€–โ€–๎€ฝโˆ’ฬƒ๐‘ฅ+maxโ€–๐‘‡(๐‘ )ฬƒ๐‘ฅโˆ’ฬƒ๐‘ฅโ€–โˆถ0โ‰ค๐‘ โ‰ค๐‘ก๐‘—๎€พ.(3.38) For all ๐‘—โˆˆโ„•, we have limsup๐‘—โ†’โˆžโ€–โ€–๐‘ฅ๐‘—โ€–โ€–โˆ’๐‘‡(๐‘ก)ฬƒ๐‘ฅโ‰คlimsup๐‘—โ†’โˆžโ€–โ€–๐‘ฅ๐‘—โ€–โ€–โˆ’ฬƒ๐‘ฅ.(3.39) By the assumption that Banach space ๐‘‹ has a weakly sequentially continuous duality mapping satisfying Opialโ€™s condition, (3.39) implies that ๐‘‡(๐‘ก)ฬƒ๐‘ฅ=ฬƒ๐‘ฅ, and we get that ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ). From (3.36), replace ๐‘ by ฬƒ๐‘ฅ to obtain ๎€ทโ€–โ€–๐‘ฅ๐›พ๐œ“๐‘—โ€–โ€–๎€ธโ€–โ€–๐‘ฅโˆ’ฬƒ๐‘ฅ๐‘—โ€–โ€–โ‰ค๎ซ๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅ๐›พ๐œ™(ฬƒ๐‘ฅ)โˆ’๐ดฬƒ๐‘ฅ,๐ฝ๐‘—โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ.(3.40) Since ๐ฝ is singlevalued and weakly sequentially continuous from ๐‘‹ to ๐‘‹โˆ—, we get that lim๐‘—โ†’โˆž๎€ท๐›พ๐œ“โ€–๐‘ฅ๐‘—๎€ธโˆ’ฬƒ๐‘ฅโ€–โ€–๐‘ฅ๐‘—โˆ’ฬƒ๐‘ฅโ€–โ‰คlim๐‘—โ†’โˆž๎€ท๐‘ฅโŸจ๐›พ๐œ™(ฬƒ๐‘ฅ)โˆ’๐ดฬƒ๐‘ฅ,๐ฝ๐‘—๎€ธโˆ’ฬƒ๐‘ฅโŸฉ=0.(3.41) Thus, ๐‘ฅ๐‘—โ†’ฬƒ๐‘ฅ as ๐‘—โ†’โˆž, namely, there is a subsequence {๐‘ฅ๐‘›๐‘—}โŠ†{๐‘ฅ๐‘›} such that ๐‘ฅ๐‘›๐‘—โ†’ฬƒ๐‘ฅ as ๐‘—โ†’โˆž.
Next, we show that ฬƒ๐‘ฅ is a solution of the variational inequality (3.31). Firstly, sinceโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–=๐›ผ๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธโˆ’๐ด๐‘ฆ๐‘›โ€–โ€–.(3.42) by condition (๐ถ1), we obtain lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–=0. Since ๐‘ฅ๐‘›๐‘—โ†’ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ), then ๐‘ฆ๐‘›๐‘—โ†’ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ).
From (3.30), we derive that ๐›พ๐œ™(๐‘ฅ๐‘›)โˆ’๐ด๐‘ฆ๐‘›=(1/๐›ผ๐‘›)(๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›). Then, for each ๐œ”โˆˆ๐น(๐’ฎ),๎ซ๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธโˆ’๐ด๐‘ฆ๐‘›๎€ท,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›=1๎€ธ๎ฌ๐›ผ๐‘›๎ซ๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›๎€ท,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›=1๎€ธ๎ฌ๐›ผ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๐‘ฅ๎€ธ๎ซ๐‘›๎€ท๐‘กโˆ’๐‘‡๐‘›๎€ธ๐‘ฅ๐‘›๎€ท,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›=1๎€ธ๎ฌ๐›ผ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๎‚ƒ๎ซ๎€ท๐‘ก๐œ”โˆ’๐‘‡๐‘›๎€ธ๐‘ฅ๐‘›๎€ท,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›โˆ’โ€–โ€–๎€ธ๎ฌ๐œ”โˆ’๐‘ฅ๐‘›โ€–โ€–2๎‚„โ‰ค1๐›ผ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๎‚ƒโ€–โ€–๐œ”โˆ’๐‘ฅ๐‘›โ€–โ€–2โˆ’โ€–โ€–๐œ”โˆ’๐‘ฅ๐‘›โ€–โ€–2๎‚„=0.(3.43) Therefore, ๎ซ๎€ท๐‘ฅ๐›พ๐œ™๐‘›๎€ธโˆ’๐ด๐‘ฆ๐‘›๎€ท,๐ฝ๐œ”โˆ’๐‘ฅ๐‘›๎€ธ๎ฌโ‰ค0.(3.44) By using the same argument and techniques as those of Theorem 3.1, we note that the variational inequality (3.31) has a unique solution. We denoted by ๐‘ฅโˆ—โˆˆ๐น(๐’ฎ) the unique solution of (3.31). Therefore, ๐‘ฅ๐‘›โ†’๐‘ฅโˆ— as ๐‘›โ†’โˆž. The proof is completed.

Remark 3.6. (1) Theorem 3.5 improves and generalizes Theorem 3.2 of Marino and Xu [9] from a nonexpansive mapping to a nonexpansive semigroup, from a contractive mapping to a weakly contractive mapping, and from Hilbert spaces to Banach spaces.
(2) Theorem 3.5 also improves and generalizes Theorem 3.1 of Chen and He [15] from a contractive mapping to a weakly contractive mapping.

Taking ๐ด=๐ผ and ๐›พ=1 in Theorem 3.5, we get the following corollary.

Corollary 3.7. Let ๐‘‹ be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping ๐ฝ from ๐‘‹ into ๐‘‹โˆ— and ๐ถ a nonempty closed convex subset of ๐‘‹ such that ๐ถยฑ๐ถโŠ‚๐ถ. Let ๐’ฎ={๐‘‡(๐‘ก)โˆถ๐‘กโˆˆโ„+} be a nonexpansive semigroup from ๐ถ into itself such that ๐น(๐’ฎ)โ‰ โˆ…. Let ๐œ™ be a weakly contractive mapping. Let {๐‘ฅ๐‘›} be a sequence defined by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‡๎€ท๐‘ก๐‘›๎€ธ๐‘ฅ๐‘›,๐‘ฅ๐‘›=๐›ผ๐‘›๐œ™๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฆ๐‘›,โˆ€๐‘›โ‰ฅ0,(3.45) where {๐›ผ๐‘›},โ€‰{๐›ฝ๐‘›} are two sequences in (0,1) and {๐‘ก๐‘›} is a positive real sequence satisfying the following conditions: (๐ถ1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (๐ถ2)lim๐‘›โ†’โˆž๐‘ก๐‘›=lim๐‘›โ†’โˆž(๐›ผ๐‘›/๐‘ก๐‘›)=0. Then the sequence {๐‘ฅ๐‘›} defined by (3.45) converges strongly to the common fixed point ๐‘ฅโˆ—โˆˆ๐น(๐’ฎ), where ๐‘ฅโˆ— is the unique solution of the variational inequality ๎ซ๐œ™๎€ท๐‘ฅโˆ—๎€ธโˆ’๐‘ฅโˆ—๎€ท,๐ฝ๐œ”โˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ค0,โˆ€๐œ”โˆˆ๐น(๐’ฎ).(3.46)

Acknowledgments

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under CSEC project no. 54000267). The first author would like to give thanks to the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand for their financial support.

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