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Journal of Applied Mathematics
Volumeย 2012ย (2012), Article IDย 641479, 19 pages
http://dx.doi.org/10.1155/2012/641479
Research Article

Strong Convergence Theorems for Nonexpansive Semigroups and Variational Inequalities in Banach Spaces

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Mathematics, Tianjin No. 8 Middle School, Tianjin 300252, China

Received 11 November 2011; Accepted 17 December 2011

Academic Editor: Rudongย Chen

Copyright ยฉ 2012 Haiqing Wang 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

Let ๐‘‹ be a uniformly convex Banach space and ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} be a nonexpansive semigroup such that โ‹‚๐น(๐’ฎ)=๐‘ >0๐น(๐‘‡(๐‘ ))โ‰ โˆ…. Consider the iterative method that generates the sequence {๐‘ฅ๐‘›} by the algorithm ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“(๐‘ฅ๐‘›)+๐›ฝ๐‘›๐‘ฅ๐‘›+(1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›)(1/๐‘ ๐‘›)โˆซ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,๐‘›โ‰ฅ0, where {๐›ผ๐‘›}, {๐›ฝ๐‘›}, and {๐‘ ๐‘›} are three sequences satisfying certain conditions, ๐‘“โˆถ๐ถโ†’๐ถ is a contraction mapping. Strong convergence of the algorithm {๐‘ฅ๐‘›} is proved assuming ๐‘‹ either has a weakly continuous duality map or has a uniformly Gรขteaux differentiable norm.

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 by ๐น(๐‘‡) the set of fixed points of ๐‘‡. One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping (Browder [1] and Reich [2]). More precisely, take ๐‘กโˆˆ(0,1) and define a contraction ๐‘‡๐‘กโˆถ๐ถโ†’๐ถ by๐‘‡๐‘ก๐‘ฅ=๐‘ก๐‘ข+(1โˆ’๐‘ก)๐‘‡๐‘ฅ,๐‘ฅโˆˆ๐ถ,(1.1)

where ๐‘ขโˆˆ๐ถ is a fixed point. Banachโ€™s contraction mapping principle guarantees that ๐‘‡๐‘ก has a unique fixed point ๐‘ฅ๐‘ก in ๐ถ. It is unclear, in general, what is the behavior of {๐‘ฅ๐‘ก} as ๐‘กโ†’0, even if ๐‘‡ has a fixed point. In 1967, in the case of ๐‘‡ having a fixed point, Browder [3] proved that if ๐‘‹ is a Hilbert space, then ๐‘ฅ๐‘ก converges strongly to the element of ๐น(๐‘‡) which is nearest to ๐‘ข in ๐น(๐‘‡) as ๐‘กโ†“0. Song and Xu [4] extended Browderโ€™s result to the setting of Banach spaces and proved that if ๐‘‹ is a uniformly smooth Banach space, then ๐‘ฅ๐‘ก converges strongly to a fixed point of ๐‘‡ and the limit defines the (unique) sunny nonexpansive retraction from ๐ถ onto ๐น(๐‘‡).

Let ๐‘“ be a contraction on ๐ป such that โ€–๐‘“๐‘ฅโˆ’๐‘“๐‘ฆโ€–โ‰ค๐›ผโ€–๐‘ฅโˆ’๐‘ฆโ€–, where ๐›ผโˆˆ[0,1) is a constant. Let ๐‘ฅโˆˆ๐ถ, ๐‘กโˆˆ(0,1) and ๐‘ฅ๐‘กโˆˆ๐ถ be the unique fixed point of the contraction ๐‘†๐‘ก๐‘ฅ=๐‘ก๐‘“(๐‘ฅ)+(1โˆ’๐‘ก)๐‘‡๐‘ฅ, that is,๐‘ฅ๐‘ก๎€ท๐‘ฅ=๐‘ก๐‘“๐‘ก๎€ธ+(1โˆ’๐‘ก)๐‘‡๐‘ฅ๐‘ก.(1.2)

Concerning the convergence problem of the net {๐‘ฅ๐‘ก}, Moudafi [5] and Xu [6] by using the viscosity approximation method proved that the net {๐‘ฅ๐‘ก} converges strongly to a fixed point ฬƒ๐‘ฅ of T in C which is the unique solution to the following variational inequality:โŸจ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡).(1.3)

Moreover, Xu [6] also studied the strong convergence of the following iterative sequence generated by๐‘ฅ๐‘›+1=๐›ฝ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‡๐‘ฅ๐‘›,๐‘›โ‰ฅ0,(1.4)

where ๐‘ฅ0โˆˆ๐ถ is arbitrary, the sequence {๐›ฝ๐‘›} in (0,1) satisfies the certain appropriate conditions.

A family {๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} of mappings of ๐ถ into itself is called a nonexpansive semigroup if it satisfies the following conditions:(i)๐‘‡(0)๐‘ฅ=๐‘ฅ for all ๐‘ฅโˆˆ๐ถ;(ii)๐‘‡(๐‘ +๐‘ก)=๐‘‡(๐‘ )๐‘‡(๐‘ก) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐‘ ,๐‘กโ‰ฅ0;(iii)โ€–๐‘‡(๐‘ )๐‘ฅโˆ’๐‘‡(๐‘ )๐‘ฆโ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€– for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐‘ โ‰ฅ0;(iv)for all ๐‘ฅโˆˆ๐ถ, ๐‘ โ†ฆ๐‘‡(๐‘ )๐‘ฅ is continuous.

We denote by ๐น(๐’ฎ) the set of all common fixed points of ๐’ฎ, that is, ๐น(๐’ฎ)={๐‘ฅโˆˆ๐ถโˆถ๐‘‡(๐‘ )๐‘ฅ=๐‘ฅ,0โ‰ค๐‘ <โˆž}. It is known that ๐น(๐’ฎ) is closed and convex.

It is an interesting problem to extend above (Moudafiโ€™s [5], Xuโ€™s [6], and so on) results to the nonexpansive semigroup case. Recently, for the nonexpansive semigroups ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž}, Plubtieng and Punpaeng [7] studied the continuous scheme {๐‘ฅ๐‘ก} defined by๐‘ฅ๐‘ก๎€ท๐‘ฅ=๐‘ก๐‘“๐‘ก๎€ธ1+(1โˆ’๐‘ก)๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๐‘‘๐‘ ,(1.5)

where ๐‘กโˆˆ(0,1) and {๐œ†๐‘ก} is a positive real divergent net, and the iterative scheme {๐‘ฅ๐‘›} defined by๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,๐‘›โ‰ฅ0,(1.6)

where ๐‘ฅ0โˆˆ๐ถ, {๐›ผ๐‘›},{๐›ฝ๐‘›} are a sequence in (0,1) and {๐‘ ๐‘›} is a positive real divergent real sequence in the setting of a real Hilbert space. They proved the continuous scheme {๐‘ฅ๐‘ก} defined by (1.5) and the iterative scheme {๐‘ฅ๐‘›} defined by (1.6) converge strongly to a fixed point ฬƒ๐‘ฅ of ๐’ฎ which is the unique solution of the variational inequality (1.3). At this stage, the following question arises naturally.

Question 1. Do Plubtieng and Punpaengโ€™s results hold for the nonexpansive semigroups in a Banach space?

The purpose of this paper is to give affirmative answers of Question 1. One result of this paper says that Plubtieng and Punpaengโ€™s results hold in a uniformly convex Banach space which has a weakly continuous duality map.

On the other hand, Chen and Song [8] proved the following implicit and explicit viscosity iteration processes defined by (1.7) to nonexpansive semigroup case,๐‘ฅ๐‘›=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘ฅ๐‘‘๐‘ ,๐‘›โ‰ฅ0,๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ1๐‘ ๐‘›๎€œs๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,๐‘›โ‰ฅ0.(1.7)

And they proved that {๐‘ฅ๐‘›} converges strongly to a common fixed point of ๐น(๐’ฎ) in a uniformly convex Banach space with a uniformly Gรขteaux differentiable norm.

Motivated by the above results, the other result of this paper says that Plubtieng and Punpaengโ€™s results hold in the framework of uniformly convex Banach space with a uniformly Gรขteaux differentiable norm. The results improve and extend the corresponding results of Plubtieng and Punpaeng [7], Chen and Song [8], Moudafiโ€™s [5], Xuโ€™s [6], and others.

2. Preliminaries

Let ๐‘‹ be a real Banach space with inner product โŸจโ‹…,โ‹…โŸฉ and norm โ€–โ‹…โ€–, respectively. Let ๐ฝ denote the normalized duality mapping from ๐‘‹ into the dual space 2๐‘‹โˆ— given by๎‚†๐‘ฅ๐ฝ(๐‘ฅ)=โˆ—โˆˆ๐‘‹โˆ—โˆถโŸจ๐‘ฅ,๐‘ฅโˆ—โŸฉ=โ€–๐‘ฅโ€–2=โ€–๐‘ฅโˆ—โ€–2๎‚‡,๐‘ฅโˆˆ๐‘‹.(2.1)

In the sequel, we will denote the single valued duality mapping by ๐‘—. When {๐‘ฅ๐‘›} is a sequence in ๐‘‹, then ๐‘ฅ๐‘›โ†’๐‘ฅ(๐‘ฅ๐‘›โ‡€๐‘ฅ) will denote strong (weak) convergence of the sequence {๐‘ฅ๐‘›} to ๐‘ฅ.

Let ๐‘†(๐‘‹)={๐‘ฅโˆˆ๐‘‹โˆถโ€–๐‘ฅโ€–=1}. Then the norm of ๐‘‹ is said to be Gรขteaux differentiable iflim๐‘กโ†’0โ€–๐‘ฅ+๐‘ก๐‘ฆโ€–โˆ’โ€–๐‘ฅโ€–๐‘ก(2.2) exists for each ๐‘ฅ,๐‘ฆโˆˆ๐‘†(๐‘‹). In this case, ๐‘‹ is called smooth. The norm of ๐‘‹ is said to be uniformly Gรขteaux differentiable if for each ๐‘ฆโˆˆ๐‘†(๐‘‹), the limit (2.2) is attained uniformly for ๐‘ฅโˆˆ๐‘†(๐‘‹). It is well known that ๐‘‹ is smooth if and only if any duality mapping on ๐‘‹ is sigle valued. Also if ๐‘‹ has a uniformly Gรขteaux differentiable norm, then the duality mapping is norm-to-weak* uniformly continuous on bounded sets. The norm of E is called Frรฉchet differentiable, if for each ๐‘ฅโˆˆ๐‘†(๐‘‹), the limit (2.2) is attained uniformly for ๐‘ฆโˆˆ๐‘†(๐‘‹). The norm of ๐‘‹ is called uniformly Frรฉchet differentiable, if the limit (2.2) is attained uniformly for ๐‘ฅ,๐‘ฆโˆˆ๐‘†(๐‘‹). It is well known that (uniformly) Frรฉchet differentiability of the norm of ๐‘‹ implies (uniformly) Gรขteaux differentiability of the norm of ๐‘‹ and ๐‘‹ is uniformly smooth if and only if the norm of ๐‘‹ is uniformly Frรฉchet differentiable.

A Banach space ๐‘‹ is said to be strictly convex ifโ€–๐‘ฅโ€–=โ€–๐‘ฆโ€–=1,๐‘ฅโ‰ ๐‘ฆimpliesโ€–๐‘ฅ+๐‘ฆโ€–2<1.(2.3)

A Banach space ๐‘‹ is said to be uniformly convex if ๐›ฟ๐‘‹(๐œ€)>0 for all ๐œ€>0, where ๐›ฟ๐‘‹(๐œ€) is modulus of convexity of ๐ธ defined by๐›ฟ๐ธ๎‚ป(๐œ€)=inf1โˆ’โ€–๐‘ฅ+๐‘ฆโ€–2โˆถ๎‚ผ[]โ€–๐‘ฅโ€–โ‰ค1,โ€–๐‘ฆโ€–โ‰ค1,โ€–๐‘ฅโˆ’๐‘ฆโ€–โ‰ฅ๐œ€,๐œ€โˆˆ0,2.(2.4)

A uniformly convex Banach space ๐ธ is reflexive and strictly convex [9, Theorem 4.1.6, Theorem 4.1.2].

Lemma 2.1 (Goebel and Reich [10], Proposition 5.3). Let ๐ถ be a nonempty closed convex subset of a strictly convex Banach space ๐‘‹ and ๐‘‡โˆถ๐ถโ†’๐ถ a nonexpansive mapping with ๐น(๐‘‡)โ‰ โˆ…. Then ๐น(๐‘‡) is closed and convex.

Lemma 2.2 (see Xu [11]). In a smooth Banach space ๐‘‹ there holds the inequality โ€–๐‘ฅ+๐‘ฆโ€–2โ‰คโ€–๐‘ฅโ€–2+2โŸจ๐‘ฆ,๐ฝ(๐‘ฅ+๐‘ฆ)โŸฉ,๐‘ฅ,๐‘ฆ,โˆˆ๐‘‹.(2.5)

Lemma 2.3 (Browder [12]). Let ๐ธ be a uniformly convex Banach space, ๐พ a nonempty closed convex subset of ๐ธ, and ๐‘‡โˆถ๐พโ†’๐ธ a nonexpansive mapping. Then ๐ผโˆ’๐‘‡ is demi closed at zero.

Lemma 2.4 (see [8, Lemma 2.7]). Let ๐ถ be a nonempty bounded closed convex subset of a uniformly convex Banach space ๐‘‹, and let ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} be a nonexpansive semigroup on ๐ถ such that ๐น(๐’ฎ)โ‰ โˆ…. For ๐‘ฅโˆˆ๐ถ and ๐‘ก>0. Then, for any 0โ‰คโ„Ž<โˆž, lim๐‘กโ†’โˆžsup๐‘ฅโˆˆ๐ถโ€–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.(2.6)

Recall that a gauge is a continuous strictly increasing function ๐œ‘โˆถ[0,โˆž)โ†’[0,โˆž) such that ๐œ‘(0)=0 and ๐œ‘(๐‘ก)โ†’โˆž as ๐‘กโ†’โˆž. Associated to a gauge ๐œ‘ is the duality map ๐ฝ๐œ‘โˆถ๐‘‹โ†’๐‘‹โˆ— defined by๐ฝ๐œ‘๎€ฝ๐‘ฅ(๐‘ฅ)=โˆ—โˆˆ๐‘‹โˆ—โˆถโŸจ๐‘ฅ,๐‘ฅโˆ—(โŸฉ=โ€–๐‘ฅโ€–๐œ‘โ€–๐‘ฅโ€–),โ€–๐‘ฅโˆ—()๎€พโ€–=๐œ‘โ€–๐‘ฅโ€–,๐‘ฅโˆˆ๐‘‹.(2.7)

Following Browder [13], we say that a Banach space ๐‘‹ has a weakly continuous duality map if there exists a gauge ๐œ‘ for which the duality map ๐ฝ๐œ‘ is single valued and weak-to-weak* sequentially continuous (i.e., if {๐‘ฅ๐‘›} is a sequence in ๐‘‹ weakly convergent to a point ๐‘ฅ, then the sequence ๐ฝ๐œ‘(๐‘ฅ๐‘›) converges weakly* to ๐ฝ๐œ‘(๐‘ฅ)). It is known that ๐‘™๐‘ has a weakly continuous duality map for all 1<๐‘<โˆž. Set๎€œฮฆ(๐‘ก)=๐‘ก0๐œ‘(๐œ)๐‘‘๐œ,๐‘กโ‰ฅ0.(2.8)

Then๐ฝ๐œ‘(๐‘ฅ)=๐œ•ฮฆ(โ€–๐‘ฅโ€–),๐‘ฅโˆˆ๐‘‹,(2.9)

where ๐œ• denotes the subdifferential in the sense of convex analysis. The next lemma is an immediate consequence of the subdifferential inequality.

Lemma 2.5 (Xu [11, Lemma 2.6]). Assume that ๐‘‹ has a weakly continuous duality map ๐ฝ๐œ‘ with gauge ๐œ‘, for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, there holds the inequality ฮฆ(๎ซโ€–๐‘ฅ+๐‘ฆโ€–)โ‰คฮฆ(โ€–๐‘ฅโ€–)+๐‘ฆ,๐ฝ๐œ‘๎ฌ.(๐‘ฅ+๐‘ฆ)(2.10)

Lemma 2.6 (Xu [6]). Assume {๐›ผ๐‘›} is a sequence of nonnegative real numbers such that ๐›ผ๐‘›+1โ‰ค๎€ท1โˆ’๐›พ๐‘›๎€ธ๐›ผ๐‘›+๐›ฟ๐‘›,๐‘›โ‰ฅ0,(2.11)where {๐›พ๐‘›} is a sequence in (0,1) and {๐›ฟ๐‘›} is a sequence in ๐‘ such that(i)โˆ‘โˆž๐‘›=1๐›พ๐‘›=โˆž; (ii)limsup๐‘›โ†’โˆž๐›ฟ๐‘›/๐›พ๐‘›โ‰ค0 or โˆ‘โˆž๐‘›=1|๐›ฟ๐‘›|<โˆž.Then lim๐‘›โ†’โˆž๐›ผ๐‘›=0.

Finally, we also need the following definitions and results [9, 14]. Let ๐œ‡ be a continuous linear functional on ๐‘™โˆž satisfying โ€–๐œ‡โ€–=1=๐œ‡(1). Then we know that ๐œ‡ is a mean on ๐‘ if and only if๎€ฝ๐‘Žinf๐‘›๎€พ๎€ฝ๐‘Ž;๐‘›โˆˆ๐‘โ‰ค๐œ‡(๐‘Ž)โ‰คsup๐‘›๎€พ;๐‘›โˆˆ๐‘,(2.12) for every ๐‘Ž=(๐‘Ž1,๐‘Ž2,โ€ฆ)โˆˆ๐‘™โˆž. Occasionally, we will use ๐œ‡๐‘›(๐‘Ž๐‘›) instead of ๐œ‡(๐‘Ž). A mean ๐œ‡ on ๐‘ is called a Banach limit if๐œ‡๐‘›๎€ท๐‘Ž๐‘›๎€ธ=๐œ‡๐‘›๎€ท๐‘Ž๐‘›+1๎€ธ,(2.13) for every ๐‘Ž=(๐‘Ž1,๐‘Ž2,โ€ฆ)โˆˆ๐‘™โˆž. Using the Hahn-Banach theorem, or the Tychonoff fixed point theorem, we can prove the existence of a Banach limit. We know that if ๐œ‡ is a Banach limit, thenliminf๐‘›โ†’โˆž๐‘Ž๐‘›โ‰ค๐œ‡๐‘›๎€ท๐‘Ž๐‘›๎€ธโ‰คlimsup๐‘›โ†’โˆž๐‘Ž๐‘›,(2.14) for every ๐‘Ž=(๐‘Ž1,๐‘Ž2,โ€ฆ)โˆˆ๐‘™โˆž. So, if ๐‘Ž=(๐‘Ž1,๐‘Ž2,โ€ฆ), ๐‘=(๐‘1,๐‘2,โ€ฆ)โˆˆ๐‘™โˆž, and ๐‘Ž๐‘›โ†’๐‘ (resp., ๐‘Ž๐‘›โˆ’๐‘๐‘›โ†’0), as ๐‘›โ†’โˆž, we have๐œ‡๐‘›๎€ท๐‘Ž๐‘›๎€ธ๎€ท=๐œ‡(๐‘Ž)=๐‘resp.,๐œ‡๐‘›๎€ท๐‘Ž๐‘›๎€ธ=๐œ‡๐‘›๎€ท๐‘๐‘›๎€ธ๎€ธ.(2.15)

Subsequently, the following result was showed in [14, Lemma 1] and [9, Lemma 4.5.4].

Lemma 2.7 (see [14, Lemma 1]). Let ๐ถ be a nonempty closed convex subset of a Banach space ๐‘‹ with a uniformly Gรขteaux differentiable norm and {๐‘ฅ๐‘›} a bounded sequence of ๐ธ. If ๐‘ง0โˆˆ๐ถ, then ๐œ‡๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ง0โ€–โ€–2=min๐‘ฅโˆˆ๐ถ๐œ‡๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ฅ2,(2.16) if and only if ๐œ‡๐‘›๎ซ๐‘ฅโˆ’๐‘ง0๎€ท๐‘ฅ,๐ฝ๐‘›โˆ’๐‘ง0๎€ธ๎ฌโ‰ค0,โˆ€๐‘ฅโˆˆ๐ถ.(2.17)

Lemma 2.8 (Song and Xu [4, Proposition 3.1]). Let ๐‘‹ be a reflexive strictly convex Banach space with a uniformly Gรขteaux differentiable norm, and ๐ถ a nonempty closed convex subset of ๐‘‹. Suppose {๐‘ฅ๐‘›} is a bounded sequence in ๐ถ such that lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–=0, an approximate fixed point of nonexpansive self-mapping ๐‘‡ on ๐ถ. Define the set ๐ถโˆ—=๎‚ป๐‘ฆโˆˆ๐ถโˆถ๐œ‡๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ฆ2=inf๐‘ฅโˆˆ๐ถ๐œ‡๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ฅ2๎‚ผ.(2.18) If ๐น(๐‘‡)โ‰ โˆ…, then ๐ถโˆ—โˆฉ๐น(๐‘‡)โ‰ โˆ….

3. Implicit Iteration Scheme

Theorem 3.1. Let ๐‘‹ be a uniformly convex Banach space that has a weakly continuous duality map ๐ฝ๐œ‘ with gauge ๐œ‘, and let ๐ถ be a nonempty closed convex subset of ๐‘‹. Let ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} be a nonexpansive semigroup from ๐ถ into itself such that โ‹‚๐น(๐’ฎ)=๐‘ >0๐น(๐‘‡(๐‘ ))โ‰ โˆ… and ๐‘“โˆถ๐ถโ†’๐ถ a contraction mapping with the contractive coefficient ๐›ผโˆˆ[0,1). Suppose {๐œ†๐‘ก}0<๐‘ก<1 is a net of positive real numbers such that lim๐‘กโ†’0+๐œ†๐‘ก=โˆž, the sequence {๐‘ฅ๐‘ก} is given by the following equation: ๐‘ฅ๐‘ก๎€ท๐‘ฅ=๐‘ก๐‘“๐‘ก๎€ธ1+(1โˆ’๐‘ก)๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(s)๐‘ฅ๐‘ก๐‘‘๐‘ .(3.1) Then {๐‘ฅ๐‘ก} converges strongly to ฬƒ๐‘ฅ as ๐‘กโ†’0+, where ฬƒ๐‘ฅ is the unique solution in ๐น(๐’ฎ) of the variational inequality โŸจ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐ฝ(๐‘ฅโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ฅ0,๐‘ฅโˆˆ๐น(๐’ฎ).(3.2)

Proof. Note that ๐น(๐’ฎ) is a nonempty closed convex set by Lemma 2.1. We first show that {๐‘ฅ๐‘ก} is bounded. Indeed, for any fixed ๐‘โˆˆ๐น(๐’ฎ), we have โ€–โ€–๐‘ฅ๐‘กโ€–โ€–โ€–โ€–๐‘“๎€ท๐‘ฅโˆ’๐‘โ‰ค๐‘ก๐‘ก๎€ธโ€–โ€–โ€–โ€–โ€–1โˆ’๐‘+(1โˆ’๐‘ก)๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘กโ€–โ€–โ€–๎€ทโ€–โ€–๐‘“๎€ท๐‘ฅ๐‘‘๐‘ โˆ’๐‘โ‰ค๐‘ก๐‘ก๎€ธโ€–โ€–๎€ธ1โˆ’๐‘“(๐‘)+โ€–๐‘“(๐‘)โˆ’๐‘โ€–+(1โˆ’๐‘ก)๐œ†๐‘ก๎€œ๐œ†๐‘ก0โ€–โ€–๐‘‡(๐‘ )๐‘ฅ๐‘กโ€–โ€–๎€ท๐›ผโ€–โ€–๐‘ฅโˆ’๐‘๐‘‘๐‘ โ‰ค๐‘ก๐‘กโ€–โ€–โ€–๎€ธ+โ€–โ€–๐‘ฅโˆ’๐‘+โ€–๐‘“(๐‘)โˆ’๐‘(1โˆ’๐‘ก)๐‘กโ€–โ€–=โ€–โ€–๐‘ฅโˆ’๐‘๐‘กโ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘โˆ’๐‘ก(1โˆ’๐›ผ)๐‘กโ€–โ€–โˆ’๐‘+๐‘กโ€–๐‘“(๐‘)โˆ’๐‘โ€–.(3.3) It follows that โ€–โ€–๐‘ฅ๐‘กโ€–โ€–โ‰ค1โˆ’๐‘1โˆ’๐›ผโ€–๐‘“(๐‘)โˆ’๐‘โ€–.(3.4) Thus {๐‘ฅ๐‘ก} is bounded, so are {๐‘“(๐‘ฅ๐‘ก)} and {๐‘‡(๐‘ )๐‘ฅ๐‘ก} for every 0โ‰ค๐‘ <โˆž. Furthermore, we note that โ€–โ€–๐‘ฅ๐‘กโˆ’๐‘‡(๐‘ )๐‘ฅ๐‘กโ€–โ€–โ‰คโ€–โ€–โ€–๐‘ฅ๐‘กโˆ’1๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘กโ€–โ€–โ€–+โ€–โ€–โ€–1๐‘‘๐‘ ๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๎‚ต1๐‘‘๐‘ โˆ’๐‘‡(๐‘ )๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๎‚ถโ€–โ€–โ€–+โ€–โ€–โ€–๎‚ต1๐‘‘๐‘ ๐‘‡(๐‘ )๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๎‚ถ๐‘‘๐‘ โˆ’๐‘‡(๐‘ )๐‘ฅ๐‘กโ€–โ€–โ€–โ€–โ€–โ€–๐‘ฅโ‰ค2๐‘กโˆ’1๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘กโ€–โ€–โ€–+โ€–โ€–โ€–1d๐‘ ๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๎‚ต1๐‘‘๐‘ โˆ’๐‘‡(๐‘ )๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๎‚ถโ€–โ€–โ€–,๐‘‘๐‘ (3.5) for every 0โ‰ค๐‘ <โˆž. On the one hand, we observe that โ€–โ€–โ€–๐‘ฅ๐‘กโˆ’1๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘กโ€–โ€–โ€–=๐‘ก๐‘‘๐‘ โ€–โ€–๐‘ฅ1โˆ’๐‘ก๐‘ก๎€ท๐‘ฅโˆ’๐‘“๐‘ก๎€ธโ€–โ€–,(3.6) for every ๐‘ก>0. On the other hand, let ๐‘ง0โˆˆ๐น(๐‘†) and ๐ท={๐‘งโˆˆ๐ถโˆถโ€–๐‘งโˆ’๐‘ง0โ€–โ‰คโ€–๐‘“(๐‘ง0)โˆ’๐‘ง0โ€–}, then ๐ท is a nonempty closed bounded convex subset of ๐ถ which is ๐‘‡(๐‘ )-invariant for each 0โ‰ค๐‘ <โˆž and contains {๐‘ฅ๐‘ก}. It follows by Lemma 2.4 that lim๐œ†๐‘กโ†’โˆžโ€–โ€–โ€–1๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๎‚ต1๐‘‘๐‘ โˆ’๐‘‡(s)๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๎‚ถโ€–โ€–โ€–๐‘‘๐‘ โ‰คlim๐œ†๐‘กโ†’โˆžsup๐‘ฅโˆˆ๐ทโ€–โ€–โ€–1๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๎‚ต1๐‘‘๐‘ โˆ’๐‘‡(๐‘ )๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๎‚ถโ€–โ€–โ€–๐‘‘๐‘ =0.(3.7) Hence, by (3.5)โ€“(3.7), we obtain โ€–โ€–๐‘ฅ๐‘กโˆ’๐‘‡(๐‘ )๐‘ฅ๐‘กโ€–โ€–โŸถ0as๐‘กโŸถ0,(3.8) for every 0โ‰ค๐‘ <โˆž. Assume {๐‘ก๐‘›}โˆž๐‘›=1โŠ‚(0,1) is such that ๐‘ก๐‘›โ†’0 as ๐‘›โ†’โˆž. Put ๐‘ฅ๐‘›โˆถ=๐‘ฅ๐‘ก๐‘›, ๐œ†๐‘›โˆถ=๐œ†๐‘ก๐‘›, we will show that {๐‘ฅ๐‘›} contains s subsequence converging strongly to ฬƒ๐‘ฅ, where ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ). Since {๐‘ฅ๐‘›} is a bounded sequence, there is a subsequence {๐‘ฅ๐‘›๐‘—} of {๐‘ฅ๐‘›} which converges weakly to ฬƒ๐‘ฅโˆˆ๐ถ. By Lemma 2.3, we have ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ). For each ๐‘›โ‰ฅ1, we have ๐‘ฅ๐‘›โˆ’ฬƒ๐‘ฅ=๐‘ก๐‘›๎€ท๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ+๎€ทโˆ’ฬƒ๐‘ฅ1โˆ’๐‘ก๐‘›๎€ธ๎‚ต1๐œ†๐‘›๎€œ๐œ†๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถ.๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ(3.9) Thus, by Lemma 2.5, we obtain ฮฆ๎€ทโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ธ๎‚ตโ€–โ€–โ€–๐‘กโˆ’ฬƒ๐‘ฅ=ฮฆ๐‘›๎€ท๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ+๎€ทโˆ’ฬƒ๐‘ฅ1โˆ’๐‘ก๐‘›๎€ธ๎‚ต1๐œ†๐‘›๎€œ๐œ†๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถโ€–โ€–โ€–๎‚ถ๎‚ตโ€–โ€–โ€–๎€ท๐‘‘๐‘ โˆ’ฬƒ๐‘ฅโ‰คฮฆ1โˆ’๐‘ก๐‘›๎€ธ๎‚ต1๐œ†๐‘›๎€œ๐œ†๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถโ€–โ€–โ€–๎‚ถ๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ+๐‘ก๐‘›๎ซ๐‘“๎€ท๐‘ฅ๐‘›๎€ธโˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›โ‰ค๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐‘ก๐‘›๎€ธฮฆ๎‚ตโ€–โ€–โ€–1๐œ†๐‘›๎€œ๐œ†๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โ€–๎‚ถ๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ+๐‘ก๐‘›๎ซ๐‘“๎€ท๐‘ฅ๐‘›๎€ธโˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›โ‰ค๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐‘ก๐‘›๎€ธฮฆ๎€ทโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ธโˆ’ฬƒ๐‘ฅ+๐‘ก๐‘›๎ซ๐‘“๎€ท๐‘ฅ๐‘›๎€ธโˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›.โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ(3.10) This implies that ฮฆ๎€ทโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ธโ‰ค๎ซ๐‘“๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅ๐‘›๎€ธโˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›.โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ(3.11) In particular, we have ฮฆ๎‚€โ€–โ€–๐‘ฅ๐‘›๐‘—โ€–โ€–๎‚โ‰ค๎‚ฌ๐‘“๎‚€๐‘ฅโˆ’ฬƒ๐‘ฅ๐‘›๐‘—๎‚โˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎‚€๐‘ฅ๐‘›๐‘—โˆ’ฬƒ๐‘ฅ๎‚๎‚ญ.(3.12) Now observing that {๐‘ฅ๐‘›๐‘—}โ‡€ฬƒ๐‘ฅ implies ๐ฝ๐œ‘(๐‘ฅ๐‘›๐‘—โˆ’ฬƒ๐‘ฅ)โ‡€0. And since ๐‘“(๐‘ฅ๐‘›๐‘—) is bounded, it follows from (3.12) that ฮฆ๎‚€โ€–โ€–๐‘ฅ๐‘›๐‘—โ€–โ€–๎‚โˆ’ฬƒ๐‘ฅโŸถ0as๐‘—โŸถโˆž.(3.13) Hence ๐‘ฅ๐‘›๐‘—โ†’ฬƒ๐‘ฅ.
Next, we show that ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ) solves the variational inequality (3.2). Indeed, for ๐‘žโˆˆ๐น(๐’ฎ), it is easy to see that ๎ƒก๐‘ฅ๐‘กโˆ’1๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๐‘‘๐‘ ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘ก๎€ธ๎ƒข๎€ทโ€–โ€–๐‘ฅโˆ’๐‘ž=ฮฆ๐‘กโ€–โ€–๎€ธ+๎ƒก1โˆ’๐‘ž๐‘žโˆ’๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๐‘‘๐‘ ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘ก๎€ธ๎ƒข๎€ทโ€–โ€–๐‘ฅโˆ’๐‘žโ‰ฅฮฆ๐‘กโ€–โ€–๎€ธโˆ’โ€–โ€–โ€–1โˆ’๐‘ž๐‘žโˆ’๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘กโ€–โ€–โ€–โ€–โ€–๐ฝ๐‘‘๐‘ ๐œ‘๎€ท๐‘ฅ๐‘ก๎€ธโ€–โ€–๎€ทโ€–โ€–๐‘ฅโˆ’๐‘žโ‰ฅฮฆ๐‘กโ€–โ€–๎€ธ๎€ทโ€–โ€–๐‘ฅโˆ’๐‘žโˆ’ฮฆ๐‘กโ€–โ€–๎€ธโˆ’๐‘ž=0.(3.14) However, we note that ๐‘ฅ๐‘กโˆ’1๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๐‘ก๐‘‘๐‘ =๎€ท๐‘“๎€ท๐‘ฅ1โˆ’๐‘ก๐‘ก๎€ธโˆ’๐‘ฅ๐‘ก๎€ธ.(3.15) Thus, we get that for ๐‘กโˆˆ(0,1) and ๐‘žโˆˆ๐น(๐’ฎ)๎ซ๐‘ฅ๐‘ก๎€ท๐‘ฅโˆ’๐‘“๐‘ก๎€ธ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘กโˆ’๐‘ž๎€ธ๎ฌโ‰ค0.(3.16) Taking the limit through ๐‘กโˆถ=๐‘ก๐‘›๐‘—โ†’0, we obtain ๎ซ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐ฝ๐œ‘๎ฌ(ฬƒ๐‘ฅโˆ’๐‘ž)โ‰ค0,โˆ€๐‘žโˆˆ๐น(๐’ฎ).(3.17) This implies that โŸจ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐ฝ(ฬƒ๐‘ฅโˆ’๐‘ž)โŸฉโ‰ค0,โˆ€๐‘žโˆˆ๐น(๐’ฎ),(3.18) since ๐ฝ๐œ‘(๐‘ฅ)=(๐œ‘(โ€–๐‘ฅโ€–)/โ€–๐‘ฅโ€–)๐ฝ(๐‘ฅ) for ๐‘ฅโ‰ 0.
Finally, we show that the net {๐‘ฅ๐‘ก} convergence strong to ฬƒ๐‘ฅ. Assume that there is a sequence {๐‘ ๐‘›}โŠ‚(0,1) such that ๐‘ฅ๐‘ ๐‘›โ†’๐‘ฅ, where ๐‘ ๐‘›โ†’0. we note by Lemma 2.3 that ๐‘ฅโˆˆ๐น(๐’ฎ). It follows from the inequality (3.18) that ๎ซ๎€ท(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐ฝฬƒ๐‘ฅโˆ’๐‘ฅ๎€ธ๎ฌโ‰ค0.(3.19) Interchange ฬƒ๐‘ฅ and ๐‘ฅ to obtain ๎ซ(๐ผโˆ’๐‘“)๎€ท๐‘ฅ,๐ฝ๐‘ฅโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ค0.(3.20) Adding (3.19) and (3.20) yields โ€–โ€–(1โˆ’๐›ผ)ฬƒ๐‘ฅโˆ’๐‘ฅโ€–โ€–2โ‰ค๎ซ๎€ท๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅโˆ’๐‘“(ฬƒ๐‘ฅ)โˆ’๐‘“๐‘ฅ๎€ท๎€ธ๎€ธ,๐ฝ๐‘ฅโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ค0.(3.21) We must have ฬƒ๐‘ฅ=๐‘ฅ and the uniqueness is proved. In a summary, we have shown that each cluster point of {๐‘ฅ๐‘ก} as ๐‘กโ†’0 equals ฬƒ๐‘ฅ. Therefore ๐‘ฅ๐‘กโ†’ฬƒ๐‘ฅ as ๐‘กโ†’0.

Theorem 3.2. Let ๐‘‹ be a uniformly convex Banach space with a uniformly Gรขteaux differentiable norm and ๐ถ be a nonempty closed convex subset of ๐‘‹. Let ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} be a nonexpansive semigroup from ๐ถ into itself such that โ‹‚๐น(๐’ฎ)=๐‘ >0๐น(๐‘‡(๐‘ ))โ‰ โˆ… and ๐‘“โˆถ๐ถโ†’๐ถ a contraction mapping with the contractive coefficient ๐›ผโˆˆ[0,1). Suppose {๐œ†๐‘ก}0<๐‘ก<1 is a net of positive real numbers such that lim๐‘กโ†’0+๐œ†๐‘ก=โˆž, the sequence {๐‘ฅ๐‘ก} is given by the following equation: ๐‘ฅ๐‘ก๎€ท๐‘ฅ=๐‘ก๐‘“๐‘ก๎€ธ1+(1โˆ’๐‘ก)๐œ†๐‘ก๎€œ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๐‘‘๐‘ .(3.22) Then {๐‘ฅ๐‘ก} converges strongly to ฬƒ๐‘ฅ as ๐‘กโ†’0+, where ฬƒ๐‘ฅ is the unique solution in ๐น(๐’ฎ) of the variational inequality โŸจ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐ฝ(๐‘ฅโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ฅ0,๐‘ฅโˆˆ๐น(๐’ฎ).(3.23)

Proof. We include only those points in this proof which are different from those already presented in the proof of Theorem 3.1. As in the proof of Theorem 3.1, we obtain that there is a subsequence {๐‘ฅ๐‘›๐‘—} of {๐‘ฅ๐‘›} which converges weakly to ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ). For each ๐‘›โ‰ฅ1, we have ๐‘ฅ๐‘›โˆ’ฬƒ๐‘ฅ=๐‘ก๐‘›๎€ท๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ+๎€ทโˆ’ฬƒ๐‘ฅ1โˆ’๐‘ก๐‘›๎€ธ๎‚ต1๐œ†๐‘›๎€œ๐œ†๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถ.๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ(3.24) Thus, we have โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2=๎ƒก๐‘ก๐‘›๎€ท๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ธ+๎€ทโˆ’ฬƒ๐‘ฅ1โˆ’๐‘ก๐‘›๎€ธ๎‚ต1๐œ†๐‘›๎€œ๐œ†๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถ๎€ท๐‘ฅ๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›๎€ธ๎ƒขโˆ’ฬƒ๐‘ฅ=๐‘ก๐‘›๎ซ๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘“(ฬƒ๐‘ฅ)+๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐‘ก๐‘›๎€ธ1๎ƒก๎‚ต๐œ†๐‘›๎€œ๐œ†๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถ๎€ท๐‘ฅ๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›๎€ธ๎ƒขโˆ’ฬƒ๐‘ฅโ‰ค๐‘ก๐‘›โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โ€–๐ฝ๎€ท๐‘ฅโˆ’๐‘“(ฬƒ๐‘ฅ)๐‘›๎€ธโ€–โ€–โˆ’ฬƒ๐‘ฅ+๐‘ก๐‘›๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐‘ก๐‘›๎€ธโ€–โ€–โ€–1๐œ†๐‘›๎€œ๐œ†๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โ€–โ€–โ€–๐ฝ๎€ท๐‘ฅ๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ๐‘›๎€ธโ€–โ€–โ‰ค๎€ทโˆ’ฬƒ๐‘ฅ1โˆ’(1โˆ’๐›ผ)๐‘ก๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+๐‘ก๐‘›๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›.โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ(3.25) Therefore, โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2โ‰ค1๎ซ๎€ท๐‘ฅ1โˆ’๐›ผ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›.โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ(3.26)
We claim that the set {๐‘ฅ๐‘›} is sequentially compact. Indeed, define the set ๐ถโˆ—=๎‚ป๐‘ฆโˆˆ๐ถโˆถ๐œ‡๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ฆ2=inf๐‘ฅโˆˆ๐ถ๐œ‡๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ฅ2๎‚ผ.(3.27) By Lemma 2.8, we found ฬƒ๐‘ฅโˆˆ๐ถโˆ—. Using Lemma 2.7 we get that ๐œ‡๐‘›๎ซ๎€ท๐‘ฅ๐‘ฅโˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ค0,โˆ€๐‘ฅโˆˆ๐ถ.(3.28) From (3.26), we get ๐œ‡๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2โ‰ค1๐œ‡1โˆ’๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ค0,(3.29) that is ๐œ‡๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ=0.(3.30) Hence, there exists a subsequence {๐‘ฅ๐‘›๐‘˜} of {๐‘ฅ๐‘›} converges strongly to ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ) as ๐‘˜โ†’โˆž.
Next we show that ฬƒ๐‘ฅ is a solution in ๐น(๐’ฎ) to the variational inequality (3.23). In fact, for any fixed ๐‘ฅโˆˆ๐น(๐’ฎ), there exists a constant ๐‘€>0 such that โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโ€–โ‰ค๐‘€, then โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ฅ2=๐‘ก๐‘›๎ซ๐‘“๎€ท๐‘ฅ๐‘›๎€ธโˆ’๐‘“(ฬƒ๐‘ฅ)+ฬƒ๐‘ฅโˆ’๐‘ฅ๐‘›๎€ท๐‘ฅ,๐ฝ๐‘›โˆ’๐‘ฅ๎€ธ๎ฌ+๐‘ก๐‘›๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›โˆ’๐‘ฅ๎€ธ๎ฌ+๐‘ก๐‘›๎ซ๐‘ฅ๐‘›๎€ท๐‘ฅโˆ’๐‘ฅ,๐ฝ๐‘›+๎€ทโˆ’๐‘ฅ๎€ธ๎ฌ1โˆ’๐‘ก๐‘›๎€ธ๎ƒก1๐œ†๐‘›๎€œ๐œ†๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎€ท๐‘ฅ๐‘‘๐‘ โˆ’๐‘ฅ,๐ฝ๐‘›๎€ธ๎ƒขโˆ’๐‘ฅโ‰ค(1+๐›ผ)๐‘ก๐‘›๐‘€โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ+๐‘ก๐‘›๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+โ€–โ€–๐‘ฅโˆ’๐‘ฅ๎€ธ๎ฌ๐‘›โ€–โ€–โˆ’๐‘ฅ2.(3.31) Therefore, ๎ซ๐‘“๎€ท(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘ฅโˆ’๐‘ฅ๐‘›โ‰คโ€–โ€–๐‘ฅ๎€ธ๎ฌ(1+๐›ผ)๐‘€๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ.(3.32) Since the duality mapping ๐ฝ is single valued and norm topology to weak* topology uniformly continuous on any bounded subset of a Banach space ๐‘‹ with a uniformly Gรขteaux differentiable norm, we have ๎ซ๐‘“๎€ท(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘ฅโˆ’๐‘ฅ๐‘›๐‘˜๎€ธ๎ฌโŸถโŸจ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ(๐‘ฅโˆ’ฬƒ๐‘ฅ)โŸฉ.(3.33) Taking limit as ๐‘—โ†’โˆž in two sides of (3.32), we get โŸจ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ(๐‘ฅโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ค0,โˆ€๐‘ฅโˆˆ๐น(๐’ฎ).(3.34)
Finally we will show that the net {๐‘ฅ๐‘ก} convergence strong to ฬƒ๐‘ฅ. This section is similar to that of Theorem 3.1.

4. Explicit Iterative Scheme

Theorem 4.1. Let ๐‘‹ be a uniformly convex Banach space that has a weakly continuous duality map ๐ฝ๐œ‘ with gauge ๐œ‘ and ๐ถ be a nonempty closed convex subset of ๐‘‹. Let ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} be a nonexpansive semigroup from ๐ถ into itself such that โ‹‚๐น(๐’ฎ)=๐‘ >0๐น(๐‘‡(๐‘ ))โ‰ โˆ… and ๐‘“โˆถ๐ถโ†’๐ถ a contraction mapping with the contractive coefficient ๐›ผโˆˆ[0,1). Let {๐›ผ๐‘›} and {๐›ฝ๐‘›} be the sequence in (0,1) which satisfies ๐›ผ๐‘›+๐›ฝ๐‘›<1, lim๐‘›โ†’โˆž๐›ผ๐‘›โ†’0, lim๐‘›โ†’โˆž๐›ฝ๐‘›โ†’0 and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, and {๐‘ ๐‘›} is a positive real divergent sequence such that lim๐‘›โ†’โˆž๐‘ ๐‘›โ†’โˆž. If the sequence {๐‘ฅ๐‘›} defined by ๐‘ฅ0โˆˆ๐ถ and ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,๐‘›โ‰ฅ0.(4.1) Then {๐‘ฅ๐‘›} converges strongly to ฬƒ๐‘ฅ as ๐‘›โ†’โˆž, where ฬƒ๐‘ฅ is the unique solution in ๐น(๐’ฎ) of the variational inequality โŸจ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐ฝ(๐‘ฅโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ฅ0,๐‘ฅโˆˆ๐น(๐’ฎ).(4.2)

Proof. Note that ๐น(๐’ฎ) is a nonempty closed convex set. We first show that {๐‘ฅ๐‘›} is bounded. Let ๐‘žโˆˆ๐น(๐’ฎ). Thus, we compute that โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–=โ€–โ€–โ€–๐›ผโˆ’๐‘ž๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โ€–๐‘‘๐‘ โˆ’๐‘žโ‰ค๐›ผ๐‘›โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โˆ’๐‘ž+๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–+๎€ทโˆ’๐‘ž1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–โ€–1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โ€–๐‘‘๐‘ โˆ’๐‘žโ‰ค๐›ผ๐‘›๎€ทโ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–๎€ธโˆ’๐‘“(๐‘ž)+โ€–๐‘“(๐‘ž)โˆ’๐‘žโ€–+๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–+๎€ทโˆ’๐‘ž1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ1๐‘ ๐‘›๎€œ๐‘ ๐‘›0โ€–โ€–๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž๐‘‘๐‘ โ‰ค๐›ผ๐‘›๐›ผโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž+๐›ผ๐‘›๎€ทโ€–๐‘“(๐‘ž)โˆ’๐‘žโ€–+1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–=๎€ทโˆ’๐‘ž1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ(1โˆ’๐›ผ)๐‘›โ€–โ€–โˆ’๐‘ž+๐›ผ๐‘›๎‚†โ€–โ€–๐‘ฅโ€–๐‘“(๐‘ž)โˆ’๐‘žโ€–โ‰คmax๐‘›โ€–โ€–,1โˆ’๐‘ž๎‚‡โ‹ฎ๎‚†โ€–โ€–๐‘ฅ1โˆ’๐›ผโ€–๐‘“(๐‘ž)โˆ’๐‘žโ€–โ‰คmax0โ€–โ€–,1โˆ’๐‘žโ€–๎‚‡.1โˆ’๐›ผโ€–๐‘“(๐‘ž)โˆ’๐‘ž(4.3) Therefore, {๐‘ฅ๐‘›} is bounded, {๐‘“(๐‘ฅ๐‘›)} and {๐‘‡(๐‘ )๐‘ฅ๐‘›} for every 0โ‰ค๐‘ <โˆž are also bounded.
Next we show โ€–๐‘ฅ๐‘›โˆ’๐‘‡(โ„Ž)๐‘ฅ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž. Notice that โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡(โ„Ž)๐‘ฅ๐‘›+1โ€–โ€–โ‰คโ€–โ€–โ€–๐‘ฅ๐‘›+1โˆ’1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โ€–+โ€–โ€–โ€–1๐‘‘๐‘ ๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ต1๐‘‘๐‘ โˆ’๐‘‡(โ„Ž)๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถโ€–โ€–โ€–+โ€–โ€–โ€–๎‚ต1๐‘‘๐‘ ๐‘‡(โ„Ž)๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถ๐‘‘๐‘ โˆ’๐‘‡(โ„Ž)๐‘ฅ๐‘›+1โ€–โ€–โ€–โ€–โ€–โ€–๐‘ฅโ‰ค2๐‘›+1โˆ’1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โ€–+โ€–โ€–โ€–1๐‘‘๐‘ ๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ต1๐‘‘๐‘ โˆ’๐‘‡(โ„Ž)๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถโ€–โ€–โ€–๐‘‘๐‘ โ‰ค2๐›ผ๐‘›โ€–โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโˆ’1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โ€–๐‘‘๐‘ +2๐›ฝ๐‘›โ€–โ€–โ€–๐‘ฅ๐‘›โˆ’1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โ€–+โ€–โ€–โ€–1๐‘‘๐‘ ๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ต1๐‘‘๐‘ โˆ’๐‘‡(โ„Ž)๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถโ€–โ€–โ€–.๐‘‘๐‘ (4.4)
Put ๐‘ง0=๐‘ƒ๐น(๐’ฎ)๐‘ฅ0 and ๐ท={๐‘งโˆˆ๐ถโˆถโ€–๐‘งโˆ’๐‘ง0โ€–โ‰คโ€–๐‘ฅ0โˆ’๐‘ง0โ€–+1/(1โˆ’๐›ผ)โ€–๐‘“(๐‘ง0)โˆ’๐‘ง0โ€–}. Then ๐ท is a nonempty closed bounded convex subset of ๐ถ which is ๐‘‡(๐‘ )-invariant for each ๐‘ โˆˆ[0,โˆž) and contains {๐‘ฅ๐‘›}. So without loss of generality, we may assume that ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} is a nonexpansive semigroup on ๐ท. By Lemma 2.4, we get lim๐‘›โ†’โˆžโ€–โ€–โ€–1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ต1๐‘‘๐‘ โˆ’๐‘‡(โ„Ž)๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถโ€–โ€–โ€–๐‘‘๐‘ =0,(4.5) for every โ„Žโˆˆ[0,โˆž). On the other hand, since {๐‘ฅ๐‘›}, {๐‘“(๐‘ฅ๐‘›)}, and {๐‘‡(๐‘ )๐‘ฅ๐‘›} are bounded, using the assumption that lim๐‘›โ†’โˆž๐›ผ๐‘›โ†’0, lim๐‘›โ†’โˆž๐›ฝ๐‘›โ†’0, and (4.5) into (4.4), we get that โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡(โ„Ž)๐‘ฅ๐‘›+1โ€–โ€–โŸถ0as๐‘›โŸถโˆž,(4.6) and hence โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡(โ„Ž)๐‘ฅ๐‘›โ€–โ€–โŸถ0as๐‘›โŸถโˆž.(4.7)
We now show that ๎ซ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ค0.(4.8) Let ๐‘ฅ๐‘ก=๐‘ก๐‘“(๐‘ฅ๐‘ก)+(1โˆ’๐‘ก)(1/๐œ†๐‘ก)โˆซ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๐‘‘๐‘ , where ๐‘ก and ๐œ†๐‘ก satisfies the condition of Theorem 3.1. Then it follows from Theorem 3.1 that ฬƒ๐‘ฅ=lim๐‘กโ†’0๐‘ฅ๐‘ก and ฬƒ๐‘ฅ be the unique solution in ๐น(๐’ฎ) of the variational inequality (3.2). Clearly ฬƒ๐‘ฅ is a unique solution of (4.2). Take a subsequence {๐‘ฅ๐‘›๐‘˜} of {๐‘ฅ๐‘›} such that limsup๐‘›โ†’โˆž๎ซ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ=lim๐‘˜โ†’โˆž๎ซ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›๐‘˜โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ.(4.9) Since ๐‘‹ is uniformly convex and hence it is reflexive, we may further assume that ๐‘ฅ๐‘›๐‘˜โ‡€๐‘. Moreover, we note that ๐‘โˆˆ๐น(๐’ฎ) by Lemma 2.3 and (4.7). Therefore, from (4.9) and (3.17), we have limsup๐‘›โ†’โˆž๎ซ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›=๎ซ๐‘“โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎ฌ(๐‘โˆ’ฬƒ๐‘ฅ)โ‰ค0.(4.10) That is (4.8) holds.
Finally we will show that ๐‘ฅ๐‘›โ†’ฬƒ๐‘ฅ. For each ๐‘›โ‰ฅ0, we have ฮฆ๎€ทโ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–๎€ธ๎‚ตโ€–โ€–โ€–๐›ผโˆ’ฬƒ๐‘ฅ=ฮฆ๐‘›๎€ท๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ธโˆ’ฬƒ๐‘ฅ+๐›ฝ๐‘›๎€ท๐‘ฅ๐‘›๎€ธ+๎€ทโˆ’ฬƒ๐‘ฅ1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ๎‚ต1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถโ€–โ€–โ€–๎‚ถ๎‚ตโ€–โ€–๐›ผ๐‘‘๐‘ โˆ’ฬƒ๐‘ฅโ‰คฮฆ๐‘›๎€ท๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ธโˆ’๐‘“(ฬƒ๐‘ฅ)+๐›ผ๐‘›(๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ)+๐›ฝ๐‘›๎€ท๐‘ฅ๐‘›๎€ธ+๎€ทโˆ’ฬƒ๐‘ฅ1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ๎‚ต1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถโ€–โ€–โ€–๎‚ถ๎‚ตโ€–โ€–โ€–๐›ผ๐‘‘๐‘ โˆ’ฬƒ๐‘ฅโ‰คฮฆ๐‘›๎€ท๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ธโˆ’๐‘“(ฬƒ๐‘ฅ)+๐›ฝ๐‘›๎€ท๐‘ฅ๐‘›๎€ธ+๎€ทโˆ’ฬƒ๐‘ฅ1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ๎‚ต1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถโ€–โ€–โ€–๎‚ถ๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ+๐›ผ๐‘›๎ซ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›+1๎‚ต๐›ผโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰คฮฆ๐‘›๐›ผโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ+๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–+๎€ทโˆ’ฬƒ๐‘ฅ1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–โ€–1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โ€–๎‚ถ๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ+๐›ผ๐‘›๎ซ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›+1โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰คฮฆ๎€ท๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ(1โˆ’๐›ผ)๐‘›โ€–โ€–๎€ธโˆ’ฬƒ๐‘ฅ+๐›ผ๐‘›๎ซ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›+1โ‰ค๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๎€ธฮฆ๎€ทโ€–โ€–๐‘ฅ(1โˆ’๐›ผ)๐‘›โ€–โ€–๎€ธโˆ’ฬƒ๐‘ฅ+๐›ผ๐‘›๎ซ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐œ‘๎€ท๐‘ฅ๐‘›+1.โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ(4.11) An application of Lemma 2.6, we can obtain ฮฆ(โ€–๐‘ฅ๐‘›โˆ’ฬƒ๐‘ฅโ€–)โ†’0, hence โ€–๐‘ฅ๐‘›โˆ’ฬƒ๐‘ฅโ€–โ†’0. That is, {๐‘ฅ๐‘›} converges strongly to a fixed point ฬƒ๐‘ฅ of ๐’ฎ. This completes the proof.

Theorem 4.2. Let ๐‘‹ be a uniformly convex Banach space with a uniformly Gรขteaux differentiable norm and ๐ถ be a nonempty closed convex subset of ๐‘‹. Let ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} be a nonexpansive semigroup from ๐ถ into itself such that โ‹‚๐น(๐’ฎ)=๐‘ >0๐น(๐‘‡(๐‘ ))โ‰ โˆ… and ๐‘“โˆถ๐ถโ†’๐ถ a contraction mapping with the contractive coefficient ๐›ผโˆˆ[0,1). Let {๐›ผ๐‘›} and {๐›ฝ๐‘›} be the sequence in (0,1) which satisfies ๐›ผ๐‘›+๐›ฝ๐‘›<1, lim๐‘›โ†’โˆž๐›ผ๐‘›โ†’0, lim๐‘›โ†’โˆž๐›ฝ๐‘›โ†’0, and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, and {๐‘ ๐‘›} is a positive real divergent sequence such that lim๐‘›โ†’โˆž๐‘ ๐‘›โ†’โˆž. If the sequence {๐‘ฅ๐‘›} defined by ๐‘ฅ0โˆˆ๐ถ and ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,๐‘›โ‰ฅ0.(4.12) Then {๐‘ฅ๐‘›} converges strongly to ฬƒ๐‘ฅ as ๐‘›โ†’โˆž, where ฬƒ๐‘ฅ is the unique solution in ๐น(๐’ฎ) of the variational inequality โŸจ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐ฝ(๐‘ฅโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ฅ0,๐‘ฅโˆˆ๐น(๐’ฎ).(4.13)

Proof. We also show only those points in this proof which are different from that already presented in the proof of Theorem 4.1. We now show that ๎ซ๐‘“๎€ท๐‘ฅ(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ค0.(4.14) Let ๐‘ฅ๐‘ก=๐‘ก๐‘“(๐‘ฅ๐‘ก)+(1โˆ’๐‘ก)(1/๐œ†๐‘ก)โˆซ๐œ†๐‘ก0๐‘‡(๐‘ )๐‘ฅ๐‘ก๐‘‘๐‘ , where ๐‘ก and ๐œ†๐‘ก satisfies the condition of Theorem 3.2. Then it follows from Theorem 3.2 that ฬƒ๐‘ฅ=lim๐‘กโ†’0๐‘ฅ๐‘ก and ฬƒ๐‘ฅ is the unique solution in ๐น(๐’ฎ) of the variational inequality (3.23). Clearly ฬƒ๐‘ฅ is a unique solution of (4.13). Take a subsequence {๐‘ฅ๐‘›๐‘˜} of {๐‘ฅ๐‘›} such that limsup๐‘›โ†’โˆž๎ซ๐‘“๎€ท๐‘ฅ(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ=lim๐‘˜โ†’โˆž๎ซ๐‘“๎€ท๐‘ฅ(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›๐‘˜โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ.(4.15) Since ๐‘‹ is uniformly convex and hence it is reflexive, we may further assume that ๐‘ฅ๐‘›๐‘˜โ‡€๐‘. Moreover, we note that ๐‘โˆˆ๐น(๐’ฎ) by Lemma 2.3 and (4.7). Therefore, from (4.15) and (3.23), we have limsup๐‘›โ†’โˆž๎ซ๐‘“๎€ท๐‘ฅ(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ=โŸจ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ(๐‘โˆ’ฬƒx)โŸฉโ‰ค0.(4.16) That is, (4.14) holds.
Finally we will show that ๐‘ฅ๐‘›โ†’ฬƒ๐‘ฅ. For each ๐‘›โ‰ฅ0, by Lemma 2.2, we have โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’ฬƒ๐‘ฅ2=โ€–โ€–โ€–๐›ผ๐‘›๎€ท๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ธโˆ’ฬƒ๐‘ฅ+๐›ฝ๐‘›๎€ท๐‘ฅ๐‘›๎€ธ+๎€ทโˆ’ฬƒ๐‘ฅ1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ๎‚ต1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถโ€–โ€–โ€–๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ2โ‰คโ€–โ€–โ€–๎€ท1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ๎‚ต1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๎‚ถ๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ+๐›ฝ๐‘›๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›๎ซ๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+1โ‰ค๎‚ต๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–โ€–1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›โ€–โ€–โ€–๐‘‘๐‘ โˆ’ฬƒ๐‘ฅ+๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎‚ถโˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›๎ซ๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+1โ‰คโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ๎€ท๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ธโˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›๎ซ๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘“(ฬƒ๐‘ฅ),๐ฝ๐‘›+1โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ+2๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+1โ‰ค๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๎€ธ2โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โ€–๐ฝ๎€ท๐‘ฅโˆ’๐‘“(ฬƒ๐‘ฅ)๐‘›+1๎€ธโ€–โ€–โˆ’ฬƒ๐‘ฅ+2๐›ผ๐‘›๎ซ๐‘“๎€ท๐‘ฅ(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+1โ‰ค๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๎€ธ2โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›๐›ผโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’ฬƒ๐‘ฅ๐‘›+1โ€–โ€–โˆ’ฬƒ๐‘ฅ+2๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+1โ‰ค๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๎€ธ2โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+๐›ผ๐‘›๐›ผ๎‚€โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’ฬƒ๐‘ฅ2๎‚+2๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+1=๎‚€๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๎€ธ2+๐›ผ๐‘›๐›ผ๎‚โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+๐›ผ๐‘›๐›ผโ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+1,โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ(4.17) which implies that โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’ฬƒ๐‘ฅ2โ‰ค1โˆ’2๐›ผ๐‘›+๐›ผ2๐‘›+๐›ผ๐‘›๐›ผ1โˆ’๐›ผ๐‘›๐›ผโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›1โˆ’๐›ผ๐‘›๐›ผ๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+1=๎‚ธโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’2(1โˆ’๐›ผ)๐›ผ๐‘›1โˆ’๐›ผ๐‘›๐›ผ๎‚นโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+๐›ผ2๐‘›1โˆ’๐›ผ๐‘›๐›ผโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›1โˆ’๐›ผ๐‘›๐›ผ๎ซ๎€ท๐‘ฅ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+1โ‰ค๎‚ธโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’2(1โˆ’๐›ผ)๐›ผ๐‘›1โˆ’๐›ผ๐‘›๐›ผ๎‚นโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+2(1โˆ’๐›ผ)๐›ผ๐‘›1โˆ’๐›ผ๐‘›๐›ผ๎‚ป๐›ผ๐‘›๐‘€+12(1โˆ’๐›ผ)๎ซ๎€ท๐‘ฅ1โˆ’๐›ผ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ๐‘›+1๎‚ผ=๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ฟ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+๐›ฟ๐‘›๐›พ๐‘›,(4.18)where ๐‘€=sup{โ€–๐‘ฅ๐‘›โˆ’ฬƒ๐‘ฅโ€–2โˆถ๐‘›โˆˆโ„•}, ๐›ฟ๐‘›โˆถ=2(1โˆ’๐›ผ)๐›ผ๐‘›/(1โˆ’๐›ผ๐‘›๐›ผ), and ๐›พ๐‘›โˆถ=(๐›ผ๐‘›๐‘€/2(1โˆ’๐›ผ))+(1/(1โˆ’๐›ผ))โŸจ๐‘“(ฬƒ๐‘ฅ)โˆ’ฬƒ๐‘ฅ,๐ฝ(๐‘ฅ๐‘›+1โˆ’ฬƒ๐‘ฅ)โŸฉ. It is easily to see that ๐›ฟ๐‘›โ†’0, โˆ‘โˆž๐‘›=1๐›ฟ๐‘›=โˆž and limsup๐‘›โ†’โˆž๐›พ๐‘›โ‰ค0 by (4.14). Finally by using Lemma 2.6, we can obtain {๐‘ฅ๐‘›} converges strongly to a fixed point ฬƒ๐‘ฅโˆˆ๐น(๐’ฎ). This completes the proof.

5. Applications

Theorem 5.1. Let ๐‘‹ be a uniformly convex Banach space that has a weakly continuous duality map ๐ฝ๐œ‘ with gauge ๐œ‘ and ๐ถ be a nonempty closed convex subset of ๐‘‹. Let ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} be a nonexpansive semigroup from ๐ถ into itself such that โ‹‚๐น(๐’ฎ)=๐‘ >0๐น(๐‘‡(๐‘ ))โ‰ โˆ… and ๐‘“โˆถ๐ถโ†’๐ถ a contraction mapping with the contractive coefficient ๐›ผโˆˆ[0,1). Let {๐›ผ๐‘›} be the sequence in (0,1) which satisfies lim๐‘›โ†’โˆž๐›ผ๐‘›โ†’0 and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, and {๐‘ ๐‘›} is a positive real divergent sequence such that lim๐‘›โ†’โˆž๐‘ ๐‘›โ†’โˆž. If the sequence {๐‘ฅ๐‘›} defined by ๐‘ฅ0โˆˆ๐ถ and ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,๐‘›โ‰ฅ0.(5.1) Then {๐‘ฅ๐‘›} converges strongly to ฬƒ๐‘ฅ as ๐‘›โ†’โˆž, where ฬƒ๐‘ฅ is the unique solution in ๐น(๐’ฎ) of the variational inequality โŸจ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐ฝ(๐‘ฅโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ฅ0,๐‘ฅโˆˆ๐น(๐’ฎ).(5.2)

Proof. Taking ๐›ฝ๐‘›=0 in the in Theorem 4.1, we get the desired conclusion easily.

Theorem 5.2. Let ๐‘‹ be a uniformly convex Banach space with a uniformly Gรขteaux differentiable norm and ๐ถ be a nonempty closed convex subset of ๐‘‹. Let ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} be a nonexpansive semigroup from ๐ถ into itself such that โ‹‚๐น(๐’ฎ)=๐‘ >0๐น(๐‘‡(๐‘ ))โ‰ โˆ… and ๐‘“โˆถ๐ถโ†’๐ถ a contraction mapping with the contractive coefficient ๐›ผโˆˆ[0,1). Let {๐›ผ๐‘›} be the sequence in (0,1) which satisfies lim๐‘›โ†’โˆž๐›ผ๐‘›โ†’0 and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, and {๐‘ ๐‘›} is a positive real divergent sequence such that lim๐‘›โ†’โˆž๐‘ ๐‘›โ†’โˆž. If the sequence {๐‘ฅ๐‘›} defined by ๐‘ฅ0โˆˆ๐ถ and ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,๐‘›โ‰ฅ0.(5.3) Then {๐‘ฅ๐‘›} converges strongly to ฬƒ๐‘ฅ as ๐‘›โ†’โˆž, where ฬƒ๐‘ฅ is the unique solution in ๐น(๐’ฎ) of the variational inequality โŸจ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐ฝ(๐‘ฅโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ฅ0,๐‘ฅโˆˆ๐น(๐’ฎ).(5.4)

Proof. Taking ๐›ฝ๐‘›=0 in the in Theorem 4.2, we get the desired conclusion easily.

When ๐‘‹ is a Hilbert space, we can get the following corollary easily.

Corollary 5.3 (Reich [2]). Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป. Let ๐’ฎ={๐‘‡(๐‘ )โˆถ0โ‰ค๐‘ <โˆž} be a strongly continuous semigroup of nonexpansive mapping on ๐ถ such that ๐น(๐’ฎ) is nonempty. Let {๐›ผ๐‘›} and {๐›ฝ๐‘›} be sequences of real numbers in (0,1) which satisfies ๐›ผ๐‘›+๐›ฝ๐‘›<1, lim๐‘›โ†’โˆž๐›ผ๐‘›โ†’0, lim๐‘›โ†’โˆž๐›ฝ๐‘›โ†’0, and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž. Let ๐‘“ be a contraction of ๐ถ into itself with a coefficient ๐›ผโˆˆ[0,1) and {๐‘ ๐‘›} be a positive real divergent sequence such that lim๐‘›โ†’โˆž๐‘ ๐‘›โ†’โˆž. Then the sequence {๐‘ฅ๐‘›} defined by ๐‘ฅ0โˆˆ๐ถ and ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›๎€ธ1๐‘ ๐‘›๎€œ๐‘ ๐‘›0๐‘‡(๐‘ )๐‘ฅ๐‘›๐‘‘๐‘ ,๐‘›โ‰ฅ0.(5.5)
Then {๐‘ฅ๐‘›} converges strongly to ฬƒ๐‘ฅ, where ฬƒ๐‘ฅ is the unique solution in ๐น(๐’ฎ) of the variational inequality โŸจ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโŸฉโ‰ฅ0,๐‘ฅโˆˆ๐น(๐’ฎ),(5.6) or equivalent ฬƒ๐‘ฅ=๐‘ƒ๐น(๐’ฎ)(๐‘“)(ฬƒ๐‘ฅ), where ๐‘ƒ is a metric projection mapping from ๐ป into ๐น(๐’ฎ).

Funding

This paper is supported by the National Science Foundation of China under Grants (10771050 and 11101305).

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