Strong convergence theorems for total quasi-ϕ-asymptotically nonexpansive multi-valued mappings in Banach spaces

Abstract

The main purpose of this article is to introduce the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and prove the strong convergence theorem in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. In order to get the theorems, the hybrid algorithms are presented and are used to approximate the fixed point. The results presented in this article improve and extend some recent results announced by some authors.

AMS (MOS) Subject Classification: 47J06, 47J25.

1 Introduction

Throughout this article, we always assume that X is a real Banach space with the dual X* and J : X → 2^Xis the normalized duality mapping defined by

$$J\left(x\right)=\left\{f^{*}\in X^{*}:<x,f^{*}>={||x||}^2={||f^{*}||}^2\right\},x\in E.$$

In the sequal, we use F(T) to denote the set of fixed points of a mapping T, and use ℛ and ℛ⁺ to denote the set of all real numbers and the set of all nonnegative real numbers, respectively. We denote by x _n → x and x _n ⇀ x the strong convergence and weak convergence of a sequence {x _n }, respectively.

A Banach space X is said to be strictly convex if $\frac{||x+y||}{2}<1$ for all x, y ∈ U = {z ∈ X : ||z|| = 1} with x ≠ y. X is said to be uniformly convex if, for each ϵ ∈ (0, 2], there exists δ > 0 such that $\frac{||x+y||}{2}<1-\delta$ for all x, y ∈ U with ||x - y|| ≥ ϵ. X is said to be smooth if the limit

$$\underset{t\to 0}{\mathsf{\text{lim}}}\frac{||x+ty||-||x||}{t}$$

exists for all x, y ∈ U. X is said to be uniformly smooth if the above limit is attained uniformly in x, y ∈ U.

Remark 1.1. The following basic properties of a Banach space X can be found in Cioranescu [1].

• (i) If X is uniformly smooth, then J is uniformly continuous on each bounded subset of X;

• (ii) If X is a reflexive and strictly convex Banach space, then J^-1 is norm-weak-continuous;

• (iii) If X is a smooth, strictly convex and reflexive Banach space, then J is single-valued, one-to-one and onto;

• (iv) A Banach space X is uniformly smooth if and only if X* is uniformly convex;

• (v) Each uniformly convex Banach space X has the Kadec-Klee property, i.e., for any sequence {x _n } ⊂ X, if x _n ⇀ x ∈ X and ||x _n || → ||x||, then x _n → x.

Let X be a smooth Banach space. We always use ϕ : X × X → ℛ⁺ to denote the Lyapunov functional defined by

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

(1.1)

It is obvious from the definition of the function ϕ that

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

(1.2)

Following Alber [2], the generalized projection Π _C : X → C is defined by

$${\Pi }_C\left(x\right)=\mathsf{\text{arg}}\underset{y\in C}{\mathsf{\text{inf}}}\varphi \left(y,x\right),\forall x\in X.$$

Lemma 1.2. [2] Let X be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of X. Then the following conclusions hold:

• (a) ϕ(x, Π _C y) + ϕ(Π _C y, y) ≤ ϕ(x, y) for all x ∈ C and y ∈ X;

• (b) If x ∈ X and z ∈ C, then

  $$z={\Pi }_{C}xiff〈z-y,Jx-Jz〉\ge 0,\forall y\in C;$$

• (c) For x, y ∈ X, ϕ(x, y) = 0 if and only if x = y.

Let X be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of X and T : C → C be a mapping. A point p ∈ C is said to be an asymptotic fixed point of T if there exists a sequence {x _n } ⊂ C such that x _n ⇀ p and ||x _n - Tx _n || → 0. We denoted the set of all asymptotic fixed points of T by $\tilde{\mathit{F}}\left(T\right)$.

Definition 1.3. (1) A mapping T : C → C is said to be relatively nonexpansive [3] if $F\left(T\right)\ne \mathrm{0̸},F\left(T\right)=F\left(\tilde{T}\right)$ and

$$\varphi \left(p,Tx\right)\le \varphi \left(p,x\right),\forall x\in C,p\in F\left(T\right).$$

• (2) A mapping T : C → C is said to be closed if, for any sequence {x _n } ⊂ C with x _n → x and Tx _n → y, then Tx = y.

Definition 1.4. (1) A mapping T : C → C is said to be quasi-ϕ-nonexpansive if $F\left(T\right)\ne \mathrm{0̸}$ and

$$\varphi \left(p,Tx\right)\le \varphi \left(p,x\right),\forall x\in C,p\in F\left(T\right).$$

• (2) A mapping T : C → C is said to be quasi-ϕ-asymptotically nonexpansive if $F\left(T\right)\ne \mathrm{0̸}$ and there exists a real sequence {k _n } ⊂ [1, ∞) with k _n → 1 such that

  $$\varphi \left(p,T^{n}x\right)\le k_n\varphi \left(p,x\right),\forall n\ge 1,x\in C,p\in F\left(T\right).$$

  (1.3)

• (3) A mapping T : C → C is said to be total quasi-ϕ-asymptotically nonexpansive if $F\left(T\right)\ne \mathrm{0̸}$ and there exist nonnegative real sequences {ν _n }, {μ _n } with ν _n → 0, μ _n → 0 (as n → ∞) and a strictly increasing continuous function ζ : ℛ⁺ → ℛ⁺ with ζ(0) = 0 such that for all x ∈ C, p ∈ F(T)

  $$\varphi \left(p,T^{n}x\right)\le \varphi \left(p,x\right)+{\nu }_n\varsigma \left(\varphi \left(p,x\right)\right)+{\mu }_n,\forall n\ge 1.$$

  (1.4)

Remark 1.5. From the definitions, it is easy to know that

• (1) Taking ζ(t) = t, t ≥ 0, ν _n = k _n - 1 and μ _n = 0, then ν _n → 0(as n → ∞) and (1.3) can be rewritten as

  $$\varphi \left(p,T^{n}x\right)\le \varphi \left(p,x\right)+{\nu }_n\varsigma \left(\varphi \left(p,x\right)\right)+{\mu }_n,\forall n\ge 1,x\in C,p\in F\left(T\right).$$

  (1.5)

This implies that the class of total quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-asymptotically nonexpansive mappings as a subclass, but the converse is not true.

• (2) The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, but the converse is not true.

• (3) The class of quasi-ϕ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true.

Let C be a nonempty closed convex subset of a Banach space X. Let N(C) be the family of nonempty subsets of C.

Definition 1.6. (1) A multi-valued mapping T : C → N(C) is said to be relatively nonexpansive [3] if $F\left(T\right)\ne \mathrm{0̸},F\left(T\right)=F\left(\tilde{T}\right)$ and

$$\varphi \left(p,w\right)\le \varphi \left(p,x\right),\forall x\in C,w\in Tx,p\in F\left(T\right).$$

• (2) A multi-valued mapping T : C → N(C) is said to be closed if, for any sequence {x _n } ⊂ C with x _n → x and w _n ∈ T(x _n ) with w _n → y, then y ∈ Tx.

Definition 1.7. (1) A multi-valued mapping T : C → N(C) is said to be quasi-ϕ-nonexpansive if $F\left(T\right)\ne \mathrm{0̸}$ and

$$\varphi \left(p,w\right)\le \varphi \left(p,x\right),\forall x\in C,w\in Tx,p\in F\left(T\right).$$

• (2) A multi-valued mapping T : C → N(C) is said to be quasi-ϕ-asymptotically non-expansive if $F\left(T\right)\ne \mathrm{0̸}$ and there exists a real sequence {k _n } ⊂ [1, ∞) with k _n → 1 such that

  $$\varphi \left(p,w_n\right)\le k_n\varphi \left(p,x\right),\forall n\ge 1,x\in C,w_n\in T^{n}x,p\in F\left(T\right).$$

  (1.6)

• (3) A multi-valued mapping T : C → N(C) is said to be total quasi-ϕ-asymptotically nonexpansive if $F\left(T\right)\ne \mathrm{0̸}$ and there exist nonnegative real sequences {ν _n }, {μ _n } with ν _n → 0, μ _n → 0(as n → ∞) and a strictly increasing continuous function ζ : ℛ⁺ → ℛ⁺ with ζ(0) = 0 such that for all x ∈ C, p ∈ F(T)

  $$\varphi \left(p,w_n\right)\le \varphi \left(p,x\right)+{\nu }_n\varsigma \left(\varphi \left(p,x\right)\right)+{\mu }_n,\forall n\ge 1,w_n\in T^{n}x.$$

  (1.7)

• (4) A total quasi-ϕ-asymptotically nonexpansive multi-valued mapping T : C → N(C) is said to be uniformly L-Lipschitz continuous if there exists a constant L > 0 such that

  $$||w_n-s_n||\le L||x-y||,\forall x,y\in C,w_n\in T^{n}x,s_n\in T^{n}y,n\ge 1.$$

Remark 1.8. From the definitions, it is easy to know that

• (1) Taking ζ(t) = t, t ≥ 0, ν _n = k _n - 1 and μ _n = 0, then ν _n → 0 (as n → ∞) and (1.6) can be rewritten as

  $$\varphi \left(p,w_n\right)\le \varphi \left(p,x\right)+{\nu }_n\varsigma \left(\varphi \left(p,x\right)\right)+{\mu }_n,\forall n\ge 1,x\in C,w_n\in T^{n}x,p\in F\left(T\right).$$

This implies that the class of total quasi-ϕ-asymptotically nonexpansive multi-valued mappings contains properly the class of quasi-ϕ-asymptotically nonexpansive multi-valued mappings as a subclass, but the converse is not true.

• (2) The class of quasi-ϕ-asymptotically nonexpansive multi-valued mappings contains properly the class of quasi-ϕ-nonexpansive multi-valued mappings as a subclass, but the converse is not true.

• (3) The class of quasi-ϕ-nonexpansive multi-valued mappings contains properly the class of relatively nonexpansive multi-valued mappings as a subclass, but the converse is not true.

In 2005, Matsushita and Takahashi [3] proved weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space X. In 2008, Plubtieng and Ungchittrakool [4] proved the strong convergence theorems to approximate a fixed point of two relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space X. In 2010, Chang et al. [5] obtained the strong convergence theorem for an infinite family of quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space X with Kadec-Klee property. In 2011, Chang et al. [6] proved some approximation theorems of common fixed points for countable families of total quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space X with Kadec-Klee property. In 2011, Homaeipour and Razani [7] proved weak and strong convergence theorems for a single relatively nonexpansive multi-valued mapping in a uniformly convex and uniformly smooth Banach space X.

Motivated and inspired by the researches going on in this direction, the purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive multivalued mapping which contains many kinds of mappings as its special cases, and then by using the hybrid iterative algorithm to prove some strong convergence theorems in uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the article improve and extend some recent results announced by some authors.

2 Preliminaries

Lemma 2.1. [6] Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed convex set of X. Let {x _n } and {y _n } be two sequences in C such that x _n → p and ϕ(x _n , y _n ) → 0, where ϕ is the function defined by (1.1), then y _n → p.

Lemma 2.2. Let X and C be as in Lemma 2.1. Let T : C → N(C) be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences {ν _n }, {μ _n } and a strictly increasing continuous function ζ : ℛ⁺ → ℛ⁺ such that ν _n → 0, μ _n → 0(as n → ∞) and ζ(0) = 0. If μ₁ = 0, then the fixed point set F(T) is a closed and convex subset of C.

Proof. Let {x _n } be a sequence in F(T) with x _n → p (as n → ∞), we prove that p ∈ F(T). In fact, by the assumption that T is total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and μ₁ = 0, we have

$$\varphi \left(x_n,u\right)\le \varphi \left(x_n,p\right)+{\nu }_1\zeta \left(\varphi \left(x_n,p\right)\right),\forall u\in Tp.$$

Furthermore, we have

$$\begin{array}{cc}\hfill \varphi \left(p,u\right)& =\underset{n\to \infty }{\mathsf{\text{lim}}}\varphi \left(x_n,u\right)\hfill \\ \le \underset{n\to \infty }{\mathsf{\text{lim}}}\left(\varphi \left(x_n,p\right)+{\nu }_1\zeta \left(\varphi \left(x_n,p\right)\right)\right)=0,\forall u\in Tp.\hfill \end{array}$$

By Lemma 1.2(c), p = u. Hence, p ∈ Tp. This implies that p ∈ F(T), i.e., F(T) is closed.

Next, we prove that F(T) is convex. For any x, y ∈ F(T), t ∈ (0, 1), putting q = tx + (1 - t)y, we prove that q ∈ F(T). Indeed, let {u _n } be a sequence generated by

$$\left\{\begin{array}{c}{u}_1\in Tq,\hfill \\ u_2\in T{u}_1\subset T^{2}q,\hfill \\ u_3\in T{u}_2\subset T^{3}q,\hfill \\ ⋮\hfill \\ u_n\in T{u}_{n-1}\subset T^{n}q,\hfill \\ ⋮\hfill \end{array}\right.$$

(2.1)

In view of the definition of ϕ(x, y), for all u _n ∈ Tu_n-1⊂ Tⁿq, we have

$$\begin{array}{cc}\hfill \varphi \left(q,u_n\right)& ={||q||}^2-2〈q,J{u}_n〉+{||u_n||}^2\hfill \\ ={||q||}^2-2t〈x,J{u}_n〉-2\left(1-t\right)〈y,J{u}_n〉+{||u_n||}^2\hfill \\ ={||q||}^2+t\varphi \left(x,u_n\right)+\left(1-t\right)\varphi \left(y,u_n\right)-t{||x||}^2-\left(1-t\right){||y||}^2\hfill \end{array}$$

(2.2)

Since

$$\begin{array}{c}t\varphi \left(x,u_n\right)+\left(1-t\right)\varphi \left(y,u_n\right)\hfill \\ \le t\left(\varphi \left(x,q\right)+{\nu }_n\zeta \left(\varphi \left(x,q\right)\right)+{\mu }_n\right)+\left(1-t\right)\left(\varphi \left(y,q\right)+{\nu }_n\zeta \left(\varphi \left(y,q\right)\right)+{\mu }_n\right)\hfill \\ =t\left({||x||}^2-2〈x,Jq〉+{||q||}^2+{\nu }_n\zeta \left(\varphi \left(x,q\right)\right)+{\mu }_n\right)\hfill \\ +\left(1-t\right)\left({||y||}^2-2〈y,Jq〉+{||q||}^2+{\nu }_n\zeta \left(\varphi \left(y,q\right)\right)+{\mu }_n\right)\hfill \\ =t{||x||}^2+\left(1-t\right){||y||}^2-{||q||}^2+t{\nu }_n\zeta \left(\varphi \left(x,q\right)\right)+\left(1-t\right){\nu }_n\zeta \left(\varphi \left(y,q\right)\right)+{\mu }_n\hfill \end{array}$$

(2.3)

Substituting (2.2) into (2.1) and simplifying it we have

$$\varphi \left(q,u_n\right)\le t{\nu }_n\zeta \left(\varphi \left(x,q\right)\right)+\left(1-t\right){\nu }_n\zeta \left(\varphi \left(y,q\right)\right)+{\mu }_n\to 0\left(n\to \infty \right).$$

By Lemma 2.1, we have u _n → q (as n → ∞). This implies that u_n+1→ q (as n → ∞). Since T is closed, we have q ∈ Tq, i.e., q ∈ F(T).

This completes the proof of Lemma2.2.

Lemma 2.3. [8] Let X be a uniformly convex Banach space, r > 0 be a positive number and B _r (0) be a closed ball of X. Then, there exists a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that

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

(2.4)

for all x, y ∈ B _r (0) and all α, β ∈ [0, 1] with α + β = 1.

3 Main results

In this section, we shall use the hybrid iterative algorithm to study the iterative solutions of nonlinear operator equations with a closed and uniformly total quasi-ϕ-asymptotically nonexpansive multi-valued mapping in Banach space.

Theorem 3.1. Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed and convex subset of X. Let T : C → N(C) be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences {ν _n },{μ _n } and a strictly increasing continuous function ζ : ℛ⁺ → ℛ⁺ such that μ₁ = 0, ν _n → 0, μ _n → 0 (as n → ∞) and ζ(0) = 0. Let x₀ ∈ C, C₀ = C and {x _n } be a sequence generated by

$$\left\{\begin{array}{c}{y}_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{z}_n\right),\hfill \\ z_n=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{w}_n\right),\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,y_n\right)\le \varphi \left(\nu ,x_n\right)+{\xi }_n\right\},\hfill \\ x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(3.1)

where w _n ∈ Tⁿx _n , ∀n ≥ 1, ξ _n = ν _n sup_p∈F(T)ζ(ϕ(p, x _n )) + μ _n , ${\Pi }_{C_{n+1}}$ is the generalized projection of X onto C_n+1, {α _n } and {β _n } are sequences in [0,1] satisfies the following conditions:

• (a) lim inf_n→∞β _n (1 - β _n ) > 0;

• (b) 0 ≤ α _n ≤ α < 1 for some α ∈ (0, 1).

If F(T) is a nonempty and bounded subset of C, then the sequence {x _n } converges strongly to Π_F(T)x₀.

Proof. We divide the proof of Theorem 3.1 into six steps.

(I) C _n is closed and convex for each n ≥ 0.

In fact, by the assumption, C₀ = C is closed and convex. Suppose that C _n is closed and convex for some n ≥ 1. Since the condition ϕ(ν, y _n ) ≤ ϕ(ν, x _n ) + ξ _n is equivalent to

$$2〈\nu ,J{x}_n-J{y}_n〉\le {||x_n||}^2-{||y_n||}^2+{\xi }_n,n=1,2,\dots ,$$

hence the set

$$C_{n+1}=\left\{\nu \in C_n:2〈\nu ,J{x}_n-J{y}_n〉\le {||x_n||}^2-{||y_n||}^2+{\xi }_n\right\}$$

is closed and convex. Therefore C _n is closed and convex for each n ≥ 0.

(II) {x _n } is bounded and {ϕ(x _n , x₀)} is a convergent sequence.

Indeed, it follows from (3.1) and Lemma 1.2(a) that for all n ≥ 0, u ∈ F(T)

$$\varphi \left(x_n,x_0\right)=\varphi \left({\Pi }_{C_n}{x}_0,x_0\right)\le \varphi \left(u,x_0\right)-\varphi \left(u,{\Pi }_{C_n}{x}_0\right)\le \varphi \left(u,x_0\right).$$

This implies that {ϕ(x _n , x₀)} is bounded. By virtue of (1.2), we know that {x _n } is bounded.

In view of structure of {C _n }, we have C_n+1⊂ C _n , $C_{n+1}\subset C_n,x_n={\Pi }_{C_n}{x}_0$ and $x_{n+1}={\Pi }_{C_{n+1}}{x}_0$. This implies that x_n+1∈ C _n and

$$\varphi \left(x_n,x_0\right)\le \varphi \left(x_{n+1},x_0\right),\forall n\ge 0.$$

Therefore {ϕ(x _n , x₀)} is a convergent sequence.

(III) F(T) ⊂ C _n for all n ≥ 0.

It is obvious that F(T) ⊂ C₀ = C. Suppose that F(T) ⊂ C _n for some n ≥ 1. Since X is uniformly smooth, X* is uniformly convex. For any given u ∈ F(T) ⊂ C _n and n ≥ 1 we have

$$\begin{array}{cc}\hfill \varphi \left(u,y_n\right)& =\varphi \left(u,J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{z}_n\right)\right)\hfill \\ ={||u||}^2-2〈u,{\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{z}_n〉+{||{\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{z}_n||}^2\hfill \\ \le {||u||}^2-2{\alpha }_n〈u,J{x}_n〉-2\left(1-{\alpha }_n\right)〈u,J{z}_n〉+{\alpha }_n{||x_n||}^2\hfill \\ +\left(1-{\alpha }_n\right){||z_n||}^2\hfill \\ ={\alpha }_n\varphi \left(u,x_n\right)+\left(1-{\alpha }_n\right)\varphi \left(u,z_n\right).\hfill \end{array}$$

(3.2)

Furthermore, it follows from Lemma 2.3 that for any u ∈ F(T), w _n ∈ Tⁿx _n we have

$$\begin{array}{cc}\hfill \varphi \left(u,z_n\right)& =\varphi \left(u,J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{w}_n\right)\right)\hfill \\ ={||u||}^2-2〈u,{\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{w}_n〉+{||{\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{w}_n||}^2\hfill \\ \le {||u||}^2-2{\beta }_n〈u,J{x}_n〉-2\left(1-{\beta }_n\right)〈u,J{w}_n〉+{\beta }_n{||x_n||}^2\hfill \\ +\left(1-{\beta }_n\right){||w_n||}^2-{\beta }_n\left(1-{\beta }_n\right)g\left(||J{x}_n-J{w}_n||\right)\hfill \\ ={\beta }_n\varphi \left(u,x_n\right)+\left(1-{\beta }_n\right)\varphi \left(u,w_n\right)-{\beta }_n\left(1-{\beta }_n\right)g\left(||J{x}_n-J{w}_n||\right)\hfill \\ \le {\beta }_n\varphi \left(u,x_n\right)+\left(1-{\beta }_n\right)\left(\varphi \left(u,x_n\right)+{\nu }_n\zeta \left(\varphi \left(u,x_n\right)\right)+{\mu }_n\right)\hfill \\ -{\beta }_n\left(1-{\beta }_n\right)g\left(||J{x}_n-J{w}_n||\right)\hfill \\ \le \varphi \left(u,x_n\right)+{\nu }_n\underset{p\in F\left(T\right)}{\mathsf{\text{sup}}}\zeta \left(\varphi \left(p,x_n\right)\right)+{\mu }_n-{\beta }_n\left(1-{\beta }_n\right)g\left(||J{x}_n-J{w}_n||\right)\hfill \\ =\varphi \left(u,x_n\right)+{\xi }_n-{\beta }_n\left(1-{\beta }_n\right)g\left(||J{x}_n-J{w}_n||\right).\hfill \end{array}$$

(3.3)

Substituting (3.3) into (3.2) and simplifying it, we have

$$\varphi \left(u,y_n\right)\le \varphi \left(u,x_n\right)+\left(1-{\alpha }_n\right){\xi }_n\le \varphi \left(u,x_n\right)+{\xi }_n,\forall u\in F\left(T\right),$$

(3.4)

i.e., u ∈ C_n+1and so F(T) ⊂ C_n+1for all n ≥ 0.

By the way, in view of the assumption on {ν _n }, {μ _n } we have

$${\xi }_n={\nu }_n\underset{p\in F\left(T\right)}{\mathsf{\text{sup}}}\zeta \left(\varphi \left(p,x_n\right)\right)+{\mu }_n\to 0\left(n\to \infty \right).$$

(IV) {x _n } converges strongly to some point p* ∈ C.

In fact, since {x _n } is bounded and X is reflexive, there exists a subsequence $\left\{x_{n_i}\right\}\subset \left\{x_n\right\}$ such that $x_{n_i}\rightharpoonup p^{*}$ (some point in C). Since C _n is closed and convex and C_n+1⊂ C _n , this implies that C _n is weakly closed and p* ∈ C _n for each n ≥ 0. In view of $x_{n_i}={\Pi }_{C_{n_i}}{x}_0$, we have

$$\varphi \left(x_{n_i},x_0\right)\le \varphi \left(p^{*},x_0\right),\forall n_i\ge 0.$$

Since the norm || · || is weakly lower semi-continuous, we have

$$\begin{array}{cc}\hfill \underset{n_i\to \infty }{\mathsf{\text{lim }}\mathsf{\text{inf}}}\varphi \left(x_{n_i},x_0\right)& =\underset{n_i\to \infty }{\mathsf{\text{lim }}\mathsf{\text{inf}}}\left({||x_{n_i}||}^2-2〈x_{n_i},J{x}_0〉+{||x_0||}^2\right)\hfill \\ \ge {||p^{*}||}^2-2〈p^{*},J{x}_0〉+{||x_0||}^2=\varphi \left(p^{*},x_0\right),\hfill \end{array}$$

and so

$$\varphi \left(p^{*},x_0\right)\le \underset{n_i\to \infty }{\mathsf{\text{lim}}\mathsf{\text{inf}}}\varphi \left(x_{n_i},x_0\right)\le \underset{n_i\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\varphi \left(x_{n_i},x_0\right)\le \varphi \left(p^{*},x_0\right).$$

This implies that ${\mathsf{\text{lim}}}_{n_i\to \infty }\varphi \left(x_{n_i},x_0\right)=\varphi \left(p^{*},x_0\right)$, and so $||x_{n_i}||\to ||p^{*}||$. Since $x_{n_i}\rightharpoonup p^{*}$, by virtue of Kadec-Klee property of X, we obtain that

$$\underset{n_i\to \infty }{\mathsf{\text{lim}}}{x}_{n_i}=p^{*}.$$

Since {ϕ(x _n , x₀)} is convergent, this together with ${\mathsf{\text{lim}}}_{n_i\to \infty }\varphi \left(x_{n_i},x_0\right)=\varphi \left(p^{*},x_0\right)$, which shows that lim_n→∞ϕ(x _n , x₀) = ϕ(p*, x₀). If there exists some sequence $\left\{x_{n_i}\right\}\subset \left\{x_n\right\}$ such that $x_{n_j}\to q$, then from Lemma 1.2(a) we have that

$$\begin{array}{cc}\hfill \varphi \left(p^{*},q\right)& =\underset{n_i,n_j\to \infty }{\mathsf{\text{lim}}}\varphi \left(x_{n_i},x_{n_j}\right)=\underset{n_i,n_j\to \infty }{\mathsf{\text{lim}}}\varphi \left(x_{n_i},{\Pi }_{C_{n_j}}{x}_0\right)\hfill \\ \le \underset{n_i,n_j\to \infty }{\mathsf{\text{lim}}}\left(\varphi \left(x_{n_i},x_0\right)-\varphi \left({\Pi }_{C_{n_j}}{x}_0,x_0\right)\right)\hfill \\ =\underset{n_i,n_j\to \infty }{\mathsf{\text{lim}}}\left(\varphi \left(x_{n_i},x_0\right)-\varphi \left(x_{n_j},x_0\right)\right)\hfill \\ =\varphi \left(p^{*},x_0\right)-\varphi \left(p^{*},x_0\right)=0.\hfill \end{array}$$

This implies that p* = q and

$$\underset{n\to \infty }{\mathsf{\text{lim}}}{x}_n=p^{*}.$$

(3.5)

(V) Now we prove that p* ∈ F(T).

In fact, since x_n+1∈ C_n+1⊂ C _n , it follows from (3.1) and (3.5) that

$$\varphi \left(x_{n+1},y_n\right)\le \varphi \left(x_{n+1},x_n\right)+{\xi }_n\to 0\left(n\to \infty \right).$$

Since x _n → p*, by the virtue of Lemma 2.1

$$\underset{n\to \infty }{\mathsf{\text{lim}}}{y}_n=p^{*}.$$

(3.6)

From (3.2) and (3.3), for any u ∈ F(T) and w _n ∈ Tⁿx _n , we have

$$\varphi \left(u,y_n\right)\le \varphi \left(u,x_n\right)+{\xi }_n-\left(1-{\alpha }_n\right){\beta }_n\left(1-{\beta }_n\right)g\left(||J{x}_n-J{w}_n||\right),$$

i.e.,

$$\left(1-{\alpha }_n\right){\beta }_n\left(1-{\beta }_n\right)g\left(||J{x}_n-J{w}_n||\right)\le \varphi \left(u,x_n\right)+{\xi }_n-\varphi \left(u,y_n\right)\to 0\left(n\to \infty \right).$$

By conditions (a) and (b) it shows that lim_n→∞g(||Jx _n - Jw _n ||) = 0. In view of property of g, we have

$$||J{x}_n-J{w}_n||\to 0\left(n\to \infty \right).$$

Since Jx _n → Jp*, this implies that Jw _n → Jp*. From Remark 1.1 (ii) it yields

$$w_n\rightharpoonup p^{*}\left(n\to \infty \right).$$

(3.7)

Again since

$$\left|||w_n||-||p^{*}||\right|=\left|||J{w}_n||-||J{p}^{*}||\right|\le ||J{w}_n-J{p}^{*}||\to 0\left(n\to \infty \right),$$

this together with (3.7) and the Kadec-Klee property of X shows that

$$\underset{n\to \infty }{\mathsf{\text{lim}}}{w}_n=p^{*}.$$

(3.8)

Let {s _n } be a sequence generated by

$$\left\{\begin{array}{c}{s}_2\in T{w}_1\subset T^2{x}_1,\hfill \\ s_3\in T{w}_2\subset T^3{x}_2,\hfill \\ ⋮\hfill \\ s_{n+1}\in T{w}_n\subset T^{n+1}{x}_n,\hfill \\ ⋮\hfill \end{array}\right.$$

By the assumption that T is uniformly L-Lipschitz continuous, hence for any w _n ∈ Tⁿx _n and s_n+1∈ Tw _n ⊂ Tⁿ⁺¹x _n we have

$$\begin{array}{cc}\hfill ||s_{n+1}-w_n||& \le ||s_{n+1}-w_{n+1}||+||w_{n+1}-x_{n+1}||+||x_{n+1}-x_n||+||x_n-w_n||\hfill \\ \le \left(L+1\right)||x_{n+1}-x_n||+||w_{n+1}-x_{n+1}||+||x_n-w_n||.\hfill \end{array}$$

(3.9)

This together with (3.5) and (3.8) shows that lim_n→∞||s_n+1- w _n || = 0 and lim_n→∞s_n+1= p*. In view of the closeness of T, it yields that p* ∈ Tp*, i.e.,

$$p^{*}\in F\left(T\right).$$

(VI) we prove that x _n → p* = Π_F(T)x₀.

Let t = Π_F(T)x₀. Since t ∈ F(T) ⊂ C _n and $x_n={\Pi }_{C_n}{x}_0$, we have

$$\varphi \left(x_n,x_0\right)\le \varphi \left(t,x_0\right),\forall n\ge 0.$$

This implies that

$$\varphi \left(p^{*},x_0\right)=\underset{n\to \infty }{\mathsf{\text{lim}}}\varphi \left(x_n,x_0\right)\le \varphi \left(t,x_0\right).$$

(3.10)

In view of the definition of Π_F(T)x₀, from (3.10) we have p* = t. Therefore, x _n → p* = Π_F(T)x₀.

This completes the proof of Theorem 3.1.

From Remark 1.8, the following theorems can be obtained from Theorem 3.1 immediately.

Theorem 3.2. Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed and convex subset of X. Let T : C → N(C) be a closed and quasi-ϕ-asymptotically nonexpansive multi-valued mapping with a real sequences {k _n } ⊂ [1, ∞) and k _n → 1(n → (∞). Let x₀ ∈ C, C₀ = C and {x _n } be a sequence generated by

$$\left\{\begin{array}{c}{y}_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{z}_n\right),\hfill \\ z_n=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{w}_n\right),w_n\in T^n{x}_n,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,y_n\right)\le \varphi \left(\nu ,x_n\right)+{\xi }_n\right\},\hfill \\ x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

where ξ _n = (k _n - 1) sup_p∈F(T)(ϕ(p, x _n )), ${\Pi }_{C_{n+1}}$ is the generalized projection of X onto C_n+1, {α _n } and {β _n } are sequences in [0,1] satisfies the following conditions:

• (a) lim inf_n→∞β _n (1 - β _n ) > 0;

• (b) 0 ≤ α _n ≤ α < 1 for some α ∈ (0, 1).

If F(T) is a nonempty and bounded subset of C, then the sequence {x _n } converges strongly to Π_F(T)x₀.

Theorem 3.3. Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed and convex subset of X. Let T : C → N(C) be a closed and quasi-ϕ nonexpansive multi-valued mapping. Let x₀ ∈ C, C₀ = C and {x _n } be a sequence generated by

$$\left\{\begin{array}{c}{y}_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{z}_n\right),\hfill \\ z_n=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{w}_n\right),w_n\in T{x}_n,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,y_n\right)\le \varphi \left(\nu ,x_n\right)\right\},\hfill \\ x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

where ${\Pi }_{C_{n+1}}$ is the generalized projection of X onto C_n+1, {α _n } and {β _n } are sequences in [0,1] satisfies the following conditions:

• (a) lim inf_n→∞β _n (1 - β _n ) > 0;

• (b) 0 ≤ α _n ≤ α < 1 for some α ∈ (0, 1).

If F(T) is a nonempty and bounded subset of C, then the sequence {x _n } converges strongly to Π_F(T)x₀.

Theorem 3.4. Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed and convex subset of X. Let T : C → N(C) be a closed and relatively nonexpansive multi-valued mapping. Let x₀ ∈ C, C₀ = C and {x _n } be a sequence generated by

$$\left\{\begin{array}{c}{y}_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{z}_n\right),\hfill \\ z_n=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{w}_n\right),w_n\in T{x}_n,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,y_n\right)\le \varphi \left(\nu ,x_n\right)\right\},\hfill \\ x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

where ${\Pi }_{C_{n+1}}$ is the generalized projection of X onto C_n+1, {α _n } and {β _n } are sequences in [0,1] satisfies the following conditions:

• (a) lim inf_n→∞β _n (1 - β _n ) > 0;

• (b) 0 ≤ α _n ≤ α < 1 for some α ∈ (0, 1).

If F(T) is a nonempty and bounded subset of C, then the sequence {x _n } converges strongly to Π_F(T)x₀.