Strong convergence theorem for totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings under relaxed conditions

Abstract

We construct a relaxed hybrid shrinking iteration algorithm for approximating common fixed points of a countable family of totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings. A strong convergence theorem for solving generalized mixed equilibrium problems is established in the framework of Banach spaces under relaxed conditions. Since there is no need to impose a uniformity assumption on the involved mappings and no need to compute complex series in the iteration process, the results improve those of the authors with related interests.

1 Introduction

Throughout this paper we assume that E is a real Banach space with its dual $E^{*}$, C is a nonempty closed convex subset of E and $J : E\rightarrow2^{E^{*}}$ is the normalized duality mapping defined by

$$Jx= \bigl\{ f\in E^{*}:\langle x,f\rangle=\|x\|^{2}=\|f \|^{2} \bigr\} , \quad\forall x\in E. $$

In the sequel, we use $F(T)$ to denote the set of fixed points of a mapping T.

Definition 1.1

[1]

(1) A multi-valued mapping $T: C\rightarrow2^{C}$ is said to be totally quasi-ϕ-asymptotically nonexpansive, if $F(T)\neq\emptyset$ and there exist nonnegative real sequences $\{\nu_{n}\}$, $\{\mu_{n}\}$ with $\nu_{n},\mu_{n}\rightarrow0$ (as $n\rightarrow\infty$) and a strictly increasing continuous function $\zeta:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}^{+}\cup\{0\}$ with $\zeta(0)=0$ such that

$$\begin{aligned} \phi(p,w_{n})\leq\phi(p,x)+\nu_{n}\zeta\bigl(\phi(p,x) \bigr)+\mu_{n},\quad \forall n\geq1, x\in C, w\in T^{n}x, p \in F(T), \end{aligned}$$

(1.1)

where $\phi:E\times E\rightarrow\mathbb{R}^{+}\cup\{0\}$ denotes the Lyapunov functional defined by

$$\begin{aligned} \phi(x,y)=\|x\|^{2}-2\langle x,Jy\rangle+\|y\|^{2},\quad \forall x, y\in E. \end{aligned}$$

(1.2)

It is obvious from the definition of ϕ that

$$\begin{aligned} \bigl(\|x\|-\|y\|\bigr)^{2}\leq\phi(x,y)\leq\bigl(\|x\|+\|y\|\bigr)^{2} \end{aligned}$$

(1.3)

and

$$\begin{aligned} \phi \bigl(x,J^{-1}\bigl(\lambda Jy+(1-\lambda)Jz\bigr) \bigr)\leq \lambda\phi(x,y)+(1-\lambda)\phi(x,z),\quad \forall x, y\in E,\lambda\in[0,1]. \end{aligned}$$

(1.4)

(2) A countable family of multi-valued mappings $\{T_{i}\}:C\rightarrow C$ said to be uniformly totally quasi-ϕ-asymptotically nonexpansive, if $F:=\bigcap^{\infty}_{i=1}F(T_{i})\neq\emptyset$ and there exist nonnegative real sequences $\{\nu_{n}\}$, $\{\mu_{n}\}$ with $\nu_{n},\mu_{n}\rightarrow0$ (as $n\rightarrow\infty$) and a strictly increasing continuous function $\zeta:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}^{+}\cup\{0\}$ with $\zeta(0)=0$ such that

$$\begin{aligned} \phi(p,w_{n,i})\leq\phi(p,x)+\nu_{n}\zeta\bigl(\phi(p,x) \bigr)+\mu_{n},\quad \forall n\geq1,w_{n,i}\in T^{n}_{i}x, i\geq1, x\in C, p \in F. \end{aligned}$$

(1.5)

(3) A totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping $T: C\rightarrow2^{C}$ is said to be uniformly L-Lipschitz continuous, if there exists a constant $L>0$ such that

$$\begin{aligned} \|w_{n}-s_{n}\|\leq L\|x-y\|, \quad\forall n\geq1, x,y \in C,w_{n}\in T^{n}x,s_{n}\in T^{n}y. \end{aligned}$$

(1.6)

Let $\theta:C\times C\rightarrow\mathbb{R}$ be a bifunction, $\psi:C\rightarrow\mathbb{R}$ a real valued function and $A:C\rightarrow E^{*}$ a nonlinear mapping. The so-called generalized mixed equilibrium problem GMEP is to find an $u\in C$ such that

$$\begin{aligned} \theta(u,y)+\langle Au,y-u\rangle+\psi(y)-\psi(u)\geq0,\quad\forall y\in C, \end{aligned}$$

(1.7)

whose set of solutions is denoted by Ω.

In 2012, Chang et al. [1] used the following hybrid shrinking iteration algorithm finding a common element of the set of solutions for a GMEP, the set of solutions for variational inequality problems, and the set of common fixed points for a countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property:

$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} x_{0}\in C;\quad C_{0}=C, \\ y_{n}=J^{-1}[\alpha_{n}Jx_{n}+(1-\alpha_{n})Jz_{n}], \\ z_{n}=J^{-1}[\beta_{n,0}Jx_{n}+\sum^{\infty}_{i=1}\beta_{n,i}Jw_{n,i}], \\ u_{n}\in C \mbox{ such that } \forall y\in C, \\ \theta(u_{n},y)+\langle Au_{n},y-u_{n}\rangle+\psi(y)-\psi(u_{n})+\frac{1}{r_{n}} \langle y-u_{n},Ju_{n}-Jy_{n}\rangle\geq0, \\ C_{n+1}=\{v\in C_{n}:\phi(v,u_{n})\leq\phi(v,x_{n})+\xi_{n}\}, \\ x_{n+1}=\Pi_{C_{n+1}}x_{0}, \quad\forall n\geq0, \end{array}\displaystyle \right . \end{aligned}$$

(1.8)

where $\{T_{i}\}:C\rightarrow2^{C}$ is a countable family of closed and uniformly totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings; $w_{n,i}\in T^{n}_{i}x_{n}$, $\forall n\geq1$, $i\geq1$, $\xi_{n}:=\nu_{n}\sup_{p\in F}\zeta(\phi(p,x_{n}))+\mu_{n}$, $\Pi_{C_{n+1}}$ is the generalized projection (see (2.1)) of E onto $C_{n+1}$. Their results not only generalized the corresponding results of [2–19] from single-valued mappings to multi-valued mappings, but they also improved and extended the main results of Homaeipour and Razani [20].

However, it is obviously a quite strong condition that the involved multi-valued mappings are assumed to be uniformly $(\{\nu_{n}\},\{\mu_{n}\},\zeta)$-totally quasi-ϕ-asymptotically nonexpansive. In addition, the accurate computation of the series $\sum^{\infty}_{i=1}\beta_{n,i}Jw_{n,i}$ at each step of the iteration process is not easily attainable, which leads to gradually increasing errors.

Inspired and motivated by the study mentioned above, in this paper, we use a relaxed hybrid iteration algorithm for approximating common fixed points of a countable family of multi-valued totally quasi-ϕ-asymptotically nonexpansive mappings and obtain a strong convergence theorem under some suitable conditions. The results improve those of Chang et al. [1].

2 Preliminaries

We say that a Banach space E is strictly convex if the following implication holds for $x, y\in E$:

$$\begin{aligned} \|x\|=\|y\|=1,\quad x\neq y\quad \Rightarrow \quad\biggl\Vert \frac{x+y}{2}\biggr\Vert < 1. \end{aligned}$$

(2.1)

E is also said to be uniformly convex if for any $\epsilon>0$, there exists $\delta>0$ such that

$$\begin{aligned} \|x\|=\|y\|=1,\quad\|x-y\|\geq\epsilon\quad\Rightarrow\quad\biggl\Vert \frac{x+y}{2} \biggr\Vert \leq1-\delta . \end{aligned}$$

(2.2)

It is well known that if E is a uniformly convex Banach space, then E is reflexive and strictly convex. A Banach space E is said to be smooth if

$$\begin{aligned} \lim_{t\rightarrow0}\frac{\|x+ty\|-\|x\|}{t} \end{aligned}$$

(2.3)

exists for each $x, y\in S(E) := \{x \in E : \|x\|= 1\}$. E is said to be uniformly smooth if the limit (2.3) is attained uniformly for $x,y \in S(E)$.

Following Alber [21], the generalized projection $\Pi_{C}:E\rightarrow C$ is defined by

$$\begin{aligned} \Pi_{C}=\arg\inf_{y\in C}\phi(y,x),\quad \forall x\in E. \end{aligned}$$

(2.4)

Lemma 2.1

[21]

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

• (1) $\phi(x,\Pi_{C}y)+\phi(\Pi_{C}y,y)\leq\phi(x,y)$ for all $x\in C$ and $y\in E$;

• (2) If $x\in E$ and $z\in C$, then $z=\Pi_{C}x\Leftrightarrow\langle z-y,Jx-Jz\rangle\geq0$, $\forall y\in C$;

• (3) For $x,y\in E$, $\phi(x,y)=0$ if and only if $x=y$.

Remark 2.2

The following basic properties for a Banach space E can be found in Cioranescu [22].

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

• (ii) If E is reflexive and strictly convex, then $J^{-1}$ is norm-weak-continuous;

• (iii) If E is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping $J : E\rightarrow2^{E^{*}}$ is single valued, one-to-one and onto;

• (iv) A Banach space E is uniformly smooth if and only if $E^{*}$ is uniformly convex;

• (v) Each uniformly convex Banach space E has the Kadec-Klee property, i.e., for any sequence $\{x_{n}\}\subset E$, if $x_{n}\rightharpoonup x\in E$ and $\|x_{n}\|\rightarrow\|x\|$, then $x_{n}\rightarrow x$ as $n\rightarrow\infty$.

Lemma 2.3

[6]

Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed convex subset of E. Let $\{x_{n}\}$ and $\{y_{n}\}$ be two sequences in C such that $x_{n}\rightarrow p$ and $\phi(x_{n},y_{n})\rightarrow0$, where ϕ is the function defined by (1.2), then $y_{n}\rightarrow p$.

Lemma 2.4

[1]

Let E and C be the same as in Lemma  2.3. Let $T:C\rightarrow C$ be a closed and totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings with nonnegative real sequences $\{\nu_{n}\}$, $\{\mu_{n}\}$ and a strictly increasing continuous function $\zeta:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}^{+}\cup\{0\}$ such that $\nu_{n},\mu_{n}\rightarrow0$ and $\zeta(0)=0$. If $\mu_{1}=0$, then the fixed point set $F(T)$ of T is a closed and convex subset of C.

Lemma 2.5

[6]

Let E be a real uniformly convex Banach space and let $B_{r}(0)$ be the closed ball of E with center at the origin and radius $r>0$. Then for any for any sequence $\{x_{i}\}\subset B_{r}(0)$ and for any sequence $\{\lambda_{i}\}$ of positive numbers with $\sum^{\infty}_{i=1}\lambda_{i}=1$, there exists a continuous strictly increasing convex function $g:[0,\infty)\rightarrow[0,\infty)$ with $g(0)=0$ such that such that for any positive integer $i\neq1$, the following hold:

$$\begin{aligned} \Biggl\Vert \sum^{\infty}_{i=1} \lambda_{i}x_{i}\Biggr\Vert ^{2}\leq\sum ^{\infty }_{i=1}\lambda_{i} \|x_{i}\|^{2}-\lambda_{1}\lambda_{i}g\bigl( \|x_{1}-x_{i}\|\bigr) \end{aligned}$$

(2.5)

and, for all $x\in E$,

$$\begin{aligned} \phi \Biggl(x,J^{-1} \Biggl(\sum^{\infty}_{i=1} \lambda_{i}Jx_{i} \Biggr) \Biggr)\leq\sum ^{\infty}_{i=1}\lambda_{i}\phi(x,x_{i})- \lambda_{1}\lambda_{i}g\bigl(\| Jx_{1}-Jx_{i} \|\bigr). \end{aligned}$$

(2.6)

Assume that, to obtain the solution of GMEP, the function $\psi:C\rightarrow\mathbb{R}$ is convex and lower semi-continuous, the nonlinear mapping $A:C\rightarrow E^{*}$ is continuous and monotone, and the bifunction $\theta:C\times C\rightarrow\mathbb{R}$ satisfies the following conditions:

(A₁):

$\theta(x,x)=0$;

(A₂):

θ is monotone, i.e., $\theta(x,y)+\theta(y,x)\leq0$;

(A₃):

$\lim \sup_{t\downarrow0}\theta(x+t(z-x),y)\leq\theta(x,y)$;

(A₄):

the mapping $y\mapsto\theta(x,y)$ is convex and lower semicontinuous.

Lemma 2.6

[16]

Let E be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty closed convex subset of E. Let $A:C\rightarrow E^{*}$ be a continuous and monotone mapping, $\psi:C\rightarrow\mathbb{R}$ a lower semi-continuous and convex function, and $\theta:C\times C\rightarrow\mathbb{R}$ a bifunction satisfying the conditions (A₁)-(A₄). Let $r>0$ and $x\in E$. Then the following hold:

• (1) There exists an $u\in C$ such that

  $$\theta(u,y)+\langle Au,y-u\rangle+\psi(y)-\psi(u)+\frac{1}{r}\langle y-u ,Ju-Jx\rangle\geq0,\quad\forall y\in C. $$

• (2) A mapping $\kappa_{r}:C\rightarrow C$ is defined by

  $$\kappa_{r}(x)= \biggl\{ u\in C:\theta(u,y)+\langle Au,y-u\rangle+ \psi(y)-\psi(u)+\frac{1}{r}\langle y-u ,Ju-Jx\rangle\geq0 \biggr\} . $$

Then the mapping $\kappa_{r}$ has the following properties:

• (i) $\kappa_{r}$ is single-valued;

• (ii) $\kappa_{r}$ a firmly nonexpansive-type mapping, i.e.,

  $$\langle\kappa_{r}z-\kappa_{r}y,J\kappa_{r}z-J \kappa_{r}y\rangle\leq\langle \kappa_{r}z- \kappa_{r}y,Jz-Jy\rangle; $$

• (iii) $F(\kappa_{r})=\Omega=\tilde{F}(\kappa_{r})$;

• (iv) Ω is a closed convex set of C;

• (v) $\phi(p,\kappa_{r}z)+\phi(\kappa_{r}z,z)\leq\phi(p,z)$, $\forall p\in F(\kappa_{r})$, $z\in E$,

where $\tilde{F}(\kappa_{r})$ denotes the set of asymptotic fixed points of $\kappa_{r}$, i.e.,

$$\tilde{F}(\kappa_{r}):=\bigl\{ x\in C:\exists \{x_{n}\} \subset C,\textit{s.t.}, x_{n}\rightharpoonup x,\|x_{n}- \kappa_{r}x_{n}\|\rightarrow0\ (n\rightarrow\infty)\bigr\} . $$

Lemma 2.7

[23]

The unique solutions to the positive integer equation

$$\begin{aligned} n=i_{n}+\frac{(m_{n}-1)m_{n}}{2},\qquad m_{n}\geq i_{n},\quad n=1,2, \ldots, \end{aligned}$$

(2.7)

are

$$\begin{aligned} i_{n}=n-\frac{(m_{n}-1)m_{n}}{2},\qquad m_{n}=- \biggl[ \frac{1}{2}-\sqrt{2n+\frac{1}{4}} \biggr],\quad n=1,2,\ldots, \end{aligned}$$

(2.8)

where $[x]$ denotes the maximal integer that is not larger than x.

3 Main results

Recall that a multi-valued mapping $T:C\rightarrow2^{C}$ is said to be closed, if for any sequence $\{x_{n}\}\subset C$ with $x_{n}\rightarrow x$ and $w_{n}\in Tx_{n}$ with $w_{n}\rightarrow y$ as $n\rightarrow\infty$, then $y\in Tx$.

Theorem 3.1

Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and C a nonempty closed convex subset of E. Let $\theta:C\times C\rightarrow\mathbb{R}$ be a bifunction satisfying the conditions (A₁)-(A₄), $A:C\rightarrow E^{*}$ a continuous and monotone mapping, and $\psi:C\rightarrow\mathbb{R}$ a lower semi-continuous and convex function. Let $\{T_{i}\}:C\rightarrow2^{C}$ be a countable family of closed and totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings with nonnegative real sequences $\{\nu^{(i)}_{n}\}$, $\{\mu^{(i)}_{n}\}$ satisfying $\nu^{(i)}_{n}\rightarrow0$ and $\mu^{(i)}_{n}\rightarrow0$ (as $n\rightarrow\infty$ and for each $i\geq1$) and a strictly increasing and continuous function $\zeta:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}^{+}\cup\{0\}$ satisfying condition (1.1) and each $T_{i}$ is uniformly $L_{i}$-Lipschitz continuous with $\mu^{(i)}_{1}=0$. Let $\{\alpha_{i}\}$ be a sequence in $[0, 1)$ and $\{\beta_{i}\}$ be a sequence in $(0, 1)$. Let $\{x_{n}\}$ be the sequence generated by

$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} x_{1}\in C; \quad C_{1}=C, \\ y_{n}=J^{-1}[\alpha_{i_{n}}Jx_{n}+(1-\alpha_{i_{n}})Jz_{n}], \\ z_{n}=J^{-1} [\beta_{i_{n}}Jx_{n}+(1-\beta_{i_{n}})Jw^{(i_{n})}_{m_{n}}], \\ u_{n}\in C\textit{ such that }\forall y\in C, \\ \theta(u_{n},y)+\langle Au_{n},y-u_{n}\rangle+\psi(y)-\psi(u_{n})+\frac{1}{r_{n}} \langle y-u_{n},Ju_{n}-Jy_{n}\rangle\geq0, \\ C_{n+1}=\{v\in C_{n}:\phi(v,u_{n})\leq\phi(v,x_{n})+\xi_{n}\}, \\ x_{n+1}=\Pi_{C_{n+1}}x_{1},\quad n\in\mathbb{N}, \end{array}\displaystyle \right . \end{aligned}$$

(3.1)

where $w^{(i_{n})}_{m_{n}}\in T^{m_{n}}_{i_{n}}x_{n}$, $\forall n\geq1$, $\xi_{n}:=\nu^{(i_{n})}_{m_{n}}\sup_{p\in F}\zeta_{i_{n}}(\phi(p,x_{n}))+\mu^{(i_{n})}_{m_{n}}$, $\Pi_{C_{n+1}}$ is the generalized projection of E onto $C_{n+1}$; and $i_{n}$ and $m_{n}$ are the solutions to the positive integer equation: $n=i_{n}+\frac{(m_{n}-1)m_{n}}{2}$ ($m_{n}\geq i_{n}$, $n=1,2,\ldots$), that is, for each $n\geq1$, there exist unique $i_{n}$ and $m_{n}$ such that

$$\begin{aligned}& i_{1}=1,\qquad i_{2}=1,\qquad i_{3}=2,\qquad i_{4}=1,\qquad i_{5}=2,\\& i_{6}=3,\qquad i_{7}=1,\qquad i_{8}=2,\qquad\ldots;\\& m_{1}=1,\qquad m_{2}=2,\qquad m_{3}=2,\qquad m_{4}=3,\qquad m_{5}=3,\\& m_{6}=3,\qquad m_{7}=4,\qquad m_{8}=4,\qquad\ldots. \end{aligned}$$

If $G:=F\cap\Omega\neq\emptyset$ and $F:=\bigcap^{\infty}_{i=1}F(T_{i})$ is bounded, then $\{x_{n}\}$ converges strongly to $\Pi_{G}x_{1}$.

Proof

Two functions $\tau:C\times C\rightarrow\mathbb{R}$ and $\kappa_{r}:C\rightarrow C$ are defined by

$$\begin{aligned}& \tau(x,y)=\theta(x,y)+\langle Ax,y-x\rangle+\psi(y)-\psi(x);\\& \kappa_{r}(x)= \biggl\{ u\in C:\tau(u,y)+\frac{1}{r}\langle y-u,Ju-Jx\rangle\geq0,\forall y\in C \biggr\} . \end{aligned}$$

By Lemma 2.6, we know that the function τ satisfies the conditions (A₁)-(A₄) and $\kappa_{r}$ has the properties (i)-(v). Therefore, (3.1) can be rewritten as

$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} x_{1}\in C; \quad C_{1}=C, \\ y_{n}=J^{-1}[\alpha_{i_{n}}Jx_{n}+(1-\alpha_{i_{n}})Jz_{n}], \\ z_{n}=J^{-1} [\beta_{i_{n}}Jx_{n}+(1-\beta_{i_{n}})Jw^{(i_{n})}_{m_{n}}], \\ u_{n}\in C\textit{ such that }\tau(u_{n},y)+\frac{1}{r_{n}} \langle y-u_{n},Ju_{n}-Jy_{n}\rangle\geq0,\quad\forall y\in C, \\ C_{n+1}=\{v\in C_{n}:\phi(v,u_{n})\leq\phi(v,x_{n})+\xi_{n}\}, \\ x_{n+1}=\Pi_{C_{n+1}}x_{1},\quad n\in\mathbb{N}. \end{array}\displaystyle \right . \end{aligned}$$

(3.2)

We divide the proof into several steps.

(I) F and $C_{n}$ ($\forall n\geq1$) both are closed and convex subsets in C.

In fact, it follows from Lemma 2.4 that each $F(T_{i})$ is a closed and convex subset of C, so is F. In addition, with $C_{1}$ (=C) being closed and convex, we may assume that $C_{n}$ is closed and convex for some $n\geq2$. In view of the definition of ϕ we have

$$C_{n+1}=\bigl\{ v\in C:\varphi(v)\leq a\bigr\} \cap C_{n}, $$

where $\varphi(v)=2\langle v,Jx_{n}-Jy_{n}\rangle$ and $a=\|x_{n}\|^{2}-\|y_{n}\|^{2}+\xi_{n}$. This shows that $C_{n+1}$ is closed and convex.

(II) G is a subset of $\bigcap^{\infty}_{n=1}C_{n}$.

It is obvious that $G\subset C_{1}$. Suppose that $G\subset C_{n}$ for some $n\geq2$. Since $u_{n}=\kappa_{r_{n}}y_{n}$, by Lemma 2.6, it is easily shown that $\kappa_{r_{n}}$ is quasi-ϕ-nonexpansive. Hence, for any $p\in G\subset C_{n}$, it follows from (1.4) that

$$\begin{aligned} \phi(p,u_{n}) =&\phi(p,\kappa_{r_{n}}y_{n})\leq \phi(p,y_{n}) =\phi \bigl(p,J^{-1} \bigl[\alpha_{n}Jx_{n}+(1- \alpha_{n})Jx_{n} \bigr] \bigr) \\ \leq&\alpha_{n}\phi(p,x_{n})+(1-\alpha_{n}) \phi(p,z_{n}). \end{aligned}$$

(3.3)

Furthermore, it follows from Lemma 2.5 that for any $p\in G\subset C_{n}$, $w^{(i_{n})}_{m_{n}} \in T^{m_{n}}_{i_{n}}x_{n}$, we have

$$\begin{aligned} \phi(p,z_{n}) =&\phi \bigl(p,J^{-1} \bigl[ \beta_{i_{n}}Jx_{n}+(1-\beta _{i_{n}})Jw^{(i_{n})}_{m_{n}} \bigr] \bigr) \\ \leq&\beta_{i_{n}}\phi(p,x_{n})+(1-\beta_{i_{n}})\phi \bigl(p,w^{(i_{n})}_{m_{n}} \bigr) -\beta_{i_{n}}(1- \beta_{i_{n}})g \bigl(\bigl\Vert Jx_{n}-Jw^{(i_{n})}_{m_{n}} \bigr\Vert \bigr) \\ \leq&\beta_{i_{n}}\phi(p,x_{n})+(1-\beta_{i_{n}}) \bigl[\phi(p,x_{n})+\nu^{(i_{n})}_{m_{n}} \zeta_{i_{n}}\bigl(\phi(p,x_{n})\bigr)+\mu ^{(i_{n})}_{m_{n}} \bigr] \\ &{}-\beta_{i_{n}}(1-\beta_{i_{n}})g \bigl(\bigl\Vert Jx_{n}-Jw^{(i_{n})}_{m_{n}}\bigr\Vert \bigr) \\ \leq&\phi(p,x_{n})+\nu^{(i_{n})}_{m_{n}}\sup _{p\in F}\zeta_{i_{n}}\bigl(\phi(p,x_{n})\bigr)+ \mu^{(i_{n})}_{m_{n}}-\beta_{i_{n}}(1-\beta _{i_{n}})g \bigl(\bigl\Vert Jx_{n}-Jw^{(i_{n})}_{m_{n}}\bigr\Vert \bigr) \\ =&\phi(p,x_{n})+\xi_{n}-\beta_{i_{n}}(1- \beta_{i_{n}})g \bigl(\bigl\Vert Jx_{n}-Jw^{(i_{n})}_{m_{n}} \bigr\Vert \bigr). \end{aligned}$$

(3.4)

Substituting (3.4) into (3.3) and simplifying it, we have

$$\begin{aligned} \phi(p,u_{n}) \leq&\phi(p,y_{n})\leq \phi(p,x_{n})+(1- \alpha_{i_{n}})\xi_{n}-(1-\alpha_{i_{n}}) \beta_{i_{n}}(1-\beta _{i_{n}})g \bigl(\bigl\Vert Jx_{n}-Jw^{(i_{n})}_{m_{n}}\bigr\Vert \bigr) \\ \leq&\phi(p,x_{n})+\xi_{n}-(1-\alpha_{i_{n}}) \beta_{i_{n}}(1-\beta_{i_{n}})g \bigl(\bigl\Vert Jx_{n}-Jw^{(i_{n})}_{m_{n}}\bigr\Vert \bigr) \\ \leq&\phi(p,x_{n})+\xi_{n}. \end{aligned}$$

(3.5)

This implies that $p\in C_{n+1}$, and so $G\subset C_{n+1}$.

(III) $x_{n}\rightarrow x^{*}\in C$ as $n\rightarrow\infty$.

In fact, since $x_{n}=\Pi_{C_{n}}x_{1}$, from Lemma 2.1(2) we have $\langle x_{n}-y,Jx_{1}-Jx_{n}\rangle\geq0$, $\forall y\in C_{n}$. Again since $F\subset\bigcap^{\infty}_{n=1}C_{n}$, we have $\langle x_{n}-p,Jx_{1}-Jx_{n}\rangle\geq0$, $\forall p\in F$. It follows from Lemma 2.1(1) that for each $p\in F$ and for each $n\geq1$,

$$\phi(x_{n},x_{1})=\phi(\Pi_{C_{n}}x_{1},x_{1}) \leq\phi(p,x_{1})-\phi(p,x_{n})\leq \phi(p,x_{1}), $$

which implies that $\{\phi(x_{n},x_{1})\}$ is bounded, so is $\{x_{n}\}$. Since for all $n\geq1$, $x_{n}=\Pi_{C_{n}}x_{1}$ and $x_{n+1}=\Pi_{C_{n+1}}x_{1}\in C_{n+1}\subset C_{n}$, we have $\phi(x_{n},x_{1})\leq\phi(x_{n+1},x_{1})$. This implies that $\{\phi(x_{n},x_{1})\}$ is nondecreasing, hence the limit

$$\lim_{n\rightarrow\infty}\phi(x_{n},x_{1}) \mbox{ exists}. $$

Since E is reflexive, there exists a subsequence $\{x_{n_{i}}\}$ of $\{x_{n}\}$ such that $x_{n_{i}}\rightharpoonup x^{*}\in C$ as $i\rightarrow\infty$. Since $C_{n}$ is closed and convex and $C_{n+1} \subset C_{n}$, this implies that $C_{n}$ is weakly closed and $x^{*}\in C_{n}$ for each $n\geq1$. In view of $x_{n_{i}}=\Pi_{C_{n_{i}}}x_{1}$, we have

$$\phi(x_{n_{i}},x_{1})\leq\phi\bigl(x^{*},x_{1} \bigr),\quad \forall i\geq1. $$

Since the norm $\|\cdot\|$ is weakly lower semi-continuous, we have

$$\begin{aligned} \liminf_{i\rightarrow\infty}\phi(x_{n_{i}},x_{1}) =&\liminf _{i\rightarrow \infty} \bigl(\|x_{n_{i}}\|^{2}-2\langle x_{n_{i}},Jx_{1}\rangle+\|x_{1}\|^{2} \bigr)\\ \geq&\bigl\| x^{*}\bigr\| ^{2}-2\bigl\langle x^{*},Jx_{1} \bigr\rangle +\|x_{1}\|^{2}\\ =&\phi\bigl(x^{*},x_{1}\bigr) \end{aligned}$$

and so

$$\begin{aligned} \phi\bigl(x^{*},x_{1}\bigr)\leq\liminf_{i\rightarrow\infty} \phi(x_{n_{i}},x_{1})\leq \limsup_{i\rightarrow\infty} \phi(x_{n_{i}},x_{1})\leq\phi\bigl(x^{*},x_{1} \bigr). \end{aligned}$$

This implies that $\lim_{i\rightarrow\infty}\phi(x_{n_{i}},x_{1})=\phi(x^{*},x_{1})$, and so $\|x_{n_{i}}\|\rightarrow\|x^{*}\|$ as $i\rightarrow\infty$. Since $x_{n_{i}}\rightharpoonup x^{*}$, by virtue of Kadec-Klee property of E, we obtain

$$\begin{aligned} \lim_{i\rightarrow\infty}x_{n_{i}}=x^{*}. \end{aligned}$$

Since $\{\phi(x_{n},x_{1})\}$ is convergent, this, together with $\lim_{i\rightarrow\infty}\phi(x_{n_{i}},x_{1})=\phi(x^{*},x_{1})$, shows that $\lim_{n\rightarrow\infty}\phi(x_{n},x_{1})=\phi(x^{*},x_{1})$. If there exists some subsequence $\{x_{n_{j}}\}$ of $\{x_{n}\}$ such that $x_{n_{j}}\rightarrow y$ as $j\rightarrow\infty$, then from Lemma 2.1(1) we have

$$\begin{aligned} \phi\bigl(x^{*},y\bigr) =&\lim_{i,j\rightarrow\infty}\phi (x_{n_{i}},x_{n_{j}})=\lim_{i,j\rightarrow\infty} \phi(x_{n_{i}},\Pi _{C_{n_{j}}}x_{1}) \\ \leq& \lim_{i,j\rightarrow\infty}\bigl(\phi(x_{n_{i}},x_{1})- \phi(\Pi _{C_{n_{j}}}x_{1},x_{1})\bigr) \\ =&\lim_{i,j\rightarrow\infty}\bigl(\phi(x_{n_{i}},x_{1})- \phi (x_{n_{j}},x_{1})\bigr) \\ =&\phi\bigl(x^{*},x_{1}\bigr)-\phi\bigl(x^{*},x_{1} \bigr)=0, \end{aligned}$$

that is, $x^{*}=y$ and so

$$\begin{aligned} \lim_{n\rightarrow\infty}x_{n}=x^{*}. \end{aligned}$$

(3.6)

(IV) $x^{*}$ is a member of F.

Set $\mathcal{K}_{i}= \{k\geq1:k=i_{k}+\frac{(m_{k}-1)m_{k}}{2},m_{k}\geq i_{k},m_{k}\in\mathbb{N} \}$ for each $i\geq1$. Note that $\nu^{(i_{k})}_{m_{k}}=\nu^{(i)}_{m_{k}}$, $\mu^{(i_{k})}_{m_{k}}=\mu^{(i)}_{m_{k}}$, and $\zeta_{i_{k}}=\zeta_{i}$ whenever $k\in\mathcal{K}_{i}$ for each $i\geq1$. For example, by Lemma 2.7 and the definition of $\mathcal{K}_{1}$, we have $\mathcal{K}_{1}=\{1,2,4,7,11,16,\ldots\}$ and $i_{1}=i_{2}=i_{4}=i_{7}=i_{11}=i_{16}=\cdots=1$. Then we have

$$\begin{aligned} \xi_{k} =&\nu^{(i)}_{m_{k}}\sup_{p\in F} \zeta_{i}\bigl(\phi(p,x_{k})\bigr)+\mu^{(i)}_{m_{k}},\quad \forall k\in \mathcal{K}_{i}. \end{aligned}$$

(3.7)

Note that $\{m_{k}\}_{k\in\mathcal{K}_{i}}=\{i,i+1,i+2,\ldots\}$, i.e., $m_{k}\uparrow\infty$ as $\mathcal{K}_{i}\ni k\rightarrow\infty$. It follows from (3.6) and (3.7) that

$$\begin{aligned} \lim_{k\rightarrow\infty}\xi_{k}=0. \end{aligned}$$

(3.8)

Since $x_{n+1}\in C_{n+1}$, it follows from (3.1), (3.6), and (3.8) that

$$\begin{aligned} \phi(x_{k+1},y_{k})\leq\phi(x_{k+1},x_{k})+ \xi_{k}\rightarrow0 \end{aligned}$$

(3.9)

as $\mathcal{K}_{i}\ni k\rightarrow\infty$. Since $x_{k}\rightarrow x^{*}$, it follows from (3.9) and Lemma 2.3 that

$$\begin{aligned} \lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}y_{k}=x^{*}. \end{aligned}$$

(3.10)

Note that $w^{(i_{k})}_{m_{k}}=w^{(i)}_{m_{k}}$, $T^{m_{k}}_{i_{k}}=T^{m_{k}}_{i}$, $\alpha _{i_{k}}=\alpha_{i}$, and $\beta_{i_{k}}=\beta_{i}$ whenever $k\in\mathcal{K}_{i}$ for each $i\geq1$. From (3.5), for any $p\in F$ and $w^{(i)}_{m_{k}}\in T^{m_{k}}_{i}x_{k}$, $\forall k\in\mathcal{K}_{i}$, we have

$$\begin{aligned} \phi(p,y_{k})\leq\phi(p,x_{k})+\xi_{k}-(1- \alpha_{i})\beta_{i}(1-\beta_{i})g \bigl(\bigl\Vert Jx_{k}-Jw^{(i)}_{m_{k}}\bigr\Vert \bigr), \end{aligned}$$

that is,

$$\begin{aligned} (1-\alpha_{i})\beta_{i}(1-\beta_{i})g \bigl( \bigl\Vert Jx_{k}-Jw^{(i)}_{m_{k}}\bigr\Vert \bigr) \leq\phi(p,x_{k})+\xi_{k}-\phi(p,y_{k}) \rightarrow0 \quad(\mathcal{K}_{i}\ni k\rightarrow\infty). \end{aligned}$$

This, together with assumption conditions imposed on the sequence $\{\alpha_{i}\}$ and $\{\beta_{i}\}$, shows that $\lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}g (\Vert Jx_{k}-Jw^{(i)}_{m_{k}}\Vert )=0$. In view of property of g, we have

$$\begin{aligned} \lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}\bigl\Vert Jx_{k}-Jw^{(i)}_{m_{k}} \bigr\Vert =0. \end{aligned}$$

In addition, $Jx_{k}\rightarrow Jx^{*}$ implies that $\lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}Jw^{(i)}_{m_{k}}=Jx^{*}$. From Remark 2.2(ii) it yields, as $\mathcal{K}_{i}\ni k\rightarrow\infty$,

$$\begin{aligned} w^{(i)}_{m_{k}}\rightharpoonup x^{*}, \quad\forall i\geq1. \end{aligned}$$

(3.11)

Again, since, for each $i\geq1$, as $\mathcal{K}_{i}\ni k\rightarrow\infty$,

$$\begin{aligned} \bigl\vert \bigl\Vert w^{(i)}_{m_{k}}\bigr\Vert - \bigl\| x^{*}\bigr\| \bigr\vert =\bigl\vert \bigl\Vert Jw^{(i)}_{m_{k}} \bigr\Vert -\bigl\| Jx^{*}\bigr\| \bigr\vert \leq\bigl\Vert Jw^{(i)}_{m_{k}}-Jx^{*}\bigr\Vert \rightarrow0, \end{aligned}$$

this, together with (3.11) and the Kadec-Klee property of E, shows that

$$\begin{aligned} \lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}w^{(i)}_{m_{k}}=x^{*},\quad \forall i\geq1. \end{aligned}$$

(3.12)

For each $i\geq1$, we now consider the sequence $\{s^{(i)}_{m_{k}} \}_{k\in\mathcal{K}_{i}}$ generated by

$$\begin{aligned} s^{(i)}_{m_{k+1}}\in T_{i}w^{(i)}_{m_{k}} \subset T^{m_{k+1}}_{i}x_{k},\quad k\in \mathcal{K}_{i}, \forall i\geq1. \end{aligned}$$

(3.13)

By the assumptions that for each $i\geq1$, $T_{i}$ is uniformly $L_{i}$-Lipschitz continuous. Noting again that $\{m_{k}\}_{k\in\mathcal{K}_{i}}=\{i,i+1,i+2,\ldots\}$, i.e., $m_{k+1}-1=m_{k}$ for all ${k\in\mathcal{K}_{i}}$, we then have

$$\begin{aligned} \bigl\Vert s^{(i)}_{m_{k+1}}-w^{(i)}_{m_{k}} \bigr\Vert \leq&\bigl\Vert s^{(i)}_{m_{k+1}}-w^{(i)}_{m_{k+1}} \bigr\Vert +\bigl\Vert w^{(i)}_{m_{k+1}}-x_{k+1}\bigr\Vert \\ &{}+\|x_{k+1}-x_{k}\|+\bigl\Vert x_{k}-w^{(i)}_{m_{k}} \bigr\Vert \\ \leq&(L_{i}+1)\|x_{k+1}-x_{k}\|+\bigl\Vert w^{(i)}_{m_{k+1}}-x_{k+1}\bigr\Vert \\ &{}+\bigl\Vert x_{k}-w^{(i)}_{m_{k}}\bigr\Vert . \end{aligned}$$

(3.14)

From (3.12) and $x_{k}\rightarrow x^{*}$ we have $\lim_{\mathcal{K}_{i}\ni k\rightarrow\infty} \Vert s^{(i)}_{m_{k+1}}-w^{(i)}_{m_{k}}\Vert =0$ and

$$\begin{aligned} \lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}s^{(i)}_{m_{k+1}}=x^{*},\quad \forall i\geq1. \end{aligned}$$

(3.15)

In view of the closedness of $T_{i}$, it follows from (3.12) and (3.13) that $x^{*}\in T_{i}x^{*}$ for each $i\geq1$, namely $x^{*}\in F$.

(V) $x^{*}$ is also a member of G.

Since $x_{n+1}=\Pi_{C_{n+1}}x_{1}$, it follows from (3.1) and (3.6) that

$$\begin{aligned} \phi(x_{k+1},u_{k})\leq\phi(x_{k+1},x_{k})+ \xi_{k}\rightarrow0 \end{aligned}$$

as $\mathcal{K}_{i}\ni k\rightarrow\infty$. Since $x_{k}\rightarrow x^{*}$, by virtue of Lemma 2.1 we have

$$\begin{aligned} \lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}u_{k}=x^{*}. \end{aligned}$$

(3.16)

This, together with (3.10), shows that $\lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}\|u_{k}-y_{k}\|=0$ and $\lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}\|Ju_{k}-Jy_{k}\|=0$. By the assumption that $\{r_{k}\}_{k\in\mathcal{K}_{i}}\subset[a,\infty)$ for some $a>0$, we have

$$\begin{aligned} \lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}\frac{\|Ju_{k}-Jy_{k}\|}{r_{k}}=0. \end{aligned}$$

(3.17)

Since $\tau(u_{k},y)+\frac{1}{r_{k}}\langle y-u_{k},Ju_{k}-Jy_{k}\rangle\geq0$, $\forall y\in C$, by condition (A₁), we have

$$\begin{aligned} \frac{1}{r_{k}}\langle y-u_{k},Ju_{k}-Jy_{k} \rangle\geq-\tau(u_{k},y)\geq\tau(y,u_{k}),\quad\forall y\in C. \end{aligned}$$

(3.18)

By the assumption that the mapping $y\mapsto\tau(x,y)$ is convex and lower semi-continuous, letting $\mathcal{K}_{i}\ni k\rightarrow\infty$ in (3.18), from (3.16) and (3.17), we have $\tau(y,x^{*})\leq0$, $\forall y\in C$.

For any $t\in(0,1]$ and any $y\in C$, set $y_{t}=ty+(1-t)x^{*}$. Then $\tau(y_{t},x^{*})\leq0$ since $y_{t}\in C$. By condition (A₁) and (A₄), we have

$$0=\tau(y_{t},y_{t})\leq t\tau(y_{t},y)+(1-t) \tau\bigl(y_{t},x^{*}\bigr)\leq t\tau(y_{t},y). $$

Dividing both sides of the above equation by t, we have $\tau(y_{t},y)\geq0$, $\forall y\in C$. Letting $t\downarrow0$, from condition (A₃), we have $\tau(x^{*},y)\geq0$, $\forall y\in C$, i.e., $x^{*}\in\Omega$ and so $x^{*}\in G$.

(VI) $x^{*}=\Pi_{G}x_{1}$, and so $x_{n}\rightarrow\Pi_{G}x_{1}$ as $n\rightarrow\infty$.

Put $u=\Pi_{G}x_{1}$. Since $u\in G\subset C_{n}$ and $x_{n}=\Pi_{C_{n}}x_{1}$, we have $\phi(x_{n},x_{1})\leq\phi(u,x_{1})$, $\forall n\geq1$. Then

$$\begin{aligned} \phi\bigl(x^{*},x_{1}\bigr)=\lim_{n\rightarrow\infty} \phi(x_{n},x_{1})\leq\phi(u,x_{1}), \end{aligned}$$

(3.19)

which implies that $x^{*}=u$ since $u=\Pi_{Gx_{1}}$, and hence $x_{n}\rightarrow x^{*}=\Pi_{F}x_{1}$ as $n\rightarrow\infty$. This completes the proof. □

A numerical result is given as follows.

Example 3.2

Let $E=\mathbb{R}^{1}$ with the standard norm $\|\cdot\|=|\cdot|$ and $C=[0,1]$. Let $\{T_{i}\}^{\infty}_{i=1}:C\rightarrow2^{C}$ be a sequence of multi-valued nonlinear mappings defined by

$$T_{i}x= \biggl\{ \frac{(\lambda{x})^{i+1}}{i+1}:\lambda\in[0,1] \biggr\} . $$

Consider the following iteration sequence generated by

$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} x_{1}\in C;\quad C_{1}=C, \\ y_{n}=J^{-1}[\alpha_{n}Jx_{n}+(1-\alpha_{n})Jz_{n}], \\ z_{n}=J^{-1}[\beta_{n}Jx_{n}+(1-\beta_{n})Jw_{i_{n}}], \\ C_{n+1}=\{v\in C_{n}:\phi(v,y_{n})\leq\phi(v,x_{n})\}, \\ x_{n+1}=\Pi_{C_{n+1}}x_{1},\quad \forall n\geq1, \end{array}\displaystyle \right . \end{aligned}$$

(3.20)

where $w_{i}:=\frac{x^{i+1}}{i+1}\in T_{i}x$, $\{\alpha_{n}\}=\{\frac{2}{3}-\frac{1}{4n}\}$, $\{\beta_{n}\}=\{\frac{4}{5}-\frac{1}{2n}\}$, and $\Pi_{C_{n+1}}(x):=\arg\inf_{y\in C_{n+1}}|y-x|$. Note that $J=I$ and $\phi(x,y)=|x-y|^{2}$ for all $x,y\in E$ since E is a Hilbert space. Moreover, it is not difficult to obtain $C_{n+1}=[0,\frac{x_{n}+y_{n}}{2}]$ for all $n\geq1$. Then (3.20) is reduced to

$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} x_{1}\in C;\quad C_{1}=C, \\ y_{n}=(\frac{2}{3}-\frac{1}{4n})x_{n}+(\frac{1}{3}+\frac{1}{4n})z_{n}, \\ z_{n}=(\frac{4}{5}-\frac{1}{2n})x_{n}+(\frac{1}{5}+\frac{1}{2n})w_{i_{n}}, \\ C_{n+1}=\{v\in C_{n}:|v-y_{n}|\leq|v-x_{n}|\}, \\ x_{n+1}=\frac{x_{n}+y_{n}}{2},\quad \forall n\geq1, \end{array}\displaystyle \right . \end{aligned}$$

(3.21)

where $i_{n}$ is the solution to the positive integer equation: $n=i_{n}+\frac{(m_{n}-1)m_{n}}{2}$ ($m_{n}\geq i_{n}$, $n=1,2,\ldots$). It is clear that $\{T_{i}\}$ is a sequence of closed and totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings with a common fixed point zero. It then can be shown by similar way of Theorem 3.1 that $\{x_{n}\}$ converges strongly to zero. The numerical experiment outcome obtained by using MATLAB 7.10.0.499 shows that, as $x_{1}=1$, the computations of $x_{100}$, $x_{200}$, $x_{300}$, and $x_{400}$ are 0.023899039, 0.00074538945, 0.000024001481, and 0.00000078318587, respectively. This example illustrates the effectiveness of the introduced algorithm for countable families of totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings.