# Weak and strong convergence theorems for relatively nonexpansive multi-valued mappings in Banach spaces

- Simin Homaeipour
^{1}Email author and - Abdolrahman Razani
^{1, 2}

**2011**:73

https://doi.org/10.1186/1687-1812-2011-73

© Homaeipour and Razani; licensee Springer. 2011

**Received: **2 June 2011

**Accepted: **31 October 2011

**Published: **31 October 2011

## Abstract

In this paper, an iterative sequence for relatively nonexpansive multi-valued mappings by using the notion of generalized projection is introduced, and then weak and strong convergence theorems are proved.

**2000 Mathematics Subject Classification:** 47H09; 47H10; 47J25.

## Keywords

## 1 Introduction and preliminaries

*D*be a nonempty closed convex subset of a real Banach space

*X*. A single-valued mapping

*T*:

*D*→

*D*is called nonexpansive if ||

*T*(

*x*) -

*T*(

*y*)|| ≤ ||

*x*-

*y*|| for all

*x*,

*y*∈

*D*. Let

*N*(

*D*) and

*CB*(

*D*) denote the family of nonempty subsets and nonempty closed bounded subsets of

*D*, respectively. The Hausdorff metric on

*CB*(

*D*) is defined by

for *A*_{1}, *A*_{2} ∈ *CB*(*D*), where *d*(*x*, *A*_{1}) = inf {||*x* - *y*||; *y* ∈ *A*_{1}}. The multi-valued mapping *T* : *D* → *CB*(*D*) is called nonexpansive if *H*(*T*(*x*), *T*(*y*)) ≤ ||*x* - *y*|| for all *x*, *y* ∈ *D*. An element *p* ∈ *D* is called a fixed point of *T* : *D* → *N*(*D*) (respectively, *T* : *D* → *D*) if *p* ∈ *F*(*T*) (respectively, *T*(*p*) = *p*). The set of fixed points of *T* is represented by *F*(*T*).

*X*be a real Banach space with dual

*X**. We denote by

*J*the normalized duality mapping from

*X*to ${2}^{{X}^{*}}$ defined by

where 〈.,.〉 denotes the generalized duality pairing.

The Banach space *X* is strictly convex if ||(*x* + *y*)/2|| < 1 for all *x*, *y* ∈ *X* with ||*x*|| = ||*y*|| = 1 and *x* ≠ *y*. The Banach space *X* is uniformly convex if lim_{
n
}_{→∞} ||*x*_{
n
} - *y*_{
n
} || = 0 for any two sequences {*x*_{
n
} }, {*y*_{
n
} } ⊆ *X* with ||*x*_{
n
} || = ||*y*_{
n
} || = 1 for all *n* ∈ ℕ and lim_{
n
}_{→∞} ||(*x*_{
n
} + *y*_{
n
} )/2|| = 1.

**Lemma 1.1**. [1]

*Let X be a uniformly convex Banach space and B*

_{ r }= {

*x*∈

*X*: ||

*x*|| ≤

*r*},

*r*> 0.

*Then, there exists a continuous, strictly increasing, and convex function g*: [0, ∞) → [0, ∞)

*with g*(0) = 0

*such that*

*for all* *x*, *y* ∈ *B*_{
r
}*and all* *α*, *β* ∈ [0, 1] *with* *α* + *β* = 1.

*X*is said to be Gâteaux differentiable if for each

*x*,

*y*∈

*S*(

*X*):= {

*x*∈

*X*: ||

*x*|| = 1} the limit

*X*is called smooth. The norm of Banach space

*X*is said to be Fréchet differentiable if for each

*x*∈

*S*(

*X*), limit (1.1) is attained uniformly for

*y*∈

*S*(

*X*) and the norm is uniformly Fréchet differentiable if limit (1.1) is attained uniformly for

*x*,

*y*∈

*S*(

*X*). In this case,

*X*is said to be uniformly smooth. The following properties of

*J*are well known [2]:

- 1.
*X*(*X**, resp.) is uniformly convex if and only if*X** (*X*, resp.) is uniformly smooth; - 2.
If

*X*is smooth, then*J*is single-valued and norm-to-weak* continuous; - 3.
If

*X*is reflexive, then*J*is onto; - 4.
If

*X*is strictly convex, then*J*(*x*) ∩*J*(*y*) = ∅ for all*x*≠*y*; - 5.
If

*X*has a Fréchet differentiable norm, then*J*is norm-to-norm continuous; - 6.
If

*X*is uniformly smooth, then*J*is uniformly norm-to-norm continuous on each bounded subset of*X*.

The normalized duality mapping *J* of a smooth Banach space *X* is called weakly sequentially continuous if *x*_{
n
} ⇀ *x* implies that $J\left({x}_{n}\right)\phantom{\rule{2.77695pt}{0ex}}\stackrel{*}{\rightharpoonup}\phantom{\rule{2.77695pt}{0ex}}J\left(x\right)$, where ⇀ denotes the weak convergence and $\stackrel{*}{\rightharpoonup}$ denotes the weak* convergence.

*X*be a smooth Banach space. The function

*ϕ*:

*X*×

*X*→ ℝ is defined by

*ϕ*that

*ϕ*has the following property:

**Lemma 1.2**. *[3, Remark 2.1] Let X be a strictly convex and smooth Banach space, then ϕ*(*x*, *y*) = 0 *if and only if x* = *y*.

**Lemma 1.3**. [4]

*Let X be a uniformly convex and smooth Banach space and r*> 0

*. Then*

*for all* *y*, *z* ∈ *B*_{
r
} = {*x* ∈ *X*; ||*x*|| ≤ *r*}, *where* *g* : [0, ∞) → [0, ∞) *is a continuous, strictly increasing* *and convex function with* *g*(0) = 0.

*D*be a nonempty closed convex subset of a smooth Banach space

*X*. A point

*p*∈

*D*is called an asymptotic fixed point of

*T*:

*D*→

*D*[5], if there exists a sequence {

*x*

_{ n }} in

*D*which converges weakly to

*p*and lim

_{ n }

_{→∞}||

*x*

_{ n }-

*T*(

*x*

_{ n })|| = 0. The set of asymptotic fixed points of

*T*is represented by $\widehat{F}\left(T\right)$. A mapping

*T*:

*D*→

*D*is called relatively nonexpansive [3, 6–8], if the following conditions are satisfied:

- 1.
*F*(*T*) is nonempty; - 2.
*ϕ*(*p*,*T*(*x*)) ≤*ϕ*(*p*,*x*), ∀*x*∈*D*,*p*∈*F*(*T*);

3.$\widehat{F}\left(T\right)=F\left(T\right)$.

*D*be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space

*X*. It is known that [4, 9] for any

*x*∈

*X*, there exists a unique point

*x*

_{0}∈

*D*such that

Following Alber [9], we denote such an element *x*_{0} by Π _{
D
}*x*. The mapping Π _{
D
} is called the generalized projection from *X* onto *D*. If *X* is a Hilbert space, then *ϕ*(*y*, *x*) = ||*y* - *x*||^{2} and Π _{
D
} is the metric projection of *X* onto *D*.

**Lemma 1.4**. [4, 9]

*Let D be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space X. Then*

**Lemma 1.5**. [4, 9]

*Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X. Let x*∈

*X and z*∈

*D, then*

*T*:

*D*→

*D*. Given

*x*

_{1}∈

*D*,

where *D* is a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space *X*, Π _{
D
} is the generalized projection onto *D* and {*α*_{
n
} } is a sequence in [0, 1].

They proved weak and strong convergence theorems in uniformly convex and uniformly smooth Banach space *X*.

Iterative methods for approximating fixed points of multi-valued mappings in Banach spaces have been studied by some authors, see for instance [11–14].

Let *D* be a nonempty closed convex subset of a smooth Banach space *X*. We define an asymptotic fixed point for a multi-valued mapping as follows.

**Definition 1.6**. *A point p* ∈ *D is called an asymptotic fixed point of T* : *D* → *N*(*D*)*, if there exists a sequence* {*x*_{
n
} } *in D which converges weakly to p and* lim_{
n
}_{→∞}*d*(*x*_{
n
} , *T*(*x*_{
n
} )) = 0.

Moreover, we define a relatively nonexpansive multi-valued mapping as follows.

**Definition 1.7**. *A multi-valued mapping T* : *D* → *N*(*D*) *is called relatively nonexpansive, if the following conditions are satisfied:*

*1*. *F*(*T*) *is nonempty*;

*2*. *ϕ*(*p*, *z*) ≤ *ϕ*(*p*, *x*), ∀*x* ∈ *D*, *z* ∈ *T*(*x*), *p* ∈ *F*(*T*);

*3*.$\widehat{F}\left(T\right)=F\left(T\right)$,

*where*$\widehat{F}\left(T\right)$*is the set of asymptotic fixed points of* *T*.

There exist relatively nonexpansive multi-valued mappings that are not nonexpansive.

**Example 1.8**. Let

*I*= [0,1],

*X*=

*L*

^{ p }(

*I*), 1 <

*p*< ∞ and

*D*= {

*f*∈

*X*;

*f*(

*x*) ≥ 0, ∀

*x*∈

*I*}. Let

*T*:

*D*→

*CB*(

*D*) be defined by

*F*(

*T*) = {0}. Let $h\in \widehat{F}\left(T\right)$. Then, there exists a sequence {

*f*

_{ n }} in

*D*which converges weakly to

*h*, and

*z*

_{ n }=

*d*(

*f*

_{ n },

*T*(

*f*

_{ n })) → 0. Let

*n*∈ ℕ, we have

*z*

_{ n }→ 0, we have ||

*f*

_{ n }||

_{ p }→ 0. Therefore,

*f*

_{ n }→ 0. Hence,

*h*= 0. Therefore,$\widehat{F}\left(T\right)=F\left(T\right)=\left\{0\right\}$. Let

*f*∈

*D*such that

*f*(

*x*) > 1 for all

*x*∈

*I*, and

*g*∈

*T*(

*f*), then

*f*∈

*D*such that there exists

*x*∈

*I*such that

*f*(

*x*) ≤ 1, then

Hence, *T* is relatively nonexpansive. However, if *f*(*x*) = 2 and *g*(*x*) = 1 for all *x* ∈ *I*, we get $H\left(T\left(f\right),T\left(g\right)\right)=\frac{7}{4}$. Then, *H*(*T*(*f*), *T*(*g*)) > ||*f* - *g*|| _{
p
} = 1. Hence, *T* is not nonexpansive.

*T*:

*D*→

*N*(

*D*). Given

*x*

_{1}∈

*D*,

where *z*_{
n
} ∈ *T*(*x*_{
n
} ) for all *n* ∈ ℕ, *D* is a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space *X*, Π _{
D
} is the generalized projection onto *D* and {*α*_{
n
} } is a sequence in [0, 1]. We prove weak and strong convergence theorems in uniformly convex and uniformly smooth Banach space *X*.

## 2 Main results

In this section, at first, concerning the fixed point set of a relatively nonexpansive multi-valued mapping, we prove the following proposition.

**Proposition 2.1**. *Let X be a strictly convex and smooth Banach space, and D a nonempty closed convex subset of X. Suppose T* : *D* → *N*(*D*) *is a relatively nonexpansive multi-valued mapping. Then, F*(*T*) *is closed and convex*.

*Proof*. First, we show

*F*(

*T*) is closed. Let {

*x*

_{ n }} be a sequence in

*F*(

*T*) such that

*x*

_{ n }→

*x**. Since

*T*is relatively nonexpansive, we have

*z*∈

*T*(

*x**) and for all

*n*∈ ℕ. Therefore,

*x** =

*z*. Hence,

*T*(

*x**) = {

*x**}. So, we have

*x** ∈

*F*(

*T*). Next, we show

*F*(

*T*) is convex. Let

*x*,

*y*∈

*F*(

*T*) and

*t*∈ (0, 1), put

*p*=

*tx*+ (1 -

*t*)

*y*. We show

*p*∈

*F*(

*T*). Let

*w*∈

*T*(

*p*), we have

By Lemma 1.2, we obtain *p* = *w*. Hence, *T*(*p*) = {*p*}. So, we have *p* ∈ *F*(*T*). Therefore, *F*(*T*) is convex. □

**Remark 2.2**. Let *X* be a strictly convex and smooth Banach space, and *D* a nonempty closed convex subset of *X*. Suppose *T* : *D* → *N*(*D*) is a relatively nonexpansive multi-valued mapping. If *p* ∈ *F*(*T*), then *T*(*p*) = {*p*}.

**Proposition 2.3**. *Let X be a uniformly convex and smooth Banach space, and D a nonempty closed convex subset of X. Suppose T* : *D* → *N*(*D*) *is a relatively nonexpansive multi-valued mapping. Let* {*α*_{
n
} } *be a sequence of real numbers such that* 0 ≤ *α*_{
n
} ≤ 1 *for all n* ∈ ℕ*. For a given x*_{1} ∈ *D, let* {*x*_{
n
} } *be the iterative sequence defined by (1.5). Then*, {Π_{
F
}_{(T)}*x*_{
n
} } *converges strongly to a fixed point of T, where* Π_{
F
}_{(T)}*is the generalized projection from D onto F*(*T*).

*Proof*. By Proposition 2.1,

*F*(

*T*) is closed and convex. So, we can define the generalized projection Π

_{ F }

_{(T)}onto

*F*(

*T*). Let

*p*∈

*F*(

*T*). From Lemma 1.4, we have

_{ n }

_{→ ∞}

*ϕ*(

*p*,

*x*

_{ n }) exists. So, {

*ϕ*(

*p*,

*x*

_{ n })} is bounded. Then, by (1.2) we have {

*x*

_{ n }} is bounded, and hence, {

*z*

_{ n }} is bounded. Let

*u*

_{ n }= Π

_{ F }

_{(T)}

*x*

_{ n }, for all

*n*∈ ℕ. Then, we have

*m*∈ ℕ. From Lemma 1.4, we obtain

*ϕ*(

*u*

_{ n },

*x*

_{ n })} converges. From

*u*

_{ n }

_{+}

_{ m }= Π

_{ F }

_{(T)}

*x*

_{ n }

_{+}

_{ m }and Lemma 1.4, we have

*m*,

*n*∈ ℕ. Let

*r*= sup

_{ n }

_{∈ℕ}||

*u*

_{ n }||. From Lemma 1.3, there exists a continuous, strictly increasing and convex function

*g*: [0, ∞) → [0, ∞) with

*g*(0) = 0 such that

for all *m*, *n* ∈ ℕ, *n* > *m*. Therefore, {*u*_{
n
} } is a Cauchy sequence. Since *X* is complete and *F*(*T*) is closed, there exists *q* ∈ *F*(*T*) such that {*u*_{
n
} } converges strongly to *q*. □

If the duality mapping *J* is weakly sequentially continuous, we have the following weak convergence theorem.

**Theorem 2.4**. *Let X be a uniformly convex and uniformly smooth Banach space, and D a nonempty closed convex subset of X. Suppose T* : *D* → *N*(*D*) *is a relatively nonexpansive multi-valued mapping. Let* {*α*_{
n
} } *be a sequence of real numbers such that* 0 ≤ *α*_{
n
} ≤ 1 *for all n* ∈ ℕ *and* lim inf_{
n
}_{→∞}*α*_{
n
} (1 - *α*_{
n
} ) > 0*. For a given x*_{1} ∈ *D, let* {*x*_{
n
} } *be the iterative sequence defined by (1.5). If J is weakly sequentially continuous, then* {*x*_{
n
} } *converges weakly to a fixed point of T*.

*Proof*. As in the proof of Proposition 2.3, {

*x*

_{ n }} and {

*z*

_{ n }} are bounded. So, there exists

*r*> 0 such that

*x*

_{ n },

*z*

_{ n }∈

*B*

_{ r }for all

*n*∈ ℕ. Since

*X*is a uniformly smooth Banach space,

*X** is a uniformly convex Banach space. Let

*p*∈

*F*(

*T*). By Lemma 1.1, there exists a continuous, strictly increasing and convex function

*g*: [0, ∞) → [0, ∞) with

*g*(0) = 0 such that

_{ n }

_{→∞}

*ϕ*(

*p*,

*x*

_{ n }) exists and lim inf

_{ n }

_{→∞}

*α*

_{ n }(1 -

*α*

_{ n }) > 0, we obtain

*J*

^{-1}is uniformly norm-to-norm continuous on bounded sets, we have

*d*(

*x*

_{ n },

*T*(

*x*

_{ n })) ≤ ||

*x*

_{ n }-

*z*

_{ n }||, we obtain

*u*

_{ n }= Π

_{ F }(

_{ T })

*x*

_{ n }. By Lemma 1.5, we have

*w*∈

*F*(

*T*). From Proposition 2.3, there exists

*p*∈

*F*(

*T*) such that {

*u*

_{ n }} converges strongly to

*p*. Let $\left\{{x}_{{n}_{j}}\right\}$ be a subsequence of {

*x*

_{ n }} such that $\left\{{x}_{{n}_{j}}\right\}$ converges weakly to

*q*. Then, by (2.11) we have

*q*∈

*F*(

*T*). It follows from (2.12) that

*j*→ ∞ in inequality (2.13), since

*J*is weakly sequentially continuous we have

*J*is monotone, we have

Since *X* is strictly convex, we have *p* = *q*. Therefore, {*x*_{
n
} } converges weakly to *p*. The proof is complete. □

**Theorem 2.5**. *Let X be a uniformly convex and uniformly smooth Banach space, and D a nonempty closed convex subset of X. Suppose T* : *D* → *N(D) is a relatively nonexpansive multi-valued mapping. Let* {*α*_{
n
} } *be a sequence of real numbers such that* 0 ≤ *α*_{
n
} ≤ 1 *for all n* ∈ ℕ *and* lim inf_{
n
}_{→∞}*α*_{
n
} (1 - *α*_{
n
} ) > 0*. For a given x*_{1} ∈ *D, let* {*x*_{
n
} } *be the iterative sequence defined by (1.5). If the interior of F*(*T*) *is nonempty, then* {*x*_{
n
} } *converges strongly to a fixed point of T*.

*Proof*. Since the interior of

*F*(

*T*) is nonempty, there exists

*p*∈

*F*(

*T*) and

*r*> 0 such that

*p*+

*rh*∈

*F*(

*T*), whenever ||

*h*|| ≤ 1. By (1.3) for any

*q*∈

*F*(

*T*) we have

*p*+

*rh*∈

*F*(

*T*), as in the proof of Proposition 2.3, we have

*h*|| ≤ 1. Therefore, we obtain

for all *m*, *n* ∈ ℕ, *n* > *m*. As in the proof of Proposition 2.3, {*ϕ*(*p*, *x*_{
n
} )} converges. Hence, {*J*(*x*_{
n
} )} is a Cauchy sequence. Since *X** is complete, {*J*(*x*_{
n
} )} converges strongly to a point in *X**. Since *X** has a Fréchet differentiable norm, then *J*^{-1} is norm-to-norm continuous on *X**. Hence, {*x*_{
n
} } converges strongly to some point *u* in *D*. As in the proof of Theorem 2.4, lim_{
n
}_{→∞}*d*(*x*_{
n
} , *T*(*x*_{
n
} )) = 0. Hence, we have *u* ∈ *F*(*T*), where *u* = lim_{
n
}_{→∞} Π_{
F
}(_{
T
})*x*_{
n
} . □

## Declarations

### Acknowledgements

AR would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Tehran, Iran, for supporting this research (Grant No. 90470122).

## Authors’ Affiliations

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