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

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.

1 Introduction and preliminaries

Let 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

$$H\left(A_1,A_2\right)=max\left\{\underset{x\in A_1}{sup}d\left(x,A_2\right),\underset{y\in A_2}{sup}d\left(y,A_1\right)\right\},$$

for A₁, A₂ ∈ CB(D), where d(x, A₁) = inf {||x - y||; y ∈ A₁}. The multi-valued mapping T : 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).

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

$$J\left(x\right):=\left\{f^{*}\in X^{*}:〈x,f^{*}〉=\parallel x{\parallel }^2=\parallel f^{*}{\parallel }^2\right\},$$

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 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

$$\parallel \alpha x+\beta y{\parallel }^2\le \alpha \parallel x{\parallel }^2+\beta \parallel y{\parallel }^2-\alpha \beta g\left(\parallel x-y\parallel \right),$$

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. 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 $J\left(x_n\right)\stackrel{*}{\rightharpoonup }J\left(x\right)$, where ⇀ denotes the weak convergence and $\stackrel{*}{\rightharpoonup }$ denotes the weak* convergence.

Let X be a smooth Banach space. The function ϕ : X × X → ℝ is defined by

$$\varphi \left(x,y\right)=\parallel x{\parallel }^2-2〈x,J\left(y\right)〉+\parallel y{\parallel }^2,\forall x,y\in X.$$

It is obvious from the definition of the function ϕ that

$${\left(\parallel x\parallel -\parallel y\parallel \right)}^2\le \varphi \left(x,y\right)\le {\left(\parallel x\parallel +\parallel y\parallel \right)}^2,\forall x,y\in X.$$

(1.2)

In addition, the function ϕ has the following property:

$$\varphi \left(y,x\right)=\varphi \left(z,x\right)+\varphi \left(y,z\right)+2〈z-y,J\left(x\right)-J\left(z\right)〉,\forall x,y,z\in X.$$

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

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

$$\varphi \left(x,{\Pi }_{D}y\right)+\varphi \left({\Pi }_{D}y,y\right)\le \varphi \left(x,y\right),\forall x\in D,y\in X.$$

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

$$z={\Pi }_{D}x\iff 〈z-y,J\left(x\right)-J\left(z\right)〉\ge 0,\forall y\in D.$$

In 2004, Matsushita and Takahashi [10] introduced the following iterative sequence for finding a fixed point of relatively nonexpansive mapping T : D → D. Given x₁ ∈ D,

$$x_{n+1}={\Pi }_D{J}^{-1}\left({\alpha }_{n}J\left(x_n\right)+\left(1-{\alpha }_n\right)J\left(T\left(x_n\right)\right)\right),$$

(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 [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

$$T\left(f\right)=\left\{\begin{array}{cc}\hfill \left\{g\in D;f\left(x\right)-\frac{3}{4}\le g\left(x\right)\le f\left(x\right)-\frac{1}{4},\forall x\in I\right\},\hfill & \hfill f\left(x\right)>1,\forall x\in I;\hfill \\ \hfill \left\{0\right\},\hfill & \hfill \mathsf{\text{otherwise}}.\hfill \end{array}\right.$$

It is clear that F(T) = {0}. Let $h\in \widehat{F}\left(T\right)$. 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

$$z_n=\left\{\begin{array}{cc}\hfill \frac{1}{4},\hfill & \hfill f_n\left(x\right)>1,\forall x\in I;\hfill \\ \hfill \parallel f_n{\parallel }_p,\hfill & \hfill \mathsf{\text{otherwise}}.\hfill \end{array}\right.$$

Since z _n → 0, we have ||f _n || _p → 0. Therefore, f _n → 0. Hence, h = 0. Therefore,$\widehat{F}\left(T\right)=F\left(T\right)=\left\{0\right\}$. Let f ∈ D such that f(x) > 1 for all x ∈ I, and g ∈ T(f), then

$$\begin{array}{cc}\hfill \varphi \left(0,g\right)\hfill & \hfill =\parallel g{\parallel }_p^2\hfill \\ \hfill \le \parallel f{\parallel }_p^2\hfill \\ \hfill =\varphi \left(0,f\right).\hfill \end{array}$$

Next, let f ∈ D such that there exists x ∈ I such that f(x) ≤ 1, then

$$\begin{array}{cc}\hfill \varphi \left(0,0\right)\hfill & \hfill =0\hfill \\ \hfill \le \parallel f{\parallel }_p^2\hfill \\ \hfill =\varphi \left(0,f\right).\hfill \end{array}$$

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 : D → N(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 ∈ 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

$$\varphi \left(x_n,z\right)\le \varphi \left(x_n,x^{*}\right),$$

for all z ∈ T(x*) and for all n ∈ ℕ. Therefore,

$$\begin{array}{ll}\hfill \varphi \left(x^{*},z\right)& ={\mathsf{\text{lim}}}_{n\to \infty }\varphi \left(x_n,z\right)\\ \le {\mathsf{\text{lim}}}_{n\to \infty }\varphi \left(x_n,x^{*}\right)\\ =\varphi \left(x^{*},x^{*}\right)\\ =0.\end{array}$$

(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

$$\begin{array}{ll}\hfill \varphi \left(p,w\right)& =\parallel p{\parallel }^2-2〈p,J\left(w\right)〉+\parallel w{\parallel }^2\\ =\parallel p{\parallel }^2-2〈tx+\left(1-t\right)y,J\left(w\right)〉+\parallel w{\parallel }^2\\ =\parallel p{\parallel }^2-2t〈x,J\left(w\right)〉-2\left(1-t\right)〈y,J\left(w\right)〉+\parallel w{\parallel }^2\\ =\parallel p{\parallel }^2+t\varphi \left(x,w\right)+\left(1-t\right)\varphi \left(y,w\right)-t\parallel x{\parallel }^2-\left(1-t\right)\parallel y{\parallel }^2\\ \le \parallel p{\parallel }^2+t\varphi \left(x,p\right)+\left(1-t\right)\varphi \left(y,p\right)-t\parallel x{\parallel }^2-\left(1-t\right)\parallel y{\parallel }^2\\ =\parallel p{\parallel }^2-2〈tx+\left(1-t\right)y,J\left(p\right)〉+\parallel p{\parallel }^2\\ =\parallel p{\parallel }^2-2〈p,J\left(p\right)〉+\parallel p{\parallel }^2\\ =0.\end{array}$$

(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 : 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 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

$$\begin{array}{ll}\hfill \varphi \left(p,x_{n+1}\right)& =\varphi \left(p,{\Pi }_D{J}^{-1}\left({\alpha }_{n}J\left(x_n\right)+\left(1-{\alpha }_n\right)J\left(z_n\right)\right)\right)\\ \le \varphi \left(p,J^{-1}\left({\alpha }_{n}J\left(x_n\right)+\left(1-{\alpha }_n\right)J\left(z_n\right)\right)\right)\\ =\parallel p{\parallel }^2-2〈p,{\alpha }_{n}J\left(x_n\right)+\left(1-{\alpha }_n\right)J\left(z_n\right)〉\\ +\parallel {\alpha }_{n}J\left(x_n\right)+\left(1-{\alpha }_n\right)J\left(z_n\right){\parallel }^2\\ \le \parallel p{\parallel }^2-2{\alpha }_n〈p,J\left(x_n\right)〉-2\left(1-{\alpha }_n\right)〈p,J\left(z_n\right)〉+{\alpha }_n\parallel x_n{\parallel }^2\\ +\left(1-{\alpha }_n\right)\parallel z_n{\parallel }^2\\ ={\alpha }_n\varphi \left(p,x_n\right)+\left(1-{\alpha }_n\right)\varphi \left(p,z_n\right)\\ \le {\alpha }_n\varphi \left(p,x_n\right)+\left(1-{\alpha }_n\right)\varphi \left(p,x_n\right)\\ =\varphi \left(p,x_n\right).\\ \hfill \text{(10)}\end{array}$$

(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 ∈ 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 ∈ ℕ. From Lemma 1.4, we obtain

$$\begin{array}{ll}\hfill \varphi \left(u_{n+1},x_{n+1}\right)& =\varphi \left({\Pi }_{F\left(T\right)}{x}_{n+1},x_{n+1}\right)\\ \le \varphi \left(u_n,x_{n+1}\right)-\varphi \left(u_n,{\Pi }_{F\left(T\right)}{x}_{n+1}\right).\end{array}$$

(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 ∈ 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

$$\begin{array}{ll}\hfill g\left(\parallel u_m-u_n\parallel \right)& \le \varphi \left(u_m,u_n\right)\\ \le \varphi \left(u_m,x_m\right)-\varphi \left(u_n,x_n\right),\end{array}$$

(2.9)

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 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 ∈ 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

$$\begin{array}{ll}\hfill \varphi \left(p,x_{n+1}\right)& =\varphi \left(p,{\Pi }_D{J}^{-1}\left({\alpha }_{n}J\left(x_n\right)+\left(1-{\alpha }_n\right)J\left(z_n\right)\right)\right)\\ \le \varphi \left(p,J^{-1}\left({\alpha }_{n}J\left(x_n\right)+\left(1-{\alpha }_n\right)J\left(z_n\right)\right)\right)\\ =\parallel p{\parallel }^2-2〈p,{\alpha }_{n}J\left(x_n\right)+\left(1-{\alpha }_n\right)J\left(z_n\right)〉\\ +\parallel {\alpha }_{n}J\left(x_n\right)+\left(1-{\alpha }_n\right)J\left(z_n\right){\parallel }^2\\ \le \parallel p{\parallel }^2-2{\alpha }_n〈p,J\left(x_n\right)〉-2\left(1-{\alpha }_n\right)〈p,J\left(z_n\right)〉+{\alpha }_n\parallel x_n{\parallel }^2\\ +\left(1-{\alpha }_n\right)\parallel z_n{\parallel }^2-{\alpha }_n\left(1-{\alpha }_n\right)g\left(\parallel J\left(x_n\right)-J\left(z_n\right)\parallel \right)\\ ={\alpha }_n\varphi \left(p,x_n\right)+\left(1-{\alpha }_n\right)\varphi \left(p,z_n\right)-{\alpha }_n\left(1-{\alpha }_n\right)g\left(\parallel J\left(x_n\right)-J\left(z_n\right)\parallel \right)\\ \le \varphi \left(p,x_n\right)-{\alpha }_n\left(1-{\alpha }_n\right)g\left(\parallel J\left(x_n\right)-J\left(z_n\right)\parallel \right).\end{array}$$

(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

$$\underset{n\to \infty }{lim}g\left(\parallel J\left(x_n\right)-J\left(z_n\right)\parallel \right)=0.$$

Therefore,

$$\underset{n\to \infty }{lim}\parallel J\left(x_n\right)-J\left(z_n\right)\parallel =0.$$

Since J^-1 is uniformly norm-to-norm continuous on bounded sets, we have

$$\underset{n\to \infty }{lim}\parallel x_n-z_n\parallel =0.$$

Since d(x _n , T(x _n )) ≤ ||x _n - z _n ||, we obtain

$$\underset{n\to \infty }{lim}d\left(x_n,T\left(x_n\right)\right)=0.$$

(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 : 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 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

$$\begin{array}{c}〈h,J\left(x_n\right)-J\left(x_{n+1}\right)〉\le \frac{1}{r}\left(〈x_{n+1}-p,J\left(x_n\right)-J\left(x_{n+1}\right)〉+\frac{1}{2}\varphi \left(x_{n+1},x_n\right)\right)\\ =\frac{1}{2r}\left(\varphi \left(p,x_n\right)-\varphi \left(p,x_{n+1}\right)\right),\end{array}$$

(2.21)

whenever ||h|| ≤ 1. Therefore, we obtain

$$\parallel J\left(x_n\right)-J\left(x_{n+1}\right)\parallel \le \frac{1}{2r}\left(\varphi \left(p,x_n\right)-\varphi \left(p,x_{n+1}\right)\right).$$

It follows that

$$\begin{array}{ll}\hfill \parallel J\left(x_m\right)-J\left(x_n\right)\parallel & \le {\Sigma }_{i=m}^{n-1}\parallel J\left(x_i\right)-J\left(x_{i+1}\right)\parallel \\ \le {\Sigma }_{i=m}^{n-1}\frac{1}{2r}\left(\varphi \left(p,x_i\right)-\varphi \left(p,x_{i+1}\right)\right)\\ =\frac{1}{2r}\left(\varphi \left(p,x_m\right)-\varphi \left(p,x_n\right)\right),\end{array}$$

(2.22)

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 . □