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

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.

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 [1–3], 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., [6–19] 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:

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

• (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., [20–43]).

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:

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

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

• (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:

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

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

• (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

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

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

• (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

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

• (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

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

• (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:

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

• (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

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

• (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) $a_{n}+b_{n}+c_{n}=1$;

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

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

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

• (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

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

• (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) $a_{n}+b_{n}+c_{n}=1$;

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

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

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

• (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

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

• (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) $a_{n}+b_{n}+c_{n}=1$;

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

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

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

• (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 $\overline{x}=P_{\mathfrak{F}}u$.