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

Open Access
Research

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.

Keywords

strong convergence generalized mixed equilibrium problems totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings 

MSC

47J06 47J25 

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, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 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. (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. (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. (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].
  1. (i)

    If E is uniformly smooth, then J is uniformly continuous on each bounded subset of E;

     
  2. (ii)

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

     
  3. (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;

     
  4. (iv)

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

     
  5. (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. (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. (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:
  1. (i)

    \(\kappa_{r}\) is single-valued;

     
  2. (ii)
    \(\kappa_{r}\) a firmly nonexpansive-type mapping, i.e.,
    $$\langle\kappa_{r}z-\kappa_{r}y,J\kappa_{r}z-J \kappa_{r}y\rangle\leq\langle \kappa_{r}z- \kappa_{r}y,Jz-Jy\rangle; $$
     
  3. (iii)

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

     
  4. (iv)

    Ω is a closed convex set of C;

     
  5. (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.

Notes

Acknowledgements

The author wishes to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions. This study is supported by the Major Science Foundation of Yunnan province education department (08Z0081).

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© Qian 2015

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Authors and Affiliations

  1. 1.School of Land Resources EngineeringKunming University of Science and TechnologyKunmingP.R. China
  2. 2.Architectural Engineering FacultyKunming Metallurgy CollegeKunmingP.R. China

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