Open Access

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

Fixed Point Theory and Applications20112011:73

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

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

multi-valued mappingrelatively nonexpansivefixed pointiterative sequence

## 1 Introduction and preliminaries

Let D be a nonempty closed convex subset of a real Banach space X. A single-valued mapping T : DD 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
$H\left({A}_{1},{A}_{2}\right)=max\left\{\underset{x\in {A}_{1}}{sup}d\left(x,{A}_{2}\right),\underset{y\in {A}_{2}}{sup}d\left(y,{A}_{1}\right)\right\},$

for A1, A2 CB(D), where d(x, A1) = inf {||x - y||; y A1}. The multi-valued mapping T : DCB(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 : DN(D) (respectively, T : DD) if p F(T) (respectively, T(p) = p). The set of fixed points of T is represented by F(T).

Let X be a real Banach space with dual X*. We denote by J the normalized duality mapping from X to ${2}^{{X}^{*}}$ defined by
$J\left(x\right):=\left\{{f}^{*}\in {X}^{*}:〈x,{f}^{*}〉\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\parallel x{\parallel }^{2}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\parallel {f}^{*}{\parallel }^{2}\right\},$

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 xy. 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 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
$\parallel \alpha x+\beta y{\parallel }^{2}\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}\alpha \parallel x{\parallel }^{2}+\phantom{\rule{2.77695pt}{0ex}}\beta \parallel y{\parallel }^{2}-\phantom{\rule{2.77695pt}{0ex}}\alpha \beta g\left(\parallel x-y\parallel \right),$

for all x, y B r and all α, β [0, 1] with α + β = 1.

The norm of Banach space X is said to be Gâteaux differentiable if for each x, y S(X):= {x X : ||x|| = 1} the limit
$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t},$
(1.1)
exists. In this case, 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. 1.

X (X*, resp.) is uniformly convex if and only if X* (X, resp.) is uniformly smooth;

2. 2.

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

3. 3.

If X is reflexive, then J is onto;

4. 4.

If X is strictly convex, then J(x) ∩ J(y) = for all xy;

5. 5.

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

6. 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 $J\left({x}_{n}\right)\phantom{\rule{2.77695pt}{0ex}}\stackrel{*}{⇀}\phantom{\rule{2.77695pt}{0ex}}J\left(x\right)$, where denotes the weak convergence and $\stackrel{*}{⇀}$ denotes the weak* convergence.

Let X be a smooth Banach space. The function ϕ : X × X is defined by
$\varphi \left(x,y\right)=\phantom{\rule{2.77695pt}{0ex}}\parallel x{\parallel }^{2}-\phantom{\rule{2.77695pt}{0ex}}2〈x,J\left(y\right)〉\phantom{\rule{2.77695pt}{0ex}}+\phantom{\rule{2.77695pt}{0ex}}\parallel y{\parallel }^{2},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall x,y\in X.$
It is obvious from the definition of the function ϕ that
${\left(\parallel x\parallel -\parallel y\parallel \right)}^{2}\le \phantom{\rule{2.77695pt}{0ex}}\varphi \left(x,y\right)\le {\left(\parallel x\parallel +\parallel y\parallel \right)}^{2},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall x,y\in X.$
(1.2)
In addition, the function ϕ has the following property:
$\varphi \left(y,x\right)=\varphi \left(z,x\right)+\varphi \left(y,z\right)+2〈z-y,J\left(x\right)-J\left(z\right)〉,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall x,y,z\in X.$
(1.3)

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
$g\left(\parallel y-z\parallel \right)\le \varphi \left(y,z\right),$

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.

Let 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 : DD[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 $\stackrel{^}{F}\left(T\right)$. A mapping T : DD is called relatively nonexpansive [3, 68], if the following conditions are satisfied:
1. 1.

F(T) is nonempty;

2. 2.

ϕ(p, T(x)) ≤ ϕ(p, x), x D, p F(T);

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

Let 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 x0 D such that
$\varphi \left({x}_{0},x\right)=\underset{y\in D}{\text{min}}\varphi \left(y,x\right).$

Following Alber [9], we denote such an element x0 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
$\varphi \left(x,{\Pi }_{D}y\right)+\varphi \left({\Pi }_{D}y,y\right)\le \varphi \left(x,y\right),\phantom{\rule{1em}{0ex}}\forall x\in D,y\in X.$
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
$z={\Pi }_{D}x\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}⇔\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}〈z-y,J\left(x\right)-J\left(z\right)〉\ge 0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall y\in D.$
In 2004, Matsushita and Takahashi [10] introduced the following iterative sequence for finding a fixed point of relatively nonexpansive mapping T : DD. Given x1 D,
${x}_{n+1}={\Pi }_{D}{J}^{-1}\left({\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left(T\left({x}_{n}\right)\right)\right),$
(1.4)

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 [1114].

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 : DN(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 : DN(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.$\stackrel{^}{F}\left(T\right)=F\left(T\right)$,

where$\stackrel{^}{F}\left(T\right)$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 : DCB(D) be defined by
$T\left(f\right)=\left\{\begin{array}{cc}\hfill \left\{g\in D;f\left(x\right)-\frac{3}{4}\le g\left(x\right)\le f\left(x\right)-\frac{1}{4},\forall x\in I\right\},\hfill & \hfill f\left(x\right)>1,\forall x\in I;\hfill \\ \hfill \left\{0\right\},\hfill & \hfill \mathsf{\text{otherwise}}.\hfill \end{array}\right\$
It is clear that F(T) = {0}. Let $h\in \stackrel{^}{F}\left(T\right)$. 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 , we have
${z}_{n}=\left\{\begin{array}{cc}\hfill \frac{1}{4},\hfill & \hfill {f}_{n}\left(x\right)>1,\forall x\in I;\hfill \\ \hfill \parallel {f}_{n}{\parallel }_{p},\hfill & \hfill \mathsf{\text{otherwise}}.\hfill \end{array}\right\$
Since z n → 0, we have ||f n || p → 0. Therefore, f n → 0. Hence, h = 0. Therefore,$\stackrel{^}{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
$\begin{array}{cc}\hfill \varphi \left(0,g\right)\hfill & \hfill =\phantom{\rule{1em}{0ex}}\parallel g{\parallel }_{p}^{2}\hfill \\ \hfill \le \phantom{\rule{1em}{0ex}}\parallel f{\parallel }_{p}^{2}\hfill \\ \hfill =\phantom{\rule{1em}{0ex}}\varphi \phantom{\rule{0.3em}{0ex}}\left(0,f\right).\hfill \end{array}$
Next, let f D such that there exists x I such that f(x) ≤ 1, then
$\begin{array}{cc}\hfill \varphi \left(0,0\right)\hfill & \hfill =\phantom{\rule{2.77695pt}{0ex}}0\hfill \\ \hfill \le \phantom{\rule{2.77695pt}{0ex}}\parallel f{\parallel }_{p}^{2}\hfill \\ \hfill =\phantom{\rule{2.77695pt}{0ex}}\varphi \left(0,f\right).\hfill \end{array}$

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.

In this article, inspired by Matsushita and Takahashi [10], we introduce the following iterative sequence for finding a fixed point of relatively nonexpansive multi-valued mapping T : DN(D). Given x1 D,
${x}_{n+1}={\Pi }_{D}{J}^{-1}\left({\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left({z}_{n}\right)\right),$
(1.5)

where z n T(x n ) for all 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 : DN(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
$\varphi \left({x}_{n},z\right)\le \varphi \left({x}_{n},{x}^{*}\right),$
for all z T(x*) and for all n . Therefore,
$\begin{array}{ll}\hfill \varphi \left({x}^{*},z\right)& ={\mathsf{\text{lim}}}_{n\to \infty }\varphi \left({x}_{n},z\right)\phantom{\rule{2em}{0ex}}\\ \le {\mathsf{\text{lim}}}_{n\to \infty }\varphi \left({x}_{n},{x}^{*}\right)\phantom{\rule{2em}{0ex}}\\ =\varphi \left({x}^{*},{x}^{*}\right)\phantom{\rule{2em}{0ex}}\\ =0.\phantom{\rule{2em}{0ex}}\end{array}$
(2.1)
By Lemma 1.2, we obtain 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
$\begin{array}{ll}\hfill \varphi \left(p,w\right)& =\parallel p{\parallel }^{2}-2〈p,J\left(w\right)〉\phantom{\rule{0.3em}{0ex}}+\parallel w{\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ =\parallel p{\parallel }^{2}-2〈tx+\left(1-t\right)y,J\left(w\right)〉\phantom{\rule{0.3em}{0ex}}+\parallel w{\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ =\parallel p{\parallel }^{2}-2t〈x,J\left(w\right)〉-2\left(1-t\right)〈y,J\left(w\right)〉\phantom{\rule{0.3em}{0ex}}+\parallel w{\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ =\parallel p{\parallel }^{2}+t\varphi \left(x,w\right)+\left(1-t\right)\varphi \left(y,w\right)-t\parallel x{\parallel }^{2}-\left(1-t\right)\parallel y{\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ \le \parallel p{\parallel }^{2}+t\varphi \left(x,p\right)+\left(1-t\right)\varphi \left(y,p\right)-t\parallel x{\parallel }^{2}-\left(1-t\right)\parallel y{\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ =\parallel p{\parallel }^{2}-2〈tx+\left(1-t\right)y,J\left(p\right)〉+\parallel p{\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ =\parallel p{\parallel }^{2}-2〈p,J\left(p\right)〉\phantom{\rule{0.3em}{0ex}}+\parallel p{\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ =0.\phantom{\rule{2em}{0ex}}\end{array}$
(2.2)

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 : DN(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 : DN(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 . For a given x1 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
$\begin{array}{ll}\hfill \varphi \left(p,{x}_{n+1}\right)& =\varphi \left(p,{\Pi }_{D}{J}^{-1}\left({\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left({z}_{n}\right)\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \varphi \left(p,{J}^{-1}\left({\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left({z}_{n}\right)\right)\right)\phantom{\rule{2em}{0ex}}\\ =\parallel p{\parallel }^{2}-2〈p,{\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left({z}_{n}\right)〉\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\parallel {\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left({z}_{n}\right){\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ \le \parallel p{\parallel }^{2}-2{\alpha }_{n}〈p,J\left({x}_{n}\right)〉-2\left(1-{\alpha }_{n}\right)〈p,J\left({z}_{n}\right)〉+{\alpha }_{n}\parallel {x}_{n}{\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\left(1-{\alpha }_{n}\right)\parallel {z}_{n}{\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ ={\alpha }_{n}\varphi \left(p,{x}_{n}\right)+\left(1-{\alpha }_{n}\right)\varphi \left(p,{z}_{n}\right)\phantom{\rule{2em}{0ex}}\\ \le {\alpha }_{n}\varphi \left(p,{x}_{n}\right)+\left(1-{\alpha }_{n}\right)\varphi \left(p,{x}_{n}\right)\phantom{\rule{2em}{0ex}}\\ =\varphi \left(p,{x}_{n}\right).\phantom{\rule{2em}{0ex}}\\ \hfill \text{(10)}\end{array}$
(2.3)
Hence, lim 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 . Then, we have
$\varphi \left({u}_{n},{x}_{n+1}\right)\le \varphi \left({u}_{n},{x}_{n}\right).$
(2.4)
Therefore
$\varphi \left({u}_{n},{x}_{n+m}\right)\le \varphi \left({u}_{n},{x}_{n}\right),$
(2.5)
for all m . From Lemma 1.4, we obtain
$\begin{array}{ll}\hfill \varphi \left({u}_{n+1},{x}_{n+1}\right)& =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\varphi \left({\Pi }_{F\left(T\right)}{x}_{n+1},{x}_{n+1}\right)\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\varphi \left({u}_{n},{x}_{n+1}\right)-\varphi \left({u}_{n},{\Pi }_{F\left(T\right)}{x}_{n+1}\right).\phantom{\rule{2em}{0ex}}\end{array}$
(2.6)
By (2.4) and (2.6) we have
$\varphi \left({u}_{n+1},{x}_{n+1}\right)\le \varphi \left({u}_{n},{x}_{n}\right).$
(2.7)
It follows that {ϕ(u n , x n )} converges. From u n + m = Π F (T)x n + m and Lemma 1.4, we have
$\varphi \left({u}_{n},{u}_{n+m}\right)+\varphi \left({u}_{n+m},{x}_{n+m}\right)\le \varphi \left({u}_{n},{x}_{n+m}\right).$
Hence, by (2.5) we obtain
$\varphi \left({u}_{n},{u}_{n+m}\right)\le \varphi \left({u}_{n},{x}_{n}\right)-\varphi \left({u}_{n+m},{x}_{n+m}\right),$
(2.8)
for all m, n . Let r = sup 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
$\begin{array}{ll}\hfill g\left(\parallel {u}_{m}-{u}_{n}\parallel \right)& \le \varphi \left({u}_{m},{u}_{n}\right)\phantom{\rule{2em}{0ex}}\\ \le \varphi \left({u}_{m},{x}_{m}\right)-\varphi \left({u}_{n},{x}_{n}\right),\phantom{\rule{2em}{0ex}}\end{array}$
(2.9)

for all m, 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 : DN(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 and lim inf n →∞α n (1 - α n ) > 0. For a given x1 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 . 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
$\begin{array}{ll}\hfill \varphi \left(p,{x}_{n+1}\right)& =\varphi \left(p,{\Pi }_{D}{J}^{-1}\left({\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left({z}_{n}\right)\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \varphi \left(p,{J}^{-1}\left({\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left({z}_{n}\right)\right)\right)\phantom{\rule{2em}{0ex}}\\ =\parallel p{\parallel }^{2}-2〈p,{\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left({z}_{n}\right)〉\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\parallel {\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left({z}_{n}\right){\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ \le \parallel p{\parallel }^{2}-2{\alpha }_{n}〈p,J\left({x}_{n}\right)〉-2\left(1-{\alpha }_{n}\right)〈p,J\left({z}_{n}\right)〉+{\alpha }_{n}\parallel {x}_{n}{\parallel }^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\left(1-{\alpha }_{n}\right)\parallel {z}_{n}{\parallel }^{2}-{\alpha }_{n}\left(1-{\alpha }_{n}\right)g\left(\parallel J\left({x}_{n}\right)-J\left({z}_{n}\right)\parallel \right)\phantom{\rule{2em}{0ex}}\\ ={\alpha }_{n}\varphi \left(p,{x}_{n}\right)+\left(1-{\alpha }_{n}\right)\varphi \left(p,{z}_{n}\right)-{\alpha }_{n}\left(1-{\alpha }_{n}\right)g\left(\parallel J\left({x}_{n}\right)-J\left({z}_{n}\right)\parallel \right)\phantom{\rule{2em}{0ex}}\\ \le \varphi \left(p,{x}_{n}\right)-{\alpha }_{n}\left(1-{\alpha }_{n}\right)g\left(\parallel J\left({x}_{n}\right)-J\left({z}_{n}\right)\parallel \right).\phantom{\rule{2em}{0ex}}\end{array}$
(2.10)
Hence
${\alpha }_{n}\left(1-{\alpha }_{n}\right)g\left(\parallel J\left({x}_{n}\right)-J\left({z}_{n}\right)\parallel \right)\le \varphi \left(p,{x}_{n}\right)-\varphi \left(p,{x}_{n+1}\right).$
Since lim n →∞ϕ(p, x n ) exists and lim inf n →∞α n (1 - α n ) > 0, we obtain
$\underset{n\to \infty }{lim}\phantom{\rule{0.3em}{0ex}}g\left(\parallel J\left({x}_{n}\right)-J\left({z}_{n}\right)\parallel \right)=0.$
Therefore,
$\underset{n\to \infty }{lim}\parallel J\left({x}_{n}\right)-J\left({z}_{n}\right)\parallel \phantom{\rule{0.3em}{0ex}}=0.$
Since J-1 is uniformly norm-to-norm continuous on bounded sets, we have
$\underset{n\to \infty }{lim}\parallel {x}_{n}-{z}_{n}\parallel \phantom{\rule{0.3em}{0ex}}=0.$
Since d(x n , T(x n )) ≤ ||x n - z n ||, we obtain
$\underset{n\to \infty }{lim}\phantom{\rule{0.3em}{0ex}}d\left({x}_{n},T\left({x}_{n}\right)\right)=0.$
(2.11)
Let u n = Π F ( T ) x n . By Lemma 1.5, we have
$〈{u}_{n}-w,J\left({x}_{n}\right)-J\left({u}_{n}\right)〉\ge 0,$
(2.12)
for each 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
$〈{u}_{{n}_{j}}-q,J\left({x}_{{n}_{j}}\right)-J\left({u}_{{n}_{j}}\right)〉\ge 0.$
(2.13)
Let j → ∞ in inequality (2.13), since J is weakly sequentially continuous we have
$〈p-q,J\left(q\right)-J\left(p\right)〉\ge 0.$
(2.14)
Since J is monotone, we have
$〈q-p,J\left(q\right)-J\left(p\right)〉\ge 0.$
(2.15)
It follows from (2.14) and (2.15) that
$〈q-p,J\left(q\right)-J\left(p\right)〉=0.$
(2.16)

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 : DN(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 and lim inf n →∞α n (1 - α n ) > 0. For a given x1 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
$\varphi \left(q,{x}_{n}\right)=\varphi \left({x}_{n+1},{x}_{n}\right)+\varphi \left(q,{x}_{n+1}\right)+2〈{x}_{n+1}-q,J\left({x}_{n}\right)-J\left({x}_{n+1}\right)〉.$
(2.17)
Therefore,
$\frac{1}{2}\left(\varphi \left(q,{x}_{n}\right)-\varphi \left(q,{x}_{n+1}\right)\right)=\frac{1}{2}\varphi \left({x}_{n+1},{x}_{n}\right)+〈{x}_{n+1}-q,J\left({x}_{n}\right)-J\left({x}_{n+1}\right)〉.$
(2.18)
Since p + rh F(T), as in the proof of Proposition 2.3, we have
$\varphi \left(p+rh,{x}_{n+1}\right)\le \varphi \left(p+rh,{x}_{n}\right).$
(2.19)
It follows from (2.18) and (2.19) that
$\frac{1}{2}\varphi \left({x}_{n+1},{x}_{n}\right)+〈{x}_{n+1}-\left(p+rh\right),J\left({x}_{n}\right)-J\left({x}_{n+1}\right)〉\ge 0.$
(2.20)
Then, by (2.18) and (2.20) we have
$\begin{array}{c}〈h,J\left({x}_{n}\right)-J\left({x}_{n+1}\right)〉\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\frac{1}{r}\left(〈{x}_{n+1}-p,J\left({x}_{n}\right)-J\left({x}_{n+1}\right)〉+\frac{1}{2}\varphi \left({x}_{n+1},{x}_{n}\right)\right)\\ =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\frac{1}{2r}\left(\varphi \left(p,{x}_{n}\right)-\varphi \left(p,{x}_{n+1}\right)\right),\end{array}$
(2.21)
whenever ||h|| ≤ 1. Therefore, we obtain
$\parallel J\left({x}_{n}\right)-J\left({x}_{n+1}\right)\parallel \phantom{\rule{0.3em}{0ex}}\le \frac{1}{2r}\left(\varphi \left(p,{x}_{n}\right)-\varphi \left(p,{x}_{n+1}\right)\right).$
It follows that
$\begin{array}{ll}\hfill \parallel J\left({x}_{m}\right)-J\left({x}_{n}\right)\parallel & \le {\Sigma }_{i=m}^{n-1}\parallel J\left({x}_{i}\right)-J\left({x}_{i+1}\right)\parallel \phantom{\rule{2em}{0ex}}\\ \le {\Sigma }_{i=m}^{n-1}\frac{1}{2r}\left(\varphi \left(p,{x}_{i}\right)-\varphi \left(p,{x}_{i+1}\right)\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{2r}\left(\varphi \left(p,{x}_{m}\right)-\varphi \left(p,{x}_{n}\right)\right),\phantom{\rule{2em}{0ex}}\end{array}$
(2.22)

for all m, 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

(1)
Department of Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran
(2)
School of Mathematics, Institute for Research in Fundamental Sciences, Tehran, Iran

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