# On the Weak Relatively Nonexpansive Multivalued Mappings in Banach Spaces

Authors: Yongfu Su
Journal: Abstract and Applied Analysis
 (2012)

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## 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 spacel2 and Lp[0,1](1<p<+∞).


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## Body

## 1. Introduction

LetE be a smooth Banach space and let C be a nonempty closed convex subset of E. We denote by ϕ the function defined by
(1.1)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,forx,y∈E.
Following Alber [1], the generalized projection ΠC from E onto C is defined by
(1.2)ΠC(x)=argminy∈Cϕ(y,x),∀x∈E.
The generalized projection ΠC from E onto C is well defined, single-value, and satisfies
(1.3)(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2,forx,y∈E.
If E is a Hilbert space, then ϕ(y,x)=∥y-x∥2 and ΠC 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 Lp[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.LetC 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,
(1.4)F(T)={x∈C:x∈Tx}.
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 {xn} which converges weakly (strongly) to p and there exists a sequence {yn} such that yn∈Txn, limn→∞∥yn-xn∥=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 mappingT 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 mappingT 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 mappingT 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.IfE 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 Jr:=(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.
LetE be a smooth Banach space and C a nonempty closed convex and balanced subset of E. Let {xn} be a sequence in C such that {xn} converges weakly to x0≠0 and ∥xn-xm∥≥r>0 for all n≠m. Define a mapping T:C→C as follows:
(2.1)T(x)={{kxn:k=nn+λ,0<λ≤M<+∞},ifx=xn(∃n≥1),-x,ifx≠xn(∀n≥1).
Then, T is a weak relatively uniformly nonexpansive multivalued mapping but not relatively uniformly nonexpansive multivalued mapping.Proof.
It is obvious thatT 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 {xn} converges weakly to x0 and
(2.2)∥Txn-xn∥=∥nn+λxn-xn∥=λn+λ∥xn∥→0,
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 {zn}⊂C such that zn→z0 and ∥zn-yn∥→0,yn∈Txn as n→∞, from the conditions of Theorem 2.1, there exist sufficiently large nature number N such that zn≠xm, for any n,m>N. Then, Tzn={-zn} for n>N, it follows from ∥zn-yn∥→0,yn∈Txn that 2zn→0 and hence zn→z0=0. On the other hand, observe that
(2.3)ϕ(0,v)=∥v∥2≤∥x∥2=ϕ(0,x),∀x∈C,∀v∈Tx.
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 numberλ0∈(0,M), we have the following result.Theorem 2.2.
LetE be a smooth Banach space and C a nonempty closed convex and balanced subset of E. Let {xn} be a sequence in C such that {xn} converges weakly to x0≠0 and ∥xn-xm∥≥r>0 for all n≠m. Define a mapping T:C→C as follows:
(2.4)T(x)={nn+λ0xn,ifx=xn(∃n≥1),-x,ifx≠xn(∀n≥1).
Then, T is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping;


## 3. An Example in Banach Spacel2

In this section, we will give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping.Example 3.1.
LetE=l2, where
(3.1)l2={ξ=(ξ1,ξ2,ξ3,…,ξn,…):∑n=1∞‍|xn|2<∞},∥ξ∥=(∑n=1∞‍|ξn|2)1/2,∀ξ∈l2,〈ξ,η〉=∑n=1∞‍ξnηn,∀ξ=(ξ1,ξ2,ξ3,…,ξn,…),η=(η1,η2,η3,…,ηn⋯.)∈l2.
It is well known that l2 is a Hilbert space, so that (l2)*=l2. Let {xn}⊂E be a sequence defined by
(3.2)x0=(1,0,0,0,…)x1=(1,1,0,0,…)x2=(1,0,1,0,0,…)x3=(1,0,0,1,0,0,…)⋮xn=(ξn,1,ξn,2,ξn,3,…,ξn,k,…)⋮,
where
(3.3)ξn,k={1,ifk=1,n+1,0ifk≠1,k≠n+1,
for all n≥1. Define a mapping T:E→E as follows:
(3.4)T(x)={{kxn:k=nn+λ,0<λ≤M<+∞},ifx=xn(∃n≥1),-x,ifx≠xn(∀n≥1).Conclusion 2.{xn}converges weakly tox0.Proof.
For anyf=(ζ1,ζ2,ζ3,…,ζk,…)∈l2=(l2)*, we have
(3.5)f(xn-x0)=〈f,xn-x0〉=∑k=2∞‍ζkξn,k=ζn+1→0,
as n→∞. That is, {xn} converges weakly to x0.The following conclusion is obvious.Conclusion 3.
∥
x
n
-
x
m
∥
=
2
for any
n
≠
m
.It follows from Theorem2.1 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 SpaceLp[0,1](1<p<+∞)

LetE=Lp[0,1],(1<p<+∞) and
(4.1)xn=1-12n,n=1,2,3,….
Define a sequence of functions in Lp[0,1] by the following expression:
(4.2)fn(x)={2xn+1-xn,ifxn≤x<xn+1+xn2,-2xn+1-xn,ifxn+1+xn2≤x<xn+10otherwise,
for all n≥1. Firstly, we can see, for any x∈[0,1], that
(4.3)∫0x‍fn(t)dt→0=∫0x‍f0(t)dt,
where f0(x)≡0. It is well known that the above relation (4.3) is equivalent to {fn(x)} which converges weakly to f0(x) in uniformly smooth Banach space Lp[0,1](1<p<+∞). On the other hand, for any n≠m, we have
(4.4)∥fn-fm∥=(∫01‍|fn(x)-fm(x)|pdx)1/p=(∫xnxn+1‍|fn(x)-fm(x)|pdx+∫xmxm+1‍|fn(x)-fm(x)|pdx)1/p=(∫xnxn+1‍|fn(x)|pdx+∫xmxm+1‍|fm(x)|pdx)1/p=((2xn+1-xn)p(xn+1-xn)+(2xm+1-xm)p(xm+1-xm))1/p=(2p(xn+1-xn)p-1+2p(xm+1-xm)p-1)1/p≥(2p+2p)1/p>0.
Let
(4.5)un(x)=fn(x)+1,∀n≥1.
It is obvious that un converges weakly to u0(x)≡1 and
(4.6)∥un-um∥=∥fn-fm∥≥(2p+2p)1/p>0,∀n≥1.
Define a mapping T:E→E as follows:
(4.7)T(x)={{kun:k=nn+λ,0<λ≤M<+∞},ifx=un(∃n≥1),-x,ifx≠un(∀n≥1).
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.



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*Source: 730619-2012-09-02.xml*