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This article has been retracted as it is found to contain a substantial amount of material, without referencing, from the paper “On the Weak Relatively Nonexpansive Mappings in Banach Spaces,” Yongchun Xu and Yongfu Su, Fixed Point Theory and Applications, Volume 2010, Article ID 189751, 7 pages. doi:10.1155/2010/189751.
Abstract and Applied Analysis
Volumeย 2012ย (2012), Article IDย 730619, 7 pages
http://dx.doi.org/10.1155/2012/730619
Review Article

On the Weak Relatively Nonexpansive Multivalued Mappings in Banach Spaces

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 12 July 2012; Accepted 7 August 2012

Academic Editor: Xiaolongย Qin

Copyright ยฉ 2012 Yongfu Su. 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

In recent years, the definition of relatively nonexpansive multivalued mapping and the definition of weak relatively nonexpansive multivalued mapping have been presented and studied by many authors. In this paper, we give some results about weak relatively nonexpansive multivalued mappings and give two examples which are weak relatively nonexpansive multivalued mappings but not relatively nonexpansive multivalued mappings in Banach space ๐‘™2 and ๐ฟ๐‘[0,1](1<๐‘<+โˆž).

1. Introduction

Let ๐ธ be a smooth Banach space and let ๐ถ be a nonempty closed convex subset of ๐ธ. We denote by ๐œ™ the function defined by ๐œ™(๐‘ฅ,๐‘ฆ)=โ€–๐‘ฅโ€–2โˆ’2โŸจ๐‘ฅ,๐ฝ๐‘ฆโŸฉ+โ€–๐‘ฆโ€–2,for๐‘ฅ,๐‘ฆโˆˆ๐ธ.(1.1) Following Alber [1], the generalized projection ฮ ๐ถ from ๐ธ onto ๐ถ is defined by ฮ ๐ถ(๐‘ฅ)=argmin๐‘ฆโˆˆ๐ถ๐œ™(๐‘ฆ,๐‘ฅ),โˆ€๐‘ฅโˆˆ๐ธ.(1.2) The generalized projection ฮ ๐ถ from ๐ธ onto ๐ถ is well defined, single-value, and satisfies ()โ€–๐‘ฅโ€–โˆ’โ€–๐‘ฆโ€–2)โ‰ค๐œ™(๐‘ฅ,๐‘ฆ)โ‰ค(โ€–๐‘ฅโ€–+โ€–๐‘ฆโ€–2,for๐‘ฅ,๐‘ฆโˆˆ๐ธ.(1.3) If ๐ธ is a Hilbert space, then ๐œ™(๐‘ฆ,๐‘ฅ)=โ€–๐‘ฆโˆ’๐‘ฅโ€–2 and ฮ ๐ถ is the metric projection of ๐ธ onto ๐ถ.

In recent years, the definition of relatively nonexpansive multivalued mapping and the definition of weak relatively nonexpansive multivalued mapping have been presented and studied by many authors (see [1]). In this paper, we give some results about weak relatively nonexpansive multivalued mappings and give two examples which are weak relatively nonexpansive multivalued mappings but not relatively nonexpansive multivalued mappings in Banach space ๐‘™2 and ๐ฟ๐‘[0,1](1<๐‘<+โˆž).

Remark 1.1. The definition of relatively nonexpansive multivalued mapping presented in this paper and the definition of [2] are different.

Let ๐ถ be a closed convex subset of ๐ธ, and let ๐‘‡ be a multivalued mapping from ๐ถ into itself. We denote by ๐น(๐‘‡) the set of fixed points of ๐‘‡, that is, ๐น(๐‘‡)={๐‘ฅโˆˆ๐ถโˆถ๐‘ฅโˆˆ๐‘‡๐‘ฅ}.(1.4) A point ๐‘ in ๐ถ is said to be an asymptotic fixed point (strong asymptotic fixed point) of ๐‘‡ [3โ€“5] if ๐ถ contains a sequence {๐‘ฅ๐‘›} which converges weakly (strongly) to ๐‘ and there exists a sequence {๐‘ฆ๐‘›} such that ๐‘ฆ๐‘›โˆˆ๐‘‡๐‘ฅ๐‘›, lim๐‘›โ†’โˆžโ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–=0. The set of asymptotic fixed point (the set of strong asymptotic fixed point) of ๐‘‡ will be denoted by ๎๎‚๐น(๐‘‡)(๐น(๐‘‡)).

A multivalued mapping ๐‘‡ of ๐ถ into itself is said to be relatively nonexpansive multivalued mapping (weak relatively nonexpansive multivalued mapping) if the following conditions are satisfied:(1)๐น(๐‘‡) is nonempty;(2)๐œ™(๐‘ข,๐‘ฃ)โ‰ค๐œ™(๐‘ข,๐‘ฅ), โˆ€๐‘ขโˆˆ๐น(๐‘‡),โˆ€๐‘ฅโˆˆ๐ถ,โˆƒ๐‘ฃโˆˆ๐‘‡๐‘ฅ;(3)๎๎‚๐น(๐‘‡)=๐น(๐‘‡)(๐น(๐‘‡)=๐น(๐‘‡)).

A multivalued mapping ๐‘‡ of ๐ถ into itself is said to be relatively uniformly nonexpansive multivalued mapping (weak relatively uniformly nonexpansive multivalued mapping) if the following conditions are satisfied:(1)๐น(๐‘‡) is nonempty;(2)๐œ™(๐‘ข,๐‘ฃ)โ‰ค๐œ™(๐‘ข,๐‘ฅ), โˆ€๐‘ขโˆˆ๐น(๐‘‡),โˆ€๐‘ฅโˆˆ๐ถ,โˆ€๐‘ฃโˆˆ๐‘‡๐‘ฅ;(3)๎๎‚๐น(๐‘‡)=๐น(๐‘‡)(๐น(๐‘‡)=๐น(๐‘‡)).

Following Matsushita and Takahashi [3], a mapping ๐‘‡ of ๐ถ into itself is said to be relatively nonexpansive mapping if the following conditions are satisfied:(1)๐น(๐‘‡) is nonempty;(2)๐œ™(๐‘ข,๐‘‡๐‘ฅ)โ‰ค๐œ™(๐‘ข,๐‘ฅ), โˆ€๐‘ขโˆˆ๐น(๐‘‡),๐‘ฅโˆˆ๐ถ;(3)๎๐น(๐‘‡)=๐น(๐‘‡). The hybrid algorithms for fixed point of relatively nonexpansive mappings and applications have been studied by many authors, for example, [3โ€“8].

In recent years, the definition of weak relatively nonexpansive mapping has been presented and studied by many authors [6โ€“9].

A mapping ๐‘‡ from ๐ถ into itself is said to be weak relatively nonexpansive mapping if(1)๐น(๐‘‡) is nonempty;(2)๐œ™(๐‘ข,๐‘‡๐‘ฅ)โ‰ค๐œ™(๐‘ข,๐‘ฅ), โˆ€๐‘ขโˆˆ๐น(๐‘‡),๐‘ฅโˆˆ๐ถ;(3)๎‚๐น(๐‘‡)=๐น(๐‘‡).

Remark 1.2. In [7], the weak relatively nonexpansive mapping is also said to be relatively weak nonexpansive mapping.

Remark 1.3. In [8], the authors have given the definition of hemirelatively nonexpansive mapping as follows. A mapping ๐‘‡ from ๐ถ into itself is called hemirelatively nonexpansive if(1)๐น(๐‘‡) is nonempty;(2)๐œ™(๐‘ข,๐‘‡๐‘ฅ)โ‰ค๐œ™(๐‘ข,๐‘ฅ), โˆ€๐‘ขโˆˆ๐น(๐‘‡),๐‘ฅโˆˆ๐ถ.

The following conclusion is obvious.

Conclusion 1. A mapping is closed hemirelatively nonexpansive if and only if it is weak relatively nonexpansive.

If ๐ธ is strictly convex and reflexive Banach space, and ๐ดโŠ‚๐ธร—๐ธโˆ— is a continuous monotone mapping with ๐ดโˆ’1(0)โ‰ โˆ…, then it is proved in [3] that ๐ฝ๐‘Ÿโˆถ=(๐ฝ+๐‘Ÿ๐ด)โˆ’1๐ฝ, for ๐‘Ÿ>0 is relatively nonexpansive. Moreover, if ๐‘‡โˆถ๐ธโ†’๐ธ is relatively nonexpansive, then using the definition of ๐œ™ one can show that ๐น(๐‘‡) is closed and convex. It is obvious that relatively nonexpansive mapping is weak relatively nonexpansive mapping. In fact, for any mapping ๐‘‡โˆถ๐ถโ†’๐ถ, we have ๎‚๎๐น(๐‘‡)โŠ‚๐น(๐‘‡)โŠ‚๐น(๐‘‡). Therefore, if ๐‘‡ is relatively nonexpansive mapping, then ๎‚๎๐น(๐‘‡)=๐น(๐‘‡)=๐น(๐‘‡).

2. Results for Weak Relatively Multivalued Nonexpansive Mappings in Banach Space

Theorem 2.1. Let ๐ธ be a smooth Banach space and ๐ถ a nonempty closed convex and balanced subset of ๐ธ. Let {๐‘ฅ๐‘›} be a sequence in ๐ถ such that {๐‘ฅ๐‘›} converges weakly to ๐‘ฅ0โ‰ 0 and โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘šโ€–โ‰ฅ๐‘Ÿ>0 for all ๐‘›โ‰ ๐‘š. Define a mapping ๐‘‡โˆถ๐ถโ†’๐ถ as follows: ๎ƒฏ๎‚†๐‘‡(๐‘ฅ)=๐‘˜๐‘ฅ๐‘›๐‘›โˆถ๐‘˜=๎‚‡๐‘›+๐œ†,0<๐œ†โ‰ค๐‘€<+โˆž,if๐‘ฅ=๐‘ฅ๐‘›(โˆƒ๐‘›โ‰ฅ1),โˆ’๐‘ฅ,if๐‘ฅโ‰ ๐‘ฅ๐‘›(โˆ€๐‘›โ‰ฅ1).(2.1) Then, ๐‘‡ is a weak relatively uniformly nonexpansive multivalued mapping but not relatively uniformly nonexpansive multivalued mapping.

Proof. It is obvious that ๐‘‡ has a unique fixed point 0, that is, ๐น(๐‘‡)={0}. Firstly, we show that ๐‘ฅ0 is an asymptotic fixed point of ๐‘‡. In fact that, since {๐‘ฅ๐‘›} converges weakly to ๐‘ฅ0 and โ€–โ€–๐‘‡๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–=โ€–โ€–โ€–๐‘›๐‘ฅ๐‘›+๐œ†๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–=๐œ†โ€–โ€–๐‘ฅ๐‘›+๐œ†๐‘›โ€–โ€–โ†’0,(2.2) as ๐‘›โ†’โˆž, so that ๐‘ฅ0 is an asymptotic fixed point of ๐‘‡. Secondly, we show that ๐‘‡ has a unique strong asymptotic fixed point 0, so that ๎‚๐น(๐‘‡)=๐น(๐‘‡). In fact that, for any strong convergent sequence {๐‘ง๐‘›}โŠ‚๐ถ such that ๐‘ง๐‘›โ†’๐‘ง0 and โ€–๐‘ง๐‘›โˆ’๐‘ฆ๐‘›โ€–โ†’0,๐‘ฆ๐‘›โˆˆ๐‘‡๐‘ฅ๐‘› as ๐‘›โ†’โˆž, from the conditions of Theorem 2.1, there exist sufficiently large nature number ๐‘ such that ๐‘ง๐‘›โ‰ ๐‘ฅ๐‘š, for any ๐‘›,๐‘š>๐‘. Then, ๐‘‡๐‘ง๐‘›={โˆ’๐‘ง๐‘›} for ๐‘›>๐‘, it follows from โ€–๐‘ง๐‘›โˆ’๐‘ฆ๐‘›โ€–โ†’0,๐‘ฆ๐‘›โˆˆ๐‘‡๐‘ฅ๐‘› that 2๐‘ง๐‘›โ†’0 and hence ๐‘ง๐‘›โ†’๐‘ง0=0. On the other hand, observe that ๐œ™(0,๐‘ฃ)=โ€–๐‘ฃโ€–2โ‰คโ€–๐‘ฅโ€–2=๐œ™(0,๐‘ฅ),โˆ€๐‘ฅโˆˆ๐ถ,โˆ€๐‘ฃโˆˆ๐‘‡๐‘ฅ.(2.3) Then, ๐‘‡ is a weak relatively uniformly nonexpansive multivalued mapping. On the other hand, since ๐‘ฅ0 is an asymptotic fixed point of ๐‘‡ but not fixed point, hence ๐‘‡ is not a relatively uniformly nonexpansive multivalued mapping.

Taking any fixed number ๐œ†0โˆˆ(0,๐‘€), we have the following result.

Theorem 2.2. Let ๐ธ be a smooth Banach space and ๐ถ a nonempty closed convex and balanced subset of ๐ธ. Let {๐‘ฅ๐‘›} be a sequence in ๐ถ such that {๐‘ฅ๐‘›} converges weakly to ๐‘ฅ0โ‰ 0 and โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘šโ€–โ‰ฅ๐‘Ÿ>0 for all ๐‘›โ‰ ๐‘š. Define a mapping ๐‘‡โˆถ๐ถโ†’๐ถ as follows: ๎ƒฏ๐‘›๐‘‡(๐‘ฅ)=๐‘›+๐œ†0๐‘ฅ๐‘›,if๐‘ฅ=๐‘ฅ๐‘›(โˆƒ๐‘›โ‰ฅ1),โˆ’๐‘ฅ,if๐‘ฅโ‰ ๐‘ฅ๐‘›(โˆ€๐‘›โ‰ฅ1).(2.4) Then, ๐‘‡ is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping;

3. An Example in Banach Space ๐‘™2

In this section, we will give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping.

Example 3.1. Let ๐ธ=๐‘™2, where ๐‘™2=๎ƒฏ๎€ท๐œ‰๐œ‰=1,๐œ‰2,๐œ‰3,โ€ฆ,๐œ‰๐‘›๎€ธโˆถ,โ€ฆโˆž๎“๐‘›=1||๐‘ฅ๐‘›||2๎ƒฐ,๎ƒฉ<โˆžโ€–๐œ‰โ€–=โˆž๎“๐‘›=1||๐œ‰๐‘›||2๎ƒช1/2,โˆ€๐œ‰โˆˆ๐‘™2,โŸจ๐œ‰,๐œ‚โŸฉ=โˆž๎“๐‘›=1๐œ‰๐‘›๐œ‚๐‘›๎€ท๐œ‰,โˆ€๐œ‰=1,๐œ‰2,๐œ‰3,โ€ฆ,๐œ‰๐‘›๎€ธ๎€ท๐œ‚,โ€ฆ,๐œ‚=1,๐œ‚2,๐œ‚3,โ€ฆ,๐œ‚๐‘›๎€ธโ‹ฏ.โˆˆ๐‘™2.(3.1) It is well known that ๐‘™2 is a Hilbert space, so that (๐‘™2)โˆ—=๐‘™2. Let {๐‘ฅ๐‘›}โŠ‚๐ธ be a sequence defined by ๐‘ฅ0๐‘ฅ=(1,0,0,0,โ€ฆ)1=๐‘ฅ(1,1,0,0,โ€ฆ)2๐‘ฅ=(1,0,1,0,0,โ€ฆ)3โ‹ฎ๐‘ฅ=(1,0,0,1,0,0,โ€ฆ)๐‘›=๎€ท๐œ‰๐‘›,1,๐œ‰๐‘›,2,๐œ‰๐‘›,3,โ€ฆ,๐œ‰๐‘›,๐‘˜๎€ธ,โ€ฆโ‹ฎ,(3.2) where ๐œ‰๐‘›,๐‘˜=๎‚ป1,if๐‘˜=1,๐‘›+1,0if๐‘˜โ‰ 1,๐‘˜โ‰ ๐‘›+1,(3.3) for all ๐‘›โ‰ฅ1. Define a mapping ๐‘‡โˆถ๐ธโ†’๐ธ as follows: ๎ƒฏ๎‚†๐‘‡(๐‘ฅ)=๐‘˜๐‘ฅ๐‘›๐‘›โˆถ๐‘˜=๎‚‡๐‘›+๐œ†,0<๐œ†โ‰ค๐‘€<+โˆž,if๐‘ฅ=๐‘ฅ๐‘›(โˆƒ๐‘›โ‰ฅ1),โˆ’๐‘ฅ,if๐‘ฅโ‰ ๐‘ฅ๐‘›(โˆ€๐‘›โ‰ฅ1).(3.4)Conclusion 2. {๐‘ฅ๐‘›} converges weakly to ๐‘ฅ0.

Proof. For any ๐‘“=(๐œ1,๐œ2,๐œ3,โ€ฆ,๐œ๐‘˜,โ€ฆ)โˆˆ๐‘™2=(๐‘™2)โˆ—, we have ๐‘“๎€ท๐‘ฅ๐‘›โˆ’๐‘ฅ0๎€ธ=โŸจ๐‘“,๐‘ฅ๐‘›โˆ’๐‘ฅ0โŸฉ=โˆž๎“๐‘˜=2๐œ๐‘˜๐œ‰๐‘›,๐‘˜=๐œ๐‘›+1โ†’0,(3.5) as ๐‘›โ†’โˆž. That is, {๐‘ฅ๐‘›} converges weakly to ๐‘ฅ0.

The following conclusion is obvious.

Conclusion 3. โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘šโˆšโ€–=2 for any ๐‘›โ‰ ๐‘š.

It follows from Theorem 2.1 and the above two conclusions that ๐‘‡ is a weak relatively uniformly nonexpansive multivalued mapping but not relatively uniformly nonexpansive multivalued mapping.

4. An Example in Banach Space ๐ฟ๐‘[0,1](1<๐‘<+โˆž)

Let ๐ธ=๐ฟ๐‘[0,1],(1<๐‘<+โˆž) and ๐‘ฅ๐‘›1=1โˆ’2๐‘›,๐‘›=1,2,3,โ€ฆ.(4.1) Define a sequence of functions in ๐ฟ๐‘[0,1] by the following expression: ๐‘“๐‘›โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ2(๐‘ฅ)=๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›,if๐‘ฅ๐‘›๐‘ฅโ‰ค๐‘ฅ<๐‘›+1+๐‘ฅ๐‘›2,โˆ’2๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›๐‘ฅ,if๐‘›+1+๐‘ฅ๐‘›2โ‰ค๐‘ฅ<๐‘ฅ๐‘›+10otherwise,(4.2) for all ๐‘›โ‰ฅ1. Firstly, we can see, for any ๐‘ฅโˆˆ[0,1], that ๎€œ๐‘ฅ0๐‘“๐‘›๎€œ(๐‘ก)๐‘‘๐‘กโ†’0=๐‘ฅ0๐‘“0(๐‘ก)๐‘‘๐‘ก,(4.3) where ๐‘“0(๐‘ฅ)โ‰ก0. It is well known that the above relation (4.3) is equivalent to {๐‘“๐‘›(๐‘ฅ)} which converges weakly to ๐‘“0(๐‘ฅ) in uniformly smooth Banach space ๐ฟ๐‘[0,1](1<๐‘<+โˆž). On the other hand, for any ๐‘›โ‰ ๐‘š, we have โ€–โ€–๐‘“๐‘›โˆ’๐‘“๐‘šโ€–โ€–=๎‚ต๎€œ10||๐‘“๐‘›(๐‘ฅ)โˆ’๐‘“๐‘š||(๐‘ฅ)๐‘๎‚ถ๐‘‘๐‘ฅ1/๐‘=๎‚ต๎€œ๐‘ฅ๐‘›+1๐‘ฅ๐‘›||๐‘“๐‘›(๐‘ฅ)โˆ’๐‘“๐‘š||(๐‘ฅ)๐‘๎€œ๐‘‘๐‘ฅ+๐‘ฅ๐‘š+1๐‘ฅ๐‘š||๐‘“๐‘›(๐‘ฅ)โˆ’๐‘“๐‘š||(๐‘ฅ)๐‘๎‚ถ๐‘‘๐‘ฅ1/๐‘=๎‚ต๎€œ๐‘ฅ๐‘›+1๐‘ฅ๐‘›||๐‘“๐‘›||(๐‘ฅ)๐‘๎€œ๐‘‘๐‘ฅ+๐‘ฅ๐‘š+1๐‘ฅ๐‘š||๐‘“๐‘š||(๐‘ฅ)๐‘๎‚ถ๐‘‘๐‘ฅ1/๐‘=2๎‚ต๎‚ต๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›๎‚ถ๐‘๎€ท๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›๎€ธ+๎‚ต2๐‘ฅ๐‘š+1โˆ’๐‘ฅ๐‘š๎‚ถ๐‘๎€ท๐‘ฅ๐‘š+1โˆ’๐‘ฅ๐‘š๎€ธ๎‚ถ1/๐‘=๎ƒฉ2๐‘๎€ท๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›๎€ธ๐‘โˆ’1+2๐‘๎€ท๐‘ฅ๐‘š+1โˆ’๐‘ฅ๐‘š๎€ธ๐‘โˆ’1๎ƒช1/๐‘โ‰ฅ(2๐‘+2๐‘)1/๐‘>0.(4.4) Let ๐‘ข๐‘›(๐‘ฅ)=๐‘“๐‘›(๐‘ฅ)+1,โˆ€๐‘›โ‰ฅ1.(4.5) It is obvious that ๐‘ข๐‘› converges weakly to ๐‘ข0(๐‘ฅ)โ‰ก1 and โ€–โ€–๐‘ข๐‘›โˆ’๐‘ข๐‘šโ€–โ€–=โ€–โ€–๐‘“๐‘›โˆ’๐‘“๐‘šโ€–โ€–โ‰ฅ(2๐‘+2๐‘)1/๐‘>0,โˆ€๐‘›โ‰ฅ1.(4.6) Define a mapping ๐‘‡โˆถ๐ธโ†’๐ธ as follows: ๎ƒฏ๎‚†๐‘‡(๐‘ฅ)=๐‘˜๐‘ข๐‘›๐‘›โˆถ๐‘˜=๎‚‡๐‘›+๐œ†,0<๐œ†โ‰ค๐‘€<+โˆž,if๐‘ฅ=๐‘ข๐‘›(โˆƒ๐‘›โ‰ฅ1),โˆ’๐‘ฅ,if๐‘ฅโ‰ ๐‘ข๐‘›(โˆ€๐‘›โ‰ฅ1).(4.7) Since (4.6) holds, by using Theorem 2.1, we know that ๐‘‡ is a weak relatively uniformly nonexpansive multivalued mapping but not relatively uniformly nonexpansive multivalued mapping.

Acknowledgment

This project is supported by the National Natural Science Foundation of China under Grant (11071279).

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