A new iterative algorithm for split solution problems of quasi-nonexpansive mappings

Abstract

Some strong convergence algorithms are introduced to solve the split common fixed point problem for quasi-nonexpansive mappings. These results develop the related ones for fixed point iterative methods in the literature.

1 Introduction and preliminaries

Throughout this paper, let H be a real Hilbert space with zero vector θ, whose inner product and norm are denoted by $\langle \cdot,\cdot\rangle$ and $\Vert\cdot\Vert$, respectively. The symbols ℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively. Let K be a nonempty closed convex subset of a Banach space E and T be a mapping from K into itself. In this paper, the set of fixed points of T is denoted by $F(T)$. The symbols → and ⇀ denote strong and weak convergence, respectively.

Let $T:K\rightarrow K$ be a mapping and K a subset of a Banach space E. T is called a nonexpansive mapping if, for all $x,y\in K$, $\Vert Tx-Ty\Vert\leq\Vert x-y\Vert$. T is called quasi-nonexpansive, if $F(T)\neq\emptyset$ and for all $x\in K$, $p\in F(T)$, $\Vert Tx-Tp\Vert\leq\Vert x-p\Vert$. For examples of quasi-nonexpansive mappings, see [1].

Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces. $T_{1}:H_{1}\rightarrow H_{1}$, $T_{2}:H_{2}\rightarrow H_{2}$ are two nonlinear operators with $F(T_{1})\neq\emptyset$ and $F(T_{2})\neq\emptyset$. $A:H_{1}\rightarrow H_{2}$ is a bounded linear operator. The split fixed point problem for $T_{1}$ and $T_{2}$ is to

$$ \mbox{find an element } x\in F(T_{1}) \mbox{ such that } Ax\in F(T_{2}). $$

(1.1)

Let $\Gamma=\{ x\in F(T_{1}): Ax\in F(T_{2})\}$ denote the solution set of the problem (1.1). The problem was proposed by Censor and Segal [2] in a finite-dimensional space firstly. Next, Moudafi [3] studied the problem (1.1) in real Hilbert spaces; this generalized the problem (1.1) from a finite-dimensional space to infinite-dimensional Hilbert spaces. More precisely, the following result was obtained.

Theorem M

(see [3])

Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces. Given a bounded linear operator $A:H_{1}\rightarrow H_{2}$, let $U:H_{1}\rightarrow H_{1}$ and $T:H_{2}\rightarrow H_{2}$ be two quasi-nonexpansive operators with $F(U)\neq\emptyset$ and $F(T)\neq\emptyset$. Assume that $U-I$ and $T-I$ are demiclosed at θ. Let $\{x_{n}\}$ be generated by

$$ \left \{ \begin{array}{l} x_{1}\in H_{1}, \\ u_{n}=x_{n}+\gamma\beta A^{*}(T-I)Ax_{n}, \\ x_{n+1}=(1-\alpha_{n})u_{n}+\alpha_{n}U(u_{n}),\quad \forall n\in \mathbb{N},\end{array} \right . $$

(1.2)

where $\beta\in(0,1)$, $\{\alpha_{n}\}\subset(\delta,1-\delta)$ for a small enough $\delta>0$, $\gamma\in(0,\frac{1}{\lambda\beta})$, and λ is the spectral radius of the operator $A^{*}A$. Then $\{x_{n}\}$ weakly converges to a split common fixed point $x^{*}\in\{x^{*}\in F(U): Ax^{*}\in F(T)\}$.

It is well known that the split feasibility problem and the convex feasibility problem are useful to some areas of applied mathematics such as image recovery, convex optimization, and so on. According to [2], the split common fixed point problem (1.1) is a generalization of both these; also see [3]. This shows the split common fixed point problem (1.1) is important. Recently, some convergence theorems for the split common solution problems were given in [4–9]. We notice that Theorem M is a weak convergence theorem, and it is well known that a strong convergence theorem is always more convenient to use. Hence, the purpose of this paper is to give some algorithms for the problem (1.1), and establishes some strong convergence theorems. At the same time, we generalize the problem (1.1) to two countable families of quasi-nonexpansive mappings.

A mapping T is said to be demiclosed if, for any sequence $\{x_{n}\}$ which weakly converges to y, and if the sequence $\{Tx_{n}\}$ strongly converges to z, we have $T(y)=z$; see [3].

Definition 1.1

(see [4, 5])

Let K be a nonempty closed convex subset of a real Hilbert space and T a mapping from K into K. The mapping T is called zero-demiclosed if $\{x_{n}\}$ in K satisfying $\Vert x_{n}-Tx_{n}\Vert\rightarrow0$ and $x_{n}\rightharpoonup z\in K$ implies $Tz=z$.

Proposition 1.1

(see [4, 5])

Let K be a nonempty closed convex subset of a real Hilbert space with zero vector θ and T a mapping from K into K. Then the following statements hold.

• (a) T is zero-demiclosed if and only if $I-T$ is demiclosed at θ.

• (b) If T is a nonexpansive mapping and there is a bounded sequence $\{x_{n}\}\subset H$ such that $\Vert x_{n}-Tx_{n}\Vert\rightarrow0$ as $n\rightarrow0$, then T is zero-demiclosed.

Example 1.1

(see [4])

Let $H=\mathbb{R}$ with the inner product defined by $\langle x,y\rangle=xy$ for all $x,y\in\mathbb{R}$ and the standard norm $|\cdot|$. Let $C:=[0,+\infty)$ and $Tx=\frac{x^{2}+2}{1+x}$ for all $x\in C$. Then T is a continuous zero-demiclosed quasi-nonexpansive mapping but not nonexpansive.

Example 1.2

(see [4])

Let $H=\mathbb{R}$ with the inner product defined by $\langle x,y\rangle=xy$ for all $x,y\in\mathbb{R}$ and the standard norm $|\cdot|$. Let $C:=[0,+\infty)$. Let T be a mapping from C into C defined by

$$ Tx=\left \{ \begin{array}{l@{\quad}l} \frac{2x}{x^{2}+1},&x\in(1,+\infty), \\ 0,& x\in[0,1]. \end{array} \right . $$

Then T is a discontinuous quasi-nonexpansive mapping but not zero-demiclosed.

The following results are important in this paper.

Let C be a closed convex subset of a real Hilbert space H. $P_{C}$ denotes a metric projection of H onto C, it is well known that $P_{C}(x)$ has the properties: for $x\in H$, and $z\in C$,

$$ z=P_{C}(x)\quad \Leftrightarrow\quad \langle x-z,z-y\rangle\geq0, \quad \forall y \in C $$

(1.3)

and

$$ \bigl\Vert y-P_{C}(x)\bigr\Vert ^{2}+\bigl\Vert x-P_{C}(x)\bigr\Vert ^{2}\leq\Vert x-y\Vert^{2}, \quad \forall y\in C, \forall x\in H. $$

(1.4)

In a real Hilbert space H, it is also well known that

$$ \bigl\Vert \lambda x+(1-\lambda)y\bigr\Vert ^{2}= \lambda \Vert x\Vert^{2}+(1-\lambda)\Vert y\Vert^{2}- \lambda(1-\lambda)\Vert x-y\Vert^{2},\quad \forall x,y\in H, \forall \lambda\in\mathbb{R} $$

(1.5)

and

$$ 2\langle x,y\rangle=\Vert x\Vert^{2}+\Vert y\Vert^{2}-\Vert x- y\Vert^{2},\quad \forall x,y\in H. $$

(1.6)

2 Strong convergence theorems

In this section, we construct some algorithms to solve the split common fixed point problem (1.1) for quasi-nonexpansive mappings.

Theorem 2.1

Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces. C is a nonempty closed convex subset of $H_{1}$ and K a nonempty closed convex subset of $H_{2}$. $T_{1}: C\rightarrow H_{1}$ and $T_{2}:H_{2}\rightarrow H_{2}$ are two quasi-nonexpansive mappings with $F(T_{1})\neq\emptyset$ and $F(T_{2})\neq\emptyset$. $A: H_{1}\rightarrow H_{2}$ is a bounded linear operator. Assume that $T_{1} -I$ and $T_{2}-I$ are demiclosed at θ. Let $x_{0}\in C$, $C_{0}=C$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})T_{1}z_{n}, \\ z_{n}=P_{C}(x_{n}+\lambda A^{*}(T_{2}-I)Ax_{n}), \\ C_{n+1}=\{x\in C_{n}:\Vert y_{n}-x\Vert\leq\Vert z_{n}-x\Vert\leq\Vert x_{n}-x\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N} \cup\{0\}, \end{array} \right . $$

(2.1)

where P is a projection operator and $A^{*}$ denotes the adjoint of A. $\{\alpha_{n}\}\subset(0,\eta]\subset(0,1)$, $\lambda\in(0,\frac{1}{\Vert A^{*}\Vert^{2}})$. Assume that $\Gamma=\{p\in F(T_{1}): Ap\in F(T_{2})\}\neq\emptyset$, then $x_{n} \rightarrow x^{*}\in \Gamma$ and $Ax_{n} \rightarrow Ax^{*}\in F(T_{2})$.

Proof

It is easy to verify that $C_{n}$ is closed for $n\in\mathbb{N}\cup\{0\}$. We verify $C_{n}$ is convex for $n\in\mathbb{N}\cup\{0\}$. In fact, let $v_{1},v_{2}\in C_{n+1}$, for each $\lambda\in(0,1)$, we have

$$\begin{aligned} \bigl\Vert y_{n}-\bigl(\lambda v_{1}+(1- \lambda)v_{2}\bigr)\bigr\Vert ^{2} =&\bigl\Vert \lambda(y_{n}-v_{1})-(1-\lambda) (y_{n}-v_{2}) \bigr\Vert ^{2} \\ =& \lambda\Vert y_{n}-v_{1}\Vert^{2}+(1-\lambda) \Vert y_{n}-v_{2}\Vert ^{2}-\lambda(1-\lambda)\Vert v_{1}-v_{2}\Vert^{2} \\ \leq&\lambda\Vert z_{n}-v_{1}\Vert^{2}+(1- \lambda)\Vert z_{n}-v_{2}\Vert^{2}-\lambda(1- \lambda)\Vert v_{1}-v_{2}\Vert^{2} \\ =&\bigl\Vert z_{n}-\bigl(\lambda v_{1}+(1- \lambda)v_{2}\bigr)\bigr\Vert ^{2}, \end{aligned}$$

namely, $\Vert y_{n}-(\lambda v_{1}+(1-\lambda)v_{2})\Vert\leq\Vert z_{n}-(\lambda v_{1}+(1-\lambda)v_{2})\Vert$. Similarly, we have $\Vert z_{n}-(\lambda v_{1}+(1-\lambda)v_{2})\Vert\leq\Vert x_{n}-(\lambda v_{1}+(1-\lambda)v_{2})\Vert$; this shows $\lambda v_{1}+(1-\lambda)v_{2}\in C_{n+1}$ and $C_{n+1}$ is a convex set for $n\in\mathbb{N}\cup\{0\}$. Now we prove $\Gamma\subset C_{n}$ for $n\in\mathbb{N}\cup\{0\}$. Let $p\in\Gamma$, then

$$\begin{aligned}& 2\lambda\bigl\langle x_{n}-p, A^{*}(T_{2}Ax_{n}-Ax_{n}) \bigr\rangle \\& \quad = 2\lambda\bigl\langle A(x_{n}-p)+(T_{2}Ax_{n}-Ax_{n})-(T_{2}Ax_{n}-Ax_{n}), T_{2}Ax_{n}-Ax_{n}\bigr\rangle \\& \quad = 2\lambda\bigl(\bigl\langle T_{2}Ax_{n}-Ap, (T_{2}Ax_{n}-Ax_{n})\bigr\rangle -\Vert T_{2}Ax_{n}-Ax_{n}\Vert^{2}\bigr) \\& \quad = 2\lambda\biggl(\frac{1}{2}\Vert T_{2}Ax_{n}-Ap \Vert^{2}+\frac{1}{2}\Vert T_{2}Ax_{n}-Ax_{n} \Vert^{2} \\& \qquad {}-\frac {1}{2}\Vert Ax_{n}-Ap\Vert^{2} -\Vert T_{2}Ax_{n}-Ax_{n}\Vert^{2}\biggr) \quad \text{by (1.6)} \\& \quad \leq 2\lambda\biggl(\frac{1}{2}\Vert T_{2}Ax_{n}-Ax_{n} \Vert^{2} -\Vert T_{2}Ax_{n}-Ax_{n} \Vert^{2}\biggr) \\& \quad = -\lambda\Vert T_{2}Ax_{n}-Ax_{n} \Vert^{2}. \end{aligned}$$

(2.2)

From (2.1) and (2.2) we have

$$\begin{aligned} \Vert z_{n}-p\Vert ^{2} =&\bigl\Vert P_{C} \bigl(x_{n}+\lambda A^{*}(T_{2}Ax_{n}-Ax_{n}) \bigr)-P_{C}(p)\bigr\Vert ^{2} \\ \leq&\bigl\Vert x_{n}+\lambda A^{*}(T_{2}Ax_{n}-Ax_{n})-p \bigr\Vert ^{2} \\ =&\Vert x_{n}-p\Vert ^{2}+\bigl\Vert \lambda A^{*}(T_{2}Ax_{n}-Ax_{n})\bigr\Vert ^{2}+ 2\lambda \bigl\langle x_{n}-p, A^{*}(T_{2}Ax_{n}-Ax_{n}) \bigr\rangle \\ \leq&\Vert x_{n}-p\Vert ^{2}+ \lambda^{2} \bigl\Vert A^{*}\bigr\Vert ^{2}\Vert T_{2}Ax_{n}-Ax_{n} \Vert ^{2}-\lambda \Vert T_{2}Ax_{n}-Ax_{n} \Vert ^{2} \\ =& \Vert x_{n}-p\Vert ^{2}-\lambda\bigl(1-\lambda\bigl\Vert A^{*}\bigr\Vert ^{2} \bigr)\Vert T_{2}Ax_{n}-Ax_{n} \Vert ^{2}. \end{aligned}$$

(2.3)

Again from $p\in\Gamma$, (2.1), and (2.3), it follows that

$$ \Vert y_{n}-p\Vert\leq\Vert z_{n}-p\Vert\leq\Vert x_{n}-p\Vert. $$

(2.4)

Hence, $p\in C_{n}$ and $\Gamma\subset C_{n}$ for $n\in \mathbb{N}\cup\{0\}$.

Notice that $\Gamma\subset C_{n+1}\subset C_{n}$ and $x_{n+1}=P_{C_{n+1}}(x_{0})\subset C_{n}$, then

$$ \Vert x_{n+1}-x_{0}\Vert\leq\Vert p-x_{0}\Vert \quad \text{for } n\in\mathbb {N} \text{ and } p\in\Gamma. $$

(2.5)

By (2.5), $\{x_{n}\}$ is bounded. For $n\in\mathbb{N}$, by (1.4), we have

$$ \Vert x_{n+1}-x_{n}\Vert ^{2}+\Vert x_{0}-x_{n}\Vert ^{2}=\bigl\Vert x_{n+1}-P_{C_{n}}(x_{0})\bigr\Vert ^{2}+ \bigl\Vert x_{0}-P_{C_{n}}(x_{0})\bigr\Vert ^{2}\leq \Vert x_{n+1}-x_{0}\Vert ^{2}, $$

which implies that $0\leq\Vert x_{n}-x_{n+1}\Vert^{2}\leq\Vert x_{n+1}-x_{0}\Vert^{2}-\Vert x_{0}-x_{n}\Vert^{2}$. Thus $\{\Vert x_{n}-x_{0}\Vert\}$ is non-decreasing. Therefore, by the boundedness of $\{ x_{n}\}$, $\lim_{n\rightarrow\infty}\Vert x_{n}-x_{0}\Vert$ exists. For $m, n\in\mathbb{N}$ with $m>n$, from $x_{m}=P_{C_{m}}(x_{0})\subset C_{n}$ and (1.4), we have

$$ \Vert x_{m}-x_{n}\Vert ^{2}+\Vert x_{0}-x_{n}\Vert ^{2}=\bigl\Vert x_{m}-P_{C_{n}}(x_{0})\bigr\Vert ^{2}+ \bigl\Vert x_{0}-P_{C_{n}}(x_{0})\bigr\Vert ^{2}\leq \Vert x_{m}-x_{0}\Vert ^{2}. $$

(2.6)

By (2.5) and (2.6), $\lim_{n\rightarrow\infty}\Vert x_{n}-x_{m}\Vert=0$. So, $\{x_{n}\}$ is a Cauchy sequence.

Let $x_{n}\rightarrow x^{*}$. Since $x_{n+1}=P_{C_{n+1}}(x_{0})\in C_{n+1}\subset C_{n}$, we have

$$\begin{aligned}& \Vert z_{n}-x_{n}\Vert\leq\Vert z_{n}-x_{n+1}\Vert+\Vert x_{n+1}-x_{n}\Vert \leq2\Vert x_{n+1}-x_{n}\Vert\rightarrow0, \\& \Vert y_{n}-x_{n}\Vert\leq\Vert y_{n}-x_{n+1} \Vert+\Vert x_{n+1}-x_{n}\Vert\leq2\Vert x_{n+1}-x_{n} \Vert\rightarrow0, \\& \Vert y_{n}-z_{n}\Vert\leq\Vert y_{n}-x_{n} \Vert+\Vert x_{n}-z_{n}\Vert \rightarrow0. \end{aligned}$$

(2.7)

Notice that $\lambda(1- \lambda\Vert A^{*}\Vert^{2})>0$, from (2.3) and (2.7),

$$\begin{aligned} \Vert T_{2}Ax_{n}-Ax_{n}\Vert^{2} \leq& \frac{ \Vert x_{n}-p\Vert^{2}-\Vert z_{n}-p\Vert^{2} }{\lambda(1- \lambda\Vert A^{*}\Vert^{2})} \\ \leq& \frac{1}{\lambda(1- \lambda\Vert A^{*}\Vert^{2})}\Vert x_{n}-z_{n}\Vert \bigl\{ \Vert x_{n}-p\Vert +\Vert z_{n}-p\Vert \bigr\} \rightarrow0. \end{aligned}$$

(2.8)

Again from (2.1) and (2.7), we have

$$ \Vert T_{1}z_{n}-z_{n}\Vert=\bigl\Vert (T_{1}-I)z_{n}\bigr\Vert \rightarrow0. $$

(2.9)

Since $x_{n}\rightarrow x^{*}$, from (2.7) we have $z_{n}\rightarrow x^{*}$, which implies that $z_{n}\rightharpoonup x^{*}$. By Proposition 1.1, we obtain $x^{*}\in F(T_{1})$.

Next, we want to show $Ax^{*}\in F(T_{2})$. Since A is a bounded linear operator, we know that $\Vert Ax_{n}-Ax^{*}\Vert\rightarrow0$ by $x_{n}\rightarrow x^{*}$. Together with $\Vert T_{2}Ax_{n}-Ax_{n}\Vert\rightarrow0$ and $T_{2}-I$ being demiclosed at θ, we have $Ax^{*}\in F(T_{2})$. Thus, $x^{*}\in \Gamma$ and $\{x_{n}\}$ converges strongly to $x^{*}\in\Gamma$. The proof is completed. □

Remark 2.1

If the quasi-nonexpansive mappings $T_{1}$ and $T_{2}$ are continuous, then the demiclosed property can be removed for the quasi-nonexpansive mappings $T_{1}$ and $T_{2}$ in Theorem 2.1.

Now, we consider the split fixed point problem for a finite family of quasi-nonexpansive mappings.

Lemma 2.1

(see [3])

Let $T:H\rightarrow H$ be a quasi-nonexpansive mapping, and set $T_{\alpha}:=(1-\alpha)I+\alpha T$ for $\alpha\in(0,1]$. Then $\Vert T_{\alpha}x-p\Vert\leq\Vert x-p\Vert-\alpha(1-\alpha)\Vert T x-x\Vert$, $p\in F(T)$ and $x\in H$. Moreover, $F(T_{\alpha})=F(T)$.

Lemma 2.2

Let $T_{1}, T_{2}:H\rightarrow H$ be two quasi-nonexpansive mappings and set $S_{\xi_{1}}:=(1-\xi_{1})I+\xi_{1}T_{1}$ and $S_{\xi_{2}}:=(1-\xi_{2})I+\xi_{2}T_{2}$ for $\xi_{1}, \xi_{2}\in(0,1)$. Again let $S=\tau S_{\xi_{1}}+(1-\tau)S_{\xi_{2}}$ for $\tau\in(0,1)$. Then S is a quasi-nonexpansive mapping, and $F(S)=\bigcap_{i=1}^{2}F(S_{\xi_{i}})=\bigcap_{i=1}^{2}F(T_{i})$.

Proof

(1) It is easy to verify that $\bigcap_{i=1}^{2}F(S_{\xi_{i}})=\bigcap_{i=1}^{2}F(T_{i})$. We only need to prove $F(S)=\bigcap_{i=1}^{2}F(S_{\xi_{i}})$. Clearly, $\bigcap_{i=1}^{2}F(S_{\xi_{i}})\subset F(S)$. On the other hand, for $p\in F(S)$ and $p_{1}\in\bigcap_{i=1}^{2}F(S_{\xi_{i}})$, we have

$$\begin{aligned} \Vert p-p_{1}\Vert ^{2} =&\bigl\Vert \tau S_{\xi_{1}}p+(1-\tau)S_{\xi_{2}} p-p_{1}\bigr\Vert ^{2}=\bigl\Vert \tau( S_{\xi_{1}}p-p_{1})+(1-\tau) (S_{\xi_{2}} p-p_{1})\bigr\Vert ^{2} \\ =&\tau \Vert S_{\xi_{1}}p-p_{1}\Vert ^{2}+(1-\tau) \Vert S_{\xi_{2}} p-p_{1}\Vert ^{2}-\tau(1-\tau) \Vert S_{\xi_{1}}p-S_{\xi_{2}} p\Vert ^{2} \\ \leq&\tau \Vert p-p_{1}\Vert ^{2}-\tau \xi_{1}(1-\xi_{1})\Vert T_{1}p-p\Vert ^{2}+(1-\tau)\Vert p-p_{1}\Vert ^{2} \\ &{}-(1-\tau) \xi_{2}(1-\xi _{2})\Vert T_{2} p-p \Vert ^{2}\quad \text{(by Lemma 2.1)} \\ =& \Vert p-p_{1}\Vert ^{2}-\tau\xi_{1}(1- \xi_{1})\Vert T_{1}p-p\Vert ^{2}-(1-\tau) \xi_{2}(1-\xi_{2})\Vert T_{2} p-p \Vert ^{2}, \end{aligned}$$

which yields $\Vert T_{1}p-p\Vert=\Vert T_{2} p-p \Vert=0$, namely, $p\in\bigcap_{i=1}^{2}F(T_{i})= \bigcap_{i=1}^{2}F(S_{\xi_{i}})$. So, $F(S)=\bigcap_{i=1}^{2}F(S_{\xi_{i}})$.

(2) Let $x\in H$ and $p\in F(S)$. Then

$$\begin{aligned} \Vert Sx-p\Vert =&\bigl\Vert \tau S_{\xi_{1}}x+(1- \tau)S_{\xi_{2}} x-p\bigr\Vert =\bigl\Vert \tau( S_{\xi_{1}}x-p)+(1- \tau) (S_{\xi_{2}} x-p)\bigr\Vert \\ \leq&\tau \Vert x-p\Vert +(1-\tau)\Vert x-p \Vert =\Vert x-p \Vert \quad \text{(by Lemma 2.1)}. \end{aligned}$$

So, S is a quasi-nonexpansive mapping. The proof is completed. □

Lemma 2.3

Let $T_{1}, T_{2},\ldots, T_{k}:H\rightarrow H$ be k quasi-nonexpansive mappings and set $S=\sum_{i=1}^{k}\tau_{i} S_{\xi_{i}}$, where $\tau_{i}\in(0,1)$ satisfies $\sum_{i=1}^{k}\tau_{i} =1$, $S_{\xi_{i}}:=(1-\xi_{i})I+\xi_{i}T_{i}$ for $\xi_{i}\in(0,1)$, $i=1,2,\ldots,k$. Then S is a quasi-nonexpansive mapping, and $F(S)=\bigcap_{i=1}^{k}F(S_{\xi_{i}})=\bigcap_{i=1}^{k}F(T_{i})$.

Proof

Using mathematical induction, Lemma 2.3 is obtained by Lemma 2.2. □

Theorem 2.2

Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces. C is a nonempty closed convex subset of $H_{1}$ and K a nonempty closed convex subset of $H_{2}$. $T_{1},\ldots, T_{k}: C\rightarrow H_{1}$ are k quasi-nonexpansive mappings with $\bigcap_{i=1}^{k}F(T_{i})\neq\emptyset$. $G_{1},\ldots, G_{l}:H_{2}\rightarrow H_{2}$ are l quasi-nonexpansive mappings with $\bigcap_{j=1}^{l}F(G_{j})\neq\emptyset$. $A: H_{1}\rightarrow H_{2}$ is a bounded linear operator. Assume that $T_{i}-I$ ($i=1,2,\ldots,k$) and $G_{j}-I$ ($j=1,2,\ldots, l$) are demiclosed at θ. Let $x_{0}\in C$, $C_{0}=C$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \textstyle \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})\sum_{i=1}^{k}\tau_{i} T_{\xi _{i}}z_{n}, \\ z_{n}=P_{C}(x_{n}+\lambda A^{*}(\sum_{i=1}^{l}\varepsilon_{j} G_{\theta_{j}} -I)Ax_{n}), \\ C_{n+1}=\{v\in C_{n}:\Vert y_{n}-v\Vert\leq\Vert z_{n}-v\Vert\leq\Vert x_{n}-v\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N}\cup\{0\},\end{array} \right . $$

(2.10)

where P is a projection operator and $A^{*}$ denotes the adjoint of A, $\{\alpha_{n}\}\subset(0,\eta]\subset(0,1)$, $\lambda\in(0,\frac{1}{\|A^{*}\|^{2}})$. $\tau_{i}\in(0,1)$ and $\varepsilon_{j}\in(0,1)$ satisfy $\sum_{i=1}^{k}\tau_{i} =1$ and $\sum_{j=1}^{l}\varepsilon_{j} =1$, $T_{\xi _{i}}:=(1-\xi_{i})I+\xi_{i}T_{i}$ for $\xi_{i}\in(0,1)$, $i=1,2,\ldots,k$, $G_{\theta_{j}}:=(1-\theta_{j})I+\theta_{j}G_{j}$ for $\theta_{j}\in(0,1)$, $j=1,2,\ldots,l$. Assume that $\Gamma=\{p\in\bigcap_{i=1}^{k}F(T_{i}): Ap\in\bigcap_{j=1}^{l}F(G_{j})\}\neq\emptyset$, then the sequence $\{x_{n}\}$ converges strongly to an element $q\in\Gamma$.

Proof

Let $T=\sum_{i=1}^{k}\tau_{i} T_{\xi_{i}}$, $S=\sum_{i=1}^{l}\varepsilon_{j} G_{\theta_{j}}$, by Lemma 2.3, $F(T)=\bigcap_{i=1}^{k}F(T_{i})\neq\emptyset$, and $F(S)=\bigcap_{j=1}^{l}F(G_{j})\neq\emptyset$. Moreover, T and S are quasi-nonexpansive mappings.

Next, we want to prove $T-I$ and $S-I$ are demiclosed at θ. By the hypothesis, $T_{i}-I$ ($i=1,2,\ldots,k$) and $G_{j}-I$ ($j=1,2,\ldots, l$) are demiclosed at θ. So, $T_{\xi_{i}}-I= \xi_{i}(T_{i}-I)$ and $G_{\theta_{j}}-I=\theta_{j}(G_{j}-I)$ are demiclosed at θ, and that $T-I=\sum_{i=1}^{k}\tau_{i} (T_{\xi_{i}}-I)$ and $S-I=\sum_{i=1}^{l}\varepsilon_{j} (G_{\theta_{j}}-I)$ are demiclosed at θ.

Thus, by Theorem 2.1, we obtain the desired result. The proof is completed. □

If $C=H_{1}$ in Theorem 2.1 and Theorem 2.2, then we have the following corollaries.

Corollary 2.1

Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces. $T_{1}: H_{1}\rightarrow H_{1}$ and $T_{2}:H_{2}\rightarrow H_{2}$ are two quasi-nonexpansive mappings with $F(T_{1})\neq\emptyset$ and $F(T_{2})\neq\emptyset$. $A: H_{1}\rightarrow H_{2}$ is a bounded linear operator. Assume that $T_{1} -I$ and $T_{2}-I$ are demiclosed at θ. Let $x_{0}\in H_{1}$, $C_{0}=H_{1}$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})T_{1}z_{n}, \\ z_{n}= x_{n}+\lambda A^{*}(T_{2}Ax_{n}-Ax_{n}) , \\ C_{n+1}=\{v\in C_{n}:\Vert y_{n}-v\Vert\leq\Vert z_{n}-v\Vert\leq\Vert x_{n}-v\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N}\cup\{0\},\end{array} \right . $$

where P is a projection operator and $A^{*}$ denotes the adjoint of A, $\{\alpha_{n}\}\subset (0,\eta]\subset(0,1)$, $\lambda\in(0,\frac{1}{\|A^{*}\|^{2}})$. Assume that $\Gamma=\{p\in F(T_{1}): Ap\in F(T_{2})\}\neq\emptyset$, then the sequence $\{x_{n}\}$ converges strongly to an element $x^{*}\in\Gamma$.

Corollary 2.2

Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces. $T_{1},\ldots, T_{k}: H_{1}\rightarrow H_{1}$ are k quasi-nonexpansive mappings with $\bigcap_{i=1}^{k}F(T_{i})\neq\emptyset$. $G_{1},\ldots, G_{l}: H_{2}\rightarrow H_{2}$ are l quasi-nonexpansive mappings with $\bigcap_{j=1}^{l}F(G_{j})\neq\emptyset$. $A: H_{1}\rightarrow H_{2}$ is a bounded linear operator. Assume that $T_{i}-I$ ($i=1,2,\ldots,k$) and $G_{j}-I$ ($j=1,2,\ldots, l$) are demiclosed at θ. Let $x_{0}\in H_{1}$, $C_{0}=H_{1}$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \textstyle \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})\sum_{i=1}^{k}\tau_{i} T_{\xi _{i}}z_{n}, \\ z_{n}= x_{n}+\lambda A^{*}(\sum_{i=1}^{k}\tau_{i} G_{\theta_{j}}Ax_{n}-Ax_{n}) , \\ C_{n+1}=\{v\in C_{n}:\Vert y_{n}-v\Vert\leq\Vert z_{n}-v\Vert\leq\Vert x_{n}-v\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N}\cup\{0\},\end{array} \right . $$

where P is a projection operator and $A^{*}$ denotes the adjoint of A, $\{\alpha_{n}\}\subset (0,\eta]\subset(0,1)$, $\lambda\in(0,\frac{1}{\|A^{*}\|^{2}})$. Here $\tau_{i}\in(0,1)$ and $\varepsilon_{j}\in(0,1)$ satisfy $\sum_{i=1}^{k}\tau_{i} =1$ and $\sum_{j=1}^{l}\varepsilon_{j} =1$, $T_{\xi _{i}}:=(1-\xi_{i})I+\xi_{i}T_{i}$ for $\xi_{i}\in(0,1)$, $i=1,2,\ldots,k$, $G_{\theta_{j}}:=(1-\theta_{j})I+\theta_{j}G_{j}$ for $\theta_{j}\in(0,1)$, $j=1,2,\ldots,l$. Assume that $\Gamma=\{p\in\bigcap_{i=1}^{k}F(T_{i}): Ap\in\bigcap_{j=1}^{l}F(G_{j})\}\neq\emptyset$, then the sequence $\{x_{n}\}$ converges strongly to an element $q\in\Gamma$.

If $H_{1}=H_{2}:=H$ and A is an identity operator, then we have the following results by Theorems 2.1 and 2.2, respectively.

Corollary 2.3

Let H be a real Hilbert space. C is a nonempty closed convex subset of H. $T_{1}: C\rightarrow H$ and $T_{2}:H\rightarrow H$ are two quasi-nonexpansive mappings with $\Gamma:=F(T_{1})\cap F(T_{2})\neq \emptyset$. Assume that $T_{1} -I$ and $T_{2}-I$ are demiclosed at θ. Let $x_{0}\in C$, $C_{0}=C$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})T_{1}z_{n}, \\ z_{n}=P_{C}((1-\lambda)x_{n}+\lambda T_{2}x_{n}), \\ C_{n+1}=\{x\in C_{n}:\Vert y_{n}-x\Vert\leq\Vert z_{n}-x\Vert\leq\Vert x_{n}-x\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N} \cup\{0\},\end{array} \right . $$

where P is a projection operator. $\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)$, $\lambda\in(0,1)$. Then $x_{n} \rightarrow x^{*}\in \Gamma$.

Corollary 2.4

Let H be a real Hilbert space. C is a nonempty closed convex subset of H. $T_{1},\ldots, T_{k}: C\rightarrow H$ are k quasi-nonexpansive mappings with $\bigcap_{i=1}^{k}F(T_{i})\neq\emptyset$. $G_{1},\ldots, G_{l}:H\rightarrow H$ are l quasi-nonexpansive mappings with $\bigcap_{j=1}^{l}F(G_{j})\neq\emptyset$. Assume that $T_{i}-I$ ($i=1,2,\ldots,k$) and $G_{j}-I$ ($j=1,2,\ldots, l$) are demiclosed at θ. Let $x_{0}\in C$, $C_{0}=C$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \textstyle \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})\sum_{i=1}^{k}\tau_{i} T_{\xi _{i}}z_{n}, \\ z_{n}=P_{C}((1-\lambda)x_{n}+\lambda \sum_{i=1}^{l}\varepsilon_{j} G_{\theta _{j}} x_{n}), \\ C_{n+1}=\{v\in C_{n}:\Vert y_{n}-v\Vert\leq\Vert z_{n}-v\Vert\leq\Vert x_{n}-v\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N}\cup\{0\} ,\end{array} \right . $$

where P is a projection operator. $\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)$, $\lambda\in(0,1)$. $\tau_{i}\in(0,1)$ and $\varepsilon_{j}\in(0,1)$ satisfy $\sum_{i=1}^{k}\tau_{i} =1$ and $\sum_{j=1}^{l}\varepsilon_{j} =1$, $T_{\xi _{i}}:=(1-\xi_{i})I+\xi_{i}T_{i}$ for $\xi_{i}\in(0,1)$, $i=1,2,\ldots,k$, $G_{\theta_{j}}:=(1-\theta_{j})I+\theta_{j}G_{j}$ for $\theta_{j}\in(0,1)$, $j=1,2,\ldots,l$. Assume that $\Gamma:=(\bigcap_{i=1}^{k}F(T_{i}))\cap( \bigcap_{j=1}^{l}F(G_{j}))\neq\emptyset$, then the sequence $\{x_{n}\}$ converges strongly to an element $q\in\Gamma$.

If $C=H:=H_{1}=H_{2}$ and A is an identity operator, then we have the following results by Corollaries 2.3 and 2.4, respectively.

Corollary 2.5

Let H be a real Hilbert space. $T_{1}, T_{2}: H\rightarrow H$ are two quasi-nonexpansive mappings with $\Gamma:=F(T_{1})\cap F(T_{2})\neq \emptyset$. Assume that $T_{1} -I$ and $T_{2}-I$ are demiclosed at θ. Let $x_{0}\in C$, $C_{0}=C$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})T_{1}z_{n}, \\ z_{n}=(1-\lambda)x_{n}+\lambda T_{2}x_{n}, \\ C_{n+1}=\{x\in C_{n}:\Vert y_{n}-x\Vert\leq\Vert z_{n}-x\Vert\leq\Vert x_{n}-x\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N} \cup\{0\},\end{array} \right . $$

where P is a projection operator. $\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)$, $\lambda\in(0,1)$. Then $x_{n} \rightarrow x^{*}\in \Gamma$.

Corollary 2.6

Let H be a real Hilbert space. $T_{1},\ldots, T_{k}: H\rightarrow H$ are k quasi-nonexpansive mappings with $\bigcap_{i=1}^{k}F(T_{i})\neq\emptyset$. $G_{1},\ldots, G_{l}:H\rightarrow H$ are l quasi-nonexpansive mappings with $\bigcap_{j=1}^{l}F(G_{j})\neq\emptyset$. Assume that $T_{i}-I$ ($i=1,2,\ldots,k$) and $G_{j}-I$ ($j=1,2,\ldots, l$) are demiclosed at θ. Let $x_{0}\in C$, $C_{0}=C$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \textstyle \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})\sum_{i=1}^{k}\tau_{i} T_{\xi _{i}}z_{n}, \\ z_{n}=(1-\lambda)x_{n}+\lambda \sum_{i=1}^{l}\varepsilon_{j} G_{\theta_{j}} x_{n}, \\ C_{n+1}=\{v\in C_{n}:\Vert y_{n}-v\Vert\leq\Vert z_{n}-v\Vert\leq\Vert x_{n}-v\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N}\cup\{0\} ,\end{array} \right . $$

where P is a projection operator. $\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)$, $\lambda\in(0,1)$. $\tau_{i}\in(0,1)$ and $\varepsilon_{j}\in(0,1)$ satisfy $\sum_{i=1}^{k}\tau_{i} =1$ and $\sum_{j=1}^{l}\varepsilon_{j} =1$, $T_{\xi _{i}}:=(1-\xi_{i})I+\xi_{i}T_{i}$ for $\xi_{i}\in(0,1)$, $i=1,2,\ldots,k$, $G_{\theta_{j}}:=(1-\theta_{j})I+\theta_{j}G_{j}$ for $\theta_{j}\in(0,1)$, $j=1,2,\ldots,l$. Assume that $\Gamma:=(\bigcap_{i=1}^{k}F(T_{i}))\cap( \bigcap_{j=1}^{l}F(G_{j}))\neq\emptyset$, then the sequence $\{x_{n}\}$ converges strongly to an element $q\in\Gamma$.

Remark 2.2

The coefficient condition that $\{\alpha_{n}\}\subset(\delta,1-\delta)$ for a small enough $\delta>0$ in Theorem M is replaced with $\{\alpha_{n}\}\subset(0,\eta]\subset (0,1)$. This shows we can let $\alpha_{n}=\frac{1}{n+1}$ in this paper, which is a natural choice.

3 Further generalization of the problem (1.1)

In Section 2, we gave a strong convergence algorithm for the problem (1.1). By the algorithm, we also considered the split solution problem for two finite families of quasi-nonexpansive mappings; see the algorithm (2.10). However, the algorithm (2.10) has an obvious drawback, in that the algorithm (2.10) will be invalid for two countable families of quasi-nonexpansive mappings. So, in this section, we introduce an algorithm for the split solution problem of two countable families of quasi-nonexpansive mappings. The following lemma can be found in [10].

Lemma

The unique solutions to the positive integer equation

$$ n=i+\frac{(m-1)m}{2},\quad m \geq i, n=1,2,3,\ldots $$

(3.1)

are

$$ i=n-\frac{(m-1)m}{2},\quad m=- \biggl[\frac{1}{2}-\sqrt {2n+ \frac{1}{2}} \biggr]\geq i, n=1,2,3,\ldots, $$

(3.2)

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

Theorem 3.1

Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces. C is a nonempty closed convex subset of $H_{1}$. $A: H_{1}\rightarrow H_{2}$ is a bounded linear operator. $\{T_{i}\}_{i=1}^{\infty}: C\rightarrow H_{1}$ and $\{G_{i}\}_{i=1}^{\infty}: H_{2}\rightarrow H_{2}$ are two countable families of quasi-nonexpansive mappings with $\Gamma=\{p\in\bigcap_{i=1}^{\infty}F(T_{i}): Ap\in\bigcap_{j=1}^{\infty}F(G_{j})\}\neq\emptyset$. Assume that $T_{i}-I$ ($i=1,2,\ldots$) and $G_{j}-I$ ($j=1,2,\ldots$) are demiclosed at θ. Let $x_{0}\in C$, $C_{0}=C$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})T_{i_{n}}z_{n}, \\ z_{n}=P_{C}(x_{n}+\lambda A^{*}( G_{i_{n}}-I)Ax_{n}), \\ C_{n+1}=\{v\in C_{n}:\Vert y_{n}-v\Vert\leq\Vert z_{n}-v\Vert\leq\Vert x_{n}-v\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N}\cup\{0\} ,\end{array} \right . $$

(3.3)

where P is a projection operator and $A^{*}$ denotes the adjoint of A, $\{\alpha_{n}\}\subset(0,\eta]\subset(0,1)$, $\lambda\in(0,\frac{1}{\|A^{*}\|^{2}})$. $i_{n}$ satisfies (3.1), i.e. $i_{n}=n-\frac{(m-1)m}{2}$ and $m\geq i_{n}$ for $n=1,2,\ldots$ . Then the sequence $\{x_{n}\}$ converges strongly to an element $q\in\Gamma$.

Proof

Just like the proof in Theorem 2.1, we can obtain the following facts (I)-(IV):

(I) For $p\in\Gamma$,

$$\begin{aligned}& 2\lambda\bigl\langle x_{n}-p, A^{*}(G_{i_{n}}-I)Ax_{n} \bigr\rangle \leq-\lambda\bigl\Vert (G_{i_{n}}-I)Ax_{n} \bigr\Vert ^{2}, \end{aligned}$$

(3.4)

$$\begin{aligned}& \Vert z_{n}-p\Vert^{2} \leq\Vert x_{n}-p \Vert^{2}-\lambda\bigl(1-\lambda\bigl\Vert A^{*}\bigr\Vert ^{2} \bigr)\bigl\Vert (G_{i_{n}}-I)Ax_{n} \bigr\Vert ^{2} \end{aligned}$$

(3.5)

and

$$ \Vert y_{n}-p\Vert\leq\Vert z_{n}-p\Vert\leq\Vert x_{n}-p\Vert. $$

(3.6)

(II) We have $\Gamma\subset C_{n}$ for $n\in \mathbb{N}\cup\{0\}$. $C_{n}$ is also closed and convex for $n\in\mathbb{N}\cup\{0\}$.

(III) $\{x_{n}\}$ is a Cauchy sequence and

$$ \lim_{n\rightarrow\infty}\Vert z_{n}-x_{n}\Vert=\lim _{n\rightarrow \infty}\Vert y_{n}-x_{n}\Vert=\lim _{n\rightarrow\infty}\Vert y_{n}-z_{n}\Vert=0. $$

(3.7)

(IV)

$$ \lim_{n\rightarrow\infty}\bigl\Vert (T_{i_{n}}-I)z_{n} \bigr\Vert =0, \qquad \lim_{n\rightarrow\infty}\bigl\Vert (G_{i_{n}}-I)Ax_{n} \bigr\Vert =0. $$

(3.8)

Now, for each $i\in\mathbb{N}$, set $K_{i}=\{k\geq1: k=i+\frac {(m-1)m}{2}, m\geq i,m\in \mathbb{N}\}$. Since $n=i_{n}+\frac {(m-1)m}{2}$, $m\geq i_{n}$, and $m\in\mathbb{N}$ for $n=1,2,\ldots$ , and the definition of $K_{i}$, we have $i_{k}\equiv i$ for $k\in K_{i}$. Obviously, $\{k\}$ is a subsequence of $\{n\}$. Thus, for $k\in K_{i}$ and $i\in\mathbb{N}$, it follows from (3.8) that

$$ \begin{aligned} &\lim_{k\rightarrow\infty}\bigl\Vert (T_{i}-I)z_{k} \bigr\Vert =\lim_{k\rightarrow\infty}\bigl\Vert (T_{i_{k}}-I)z_{k} \bigr\Vert =0, \\ &\lim_{k\rightarrow\infty}\bigl\Vert (G_{i}-I)Ax_{k} \bigr\Vert =\lim_{k\rightarrow\infty} \bigl\Vert (G_{i_{k}}-I)Ax_{k} \bigr\Vert =0. \end{aligned} $$

(3.9)

Let $x_{n}\rightarrow x^{*}$. From (3.7) we have $z_{n}\rightarrow x^{*}$. By (3.9), we obtain $x^{*}\in F(T_{i})$.

Next, we want to prove $Ax^{*}\in F(G_{i})$. Since A is a bounded linear operator, $\Vert Ax_{n}-Ax^{*}\Vert\rightarrow0$ by $x_{n}\rightarrow x^{*}$. Together with $\Vert(G_{i}-I)Ax_{k} \Vert\rightarrow0$, we have $Ax_{n}\rightarrow Ax^{*}\in F(G_{i})$. Thus, $x^{*}\in \Gamma$ and $\{x_{n}\}$ converges strongly to $x^{*}\in\Gamma$. The proof is completed. □

If $C=H_{1}$, then we have the following result by Theorem 3.1.

Corollary 3.1

Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces. $A: H_{1}\rightarrow H_{2}$ is a bounded linear operator. $\{T_{i}\}_{i=1}^{\infty}: H_{1}\rightarrow H_{1}$ and $\{G_{i}\}_{i=1}^{\infty}: H_{2}\rightarrow H_{2}$ are two countable families of quasi-nonexpansive mappings with $\Gamma=\{p\in\bigcap_{i=1}^{\infty}F(T_{i}): Ap\in\bigcap_{j=1}^{\infty}F(G_{j})\}\neq\emptyset$. Assume that $T_{i}-I$ ($i=1,2,\ldots$) and $G_{j}-I$ ($j=1,2,\ldots$) are demiclosed at θ. Let $x_{0}\in C$, $C_{0}=H_{1}$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})T_{i_{n}}z_{n}, \\ z_{n}=x_{n}+\lambda A^{*}( G_{i_{n}}-I)Ax_{n}, \\ C_{n+1}=\{v\in C_{n}:\Vert y_{n}-v\Vert\leq\Vert z_{n}-v\Vert\leq\Vert x_{n}-v\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N}\cup\{0\} ,\end{array} \right . $$

(3.10)

where P is a projection operator and $A^{*}$ denotes the adjoint of A, $\{\alpha_{n}\}\subset(0,\eta]\subset(0,1)$, $\lambda\in(0,\frac{1}{\|A^{*}\|^{2}})$. $i_{n}$ satisfies (3.1), i.e. $i_{n}=n-\frac{(m-1)m}{2}$ and $m\geq i_{n}$ for $n=1,2,\ldots$ . Then the sequence $\{x_{n}\}$ converges strongly to an element $q\in\Gamma$.

If $H_{1}=H_{2}:=H$ and A is an identity operator, then we have the following results by Theorem 3.1 and Corollary 3.1, respectively.

Corollary 3.2

Let H be a real Hilbert space. C is a nonempty closed convex subset of H. $\{T_{i}\}_{i=1}^{\infty}: C\rightarrow H$ and $\{G_{i}\}_{i=1}^{\infty}: H\rightarrow H$ are two countable families of quasi-nonexpansive mappings with $\Gamma:= (\bigcap_{i=1}^{\infty}F(T_{i}))\cap( \bigcap_{j=1}^{\infty}F(G_{j}))\neq\emptyset$. Assume that $T_{i}-I$ ($i=1,2,\ldots$) and $G_{j}-I$ ($j=1,2,\ldots$) are demiclosed at θ. Let $x_{0}\in C$, $C_{0}=C$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})T_{i_{n}}z_{n}, \\ z_{n}=P_{C}((1-\lambda)x_{n}+\lambda G_{i_{n}}x_{n}), \\ C_{n+1}=\{v\in C_{n}:\Vert y_{n}-v\Vert\leq\Vert z_{n}-v\Vert\leq\Vert x_{n}-v\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N}\cup\{0\} ,\end{array} \right . $$

(3.11)

where P is a projection operator. $\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)$, $\lambda\in(0,1)$. $i_{n}$ satisfies (3.1), i.e. $i_{n}=n-\frac{(m-1)m}{2}$ and $m\geq i_{n}$ for $n=1,2,\ldots$ . Then the sequence $\{x_{n}\}$ converges strongly to an element $q\in\Gamma$.

Corollary 3.3

Let H be a real Hilbert space. $\{T_{i}\}_{i=1}^{\infty}: H\rightarrow H$ and $\{G_{i}\}_{i=1}^{\infty}: H\rightarrow H$ are two countable families of quasi-nonexpansive mappings with $\Gamma=\{p\in(\bigcap_{i=1}^{\infty}F(T_{i}))\cap( \bigcap_{j=1}^{\infty}F(G_{j}))\}\neq\emptyset$. Assume that $T_{i}-I$ ($i=1,2,\ldots$) and $G_{j}-I$ ($j=1,2,\ldots$) are demiclosed at θ. Let $x_{0}\in H$, $C_{0}=H$, and $\{x_{n}\}$ be a sequence generated in the following manner:

$$ \left \{ \begin{array}{l} y_{n}=\alpha_{n}z_{n}+(1-\alpha_{n})T_{i_{n}}z_{n}, \\ z_{n}=(1-\lambda)x_{n}+\lambda G_{i_{n}}x_{n}, \\ C_{n+1}=\{v\in C_{n}:\Vert y_{n}-v\Vert\leq\Vert z_{n}-v\Vert\leq\Vert x_{n}-v\Vert\}, \\ x_{n+1}=P_{C_{n+1}}(x_{0}),\quad \forall n\in \mathbb{N}\cup\{0\} ,\end{array} \right . $$

(3.12)

where P is a projection operator. $\{\alpha_{n}\}\subset(0,\eta ]\subset(0,1)$, $\lambda\in(0,1)$. $i_{n}$ satisfies (3.1), i.e. $i_{n}=n-\frac{(m-1)m}{2}$ and $m\geq i_{n}$ for $n=1,2,\ldots$ . Then the sequence $\{x_{n}\}$ converges strongly to an element $q\in\Gamma$.

4 Conclusion

• (1) We give strong convergence algorithms for the split common fixed point problem of quasi-nonexpansive mappings. Our results improve and generalize some well-known results in [3, 11] and so on.

• (2) Although Theorem 3.1 gives a strong convergence algorithm for two countable families of quasi-nonexpansive mappings, the condition that each mapping must be demiclosed at θ is very strong. In addition, we guess the speed of convergence is not too fast for the algorithm (3.3). Therefore, the algorithm (3.3) should be improved further in the future.

• (3) The split common solution problem is a very interesting topic. It has received attention by many scholars. Many research articles have been published, for example, [12–21] and references therein.