Strong convergence theorems for modifying Halpern-Mann iterations for a quasi-ϕ-asymptotically nonexpansive multi-valued mapping in Banach spaces

Abstract

The purpose of this paper is to introduce modifying Halpern-Mann’s iterations sequence for a quasi-ϕ-asymptotically nonexpansive multi-valued mapping. Under suitable limit conditions, some strong convergence theorems are proved. The results presented in the paper improve and extend the corresponding results of Chang (Appl. Math. Comput. 218:6489-6497, 2012).

MSC:47J05, 47H09, 49J25.

1 Introduction

Throughout this paper, we denote by N and R the sets of positive integers and real numbers, respectively. Let D be a nonempty closed subset of a real Banach space X. A mapping $T:D\to D$ is said to be nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in D$. Let $N(D)$ and $CB(D)$ denote the families of nonempty subsets and nonempty closed bounded subsets of D, respectively. The Hausdorff metric on $CB(D)$ is defined by

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

for $A_1,A_2\in CB(D)$, where $d(x,A_1)=inf\{\parallel x-y\parallel ,y\in A_1\}$. The multi-valued mapping $T:D\to CB(D)$ is called nonexpansive if $H(Tx,Ty)\le \parallel x-y\parallel$ for all $x,y\in D$. An element $p\in D$ is called a fixed point of $T:D\to N(D)$ if $p\in T(p)$. The set of fixed points of T is represented by $F(T)$.

In the sequel, let $S(X)=\{x\in X:\parallel x\parallel =1\}$. A Banach space X is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel \le 1$ for all $x,y\in S(X)$ and $x\ne y$. A Banach space is said to be uniformly convex if ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$ for any two sequences $\{x_n\},\{y_n\}\subset S(X)$ and ${lim}_{n\to \mathrm{\infty }}\parallel \frac{x_n+y_n}{2}\parallel =0$. The norm of the Banach space X is said to be Gâteaux differentiable if for each $x,y\in S(X)$, the limit

$$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

(1.1)

exists. In this case, X is said to be smooth. The norm of the Banach space X is said to be Fréchet differentiable if for each $x\in S(X)$, the limit (1.1) is attained uniformly for $y\in S(x)$, and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for $x,y\in S(X)$. In this case, X is said to be uniformly smooth.

Remark 1.1 Let X be a real Banach space with dual $X^{\ast }$. We denote by J the normalized duality mapping from X to $2^{X^{\ast }}$, which is defined by

$$J(x)=\{x^{\ast }\in X^{\ast }:〈x,x^{\ast }〉={\parallel x\parallel }^2={\parallel x^{\ast }\parallel }^2\},x\in X,$$

where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing.

The following basic properties of the normalized duality mapping J in a Banach space X can be found in Cioranescu [1].

• (1) X ($X^{\ast }$, resp.) is uniformly convex if and only if $X^{\ast }$ (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 $Jx\cap Jy\ne \mathrm{\Phi }$ for all $x,y\in X$;

• (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;

• (7) Each uniformly convex Banach space X has the Kadec-Klee property, i.e., for any sequence $\{x_n\}\subset X$, if $x_n\rightharpoonup x\in X$ and $\parallel x_n\parallel \to \parallel x\parallel$, then $x_n\to x\in X$;

• (8) If X is a reflexive and strictly convex Banach space with a strictly convex dual $X^{\ast }$ and $J^{\ast }:X^{\ast }\to X$ is the normalized duality mapping in $X^{\ast }$, then $J^{-1}=J^{\ast }$, $J{J}^{\ast }=I_{x^{\ast }}$ and $J^{\ast }J=I_x$.

Next we assume that X is a smooth, strictly convex, and reflexive Banach space and D is a nonempty closed convex subset of X. In the sequel, we always use $\varphi :X\times X\to \overline{R^{-}}$ to denote the Lyapunov bifunction defined by

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

(1.2)

It is obvious from the definition of the function ϕ that

$${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^2,$$

(1.3)

$$\varphi (y,x)=\varphi (y,z)+\varphi (z,x)+2〈z-y,Jx-Jz〉,x,y,z\in X,$$

(1.4)

and

$$\varphi (x,J^{-1}(\alpha Jy+(1-\alpha )Jz))\le \alpha \varphi (x,y)+(1-\alpha )\varphi (x,z)$$

(1.5)

for all $\alpha \in [0,1]$ and $x,y,z\in X$.

Following Alber [2], the generalized projection ${\mathrm{\Pi }}_D:X\to D$ is defined by

$${\mathrm{\Pi }}_{D}x=arg\underset{y\in D}{inf}\varphi (y,x),\mathrm{\forall }x\in X.$$

Remark 1.2 (see [3])

Let ${\mathrm{\Pi }}_D$ be the generalized projection from a smooth, reflexive, and strictly convex Banach space X onto a nonempty closed convex subset D of X, then ${\mathrm{\Pi }}_D$ is closed and quasi-ϕ-nonexpansive from X onto D.

Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping. In 1953, Mann [4] introduced the following iterative sequence $\{x_n\}$:

$$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n,$$

where the initial guess $x_1\in D$ is arbitrary and $\{{\alpha }_n\}$ is a real sequence in $[0,1]$. It is known that under appropriate settings the sequence $\{x_n\}$ converges weakly to a fixed point of T. However, even in a Hilbert space, the Mann iteration may fail to converge strongly [5]. Some attempts to construct an iteration method guaranteeing the strong convergence have been made. For example, Halpern [6] proposed the following so-called Halpern iteration:

$$x_{n+1}={\alpha }_{n}u+(1-{\alpha }_n)T{x}_n,$$

where $u,x_1\in D$ are arbitrarily given and $\{{\alpha }_n\}$ is a real sequence in $[0,1]$. Another approach was proposed by Nakajo and Takahashi [7]. They generated a sequence as follows:

$$\{\begin{array}{c}{x}_1\in X\text{is arbitrary},\hfill \\ y_n={\alpha }_{n}u+(1-{\alpha }_n)T{x}_n,\hfill \\ C_n=\{z\in D:\parallel y_n-z\parallel \le \parallel x_n-z\parallel \},\hfill \\ Q_n=\{z\in D:〈x_n-z,x_1-x_n〉\ge 0\},\hfill \\ x_{n+1}=P_{C_n\cap Q_n}{x}_1(n=1,2,\dots ),\hfill \end{array}$$

(1.6)

where $\{{\alpha }_n\}$ is a real sequence in $[0,1]$ and $P_K$ denotes the metric projection from a Hilbert space H onto a closed convex subset K of H. It should be noted here that the iteration above works only in a Hilbert space setting. To extend this iteration to a Banach space, the concepts of relatively nonexpansive mappings and quasi-ϕ-nonexpansive mappings have been introduced (see [8–11] and [12]).

Inspired by Matsushita and Takahashi, in this paper, we introduce modifying Halpern-Mann iterations sequence for finding a fixed point of a quasi-ϕ-nonexpansive mappings multi-valued mapping $T:D\to CB(D)$ and prove some strong convergence theorems. The results presented in the paper improve and extend the corresponding results in [13] and other.

2 Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence $\{x_n\}$ by $x_n\to x$ and $x_n\rightharpoonup x$, respectively.

Lemma 2.1 (see [2])

Let X be a smooth, strictly convex and reflexive Banach space, and let D be a nonempty closed convex subset of X. Then the following conclusions hold:

• (a) $\varphi (x,y)=0$ if and only if $x=y$;

• (b) $\varphi (x,{\mathrm{\Pi }}_{D}y)+\varphi ({\mathrm{\Pi }}_{D}y,y)\le \varphi (x,y)$, $\mathrm{\forall }x,y\in D$;

• (c) If $x\in X$ and $z\in D$, then $z={\mathrm{\Pi }}_{D}x\iff 〈z-y,Jx-Jz〉\ge 0$, $\mathrm{\forall }y\in D$.

Remark 2.1 If H is a real Hilbert space, then $\varphi (x,y)={\parallel x-y\parallel }^2$ and ${\mathrm{\Pi }}_D$ is the metric projection $P_D$ of H onto D.

Lemma 2.2 (see [13])

Let X be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and let D be a nonempty closed convex subset of X. Let $\{x_n\}$ and $\{y_n\}$ be two sequences in D such that $x_n\to p$ and $\varphi (x_n,y_n)\to 0$, where ϕ is the function defined by (1.2), then $y_n\to p$.

Definition 2.1 A point $x\in D$ is said to be an asymptotic fixed point of $T:D\to CB(D)$ if there exists a sequence $\{x_n\}\subset D$ such that $x_n\rightharpoonup x\in X$ and $d(x_n,T(x_n))\to 0$. Denote the set of all asymptotic fixed points of T by $\hat{F}(T)$.

Definition 2.2 A multi-valued mapping $T:D\to CB(D)$ is said to be closed if for any sequence $\{x_n\}\subset D$ with $x_n\to x\in X$ and $d(y,T(x_n))\to 0$, then $d(y,T(x))=0$.

Definition 2.3 (1) A multi-valued mapping $T:D\to CB(D)$ is said to be relatively nonexpansive if $F(T)\ne \mathrm{\Phi }$, $\hat{F}(T)=F(T)$ and $\varphi (p,z)\le \varphi (p,x)$, $\mathrm{\forall }x\in D$, $p\in F(T)$, $z\in T(x)$.

(2) A multi-valued mapping $T:D\to CB(D)$ is said to be quasi-ϕ-nonexpansive if $F(T)\ne \mathrm{\Phi }$, and $\varphi (p,z)\le \varphi (p,x)$, $\mathrm{\forall }x\in D$, $p\in F(T)$, $z\in Tx$.

(3) A multi-valued mapping $T:D\to CB(D)$ is said to be quasi-ϕ-asymptotically nonexpansive if $F(T)\ne \mathrm{\Phi }$ and there exists a real sequence $k_n\subset [1,+\mathrm{\infty })$, $k_n\to 1$ such that

$$\varphi (p,z_n)\le k_n\varphi (p,x),\mathrm{\forall }x\in D,p\in F(T),z_n\in T^{n}x.$$

(2.1)

Definition 2.4 A mapping $T:D\to CB(D)$ is said to be uniformly L-Lipschitz continuous if there exists a constant $L>0$ such that $\parallel x_n-y_n\parallel \le L\parallel x-y\parallel$, where $x,y\in D$, $x_n\in T^{n}x$, $y_n\in T^{n}y$.

Next, we present an example of a relatively nonexpansive multi-valued mapping.

Example 2.1 (see [13])

Let $I=[0,1]$, $X=C(I)$ (the Banach space of continuous functions defined on I with the uniform convergence norm ${\parallel f\parallel }_C={sup}_{t\in I}|f(t)|$), $D=\{f\in X:f(x)\ge 0,x\in I\}$ and let a, b be two constants in $(0,1)$ with $a<b$. Let $T:D\to N(D)$ be a multi-valued mapping defined by

$$T(f)=\{\begin{array}{c}\{g\in D:a\le f(x)-g(x)\le b,\mathrm{\forall }x\in I\},\hfill \\ \{0\}\text{otherwise}.\hfill \end{array}$$

(2.2)

It is easy to see that $F(T)=\{0\}$, therefore $F(T)$ is nonempty.

From the example in [13], we can see that $T:D\to N(D)$ is a closed quasi-ϕ-asymptotically nonexpansive multi-valued mapping.

Remark 2.2 From the definitions, it is obvious that a relatively nonexpansive multi-valued mapping is a quasi-ϕ-nonexpansive multi-valued mapping, and a quasi-ϕ-nonexpansive multi-valued mapping is a quasi-ϕ-asymptotically nonexpansive multi-valued mapping, and a quasi-ϕ-asymptotically nonexpansive multi-valued mapping is a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping, but the converse is not true.

Lemma 2.3 Let X and D be as in Lemma 2.2. Let $T:D\to CB(D)$ be a closed and quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences $\{k_n\}$ with $\{k_n\}\subset [1,\mathrm{\infty })$ and $k_n\to 1$ (as $n\to \mathrm{\infty }$), then $F(T)$ is a closed and convex subset of D.

Proof Let $\{x_n\}$ be a sequence in $F(T)$ such that $x_n\to x^{\ast }$. Since T is a quasi-ϕ-asymptotically nonexpansive multi-valued mapping, we have

$$\varphi (x_n,z)\le k_1\varphi (x_n,x^{\ast })$$

for all $z\in T{x}^{\ast }$ and for all $n\in N$. Therefore,

$$\varphi (x^{\ast },z)=\underset{n\to \mathrm{\infty }}{lim}\varphi (x_n,z)\le \underset{n\to \mathrm{\infty }}{lim}{k}_1\varphi (x_n,x^{\ast })=k_1\varphi (x^{\ast },x^{\ast })=0.$$

By Lemma 2.1(a), we obtain $z=x^{\ast }$. Hence, $T{x}^{\ast }=\{x^{\ast }\}$. So, we have $x^{\ast }\in F(T)$. This implies $F(T)$ is closed.

Let $p,q\in F(T)$ and $t\in (0,1)$, and put $w=tp+(1-t)q$. Next we prove that $w\in F(T)$. Indeed, in view of the definition of ϕ, let $z_n\in T^{n}w$, we have

$$\begin{array}{rl}\varphi (w,z_n)& ={\parallel w\parallel }^2-2〈w,J{z}_n〉+{\parallel z_n\parallel }^2\\ ={\parallel w\parallel }^2-2〈tp+(1-t)q,J{z}_n〉+{\parallel z_n\parallel }^2\\ ={\parallel w\parallel }^2+t\varphi (p,z_n)+(1-t)\varphi (q,z_n)-t{\parallel p\parallel }^2-(1-t){\parallel q\parallel }^2.\end{array}$$

(2.3)

Since

$$\begin{array}{r}t\varphi (p,z_n)+(1-t)\varphi (q,z_n)\\ \le t{k}_n\varphi (p,w)+(1-t)k_n\varphi (q,w)\\ =t\{{\parallel p\parallel }^2-2〈p,Jw〉+{\parallel w\parallel }^2+(k_n-1)\varphi (p,w)\}\\ +(1-t)\{{\parallel q\parallel }^2-2〈q,Jw〉+{\parallel w\parallel }^2+(k_n-1)\varphi (q,w)\}\\ =t{\parallel p\parallel }^2+(1-t){\parallel q\parallel }^2-{\parallel w\parallel }^2+t(k_n-1)\varphi (p,w)+(1-t)(k_n-1)\varphi (q,w).\end{array}$$

(2.4)

Substituting (2.3) into (2.4) and simplifying it, we have

$$\varphi (w,z_n)\le t(k_n-1)\varphi (p,w)+(1-t)(k_n-1)\varphi (q,w)\to 0(\text{as }n\to \mathrm{\infty }).$$

Hence, by Lemma 2.2, we have $z_n\to w$. This implies that $z_{n+1}(\in T{T}^{n}w)\to w$. Since T is closed, we have $w\in Tw$, i.e., $w\in F(T)$. This completes the proof of Lemma 2.3. □

Lemma 2.4 ([13])

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, for any given sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}\subset B_r(0)$ and for any given sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ of positive numbers with ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n=1$, there exists a continuous, strictly increasing, and convex function $g:[0,2r)\to [0,\mathrm{\infty })$ with $g(0)=0$ such that for any positive integers i, j with $i<j$,

$${\parallel \sum _{n=1}^{\mathrm{\infty }}{\lambda }_n{x}_n\parallel }^2\le \sum _{n=1}^{\mathrm{\infty }}{\lambda }_n{\parallel x_n\parallel }^2-{\lambda }_i{\lambda }_{j}g{\parallel x_i-x_j\parallel }^2.$$

(2.5)

Lemma 2.5 ([14])

Let X be a uniformly convex and smooth Banach space and let $\{x_n\}$ and $\{y_n\}$ be two sequences of X such that $\{x_n\}$ or $\{y_n\}$ is bounded. If ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,y_n)=0$, then ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$.

Let X be a reflexive, strictly convex, and smooth Banach space. The duality mapping $J^{\ast }$ from $X^{\ast }$ onto $X^{\ast \ast }=X$ coincides with the inverse of the duality mapping J from E onto $E^{\ast }$. We make use the following mapping $v:X\times X^{\ast }\to R$ studied in Alber [2]:

$$v(x,x^{\ast })={\parallel x\parallel }^2-2〈x,x^{\ast }〉+{\parallel x^{\ast }\parallel }^2$$

(2.6)

for all $x\in X$, $x^{\ast }\in X^{\ast }$. Obviously, $v(x,x^{\ast })=\varphi (x,J^{-1}{x}^{\ast })$.

Lemma 2.6 ([15])

Let X be a reflexive, strictly convex, and smooth Banach space, and let v as in (2.6). Then

$$v(x,x^{\ast })+2〈J^{-1}{x}^{\ast }-x,y^{\ast }〉\le v(x,x^{\ast }+y^{\ast })$$

(2.7)

for all $x\in X$, $x^{\ast },y^{\ast }\in X^{\ast }$.

Lemma 2.7 ([16])

Assume that $\{{\alpha }_n\}$ is a sequence of nonnegative real numbers such that

$${\alpha }_{n+1}\le (1-{\gamma }_n){\alpha }_n+{\gamma }_n{\delta }_n,$$

where $\{{\gamma }_n\}$ is a sequence in $(0,1)$ and $\{{\delta }_n\}$ is a sequence such that

• (a) ${lim}_{n\to \mathrm{\infty }}{\gamma }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n=\mathrm{\infty }$;

• (b) $lim{sup}_{n\to \mathrm{\infty }}{\delta }_n\le 0$.

Then ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$.

Lemma 2.8 ([17])

Let $\{{\alpha }_n\}$ be a sequence of real numbers such that there exists a subsequence $\{n_i\}$ of $\{n\}$ such that ${\alpha }_{n_i}\le {\alpha }_{n_i+1}$ for all $i\in N$. Then there exists a nondecreasing $\{m_k\}\subset N$ such that $m_k\to \mathrm{\infty }$ and the following properties are satisfied for all (sufficiently large) numbers sequence $k\subset N$:

$${\alpha }_{m_k}\le {\alpha }_{m_k+1}\mathit{\text{and}}{\alpha }_k\le {\alpha }_{m_k+1}.$$

In fact, $m_k=max\{j\le k:{\alpha }_j\le {\alpha }_{j+1}\}$.

3 Main results

Theorem 3.1 Let X be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let D be a nonempty closed convex subset of X, and let $T:D\to CB(D)$ be a closed and uniformly L-Lipschitz continuous quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences $\{k_n\}\subset [0,+\mathrm{\infty })$, $k_n\to 1$ (as $n\to \mathrm{\infty }$) such that condition (2.1) and ${\prod }_{n=1}^{\mathrm{\infty }}{k}_n<+\mathrm{\infty }$. Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be two sequences in $(0,1)$ satisfying

• (R1) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{k_n-1}{{\alpha }_n}=0$;

• (R2) ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (R3) $0<lim{inf}_{n\to \mathrm{\infty }}{\beta }_n\le lim{sup}_{n\to \mathrm{\infty }}{\beta }_n<1$.

If $\{x_n\}$ is the sequence generated by

$$x_{n+1}={\mathrm{\Pi }}_D{J}^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n)],z_n\in T^n{x}_n,$$

(3.1)

where $x_1\in X$ is arbitrary, $F(T)$ is the fixed point set of T, and ${\mathrm{\Pi }}_D$ is the generalized projection of X onto D. If $I-T$ is demi-closed at zero and $F(T)\ne \mathrm{\Phi }$, then ${lim}_{n\to \mathrm{\infty }}{x}_n={\mathrm{\Pi }}_{F(T)}{x}_1$.

Remark 3.1 We can present an example of $\{k_n\}$ satisfying the conditions $\{k_n\}\subset [0,+\mathrm{\infty })$, $k_n\to 1$ (as $n\to \mathrm{\infty }$) and ${\prod }_{n=1}^{\mathrm{\infty }}{k}_n<+\mathrm{\infty }$. For instance, if $k_n=1+\frac{1}{2^{2^{n-1}}}$, then ${\prod }_{n=1}^{\mathrm{\infty }}{k}_n=2<+\mathrm{\infty }$.

Proof First, we prove that $\{x_n\}$ is a bounded sequence in D.

Let $p\in F(T)$ and $y_n=J^{-1}[{\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n]$ for any $n\in N$. Then

$$x_{n+1}={\mathrm{\Pi }}_D{J}^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n]$$

for any $n\in N$. Using (2.1) and (1.5), we have

$$\begin{array}{rcl}\varphi (p,y_n)& =& \varphi (p,J^{-1}[{\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n])\\ \le & {\beta }_n\varphi (p,x_n)+(1-{\beta }_n)\varphi (p,z_n)\\ \le & {\beta }_n\varphi (p,x_n)+(1-{\beta }_n)k_n\varphi (p,x_n)\\ \le & k_n\varphi (p,x_n),\end{array}$$

and

$$\begin{array}{rcl}\varphi (p,x_{n+1})& =& \varphi (p,{\mathrm{\Pi }}_D{J}^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n])\\ \le & \varphi (p,J^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n])\\ \le & {\alpha }_n\varphi (p,x_1)+(1-{\alpha }_n)\varphi (p,y_n)\\ \le & {\alpha }_n\varphi (p,x_1)+(1-{\alpha }_n)k_n\varphi (p,x_n)\\ \le & max\{\varphi (p,x_1),k_n\varphi (p,x_n)\}\\ \le & max\{\varphi (p,x_1),k_n{k}_{n-1}\varphi (p,x_{n-1})\}.\end{array}$$

By induction, we have

$$\varphi (p,x_{n+1})\le k_n{k}_{n-1}\cdots k_1\varphi (p,x_1).$$

Since ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ and ${\prod }_{n=1}^{\mathrm{\infty }}{k}_n<+\mathrm{\infty }$, then $k_n{k}_{n-1}\cdots k_1$ is bounded, and we get $\varphi (p,x_{n+1})$ is bounded. This implies that $\{x_n\}$ is bounded, so is $\{z_n\}$.

Next, let $g:[0,2r]\to [0,\mathrm{\infty })$ be a function satisfying the properties of Lemma 2.5, where $r=sup\{\parallel x_1\parallel ,\parallel x_n\parallel ,\parallel z_n\parallel \}$. Put $p={\mathrm{\Pi }}_{F(T)}{x}_1$ and $y_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n)$. Then

$$\begin{array}{rcl}\varphi (p,y_n)& =& \varphi (p,J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n))\\ \le & {\beta }_n\varphi (p,x_n)+(1-{\beta }_n)k_n\varphi (p,x_n)-{\beta }_n(1-{\beta }_n)g(\parallel J{x}_n-J{z}_n\parallel )\\ \le & k_n\varphi (p,x_n)-{\beta }_n(1-{\beta }_n)g(\parallel J{x}_n-J{z}_n\parallel ),\end{array}$$

and

$$\begin{array}{rcl}\varphi (p,x_{n+1})& =& \varphi (p,{\mathrm{\Pi }}_D{J}^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n])\\ \le & \varphi (p,J^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n])\\ \le & {\alpha }_n\varphi (p,x_1)+(1-{\alpha }_n)\varphi (p,y_n)\\ \le & {\alpha }_n\varphi (p,x_1)+(1-{\alpha }_n)[k_n\varphi (p,x_n)-{\beta }_n(1-{\beta }_n)g(\parallel J{x}_n-J{z}_n\parallel )]\\ \le & {\alpha }_n{k}_n(\varphi (p,x_1)-\varphi (p,x_n))+(k_n-1)\varphi (p,x_n)+\varphi (p,x_n)\\ -(1-{\alpha }_n){\beta }_n(1-{\beta }_n)g(\parallel J{x}_n-J{z}_n\parallel ).\end{array}$$

(3.2)

Letting

$$M=\underset{n\in N}{sup}\{|k_n(\varphi (p,x_1)-\varphi (p,x_n))+\frac{k_n-1}{{\alpha }_n}\varphi (p,x_n)|+{\beta }_n(1-{\beta }_n)g(\parallel J{x}_n-J{z}_n\parallel )\},$$

by (3.2), we have

$${\beta }_n(1-{\beta }_n)g(\parallel J{x}_n-J{z}_n\parallel )\le \varphi (p,x_n)-\varphi (p,x_{n+1})+{\alpha }_{n}M.$$

Let $u_n=J^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n]$. Then $x_{n+1}={\mathrm{\Pi }}_D{u}_n$ for all $n\in N$. It follows from (2.6) and (2.7) that

$$\begin{array}{rcl}\varphi (p,x_{n+1})& \le & \varphi (p,J^{-1}({\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n))=v(p,{\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n)\\ \le & v(p,J^{-1}({\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n)-{\alpha }_n(J{x}_1-Jp))\\ -2〈J^{-1}({\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n)-p,-{\alpha }_n(J{x}_1-Jp)〉\\ =& v(p,{\alpha }_{n}Jp+(1-{\alpha }_n)J{y}_n)+2{\alpha }_n〈u_n-p,J{x}_1-Jp〉\\ =& \varphi (p,J^{-1}[{\alpha }_{n}Jp+(1-{\alpha }_n)J{y}_n])+2{\alpha }_n〈u_n-p,J{x}_1-Jp〉\\ \le & {\parallel p\parallel }^2-2〈p,{\alpha }_{n}Jp+(1-{\alpha }_n)J{y}_n〉+{\parallel {\alpha }_{n}Jp+(1-{\alpha }_n)J{y}_n\parallel }^2\\ +2{\alpha }_n〈u_n-p,J{x}_1-Jp〉\\ \le & {\parallel p\parallel }^2-2{\alpha }_n〈p,Jp〉-2(1-{\alpha }_n)〈p,y_n〉+{\alpha }_n{\parallel Jp\parallel }^2+(1-{\alpha }_n){\parallel J{y}_n\parallel }^2\\ +2{\alpha }_n〈u_n-p,J{x}_1-Jp〉\\ =& {\alpha }_n\varphi (p,p)+(1-{\alpha }_n)\varphi (p,y_n)+2{\alpha }_n〈u_n-p,J{x}_1-Jp〉\\ \le & (1-{\alpha }_n)k_n\varphi (p,x_n)+2{\alpha }_n〈u_n-p,J{x}_1-Jp〉.\end{array}$$

(3.3)

The rest of the proof will be divided into two parts.

Case (1). Suppose that there exists $n_0$ such that ${\{\varphi (p,x_n)\}}_{n_0}^{\mathrm{\infty }}$ is nonincreasing. In this situation, ${\{\varphi (p,x_n)\}}_{n_0}^{\mathrm{\infty }}$ is convergent. Together with (R1), (R3), and (3.2), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}g(\parallel J{x}_n-J{z}_n\parallel )=0.$$

(3.4)

Therefore, ${lim}_{n\to \mathrm{\infty }}\parallel J{x}_n-J{z}_n\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel x_n-z_n\parallel =0$. Since $d(x_n,T{x}_n)\le \parallel x_n-z_n\parallel$, we obtain

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,T{x}_n)=0.$$

(3.5)

Then

$$\begin{array}{rcl}\varphi (z_n,y_n)& =& \varphi (z_n,J^{-1}[{\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n])\\ \le & {\beta }_n\varphi (z_n,x_n)+(1-{\beta }_n)\varphi (z_n,z_n)\\ =& {\beta }_n\varphi (z_n,x_n)\to 0\end{array}$$

(3.6)

and

$$\begin{array}{rcl}\varphi (y_n,u_n)& =& \varphi (y_n,J^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{y}_n])\\ \le & {\alpha }_n\varphi (y_n,x_1)+(1-{\alpha }_n)\varphi (y_n,y_n)\\ =& {\alpha }_n\varphi (y_n,x_1)\to 0.\end{array}$$

(3.7)

From (3.6), (3.7) and Lemma 2.5, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-z_n\parallel =0\text{and}\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-u_n\parallel =0.$$

From (3.4), we have

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

(3.8)

Since $I-T$ is demi-closed at zero, we choose a subsequence $\{x_{n_i}\}\subset \{x_n\}$ such that $x_{n_i}\rightharpoonup p^{\ast }\in F(T)$. By Lemma 2.2(c), we have

$$\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u_n-p,J{x}_1-Jp〉\\ =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈x_n-p,J{x}_1-Jp〉\\ =\underset{i\to \mathrm{\infty }}{lim}〈x_{n_i}-p,J{x}_1-Jp〉=〈p^{\ast }-p,J{x}_1-Jp〉\le 0.\end{array}$$

(3.9)

Hence the conclusion follows.

Case (2). Suppose that there exists a subsequence $\{x_{n_i}\}\subset \{x_n\}$ such that $\varphi (p,x_{n_i})\le \varphi (p,x_{n_i+1})$. Then, by Lemma 2.8, there exists a nondecreasing sequence $\{m_k\}\subset N$, $m_k\to \mathrm{\infty }$ such that

$$\varphi (p,x_{m_k})\le \varphi (p,x_{m_k+1})\text{and}\varphi (p,x_k)\le \varphi (p,x_{m_k+1}).$$

This together with (3.2) gives

$${\beta }_{m_k}(1-{\beta }_{m_k})g(\parallel J{x}_{m_k}-J{z}_{m_k}\parallel )\le \varphi (p,x_{m_k})-\varphi (p,x_{m_k+1})+{\alpha }_{m_k}M$$

for all $k\in N$. Then, by conditions (R1) and (R3),

$$\underset{k\to \mathrm{\infty }}{lim}g(\parallel J{x}_{m_k}-J{z}_{m_k}\parallel )=0.$$

By the same argument as Case (1), we get

$$\underset{k\to \mathrm{\infty }}{lim}sup〈u_{m_k}-p,J{x}_1-Jp〉\le 0.$$

(3.10)

From (3.3), we get

$$\varphi (p,x_{m_k+1})\le (1-{\alpha }_{m_k})k_{m_k}\varphi (p,x_{m_k})+2{\alpha }_{m_k}〈u_{m_k}-p,J{x}_1-Jp〉$$

and

$${\alpha }_{m_k}{k}_{m_k}\varphi (p,x_{m_k})\le k_{m_k}\varphi (p,x_{m_k})-\varphi (p,x_{m_k+1})+2{\alpha }_{m_k}〈u_{m_k}-p,J{x}_1-Jp〉.$$

Since $\varphi (p,x_{m_k})-\varphi (p,x_{m_k+1})\le 0$, we have

$${\alpha }_{m_k}{k}_{m_k}\varphi (p,x_{m_k})\le (k_{m_k}-1)\varphi (p,x_{m_k})+2{\alpha }_{m_k}〈u_{m_k}-p,J{x}_1-Jp〉.$$

This implies that

$$\varphi (p,x_{m_k})\le \frac{k_{m_k}-1}{{\alpha }_{m_k}}\varphi (p,x_{m_k})+2〈u_{m_k}-p,J{x}_1-Jp〉.$$

From (3.10) and (R1), we get ${lim}_{k\to \mathrm{\infty }}\varphi (p,x_{m_k})\le 0$ and $x_{m_k}\to p$. This implies that ${lim}_{n\to \mathrm{\infty }}{x}_n=p$, which yields that $p=w={\mathrm{\Pi }}_{F(T)}{x}_1$. Therefore, $x_n\to {\mathrm{\Pi }}_{F(T)}{x}_1$. The proof of Theorem 3.1 is completed. □

By Remark 2.2, the following corollaries are obtained.

Corollary 3.1 Let X and D be as in Theorem 3.1, and let $T:D\to CB(D)$ be a closed and uniformly L-Lipschitz continuous relatively nonexpansive multi-valued mapping. Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be two sequences in $(0,1)$ satisfying

• (R1) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$;

• (R2) ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (R3) $0<lim{inf}_{n\to \mathrm{\infty }}{\beta }_n\le lim{sup}_{n\to \mathrm{\infty }}{\beta }_n<1$.

Let $\{x_n\}$ be the sequence generated by (3.1), where $F(T)$ is the set of fixed points of T, and ${\mathrm{\Pi }}_D$ is the generalized projection of X onto D, then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{F(T)}{x}_1$.

If we take ${\beta }_n=\beta$, the following result is obtained.

Corollary 3.2 Let X be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let D be a nonempty closed convex subset of X and let $T:D\to CB(D)$ be a closed and uniformly L-Lipschitz continuous quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences $\{k_n\}\subset [0,+\mathrm{\infty })$, $k_n\to 1$ (as $n\to \mathrm{\infty }$) satisfying condition (2.1). Let $\{{\alpha }_n\}$ be a sequence in $(0,1)$ satisfying

• (Q1) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{k_n-1}{{\alpha }_n}=0$;

• (Q2) ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$.

If $\beta \in (0,1)$ and $\{x_n\}$ is the sequence generated by

$$x_{n+1}={\mathrm{\Pi }}_D{J}^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)J(\beta J{x}_n+(1-\beta )J{z}_n)],z_n\in T^n{x}_n,$$

(3.11)

where $x_1\in X$ is arbitrary, $F(T)$ is the fixed point set of T, and ${\mathrm{\Pi }}_D$ is the generalized projection of X onto D; if $I-T$ is demi-closed at zero and $F(T)$ is nonempty, then ${lim}_{n\to \mathrm{\infty }}{x}_n={\mathrm{\Pi }}_{F(T)}{x}_1$.