[Retracted] On the Weak Relatively Nonexpansive Multivalued Mappings in Banach Spaces

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 𝑙² and 𝐿^𝑝[0,1](1<𝑝<+∞).

1. Introduction

Let 𝐸 be a smooth Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸. We denote by 𝜙 the function defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,for𝑥,𝑦∈𝐸.(1.1) Following Alber [1], the generalized projection Π_𝐶 from 𝐸 onto 𝐶 is defined by Π_𝐶(𝑥)=argmin_𝑦∈𝐶𝜙(𝑦,𝑥),∀𝑥∈𝐸.(1.2) The generalized projection Π_𝐶 from 𝐸 onto 𝐶 is well defined, single-value, and satisfies (‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)²,for𝑥,𝑦∈𝐸.(1.3) If 𝐸 is a Hilbert space, then 𝜙(𝑦,𝑥)=‖𝑦−𝑥‖² and Π_𝐶 is the metric projection of 𝐸 onto 𝐶.

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 𝑙² and 𝐿^𝑝[0,1](1<𝑝<+∞).

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

Let 𝐶 be a closed convex subset of 𝐸, and let 𝑇 be a multivalued mapping from 𝐶 into itself. We denote by 𝐹(𝑇) the set of fixed points of 𝑇, that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑥∈𝑇𝑥}.(1.4) A point 𝑝 in 𝐶 is said to be an asymptotic fixed point (strong asymptotic fixed point) of 𝑇 [3–5] if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly (strongly) to 𝑝 and there exists a sequence {𝑦_𝑛} such that 𝑦_𝑛∈𝑇𝑥_𝑛, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0. The set of asymptotic fixed point (the set of strong asymptotic fixed point) of 𝑇 will be denoted by 𝐹(𝑇)(𝐹(𝑇)).

A multivalued mapping 𝑇 of 𝐶 into itself is said to be relatively nonexpansive multivalued mapping (weak relatively nonexpansive multivalued mapping) if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty;(2)𝜙(𝑢,𝑣)≤𝜙(𝑢,𝑥), ∀𝑢∈𝐹(𝑇),∀𝑥∈𝐶,∃𝑣∈𝑇𝑥;(3)𝐹(𝑇)=𝐹(𝑇)(𝐹(𝑇)=𝐹(𝑇)).

A multivalued mapping 𝑇 of 𝐶 into itself is said to be relatively uniformly nonexpansive multivalued mapping (weak relatively uniformly nonexpansive multivalued mapping) if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty;(2)𝜙(𝑢,𝑣)≤𝜙(𝑢,𝑥), ∀𝑢∈𝐹(𝑇),∀𝑥∈𝐶,∀𝑣∈𝑇𝑥;(3)𝐹(𝑇)=𝐹(𝑇)(𝐹(𝑇)=𝐹(𝑇)).

Following Matsushita and Takahashi [3], a mapping 𝑇 of 𝐶 into itself is said to be relatively nonexpansive mapping if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty;(2)𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥), ∀𝑢∈𝐹(𝑇),𝑥∈𝐶;(3)𝐹(𝑇)=𝐹(𝑇). The hybrid algorithms for fixed point of relatively nonexpansive mappings and applications have been studied by many authors, for example, [3–8].

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

A mapping 𝑇 from 𝐶 into itself is said to be weak relatively nonexpansive mapping if(1)𝐹(𝑇) is nonempty;(2)𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥), ∀𝑢∈𝐹(𝑇),𝑥∈𝐶;(3)𝐹(𝑇)=𝐹(𝑇).

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 𝑇 from 𝐶 into itself is called hemirelatively nonexpansive if(1)𝐹(𝑇) is nonempty;(2)𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥), ∀𝑢∈𝐹(𝑇),𝑥∈𝐶.

The following conclusion is obvious.

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

If 𝐸 is strictly convex and reflexive Banach space, and 𝐴⊂𝐸×𝐸^∗ is a continuous monotone mapping with 𝐴⁻¹(0)≠∅, then it is proved in [3] that 𝐽_𝑟∶=(𝐽+𝑟𝐴)⁻¹𝐽, for 𝑟>0 is relatively nonexpansive. Moreover, if 𝑇∶𝐸→𝐸 is relatively nonexpansive, then using the definition of 𝜙 one can show that 𝐹(𝑇) is closed and convex. It is obvious that relatively nonexpansive mapping is weak relatively nonexpansive mapping. In fact, for any mapping 𝑇∶𝐶→𝐶, we have 𝐹(𝑇)⊂𝐹(𝑇)⊂𝐹(𝑇). Therefore, if 𝑇 is relatively nonexpansive mapping, then 𝐹(𝑇)=𝐹(𝑇)=𝐹(𝑇).

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

Theorem 2.1. Let 𝐸 be a smooth Banach space and 𝐶 a nonempty closed convex and balanced subset of 𝐸. Let {𝑥_𝑛} be a sequence in 𝐶 such that {𝑥_𝑛} converges weakly to 𝑥₀≠0 and ‖𝑥_𝑛−𝑥_𝑚‖≥𝑟>0 for all 𝑛≠𝑚. Define a mapping 𝑇∶𝐶→𝐶 as follows: 𝑇(𝑥)=𝑘𝑥^𝑛∶𝑘=𝑛𝑛+𝜆,0<𝜆≤𝑀<+∞,if𝑥=𝑥^𝑛−𝑥,if𝑥≠𝑥_𝑛(∃𝑛≥1),(∀𝑛≥1).(2.1) Then, 𝑇 is a weak relatively uniformly nonexpansive multivalued mapping but not relatively uniformly nonexpansive multivalued mapping.

Proof. It is obvious that 𝑇 has a unique fixed point 0, that is, 𝐹(𝑇)={0}. Firstly, we show that 𝑥₀ is an asymptotic fixed point of 𝑇. In fact that, since {𝑥_𝑛} converges weakly to 𝑥₀ and ‖‖𝑇𝑥_𝑛−𝑥_𝑛‖‖=‖‖‖𝑛𝑛+𝜆𝑥^𝑛−𝑥_𝑛‖‖‖=𝜆𝑛+𝜆‖‖𝑥^𝑛‖‖→0,(2.2) as 𝑛→∞, so that 𝑥₀ is an asymptotic fixed point of 𝑇. Secondly, we show that 𝑇 has a unique strong asymptotic fixed point 0, so that 𝐹(𝑇)=𝐹(𝑇). In fact that, for any strong convergent sequence {𝑧_𝑛}⊂𝐶 such that 𝑧_𝑛→𝑧₀ and ‖𝑧_𝑛−𝑦_𝑛‖→0,𝑦_𝑛∈𝑇𝑥_𝑛 as 𝑛→∞, from the conditions of Theorem 2.1, there exist sufficiently large nature number 𝑁 such that 𝑧_𝑛≠𝑥_𝑚, for any 𝑛,𝑚>𝑁. Then, 𝑇𝑧_𝑛={−𝑧_𝑛} for 𝑛>𝑁, it follows from ‖𝑧_𝑛−𝑦_𝑛‖→0,𝑦_𝑛∈𝑇𝑥_𝑛 that 2𝑧_𝑛→0 and hence 𝑧_𝑛→𝑧₀=0. On the other hand, observe that 𝜙(0,𝑣)=‖𝑣‖²≤‖𝑥‖²=𝜙(0,𝑥),∀𝑥∈𝐶,∀𝑣∈𝑇𝑥.(2.3) Then, 𝑇 is a weak relatively uniformly nonexpansive multivalued mapping. On the other hand, since 𝑥₀ is an asymptotic fixed point of 𝑇 but not fixed point, hence 𝑇 is not a relatively uniformly nonexpansive multivalued mapping.

Taking any fixed number 𝜆₀∈(0,𝑀), we have the following result.

Theorem 2.2. Let 𝐸 be a smooth Banach space and 𝐶 a nonempty closed convex and balanced subset of 𝐸. Let {𝑥_𝑛} be a sequence in 𝐶 such that {𝑥_𝑛} converges weakly to 𝑥₀≠0 and ‖𝑥_𝑛−𝑥_𝑚‖≥𝑟>0 for all 𝑛≠𝑚. Define a mapping 𝑇∶𝐶→𝐶 as follows: 𝑇(𝑥)=𝑛+𝑛𝜆₀−𝑥,𝑥_𝑛,iiff𝑥𝑥≠=𝑥𝑥_𝑛𝑛(∃𝑛≥1),(∀𝑛≥1).(2.4) Then, 𝑇 is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping;

3. An Example in Banach Space 𝑙²

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 𝐸=𝑙², where 𝑙²=𝜉=𝜉₁,𝜉₂,𝜉₃,…,𝜉_𝑛,…∶^∞_𝑛=1‖𝜉‖=_∞||𝑥_𝑛||²<∞,_𝑛=1||𝜉_𝑛||²^1/2⟨𝜉,𝜂⟩=_∞,∀𝜉∈𝑙²,_𝑛=1𝜉_𝑛𝜂_𝑛,∀𝜉=𝜉₁,𝜉₂,𝜉₃,…,𝜉_𝑛,…,𝜂=𝜂₁,𝜂₂,𝜂₃,…,𝜂_𝑛⋯.∈𝑙².(3.1) It is well known that 𝑙² is a Hilbert space, so that (𝑙²)^∗=𝑙². Let {𝑥_𝑛}⊂𝐸 be a sequence defined by 𝑥𝑥₀=(1,0,0,0,…)₁𝑥=(1,1,0,0,…)₂𝑥=(1,0,1,0,0,…)₃𝑥⋮=(1,0,0,1,0,0,…)_𝑛⋮,=𝜉_𝑛,1,𝜉_𝑛,2,𝜉_𝑛,3,…,𝜉_𝑛,𝑘,…(3.2) where 𝜉_𝑛,𝑘=10,iiff𝑘𝑘≠=11,,𝑘𝑛≠+𝑛1+,1,(3.3) for all 𝑛≥1. Define a mapping 𝑇∶𝐸→𝐸 as follows: 𝑇(𝑥)=𝑘𝑥^𝑛∶𝑘=𝑛𝑛+𝜆,0<𝜆≤𝑀<+∞,if𝑥=𝑥^𝑛−𝑥,if𝑥≠𝑥_𝑛(∃𝑛≥1),(∀𝑛≥1).(3.4)Conclusion 2. {𝑥_𝑛} converges weakly to 𝑥₀.

Proof. For any 𝑓=(𝜁₁,𝜁₂,𝜁₃,…,𝜁_𝑘,…)∈𝑙²=(𝑙²)^∗, we have 𝑓𝑥_𝑛−𝑥₀=⟨𝑓,𝑥_𝑛−𝑥₀⟩=^∞_𝑘=2𝜁_𝑘𝜉_𝑛,𝑘=𝜁_𝑛+1→0,(3.5) as 𝑛→∞. That is, {𝑥_𝑛} converges weakly to 𝑥₀.

The following conclusion is obvious.

Conclusion 3. ‖𝑥_𝑛−𝑥_𝑚‖=√2 for any 𝑛≠𝑚.

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

4. An Example in Banach Space 𝐿^𝑝[0,1](1<𝑝<+∞)

Let 𝐸=𝐿^𝑝[0,1],(1<𝑝<+∞) and 𝑥_𝑛=1−12^𝑛,𝑛=1,2,3,….(4.1) Define a sequence of functions in 𝐿^𝑝[0,1] by the following expression: 𝑓_𝑛(𝑥)=⎧⎪⎪⎨⎪⎪⎩2𝑥_𝑛+1−𝑥_𝑛,if𝑥_𝑛≤𝑥<𝑥^𝑛+1+𝑥_𝑛−22,𝑥_𝑛+1−𝑥_𝑛,if𝑥^𝑛+1+𝑥_𝑛2≤𝑥<𝑥^𝑛+10otherwise,(4.2) for all 𝑛≥1. Firstly, we can see, for any 𝑥∈[0,1], that ^𝑥0𝑓_𝑛(𝑡)𝑑𝑡→0=^𝑥0𝑓₀(𝑡)𝑑𝑡,(4.3) where 𝑓₀(𝑥)≡0. It is well known that the above relation (4.3) is equivalent to {𝑓_𝑛(𝑥)} which converges weakly to 𝑓₀(𝑥) in uniformly smooth Banach space 𝐿^𝑝[0,1](1<𝑝<+∞). On the other hand, for any 𝑛≠𝑚, we have ‖‖𝑓_𝑛−𝑓_𝑚‖‖=¹⁰||𝑓_𝑛(𝑥)−𝑓_𝑚(𝑥)||^𝑝𝑑𝑥^1/𝑝=^𝑥𝑛+1_𝑥𝑛||𝑓_𝑛(𝑥)−𝑓_𝑚(𝑥)||^𝑝𝑑𝑥+^𝑥𝑚+1_𝑥𝑚||𝑓_𝑛(𝑥)−𝑓_𝑚(𝑥)||^𝑝𝑑𝑥^1/𝑝=^𝑥𝑛+1_𝑥𝑛||𝑓_𝑛(𝑥)||^𝑝𝑑𝑥+^𝑥𝑚+1_𝑥𝑚||𝑓_𝑚(𝑥)||^𝑝𝑑𝑥^1/𝑝=2𝑥_𝑛+1−𝑥_𝑛^𝑝𝑥_𝑛+1−𝑥_𝑛+2𝑥_𝑚+1−𝑥_𝑚^𝑝𝑥_𝑚+1−𝑥_𝑚^1/𝑝=2^𝑝𝑥_𝑛+1−𝑥_𝑛^𝑝−1+2^𝑝𝑥_𝑚+1−𝑥_𝑚^𝑝−1^1/𝑝≥(2^𝑝+2^𝑝)^1/𝑝>0.(4.4) Let 𝑢_𝑛(𝑥)=𝑓_𝑛(𝑥)+1,∀𝑛≥1.(4.5) It is obvious that 𝑢_𝑛 converges weakly to 𝑢₀(𝑥)≡1 and ‖‖𝑢_𝑛−𝑢_𝑚‖‖=‖‖𝑓_𝑛−𝑓_𝑚‖‖≥(2^𝑝+2^𝑝)^1/𝑝>0,∀𝑛≥1.(4.6) Define a mapping 𝑇∶𝐸→𝐸 as follows: 𝑇(𝑥)=𝑘𝑢^𝑛∶𝑘=𝑛𝑛+𝜆,0<𝜆≤𝑀<+∞,if𝑥=𝑢^𝑛−𝑥,if𝑥≠𝑢_𝑛(∃𝑛≥1),(∀𝑛≥1).(4.7) Since (4.6) holds, by using Theorem 2.1, we know that 𝑇 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).