Open Access

Strong convergence theorem for a common fixed point of a finite family of strictly pseudo-contractive mappings and a strictly pseudononspreading mapping

Fixed Point Theory and Applications20152015:116

https://doi.org/10.1186/s13663-015-0366-6

Received: 9 February 2015

Accepted: 25 June 2015

Published: 16 July 2015

Abstract

In this paper, we introduce a new mapping in a real Hilbert space to prove a strong convergence theorem for finding a common fixed point of a finite family of strictly pseudo-contractive mappings and a strictly pseudononspreading mapping. Moreover, we also obtain a strong convergence theorem for a finite family of inverse-strongly monotone mappings and a strictly pseudononspreading mapping.

Keywords

strictly pseudo-contractive mappingstrictly pseudononspreading mappinginverse-strongly monotone mappingstrong convergence

MSC

47H0947H1049J40

1 Introduction

In this paper, we assume that H is a real Hilbert space with the inner product \(\langle\cdot,\cdot\rangle\) and the induced norm \(\|\cdot\|\), and C is a nonempty closed convex subset of H. Let \(T: C\rightarrow C\) be a mapping. \(F(T )\) denotes the set of fixed points of the mapping T, i.e., \(F(T )=\{x\in C:Tx=x\}\).

Recall that a mapping \(T:C\rightarrow C\) is nonexpansive if
$$ \| Tx-Ty \|\leq\| x-y \|, \quad\forall x, y\in C. $$
(1.1)
A mapping \(T:C\rightarrow C\) is κ-strictly pseudo-contractive if there exists a constant \(\kappa\in[0, 1)\) such that
$$ \| Tx-Ty \|^{2}\leq\| x-y \|^{2}+ \kappa\bigl\| (I-T)x-(I-T)y\bigr\| ^{2},\quad \forall x, y\in C. $$
(1.2)
A mapping \(T:C\rightarrow C\) is ρ-strictly pseudononspreading if there exists a constant \(\rho\in[0, 1)\) such that
$$ \| Tx-Ty \|^{2}\leq\| x-y \|^{2}+ \rho\bigl\| (I-T)x-(I-T)y \bigr\| ^{2}+2\langle x-Tx,y-Ty \rangle,\quad \forall x, y\in C. $$
(1.3)
It is obvious that the 0-strictly pseudo-contractive mapping T is a nonexpansive mapping. Note that (1.2) is equivalent to
$$ \langle Tx-Ty,x-y\rangle\leq\| x-y\|^{2}- \frac{1-\kappa}{2}\bigl\| (I-T)x-(I-T)y\bigr\| ^{2}, \quad\forall x, y\in C, $$
(1.4)
and the κ-strictly pseudo-contractive mapping T is Lipschitz continuous with constant \(\frac{1+\kappa}{1-\kappa}\), that is,
$$ \| Tx-Ty\| \leq\frac{1+\kappa}{1-\kappa} \| x-y\|, \quad\forall x, y\in C. $$
(1.5)
A mapping \(T : C \rightarrow H \) is said to be ξ-inverse-strongly monotone if there exists a positive real number ξ such that
$$ \langle Tx-Ty,x-y\rangle\geq\xi\| Tx-Ty\|^{2},\quad \forall x, y\in C. $$
(1.6)

Finding the fixed points of nonexpansive mappings is an important topic in the theory of nonexpansive mappings, and it has wide applications in a number of applied areas such as the convex feasibility problem [13], the split feasibility problem [4], image recovery and signal processing [5]. After that, as an important generalization of nonexpansive mappings, strictly pseudo-contractive, strictly pseudononspreading and inverse-strongly monotone mappings became one of the most interesting studied classes of nonexpansive mappings. Iterative methods for them have been extensively investigated (see, e.g., [619] and the references contained therein).

In 2000, Takahashi and Shimoji [20] introduced a W-mapping generated by \(T_{1},T_{2}, \ldots,T_{r}\) and \(\alpha_{1}, \alpha_{2},\ldots,\alpha _{r}\) as follows.

Definition 1.1

[20]

Let C be a convex subset of a Banach space E. Let \(T_{1}, T_{2},\ldots ,T_{r}\) be finite mappings of C into itself, and let \(\alpha_{1}, \alpha_{2},\ldots,\alpha_{r}\) be real numbers such that \(0 \leq\alpha_{i}\leq1\) for every \(i = 1,2,\ldots,r\). Then we define a mapping W of C into itself as follows:
$$\begin{aligned}& U_{1}=\alpha_{1} T_{1}+(1-\alpha_{1})I, \\& U_{2}=\alpha_{2} T_{2}U_{1}+(1- \alpha_{2})I, \\& U_{3}=\alpha_{3} T_{3}U_{2}+(1- \alpha_{3})I, \\& \vdots \\& U_{r-1}=\alpha_{r-1} T_{r-1}U_{r-2}+(1- \alpha_{r-1})I, \\& W=U_{r}=\alpha_{r} T_{r}U_{r-1}+(1- \alpha_{r})I. \end{aligned}$$
Such a mapping W is called the W-mapping generated by \(T_{1},T_{2},\ldots,T_{r}\) and \(\alpha_{1}, \alpha_{2},\ldots,\alpha_{r}\).

Lemma 1.1

[20]

Let C be a closed convex subset of a Banach space E. Let \(T_{1},T_{2},\ldots,T_{r}\) be nonexpansive mappings of C into itself such that \(\bigcap_{i=1}^{r}F(T_{i})\) is nonempty, and let \(\alpha_{1}, \alpha _{2},\ldots,\alpha_{r}\) be real numbers such that \(0<\alpha_{i}<1\) for every \(i = 1,2,\ldots,r\). Let W be the W-mapping of C into itself generated by \(T_{1},T_{2},\ldots,T_{r}\) and \(\alpha_{1}, \alpha_{2},\ldots ,\alpha_{r}\). Then W is asymptotically regular. Further, if E is strictly convex, then \(F(W) =\bigcap_{i=1}^{r} F(T_{i})\).

In 2009, Kangtunyakarn and Suantai [21] gave a K-mapping generated by \(T_{1},T_{2},\ldots,T_{N}\) and \(\lambda_{1}, \lambda_{2},\ldots ,\lambda_{N}\) as follows.

Definition 1.2

[21]

Let C be a nonempty convex subset of a real Banach space. Let \(\{T_{i}\} ^{N}_{i=1}\) be a finite family of mappings of C into itself, and let \(\lambda_{1},\lambda_{2},\ldots, \lambda_{N}\) be real numbers such that \(0\leq\lambda_{i}\leq1\) for every \(i=1,2,\ldots, N\). We define a mapping \(K : C\rightarrow C\) as follows:
$$\begin{aligned}& U_{1}=\lambda_{1} T_{1}+(1-\lambda_{1})I, \\& U_{2}=\lambda_{2} T_{2}U_{1}+(1- \lambda_{2})U_{1}, \\& U_{3}=\lambda_{3} T_{3}U_{2}+(1- \lambda_{3})U_{2}, \\& \vdots \\& U_{N-1}=\lambda_{N-1} T_{N-1}U_{N-2}+(1- \lambda_{N-1})U_{N-2}, \\& K=U_{N}=\lambda_{N} T_{N}U_{N-1}+(1- \lambda_{N})U_{N-1}. \end{aligned}$$
Such a mapping K is called the K-mapping generated by \(T_{1}, T_{2},\ldots, T_{N}\) and \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{N}\).

In 2014, Suwannaut and Kangtunyakarn [22] established the following main result for the K-mapping generated by \(T_{1}, T_{2},\ldots T_{N}\) and \(\lambda_{1}, \lambda_{2},\ldots, \lambda_{N}\).

Lemma 1.2

[22]

Let C be a nonempty closed convex subset of a real Hilbert space H. Let \(\{T_{i}\}^{N}_{i=1}\) be a finite family of \(\kappa_{i}\)-strictly pseudo-contractive mappings of C into itself with \(\kappa_{i}\leq\gamma_{1}\) for all \(i=1, 2,\ldots, N\), and \(\bigcap_{i=1}^{N}F(T_{i})\neq\emptyset\). Let \(\lambda_{1},\lambda_{2},\ldots, \lambda_{N}\) be real numbers with \(0<\lambda _{i}<\gamma_{2}\) for all \(i=1, 2,\ldots, N\) and \(\gamma_{1}+\gamma_{2}<1\). Let K be the K-mapping generated by \(T_{1}, T_{2}, \ldots, T_{N}\) and \(\lambda_{1}, \lambda_{2},\ldots, \lambda_{N}\). Then the following properties hold:
  1. (i)

    \(F(K)=\bigcap_{i=1}^{N}F(T_{i})\);

     
  2. (ii)

    K is a nonexpansive mapping.

     

In 2009, Kangtunyakarn and Suantai [23] also introduced an S-mapping generated by \(T_{1},T_{2},\ldots,T_{N}\) and \(\alpha_{1}, \alpha _{2},\ldots,\alpha_{N}\) as follows.

Definition 1.3

[23]

Let C be a nonempty convex subset of a real Banach space. Let \(\{T_{i}\} ^{N}_{i=1}\) be a finite family of mappings of C into itself. For each \(j=1,2,\ldots,N\), let \(\alpha _{j}=(\alpha_{1}^{j},\alpha_{2}^{j},\alpha_{3}^{j})\), where \(\alpha_{1}^{j},\alpha _{2}^{j},\alpha_{3}^{j}\in[0,1]\) and \(\alpha_{1}^{j}+\alpha_{2}^{j}+\alpha_{3}^{j}=1\). We define the mapping \(S: C \rightarrow C\) as follows:
$$\begin{aligned}& U_{0}=I, \\& U_{1}=\alpha^{1}_{1} T_{1}U_{0}+ \alpha^{1}_{2}U_{0}+\alpha^{1}_{3} I, \\& U_{2}=\alpha^{2}_{1} T_{2}U_{1}+ \alpha^{2}_{2}U_{1}+\alpha^{2}_{3} I, \\& U_{3}=\alpha^{3}_{1} T_{3}U_{2}+ \alpha^{3}_{2}U_{2}+\alpha^{3}_{3} I, \\& \vdots \\& U_{N-1}=\alpha^{N-1}_{1} T_{N-1}U_{N-2}+ \alpha^{N-1}_{2}U_{N-2}+\alpha ^{N-1}_{3} I, \\& S=U_{N}=\alpha^{N}_{1} T_{N}U_{N-1}+ \alpha^{N}_{2}U_{N-1}+\alpha^{N}_{3} I. \end{aligned}$$
This mapping is called the S-mapping generated by \(T_{1},T_{2},\ldots ,T_{N}\) and \(\alpha_{1}, \alpha_{2},\ldots,\alpha_{N}\).

In 2010, Kangtunyakarn and Suantai [24] gave the following lemma for the S-mapping generated by \(T_{1},T_{2},\ldots,T_{N}\) and \(\alpha_{1}, \alpha_{2},\ldots,\alpha_{N}\).

Lemma 1.3

[24]

Let C be a nonempty closed convex subset of a real Hilbert space. Let \(\{T_{i}\}^{N}_{i=1}\) be a finite family of \(\kappa_{i}\)-strict pseudocontractive mappings of C into C with \(\bigcap_{i=1}^{N} F(T_{i})\neq\emptyset\) and \(\kappa=\max\{\kappa_{i}: i=1,2,\ldots,N\}\), and let \(\alpha_{j}=(\alpha_{1}^{j},\alpha_{2}^{j},\alpha_{3}^{j})\in I \times I\times I\), \(j=1,2,\ldots, N\), where \(I=[0, 1]\), \(\alpha_{1}^{j}+\alpha _{2}^{j}+\alpha_{3}^{j}=1\), \(\alpha_{1}^{j},\alpha_{3}^{j}\in(\kappa,1)\) for all \(j = 1, 2,\ldots, N-1\) and \(\alpha_{1}^{N}\in(\kappa,1]\), \(\alpha_{3}^{N}\in[\kappa ,1)\), \(\alpha_{2}^{j}\in[\kappa,1)\) for all \(j=1,2,\ldots,N\). Let S be the mapping generated by \(T_{1},T_{2},\ldots,T_{N}\) and \(\alpha_{1}, \alpha_{2},\ldots,\alpha_{N}\). Then \(F(S) = \bigcap_{i=1}^{N} F(T_{i})\) and S is a nonexpansive mapping.

Let \(T: C\rightarrow H\). The variational inequality problem is to find a point \(x\in C\) such that
$$ \langle Ax,y-x\rangle\geq0,\quad \forall y\in C. $$
(1.7)
The set of solutions of (1.7) is denoted by \(VI(C,A)\).

In the recent years, there have been many research works concerning the problem of approximating a common fixed point of various classes of nonlinear mappings by using W-mappings, K-mappings and S-mappings (see, e.g., [2043]).

Recently, Kangtunyakarn [44] proposed an iterative algorithm for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and a finite family of the set of solutions of variational inequality problems as follows.

Theorem 1.1

[44]

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. For every \(i=1,2,\ldots, N\), let \(B_{i} : C\rightarrow H\) be \(\delta_{i}\)-inverse strongly monotone mappings and let \(T : C \rightarrow C\) be a κ-strictly pseudononspreading mapping for some \(\kappa\in[0,1)\). Let \(G_{i} : C \rightarrow C\) be defined by \(G_{i}x=P_{C}(I-\eta B_{i})x\) for every \(x\in C\) and \(\eta\in(0,2\delta_{i})\) for every \(i=1,2,\ldots, N\), and let \(\delta_{j}=(\alpha_{1}^{j}, \alpha_{2}^{j}, \alpha_{3}^{j})\in I\times I\times I\), \(j=1,2,\ldots,N\), where \(I=[0,1]\), \(\alpha_{1}^{j}+\alpha_{2}^{j}+ \alpha_{3}^{j}=1\), \(\alpha_{1}^{j}\in(0,1)\) for all \(j=1,2,\ldots,N-1\), \(\alpha_{1}^{N}\in (0,1]\), \(\alpha_{2}^{j}, \alpha_{3}^{j}\in[0,1)\) for all \(j=1,2,\ldots,N\). Let \(S: C \rightarrow C\) be the S-mapping generated by \(G_{1},G_{2},\ldots,G_{N}\) and \(\delta_{1}, \delta_{2},\ldots, \delta_{N}\). Assume that \(\mathfrak{F} = F(T)\cap \bigcap^{N}_{i=1}VI(C, B_{i})\neq\emptyset\). For every \(n\in\mathbb{N}\), \(i=1,2,\ldots, N\), let \(x_{1}, u \in C\) and \(\{ x_{n}\}\) be a sequence generated by
$$ x_{n+1}=\alpha_{n} u+\beta_{n} P_{C} \bigl(I-\lambda_{n}(I-T) \bigr)x_{n}+\gamma_{n} Sx_{n},\quad \forall n\in\mathbb{N}, $$
(1.8)
where \(\{\alpha_{n}\}, \{\beta_{n}\}, \{\gamma_{n}\}, \{\lambda_{n}\} \subset(0,1)\) such that \(\alpha_{n} + \beta_{n} + \gamma_{n} =1\), \(\beta_{n}\in[c,d] \subset(0,1)\), \(\{\lambda_{n}\}\subset(0,1-\kappa)\) and suppose the following conditions hold:
  1. (i)

    \(\lim_{n\rightarrow\infty} \alpha_{n}=0\) and \(\sum_{n=0}^{\infty}\alpha_{n}=\infty\);

     
  2. (ii)

    \(\sum_{n=1}^{\infty}\lambda_{n}< \infty\);

     
  3. (iii)

    \(\sum_{n=1}^{\infty}|\lambda_{n+1}-\lambda_{n}|, \sum_{n=1}^{\infty}|\gamma_{n+1}-\gamma_{n}|, \sum_{n=1}^{\infty}|\alpha _{n+1}-\alpha_{n}|, \sum_{n=1}^{\infty}|\beta_{n+1}-\beta_{n}|<\infty\).

     
Then the sequence \(\{x_{n}\}\) converges strongly to \(z=P_{\mathfrak{F}}u\).

Motivated and inspired by the above facts, we define a new mapping for the common fixed point set of a finite family of strict pseudo-contractive mappings. Moreover, by using our main result, we also obtain a new strong convergence theorem for the common fixed point of a finite family of strict pseudo-contractive mappings and a strictly pseudononspreading mapping.

2 Preliminaries

Lemma 2.1

In the real Hilbert space H, the following relations hold:
  1. (i)

    \(\|x+y\|^{2}=\|x\|^{2}+2\langle x,y\rangle+\|y\|^{2}\);

     
  2. (ii)

    \(\|x+y\|^{2}\leq\|x\|^{2}+2 \langle y, x+y\rangle\);

     
  3. (iii)

    \(\|\sum_{i=1}^{m} \alpha_{i} x_{i}\|^{2}=\sum_{i=1}^{m} \alpha_{i}\| x_{i}\|^{2}-\sum_{i\neq j} \alpha_{i} \alpha_{j}\|x_{i}-x_{j}\|^{2}\)

     
for \(\sum_{i=1}^{m} \alpha_{i}=1\), \(\alpha_{i}\in[0,1]\), \(\forall i\in\{ 1,2,\ldots,m\}\).

Definition 2.1

\(P_{C}: H \rightarrow C\) is called a metric projection if for every point \(x\in H\), there exists a unique nearest point in C, denoted by \(P_{C}x\), such that
$$ \| x-P_{C}x \|\leq\| x-y \|,\quad \forall y\in C. $$
(2.1)

Lemma 2.2

Let C be a nonempty closed convex subset of H and \(P_{C}: H \rightarrow C\) be a metric projection. Then
  1. (i)

    \(\| P_{C}x-P_{C}y\|^{2} \leq\langle x-y,P_{C}x-P_{C}y\rangle\), \(\forall x, y\in H\);

     
  2. (ii)

    \(P_{C}\) is a nonexpansive mapping, i.e., \(\| P_{C}x-P_{C}y\|\leq\| x-y\|\), \(\forall x, y\in H\);

     
  3. (iii)

    \(\langle x-P_{C}x,y-P_{C}x\rangle\leq0\), \(\forall x\in H\), \(y\in C \).

     

From the proof of Theorem 3.1 in [44], we have the following results.

Lemma 2.3

[44]

Let C be a nonempty closed convex subset of H and \(T : C \rightarrow C\) be a ρ-strictly pseudononspreading mapping with \(F(T)\neq\emptyset\). Then
$$ \bigl\| P_{C} \bigl(I-\lambda(I-T) \bigr)x-x^{*}\bigr\| \leq\bigl\| x-x^{*}\bigr\| $$
(2.2)
for any \(\lambda\in(0,1-\rho)\), \(x^{*}\in F(T)\).

Lemma 2.4

[44]

Let C be a nonempty closed convex subset of H and \(T : C \rightarrow C\) be a ρ-strictly pseudononspreading mapping with \(F(T)\neq\emptyset\). Then
$$ \bigl\| Tx-x^{*}\bigr\| \leq\frac{1+\rho}{1-\rho}\bigl\| x-x^{*}\bigr\| $$
(2.3)
for any \(x^{*}\in F(T)\).

Lemma 2.5

[45]

Let \(\{s_{n}\}\) be a sequence of nonnegative real numbers such that
$$ s_{n+1}\leq(1-\alpha_{n})s_{n}+ \beta_{n}, \quad\forall n\geq0, $$
(2.4)
where \(\{\alpha_{n}\}\) is a sequence in \((0,1)\) and \(\{\beta_{n}\}\) is a sequence such that
  1. (i)

    \(\sum_{n=0}^{\infty}\alpha_{n}=\infty\);

     
  2. (ii)

    \(\limsup_{n\rightarrow\infty}\frac{\beta_{n}}{\alpha_{n}}\leq 0\) or \(\sum_{n=0}^{\infty}|\beta_{n}|<\infty\).

     
Then \(\lim_{n\rightarrow\infty}s_{n}=0\).

Lemma 2.6

[45]

Let \(\{s_{n}\}\) be a sequence of nonnegative numbers such that
$$ s_{n+1}\leq(1-\alpha_{n})s_{n}+ \alpha_{n}\beta_{n}, \quad\forall n\geq0, $$
(2.5)
where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are sequences of real numbers such that
  1. (i)

    \(\{\alpha_{n}\}\subset[0,1]\) and \(\sum_{n=0}^{\infty}\alpha _{n}=\infty\);

     
  2. (ii)

    \(\limsup_{n\rightarrow\infty}\beta_{n}\leq0\) or \(\sum_{n=0}^{\infty}\alpha_{n}\beta_{n}<\infty\).

     
Then \(\lim_{n\rightarrow\infty}s_{n}=0\).
Let C be a nonempty subset of H and \(T : C \rightarrow H\) be a mapping. Then T is said to be demi-closed at \(v\in H\) if for any sequence \(\{x_{n}\}\subseteq C\), the following implication holds:
$$ x_{n} \rightharpoonup u\in C \quad\mbox{and}\quad Tx_{n} \rightarrow v \quad\mbox{imply}\quad Tu=v, $$
(2.6)
where → (resp. ) denotes strong (resp. weak) convergence.

Lemma 2.7

[46]

Let C be a nonempty closed convex subset of H and \(T: C\rightarrow H\) be a nonexpansive mapping. Then the mapping \(I-T\) is demi-closed at zero.

Lemma 2.8

(Opial’s property [47])

If \(x_{n}\rightharpoonup u\), then the following inequality holds:
$$ \liminf_{n\rightarrow\infty}\|x_{n}-y\|>\liminf _{n\rightarrow\infty}\| x_{n}-u\|, \quad\forall y\in H, y\neq u. $$
(2.7)

We define a new mapping as follows.

Definition 2.2

Let C be a nonempty convex subset of a Banach space E. Let \(\{T_{i}\} ^{N}_{i=1}\) be a finite family of mappings of C into itself. For each \(i=1,2,\ldots,N\), let \(\pi _{i}=(\alpha_{i},\beta_{i},\gamma_{i}, \delta_{i})\), where \(\alpha_{i},\beta _{i},\gamma_{i}, \delta_{i}\in[0,1]\) and \(\alpha_{i}+\beta_{i}+\gamma_{i}+\delta_{i}=1\). We define the mapping \(G: C \rightarrow C\) as follows:
$$\begin{aligned}& U_{0}=I, \\& U_{1}= \alpha_{1} T_{1}^{2} U_{0}+\beta_{1} T_{1} U_{0}+ \gamma_{1} U_{0}+\delta_{1} I, \\& U_{2}= \alpha_{2} T_{2}^{2} U_{1}+\beta_{2} T_{2} U_{1}+ \gamma_{2} U_{1}+\delta_{2} I, \\& U_{3}= \alpha_{3} T_{3}^{2} U_{2}+\beta_{3} T_{3} U_{2}+ \gamma_{3} U_{2}+\delta_{3} I, \\& \vdots \\& U_{N-1}=\alpha_{N-1} T_{N-1}^{2} U_{N-2}+\beta_{N-1} T_{N-1} U_{N-2}+ \gamma_{N-1} U_{N-2}+\delta_{N-1} I, \\& G=U_{N}=\alpha_{N} T_{N}^{2} U_{N-1}+\beta_{N} T_{N} U_{N-1}+ \gamma_{N} U_{N-1}+\delta_{N} I. \end{aligned}$$
This mapping is called the G-mapping generated by \(T_{1},T_{2},\ldots ,T_{N}\) and \(\pi_{1}, \pi_{2},\ldots,\pi_{N}\).

We remark that (i) if \(\alpha_{i}=0\) for every \(i=1,2,\ldots,N\), then G-mapping is reduced to S-mapping; (ii) if \(\alpha_{i}=0\) and \(\gamma_{i}=0\) for every \(i=1,2,\ldots, N\), then G-mapping is reduced to W-mapping; (iii) if \(\alpha_{i}=0\) and \(\delta_{i}=0\) for every \(i=1,2,\ldots, N\), then G-mapping is reduced to K-mapping.

Lemma 2.9

Let C be a nonempty closed convex subset of the real Hilbert space H. For every \(i=1,2,\ldots, N\), let \(T_{i}: C \rightarrow C\) be \(\kappa _{i}\)-strict pseudo-contractive mappings with \(\bigcap_{i=1}^{N}F( T_{i})\neq\emptyset\), and let \(\pi_{i}=(\alpha _{i},\beta_{i},\gamma_{i}, \delta_{i})\), where \(\alpha_{i},\beta_{i},\gamma_{i}, \delta_{i}\in[0,1]\) and \(\alpha_{i}+\beta_{i}+\gamma_{i}+\delta_{i}=1\). Let G be the G-mapping generated by \(T_{1},T_{2},\ldots,T_{N}\) and \(\pi _{1}, \pi_{2},\ldots,\pi_{N}\). If the following conditions hold:
  1. (i)

    \(\kappa_{1}\leq\beta_{1}<1-\kappa_{1}\) and \(\alpha_{1}(\kappa _{1}+\beta_{1})<\beta_{1}(1-\beta_{1}-\kappa_{1})\);

     
  2. (ii)

    \(\beta_{i}\geq\kappa_{i}\), \(\kappa_{i}<\gamma_{i}<1\) and \(\kappa _{i}\alpha_{i}\leq\beta_{i}\gamma_{i}-\beta_{i}\kappa_{i}\) for \(i=2,3,\ldots, N\).

     
Then \(F(G) = \bigcap_{i=1}^{N} F(T_{i})\) and G is a nonexpansive mapping.

Proof

It is clear that \(\bigcap_{i=1}^{N} F(T_{i}) \subseteq F(G)\). Next, we will show that \(F(G)\subseteq\bigcap_{i=1}^{N} F(T_{i})\).

Let \(x_{0}\in F(G)\) and \(x^{*}\in\bigcap_{i=1}^{N} F(T_{i})\), then we have
$$\begin{aligned} & \bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ &\quad=\bigl\| Gx_{0}-x^{*} \bigr\| ^{2} \\ &\quad=\bigl\| \alpha_{N} \bigl(T_{N}^{2} U_{N-1} x_{0}-x^{*} \bigr)+\beta_{N} \bigl(T_{N} U_{N-1} x_{0}-x^{*} \bigr)+ \gamma_{N} \bigl(U_{N-1} x_{0}-x^{*} \bigr)+ \delta_{N} \bigl(x_{0}-x^{*} \bigr)\bigr\| ^{2} \\ &\quad=\alpha_{N}\bigl\| T_{N}^{2} U_{N-1} x_{0}-x^{*}\bigr\| ^{2}+\beta_{N} \bigl\| T_{N} U_{N-1} x_{0}-x^{*}\bigr\| ^{2} \\ &\qquad{}+\gamma_{N} \bigl\| U_{N-1} x_{0}-x^{*}\bigr\| ^{2}+\delta_{N} \bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ &\qquad{} -\alpha_{N}\beta_{N} \bigl\| T_{N}^{2} U_{N-1} x_{0}-T_{N} U_{N-1} x_{0} \bigr\| ^{2}- \alpha_{N} \gamma_{N}\bigl\| T_{N}^{2} U_{N-1} x_{0}-U_{N-1} x_{0} \bigr\| ^{2} \\ &\qquad{}-\alpha_{N} \delta_{N} \bigl\| T_{N}^{2} U_{N-1} x_{0}-x_{0}\bigr\| ^{2} -\beta_{N} \gamma_{N}\|T_{N} U_{N-1} x_{0}-U_{N-1} x_{0} \|^{2} \\ &\qquad{}-\beta _{N}\delta_{N}\|T_{N} U_{N-1} x_{0}-x_{0}\|^{2} - \gamma_{N} \delta_{N}\|U_{N-1} x_{0}-x_{0} \|^{2} \\ &\quad\leq\alpha_{N}\bigl\| T_{N}^{2} U_{N-1} x_{0}-x^{*}\bigr\| ^{2}+\beta_{N} \bigl\| T_{N} U_{N-1} x_{0}-x^{*}\bigr\| ^{2} \\ &\qquad{}+\gamma_{N} \bigl\| U_{N-1} x_{0}-x^{*}\bigr\| ^{2}+\delta_{N}\bigl\| x_{0}-x^{*}\bigr\| ^{2} -\alpha_{N}\beta_{N} \bigl\| T_{N}^{2} U_{N-1} x_{0}-T_{N} U_{N-1} x_{0} \bigr\| ^{2} \\ &\qquad{}-\beta_{N} \gamma_{N}\|T_{N} U_{N-1} x_{0}-U_{N-1} x_{0}\|^{2} \\ &\quad \leq\alpha_{N} \bigl(\bigl\| T_{N} U_{N-1} x_{0}-x^{*}\bigr\| ^{2}+\kappa_{N}\bigl\| (I-T_{N})T_{N} U_{N-1} x_{0}\bigr\| ^{2} \bigr) \\ &\qquad{}+\beta_{N} \bigl\| T_{N} U_{N-1} x_{0}-x^{*}\bigr\| ^{2}+ \gamma_{N}\bigl\| U_{N-1} x_{0}-x^{*}\bigr\| ^{2}+\delta_{N}\bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ &\qquad{} - \alpha_{N}\beta_{N} \bigl\| T_{N}^{2} U_{N-1} x_{0}-T_{N} U_{N-1} x_{0} \bigr\| ^{2}-\beta_{N} \gamma_{N}\|T_{N} U_{N-1} x_{0}-U_{N-1} x_{0}\|^{2} \\ & \quad=(\alpha_{N}+\beta_{N})\bigl\| T_{N} U_{N-1} x_{0}-x^{*}\bigr\| ^{2}+\alpha_{N}(\kappa _{N}- \beta_{N})\bigl\| T_{N}^{2} U_{N-1} x_{0}-T_{N} U_{N-1} x_{0} \bigr\| ^{2} \\ &\qquad{}+\gamma_{N}\bigl\| U_{N-1} x_{0}-x^{*} \bigr\| ^{2}+ \delta_{N}\bigl\| x_{0}-x^{*}\bigr\| ^{2}-\beta _{N} \gamma_{N}\|T_{N} U_{N-1} x_{0}-U_{N-1} x_{0}\|^{2} \\ & \quad\leq(\alpha_{N}+\beta_{N}) \bigl(\bigl\| U_{N-1} x_{0}-x^{*}\bigr\| ^{2}+\kappa_{N}\bigl\| (I-T_{N}) U_{N-1} x_{0} \bigr\| ^{2} \bigr) \\ &\qquad{}+\alpha_{N}( \kappa_{N}-\beta_{N})\bigl\| T_{N}^{2} U_{N-1} x_{0}-T_{N} U_{N-1} x_{0} \bigr\| ^{2} \\ &\qquad{}+\gamma_{N}\bigl\| U_{N-1} x_{0}-x^{*} \bigr\| ^{2}+ \delta_{N}\bigl\| x_{0}-x^{*}\bigr\| ^{2}-\beta _{N} \gamma_{N}\|T_{N} U_{N-1} x_{0}-U_{N-1} x_{0}\|^{2} \\ & \quad=(1-\delta_{N})\bigl\| U_{N-1} x_{0}-x^{*} \bigr\| ^{2}+ \bigl(1-(1-\delta_{N}) \bigr)\bigl\| x_{0}-x^{*} \bigr\| ^{2} \\ &\qquad{}+\alpha_{N}(\kappa_{N}-\beta_{N}) \bigl\| T_{N}^{2} U_{N-1} x_{0}-T_{N} U_{N-1} x_{0}\bigr\| ^{2} \\ &\qquad{} + \bigl((\alpha_{N}+\beta_{N}) \kappa_{N}- \beta_{N} \gamma_{N} \bigr)\| T_{N} U_{N-1} x_{0}-U_{N-1} x_{0} \|^{2} \\ & \quad\leq(1-\delta_{N})\bigl\| U_{N-1} x_{0}-x^{*} \bigr\| ^{2}+ \bigl(1-(1-\delta_{N}) \bigr)\bigl\| x_{0}-x^{*} \bigr\| ^{2} \\ &\quad\vdots \\ &\quad\leq(1-\delta_{N}) \bigl[(1-\delta_{N-1})\bigl\| U_{N-2} x_{0}-x^{*}\bigr\| ^{2}+ \bigl(1-(1- \delta_{N-1}) \bigr)\bigl\| x_{0}-x^{*}\bigr\| ^{2} \bigr] \\ &\qquad{}+ \bigl(1-(1-\delta_{N}) \bigr)\bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ &\quad=(1-\delta_{N}) (1-\delta_{N-1})\bigl\| U_{N-2} x_{0}-x^{*}\bigr\| ^{2}+ \bigl(1-(1-\delta _{N}) (1- \delta_{N-1}) \bigr)\bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ &\quad\vdots \\ & \quad\leq \prod_{i=3}^{N} (1- \delta_{i})\bigl\| U_{2} x_{0}-x^{*}\bigr\| ^{2}+ \Biggl(1-\prod_{i=3}^{N} (1- \delta_{i}) \Biggr)\bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ & \quad\leq \prod_{i=3}^{N} (1- \delta_{i}) \bigl[(1-\delta_{2}) \bigl\| U_{1} x_{0}-x^{*}\bigr\| ^{2}+\delta_{2}\bigl\| x_{0}-x^{*} \bigr\| ^{2} \\ &\qquad{}+ \alpha_{2}(\kappa_{2}-\beta_{2} )\bigl\| T_{2}^{2} U_{1} x_{0}-T_{2} U_{1} x_{0}\bigr\| ^{2} + \bigl((\alpha_{2}+\beta_{2}) \kappa_{2} - \beta_{2} \gamma_{2} \bigr)\| T_{2} U_{1} x_{0}-U_{1} x_{0} \|^{2} \bigr] \\ &\qquad{}+ \Biggl(1-\prod _{i=3}^{N} (1-\delta_{i}) \Biggr) \bigl\| x_{0}-x^{*}\bigr\| ^{2} \end{aligned}$$
(2.8)
$$\begin{aligned} & \quad\leq \prod_{i=2}^{N} (1- \delta_{i})\bigl\| U_{1} x_{0}-x^{*} \bigr\| ^{2}+ \Biggl(1-\prod_{i=2}^{N} (1- \delta_{i}) \Biggr)\bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ & \quad= \prod_{i=2}^{N} (1- \delta_{i})\bigl\| \alpha_{1} \bigl(T_{1}^{2} x_{0}-x^{*} \bigr)+\beta_{1} \bigl( T_{1} x_{0}-x^{*} \bigr)+(1-\alpha_{1}-\beta_{1}) \bigl(x_{0}-x^{*} \bigr)\bigr\| ^{2} \\ &\qquad{}+ \Biggl(1-\prod _{i=2}^{N} (1-\delta_{i}) \Biggr) \bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ & \quad= \prod_{i=2}^{N} (1- \delta_{i}) \bigl[\alpha_{1}\bigl\| T_{1}^{2} x_{0}-x^{*}\bigr\| ^{2}+\beta _{1}\bigl\| T_{1} x_{0}-x^{*}\bigr\| ^{2}+(1-\alpha_{1}-\beta_{1}) \bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ &\qquad{}-\alpha_{1}\beta_{1}\bigl\| T_{1}^{2} x_{0}-T_{1} x_{0} \bigr\| ^{2} -\alpha_{1}(1-\alpha_{1}-\beta_{1}) \bigl\| T_{1}^{2} x_{0}-x_{0}\bigr\| ^{2} \\ &\qquad{}- \beta_{1}(1-\alpha _{1}-\beta_{1})\|T_{1} x_{0}-x_{0}\|^{2} \bigr]+ \Biggl(1-\prod _{i=2}^{N} (1-\delta_{i}) \Biggr) \bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ & \quad\leq\prod_{i=2}^{N} (1- \delta_{i}) \bigl[\alpha_{1}\bigl\| T_{1}^{2} x_{0}-x^{*}\bigr\| ^{2}+\beta _{1}\bigl\| T_{1} x_{0}-x^{*}\bigr\| ^{2}+(1-\alpha_{1}-\beta_{1}) \bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ &\qquad{}-\alpha_{1}\beta_{1}\bigl\| T_{1}^{2} x_{0}-T_{1} x_{0} \bigr\| ^{2} -\beta_{1}(1-\alpha_{1}-\beta_{1}) \|T_{1} x_{0}-x_{0}\|^{2} \bigr] \\ &\qquad{}+ \Biggl(1-\prod_{i=2}^{N} (1- \delta_{i}) \Biggr)\bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ & \quad\leq\prod_{i=2}^{N} (1- \delta_{i}) \bigl[\alpha_{1} \bigl(\bigl\| T_{1} x_{0}-x^{*}\bigr\| ^{2}+\kappa_{1}\bigl\| (I-T_{1})T_{1} x_{0}\bigr\| ^{2} \bigr)+\beta_{1}\bigl\| T_{1} x_{0}-x^{*}\bigr\| ^{2} \\ &\qquad{}+(1-\alpha _{1}- \beta_{1}) \bigl\| x_{0}-x^{*}\bigr\| ^{2} -\alpha_{1}\beta_{1}\bigl\| T_{1}^{2} x_{0}-T_{1} x_{0}\bigr\| ^{2}- \beta_{1}(1-\alpha_{1}-\beta _{1})\|T_{1} x_{0}-x_{0}\|^{2} \bigr] \\ &\qquad{}+ \Biggl(1-\prod _{i=2}^{N} (1-\delta_{i}) \Biggr)\bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ & \quad= \prod_{i=2}^{N} (1- \delta_{i}) \bigl[(\alpha_{1}+\beta_{1}) \bigl\| T_{1} x_{0}-x^{*}\bigr\| ^{2}+\alpha_{1}( \kappa_{1}-\beta_{1})\bigl\| T_{1}^{2} x_{0}-T_{1} x_{0}\bigr\| ^{2} \\ &\qquad{}+(1- \alpha_{1}-\beta _{1}) \bigl\| x_{0}-x^{*}\bigr\| ^{2} -\beta_{1}(1-\alpha_{1}-\beta_{1}) \|T_{1} x_{0}-x_{0}\|^{2} \bigr] \\ &\qquad{}+ \Biggl(1-\prod_{i=2}^{N} (1- \delta_{i}) \Biggr)\bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ & \quad\leq\prod_{i=2}^{N} (1- \delta_{i}) \bigl[(\alpha_{1}+\beta_{1}) \bigl(\bigl\| x_{0}-x^{*}\bigr\| ^{2}+\kappa_{1} \bigl\| (I-T_{1})x_{0} \bigr\| ^{2} \bigr)+\alpha_{1}(\kappa_{1}- \beta_{1})\bigl\| T_{1}^{2} x_{0}-T_{1} x_{0}\bigr\| ^{2} \\ &\qquad{} +(1-\alpha_{1}-\beta_{1}) \bigl\| x_{0}-x^{*} \bigr\| ^{2}-\beta_{1}(1-\alpha_{1}-\beta_{1})\| T_{1} x_{0}-x_{0}\|^{2} \bigr] \\ &\qquad{}+ \Biggl(1-\prod_{i=2}^{N} (1- \delta_{i}) \Biggr)\bigl\| x_{0}-x^{*}\bigr\| ^{2} \\ & \quad= \prod_{i=2}^{N} (1- \delta_{i}) \bigl[\bigl\| x_{0}-x^{*}\bigr\| ^{2}+ \alpha_{1}(\kappa _{1}-\beta_{1}) \bigl\| T_{1}^{2} x_{0}-T_{1} x_{0} \bigr\| ^{2} \\ &\qquad{}+ \bigl((\alpha_{1}+\beta_{1})\kappa_{1}- \beta_{1}(1-\alpha_{1}-\beta_{1}) \bigr)\| T_{1} x_{0}-x_{0}\|^{2} \bigr] \\ &\qquad{}+ \Biggl(1-\prod_{i=2}^{N} (1- \delta_{i}) \Biggr)\bigl\| x_{0}-x^{*}\bigr\| ^{2}. \end{aligned}$$
(2.9)
By the condition (i), we have
$$ \alpha_{1}(\kappa_{1}-\beta_{1}) \bigl\| T_{1}^{2} x_{0}-T_{1} x_{0} \bigr\| ^{2} + \bigl((\alpha_{1}+\beta_{1}) \kappa_{1}-\beta_{1}(1-\alpha_{1}- \beta_{1}) \bigr)\|T_{1} x_{0}-x_{0} \|^{2}\leq0. $$
(2.10)
From (2.9) and \(\delta_{i}<1\) for \(i=2,3,\ldots,N\), it yields
$$ \alpha_{1}(\kappa_{1}-\beta_{1}) \bigl\| T_{1}^{2} x_{0}-T_{1} x_{0} \bigr\| ^{2} + \bigl((\alpha_{1}+\beta_{1}) \kappa_{1}-\beta_{1}(1-\alpha_{1}- \beta_{1}) \bigr)\|T_{1} x_{0}-x_{0} \|^{2}\geq0. $$
(2.11)
This implies that
$$ \|T_{1} x_{0}-x_{0}\|=0. $$
(2.12)
Therefore, \(T_{1} x_{0}=x_{0}\), that is, \(x_{0}\in F(T_{1})\). By the definition of \(U_{1}\), we have
$$\begin{aligned} U_{1}x_{0} =&\alpha_{1} T_{1}^{2} U_{0} x_{0}+ \beta_{1} T_{1} U_{0} x_{0}+ \gamma_{1} U_{0} x_{0}+\delta_{1} x_{0} \\ =&\alpha_{1} T_{1}^{2} x_{0}+ \beta_{1} T_{1} x_{0}+\gamma_{1} x_{0}+\delta_{1} x_{0} \\ =&\alpha_{1} T_{1} x_{0}+\beta_{1} x_{0}+\gamma_{1} x_{0}+\delta_{1} x_{0} \\ =&x_{0}. \end{aligned}$$
(2.13)
Again, by (2.8), (2.13) and \(\delta_{i}<1\) for \(i=3,4,\ldots,N\), we have
$$\begin{aligned} & \alpha_{2}(\kappa_{2}-\beta_{2} ) \bigl\| T_{2}^{2} U_{1} x_{0}-T_{2} U_{1} x_{0}\bigr\| ^{2} + \bigl((\alpha_{2}+ \beta_{2})\kappa_{2} -\beta_{2} \gamma_{2} \bigr)\| T_{2} U_{1} x_{0}-U_{1} x_{0}\|^{2} \\ & \quad=\alpha_{2}(\kappa_{2}-\beta_{2} ) \bigl\| T_{2}^{2} x_{0}-T_{2} x_{0}\bigr\| ^{2} + \bigl((\alpha_{2}+ \beta_{2})\kappa_{2} -\beta_{2} \gamma_{2} \bigr)\|T_{2} x_{0}- x_{0} \|^{2} \\ & \quad\geq0. \end{aligned}$$
(2.14)
From the condition (ii), this implies
$$ \|T_{2}x_{0}-x_{0}\|=0. $$
(2.15)
Therefore, \(T_{2} x_{0}=x_{0}\), that is, \(x_{0}\in F(T_{2})\). By the definition of \(U_{2}\), we also have
$$ U_{2}x_{0}=x_{0}. $$
(2.16)
Using the same argument, we can conclude that
$$ x_{0}\in F(T_{i}), \quad i=3,4,\ldots, N. $$
(2.17)
Hence, \(F(G)\subseteq\bigcap_{i=1}^{N} F(T_{i})\).
Now, we show that G is nonexpansive. Let any \(x,y\in C\). Then
$$\begin{aligned} & \|Gx-Gy\|^{2} \\ & \quad=\bigl\| \alpha_{N} \bigl(T_{N}^{2} U_{N-1} x-T_{N}^{2} U_{N-1} y \bigr)+ \beta_{N} (T_{N} U_{N-1} x-T_{N} U_{N-1} y) \\ &\qquad{}+ \gamma_{N} (U_{N-1} x-U_{N-1} y)+ \delta_{N}( x-y)\bigr\| ^{2} \\ & \quad\leq\alpha_{N}\bigl\| T_{N}^{2} U_{N-1} x-T_{N}^{2} U_{N-1} y\bigr\| ^{2}+ \beta_{N} \| T_{N} U_{N-1} x-T_{N} U_{N-1} y\|^{2} \\ &\qquad{}+\gamma_{N}\|U_{N-1} x-U_{N-1} y\| ^{2}+\delta_{N}\| x-y\|^{2} \\ &\qquad{} -\alpha_{N}\beta_{N} \bigl\| (I-T_{N})T_{N} U_{N-1} x-(I-T_{N})T_{N} U_{N-1} y \bigr\| ^{2} \\ &\qquad{}-\beta_{N} \gamma_{N}\bigl\| (I-T_{N}) U_{N-1} x-(I-T_{N}) U_{N-1} y\bigr\| ^{2} \\ & \quad\leq\alpha_{N} \bigl(\|T_{N} U_{N-1} x-T_{N} U_{N-1} y\|^{2}+\kappa_{N}\bigl\| (I-T_{N})T_{N} U_{N-1} x-(I-T_{N})T_{N} U_{N-1} y\bigr\| ^{2} \bigr) \\ &\qquad{} +\beta_{N} \|T_{N} U_{N-1} x-T_{N} U_{N-1} y\|^{2}+\gamma_{N}\| U_{N-1} x-U_{N-1} y\|^{2}+\delta_{N}\| x-y \|^{2} \\ &\qquad{} -\alpha_{N}\beta_{N} \bigl\| (I-T_{N})T_{N} U_{N-1} x-(I-T_{N})T_{N} U_{N-1} y \bigr\| ^{2} \\ &\qquad{}-\beta_{N} \gamma_{N}\bigl\| (I-T_{N}) U_{N-1} x-(I-T_{N}) U_{N-1} y\bigr\| ^{2} \\ & \quad=(\alpha_{N}+\beta_{N})\bigl\| T_{N} U_{N-1} x-T_{N} U_{N-1} y\bigr\| ^{2} \\ &\qquad{}+\alpha _{N}( \kappa_{N}-\beta_{N})\bigl\| (I-T_{N})T_{N} U_{N-1} x-(I-T_{N})T_{N} U_{N-1} y\bigr\| ^{2} \\ &\qquad{} +\gamma_{N}\|U_{N-1} x-U_{N-1} y \|^{2}+ \delta_{N}\| x-y\|^{2} \\ &\qquad{}-\beta _{N} \gamma_{N}\bigl\| (I-T_{N}) U_{N-1} x-(I-T_{N}) U_{N-1} y\bigr\| ^{2} \\ & \quad\leq(\alpha_{N}+\beta_{N}) \bigl(\| U_{N-1} x-U_{N-1} y\|^{2}+\kappa_{N}\bigl\| (I-T_{N})U_{N-1} x-(I-T_{N})U_{N-1} y\bigr\| ^{2} \bigr) \\ &\qquad{} +\alpha_{N}(\kappa_{N}-\beta_{N}) \bigl\| (I-T_{N})T_{N} U_{N-1} x-(I-T_{N})T_{N} U_{N-1} y\bigr\| ^{2} \\ &\qquad{}+\gamma_{N}\|U_{N-1} x-U_{N-1} y \|^{2}+ \delta_{N}\| x-y\|^{2} \\ &\qquad{}-\beta _{N} \gamma_{N}\bigl\| (I-T_{N}) U_{N-1} x-(I-T_{N}) U_{N-1} y\bigr\| ^{2} \\ & \quad=(1-\delta_{N})\| U_{N-1} x-U_{N-1} y \|^{2}+ \bigl(1-(1-\delta_{N}) \bigr)\| x-y\|^{2} \\ &\qquad{}+\alpha_{N}(\kappa_{N}-\beta_{N}) \bigl\| (I-T_{N})T_{N} U_{N-1} x-(I-T_{N})T_{N} U_{N-1} y\bigr\| ^{2} \\ &\qquad{} + \bigl((\alpha_{N}+\beta_{N}) \kappa_{n}- \beta_{N}\gamma_{N} \bigr) \bigl\| (I-T_{N}) U_{N-1} x-(I-T_{N}) U_{N-1} y \bigr\| ^{2} \\ & \quad\leq(1-\delta_{N})\| U_{N-1} x-U_{N-1} y \|^{2}+ \bigl(1-(1-\delta_{N}) \bigr)\| x-y\|^{2} \\ & \quad \vdots \\ & \quad\leq(1-\delta_{N}) \bigl[(1-\delta_{N-1})\| U_{N-2} x-U_{N-2} y\|^{2}+ \bigl(1-(1- \delta_{N-1}) \bigr)\| x-y\|^{2} \bigr] \\ &\qquad{}+ \bigl(1-(1- \delta_{N}) \bigr)\| x-y\|^{2} \\ & \quad=(1-\delta_{N}) (1-\delta_{N-1})\| U_{N-2} x-U_{N-2} y\|^{2}+ \bigl(1-(1-\delta_{N}) (1- \delta_{N-1}) \bigr)\| x-y\|^{2} \\ & \quad \vdots \\ & \quad\leq \prod_{i=2}^{N} (1- \delta_{i}) \| U_{1} x-U_{1} y\|^{2}+ \Biggl(1-\prod_{i=2}^{N} (1- \delta_{i}) \Biggr)\|x-y\|^{2} \\ & \quad= \prod_{i=2}^{N} (1- \delta_{i})\bigl\| \alpha_{1} \bigl(T_{1}^{2} x-T_{1}^{2} y \bigr)+\beta_{1} (T_{1} x-T_{1} y)+(1-\alpha_{1}-\beta_{1}) (x-y) \bigr\| ^{2} \\ &\qquad{}+ \Biggl(1-\prod_{i=2}^{N} (1-\delta_{i}) \Biggr)\|x-y\|^{2} \\ & \quad\leq \prod_{i=2}^{N} (1- \delta_{i}) \bigl[\alpha_{1}\bigl\| T_{1}^{2} x-T_{1}^{2} y\bigr\| ^{2}+\beta_{1} \|T_{1} x-T_{1} y\|^{2}+(1-\alpha_{1}- \beta_{1}) \|x-y\|^{2} \\ &\qquad{} -\alpha_{1}\beta_{1}\bigl\| (I-T_{1})T_{1} x-(I-T_{1})T_{1} y \bigr\| ^{2}-\beta_{1}(1- \alpha _{1}-\beta_{1})\bigl\| (I-T_{1}) x-(I-T_{1})y\bigr\| ^{2} \bigr] \\ &\qquad{} + \Biggl(1-\prod_{i=2}^{N} (1- \delta_{i}) \Biggr)\|x-y\|^{2} \\ & \quad\leq \prod_{i=2}^{N} (1- \delta_{i}) \bigl[\alpha_{1} \bigl(\|T_{1} x-T_{1} y\| ^{2}+\kappa_{1}\bigl\| (I-T_{1})T_{1} x-(I-T_{1})T_{1} y\bigr\| ^{2} \bigr)+\beta_{1} \|T_{1} x-T_{1} y\| ^{2} \\ &\qquad{} +(1-\alpha_{1}-\beta_{1}) \|x-y\|^{2} - \alpha_{1}\beta_{1}\bigl\| (I-T_{1})T_{1} x-(I-T_{1})T_{1} y \bigr\| ^{2} \\ &\qquad{} -\beta_{1}(1-\alpha_{1}-\beta_{1}) \bigl\| (I-T_{1}) x-(I-T_{1})y\bigr\| ^{2} \bigr] \\ &\qquad{}+ \Biggl(1- \prod_{i=2}^{N} (1-\delta_{i}) \Biggr)\|x-y\|^{2} \\ & \quad= \prod_{i=2}^{N} (1- \delta_{i}) \bigl[(\alpha_{1}+\beta_{1}) \|T_{1} x-T_{1} y\| ^{2}+\alpha_{1}( \kappa_{1}-\beta_{1})\bigl\| (I-T_{1})T_{1} x-(I-T_{1})T_{1} y\bigr\| ^{2} \\ &\qquad{} +(1-\alpha_{1}-\beta_{1}) \|x-y\|^{2}- \beta_{1}(1-\alpha_{1}-\beta_{1})\bigl\| (I-T_{1}) x-(I-T_{1})y\bigr\| ^{2} \bigr] \\ &\qquad{}+ \Biggl(1-\prod _{i=2}^{N} (1-\delta_{i}) \Biggr) \| x-y\|^{2} \\ & \quad\leq \prod_{i=2}^{N} (1- \delta_{i}) \bigl[(\alpha_{1}+\beta_{1}) \bigl(\| x- y\|^{2}+\kappa_{1}\bigl\| (I-T_{1})x-(I-T_{1})y \bigr\| ^{2} \bigr) \\ &\qquad{} +\alpha_{1}(\kappa_{1}-\beta_{1}) \bigl\| (I-T_{1})T_{1} x-(I-T_{1})T_{1} y\bigr\| ^{2}+(1-\alpha_{1}-\beta_{1}) \|x-y\|^{2} \\ &\qquad{} -\beta_{1}(1-\alpha_{1}-\beta_{1}) \bigl\| (I-T_{1}) x-(I-T_{1})y\bigr\| ^{2} \bigr]+ \Biggl(1- \prod_{i=2}^{N} (1-\delta_{i}) \Biggr)\|x-y\|^{2} \\ & \quad= \prod_{i=2}^{N} (1- \delta_{i}) \bigl[\| x-y\|^{2}+\alpha_{1}( \kappa_{1}-\beta _{1})\bigl\| (I-T_{1})T_{1} x-(I-T_{1})T_{1} y\bigr\| ^{2} \\ &\qquad{} + \bigl((\alpha_{1}+\beta_{1}) \kappa_{1}- \beta_{1}(1-\alpha_{1}- \beta_{1}) \bigr)\bigl\| (I-T_{1}) x-(I-T_{1})y \bigr\| ^{2} \bigr] \\ &\qquad{}+ \Biggl(1-\prod_{i=2}^{N} (1-\delta_{i}) \Biggr) \| x-y\|^{2} \\ & \quad \leq\prod_{i=2}^{N} (1- \delta_{i})\| x-y\|^{2}+ \Biggl(1-\prod _{i=2}^{N} (1-\delta_{i}) \Biggr)\|x-y \|^{2} \\ & \quad=\|x-y\|^{2}. \end{aligned}$$
(2.18)
This completes the proof. □

Remark 2.1

From the above proof, we can see that the mapping G is quasi-nonexpansive under the conditions in Lemma 2.9, that is,
$$ \bigl\| Gx-x^{*}\bigr\| \leq\bigl\| x-x^{*}\bigr\| , \quad\forall x\in C, x^{*}\in F(G). $$
(2.19)

Example 2.1

Let \(T_{1},T_{2}: \mathbb{R}\rightarrow\mathbb{R}\) be defined by
$$ T_{1}x= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} x,& x\in(-\infty,0],\\ -\frac{3}{2}x,&x\in[0,+\infty); \end{array}\displaystyle \right . $$
and
$$ T_{2}x= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} -2x,& x\in(-\infty,0],\\ x,&x\in[0,+\infty). \end{array}\displaystyle \right . $$
Then we observe that \(F(T_{1})=(-\infty,0]\) and \(F(T_{2})=[0,+\infty)\). Hence, \(F(T_{1})\cap F(T_{2})=\{0\}\).

Firstly, we show that \(T_{1}\) is a \(\frac{1}{5}\)-strictly pseudo-contractive mapping.

(1) If \(x,y\in(-\infty,0]\), then we have
$$\|T_{1}x-T_{1}y\|^{2}=(x-y)^{2} $$
and
$$\bigl\| (I-T_{1})x-(I-T_{1})y\bigr\| ^{2}=0. $$
From the above, then there exists \(\kappa_{1}\in[0,1)\) such that
$$\| T_{1}x-T_{1}y \|^{2}\leq\| x-y \|^{2}+ \kappa_{1} \bigl\| (I-T_{1})x-(I-T_{1})y \bigr\| ^{2}. $$
(2) If \(x,y\in[0,+\infty)\), then we have
$$\|T_{1}x-T_{1}y\|^{2}= \biggl(-\frac{3}{2}x+ \frac{3}{2}y \biggr)^{2}=\frac{9}{4}(x-y)^{2} $$
and
$$\bigl\| (I-T_{1})x-(I-T_{1})y\bigr\| ^{2}= \biggl( \biggl(x+ \frac{3}{2}x \biggr)- \biggl(y+\frac{3}{2}y \biggr) \biggr)^{2}= \frac{25}{4}(x-y)^{2}. $$
From the above, then there exists \(\kappa_{1}\in[\frac{1}{5},1)\) such that
$$\| T_{1}x-T_{1}y \|^{2}\leq\| x-y \|^{2}+ \kappa_{1} \bigl\| (I-T_{1})x-(I-T_{1})y \bigr\| ^{2}. $$
(3) If \(x\in(-\infty,0]\) and \(y\in[0,+\infty)\), then we have
$$\|T_{1}x-T_{1}y\|^{2}= \biggl( x+ \frac{3}{2}y \biggr)^{2} $$
and
$$\bigl\| (I-T_{1})x-(I-T_{1})y\bigr\| ^{2}= \biggl((x-x)- \biggl(y+\frac{3}{2}y \biggr) \biggr)^{2}=\frac{25}{4}y^{2}. $$
Note that
$$\begin{aligned} \biggl( x+\frac{3}{2}y \biggr)^{2}-(x-y)^{2}- \kappa_{1}\frac{25}{4}y^{2}= \biggl(\frac{5}{4}- \frac {25}{4}\kappa_{1} \biggr)y^{2}+5xy. \end{aligned}$$
From the above, then there exists \(\kappa_{1}\in[\frac{1}{5},1)\) such that
$$\| T_{1}x-T_{1}y \|^{2}\leq\| x-y \|^{2}+ \kappa_{1} \bigl\| (I-T_{1})x-(I-T_{1})y \bigr\| ^{2}. $$

Next, we show that \(T_{2}\) is a \(\frac{1}{3}\)-strictly pseudo-contractive mapping.

(1) If \(x,y\in(-\infty,0]\), then we have
$$\|T_{2}x-T_{2}y\|^{2}=(-2x+2y)^{2}=4(x-y)^{2} $$
and
$$\bigl\| (I-T_{2})x-(I-T_{2})y\bigr\| ^{2}= \bigl((x+2x)-(y+2y) \bigr)^{2}=9(x-y)^{2}. $$
From the above, then there exists \(\kappa_{2}\in[\frac{1}{3},1)\) such that
$$\| T_{2}x-T_{2}y \|^{2}\leq\| x-y \|^{2}+ \kappa_{2} \bigl\| (I-T_{2})x-(I-T_{2})y \bigr\| ^{2}. $$
(2) If \(x,y\in[0,+\infty)\), then we have
$$\|T_{2}x-T_{2}y\|^{2}=(x-y)^{2} $$
and
$$\bigl\| (I-T_{2})x-(I-T_{2})y\bigr\| ^{2}=0. $$
From the above, then there exists \(\kappa_{2}\in[0,1)\) such that
$$\| T_{2}x-T_{2}y \|^{2}\leq\| x-y \|^{2}+ \kappa_{2} \bigl\| (I-T_{2})x-(I-T_{2})y \bigr\| ^{2}. $$
(3) If \(x\in(-\infty,0]\) and \(y\in[0,+\infty)\), then we have
$$\|T_{2}x-T_{2}y\|^{2}=( -2x-y)^{2} $$
and
$$\bigl\| (I-T_{2})x-(I-T_{2})y\bigr\| ^{2}= \bigl((x+2x)-(y-y) \bigr)^{2}=9x^{2}. $$
Note that
$$\begin{aligned} ( -2x-y)^{2}-(x-y)^{2}-9\kappa_{2}x^{2}=(3-9 \kappa_{2})x^{2}+6xy. \end{aligned}$$
From the above, then there exists \(\kappa_{2}\in[\frac{1}{3},1)\) such that
$$\| T_{2}x-T_{2}y \|^{2}\leq\| x-y \|^{2}+ \kappa_{2} \bigl\| (I-T_{2})x-(I-T_{2})y \bigr\| ^{2}. $$
Let
$$\pi_{1}= \biggl(\frac{1}{5},\frac{1}{5}, \frac{2}{5}, \frac{1}{5} \biggr), $$
which satisfies condition (i) in Lemma 2.9. And
$$ T_{1}^{2}x= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} x,& x\in(-\infty,0],\\ -\frac{3}{2}x,&x\in[0,+\infty). \end{array}\displaystyle \right . $$
Then
$$\begin{aligned} U_{1}x=\frac{1}{5}T_{1}^{2}x+ \frac{1}{5}T_{1} x+\frac{2}{5}x+\frac{1}{5}x =\left \{ \textstyle\begin{array}{@{}l@{\quad}l} x,& x\in(-\infty,0],\\ 0,&x\in[0,+\infty). \end{array}\displaystyle \right . \end{aligned}$$
Let
$$\pi_{2}= \biggl(\frac{1}{7},\frac{1}{3}, \frac{1}{2},\frac{1}{42} \biggr), $$
which satisfies condition (ii) in Lemma 2.9. Again, we have
$$ T_{2}U_{1}x= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} -2x,& x\in(-\infty,0],\\ 0,&x\in[0,+\infty); \end{array}\displaystyle \right . $$
and
$$ T_{2}^{2}U_{1}x= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} -2x,& x\in(-\infty,0],\\ 0,&x\in[0,+\infty). \end{array}\displaystyle \right . $$
Then
$$\begin{aligned} Gx =&U_{2}x=\frac{1}{7}T_{2}^{2}U_{1}x+ \frac{1}{3}T_{2}U_{1} x+\frac{1}{2}U_{1}x+ \frac {1}{42}x \\ =&\left \{ \textstyle\begin{array}{@{}l@{\quad}l} -\frac{3}{7}x,& x\in(-\infty,0],\\ \frac{1}{42}x,&x\in[0,+\infty). \end{array}\displaystyle \right . \end{aligned}$$
From the above, we can get \(F(G)=\{0\}\), that is, \(F(G)=F(T_{1})\cap F(T_{2})\).

Finally, we show that G is nonexpansive.

(1) If \(x,y\in(-\infty,0]\), it is easy to see that
$$\biggl|-\frac{3}{7}x+\frac{3}{7}y \biggr|\leq|x-y|. $$
(2) If \(x,y\in[0,+\infty)\), we have
$$\biggl|\frac{1}{42}x-\frac{1}{42}y \biggr|\leq|x-y|. $$
(3) If \(x\in(-\infty,0]\) and \(y\in[0,+\infty)\), then
$$\begin{aligned} & \biggl|-\frac{3}{7}x-\frac{1}{42}y \biggr|^{2}- |x-y|^{2} \\ & \quad=-\frac{40}{49}x^{2}-\frac{1{,}763}{1{,}764}y^{2}+ \frac{99}{49}xy \\ & \quad\leq0 \quad\biggl(\mbox{since } x\leq0 \mbox{ and } y\geq0, \mbox{then } \frac{99}{49}xy\leq0 \biggr). \end{aligned}$$
Hence,
$$\biggl|-\frac{3}{7}x-\frac{1}{42}y \biggr| \leq |x-y|. $$

3 Main results

Theorem 3.1

Let C be a nonempty closed convex subset of the real Hilbert space H. For every \(i=1,2,\ldots, N\), let \(T_{i}: C \rightarrow C\) be \(\kappa _{i}\)-strict pseudo-contractive mappings and \(T: C \rightarrow C\) be a ρ-strictly pseudononspreading mapping for some \(\rho\in[0,1)\). For \(i=1,2,\ldots, N\), let \(\pi_{i}=(\alpha_{i},\beta_{i},\gamma_{i}, \delta _{i})\), where \(\alpha_{i},\beta_{i},\gamma_{i}, \delta_{i}\in[0,1]\), \(\alpha _{i}+\beta_{i}+\gamma_{i}+\delta_{i}=1\) and satisfy
  1. (i)

    \(\kappa_{1}\leq\beta_{1}<1-\kappa_{1}\) and \(\alpha_{1}(\kappa _{1}+\beta_{1})<\beta_{1}(1-\beta_{1}-\kappa_{1})\);

     
  2. (ii)

    \(\beta_{i}\geq\kappa_{i}\), \(\kappa_{i}<\gamma_{i}<1\) and \(\kappa _{i}\alpha_{i}\leq\beta_{i}\gamma_{i}-\beta_{i}\kappa_{i}\) for \(i=2,3,\ldots, N\).

     
Let G be the G-mapping generated by \(T_{1},T_{2},\ldots,T_{N}\) and \(\pi _{1}, \pi_{2},\ldots,\pi_{N}\). Assume that \(\mathfrak{F}=F(T)\cap\bigcap_{i=1}^{N}F(T_{i})\neq\emptyset\). Pick any \(u, x_{0} \in C\), let \(\{x_{n}\}\) be a sequence generated by
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=(1-s_{n})x_{n}+s_{n} P_{C}(I-\lambda_{n}(I-T))x_{n}, \\ z_{n}=(1-t_{n})x_{n}+t_{n} P_{C}(I-\lambda_{n}(I-T))y_{n}, \\ x_{n+1}=a_{n} u+b_{n} z_{n}+c_{n} Gz_{n}, \end{array}\displaystyle \right . $$
(3.1)
where \(\{s_{n}\}, \{t_{n}\}, \{a_{n}\}, \{b_{n}\}, \{c_{n}\}\subset[0,1]\) and \(\{\lambda_{n}\}\subset(0,1-\rho)\) satisfy the following conditions:
  1. (1)

    \(a_{n}+b_{n}+c_{n}=1\);

     
  2. (2)

    \(\lim_{n\rightarrow\infty}a_{n}=0\) and \(\sum_{n=0}^{\infty}a_{n}=\infty\);

     
  3. (3)

    \(\liminf_{n\rightarrow\infty} b_{n}>0\) and \(\liminf_{n\rightarrow\infty} c_{n}>0\);

     
  4. (4)

    \(\sum_{n=0}^{\infty}\lambda_{n}< \infty\);

     
  5. (5)

    \(\sum_{n=0}^{\infty}|\lambda_{n+1}-\lambda_{n}|, \sum_{n=0}^{\infty}|s_{n+1}-s_{n}|, \sum_{n=0}^{\infty}|t_{n+1}-t_{n}|, \sum_{n=0}^{\infty}|a_{n+1}-a_{n}|, \sum_{n=0}^{\infty}|b_{n+1}-b_{n}|, \sum_{n=0}^{\infty}|c_{n+1}-c_{n}|< \infty\).

     
Then \(\{x_{n}\}\) converges strongly to \(\overline{x}=P_{\mathfrak{F}}u\).

Proof

Step 1. Firstly, we show that L is bounded, where
$$\begin{aligned} L =&\max_{n\in\mathbb{N}} \bigl\{ \|u\|, \|x_{n}\|, \|z_{n}\|,\|Gz_{n}\|, \bigl\| P_{C} \bigl(I- \lambda_{n}(I-T) \bigr)x_{n}\bigr\| ,\bigl\| P_{C} \bigl(I- \lambda_{n}(I-T) \bigr)y_{n}\bigr\| , \\ &{} \bigl\| (I-T)x_{n}-(I-T)x_{n-1}\bigr\| , \bigl\| (I-T)y_{n}-(I-T)y_{n-1} \bigr\| , \\ &{} \bigl\| (I-T)x_{n}\bigr\| , \bigl\| (I-T)y_{n}\bigr\| \bigr\} . \end{aligned}$$
(3.2)
Indeed, take \(p \in\mathfrak{F}\) arbitrarily. From (3.1), we have
$$\begin{aligned} \|x_{n+1}-p\| =&\|a_{n} u+b_{n} z_{n}+c_{n} Gz_{n}-p\| \\ =&\bigl\| a_{n} (u-p)+b_{n}( z_{n}-p)+c_{n} (Gz_{n}-p)\bigr\| \\ \leq&a_{n}\|u-p\|+b_{n}\| z_{n}-p \|+c_{n} \|Gz_{n}-p\| \\ \leq&a_{n}\|u-p\|+b_{n}\| z_{n}-p \|+c_{n} \|z_{n}-p\| \\ =&a_{n}\|u-p\|+(1-a_{n})\| z_{n}-p\|. \end{aligned}$$
(3.3)
From Lemma 2.3 and (3.1), we have
$$\begin{aligned} \| z_{n}-p\| =&\bigl\| (1-t_{n})x_{n}+t_{n} P_{C} \bigl(I-\lambda_{n}(I-T) \bigr)y_{n}-p\bigr\| \\ \leq& (1-t_{n})\|x_{n}-p\|+t_{n}\bigl\| P_{C} \bigl(I-\lambda_{n}(I-T) \bigr)y_{n}-p\bigr\| \\ \leq& (1-t_{n})\|x_{n}-p\|+t_{n}\| y_{n}-p\|, \end{aligned}$$
(3.4)
and
$$\begin{aligned} \| y_{n}-p\| =&\bigl\| (1-s_{n})x_{n}+s_{n} P_{C} \bigl(I-\lambda_{n}(I-T) \bigr)x_{n}-p\bigr\| \\ \leq& (1-s_{n})\|x_{n}-p\|+s_{n}\bigl\| P_{C} \bigl(I-\lambda_{n}(I-T) \bigr)x_{n}-p\bigr\| \\ \leq& (1-s_{n})\|x_{n}-p\|+s_{n}\| x_{n}-p\| \\ =&\|x_{n}-p\|. \end{aligned}$$
(3.5)
Substituting (3.4) and (3.5) into (3.3), we obtain that
$$ \|x_{n+1}-p\|\leq a_{n}\|u-p \|+(1-a_{n})\| x_{n}-p\|. $$
(3.6)
From (3.6), we can see by induction that
$$ \|x_{n+1}-p\|\leq\max \bigl\{ \|u-p\|, \| x_{0}-p\| \bigr\} ,\quad \forall n \geq0. $$
(3.7)
This implies that \(\{x_{n}\}\) is bounded. Then \(\{y_{n}\}\), \(\{z_{n}\}\) and \(\{Gz_{n}\}\) are bounded. From Lemma 2.3 and the boundedness of \(\{x_{n}\}\) and \(\{y_{n}\}\), it can be seen that \(\{P_{C}(I-\lambda_{n}(I-T))x_{n}\}\) and \(\{P_{C}(I-\lambda _{n}(I-T))y_{n}\}\) are bounded. And from Lemma 2.4, we also have that \(\{(I-T)x_{n}-(I-T)x_{n-1}\}\) and \(\{ (I-T)y_{n}-(I-T)y_{n-1}\}\) are bounded. Hence, L is bounded.

Step 2. Next, we prove that \(\lim_{n\rightarrow\infty}\| x_{n+1}-x_{n}\|=0\).

From (3.1), it follows that
$$\begin{aligned} & \|x_{n+1}-x_{n}\| \\ & \quad=\bigl\| a_{n} u+b_{n} z_{n}+c_{n} Gz_{n}-(a_{n-1} u+b_{n-1} z_{n-1}+c_{n-1} Gz_{n-1})\bigr\| \\ & \quad=\bigl\| (a_{n}-a_{n-1}) u+b_{n}( z_{n}- z_{n-1})+(b_{n}-b_{n-1})z_{n-1}+c_{n} (Gz_{n}-Gz_{n-1}) \\ &\qquad{}+(c_{n}-c_{n-1})Gz_{n-1} \bigr\| \\ & \quad\leq|a_{n}-a_{n-1}|\|u\|+b_{n} \|z_{n}- z_{n-1}\|+|b_{n}-b_{n-1}| \|z_{n-1}\| +c_{n}\|Gz_{n}-Gz_{n-1} \| \\ &\qquad{}+|c_{n}-c_{n-1}|\|Gz_{n-1}\| \\ & \quad\leq|a_{n}-a_{n-1}|L+b_{n} \|z_{n}- z_{n-1}\|+|b_{n}-b_{n-1}|L+c_{n} \| z_{n}-z_{n-1}\|+|c_{n}-c_{n-1}|L \\ & \quad=(1-a_{n})\|z_{n}- z_{n-1} \|+|a_{n}-a_{n-1}|L+|b_{n}-b_{n-1}|L+|c_{n}-c_{n-1}|L, \end{aligned}$$
(3.8)
$$\begin{aligned} &\|z_{n+1}-z_{n}\| \\ & \quad=\bigl\| (1-t_{n})x_{n}+t_{n} P_{C} \bigl(I-\lambda _{n}(I-T) \bigr)y_{n}- \bigl((1-t_{n-1})x_{n-1} \\ &\qquad{}+t_{n-1} P_{C} \bigl(I-\lambda _{n-1}(I-T) \bigr)y_{n-1} \bigr) \bigr\| \\ & \quad\leq\bigl\| (1-t_{n})x_{n}-(1-t_{n-1})x_{n-1} \bigr\| +\bigl\| t_{n}P_{C} \bigl(I-\lambda _{n}(I-T) \bigr)y_{n} \\ &\qquad{}-t_{n-1}P_{C} \bigl(I- \lambda_{n-1}(I-T) \bigr)y_{n-1}\bigr\| \\ & \quad\leq(1-t_{n})\|x_{n}-x_{n-1} \|+|t_{n}-t_{n-1}|\|x_{n-1}\|+t_{n}\bigl\| P_{C} \bigl(I-\lambda_{n}(I-T) \bigr)y_{n} \\ &\qquad{}-P_{C} \bigl(I-\lambda_{n-1}(I-T) \bigr)y_{n-1}\bigr\| \\ & \qquad{}+ |t_{n}-t_{n-1}|\bigl\| P_{C} \bigl(I- \lambda_{n-1}(I-T) \bigr)y_{n-1}\bigr\| \\ & \quad\leq(1-t_{n})\|x_{n}-x_{n-1} \|+|t_{n}-t_{n-1}|L+t_{n}\bigl\| \bigl(I-\lambda _{n}(I-T) \bigr)y_{n}- \bigl(I-\lambda_{n-1}(I-T) \bigr)y_{n-1}\bigr\| \\ & \qquad{} +|t_{n}-t_{n-1}|L \\ & \quad\leq(1-t_{n})\|x_{n}-x_{n-1} \|+2|t_{n}-t_{n-1}|L+t_{n}\|y_{n}-y_{n-1} \| \\ & \qquad{} +t_{n}\bigl\| \lambda_{n}(I-T)y_{n}- \lambda_{n}(I-T)y_{n-1}+\lambda_{n}(I-T)y_{n-1}- \lambda_{n-1}(I-T)y_{n-1}\bigr\| \\ & \quad\leq(1-t_{n})\|x_{n}-x_{n-1} \|+2|t_{n}-t_{n-1}|L+t_{n}\|y_{n}-y_{n-1} \| \\ & \qquad{} +t_{n}\lambda_{n}\bigl\| (I-T)y_{n}-(I-T)y_{n-1} \bigr\| +t_{n}|\lambda_{n}-\lambda _{n-1}| \bigl\| (I-T)y_{n-1}\bigr\| \\ & \quad\leq(1-t_{n})\|x_{n}-x_{n-1} \|+t_{n}\|y_{n}-y_{n-1}\|+2|t_{n}-t_{n-1}|L +t_{n}\lambda_{n}L+t_{n}|\lambda_{n}- \lambda_{n-1}|L, \end{aligned}$$
(3.9)
and
$$\begin{aligned} & \|y_{n+1}-y_{n}\| \\ & \quad=\bigl\| (1-s_{n})x_{n}+s_{n} P_{C} \bigl(I-\lambda _{n}(I-T) \bigr)x_{n}- \bigl((1-s_{n-1})x_{n-1} \\ &\qquad{}+s_{n-1} P_{C} \bigl(I-\lambda _{n-1}(I-T) \bigr)x_{n-1} \bigr) \bigr\| \\ & \quad\leq\bigl\| (1-s_{n})x_{n}-(1-s_{n-1})x_{n-1} \bigr\| +\bigl\| s_{n}P_{C} \bigl(I-\lambda _{n}(I-T) \bigr)x_{n} \\ &\qquad{}-s_{n-1}P_{C} \bigl(I- \lambda_{n-1}(I-T) \bigr)x_{n-1}\bigr\| \\ & \quad\leq(1-s_{n})\|x_{n}-x_{n-1} \|+|s_{n}-s_{n-1}|\|x_{n-1}\|+s_{n}\bigl\| P_{C} \bigl(I-\lambda_{n}(I-T) \bigr)x_{n} \\ &\qquad{}-P_{C} \bigl(I-\lambda_{n-1}(I-T) \bigr)x_{n-1}\bigr\| \\ & \qquad{}+ |s_{n}-s_{n-1}|\bigl\| P_{C} \bigl(I- \lambda_{n-1}(I-T) \bigr)x_{n-1}\bigr\| \\ & \quad\leq(1-s_{n})\|x_{n}-x_{n-1} \|+|s_{n}-s_{n-1}|L+s_{n}\bigl\| \bigl(I-\lambda _{n}(I-T) \bigr)x_{n}- \bigl(I-\lambda_{n-1}(I-T) \bigr)x_{n-1}\bigr\| \\ & \qquad{} +|s_{n}-s_{n-1}|L \\ & \quad\leq(1-s_{n})\|x_{n}-x_{n-1} \|+2|s_{n}-s_{n-1}|L+s_{n}\|x_{n}-x_{n-1} \| \\ & \qquad{} +s_{n}\bigl\| \lambda_{n}(I-T)x_{n}- \lambda_{n}(I-T)x_{n-1}+\lambda_{n}(I-T)x_{n-1}- \lambda_{n-1}(I-T)x_{n-1}\bigr\| \\ & \quad\leq\|x_{n}-x_{n-1}\|+2|s_{n}-s_{n-1}|L+s_{n} \lambda_{n}\bigl\| (I-T)x_{n}-(I-T)x_{n-1} \bigr\| \\ &\qquad{}+s_{n}|\lambda_{n}-\lambda_{n-1}| \bigl\| (I-T)x_{n-1}\bigr\| \\ & \quad\leq\|x_{n}-x_{n-1}\|+2|s_{n}-s_{n-1}|L+s_{n} \lambda_{n}L+s_{n}|\lambda _{n}- \lambda_{n-1}|L. \end{aligned}$$
(3.10)
Substituting (3.9) and (3.10) into (3.8), we can get that
$$\begin{aligned} & \|x_{n+1}-x_{n}\| \\ & \quad\leq(1-a_{n})\|z_{n}- z_{n-1}\| +|a_{n}-a_{n-1}|L+|b_{n}-b_{n-1}|L+|c_{n}-c_{n-1}|L \\ & \quad\leq(1-a_{n}) \bigl[(1-t_{n})\|x_{n}-x_{n-1} \|+t_{n}\|y_{n}-y_{n-1}\|+2|t_{n}-t_{n-1}|L \\ &\qquad{}+t_{n}\lambda_{n}L+t_{n}|\lambda_{n}- \lambda_{n-1}|L \bigr] +|a_{n}-a_{n-1}|L+|b_{n}-b_{n-1}|L+|c_{n}-c_{n-1}|L \\ & \quad\leq(1-a_{n}) \bigl[(1-t_{n})\|x_{n}-x_{n-1} \|+t_{n} \bigl(\|x_{n}-x_{n-1}\| +2|s_{n}-s_{n-1}|L \\ &\qquad{}+s_{n} \lambda_{n}L+s_{n}|\lambda_{n}- \lambda_{n-1}|L \bigr) \bigr] \\ & \qquad{} +2(1-a_{n})|t_{n}-t_{n-1}|L +(1-a_{n})t_{n}\lambda_{n}L+(1-a_{n})t_{n}| \lambda_{n}-\lambda_{n-1}|L \\ & \qquad{} +|a_{n}-a_{n-1}|L+|b_{n}-b_{n-1}|L+|c_{n}-c_{n-1}|L \\ & \quad=(1-a_{n})\|x_{n}-x_{n-1}\| +2(1-a_{n})t_{n}|s_{n}-s_{n-1}|L+(1-a_{n})t_{n}s_{n} \lambda_{n}L \\ &\qquad{}+(1-a_{n})t_{n}s_{n}| \lambda _{n}-\lambda_{n-1}|L \\ & \qquad{} +2(1-a_{n})|t_{n}-t_{n-1}|L +(1-a_{n})t_{n}\lambda_{n}L+(1-a_{n})t_{n}| \lambda_{n}-\lambda_{n-1}|L \\ & \qquad{} +|a_{n}-a_{n-1}|L+|b_{n}-b_{n-1}|L+|c_{n}-c_{n-1}|L \\ & \quad=(1-a_{n})\|x_{n}-x_{n-1}\|+ \theta_{n}, \end{aligned}$$
(3.11)
where
$$\begin{aligned} \theta_{n} =&2(1-a_{n})t_{n}|s_{n}-s_{n-1}|L+(1-a_{n})t_{n}s_{n} \lambda _{n}L+(1-a_{n})t_{n}s_{n}| \lambda_{n}-\lambda_{n-1}|L \\ &{}+2(1-a_{n})|t_{n}-t_{n-1}|L +(1-a_{n})t_{n}\lambda_{n}L+(1-a_{n})t_{n}| \lambda_{n}-\lambda_{n-1}|L \\ &{}+|a_{n}-a_{n-1}|L+|b_{n}-b_{n-1}|L+|c_{n}-c_{n-1}|L. \end{aligned}$$
(3.12)
By the conditions in Theorem 3.1, we can get that
$$ \sum_{n=0}^{\infty}\theta_{n}< \infty. $$
(3.13)
Thus, from Lemma 2.5 and (3.11), we have
$$ \lim_{n\rightarrow\infty}\| x_{n+1}-x_{n}\|=0. $$
(3.14)

Step 3. In this step, we will show that \(\lim_{n\rightarrow \infty}\|Gz_{n}-z_{n}\|=0\) and \(\lim_{n\rightarrow\infty}\|x_{n}-z_{n}\|=0\).

From Lemma 2.1, (3.1), (3.4) and (3.5), we have
$$\begin{aligned} \|x_{n+1}-p\|^{2} =&\|a_{n} u+b_{n} z_{n}+c_{n} Gz_{n}-p\|^{2} \\ =&\bigl\| a_{n} (u-p)+b_{n}( z_{n}-p)+c_{n} (Gz_{n}-p)\bigr\| ^{2} \\ =&a_{n}\|u-p\|^{2}+b_{n}\|z_{n}-p \|^{2}+c_{n}\|Gz_{n}-p\|^{2} \\ &{} -a_{n}b_{n}\|u-z_{n}\|^{2}-a_{n}c_{n} \|u-Gz_{n}\|^{2}-b_{n}c_{n} \|Gz_{n}-z_{n}\| ^{2} \\ \leq& a_{n}\|u-p\|^{2}+b_{n} \|z_{n}-p\|^{2}+c_{n}\|Gz_{n}-p \|^{2}-b_{n}c_{n}\|Gz_{n}-z_{n} \| ^{2} \\ \leq& a_{n}\|u-p\|^{2}+b_{n} \|z_{n}-p\|^{2}+c_{n}\|z_{n}-p \|^{2}-b_{n}c_{n}\|Gz_{n}-z_{n} \| ^{2} \\ \leq& a_{n}\|u-p\|^{2}+(1-a_{n}) \|x_{n}-p\|^{2}-b_{n}c_{n} \|Gz_{n}-z_{n}\|^{2} \\ \leq& a_{n}\|u-p\|^{2}+\|x_{n}-p \|^{2}-b_{n}c_{n}\|Gz_{n}-z_{n} \|^{2}, \end{aligned}$$
(3.15)
which implies that
$$\begin{aligned} b_{n}c_{n}\|Gz_{n}-z_{n} \|^{2} \leq&a_{n}\|u-p\|^{2}+\|x_{n}-p \|^{2}-\|x_{n+1}-p\| ^{2} \\ \leq&a_{n}\|u-p\|^{2}+\bigl(\|x_{n}-p\|+ \|x_{n+1}-p\|\bigr)\|x_{n+1}-x_{n}\|. \end{aligned}$$
(3.16)
Since \(\liminf_{n\rightarrow\infty} b_{n}>0\), \(\liminf_{n\rightarrow \infty} c_{n}>0\), \(\lim_{n\rightarrow\infty} a_{n}=0\), \(\lim_{n\rightarrow \infty}\|x_{n+1}-x_{n}\|=0\) and by the boundedness of \(\|u-p\|\) and \(\{ x_{n}\}\), we have
$$ \lim_{n\rightarrow\infty}\|Gz_{n}-z_{n} \|=0. $$
(3.17)
Again,
$$\begin{aligned} \|x_{n}-z_{n}\| \leq&\|x_{n}-x_{n+1} \|+\|x_{n+1}-z_{n}\| \\ \leq&\|x_{n}-x_{n+1}\|+\|a_{n} u+b_{n} z_{n}+c_{n} Gz_{n}-z_{n} \| \\ \leq&\|x_{n}-x_{n+1}\|+a_{n}\| u-z_{n}\|+c_{n} \|Gz_{n}-z_{n}\|. \end{aligned}$$
(3.18)
Thus,
$$ \lim_{n\rightarrow\infty}\|x_{n}-z_{n} \|=0. $$
(3.19)

Step 4. Now, we prove that \(\limsup_{n\rightarrow\infty}\langle u-\overline{x}\), \(x_{n}-\overline{x}\rangle\leq0\), where \(\overline{x}=P_{\mathfrak{F}}u\).

Take a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that
$$\begin{aligned} \limsup_{n\rightarrow\infty}\langle u-\overline{x},x_{n}-\overline {x}\rangle=\lim_{n\rightarrow\infty}\langle u-\overline {x},x_{n_{i}}- \overline{x}\rangle. \end{aligned}$$
(3.20)
Since \(\{x_{n}\}\) is bounded, there exists a subsequence of \(\{x_{n}\}\), which converges weakly to \(x^{*}\). Without loss of generality, we may assume that \(x_{n_{i}}\rightharpoonup x^{*}\). From (3.19), we have \(z_{n_{i}}\rightharpoonup x^{*}\). From (3.17) and Lemma 2.7, we have \(x^{*}=Gx^{*}\), that is, \(x^{*}\in F(G)\). Since \(x_{n_{i}}\rightharpoonup x^{*}\), then \(x^{*}\in F(T)\). In fact, if \(x^{*} \notin F(T)\), then \(Tx^{*}\neq x^{*}\). Thus,
$$ \bigl(I-\lambda_{n_{i}}(I-T) \bigr)x^{*}\neq x^{*}. $$
(3.21)
By Lemma 2.8, we have
$$\begin{aligned} \liminf_{i\rightarrow\infty} \bigl\| x_{n_{i}}-x^{*}\bigr\| < &\liminf _{i\rightarrow \infty} \bigl\| x_{n_{i}}- \bigl(I-\lambda_{n_{i}}(I-T) \bigr)x^{*}\bigr\| \\ \leq&\liminf_{i\rightarrow\infty} \bigl( \bigl\| x_{n_{i}}-x^{*}\bigr\| + \lambda_{n_{i}}\bigl\| (I-T) x^{*}\bigr\| \bigr) \\ \leq&\liminf_{i\rightarrow\infty} \bigl\| x_{n_{i}}-x^{*}\bigr\| . \end{aligned}$$
(3.22)
This is a contradiction. Therefore,
$$ x^{*}\in\mathfrak{F}=F(T)\cap\bigcap_{i=1}^{N}F(T_{i}). $$
(3.23)
This together with the property of metric projection implies that
$$\begin{aligned} \limsup_{n\rightarrow\infty}\langle u-\overline {x},x_{n}- \overline{x}\rangle=\lim_{n\rightarrow\infty}\langle u- \overline{x},x_{n_{i}}- \overline{x}\rangle= \bigl\langle u-\overline {x},x^{*}-\overline{x} \bigr\rangle \leq0. \end{aligned}$$
(3.24)
Step 5. Finally, we will show that \(x_{n}\rightarrow\overline {x}\) as \(n\rightarrow\infty\).
$$\begin{aligned} & \|x_{n+1}-\overline{x}\|^{2} \\ & \quad=\langle a_{n} u+b_{n} z_{n}+c_{n} Gz_{n}-\overline{x},x_{n+1}-\overline {x}\rangle \\ & \quad=a_{n}\langle u-\overline{x},x_{n+1}-\overline{x} \rangle+b_{n} \langle z_{n}-\overline{x},x_{n+1}- \overline{x}\rangle+ c_{n}\langle Gz_{n}- \overline{x},x_{n+1}-\overline{x}\rangle \\ & \quad\leq a_{n}\langle u-\overline{x},x_{n+1}-\overline{x} \rangle+b_{n} \| z_{n}-\overline{x}\|\|x_{n+1}- \overline{x}\|+ c_{n}\| Gz_{n}-\overline{x}\| \|x_{n+1}-\overline{x}\| \\ & \quad\leq a_{n}\langle u-\overline{x},x_{n+1}-\overline{x} \rangle+b_{n} \| z_{n}-\overline{x}\|\|x_{n+1}- \overline{x}\|+ c_{n}\| z_{n}-\overline{x}\| \|x_{n+1}-\overline{x}\| \\ & \quad\leq a_{n}\langle u-\overline{x},x_{n+1}-\overline{x} \rangle+b_{n} \| x_{n}-\overline{x}\|\|x_{n+1}- \overline{x}\|+ c_{n}\| x_{n}-\overline{x}\| \|x_{n+1}-\overline{x}\| \\ & \quad\leq a_{n}\langle u-\overline{x},x_{n+1}-\overline{x} \rangle+\frac {b_{n}}{2} \bigl(\|x_{n}-\overline{x}\|^{2}+ \|x_{n+1}-\overline{x}\|^{2} \bigr) \\ &\qquad{}+ \frac{c_{n}}{2} \bigl( \|x_{n}-\overline{x}\|^{2}+\|x_{n+1}-\overline{x} \|^{2} \bigr), \end{aligned}$$
(3.25)
that is,
$$\begin{aligned} \| x_{n+1}-\overline{x} \|^{2}\leq \biggl(1- \frac{2a_{n}}{1+a_{n}} \biggr)\| x_{n}-\overline {x}\|^{2}+ \frac{2a_{n}}{1+a_{n}}\langle u-\overline{x},x_{n+1}-\overline{x}\rangle. \end{aligned}$$
(3.26)
It is clear that \(\sum_{n=0}^{\infty}\frac{2a_{n}}{1+a_{n}}=\infty\). Hence, applying (3.24), (3.26) and Lemma 2.6, we obtain immediately that
$$ \lim_{n\rightarrow\infty}\| x_{n+1}-\overline{x} \|^{2}=0, $$
(3.27)
that is, \(x_{n}\rightarrow\overline{x}\) as \(n\rightarrow\infty\). This completes the proof. □

4 Application

From Theorem 3.1, we can obtain the following theorem.

Theorem 4.1

Let C be a nonempty closed convex subset of the real Hilbert space H. For every \(i=1,2,\ldots, N\), let \(T_{i}: C \rightarrow C\) be nonexpansive mappings and \(T: C \rightarrow C\) be a ρ-strictly pseudononspreading mapping for some \(\rho\in[0,1)\). For \(i=1,2,\ldots, N\), let \(\pi_{i}=(\alpha_{i},\beta_{i},\gamma_{i}, \delta _{i})\), where \(\alpha_{i},\beta_{i},\gamma_{i}, \delta_{i}\in[0,1]\), \(\alpha _{i}+\beta_{i}+\gamma_{i}+\delta_{i}=1\) and satisfy
  1. (i)

    \(0< \beta_{1}<1\) and \(\alpha_{1}<1-\beta_{1}\);

     
  2. (ii)

    \(0<\gamma_{i}<1\) for \(i=2,3,\ldots, N\).

     
Let G be the G-mapping generated by \(T_{1},T_{2},\ldots,T_{N}\) and \(\pi _{1}, \pi_{2},\ldots,\pi_{N}\). Assume that \(\mathfrak{F}=F(T)\cap\bigcap_{i=1}^{N}F(T_{i})\neq\emptyset\). Pick any \(u, x_{0} \in C\), let \(\{x_{n}\}\) be a sequence generated by
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=(1-s_{n})x_{n}+s_{n} P_{C}(I-\lambda_{n}(I-T))x_{n}, \\ z_{n}=(1-t_{n})x_{n}+t_{n} P_{C}(I-\lambda_{n}(I-T))y_{n}, \\ x_{n+1}=a_{n} u+b_{n} z_{n}+c_{n} Gz_{n}, \end{array}\displaystyle \right . $$
(4.1)
where \(\{s_{n}\}, \{t_{n}\}, \{a_{n}\}, \{b_{n}\}, \{c_{n}\}\subset[0,1]\) and \(\{\lambda_{n}\}\subset(0,1-\rho)\) satisfy the following conditions:
  1. (1)

    \(a_{n}+b_{n}+c_{n}=1\);

     
  2. (2)

    \(\lim_{n\rightarrow\infty}a_{n}=0\) and \(\sum_{n=0}^{\infty}a_{n}=\infty\);

     
  3. (3)

    \(\liminf_{n\rightarrow\infty} b_{n}>0\) and \(\liminf_{n\rightarrow\infty} c_{n}>0\);

     
  4. (4)

    \(\sum_{n=0}^{\infty}\lambda_{n}< \infty\);

     
  5. (5)

    \(\sum_{n=0}^{\infty}|\lambda_{n+1}-\lambda_{n}|, \sum_{n=0}^{\infty}|s_{n+1}-s_{n}|, \sum_{n=0}^{\infty}|t_{n+1}-t_{n}|, \sum_{n=0}^{\infty}|a_{n+1}-a_{n}|, \sum_{n=0}^{\infty}|b_{n+1}-b_{n}|, \sum_{n=0}^{\infty}|c_{n+1}-c_{n}|< \infty\).

     
Then \(\{x_{n}\}\) converges strongly to \(\overline{x}=P_{\mathfrak{F}}u\).

Lemma 4.1

[48]

Let C be a nonempty closed convex subset of H and \(T : C \rightarrow H\) be a ξ-inverse-strongly monotone mapping, then for all \(x,y\in C\) and \(\eta>0\), we have
$$\begin{aligned} \bigl\| (I-\eta T)x-(I-\eta T)y\bigr\| ^{2} =&\bigl\| (x-y)-\eta(Tx-Ty)\bigr\| ^{2} \\ =&\| x-y \|^{2}-2\eta\langle Tx-Ty,x-y\rangle+\eta^{2}\| Tx-Ty\|^{2} \\ \leq&\| x-y\|^{2}+\eta(\eta-2\xi)\| Tx-Ty\|^{2}. \end{aligned}$$
(4.2)
So, if \(0<\eta\leq2\xi\), then \(I-\eta T\) is a nonexpansive mapping from C to H.

From Theorem 4.1, Lemmas 2.2 and 4.1, we have the following result.

Theorem 4.2

Let C be a nonempty closed convex subset of the real Hilbert space H. For every \(i=1,2,\ldots, N\), let \(B_{i}: C \rightarrow H\) be \(\xi _{i}\)-inverse-strongly monotone mappings and \(T: C \rightarrow C\) be a ρ-strictly pseudononspreading mapping for some \(\rho\in[0,1)\). For \(i=1,2,\ldots, N\), let \(T_{i}: C \rightarrow C\) be defined by \(T_{i}x=P_{C}(I-\eta_{i}B_{i})x\) for every \(x\in C\) and \(\eta_{i}\in(0,2\xi_{i})\), and let \(\pi_{i}=(\alpha_{i},\beta_{i},\gamma_{i}, \delta_{i})\), where \(\alpha _{i},\beta_{i},\gamma_{i}, \delta_{i}\in[0,1]\), \(\alpha_{i}+\beta_{i}+\gamma _{i}+\delta_{i}=1\) and satisfy
  1. (i)

    \(0< \beta_{1}<1\) and \(\alpha_{1}<1-\beta_{1}\);

     
  2. (ii)

    \(0<\gamma_{i}<1\) for \(i=2,3,\ldots, N\).

     
Let G be the G-mapping generated by \(T_{1},T_{2},\ldots,T_{N}\) and \(\pi _{1}, \pi_{2},\ldots,\pi_{N}\). Assume that \(\mathfrak{F}=F(T)\cap\bigcap_{i=1}^{N}F(T_{i})\neq\emptyset\). Pick any \(u, x_{0} \in C\), let \(\{x_{n}\}\) be a sequence generated by
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=(1-s_{n})x_{n}+s_{n} P_{C}(I-\lambda_{n}(I-T))x_{n}, \\ z_{n}=(1-t_{n})x_{n}+t_{n} P_{C}(I-\lambda_{n}(I-T))y_{n}, \\ x_{n+1}=a_{n} u+b_{n} z_{n}+c_{n} Gz_{n}, \end{array}\displaystyle \right . $$
(4.3)
where \(\{s_{n}\}, \{t_{n}\}, \{a_{n}\}, \{b_{n}\}, \{c_{n}\}\subset[0,1]\) and \(\{\lambda_{n}\}\subset(0,1-\rho)\) satisfy the following conditions:
  1. (1)

    \(a_{n}+b_{n}+c_{n}=1\);

     
  2. (2)

    \(\lim_{n\rightarrow\infty}a_{n}=0\) and \(\sum_{n=0}^{\infty}a_{n}=\infty\);

     
  3. (3)

    \(\liminf_{n\rightarrow\infty} b_{n}>0\) and \(\liminf_{n\rightarrow\infty} c_{n}>0\);

     
  4. (4)

    \(\sum_{n=0}^{\infty}\lambda_{n}< \infty\);

     
  5. (5)

    \(\sum_{n=0}^{\infty}|\lambda_{n+1}-\lambda_{n}|, \sum_{n=0}^{\infty}|s_{n+1}-s_{n}|, \sum_{n=0}^{\infty}|t_{n+1}-t_{n}|, \sum_{n=0}^{\infty}|a_{n+1}-a_{n}|, \sum_{n=0}^{\infty}|b_{n+1}-b_{n}|, \sum_{n=0}^{\infty}|c_{n+1}-c_{n}|< \infty\).

     
Then \(\{x_{n}\}\) converges strongly to \(\overline{x}=P_{\mathfrak{F}}u\).

Declarations

Acknowledgements

The authors are most grateful to the anonymous referees for their constructive comments and helpful suggestions, which greatly improved the original paper. The work is supported by the National Natural Science Foundation of China (11071041, 11201074), Fujian Natural Science Foundation (2013J01006, 2015J01578) and R&D of Key Instruments and Technologies for Deep Resources Prospecting (the National R&D Projects for Key Scientific Instruments) under grant number ZDYZ2012-1-02-04.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Computer Science, Fujian Normal University

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© Ke and Ma 2015