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The split common fixed point problem for infinite families of demicontractive mappings

Fixed Point Theory and Applications20182018:14

https://doi.org/10.1186/s13663-018-0639-y

  • Received: 5 September 2017
  • Accepted: 7 January 2018
  • Published:

Abstract

In this paper, we propose a new algorithm for solving the split common fixed point problem for infinite families of demicontractive mappings. Strong convergence of the proposed method is established under suitable control conditions. We apply our main results to study the split common null point problem, the split variational inequality problem, and the split equilibrium problem in the framework of a real Hilbert space. A numerical example supporting our main result is also given.

Keywords

  • Fixed point problem
  • Demicontractive mappings
  • Null point problem
  • Variational inequality problem
  • Equilibrium problem

MSC

  • 47H09
  • 47J05
  • 47J25
  • 47N10

1 Introduction

Let H be a real Hilbert space with inner product \(\langle\cdot ,\cdot\rangle\) and norm \(\|\cdot\|\). Let I denote the identity mapping. Let C and Q be nonempty closed convex subsets of real Hilbert spaces \(H_{1} \) and \(H_{2}\), respectively. Let \(A:H_{1}\to H_{2} \) be a bounded linear operator with adjoint operator \(A^{*}\).

The split feasibility problem (SFP), which was first introduced by Censor and Elfving [1], is to find
$$ v^{*}\in C\quad \text{such that}\quad Av^{*}\in Q. $$
(1)
Let \(P_{C} \) and \(P_{Q} \) be the orthogonal projections onto the sets C and Q, respectively. Assume that (1) has a solution. It known that \(v^{*}\in H_{1} \) solves (1) if and only if it solves the fixed point equation
$$v^{*}=P_{C}\bigl(I+\gamma A^{*}(P_{Q}-I)A \bigr)v^{*}, $$
where \(\gamma>0 \) is any positive constant.

SFP has been used to model significant real-world inverse problems in sensor networks, radiation therapy treatment planning, antenna design, immaterial science, computerized tomography, etc. (see [24]).

The split common fixed point problem (SCFP) for mappings T and S, which was first introduced by Censor and Segal [5], is to find
$$ v^{*}\in F(T) \quad\text{such that}\quad Av^{*}\in F(S), $$
(2)
where \(T:H_{1}\to H_{1} \) and \(S:H_{2}\to H_{2} \) are two mappings satisfying \(F(T)=\{x\in H_{1} : Tx=x\}\neq\emptyset\) and \(F(S)=\{ x\in H_{2} : Sx=x\}\neq\emptyset\), respectively. Since each closed and convex subset may be considered as a fixed point set of a projection onto the subset, the SCFP is a generalization of the SFP. Recently, the SFP and SCFP have been studied by many authors; see, for example, [611].
In 2010, Moudafi [11] introduced the following algorithm for solving (2) for two demicontractive mappings:
$$ \left \{ \textstyle\begin{array}{l} x_{1}\in H_{1} \text{ choose arbitrarily,}\\ u_{n}= x_{n}+\gamma\alpha A^{*}(S-I)Ax_{n}, \\ x_{n+1}=(1-\beta_{n})u_{n}+\beta_{n}Tu_{n}, \quad n\in\mathbb{N}. \end{array}\displaystyle \right . $$
(3)
He proved that \(\{x_{n}\} \) converges weakly to some solution of SCFP.
The multiple set split feasibility problem (MSSFP), which was first introduced by Censor et al. [4], is to find
$$ v^{*}\in\bigcap_{i=1}^{m} C_{i} \quad\text{such that}\quad Av^{*}\in\bigcap _{i=1}^{r} Q_{i}, $$
(4)
where \(\{C_{i}\}_{i=1}^{m} \) and \(\{Q_{i}\}_{i=1}^{r} \) are families of nonempty closed convex subsets of real Hilbert spaces \(H_{1} \) and \(H_{2}\), respectively. We see that if \(m=r=1\), then problem (4) reduces to problem (1).
Recently, Eslamian [12] considered the problem of finding a point
$$ v^{*}\in\bigcap_{i=1}^{m} F(U_{i}) \quad\text{such that} \quad A_{1}v^{*} \in\bigcap_{i=1}^{m}F(S_{i}) \quad\text{and}\quad A_{2}v^{*}\in\bigcap_{i=1}^{m}F(T_{i}), $$
(5)
where \(A_{1}, A_{2}:H_{1}\to H_{2} \) are bounded linear operators, and \(U_{i}:H_{1}\to H_{1}\), \(T_{i}:H_{2}\to H_{2} \) and \(S_{i}:H_{2}\to H_{2}\), \(i=1,2,\ldots,m\). He also presented a new algorithm to solve (5) for finite families of quasi-nonexpansive mappings:
$$ \left \{ \textstyle\begin{array}{l} x_{1}\in H_{1} \text{ choose arbitrarily,}\\ u_{n} = x_{n} + \sum_{i=1}^{m}\frac{1}{m}\eta\beta A_{1}^{*}(S_{i}-I)A_{1}x_{n},\\ y_{n} = u_{n} + \sum_{i=1}^{m}\frac{1}{m}\eta'\beta' A_{2}^{*}(T_{i}-I)A_{2}u_{n},\\ z_{n}= \alpha_{n,0}y_{n} + \sum_{i=1}^{m}\alpha_{n,i}U_{i}y_{n},\\ x_{n+1}=\theta_{n}\gamma f(x_{n}) + (I-\theta_{n}B)z_{n}, \quad n\in \mathbb{N}. \end{array}\displaystyle \right . $$
(6)
He proved that \(\{x_{n}\} \) converges strongly to some solution of (5) under some control conditions.
Question. Can we modify algorithm (6) to a simple one for solving the problem of finding
$$ v^{*}\in\bigcap_{i=1}^{\infty} F(U_{i}) \quad\text{such that} \quad A_{1}v^{*} \in\bigcap_{i=1}^{\infty}F(S_{i}) \quad\text{and}\quad A_{2}v^{*}\in \bigcap_{i=1}^{\infty}F(T_{i}), $$
(7)
where \(A_{1}, A_{2}:H_{1}\to H_{2} \) are bounded linear operators, and \(\{U_{i}:H_{1}\to H_{1}: i\in\mathbb{N}\}\), \(\{T_{i}:H_{2}\to H_{2}: i\in\mathbb{N}\} \) and \(\{S_{i}:H_{2}\to H_{2}: i\in\mathbb {N}\} \) are infinite families of \(k_{3}\)-, \(k_{2}\)-, and \(k_{1}\)-demicontractive mappings, respectively.

In this work, we introduce a new algorithm for solving problem (7) for infinite families of demicontractive mappings and prove its strong convergence to a solution of problem (7).

2 Preliminaries

Throughout this paper, we adopt the following notations.
  1. (i)

    “→” and “” denote the strong and weak convergence, respectively.

     
  2. (ii)

    \(\omega_{\omega}(x_{n}) \) denotes the set of the cluster points of \(\{x_{n}\} \) in the weak topology, that is, \(\exists\{x_{n_{i}}\} \) of \(\{x_{n}\} \) such that \(x_{n_{i}}\rightharpoonup x\).

     
  3. (iii)
    Γ is the solution set of problem (7), that is,
    $$\Gamma= \Biggl\{ v^{*}\in\bigcap_{i=1}^{\infty} F(U_{i}) : A_{1}v^{*}\in \bigcap _{i=1}^{\infty}F(S_{i}) \text{ and } A_{2}v^{*}\in\bigcap_{i=1}^{\infty}F(T_{i}) \Biggr\} . $$
     
A mapping \(P_{C} \) is said to be a metric projection of H onto C if for every \(x\in H\), there exists a unique nearest point in C, denoted by \(P_{C}x \), such that
$$\Vert x-P_{C}x \Vert \leq \Vert x-z \Vert ,\quad \forall z\in C. $$
It is known that \(P_{C} \) is a firmly nonexpansive mapping. Moreover, \(P_{C} \) is characterized by the following property: \(\langle x-P_{C}x,y-P_{C}x\rangle\leq0\) for all \(x\in H\), \(y\in C\). A bounded linear operator \(B:H\to H \) is said to be strongly positive if there is a constant \(\xi>0\) such that
$$\langle Bx,x\rangle\geq\xi \Vert x \Vert ^{2} \quad\text{for all } x\in H. $$

Definition 2.1

The mapping \(T:H\to H \) is said to be
  1. (i)
    L-Lipschitzian if there exists \(L>0 \) such that
    $$\Vert Tu-Tv \Vert \leq L \Vert u-v \Vert \quad \text{for all } u,v\in H; $$
     
  2. (ii)
    α-contraction if T is α-Lipschitzian with \(\alpha\in[0,1)\), that is,
    $$\Vert Tu-Tv \Vert \leq\alpha \Vert u-v \Vert \quad \text{for all } u,v \in H; $$
     
  3. (iii)

    nonexpansive if T is 1-Lipschitzian;

     
  4. (iv)
    quasi-nonexpansive if \(F(T)\neq\emptyset\) and
    $$\Vert Tu-v \Vert \leq \Vert u-v \Vert \quad \text{for all } u\in H, v\in F(T); $$
     
  5. (v)
    firmly nonexpansive if
    $$\Vert Tu-Tv \Vert ^{2}\leq \Vert u-v \Vert ^{2} - \bigl\Vert (u-v)-(Tu-Tv) \bigr\Vert ^{2}\quad \text{for all } u,v\in H; $$
    or equivalently, for all \(u,v\in H\),
    $$\Vert Tu-Tv \Vert ^{2}\leq\langle Tu-Tv,u-v\rangle; $$
     
  6. (vi)
    λ-inverse strongly monotone if there exists \(\lambda>0 \) such that
    $$\langle u-v,Tu-Tv\rangle\geq\lambda \Vert Tu-Tv \Vert ^{2} \quad \text{for all } u,v\in H; $$
     
  7. (vii)
    k-demicontractive if \(F(T)\neq\emptyset\) and there exists \(k\in[0,1) \) such that
    $$\Vert Tu-v \Vert ^{2}\leq \Vert u-v \Vert ^{2} +k \Vert u-Tu \Vert ^{2}\quad \text{for all } u\in H, v\in F(T). $$
     

The following example is an infinite family of k-demicontractive mappings in \(\mathbb{R}^{2} \).

Example 2.2

For \(i\in\mathbb{N}\), let \(U_{i}:\mathbb{R}^{2}\to\mathbb{R}^{2} \) be defined for all \(x_{1},x_{2}\in\mathbb{R} \) by
$$U_{i}(x_{1},x_{2})= \biggl(\frac{-2i}{i+1}x_{1},x_{2} \biggr), $$
and \(\|\cdot\| \) is the Euclidean norm on \(\mathbb{R}^{2}\). Observe that \(F(U_{i})=0\times\mathbb{R} \) for all \(i\in\mathbb{N} \), that is, if \(x=(x_{1},x_{2})\in\mathbb{R}\times\mathbb{R} \) and \(p=(0,p_{2}) \in F(U_{i})\), then
$$\begin{aligned} \Vert U_{i}x-p \Vert ^{2}&= \biggl\Vert \biggl( \frac {-2i}{i+1}x_{1},x_{2}\biggr)-(0,p_{2}) \biggr\Vert ^{2} \\ &= \biggl(\frac{-2i}{i+1} \biggr)^{2} \vert x_{1} \vert ^{2}+ \vert x_{2}-p_{2} \vert ^{2} \\ &\leq4 \vert x_{1} \vert ^{2}+ \vert x_{2}-p_{2} \vert ^{2} \\ &= \vert x_{1} \vert ^{2} + \frac{3}{4}(1+1)^{2} \vert x_{1} \vert ^{2}+ \vert x_{2}-p_{2} \vert ^{2} \\ &\leq \Vert x-p \Vert ^{2}+ \frac{3}{4} \biggl(1+ \frac{2i}{i+1} \biggr)^{2} \vert x_{1} \vert ^{2} \\ &= \Vert x-p \Vert ^{2}+ \frac{3}{4} \Vert U_{i}x-x \Vert ^{2}.\end{aligned} $$
So, \(U_{i} \) are \(\frac{3}{4} \)-demicontractive mappings for all \(i\in\mathbb{N}\).

Definition 2.3

The mapping \(T:H\to H \) is said to be demiclosed at zero if for any sequence \(\{u_{n}\}\subset H \) with \(u_{n}\rightharpoonup u \) and \(Tu_{n}\to0\), we have \(Tu=0\).

Lemma 2.4

([13])

Assume that B is a self-adjoint strongly positive bounded linear operator on a Hilbert space H with coefficient \(\xi>0 \) and \(0<\mu \leq\|B\|^{-1}\). Then \(\|I-\mu B\|\leq1-\xi\mu\).

Lemma 2.5

([14])

Let H be a real Hilbert space. Then the following results hold:
  1. (i)

    \(\|u+v\|^{2}=\|u\|^{2}+2\langle u,v\rangle+\|v\|^{2} \) \(\forall u,v\in H\);

     
  2. (ii)

    \(\|u+v\|^{2}\leq\|u\|^{2}+2\langle v,u+v\rangle\) \(\forall u,v\in H\).

     

Lemma 2.6

([15])

Let \(\{a_{n}\} \) be a sequence of nonnegative real numbers satisfying the following relation:
$$ a_{n+1}\leq(1-\gamma_{n})a_{n}+ \delta_{n}, \quad n\in\mathbb{N}, $$
where
  1. (i)

    \(\{\gamma_{n}\}\subset(0,1)\), \(\sum_{n=1}^{\infty}\gamma_{n} =\infty\);

     
  2. (ii)

    \(\limsup_{n\to\infty}\frac{\delta_{n}}{\gamma_{n}} \leq0\) or \(\sum_{n=1}^{\infty}|\delta_{n}|< \infty\).

     
Then \(\lim_{n\to\infty}a_{n}=0\).

Lemma 2.7

([16])

Let \(\{\kappa_{n}\} \) be a sequence of real numbers that does not decrease at infinity, that is, there exists at a subsequence \(\{\kappa _{n_{i}}\} \) of \(\{\kappa_{n}\} \) that satisfies \(\kappa _{n_{i}}<\kappa_{n_{i}+1} \) for all \(i\in\mathbb{N}\). For every \(n\geq n_{o}\), define the integer sequence \(\{\tau(n)\} \) as follows:
$$\tau(n)=\max\{l\in\mathbb{N} : l\leq n, \kappa_{l}< \kappa_{l+1}\}, $$
where \(n_{o}\in\mathbb{N} \) is such that \(\{l\leq n_{o} : \kappa_{l}< \kappa_{l+1}\}\neq\emptyset\). Then:
  1. (i)

    \(\tau(n_{o})\leq \tau(n_{o}+1)\leq\cdots\), and \(\tau(n)\to \infty\);

     
  2. (ii)

    for all \(n\geq n_{o}\), \(\max\{\kappa_{n},\kappa_{\tau(n)}\} \leq\kappa_{\tau(n)+1}\).

     

3 Results and discussion

In this section, we propose a new algorithm, which is a modification of (6) and prove its strong convergence under some suitable conditions. We start with the following important lemma.

Lemma 3.1

For two real Hilbert spaces \(H_{1} \) and \(H_{2}\), let \(A:H_{1}\to H_{2} \) be a bounded linear operator with adjoint operator \(A^{*}\). If \(T:H_{2}\to H_{2} \) is a k-demicontractive mapping, then
$$\bigl\Vert x + \delta A^{*}(T-I)Ax -x^{*} \bigr\Vert ^{2}\leq \bigl\Vert x - x^{*} \bigr\Vert ^{2}- \delta_{n}\bigl(1-k-\delta \Vert A \Vert ^{2}\bigr) \bigl\Vert (T-I)Ax \bigr\Vert ^{2} $$
for all \(x^{*}\in H_{1} \) such that \(Ax^{*}\in F(T)\).

Proof

Suppose that \(T:H_{2}\to H_{2} \) is a k-demicontractive mapping and let \(x^{*}\in H_{1} \) be such that \(Ax^{*}\in F(T)\). Then we have
$$ \begin{aligned}[b] \bigl\Vert x-x^{*}+\delta A^{*}(T-I)Ax \bigr\Vert ^{2} &\leq \bigl\Vert x-x^{*} \bigr\Vert ^{2}+2\delta\bigl\langle x-x^{*}, A^{*}(T-I)Ax\bigr\rangle \\ &\quad +\delta^{2} \Vert A \Vert ^{2} \bigl\Vert (T-I)Ax \bigr\Vert ^{2}.\end{aligned} $$
(8)
Since A is a bounded linear operator with adjoint operator \(A^{*}\) and T is a k-demicontractive mapping, by Lemma 2.5(ii) we deduce that
$$ \begin{aligned}[b] \bigl\langle x-x^{*}, A^{*}(T-I)Ax\bigr\rangle &=\bigl\langle Ax-Ax^{*}, (T-I)Ax\bigr\rangle \\ &=\bigl\langle TAx-Ax^{*},TAx-Ax\bigr\rangle - \bigl\Vert (T-I)Ax \bigr\Vert ^{2} \\ &=\frac{1}{2} \bigl[ \bigl\Vert TAx-Ax^{*} \bigr\Vert ^{2}+ \Vert TAx-Ax \Vert ^{2} - \bigl\Vert Ax-Ax^{*} \bigr\Vert ^{2} \bigr] \\ & \quad- \bigl\Vert (T-I)Ax \bigr\Vert ^{2} \\ &\leq\frac{1}{2} \bigl[ \bigl\Vert Ax-Ax^{*} \bigr\Vert ^{2}+k \Vert TAx-Ax \Vert ^{2} \\ & \quad+ \Vert TAx-Ax \Vert ^{2} - \bigl\Vert Ax-Ax^{*} \bigr\Vert ^{2} \bigr]- \bigl\Vert (T-I)Ax \bigr\Vert ^{2} \\ &=\frac{k-1}{2} \bigl\Vert (T-I)Ax \bigr\Vert ^{2}.\end{aligned} $$
(9)
From (8) and (9) we get
$$ \bigl\Vert x+\delta A^{*}(T-I)Ax-x^{*} \bigr\Vert ^{2} \leq \bigl\Vert x-x^{*} \bigr\Vert ^{2}- \delta\bigl(1-k-\delta \Vert A \Vert ^{2}\bigr) \bigl\Vert (T-I)Ax \bigr\Vert ^{2}. $$
This completes the proof. □

Lemma 3.2

For two real Hilbert spaces \(H_{1} \) and \(H_{2}\), let \(A:H_{1}\to H_{2} \) be a bounded linear operator with adjoint operator \(A^{*}\), and let \(\{T_{i}:H_{2}\to H_{2}:i\in\mathbb{N}\} \) be an infinite family of k-demicontractive mappings. Let \(\{x_{n}\} \) be sequence in \(H_{1} \), and let
$$ u_{n} = x_{n} + \sum _{i=1}^{n}\alpha_{n,i}\delta _{n}A^{*}(T_{i}-I)Ax_{n}, \quad\forall n \in\mathbb{N,} $$
(10)
where \(\{\alpha_{n,i}\} \) is a real sequence in \([0,1] \) satisfying \(\sum_{i=1}^{n}\alpha_{n,i}=1\). Then we have
$$\bigl\Vert u_{n} -x^{*} \bigr\Vert ^{2}\leq \bigl\Vert x_{n} - x^{*} \bigr\Vert ^{2}-\sum _{i=1}^{n}\alpha_{n,i} \delta_{n}\bigl(1-k-\delta_{n} \Vert A \Vert ^{2}\bigr) \bigl\Vert (T_{i}-I)Ax_{n} \bigr\Vert ^{2} $$
for all \(x^{*}\in H_{1} \) such that \(Ax^{*}\in\bigcap_{i=1}^{\infty }F(T_{i})\).

Proof

Let \(x^{*}\in H_{1} \) be such that \(Ax^{*}\in\bigcap_{i=1}^{\infty }F(T_{i})\). From (10) and Lemma 3.1 we obtain
$$\begin{aligned} \bigl\Vert u_{n}-x^{*} \bigr\Vert ^{2}&\leq \sum_{i=1}^{n}\alpha_{n,i} \bigl\Vert x_{n}-x^{*}+\delta_{n} A^{*}(T_{i}-I)Ax_{n} \bigr\Vert ^{2} \\ &\leq\sum_{i=1}^{n}\alpha_{n,i} \bigl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \delta_{n}\bigl(1-k-\delta_{n} \Vert A \Vert ^{2}\bigr) \bigl\Vert (T_{i}-I)Ax_{n} \bigr\Vert ^{2} \bigr] \\ &= \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\sum _{i=1}^{n}\alpha_{n,i}\delta _{n}\bigl(1-k-\delta_{n} \Vert A \Vert ^{2} \bigr) \bigl\Vert (T_{i}-I)Ax_{n} \bigr\Vert ^{2}.\end{aligned} $$
This completes the proof. □

Lemma 3.3

Let \(\{T_{i}:H_{1}\to H_{1}:i\in\mathbb{N}\} \) be an infinite family of k-demicontractive mappings from a Hilbert space \(H_{1} \) to itself. Let \(\{x_{n}\} \) be sequence in \(H_{1} \), and let
$$ u_{n} = x_{n} + \sum _{i=1}^{n}\alpha_{n,i}\delta_{n}(T_{i}-I)x_{n}, \quad\forall n\in\mathbb{N,} $$
(11)
where \(\{\alpha_{n,i}\} \) is a real sequence in \([0,1] \) satisfying \(\sum_{i=1}^{n}\alpha_{n,i}=1\). Then we have
$$\bigl\Vert u_{n} -x^{*} \bigr\Vert ^{2}\leq \bigl\Vert x_{n} - x^{*} \bigr\Vert ^{2}-\sum _{i=1}^{n}\alpha_{n,i} \delta_{n}(1-k-\delta_{n}) \bigl\Vert (T_{i}-I)x_{n} \bigr\Vert ^{2} $$
for all \(x^{*}\in\bigcap_{i=1}^{\infty}F(T_{i})\).

Proof

The statement directly follows from Lemma 3.2 by putting \(H_{1}=H_{2} \) and \(A=I \). □

Now, we introduce a new algorithm for solving problem (7) for an infinite family of demicontractive mappings and then prove its strong convergence.

Theorem 3.4

Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces, and let \(A_{1}, A_{2}: H_{1}\to H_{2} \) be two bounded linear operators with adjoint operators \(A_{1}^{*} \) and \(A_{2}^{*}\), respectively. Let \(f : H_{1} \to H_{1} \) be a ρ-contraction mapping, and let B be a self-adjoint strongly positive bounded linear operator on \(H_{1} \) with coefficient \(\xi>2\rho\) and \(\|B\|=1\). Let \(\{ S_{i}:H_{2}\to H_{2}:i\in\mathbb{N}\}\), \(\{ T_{i}:H_{2}\to H_{2}:i\in\mathbb{N}\} \), and \(\{U_{i}:H_{1}\to H_{1}:i\in\mathbb{N}\} \) be infinite families of \(k_{1}\)-, \(k_{2}\)-, and \(k_{3} \)-demicontractive mappings such that \(S_{i}-I\), \(T_{i}-I \), and \(U_{i}-I\) are demiclosed at zero, respectively. Suppose that \(\Gamma = \{v^{*}\in\bigcap_{i=1}^{\infty} F(U_{i}) : A_{1}v^{*}\in\bigcap_{i=1}^{\infty}F(S_{i}) \textit{ and } A_{2}v^{*}\in\bigcap_{i=1}^{\infty }F(T_{i}) \}\neq\emptyset\). For arbitrary \(x_{1}\in H_{1} \), let \(\{u_{n}\}\), \(\{v_{n}\}\), \(\{y_{n}\} \), and \(\{x_{n}\} \) be generated by
$$ \left \{ \textstyle\begin{array}{l} u_{n}= x_{n} + \sum_{i=1}^{n}\alpha_{n,i}\delta_{n} A_{1}^{*}(S_{i}-I)A_{1}x_{n}, \\ v_{n} = u_{n} + \sum_{i=1}^{n}\beta_{n,i}\theta_{n} A_{2}^{*}(T_{i}-I)A_{2}u_{n},\\ y_{n}= v_{n} + \sum_{i=1}^{n}\gamma_{n,i}\tau_{n} (U_{i}-I)v_{n},\\ x_{n+1}=\sigma_{n}f(y_{n}) + (I-\sigma_{n}B)y_{n}, \quad n\in\mathbb{N}, \end{array}\displaystyle \right . $$
(12)
where \(\{\delta_{n}\}\), \(\{\theta_{n}\}\), \(\{\tau_{n}\}\), \(\{\sigma_{n}\}\), \(\{ \alpha_{n,i}\}\), \(\{\beta_{n,i}\} \), and \(\{\gamma_{n,i}\} \) are sequences in \([0,1] \) satisfying the following conditions:
  1. (C1)

    \(\sum_{i=1}^{n}\alpha_{n,i}= \sum_{i=1}^{n}\beta _{n,i} = \sum_{i=1}^{n}\gamma_{n,i}= 1\) for all \(n\in\mathbb{N}\);

     
  2. (C2)

    \(\liminf_{n\to\infty}\alpha_{n,i}>0\), \(\liminf_{n\to \infty}\beta_{n,i}>0 \), and \(\liminf_{n\to\infty}\gamma_{n,i}>0 \) for all \(i\in\mathbb{N}\);

     
  3. (C3)

    \(\lim_{n\to\infty}\sigma_{n}=0 \) and \(\sum_{n=1}^{\infty} \sigma_{n} = \infty\);

     
  4. (C4)

    \(0< a_{1}\leq\delta_{n}\leq a_{2}<\frac{1-k_{1}}{\| A_{1}\|^{2}}\);

     
  5. (C5)

    \(0< b_{1}\leq\theta_{n}\leq b_{2}<\frac{1-k_{2}}{\| A_{2}\|^{2}}\);

     
  6. (C6)

    \(0< c_{1}\leq\tau_{n}\leq c_{2}<1-k_{3}\).

     
Then the sequence \(\{x_{n}\} \) converges strongly to \(x^{*}=P_{\Gamma }(f+I-B)x^{*}\).

Proof

For any \(u,v\in H_{1}\), by Lemma 2.4 we have
$$\begin{aligned} \bigl\Vert P_{\Gamma}(f+I-B)u-P_{\Gamma}(f+I-B)v \bigr\Vert & \leq \bigl\Vert (f+I-B)u-(f+I-B)v \bigr\Vert \\ &\leq \bigl\Vert f(u)-f(v) \bigr\Vert + \Vert I-B \Vert \Vert u-v \Vert \\ &\leq\rho \Vert u-v \Vert + (1-\xi) \Vert u-v \Vert \\ &\leq(1-\rho) \Vert u-v \Vert ,\end{aligned} $$
that is, the mapping \(P_{\Gamma}(f+I-B) \) is a contraction. So, by the Banach contraction principle there is a unique element \(x^{*}\in H_{1} \) such that \(x^{*}=P_{\Gamma}(f+I-B)x^{*}\).
Let \(x^{*}=P_{\Gamma}(f+I-B)x^{*}\), that is, \(x^{*}\in\bigcap_{i=1}^{\infty} F(U_{i})\) is such that \(A_{1}x^{*}\in\bigcap_{i=1}^{\infty}F(S_{i})\) and \(A_{2}x^{*}\in\bigcap_{i=1}^{\infty }F(T_{i})\). From Lemmas 3.2 and 3.3 and from (12) we obtain
$$\begin{aligned}& \bigl\Vert u_{n} -x^{*} \bigr\Vert ^{2}\leq \bigl\Vert x_{n} - x^{*} \bigr\Vert ^{2}-\sum _{i=1}^{n}\alpha_{n,i} \delta_{n}\bigl(1-k_{1}-\delta_{n} \Vert A_{1} \Vert ^{2}\bigr) \bigl\Vert (S_{i}-I)A_{1}x_{n} \bigr\Vert ^{2}, \end{aligned}$$
(13)
$$\begin{aligned}& \bigl\Vert v_{n} -x^{*} \bigr\Vert ^{2}\leq \bigl\Vert u_{n} - x^{*} \bigr\Vert ^{2}-\sum _{i=1}^{n}\beta_{n,i} \theta_{n}\bigl(1-k_{2}-\theta_{n} \Vert A_{2} \Vert ^{2}\bigr) \bigl\Vert (T_{i}-I)A_{2}u_{n} \bigr\Vert ^{2}, \end{aligned}$$
(14)
and
$$ \bigl\Vert y_{n} -x^{*} \bigr\Vert ^{2}\leq \bigl\Vert v_{n} - x^{*} \bigr\Vert ^{2}-\sum _{i=1}^{n}\lambda_{n,i} \tau_{n}(1-k_{3}-\tau_{n}) \bigl\Vert (U_{i}-I)v_{n} \bigr\Vert ^{2}. $$
(15)
Therefore
$$ \begin{aligned}[b]\bigl\Vert y_{n} -x^{*} \bigr\Vert ^{2}&\leq \bigl\Vert x_{n} - x^{*} \bigr\Vert ^{2} -\sum_{i=1}^{n} \alpha_{n,i}\delta_{n}\bigl(1-k_{1}- \delta_{n} \Vert A_{1} \Vert ^{2}\bigr) \bigl\Vert (S_{i}-I)A_{1}x_{n} \bigr\Vert ^{2} \\ & \quad-\sum_{i=1}^{n}\beta_{n,i} \theta_{n}\bigl(1-k_{2}-\theta_{n} \Vert A_{2} \Vert ^{2}\bigr) \bigl\Vert (T_{i}-I)A_{2}u_{n} \bigr\Vert ^{2} \\ &\quad -\sum_{i=1}^{n}\lambda_{n,i} \tau_{n}(1-k_{3}-\tau_{n}) \bigl\Vert (U_{i}-I)v_{n} \bigr\Vert ^{2}.\end{aligned} $$
(16)
By conditions (C4), (C5), and (C6) we have
$$ \bigl\Vert y_{n} -x^{*} \bigr\Vert \leq \bigl\Vert x_{n} - x^{*} \bigr\Vert . $$
(17)
By condition (C3) we may assume that \(\sigma_{n}\in(0,\|B\| ^{-1}) \) for all \(n\in\mathbb{N}\). By Lemma 2.4 we get \(\| I-\sigma_{n}B\|\leq1- \sigma_{n}\xi\). From (12) and (17) we get
$$\begin{aligned}[b] \bigl\Vert x_{n+1}-x^{*} \bigr\Vert &= \bigl\Vert \sigma_{n}f(y_{n})+(I-\sigma _{n}B)y_{n}-x^{*} \bigr\Vert \\ &= \bigl\Vert \sigma_{n}\bigl(f(y_{n})- Bx^{*} \bigr)+(I-\sigma_{n}B) \bigl(y_{n}-x^{*}\bigr) \bigr\Vert \\ &\leq\sigma_{n} \bigl[ \bigl\Vert f(y_{n})-f \bigl(x^{*}\bigr) \bigr\Vert + \bigl\Vert f\bigl(x^{*} \bigr)-Bx^{*} \bigr\Vert \bigr]+ \Vert I-\sigma_{n}B \Vert \bigl\Vert y_{n}-x^{*} \bigr\Vert \\ &\leq\sigma_{n}\rho \bigl\Vert y_{n}-x^{*} \bigr\Vert +\sigma_{n} \bigl\Vert f\bigl(x^{*} \bigr)-Bx^{*} \bigr\Vert +(1-\sigma_{n}\xi) \bigl\Vert y_{n}-x^{*} \bigr\Vert \\ &\leq\bigl(1-\sigma_{n}(\xi-\rho)\bigr) \bigl\Vert x_{n}-x^{*} \bigr\Vert +\sigma_{n}(\xi -\rho) \frac{ \Vert f(x^{*})-Bx^{*} \Vert }{\xi-\rho} \\ &\leq\max \biggl\{ \bigl\Vert x_{n}-x^{*} \bigr\Vert , \frac{ \Vert f(x^{*})-Bx^{*} \Vert }{\xi-\rho} \biggr\} \\ & \vdots \\ &\leq\max \biggl\{ \bigl\Vert x_{1}-x^{*} \bigr\Vert , \frac{ \Vert f(x^{*})-Bx^{*} \Vert }{\xi-\rho} \biggr\} .\end{aligned} $$
(18)
Therefore \(\{x_{n}\} \) is bounded, and we also have that \(\{y_{n}\} \) and \(\{f(y_{n})\} \) are bounded. To this end, we consider the following two cases.
Case 1. Suppose that \(\{\|x_{n}-x^{*}\|\} _{n=n_{o}}^{\infty} \) is nonincreasing for some \(n_{o}\in\mathbb{N}\). Then we get that \(\lim_{n\to\infty} \|x_{n}-x^{*}\|\) exists. By (16), (17), and Lemma 2.5(i) we get
$$\begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}&= \bigl\Vert \sigma_{n}\bigl(f(y_{n})- Bx^{*} \bigr)+(I-\sigma_{n}B) \bigl(y_{n}-x^{*}\bigr) \bigr\Vert ^{2} \\ &\leq\sigma_{n} \bigl\Vert f(y_{n})- Bx^{*} \bigr\Vert ^{2} + (1-\sigma_{n}\xi ) \bigl\Vert y_{n}-x^{*} \bigr\Vert ^{2} \\ & \quad+2\sigma_{n}(1-\sigma_{n}\xi) \bigl\Vert f(y_{n})- Bx^{*} \bigr\Vert \bigl\Vert y_{n}-x^{*} \bigr\Vert \\ &\leq\sigma_{n}M + \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad -\sum_{i=1}^{n}\alpha_{n,i} \delta_{n}\bigl(1-k_{1}-\delta_{n} \Vert A_{1} \Vert ^{2}\bigr) \bigl\Vert (S_{i}-I)A_{1}x_{n} \bigr\Vert ^{2} \\ &\quad -\sum_{i=1}^{n}\beta_{n,i} \theta_{n}\bigl(1-k_{2}-\theta_{n} \Vert A_{2} \Vert ^{2}\bigr) \bigl\Vert (T_{i}-I)A_{2}u_{n} \bigr\Vert ^{2} \\ &\quad -\sum_{i=1}^{n}\lambda_{n,i} \tau_{n}(1-k_{3}-\tau_{n}) \bigl\Vert (U_{i}-I)v_{n} \bigr\Vert ^{2}, \end{aligned}$$
where
$$ M=\sup_{n} \bigl\{ \bigl\Vert f(y_{n})- Bx^{*} \bigr\Vert ^{2} + 2 \bigl\Vert f(y_{n})- Bx^{*} \bigr\Vert \bigl\Vert x_{n}-x^{*} \bigr\Vert \bigr\} . $$
This implies, for \(j=1,2,\ldots,n\),
$$\begin{aligned}& \begin{aligned}[b] &\alpha_{n,j}\delta_{n}\bigl(1-k_{1}- \delta_{n} \Vert A_{1} \Vert ^{2}\bigr) \bigl\Vert (S_{j}-I)A_{1}x_{n} \bigr\Vert ^{2}\\&\quad\leq\sum_{i=1}^{n} \alpha_{n,i}\delta _{n}\bigl(1-k_{1}- \delta_{n} \Vert A_{1} \Vert ^{2}\bigr) \bigl\Vert (S_{i}-I)A_{1}x_{n} \bigr\Vert ^{2} \\ &\quad\leq\sigma_{n}M + \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} - \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2},\end{aligned} \end{aligned}$$
(19)
$$\begin{aligned}& \begin{aligned}[b]&\beta_{n,j}\theta_{n}\bigl(1-k_{2}- \theta_{n} \Vert A_{2} \Vert ^{2}\bigr) \bigl\Vert (T_{j}-I)A_{2}u_{n} \bigr\Vert ^{2}\\&\quad\leq\sum_{i=1}^{n} \beta_{n,i}\theta _{n}\bigl(1-k_{2}- \theta_{n} \Vert A_{2} \Vert ^{2}\bigr) \bigl\Vert (T_{i}-I)A_{2}u_{n} \bigr\Vert ^{2} \\ &\quad\leq\sigma_{n}M + \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} - \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2},\end{aligned} \end{aligned}$$
(20)
and
$$ \begin{aligned}[b]\lambda_{n,j}\tau_{n}(1-k_{3}- \tau_{n}) \bigl\Vert (U_{j}-I)v_{n} \bigr\Vert ^{2}&\leq\sum_{i=1}^{n} \lambda_{n,i}\tau_{n}(1-k_{3}-\tau_{n}) \bigl\Vert (U_{i}-I)v_{n} \bigr\Vert ^{2} \\ &\leq\sigma_{n}M + \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} - \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}.\end{aligned} $$
(21)
From (19), (20), (21), and conditions (C2)–(C6) we obtain
$$\begin{aligned}& \lim_{n\to\infty} \bigl\Vert (S_{j}-I)A_{1}x_{n} \bigr\Vert =\lim_{n\to\infty }\sum_{i=1}^{n} \alpha_{n,i} \bigl\Vert (S_{i}-I)A_{1}x_{n} \bigr\Vert ^{2}=0, \end{aligned}$$
(22)
$$\begin{aligned}& \lim_{n\to\infty} \bigl\Vert (T_{j}-I)A_{2}u_{n} \bigr\Vert =\lim_{n\to\infty }\sum_{i=1}^{n} \beta_{n,i} \bigl\Vert (T_{i}-I)A_{2}u_{n} \bigr\Vert ^{2}=0, \end{aligned}$$
(23)
and
$$ \lim_{n\to\infty} \bigl\Vert (U_{j}-I)v_{n} \bigr\Vert =\lim_{n\to\infty}\sum_{i=1}^{n} \lambda_{n,i} \bigl\Vert (U_{i}-I)v_{n} \bigr\Vert ^{2}=0. $$
(24)
Next, we show that
$$ \limsup_{n\to\infty}\bigl\langle f\bigl(x^{*} \bigr)-Bx^{*},x_{n}-x^{*}\bigr\rangle \leq0, \quad\text{where } x^{*}=P_{\Gamma}(f+I-B)x^{*}. $$
To see this, choose a subsequence \(\{x_{n_{p}}\} \) of \(\{x_{n}\} \) such that
$$ \limsup_{n\to\infty}\bigl\langle f\bigl(x^{*} \bigr)-Bx^{*},x_{n}-x^{*}\bigr\rangle =\lim _{p\to\infty}\bigl\langle f\bigl(x^{*}\bigr)-Bx^{*},x_{n_{p}}-x^{*} \bigr\rangle . $$
Since the sequence \(\{x_{n_{p}}\} \) is bounded, there exists a subsequence \(\{x_{n_{p_{j}}}\} \) of \(\{x_{n_{p}}\} \) such that \(x_{n_{p_{j}}}\rightharpoonup z\in H_{1}\). Without loss of generality, we may assume that \(x_{n_{p}}\rightharpoonup z\in H_{1}\). Since \(A_{1} \) is a bounded linear operator, this yields that \(A_{1}x_{n_{p}}\rightharpoonup A_{1}z\). By the demiclosedness principle of \(S_{i}-I \) at zero and (22) we get \(A_{1}z\in\bigcap_{i=1}^{\infty} F(S_{i})\). By (12) and (22) we have
$$\begin{aligned} \Vert u_{n}-x_{n} \Vert ^{2}&= \Biggl\Vert x_{n} + \sum_{i=1}^{n}\alpha _{n,i}\delta_{n} A_{1}^{*}(S_{i}-I)A_{1}x_{n}-x_{n} \Biggr\Vert ^{2} \\ &\leq\sum_{i=1}^{n}\alpha_{n,i} \delta_{n} \Vert A_{1} \Vert ^{2} \bigl\Vert (S_{i}-I)A_{1}x_{n} \bigr\Vert ^{2} \to0 \quad\text{as } n\to\infty.\end{aligned} $$
Similarly, we also have \(\|v_{n}-u_{n}\|\to0 \) as \(n\to\infty\). Using the fact that \(x_{n_{p}}\rightharpoonup z \) and \(\|u_{n}-x_{n}\| \to0\), we conclude that \(u_{n_{p}}\rightharpoonup z\). Since \(A_{2} \) is a bounded linear operator, we get that \(A_{2}u_{n_{p}}\rightharpoonup A_{2}z\). By the demiclosedness principle of \(T_{i}-I \) at zero and (23) we get \(A_{2}z\in\bigcap_{i=1}^{\infty} F(T_{i})\). Again, since \(u_{n_{p}}\rightharpoonup z \) and \(\|v_{n}-u_{n}\|\to0\), we conclude that \(v_{n_{p}}\rightharpoonup z\). By the demiclosedness principle of \(U_{i}-I \) at zero and (24) we also have \(z\in\bigcap_{i=1}^{\infty} F(U_{i})\). Therefore \(z\in\Gamma\).
Since \(x^{*}=P_{\Gamma}(f+I-B)x^{*} \) and \(z\in\Gamma\), we get
$$ \begin{aligned}[b] \limsup_{n\to\infty}\bigl\langle f\bigl(x^{*} \bigr)-Bx^{*},x_{n}-x^{*}\bigr\rangle &=\lim _{p\to\infty}\bigl\langle f\bigl(x^{*}\bigr)-Bx^{*},x_{n_{p}}-x^{*} \bigr\rangle \\ &=\bigl\langle f\bigl(x^{*}\bigr)-Bx^{*},z-x^{*} \bigr\rangle \leq0.\end{aligned} $$
(25)
Using Lemma 2.5 and (17), we have
$$\begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}&= \bigl\Vert \sigma_{n}\bigl(f(y_{n})- Bx^{*} \bigr)+(I-\sigma_{n}B) \bigl(y_{n}-x^{*}\bigr) \bigr\Vert ^{2} \\ &\leq(1-\sigma_{n}\xi) \bigl\Vert y_{n}-x^{*} \bigr\Vert ^{2}+2\sigma _{n}\bigl\langle f(y_{n})- Bx^{*}, x_{n+1}-x^{*}\bigr\rangle \\ &\leq(1-\sigma_{n}\xi) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} + 2\rho\sigma _{n} \bigl\Vert y_{n}- x^{*} \bigr\Vert \bigl\Vert x_{n+1}-x^{*} \bigr\Vert \\ & \quad+2\sigma_{n}\bigl\langle f\bigl(x^{*}\bigr)- Bx^{*}, x_{n+1}-x^{*}\bigr\rangle \\ &\leq(1-\sigma_{n}\xi) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+ \rho\sigma _{n} \bigl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} + \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2} \bigr] \\ & \quad+2\sigma_{n}\bigl\langle f\bigl(x^{*}\bigr)- Bx^{*}, x_{n+1}-x^{*}\bigr\rangle \\ &= \bigl(1-\sigma_{n}(\xi-\rho)\bigr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} + \rho\sigma _{n} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2} \\ & \quad+2\sigma_{n}\bigl\langle f\bigl(x^{*}\bigr)- Bx^{*}, x_{n+1}-x^{*}\bigr\rangle .\end{aligned} $$
This implies that
$$ \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}\leq \biggl[ 1-\frac{\sigma_{n}(\xi-\rho )}{1-\sigma_{n}\rho} \biggr] \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+\frac{2\sigma _{n}}{1-\sigma_{n}\rho} \bigl\langle f\bigl(x^{*}\bigr)- Bx^{*}, x_{n+1}-x^{*} \bigr\rangle . $$
(26)
By (25), (26), and Lemma 2.6 we conclude that \(x_{n}\to x^{*} \) as \(n\to\infty\).
Case 2. Suppose that there exists an integer \(m_{o} \) such that
$$\bigl\Vert x_{m_{o}}-x^{*} \bigr\Vert \leq \bigl\Vert x_{m_{o}+1}-x^{*} \bigr\Vert . $$
Put \(\kappa_{n}= \|x_{n}-x^{*}\|\) for all \(n\geq m_{o}\). Then we have \(\kappa_{m_{o}}\leq\kappa_{m_{o}+1}\). Let \(\{\mu(n)\} \) be the sequence defined by
$$\mu(n)=\max\{l\in\mathbb{N} : l\leq n, \kappa_{l}\leq \kappa_{l+1}\} $$
for all \(n\geq m_{o}\). By Lemma 2.7 we obtain that \(\{\mu (n)\} \) is a nondecreasing sequence such that
$$\lim_{n\to\infty}\mu(n)=\infty \quad\text{and}\quad \kappa_{\mu(n)}\leq \kappa _{\mu(n)+1} \quad\text{for all } n\geq m_{o}. $$
By the same argument as in case 1 we obtain
$$ \lim_{n\to\infty} \bigl\Vert (S_{i}-I)A_{1}x_{\mu(n)} \bigr\Vert =0, \qquad\lim_{n\to\infty} \bigl\Vert (T_{i}-I)A_{2}u_{\mu(n)} \bigr\Vert =0, $$
and
$$ \lim_{n\to\infty} \bigl\Vert (U_{i}-I)v_{\mu(n)} \bigr\Vert =0. $$
By the demiclosedness principle of \(S_{i}-I\), \(T_{i}-I \), and \(U_{i}-I \) at zero, we have \(\omega_{\omega}(x_{\mu(n)})\subset\Gamma\). This implies that
$$\limsup_{n\to\infty}\bigl\langle f\bigl(x^{*} \bigr)-Bx^{*},x_{\mu(n)}-x^{*}\bigr\rangle \leq 0. $$
By a similar argument from (26) we also have
$$ \kappa_{\mu(n)+1}^{2}\leq \biggl[ 1-\frac{\sigma_{\mu(n)}(\xi-\rho )}{1-\sigma_{\mu(n)}\rho} \biggr] \kappa_{\mu(n)}^{2}+\frac{2\sigma_{\mu (n)}}{1-\sigma_{\mu(n)}\rho}\bigl\langle f \bigl(x^{*}\bigr)- Bx^{*}, x_{\mu (n)+1}-x^{*} \bigr\rangle . $$
So, we get \(\lim_{n\to\infty}\kappa_{\mu(n)} =0 \) and also have \(\lim_{n\to\infty}\kappa_{\mu(n)+1} =0\). By Lemma 2.7 we have
$$0\leq\kappa_{n}\leq\max\{ \kappa_{n}, \kappa_{\mu(n)} \}\leq\kappa _{\mu(n)+1}. $$
Therefore \(x_{n}\to x^{*} \) as \(n\to\infty\). This completes the proof. □

By setting \(T_{i}=I \) for all \(i\in\mathbb{N} \) in Theorem 3.4 we obtain the following result.

Corollary 3.5

Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces, let \(A_{1}: H_{1}\to H_{2} \) be a bounded linear operator with adjoint operator \(A_{1}^{*}\). Let \(f : H_{1} \to H_{1} \) be a ρ-contraction mapping, and let B be a self-adjoint strongly positive bounded linear operator on \(H_{1} \) with coefficient \(\xi>2\rho\) and \(\|B\|=1\). Let \(\{S_{i}:H_{2}\to H_{2}:i\in\mathbb{N}\}\) and \(\{ U_{i}:H_{1}\to H_{1}:i\in\mathbb{N} \}\) be infinite families of \(k_{1}\)- and \(k_{3} \)-demicontractive mappings such that \(S_{i}-I\) and \(U_{i}-I\) are demiclosed at zero, respectively. Suppose that \(\Omega = \{v^{*}\in\bigcap_{i=1}^{\infty} F(U_{i}) : A_{1}v^{*}\in\bigcap_{i=1}^{\infty}F(S_{i}) \}\neq\emptyset\). For arbitrary \(x_{1}\in H_{1} \), let \(\{u_{n}\}\), \(\{y_{n}\} \), and \(\{x_{n}\} \) be generated by
$$ \left \{ \textstyle\begin{array}{l} u_{n}= x_{n} + \sum_{i=1}^{n}\alpha_{n,i}\delta_{n} A_{1}^{*}(S_{i}-I)A_{1}x_{n}, \\ y_{n}= u_{n} + \sum_{i=1}^{n}\gamma_{n,i}\tau_{n} (U_{i}-I)u_{n},\\ x_{n+1}=\sigma_{n}f(y_{n}) + (I-\sigma_{n}B)y_{n}, \quad n\in\mathbb{N}, \end{array}\displaystyle \right . $$
(27)
where \(\{\delta_{n}\}\), \(\{\tau_{n}\}\), \(\{\sigma_{n}\}\), \(\{\alpha_{n,i}\} \), and \(\{\gamma_{n,i}\} \) are sequences in \([0,1] \) satisfying the following conditions:
  1. (C1)

    \(\sum_{i=1}^{n}\alpha_{n,i}=\sum_{i=1}^{n}\gamma_{n,i}= 1\) for all \(n\in\mathbb{N}\);

     
  2. (C2)

    \(\liminf_{n\to\infty}\alpha_{n,i}>0 \) and \(\liminf_{n\to\infty }\gamma_{n,i}>0 \) for all \(i\in\mathbb{N}\);

     
  3. (C3)

    \(\lim_{n\to\infty}\sigma_{n}=0 \) and \(\sum_{n=1}^{\infty} \sigma _{n} = \infty\);

     
  4. (C4)

    \(0< a_{1}\leq\delta_{n}\leq a_{2}<\frac{1-k_{1}}{\|A_{1}\|^{2}}\);

     
  5. (C5)

    \(0< c_{1}\leq\tau_{n}\leq c_{2}<1-k_{3}\).

     
Then the sequence \(\{x_{n}\} \) converges strongly to \(x^{*}=P_{\Omega }(f+I-B)x^{*}\).

Remark 3.6

By the same setting as in Corollary 3.5, Eslamian [17] used another algorithm for solving the same problem as in Corollary 3.5; see [17], Theorem 3.3. Note that each step of our algorithm is much easier for computation than that of Eslamian [17] because our algorithm concerns only the finite sum.

By setting \(f(y)=v \) for all \(y\in H_{1}\) and \(B=I \) in Theorem 3.4 we obtain the following result.

Corollary 3.7

Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces, and let \(A_{1}, A_{2}: H_{1}\to H_{2} \) be two bounded linear operators with adjoint operators \(A_{1}^{*} \) and \(A_{2}^{*}\), respectively. Let \(\{ S_{i}:H_{2}\to H_{2}:i\in\mathbb{N}\}\), \(\{ T_{i}:H_{2}\to H_{2}:i\in \mathbb{N}\} \), and \(\{U_{i}:H_{1}\to H_{1}:i\in\mathbb{N}\} \) be infinite families of \(k_{1}\)-, \(k_{2}\)-, and \(k_{3} \)-demicontractive mappings such that \(S_{i}-I\), \(T_{i}-I \), and \(U_{i}-I\) are demiclosed at zero, respectively. Suppose that \(\Gamma= \{v^{*}\in \bigcap_{i=1}^{\infty} F(U_{i}) : A_{1}v^{*}\in\bigcap_{i=1}^{\infty }F(S_{i}) \textit{ and } A_{2}v^{*}\in\bigcap_{i=1}^{\infty}F(T_{i}) \}\neq\emptyset\). For arbitrary \(x_{1}\in H_{1} \), let \(\{u_{n}\}\), \(\{ v_{n}\}\), \(\{y_{n}\} \), and \(\{x_{n}\} \) be generated by
$$ \left \{ \textstyle\begin{array}{l} u_{n}= x_{n} + \sum_{i=1}^{n}\alpha_{n,i}\delta_{n} A_{1}^{*}(S_{i}-I)A_{1}x_{n}, \\ v_{n} = u_{n} + \sum_{i=1}^{n}\beta_{n,i}\theta_{n} A_{2}^{*}(T_{i}-I)A_{2}u_{n},\\ y_{n}= v_{n} + \sum_{i=1}^{n}\gamma_{n,i}\tau_{n} (U_{i}-I)v_{n},\\ x_{n+1}=\sigma_{n}v + (1-\sigma_{n})y_{n}, \quad n\in\mathbb{N}, \end{array}\displaystyle \right . $$
(28)
where \(\{\delta_{n}\}\), \(\{\theta_{n}\}\), \(\{\tau_{n}\}\), \(\{\sigma_{n}\}\), \(\{ \alpha_{n,i}\}\), \(\{\beta_{n,i}\} \), and \(\{\gamma_{n,i}\} \) are sequences in \([0,1] \) satisfying the following conditions:
  1. (C1)

    \(\sum_{i=1}^{n}\alpha_{n,i}= \sum_{i=1}^{n}\beta_{n,i} = \sum_{i=1}^{n}\gamma_{n,i}= 1\) for all \(n\in\mathbb{N}\);

     
  2. (C2)

    \(\liminf_{n\to\infty}\alpha_{n,i}>0\), \(\liminf_{n\to\infty}\beta _{n,i}>0 \), and \(\liminf_{n\to\infty}\gamma_{n,i}>0 \) for all \(i\in \mathbb{N}\);

     
  3. (C3)

    \(\lim_{n\to\infty}\sigma_{n}=0 \) and \(\sum_{n=1}^{\infty} \sigma _{n} = \infty\);

     
  4. (C4)

    \(0< a_{1}\leq\delta_{n}\leq a_{2}<\frac{1-k_{1}}{\|A_{1}\|^{2}}\);

     
  5. (C5)

    \(0< b_{1}\leq\theta_{n}\leq b_{2}<\frac{1-k_{2}}{\|A_{2}\|^{2}}\);

     
  6. (C6)

    \(0< c_{1}\leq\tau_{n}\leq c_{2}<1-k_{3}\).

     
Then the sequence \(\{x_{n}\} \) converges strongly to \(x^{*}=P_{\Gamma }(v)\).

It is known that every quasi-nonexpansive mapping is 0-demicontractive mapping, so the following result is directly obtained by Theorem 3.2.

Corollary 3.8

Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces, and let \(A_{1}, A_{2}: H_{1}\to H_{2} \) be two bounded linear operators with adjoint operators \(A_{1}^{*} \) and \(A_{2}^{*}\), respectively. Let \(\{ S_{i}:H_{2}\to H_{2}:i\in\mathbb{N}\}\), \(\{ T_{i}:H_{2}\to H_{2}:i\in \mathbb{N}\} \), and \(\{U_{i}:H_{1}\to H_{1}:i\in\mathbb{N}\} \) be infinite families of quasi-nonexpansive mappings such that \(S_{i}-I\), \(T_{i}-I \), and \(U_{i}-I\) are demiclosed at zero, respectively. Suppose that \(\Gamma= \{v^{*}\in\bigcap_{i=1}^{\infty} F(U_{i}) : A_{1}v^{*}\in\bigcap_{i=1}^{\infty}F(S_{i}) \textit{ and } A_{2}v^{*}\in \bigcap_{i=1}^{\infty}F(T_{i}) \}\neq\emptyset\). For arbitrary \(x_{1}\in H_{1} \), let \(\{u_{n}\}\), \(\{v_{n}\}\), \(\{y_{n}\} \), and \(\{ x_{n}\} \) be generated by
$$ \left \{ \textstyle\begin{array}{l} u_{n}= x_{n} + \sum_{i=1}^{n}\alpha_{n,i}\delta_{n} A_{1}^{*}(S_{i}-I)A_{1}x_{n}, \\ v_{n} = u_{n} + \sum_{i=1}^{n}\beta_{n,i}\theta_{n} A_{2}^{*}(T_{i}-I)A_{2}u_{n},\\ y_{n}= v_{n} + \sum_{i=1}^{n}\gamma_{n,i}\tau_{n} (U_{i}-I)v_{n},\\ x_{n+1}=\sigma_{n}v + (1-\sigma_{n})y_{n}, \quad n\in\mathbb{N}, \end{array}\displaystyle \right . $$
(29)
where \(\{\delta_{n}\}\), \(\{\theta_{n}\}\), \(\{\tau_{n}\}\), \(\{\sigma_{n}\}\), \(\{ \alpha_{n,i}\}\), \(\{\beta_{n,i}\} \), and \(\{\gamma_{n,i}\} \) are sequences in \([0,1] \) satisfying the following conditions:
  1. (C1)

    \(\sum_{i=1}^{n}\alpha_{n,i}= \sum_{i=1}^{n}\beta_{n,i} = \sum_{i=1}^{n}\gamma_{n,i}= 1\) for all \(n\in\mathbb{N}\);

     
  2. (C2)

    \(\liminf_{n\to\infty}\alpha_{n,i}>0\), \(\liminf_{n\to\infty}\beta _{n,i}>0 \), and \(\liminf_{n\to\infty}\gamma_{n,i}>0 \) for all \(i\in \mathbb{N}\);

     
  3. (C3)

    \(\lim_{n\to\infty}\sigma_{n}=0 \) and \(\sum_{n=1}^{\infty} \sigma _{n} = \infty\);

     
  4. (C4)

    \(0< a_{1}\leq\delta_{n}\leq a_{2}<\frac{1}{\|A_{1}\|^{2}}\);

     
  5. (C5)

    \(0< b_{1}\leq\theta_{n}\leq b_{2}<\frac{1}{\|A_{2}\|^{2}}\);

     
  6. (C6)

    \(0< c_{1}\leq\tau_{n}\leq c_{2}<1\).

     
Then the sequence \(\{x_{n}\} \) converges strongly to \(x^{*}=P_{\Gamma }(v)\).

4 Applications

4.1 The split common null point problem

Let M be the set-valued mapping of H into \(2^{H}\). The effective domain of M is denoted by \(D(M)\), that is, \(D(M)=\{x\in H : Mx\neq\emptyset\}\). The mapping M is said to be monotone if
$$\langle x-y,u-v\rangle\geq0,\quad \forall x,y\in D(M), u\in Mx,v\in My. $$
A monotone mapping M is said to be maximal if the graph \(G(M) \) is not properly contained in the graph of any other monotone map, where \(G(M)=\{(x,y)\in H\times H : y\in Mx\}\). It is known that M is maximal if and only if for \((x,u)\in H\times H\), \(\langle x-y,u-v\rangle\geq0 \) for every \((y,v)\in G(M) \) implies \(u\in Mx\). For the maximal monotone operator M, we can associate its resolvent \(J_{\delta}^{M} \) defined by
$$J_{\delta}^{M}\equiv(I+\delta M)^{-1}:H\to D(M), \quad\text{where } \delta >0. $$
It is known that if M is a maximal monotone operator, then the resolvent \(J_{\delta}^{M} \) is firmly nonexpansive, and \(F(J_{\delta }^{M})=M^{-1}0\equiv\{x\in H : 0\in Mx\} \) for every \(\delta>0\).
Let \(H_{1} \) and \(H_{2} \) be two real Hilbert spaces. Let \(M_{i}:H_{1}\to2^{H_{1}}\), \(O_{i}:H_{2}\to2^{H_{2}}\), and \(P_{i}:H_{2}\to2^{H_{2}} \) be multivalued mappings. The split common null point problem (SCNPP) [18] is to find a point \(u^{*}\in H_{1} \) such that
$$ 0\in\bigcap_{i=1}^{p}M_{i}u^{*} $$
(30)
and the points \(v_{j}^{*}=A_{j}u^{*}\in H_{2} \) satisfy
$$ 0\in\bigcap_{j=1}^{q}O_{j}v_{j}^{*}, $$
(31)
where \(A_{j}:H_{1}\to H_{2} \) (\(1\leq j\leq q\)) are bounded linear operators.
Now, we apply Theorem 3.4 to solve the problem of finding a point \(u^{*}\in H_{1} \) such that
$$ 0\in\bigcap_{i=1}^{\infty}M_{i}u^{*} $$
(32)
and the points \(v^{*}=A_{1}u^{*}\in H_{2}\) and \(s^{*}=A_{2}u^{*}\in H_{2} \) satisfy
$$ 0\in\bigcap_{i=1}^{\infty}O_{i}v^{*} \quad\text{and}\quad0\in\bigcap_{i=1}^{\infty}P_{i}s^{*}, $$
(33)
where \(A_{1}, A_{2}:H_{1}\to H_{2} \) are bounded linear operators.

Since every firmly nonexpansive mapping is a 0-demicontractive mapping, we obtain the following theorem for problem (32)–(33).

Theorem 4.1

Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces, and let \(A_{1}, A_{2}: H_{1}\to H_{2} \) be two bounded linear operators with adjoint operators \(A_{1}^{*} \) and \(A_{2}^{*}\), respectively. Let \(f : H_{1} \to H_{1} \) be a ρ-contraction mapping, and let B be a self-adjoint strongly positive bounded linear operator on \(H_{1} \) with coefficient \(\xi>2\rho\) and \(\|B\|=1\). Let \(\{M_{i}:H_{1}\to 2^{H_{1}}: i\in\mathbb{N}\}\), \(\{O_{i}: H_{2}\to2^{H_{2}}: i\in\mathbb {N} \}\), and \(\{P_{i}:H_{2}\to2^{H_{2}}: i\in\mathbb{N}\} \) be maximal monotone mappings. Suppose that \(\Omega= \{v^{*}\in\bigcap_{i=1}^{\infty} M_{i}^{-1}0 : A_{1}v^{*}\in\bigcap_{i=1}^{\infty }O_{i}^{-1}0 \textit{ and } A_{2}v^{*}\in\bigcap_{i=1}^{\infty }P_{i}^{-1}0 \}\neq\emptyset\). For arbitrary \(x_{1}\in H_{1} \), let \(\{u_{n}\}\), \(\{v_{n}\}\), \(\{y_{n}\} \), and \(\{x_{n}\} \) be generated by
$$ \left \{ \textstyle\begin{array}{l} u_{n}= x_{n} + \sum_{i=1}^{n}\alpha_{n,i}\delta_{n} A_{1}^{*}(J_{r_{1}}^{O_{i}}-I)A_{1}x_{n}, \\ v_{n} = u_{n} + \sum_{i=1}^{n}\beta_{n,i}\theta_{n} A_{2}^{*}(J_{r_{2}}^{P_{i}}-I)A_{2}u_{n},\\ y_{n}= v_{n} + \sum_{i=1}^{n}\gamma_{n,i}\tau_{n} (J_{r_{3}}^{M_{i}}-I)v_{n},\\ x_{n+1}=\sigma_{n}f(y_{n}) + (I-\sigma_{n}B)y_{n}, \quad n\in\mathbb{N}, \end{array}\displaystyle \right . $$
(34)
where \(r_{1}, r_{2}, r_{3}>0 \) and \(\{\delta_{n}\}\), \(\{\theta_{n}\}\), \(\{\tau _{n}\}\), \(\{\sigma_{n}\}\), \(\{\alpha_{n,i}\}\), \(\{\beta_{n,i}\}\), \(\{\gamma _{n,i}\} \) are sequences in \([0,1] \) satisfying the following conditions:
  1. (C1)

    \(\sum_{i=1}^{n}\alpha_{n,i}= \sum_{i=1}^{n}\beta_{n,i} = \sum_{i=1}^{n}\gamma_{n,i}= 1\) for all \(n\in\mathbb{N}\);

     
  2. (C2)

    \(\liminf_{n\to\infty}\alpha_{n,i}>0\), \(\liminf_{n\to\infty}\beta _{n,i}>0 \), and \(\liminf_{n\to\infty}\gamma_{n,i}>0 \) for all \(i\in \mathbb{N}\);

     
  3. (C3)

    \(\lim_{n\to\infty}\sigma_{n}=0 \) and \(\sum_{n=1}^{\infty} \sigma _{n} = \infty\);

     
  4. (C4)

    \(0< a_{1}\leq\delta_{n}\leq a_{2}<\frac{1}{\|A_{1}\|^{2}}\);

     
  5. (C5)

    \(0< b_{1}\leq\theta_{n}\leq b_{2}<\frac{1}{\|A_{2}\|^{2}}\);

     
  6. (C6)

    \(0< c_{1}\leq\tau_{n}\leq c_{2}<1\).

     
Then the sequence \(\{x_{n}\} \) converges strongly to \(x^{*}=P_{\Omega }(f+I-B)x^{*}\).

4.2 The split variational inequality problem

Let C and Q be nonempty closed convex subsets of two real Hilbert spaces \(H_{1} \) and \(H_{2}\), respectively. Let \(A:H_{1}\to H_{2} \) be a bounded linear operator, \(g:H_{1}\to H_{1} \), and \(h:H_{2}\to H_{2}\). The split variational inequality problem (SVIP) is to find a point \(u^{*}\in C \) such that
$$ \bigl\langle g\bigl(u^{*}\bigr),x-u^{*}\bigr\rangle \geq0, \quad\forall x\in C, $$
(35)
and the point \(v^{*}=Au^{*}\in Q \) satisfy
$$ \bigl\langle h\bigl(v^{*}\bigr),y-v^{*}\bigr\rangle \geq0, \quad\forall y\in Q. $$
(36)
We denote the solution set of the SVIP by \(\varOmega=\operatorname{SVIP}(C,Q,g,h,A)\). The set of all solutions of variational inequality problem (35) is denoted by \(\operatorname{VIP}(C,g) \), and it is known that \(\operatorname{VIP}(C,g)=F(P_{C}(I-\lambda g)) \) for all \(\lambda>0\).
Let \(A_{1}, A_{2}:H_{1}\to H_{2} \) be two bounded linear operators, \(g_{i}:H_{1}\to H_{1} \), and \(h_{i}, l_{i}:H_{2}\to H_{2}\). In this section, we apply Theorem 3.4 to solve the problem of finding a point \(u^{*}\in\bigcap_{i=1}^{\infty}C_{i} \) such that
$$ \bigl\langle g_{i}\bigl(u^{*} \bigr),x-u^{*}\bigr\rangle \geq0, \quad\forall x\in\bigcap _{i=1}^{\infty}C_{i}, $$
(37)
and the point \(v^{*}=A_{1}u^{*}\in\bigcap_{i=1}^{\infty}Q_{i}\), \(s^{*}=A_{2}u^{*}\in\bigcap_{i=1}^{\infty}K_{i} \) satisfy
$$ \bigl\langle h_{i}\bigl(v^{*} \bigr),y-v^{*}\bigr\rangle \geq0, \quad\forall y\in\bigcap _{i=1}^{\infty}Q_{i},\quad\text{and}\quad\bigl\langle l_{i}\bigl(s^{*}\bigr),z-s^{*}\bigr\rangle \geq0, \quad\forall z\in\bigcap_{i=1}^{\infty}K_{i}, $$
(38)
where \(\{C_{i}\}_{i\in\mathbb{N}} \) is a family of nonempty closed convex subsets of a real Hilbert space \(H_{1} \), and \(\{Q_{i}\}_{i\in \mathbb{N}}\) and \(\{K_{i}\}_{i\in\mathbb{N}} \) are two families of nonempty closed convex subsets of a real Hilbert space \(H_{2}\). We now prove a strong convergence theorem for problem (37)–(38).

Theorem 4.2

Let \(\{C_{i}\}_{i\in\mathbb{N}} \) be the family of nonempty closed convex subsets of a real Hilbert space \(H_{1} \), let \(\{Q_{i}\}_{i\in \mathbb{N}}\) and \(\{K_{i}\}_{i\in\mathbb{N}} \) be two families of nonempty closed convex subsets of a real Hilbert space \(H_{2}\), and let \(A_{1}, A_{2}: H_{1}\to H_{2} \) be two bounded linear operators with adjoint operators \(A_{1}^{*} \) and \(A_{2}^{*}\), respectively. Let \(f : H_{1} \to H_{1} \) be a ρ-contraction mapping, and let B be a self-adjoint strongly positive bounded linear operator on \(H_{1} \) with coefficient \(\xi>2\rho\) and \(\|B\|=1\). Let \(\{ g_{i}:H_{1}\to H_{1}:i\in\mathbb{N}\}\), \(\{h_{i}:H_{2}\to H_{2};i\in \mathbb{N}\} \), and \(\{ l_{i}:H_{2}\to H_{2};i\in\mathbb{N}\} \) be \(r_{1}\)-, \(r_{2}\)-, and \(r_{3} \)-inverse strongly monotone mappings, respectively. Let \(r = \min\{r_{1},r_{2}, r_{3}\}\) and \(\mu\in (0,2r)\). Suppose that \(\Omega= \{v^{*}\in\bigcap_{i=1}^{\infty} \operatorname{VIP}(C_{i},g_{i}) : A_{1}v^{*}\in\bigcap_{i=1}^{\infty}\operatorname{VIP}(Q_{i},h_{i}) \textit{ and } A_{2}v^{*}\in\bigcap_{i=1}^{\infty}\operatorname{VIP}(K_{i},l_{i}) \} \neq\emptyset\). For arbitrary \(x_{1}\in H_{1} \), let \(\{u_{n}\}\), \(\{ v_{n}\}\), \(\{y_{n}\} \), and \(\{x_{n}\} \) be generated by
$$ \left \{ \textstyle\begin{array}{l} u_{n}= x_{n} + \sum_{i=1}^{n}\alpha_{n,i}\delta_{n} A_{1}^{*}(P_{Q_{i}}(I-\mu h_{i})-I)A_{1}x_{n}, \\ v_{n} = u_{n} + \sum_{i=1}^{n}\beta_{n,i}\theta_{n} A_{2}^{*}(P_{K_{i}}(I-\mu l_{i})-I)A_{2}u_{n},\\ y_{n}= v_{n} + \sum_{i=1}^{n}\gamma_{n,i}\tau_{n} (P_{C_{i}}(I-\mu g_{i})-I)v_{n},\\ x_{n+1}=\sigma_{n}f(y_{n}) + (I-\sigma_{n}B)y_{n}, \quad n\in\mathbb{N}, \end{array}\displaystyle \right . $$
(39)
where \(\{\delta_{n}\}\), \(\{\theta_{n}\}\), \(\{\tau_{n}\}\), \(\{\sigma_{n}\}\), \(\{ \alpha_{n,i}\}\), \(\{\beta_{n,i}\}\), \(\{\gamma_{n,i}\} \) are sequences in \([0,1] \) satisfying the following conditions:
  1. (C1)

    \(\sum_{i=1}^{n}\alpha_{n,i}= \sum_{i=1}^{n}\beta_{n,i} = \sum_{i=1}^{n}\gamma_{n,i}= 1\) for all \(n\in\mathbb{N}\);

     
  2. (C2)

    \(\liminf_{n\to\infty}\alpha_{n,i}>0\), \(\liminf_{n\to\infty}\beta _{n,i}>0 \), and \(\liminf_{n\to\infty}\gamma_{n,i}>0 \) for all \(i\in \mathbb{N}\);

     
  3. (C3)

    \(\lim_{n\to\infty}\sigma_{n}=0 \) and \(\sum_{n=1}^{\infty} \sigma _{n} = \infty\);

     
  4. (C4)

    \(0< a_{1}\leq\delta_{n}\leq a_{2}<\frac{1}{\|A_{1}\|^{2}}\);

     
  5. (C5)

    \(0< b_{1}\leq\theta_{n}\leq b_{2}<\frac{1}{\|A_{2}\|^{2}}\);

     
  6. (C6)

    \(0< c_{1}\leq\tau_{n}\leq c_{2}<1\).

     
Then the sequence \(\{x_{n}\} \) converges strongly to \(x^{*}=P_{\Omega }(f+I-B)x^{*}\).

Proof

It is known that \(S_{i}:=P_{Q_{i}}(I-\mu h_{i})\), \(T_{i}=: P_{K_{i}}(I-\mu l_{i}) \), and \(U_{i}:=P_{C_{i}}(I-\mu g_{i}) \) are nonexpensive mappings for all \(\mu\in(0,2r) \), and hence they are 0-demicontractive mappings. We obtain the desired result from Theorem 3.4. □

4.3 The split equilibrium problem

Let \(H_{1} \) and \(H_{2} \) be two real Hilbert spaces, and let C and Q be nonempty closed convex subsets of \(H_{1} \) and \(H_{2}\), respectively. Let \(A:H_{1}\to H_{2} \) be a bounded linear operator, and let \(g:C\times C\to\mathbb{R} \) and \(h:Q\times Q\to\mathbb{R} \) be two bifunctions. The split equilibrium problem (SEP) is to find a point \(u^{*}\in C \) such that
$$ g\bigl(u^{*},x\bigr) \geq0, \quad\forall x\in C, $$
(40)
and \(Au^{*}\in Q \) satisfy
$$ h\bigl(Au^{*},y\bigr) \geq0, \quad\forall y\in Q. $$
(41)
The set of all solutions of equilibrium problem (40) is denoted by \(\operatorname{EP}(g)\).

Lemma 4.3

([19])

Let C be a nonempty closed convex subset of H, and let g be a bifunction of \(C\times C \) into \(\mathbb{R} \) satisfying the following conditions:
  1. (A1)

    \(g(x,x)=0 \) for all \(x\in C\);

     
  2. (A2)

    g is monotone, that is, \(g(x,y)+g(y,x)\leq0 \) for all \(x,y \in C\);

     
  3. (A3)
    for all \(x,y,z\in C\),
    $$\limsup_{t\downarrow0}g\bigl(tz+(1-t)x,y\bigr)\leq g(x,y); $$
     
  4. (A4)

    \(g(x,\cdot) \) is convex and lower semicontinuous for all \(x\in C\).

     
Let \(g:C\times C\to\mathbb{R} \) be a bifunction satisfying conditions (A1)(A4), and let \(r>0 \) and \(x\in H\). Then there exists \(z\in C \) such that
$$g(z,y)+\frac{1}{r}\langle y-z,z-x\rangle\geq0 \quad\textit{for all } y\in C. $$

Lemma 4.4

([20])

Let C be a nonempty closed convex subset of H, and let g be a bifunction of \(C\times C \) into \(\mathbb{R} \) satisfying conditions (A1)(A4). For \(r>0 \) and \(x\in H\), define the mapping \(T_{r}^{g}:H\to C \) of g by
$$T_{r}^{g}x= \biggl\{ z\in C : g(z,y)+\frac{1}{r} \langle y-z,z-x\rangle \geq0, \forall y\in C \biggr\} ,\quad \forall x\in H. $$
Then the following hold:
  1. (i)

    \(T_{r}^{g} \) is single-valued;

     
  2. (ii)

    \(T_{r}^{g} \) is firmly nonexpansive;

     
  3. (iii)

    \(F(T_{r}^{g})=\operatorname{EP}(g)\);

     
  4. (iv)

    \(\operatorname{EP}(g) \) is closed and convex.

     
Let \(A_{1}, A_{2}:H_{1}\to H_{2} \) be two bounded linear operators, and let \(g_{i}:C_{i}\times C_{i}\to\mathbb{R} \) and \(h_{i}, l_{i}:Q_{i}\times Q_{i}\to\mathbb{R} \) be bifunctions for all \(i\in \mathbb{N}\). In this section, we apply Theorem 3.4 to solve the problem of finding a point
$$ u^{*}\in\bigcap_{i=1}^{\infty} \operatorname{EP}(g_{i}) \quad\text{such that} \quad A_{1}v^{*} \in\bigcap_{i=1}^{\infty}\operatorname{EP}(h_{i}) \quad\text{and}\quad A_{2}v^{*}\in \bigcap_{i=1}^{\infty}\operatorname{EP}(l_{i}). $$
(42)

By Lemma 4.4(iii) we have that \(T_{r_{1}}^{h_{i}}\), \(T_{r_{2}}^{l_{i}} \), and \(T_{r_{3}}^{g_{i}} \) are firmly nonexpansive mappings, and hence they are 0-demicontractive mappings. We obtain the following result from Theorem 3.4.

Theorem 4.5

Let \(\{C_{i}\}_{i\in\mathbb{N}} \) be a family of nonempty closed convex subsets of a real Hilbert space \(H_{1} \), let \(\{Q_{i}\}_{i\in \mathbb{N}} \) and \(\{K_{i}\}_{i\in\mathbb{N}} \) be two families of nonempty closed convex subsets of a real Hilbert space \(H_{2}\), and let \(A_{1}, A_{2}: H_{1}\to H_{2} \) be two bounded linear operators with adjoint operators \(A_{1}^{*} \) and \(A_{2}^{*}\), respectively. Let \(f : H_{1} \to H_{1} \) be a ρ-contraction mapping, and let B be a self-adjoint strongly positive bounded linear operator on \(H_{1} \) with coefficient \(\xi>2\rho\) and \(\|B\|=1\). Let \(g_{i}:C_{i}\times C_{i}\to\mathbb{R} \) and \(h_{i}, l_{i}:Q_{i}\times Q_{i}\to\mathbb{R} \) be bifunctions satisfying conditions (A1)(A4) for all \(i\in\mathbb{N}\). Suppose that \(\Omega=\{v^{*}\in\bigcap_{i=1}^{\infty}\operatorname{EP}(g_{i}) : A_{1}v^{*}\in\bigcap_{i=1}^{\infty }\operatorname{EP}(h_{i}) \textit{ and } A_{2}v^{*}\in\bigcap_{i=1}^{\infty }\operatorname{EP}(l_{i})\}\neq\emptyset\). For arbitrary \(x_{1}\in H_{1} \), let \(\{ u_{n}\}\), \(\{v_{n}\}\), \(\{y_{n}\} \), and \(\{x_{n}\} \) be generated by
$$ \left \{ \textstyle\begin{array}{l} u_{n}= x_{n} + \sum_{i=1}^{n}\alpha_{n,i}\delta_{n} A_{1}^{*}(T_{r_{1}}^{h_{i}}-I)A_{1}x_{n}, \\ v_{n} = u_{n} + \sum_{i=1}^{n}\beta_{n,i}\theta_{n} A_{2}^{*}(T_{r_{2}}^{l_{i}}-I)A_{2}u_{n},\\ y_{n}= v_{n} + \sum_{i=1}^{n}\gamma_{n,i}\tau_{n} (T_{r_{3}}^{g_{i}}-I)v_{n},\\ x_{n+1}=\sigma_{n}f(y_{n}) + (I-\sigma_{n}B)y_{n}, \quad n\in\mathbb{N}, \end{array}\displaystyle \right . $$
(43)
where \(r_{1}, r_{2}, r_{3}>0 \) and \(\{\delta_{n}\}\), \(\{\theta_{n}\}\), \(\{\tau _{n}\}\), \(\{\sigma_{n}\}\), \(\{\alpha_{n,i}\}\), \(\{\beta_{n,i}\}\), \(\{\gamma _{n,i}\} \) are sequences in \([0,1] \) satisfying the following conditions:
  1. (C1)

    \(\sum_{i=1}^{n}\alpha_{n,i}= \sum_{i=1}^{n}\beta_{n,i} = \sum_{i=1}^{n}\gamma_{n,i}= 1\) for all \(n\in\mathbb{N}\);

     
  2. (C2)

    \(\liminf_{n\to\infty}\alpha_{n,i}>0\), \(\liminf_{n\to\infty}\beta _{n,i}>0 \), and \(\liminf_{n\to\infty}\gamma_{n,i}>0 \) for all \(i\in \mathbb{N}\);

     
  3. (C3)

    \(\lim_{n\to\infty}\sigma_{n}=0 \) and \(\sum_{n=1}^{\infty} \sigma _{n} = \infty\);

     
  4. (C4)

    \(0< a_{1}\leq\delta_{n}\leq a_{2}<\frac{1}{\|A_{1}\|^{2}}\);

     
  5. (C5)

    \(0< b_{1}\leq\theta_{n}\leq b_{2}<\frac{1}{\|A_{2}\|^{2}}\);

     
  6. (C6)

    \(0< c_{1}\leq\tau_{n}\leq c_{2}<1\).

     
Then the sequence \(\{x_{n}\} \) converges strongly to \(x^{*}=P_{\Omega }(f+I-B)x^{*}\).

5 Numerical example for the main result

We now give a numerical example of the studied method. Let \(H_{1}=H_{2}=(\mathbb{R}^{2},\|\cdot\|_{2})\). Define the mappings \(S_{i}:\mathbb{R}^{2}\to\mathbb{R}^{2}\), \(U_{i}:\mathbb{R}^{2}\to\mathbb{R}^{2} \), and \(T_{i}:\mathbb{R}^{2}\to\mathbb{R}^{2} \) by
$$S_{i}(x_{1},x_{2})= \frac{-3i}{i+1}(x_{1},x_{2}), \qquad U_{i}(x_{1},x_{2})= \biggl(\frac{-2i}{i+1}x_{1},x_{2}\biggr), \quad i\in\mathbb {N}, $$
and
$$ T_{i}(x_{1},x_{2}) = \left \{ \textstyle\begin{array}{l@{\quad}l} (x_{1},\frac{x_{2}}{3i}\sin\frac{1}{x_{2}}) & \text{if $x_{2}\neq0$,}\\ (x_{1},0) & \text{if $x_{2}=0$,} \end{array}\displaystyle \right . \quad i\in\mathbb{N} $$
for all \(x_{1}, x_{2}\in\mathbb{R}\). Then \(S_{i} \) are \(\frac {12}{25} \)-demicontractive mappings for all \(i\in\mathbb{N} \) and \(\bigcap_{i=1}^{\infty}F(S_{i})=\{(0,0)\}\), \(U_{i} \) are \(\frac{3}{4} \)-demicontractive mappings for all \(i\in\mathbb{N} \) and \(\bigcap_{i=1}^{\infty}F(U_{i})=0\times\mathbb{R} \), and \(T_{i} \) are 0-demicontractive mappings for all \(i\in\mathbb{N} \) and \(\bigcap_{i=1}^{\infty}F(T_{i})=\mathbb{R}\times0\). Next, we define the mappings \(f:\mathbb{R}^{2}\to\mathbb{R}^{2}\), \(B:\mathbb{R}^{2}\to\mathbb {R}^{2}\), \(A_{1}:\mathbb{R}^{2}\to\mathbb{R}^{2} \), and \(A_{2}:\mathbb {R}^{2}\to\mathbb{R}^{2} \) by
$$f(x_{1},x_{2})= \biggl(\frac{x_{1}}{8},\frac{x_{2}}{8}\biggr),\qquad B(x_{1},x_{2})= \biggl(x_{1},\frac{x_{2}}{2}\biggr), \qquad A_{1}(x_{1},x_{2})= (x_{1},2x_{1}), $$
and
$$A_{2}(x_{1},x_{2})= (x_{2}-x_{1},2x_{1}) $$
for all \(x_{1}, x_{2}\in\mathbb{R}\). Then f is a \(\frac{1}{8} \)-contraction, B is a self-adjoint strongly positive bounded linear operator with coefficient \(\xi=\frac{1}{2} \), and \(A_{1}\), \(A_{2} \) are bounded linear operators. Define the real sequence \(\{\alpha_{n,i}\}\), \(\{\beta_{n,i}\} \), and \(\{\gamma_{n,i}\} \) as follows:
$$\begin{gathered} \alpha_{n,i} = \left \{ \textstyle\begin{array}{l@{\quad}l} 1 & \text{if $n=i=1$,}\\ \frac{1}{2^{i}} (\frac{n}{n+1} ) & \text{if $n>i$,}\\ 1-\sum_{i=1}^{n-1}\frac{1}{2^{i}} (\frac{n}{n+1} ) & \text{if $n=i>1$,}\\ 0 & \text{otherwise,} \end{array}\displaystyle \right . \\ \beta_{n,i} = \left \{ \textstyle\begin{array}{l@{\quad}l} 1 & \text{if $n=i=1$,}\\ \frac{1}{3^{i}} (\frac{n}{n+1} ) & \text{if $n>i$,}\\ 1-\sum_{i=1}^{n-1}\frac{1}{3^{i}} (\frac{n}{n+1} ) & \text{if $n=i>1$,}\\ 0 & \text{otherwise,} \end{array}\displaystyle \right .\end{gathered} $$
and
$$ \gamma_{n,i} = \left \{ \textstyle\begin{array}{l@{\quad}l} 1 & \text{if $n=i=1$,}\\ \frac{1}{4^{i+1}} (\frac{n}{2n+1} ) & \text{if $n>i$,}\\ 1-\sum_{i=1}^{n-1}\frac{1}{4^{i+1}} (\frac{n}{2n+1} ) & \text{if $n=i>1$,}\\ 0 & \text{otherwise,} \end{array}\displaystyle \right . $$
that is,
$$\begin{aligned}& \alpha_{n,i}= \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots \\ 1/3 & 2/3 & 0 & 0 & 0 & 0 & 0 &0 &\ldots \\ 3/8 & 3/16 & 7/16 & 0 & 0 & 0 & 0 &0 &\ldots \\ 2/5 & 1/5 & 1/10 & 3/10 & 0 & 0 & 0 &0 &\ldots \\ 5/12 & 5/24 & 5/48 & 5/96 & 7/32 & 0 & 0 &0 &\ldots \\ 3/7 & 3/14 & 3/28 & 3/56 & 3/112 & 19/112 & 0 &0 &\ldots \\ 7/16 & 7/32 & 7/64 & 7/128 & 7/256 & 7/512 & 71/512 &0 &\ldots\\ \vdots& \vdots& \vdots& \vdots& \vdots& \vdots& \vdots& \vdots \end{pmatrix} , \\& \beta_{n,i}= \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 & \ldots\\ 2/9 & 7/9 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots\\ 1/4 & 1/12 & 2/3 & 0 & 0 & 0 & 0 &0 &\ldots\\ 4/15 & 4/45 & 4/135 & 83/135 & 0 & 0 & 0 &0 &\ldots\\ 5/18 & 5/54 & 5/162 & 5/486 & 143/243 & 0 & 0 & 0 &\ldots\\ 2/7 & 2/21 & 2/63 & 2/189 & 1/284 & 325/567 & 0 &0 &\ldots\\ 7/24 & 7/72 & 7/216 & 7/648 & 1/278 & 1/833 & 58/103 &0 &\ldots\\ \vdots& \vdots& \vdots& \vdots& \vdots& \vdots& \vdots& \vdots \end{pmatrix} , \end{aligned}$$
and
$$ \gamma_{n,i}= \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots\\ 1/40 & 39/40 & 0 & 0 & 0 & 0 & 0 & 0 &\ldots\\ 3/112 & 3/448 & 433/448 & 0 & 0 & 0 & 0 & 0 &\ldots\\ 1/36 & 1/144 & 1/576 & 185/192 & 0 & 0 & 0 & 0 &\ldots\\ 5/176 & 5/704 & 1/563 & 1/2253 & 51/53 & 0 & 0 & 0 &\ldots\\ 3/104 & 3/416 & 1/555 & 1/2219 & 1/8875 & 976/1015 & 0 & 0 &\ldots\\ 7/240 & 7/960 & 1/549 & 1/2194 & 1/8777 & 1/35\mbox{,}109 & 618/643 & 0 &\ldots \\ \vdots& \vdots& \vdots& \vdots& \vdots& \vdots& \vdots& \vdots \end{pmatrix} . $$
We see that \(\lim_{n\to\infty}\alpha_{n,i}= \frac{1}{2^{i}}\), \(\lim_{n\to\infty}\beta_{n,i}= \frac{1}{3^{i}} \), and \(\lim_{n\to\infty }\gamma_{n,i}= \frac{1}{2^{2i+3}} \) for \(i\in\mathbb{N}\). Now, we start with the initial point \(x_{1}=(1,1) \) and let \(\{ x_{n}\}\) be the sequence generated by (12). Suppose that \(x_{n} \) is of the form \(x_{n}=(a_{n}, b_{n})\). where \(a_{n}, b_{n} \in\mathbb{R}\). The criterion for stopping our testing method is taken as \(\| x_{n-1}-x_{n}\|_{2}<10^{-6}\). Choose \(\delta_{n}= \frac{n}{11n-1}\), \(\theta_{n}=\frac{n}{30n-1}\), \(\tau _{n}=\frac{n}{2n-1}\), and \(\sigma_{n}=\frac{1}{n^{0.01}} \) for all \(n\in\mathbb{N}\). Figure 1 shows the errors \(\| x_{n-1} -x_{n}\|_{2} \) of our proposed method. The values of \(x_{n} \) and \(\| x_{n-1}-x_{n}\|_{2} \) are shown in Table 1.
Figure 1
Figure 1

Graph for errors

Table 1

Numerical experiment for \(x_{n}\)

n

\(a_{n} \)

\(b_{n} \)

\(\| x_{n-1} -x_{n}\|_{2} \)

1

1.00000000

1.00000000

2

0.12500000

0.62500000

0.95197164

3

0.01751567

0.39224395

0.25637524

4

0.00414010

0.24675959

0.14609793

5

0.00202951

0.15549870

0.09128529

6

0.00140947

0.09811767

0.05738438

7

0.00107109

0.06197693

0.03614232

8

0.00063002

0.03918347

0.02279773

9

0.00047832

0.02479206

0.01439221

10

0.00030270

0.01569709

0.00909667

11

0.00022553

0.00994467

0.00575293

12

0.00014616

0.00630378

0.00364176

13

0.00008740

0.00399788

0.00230665

14

0.00005861

0.00253664

0.00146152

28

0.00000009

0.00000450

0.00000257

29

0.00000007

0.00000287

0.00000163

30

0.00000007

0.00000183

0.00000104

31

0.00000003

0.00000117

0.00000066

We observe from Table 1 that \(x_{n}\to(0,0)\in\Gamma\). We also note that the error is bounded by \(\| x_{30} -x_{31}\|_{2} <10^{-6}\), and we can use \(x_{31}=( 0.00000003, 0.00000117) \) to approximate the solution of (7) with accuracy at least 6 D.P.

6 Conclusion

We introduce a new algorithm for solving the split common fixed point problem (7) of the infinite families of demicontractive mappings in Hilbert spaces. Strong convergence of the proposed algorithm is obtained under some suitable control conditions. The main results of this paper can be considered as an extension of work by Eslamian [12] by providing an algorithm for finding a solution of problem (7), which is a generalization of problem (5).

Declarations

Acknowledgements

The authors would like to thank Chiang Mai University and Center of Excellence in Mathematics, CHE, Bangkok 10400, Thailand, for the financial support.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Authors’ contributions

Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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

Authors’ Affiliations

(1)
Ph.D. Degree Program in Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand
(2)
Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand
(3)
Center of Excellence in Mathematics, CHE, Bangkok, Thailand

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