Skip to main content

Advertisement

Parallel hybrid viscosity method for fixed point problems, variational inequality problems and split generalized equilibrium problems

Article metrics

  • 286 Accesses

Abstract

In this paper, we first propose a new parallel hybrid viscosity iterative method for finding a common element of three solution sets: (i) finite split generalized equilibrium problems; (ii) finite variational inequality problems; and (iii) fixed point problem of a finite collection of demicontractive operators. And we prove that the sequence generated by the iterative scheme strongly converges to a common solution of the above-mentioned problems. Also, we present numerical examples to demonstrate the effectiveness of our algorithm. Our results presented in this paper improve and extend many recent results in the literature.

Introduction

Let \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) be two infinite dimensional real Hilbert space with inner product and norm denoted by \(\langle \cdot , \cdot \rangle \) and \(\|\cdot \|\), respectively. Let C and Q be a nonempty closed convex subset of \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\), respectively. Let \(T: C\rightarrow C\) be a mapping. The set of fixed points of T is denoted by \(F(T)\), that is, \(F(T)=\{x\in C: Tx=x\}\).

In what follows, we recall some definitions of classes of operators often used in fixed point theory.

Definition 1.1

Let \(T: C\rightarrow C\) be a mapping. Then

  1. (i)

    T is ρ-Lipschitzian with \(\rho >0\) if

    $$ \Vert Tx-Ty \Vert \leq \rho \Vert x-y \Vert , \quad \forall x, y\in C; $$

    If \(\rho \in (0, 1)\), then T is ρ-contractive and if \(\rho =1\), then T is nonexpansive.

  2. (ii)

    T is firmly nonexpansive if

    $$ \Vert Tx-Ty \Vert ^{2}\leq \Vert x-y \Vert ^{2}- \bigl\Vert (I-T)x-(I-T)y \bigr\Vert ^{2}, \quad \forall x, y \in C; $$
  3. (iii)

    T is κ-strictly pseudo-contractive with \(\kappa \in [0, 1)\) if

    $$ \Vert Tx-Ty \Vert ^{2}\leq \Vert x-y \Vert ^{2}+ \kappa \bigl\Vert (I-T)x-(I-T)y \bigr\Vert ^{2}, \quad \forall x, y \in C. $$

Definition 1.2

Let \(T: C\rightarrow C\) be a mapping with \(F(T)\neq \emptyset \). Then

  1. (i)

    T is directed if

    $$ \Vert Tx-z \Vert ^{2}\leq \Vert x-z \Vert ^{2}- \Vert x-Tx \Vert ^{2}, \quad \forall z\in F(T), x\in C; $$
  2. (ii)

    T is quasi-nonexpansive if

    $$ \Vert Tx-z \Vert \leq \Vert x-z \Vert , \quad \forall z\in F(T), x\in C; $$
  3. (iii)

    T is β-demicontractive with \(\beta <1\) if

    $$ \Vert Tx-z \Vert ^{2}\leq \Vert x-z \Vert ^{2}+\beta \Vert x-Tx \Vert ^{2}, \quad \forall z\in F(T), x\in C. $$

Note that the class of demicontractive operators contains important classes of operators: directed operator (firmly nonexpansive operator with nonempty fixed points set) for \(\beta =-1\), quasi-nonexpansive operator (nonexpansive operator with nonempty fixed points set) for \(\beta =0\), and strictly pseudo-contractive operator with nonempty fixed points set for \(\beta \in (0, 1)\); the class of quasi-nonexpansive operators also contains nonspreading mappings with nonempty fixed point set and N-generalized hybrid mappings with nonempty fixed point set.

It is well known that every nonexpansive operator \(T : \mathcal{H} _{1} \rightarrow \mathcal{H}_{1}\) satisfies the following inequality;

$$ \bigl\langle (I-T)x-(I-T)y, Ty -Tx \bigr\rangle \leq \frac{1}{2} \bigl\Vert (I-T)y-(I-T)x \bigr\Vert ^{2}, $$

for all \(x, y \in \mathcal{H}_{1}\). Therefore, for all \(x\in \mathcal{H}_{1}\), \(y\in F(T)\),

$$ \bigl\langle (I-T)x, y -Tx \bigr\rangle \leq \frac{1}{2} \bigl\Vert (T-I)x \bigr\Vert ^{2}. $$
(1.1)

We also know that \(F(T)\) of nonexpansive mapping T is closed and convex.

The fixed point problem (FPP) for the mapping T is to find \(x\in C\) such that

$$ Tx=x. $$

Many iterative algorithms has been introduced for finding fixed points of nonexpansive mappings, quasi-nonexpansive mappings, firmly nonexpansive mappings, demicontractive mappings (see [1,2,3,4,5,6]), including the since recently popular viscosity iterative algorithms, which formally consist of the sequence \(\{x_{n}\}\) given by the iteration

$$ x_{n+1}=\alpha _{n}f(x_{n})+(1-\alpha _{n})Tx_{n},\quad \forall n\geq 0, $$
(1.2)

where f is a contraction, \(\{\alpha _{n}\}\subset (0, 1)\) is a slowly vanishing sequence, i.e., \(\lim_{n\rightarrow \infty }\alpha _{n}=0\) and \(\sum_{n}\alpha _{n}=\infty \). The above method was first considered with regard to the special case when \(f=\mu \) (μ being any given element), in 1967 by Halpern [7] (for \(\mu =0\)). There is an extensive literature regarding the convergence analysis of (1.2), with several types of operator T, in the setting of Hilbert spaces and Banach spaces. This procedure can be regarded as a regularization process for fixed point iterations which is supposed to induce the convergence in norm of the iterates. Another advantage of this method is that it allows one to select a particular fixed point of T which satisfies some variational inequality.

Given a nonlinear mapping \(B : C\rightarrow \mathcal{H}_{1}\). Recall that B is said to be monotone if

$$ \langle x-y,Bx-By\rangle \geq 0, \quad \forall x, y\in C; $$

B is said to be α-strongly monotone if there exists \(\alpha > 0\) such that

$$ \langle x-y,Bx-By\rangle \geq \alpha \Vert x-y \Vert ^{2}, \quad \forall x, y \in C; $$

B is said to be α-inverse strongly monotone (for short, α-ism) if there exists \(\alpha >0\) such that

$$ \langle x-y,Bx-By\rangle \geq \alpha \Vert Bx-By \Vert ^{2}, \quad \forall x, y \in C. $$

We can easily see that

  1. (i)

    if B is nonexpansive, then \(I-B\) is monotone;

  2. (ii)

    if B is an α-inverse-strongly monotone mapping, then it must be a \(\frac{1}{\alpha }\)-Lipschitz operator. Moreover, \(I-rB\) is nonexpansive when \(0< r\leq 2\alpha \).

The variational inequality problem (VIP) is to find \(x\in C\) such that

$$ \langle Bx, y-x\rangle \geq 0, \quad \forall y\in C. $$
(1.3)

The solution set of (1.3) is denoted by \(\operatorname{VI}(C, B)\). In fact,

$$\begin{aligned}& x^{\ast }\in \operatorname{VI}(C, B) \\& \qquad \Updownarrow \\& \bigl\langle Bx^{\ast }, y-x^{\ast } \bigr\rangle \geq 0, \quad \forall y\in C \\& \qquad \Updownarrow \\& \bigl\langle -\lambda Bx^{\ast }, y-x^{\ast } \bigr\rangle \leq 0, \quad \forall \lambda >0, \forall y\in C \\& \qquad \Updownarrow \\& \bigl\langle \bigl(I-\lambda Bx^{\ast } \bigr)x^{\ast }-x^{\ast }, y-x^{\ast } \bigr\rangle \leq 0, \quad \forall \lambda >0, \forall y\in C \\& \qquad \Updownarrow \\& \bigl\langle \bigl(I-\lambda Bx^{\ast } \bigr)x^{\ast }-x^{\ast }, x^{\ast }-y \bigr\rangle \geq 0, \quad \forall \lambda >0, \forall y\in C \\& \qquad \Updownarrow \\& x^{\ast }=P_{C}(I-\lambda B)x^{\ast }, \quad \forall \lambda >0. \end{aligned}$$

It is well known that if B is strongly monotone and Lipschitz continuous mapping on C, then (1.3) has a unique solution. There are several different approaches towards solving this problem in finite dimensional and infinite dimensional spaces see [8,9,10,11,12,13,14] and the research in this direction is intensively continued.

The equilibrium problem for a bifunction \(f : C \times C\rightarrow R\) is to find a point \(x\in C\) such that

$$ f(x, y)\geq 0,\quad \forall y\in C. $$
(1.4)

We denote \(\operatorname{EP}(f)\) by the solution set of (1.4). It is easy to see that \(\operatorname{EP}(f)=\operatorname{VI}(C, B)\) when \(f(x, y)=\langle Bx, y-x\rangle \) for all \(x, y\in C\). Let \(h : C\times C \rightarrow R\) be a nonlinear bifunction, then the generalized equilibrium problem (GEP) is to find \(x^{\ast }\in C\) such that

$$ f \bigl(x^{\ast }, x \bigr)+h \bigl(x^{\ast }, x \bigr) \geq 0,\quad \forall x \in C. $$
(1.5)

We denote the solution set of generalized equilibrium problem (1.5) by \(\operatorname{GEP}(f, h)\). Note that this problem reduces to the equilibrium problem when the bifunction h is a zero mapping; this problem reduces to the mixed equilibrium problem when the bifunction \(h(x^{\ast }, x)=\varphi (x)-\varphi (x^{\ast })\), where \(\varphi : C \rightarrow R\cup \{+\infty \}\) is for proper lower semicontinuous and convex functions.

The split generalized equilibrium problem (SGEP) introduced by Kazmi and Rizvi [15] in 2013 is the following problem: find \(x^{\ast }\in C\)

$$ f \bigl(x^{\ast }, x \bigr)+h \bigl(x^{\ast }, x \bigr)\geq 0, \quad \forall x \in C, $$

such that

$$ y^{\ast }=Ax^{\ast }\in Q \quad \mbox{solves}\quad F \bigl(y^{\ast }, y \bigr)+H \bigl(y^{\ast }, y \bigr) \geq 0, \quad \forall y \in Q, $$

where \(f, h : C\times C\rightarrow R\) and \(F, H : Q\times Q\rightarrow R\) are four nonlinear bifunctions and \(A : \mathcal{H}_{1}\rightarrow \mathcal{H}_{2}\) is a bounded linear operator. The solution set of the split generalized equilibrium problem SGEP is denoted by

$$ \varOmega = \bigl\{ x^{\ast } \in \operatorname{GEP}(f, h) : Ax^{\ast } \in \operatorname{GEP}(F, H) \bigr\} . $$

If \(H=0\) and \(F=0\), then the split generalized equilibrium problem reduces to the generalized equilibrium problem considered by Cianciaruso et al. [16]; If \(h=0\) and \(H=0\), then the split generalized equilibrium problem reduces to the split equilibrium problem introduced in 2011 by Moudafi [17]; if \(h=\varphi (\cdot , \cdot )\) and \(H=\phi (\cdot , \cdot )\), where \(\varphi : C\rightarrow R\cup \{+\infty \}\) and \(\phi : Q\rightarrow R\cup \{+\infty \}\) are proper lower semicontinuous and convex functions, then the split generalized equilibrium problem reduces to the split mixed equilibrium problem (SEP).

In this paper, we are interested in finding the common solution for a finite family of the split generalized equilibrium problems, that is, find a \(x^{\ast }\in C\),

$$ f_{i} \bigl(x^{\ast }, x \bigr)+h_{i} \bigl(x^{\ast }, x \bigr)\geq 0, \quad \forall x \in C, $$

such that

$$ y^{\ast }=A_{i}x^{\ast }\in Q\quad \mbox{solves}\quad F_{i} \bigl(y^{\ast }, y \bigr)+H_{i} \bigl(y ^{\ast }, y \bigr)\geq 0, \quad \forall y \in Q, $$

where \(f_{i}, h_{i} : C\times C\rightarrow R\) and \(F_{i}, H_{i} : Q \times Q\rightarrow R\) are nonlinear bifunctions, \(A_{i} : \mathcal{H}_{1}\rightarrow \mathcal{H}_{2}\) is a bounded linear operator, for \(1\leq i\leq N_{1}\).

In 2017, Majee and Nahak [18] introduced a hybrid viscosity iterative method to approximate a common solution of a split equilibrium problem and a fixed point problem of a finite collection of nonexpansive mappings; Onjai-uea and Phuengrattana [19] studied iterative algorithms for solving split mixed equilibrium problems and fixed point problems of hybrid multivalued mappings in real Hilbert spaces; Sitthithakerngkiet et al. [20] proposed an iterative method for finding a common solution of a single split generalized equilibrium problem, variational inequality problem and fixed point problem of nonexpansive mapping in Hilbert spaces. For recent developments in the analysis technique and algorithm design, see [21,22,23,24,25] and the references therein.

Motivated by the above related results in this field, in this paper, we first propose a new parallel hybrid viscosity method for finding a common element of the set of solutions of a finite family of split generalized equilibrium problems, variational inequality problems and the set of common fixed points of a finite family of demicontractive operators in Hilbert spaces.

The rest of the paper is organized as follows: Sect. 2 describes several definitions and lemmas which will be used in proving our main results; Sect. 3 presents a new parallel hybrid viscosity method for finding a common element of the set of solutions of a finite family of split generalized equilibrium problems, variational inequality problems and the set of common fixed points of a finite family of demicontractive operators in Hilbert spaces, and establish the corresponding strong convergence theorem under suitable conditions. Section 4 gives numerical examples to demonstrate the convergence of our algorithm.

Preliminaries

Throughout the paper, let the symbol → and denote strong convergence and weak convergence, respectively. In addition, \(F(T)\) and \(\omega _{w}(x_{n})\) denote the fixed point set of T and the weak ω-limit set of the sequence \(\{x_{n}\}\), respectively, that is, \(F(T)=\{x: Tx=x\}\) and \(\omega _{w}(x_{n})=\{u: \exists x_{n_{j}}\rightharpoonup u\}\). In order to prove our main results, we recall some basic definitions and lemmas, which will be needed in the sequel.

Definition 2.1

([26])

Assume that \(T : \mathcal{H}\rightarrow \mathcal{H}\) is a nonlinear operator, then \(I-T\) is said to be demiclosed at zero if for any sequence \(\{x_{n}\}\) in \(\mathcal{H}\), the following implication holds:

$$ x_{n}\rightharpoonup x \quad \mbox{and}\quad (I-T)x_{n} \rightarrow 0 \quad \Rightarrow\quad x \in F(T). $$

Lemma 2.2

([27])

Let C be a nonempty closed convex subset of a real Hilbert space \(\mathcal{H}\), and let \(U : C\rightarrow C\) be a β-strict pseudo-contractive. Then \(I-U\) is demiclosed at 0.

Lemma 2.3

([28])

Suppose that \(U : \mathcal{H}\rightarrow \mathcal{H}\) is a β-demicontractive mapping. Then the fixed point set \(F(U)\) of U is closed and convex.

Recall that \(P_{C}\) is the metric projection from \(\mathcal{H}\) into C, then, for each point \(x\in \mathcal{H}\), the unique point \(P_{C}x \in C\) satisfies the property:

$$ \Vert x-P_{C}x \Vert =\inf_{y\in C} \Vert x-y \Vert =: d(x, C). $$

Lemma 2.4

([29])

For a given \(x\in \mathcal{H}\):

  1. (i)

    \(z=P_{C}x\) if and only if \(\langle x-z, z-y\rangle \geq 0\), \(\forall y\in C\);

  2. (ii)

    \(z=P_{C}x\) if and only if \(\|x-z\|^{2}\leq \|x-y\|^{2}- \|y-z\|^{2}\), \(\forall x, y\in C\);

  3. (iii)

    \(\langle P_{C}x-P_{C}y, x-y\rangle \geq \|P_{C}x-P_{C}y \|^{2}\), \(\forall x, y\in \mathcal{H}\).

It is obvious that \(P_{C}\) is nonexpansive and monotone.

Lemma 2.5

([30])

Let \(f: C \times C\rightarrow R\) be a bifunction satisfying the following assumptions:

  1. (i)

    \(f(x,x)\geq 0\), \(\forall x\in C\);

  2. (ii)

    f is monotone, that is, \(f(x,y)+f(y,x)\leq 0\), \(\forall x,y\in C\);

  3. (iii)

    f is upper hemicontinuous, that is, for each \(\forall x, y, z\in C\),

    $$ \limsup_{t\rightarrow 0}f \bigl(tz+(1-t)x \bigr),y)\leq f(x,y); $$
  4. (iv)

    For each \(x\in C\) fixed, the function \(y\mapsto f(x,y)\) is convex and lower semicontinuous.

Suppose that \(h: C \times C\rightarrow R\) is a bifunction satisfying the following assumptions:

  1. (i)

    \(h(x,x)\geq 0\), \(\forall x\in C\);

  2. (ii)

    for each \(y\in C\) fixed, the function \(x\mapsto h(x,y)\) is upper semicontinuous;

  3. (iii)

    for each \(x\in C\) fixed, the function \(y\mapsto h(x,y)\) is convex and lower semicontinuous.

Then, for fixed \(r>0\) and \(z\in C\), there exist a nonempty compact convex subset K of \(\mathcal{H}_{1}\) and \(x\in C \cap K\) such that

$$ f(y, x)+h(y, x)+\frac{1}{r}\langle y-x, x-z\rangle < 0, \quad \forall y \in C\setminus K. $$

Lemma 2.6

Assume that \(f, h: C\times C\rightarrow R\) satisfying Lemma 2.5. Let \(r>0\) and \(x\in \mathcal{H}_{1}\), Then there exists \(z\in C\) such that

$$ f(z,y)+h(z,y)+\frac{1}{r}\langle y-z,z-x\rangle \geq 0, \quad \forall y\in C. $$

Lemma 2.7

([31])

Assume that \(f, h : C\times C\rightarrow R\) satisfying Lemma 2.5 and h is monotone. For \(r>0\) and \(x\in \mathcal{H}_{1}\), define the mapping \(T_{r}^{f, h}: H_{1}\rightarrow C\) as follows:

$$ T_{r}^{f, h}(x):= \biggl\{ z\in C: f(z,y)+h(z,y)+ \frac{1}{r}\langle y-z,z-x \rangle \geq 0, \forall y\in C \biggr\} . $$

Then the following statements hold:

  1. (i)

    \(T_{r}^{f, h}\) is single-valued;

  2. (ii)

    \(T_{r}^{f, h}\) is firmly nonexpansive, that is,

    $$ \bigl\Vert T_{r}^{f, h}(x)-T_{r}^{f, h}(y) \bigr\Vert ^{2}\leq \bigl\langle T_{r}^{f, h}(x)-T _{r}^{f, h}(y),x-y \bigr\rangle , \quad \forall x, y \in \mathcal{H}_{1}; $$
  3. (iii)

    \(F(T_{r}^{f, h})=\operatorname{GEP}(f, h)\);

  4. (iv)

    \(\operatorname{GEP}(f, h)\) is compact and convex.

Let \(F, H : Q\times Q\rightarrow R\) satisfying Lemma 2.5. From the previous lemma, we can define a mapping \(T_{s}^{F, H}: H_{2} \rightarrow Q\) as follows:

$$ T_{s}^{F, H}(w):= \biggl\{ d\in Q: F(d,e)+H(d,e)+ \frac{1}{s}\langle e-d,d-w \rangle \geq 0,\forall e\in Q \biggr\} , $$

where \(s>0\) and \(w\in \mathcal{H}_{2}\), Then \(T_{s}^{F, H}: \mathcal{H}_{2}\rightarrow Q\) also satisfies the same properties in Lemma 2.7. Further, it is easy to prove that Ω is a closed and convex set. We see that Lemma 3.5 in [16] is a special case of Lemmas 2.6 and 2.7; for more details see [32].

Lemma 2.8

([33])

Let \(\{B_{i}\}_{i=1}^{N}\) be a finite family of inverse strongly monotone mappings from C to \(\mathcal{H}\) with the constants \(\{\beta _{i}\}_{i=1}^{N}\) and assume that \(\bigcap_{i=1}^{N}\operatorname{VI}(C,B _{i})\neq \emptyset \). Let \(B=\sum_{i=1}^{N}\alpha _{i} B_{i}\), \(\{\alpha _{i}\}_{i=1}^{N}\subset (0, 1)\) and \(\sum_{i=1}^{N}\alpha _{i}=1\). Then \(B : C\rightarrow \mathcal{H}\) is a β-inverse strongly monotone mapping with \(\beta =\min \{\beta _{1}, \ldots ,\beta _{N}\}\) and \(\operatorname{VI}(C,B)=\bigcap_{i=1}^{N}\operatorname{VI}(C,B_{i})\).

A linear bounded operator \(A : \mathcal{H}\rightarrow \mathcal{H}\) is called strongly positive if and only if there exists \(\overline{\gamma }>0\) such that \(\langle Ax, x\rangle \geq \overline{ \gamma }\|x\|^{2}\) for all \(x\in \mathcal{H}\). and we call such an A a strongly positive operator with coefficient γ̅.

Lemma 2.9

([34])

Let \(\mathcal{H}\) be a Hilbert space and let A be a strongly positive bounded linear operator on \(\mathcal{H}\) with coefficient \(\overline{ \gamma }>0\). If \(0<\delta \leq \|A\|^{-1}\), then \(\|I-\delta A\| \leq 1-\delta \overline{\gamma }\).

Lemma 2.10

([18])

Let \(\mathcal{H}\) be a Hilbert space. Let \(f : C \rightarrow C\) be a ρ-Lipschitzian mapping and \(A : \mathcal{H} \rightarrow \mathcal{H}\) be a strongly positive bounded linear operator with coefficient \(\delta > 0\). If \(\mu \delta > \eta \rho \), then

$$ \bigl\langle (\mu A-\eta f)x-(\mu A-\eta f)y, x-y \bigr\rangle \geq (\mu \delta - \eta \rho ) \Vert x-y \Vert ^{2} $$

for all \(x, y\in \mathcal{H}\). That is, \(\mu A-\eta f\) is strongly monotone with coefficient \(\mu \delta -\eta \rho \).

Lemma 2.11

([35])

The following inequality holds in a Hilbert space \(\mathcal{H}\):

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

Lemma 2.12

([36])

For each \(x_{1}, \ldots , x_{m}\in \mathcal{H}\) and \(\alpha _{1}, \ldots , \alpha _{m}\in [0, 1]\) with \(\sum_{i=1}^{n}\alpha _{i}=1\), we have the equality

$$ \Vert \alpha _{1} x_{1}+\cdots +\alpha _{m} x_{m} \Vert ^{2}=\sum_{i=1}^{m} \alpha _{i} \Vert x_{i} \Vert ^{2}-\sum _{1\leq i< j\leq m}\alpha _{i}\alpha _{j} \Vert x_{i}-x _{j} \Vert ^{2}. $$

Lemma 2.13

([37])

Let \(\{a_{n}\}\) be a sequence of non-negative real numbers, such that there exists a subsequence \(\{a_{n_{j}}\}\) of \(\{a_{n}\}\), such that \(a_{n_{j}}< a_{n_{j}+1}\) for all \(j\in N\). Then there exists a nondecreasing sequence \(\{m_{k}\}\) of N, such that \(\lim_{k\rightarrow \infty }m_{k}=\infty \), and the following properties are satisfied by all (sufficiently large) numbers \(k\in N\):

$$ a_{m_{k}}\leq a_{m_{k}+1} \quad \textit{and}\quad a_{k} \leq a_{m_{k}+1}. $$

In fact, \(m_{k}\) is the largest number n in the set \(\{1, 2, \ldots , k\}\), such that \(a_{n}\leq a_{n+1}\).

Lemma 2.14

([38])

Let \(\{s_{n}\}\) be a sequence of non-negative real numbers satisfying

$$ s_{n+1}\leq (1-\alpha _{n})s_{n}+\alpha _{n}\beta _{n}+\gamma _{n}, \quad n\geq 0, $$

where \(\{\alpha _{n}\}\), \(\{\beta _{n}\}\) and \(\{\gamma _{n}\}\) satisfy the conditions:

  1. (i)

    \(\{\alpha _{n}\}\subset [0, 1]\), \(\sum_{n=1}^{\infty } \alpha _{n}=\infty \), or equivalently, \(\prod_{n=1}^{\infty }(1-\alpha _{n})=0\);

  2. (ii)

    \(\limsup_{n\rightarrow \infty }\beta _{n}\leq 0\);

  3. (iii)

    \(\gamma _{n}\geq 0\) (\(n\geq 0\)), \(\sum_{n=1}^{\infty } \gamma _{n}<\infty \).

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

Main results

Theorem 3.1

Let \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) be two real Hilbert space. Let C and Q be nonempty, closed and convex subsets of \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\), respectively. Let \(A_{i} : \mathcal{H}_{1}\rightarrow \mathcal{H}_{2}\) be a bounded linear operator and \(A_{i}^{\ast } : \mathcal{H}_{2} \rightarrow \mathcal{H}_{1}\) be the adjoint of \(A_{i}\). Assume that \(f_{i}, h_{i} : C\times C \rightarrow R\) and \(F_{i}, H_{i} : Q\times Q\rightarrow R\) are bifunctions satisfying Lemma 2.5; \(h_{i}\), \(H_{i}\) are monotone and \(F_{i}\) is upper semicontinuous for \(1\leq i\leq N_{1}\). Let \(S_{j}: C\rightarrow C\) be a \(\kappa _{j}\)-demicontractive mappings such that \(S_{j}-I\) is demiclosed at 0 for all \(1 \leq j \leq N_{2}\). \(B_{l} : C \rightarrow \mathcal{H}_{1}\) is a \(\sigma _{l}\)-inverse strongly monotone operator for all \(1 \leq l\leq N_{3}\). Suppose that \(\varGamma =\bigcap_{i=1}^{N_{1}}\varOmega _{i}\cap (\bigcap_{j=1}^{N_{2}}F(S _{j}))\cap (\bigcap_{l=1}^{N_{3}}\operatorname{VI}(C, B_{l}))\neq \emptyset \). Let \(f : \mathcal{H}_{1}\rightarrow \mathcal{H}_{1}\) be a Lipschitzian mapping with coefficient \(\rho \geq 0\). Let \(L : \mathcal{H}_{1} \rightarrow \mathcal{H}_{1}\) be a strongly positive bounded linear operator with coefficient \(\delta > 0\). Let \(\{x_{n}\}\) be a sequence generated from an arbitrary \(x_{1}\in \mathcal{H}_{1}\) by the following algorithm:

$$ \textstyle\begin{cases} \omega _{n, i}=T_{r_{n, i}}^{f_{i}, h_{i}}(x_{n}+\xi _{n, i}A_{i}^{ \ast }(T_{r_{n, i}}^{F_{i}, H_{i}}-I)A_{i}x_{n}), \\ \omega _{n}=\omega _{n, i_{n}},\quad i_{n}=\arg \max_{1\leq i\leq N_{1}}\{ \Vert \omega _{n, i}-x_{n} \Vert \}, \\ z_{n, j}=\beta _{n, j}\omega _{n}+(1-\beta _{n, j})S_{j}\omega _{n}, \\ y_{n}=P_{C}(I-\lambda _{n}(\sum_{l=1}^{N_{3}}\mu _{l}B_{l}))(\sum_{j=1} ^{N_{2}}\nu _{n, j}z_{n, j}), \\ x_{n+1}=\alpha _{n}\gamma f(x_{n})+(I-\alpha _{n} \eta L)y_{n}. \end{cases} $$
(3.1)

Also, the following conditions are satisfied:

  1. (i)

    \(\eta \delta >\gamma \rho \);

  2. (ii)

    \(\mu _{l}\in (0, 1)\), \(\sum_{l=1}^{N_{3}}\mu _{l}=1\);

  3. (iii)

    \(\{\alpha _{n}\}\subset (0, 1)\), \(\lim_{n\rightarrow \infty }\alpha _{n}=0\), \(\sum_{n=1}^{\infty }\alpha _{n}=\infty \);

  4. (iv)

    \(r_{n, i}\subset (0, \infty )\), \(\liminf_{n\rightarrow \infty }r_{n, i}>0\);

  5. (v)

    \(\{\beta _{n, j}\}\subset (0, 1)\), \(\{\nu _{n, j}\}\subset (0, 1)\), \(\sum_{j=1}^{N_{2}}\nu _{n, j}=1\), for \(\forall n\geq 1\). \(\liminf_{n\rightarrow \infty }\nu _{n, j}(1-\beta _{n, j})(\beta _{n, j}- \kappa _{j})>0\) for \(\forall j\in \{1, \ldots , N_{2}\}\);

  6. (vi)

    \(0<\liminf_{n\rightarrow \infty }\xi _{n, i}\leq \limsup_{n\rightarrow \infty }\xi _{n, i}<\frac{2}{\|A\|^{2}}\) for \(1\leq i\leq N_{1}\);

  7. (vii)

    \(0<\liminf_{n\rightarrow \infty }\lambda _{n}\leq \limsup_{n\rightarrow \infty }\lambda _{n}<2\sigma \), \(\sigma = \max_{1\leq l\leq N_{3}}\{\sigma _{l}\}\).

Then \(\{x_{n}\}\) converges strongly to a point \(x^{\ast }\in \varGamma \), which is the unique solution of the following variational inequality:

$$ \bigl\langle (\eta L-\gamma f)x^{\ast }, x^{\ast }-x \bigr\rangle \leq 0, \quad \forall x\in \varGamma . $$
(3.2)

Equivalently, we have \(x^{\ast }=P_{\varGamma }(I-\eta L+\gamma f)x^{ \ast }\).

Proof

First, we show the uniqueness of the solution of the variational inequality(3.2). We show it by contradiction. Suppose \(\hat{x}\in \varGamma \) and \(\tilde{x}\in \varGamma \) be two solution of (3.2) with \(\hat{x}\neq \tilde{x}\). Then we have

$$ \bigl\langle (\eta L-\gamma f)\hat{x}, \hat{x}-\tilde{x} \bigr\rangle \leq 0 $$

and

$$ \bigl\langle (\eta L-\gamma f)\tilde{x}, \tilde{x}-\hat{x} \bigr\rangle \leq 0. $$

We can obtain

$$ \bigl\langle (\eta L-\gamma f)\hat{x}-(\eta L-\gamma f)\tilde{x}, \hat{x}- \tilde{x} \bigr\rangle \leq 0. $$

From \(\eta \delta >\gamma \rho \) and Lemma 2.9, we can get

$$ \bigl\langle (\eta L-\gamma f)\hat{x}-(\eta L-\gamma f)\tilde{x}, \hat{x}- \tilde{x} \bigr\rangle \geq (\eta \delta -\gamma \rho ) \Vert \hat{x}-\tilde{x} \Vert ^{2}\geq 0. $$

This leads to a contradiction. Hence, the variational inequality problem (3.2) has a unique solution and we denote it by \(x^{\ast } \in \varGamma \).

We have

$$ \bigl\langle (\eta L-\gamma f)x^{\ast }, x^{\ast }-x \bigr\rangle \leq 0 \quad \Leftrightarrow\quad \bigl\langle \hat{x}-(I-\eta L+\gamma f)x^{\ast }, x^{\ast }-x \bigr\rangle \leq 0,\quad \forall x\in \varGamma . $$

It is easy to verify Γ is closed and convex. From Lemma 2.4, we obtain \(x^{\ast }=P_{C}(I-\eta L+\gamma f)x^{\ast }\).

Next, we show that the sequence \(\{x_{n}\}\) is bounded. Let \(p\in \varGamma \), that is, \(p\in \varOmega _{i}\), for \(\forall i\in \{1, 2, \ldots , N_{1}\}\), and we have \(p=T_{r_{n, i}}^{f_{i}, h_{i}}p\) and \(A_{i}p=T_{r_{n, i}}^{F_{i}, H_{i}}A_{i}p\). Observe that

$$\begin{aligned} \Vert \omega _{n, i}-p \Vert ^{2} =& \bigl\Vert T_{r_{n, i}}^{f_{i}, h_{i}} \bigl(x_{n}+\xi _{n, i}A_{i}^{\ast } \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr)-p \bigr\Vert ^{2} \\ \leq & \bigl\Vert x_{n}+\xi _{n, i}A_{i}^{\ast } \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A _{i}x_{n}-p \bigr\Vert ^{2} \\ =& \Vert x_{n}-p \Vert ^{2}+2\xi _{n, i} \bigl\langle A_{i}^{\ast } \bigl(T_{r_{n, i}}^{F _{i}, H_{i}}-I \bigr)A_{i}x_{n}, x_{n}-p \bigr\rangle \\ &{}+\xi _{n, i}^{2} \bigl\Vert A_{i}^{\ast } \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A _{i}x_{n} \bigr\Vert ^{2} \\ \leq & \Vert x_{n}-p \Vert ^{2}+2\xi _{n, i} \bigl\langle \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A _{i}x_{n}, A_{i}x_{n}-A_{i}p \bigr\rangle \\ &{}+\xi _{n, i}^{2} \Vert A_{i} \Vert ^{2} \bigl\Vert \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A _{i}x_{n} \bigr\Vert ^{2}. \end{aligned}$$

Denoting \(\varLambda =2\xi _{n, i}\langle (T_{r_{n, i}}^{F_{i}, H_{i}}-I)A _{i}x_{n}, A_{i}x_{n}-A_{i}p\rangle \) and using (1.1), we get

$$\begin{aligned} \varLambda =&2\xi _{n, i} \bigl\langle \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x _{n}, A_{i}x_{n}-A_{i}p \bigr\rangle \\ =&2\xi _{n, i} \bigl\langle \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n}, A _{i}x_{n}-A_{i}p \\ &{}+ \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n}- \bigl(T_{r_{n, i}}^{F_{i}, H _{i}}-I \bigr)A_{i}x_{n} \bigr\rangle \\ =&2\xi _{n, i} \bigl\{ \bigl\langle \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n}, T _{r_{n, i}}^{F_{i}, H_{i}}A_{i}x_{n}-A_{i}p \bigr\rangle - \bigl\Vert \bigl(T_{r_{n, i}} ^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert ^{2} \bigr\} \\ \leq &2\xi _{n, i} \biggl\{ \frac{1}{2} \bigl\Vert \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x _{n} \bigr\Vert ^{2}- \bigl\Vert \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert ^{2} \biggr\} \\ \leq &-\xi _{n, i} \bigl\Vert \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert ^{2}. \end{aligned}$$

Therefore, for \(\forall i\in \{1, 2, \ldots , N_{1}\}\), we have

$$ \Vert \omega _{n, i}-p \Vert ^{2} \leq \Vert x_{n}-p \Vert ^{2}+ \bigl(\xi _{n, i}^{2} \Vert A_{i} \Vert ^{2}-\xi _{n, i} \bigr) \bigl\Vert \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert ^{2}. $$
(3.3)

From the condition (v), we obtain

$$ \Vert \omega _{n}-p \Vert = \Vert \omega _{n, i_{n}}-p \Vert \leq \Vert x_{n}-p \Vert . $$
(3.4)

From Lemma 2.12 and the definition of \(S_{j}\), we have

$$\begin{aligned} \Vert z_{n, j}-p \Vert ^{2} =& \bigl\Vert \beta _{n, j}\omega _{n}+(1-\beta _{n, j})S_{j} \omega _{n}-p \bigr\Vert ^{2} \\ =&\beta _{n, j} \Vert \omega _{n}-p \Vert ^{2}+(1-\beta _{n, j}) \Vert S_{j}\omega _{n}-p \Vert ^{2} \\ &{}-\beta _{n, j}(1-\beta _{n, j}) \Vert \omega _{n}-S_{j}\omega _{n} \Vert ^{2} \\ \leq &\beta _{n, j} \Vert \omega _{n}-p \Vert ^{2}+(1-\beta _{n, j}) \bigl[ \Vert \omega _{n}-p \Vert ^{2}+\kappa _{j} \Vert \omega _{n}-S_{j} \omega _{n} \Vert ^{2} \bigr] \\ &{}-\beta _{n, j}(1-\beta _{n, j}) \Vert \omega _{n}-S_{j}\omega _{n} \Vert ^{2} \\ =& \Vert \omega _{n}-p \Vert ^{2}+(1-\beta _{n, j}) (\kappa _{j}-\beta _{n, j}) \Vert \omega _{n}-S_{j}\omega _{n} \Vert ^{2}. \end{aligned}$$

It follows from the condition (iv) that

$$ \Vert z_{n, j}-p \Vert \leq \Vert \omega _{n}-p \Vert \leq \Vert x_{n}-p \Vert , \quad \forall j \in \{1, 2, \ldots , N_{2}\}. $$
(3.5)

Let \(B=\sum_{l=1}^{N_{3}}\mu _{l}B_{l}\) and \(\sigma =\min \{\sigma _{1}, \ldots , \sigma _{N_{3}}\}\), by Lemma 2.8, we know that B is σ-ism, and from the condition \(0<\lambda _{n}<2\sigma \), we see that \(I-\lambda _{n}B\) is nonexpansive, and \(P_{C}(I-\lambda _{n}B)\) is also nonexpansive. We have \(p\in \varGamma \), that is, \(p\in \bigcap_{l=1} ^{N_{3}}\operatorname{VI}(C, B_{l})=\operatorname{VI}(C, B)\). Then from Lemma 2.12, we have

$$\begin{aligned} \Vert y_{n}-p \Vert ^{2} =& \Biggl\Vert P_{C}(I-\lambda _{n}B) \Biggl(\sum _{j=1}^{N_{2}}\nu _{n, j}z _{n, j} \Biggr)-p \Biggr\Vert ^{2} \\ =& \Biggl\Vert P_{C}(I-\lambda _{n}B) \Biggl(\sum _{j=1}^{N_{2}}\nu _{n, j}z_{n, j} \Biggr)-P _{C}(I-\lambda _{n}B)p \Biggr\Vert ^{2} \\ \leq & \Biggl\Vert \sum_{j=1}^{N_{2}}\nu _{n, j}z_{n, j}-p \Biggr\Vert ^{2} \\ \leq &\sum_{j=1}^{N_{2}}\nu _{n, j} \Vert z_{n, j}-p \Vert ^{2} \\ \leq & \Vert x_{n}-p \Vert ^{2}. \end{aligned}$$
(3.6)

From the condition \(\lim_{n\rightarrow \infty }\alpha _{n}=0\), we may assume, with no loss of generality, that \(\alpha _{n} < \frac{1}{ \eta \|L\|}\) for all n. It follows from Lemma 2.9 that

$$\begin{aligned} \Vert x_{n+1}-p \Vert =& \bigl\Vert \alpha _{n}\gamma f(x_{n})+(I-\alpha _{n}\eta L)y_{n}-p \bigr\Vert \\ =& \bigl\Vert \alpha _{n} \bigl(\gamma f(x_{n})-\eta Lp \bigr)+(I-\alpha _{n}\eta L) (y_{n}-p) \bigr\Vert \\ \leq &\alpha _{n} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert + \Vert I-\alpha _{n}\eta L \Vert \Vert y _{n}-p \Vert \\ =&\alpha _{n} \bigl\Vert \gamma f(x_{n})-\gamma f(p)+ \gamma f(p)-\eta Lp \bigr\Vert + \Vert 1- \alpha _{n}\eta L \Vert \Vert y_{n}-p \Vert \\ \leq &\alpha _{n}\gamma \rho \Vert x_{n}-p \Vert + \alpha _{n} \bigl\Vert \gamma f(p)- \eta Lp \bigr\Vert +(1-\alpha _{n}\eta \delta ) \Vert x_{n}-p \Vert \\ =& \bigl[1-\alpha _{n}(\eta \delta -\gamma \rho ) \bigr] \Vert x_{n}-p \Vert +\alpha _{n} \bigl\Vert \gamma f(p)-\eta Lp \bigr\Vert \\ =& \bigl[1-\alpha _{n}(\eta \delta -\gamma \rho ) \bigr] \Vert x_{n}-p \Vert +\alpha _{n}( \eta \delta -\gamma \rho ) \frac{ \Vert \gamma f(p)-\eta Lp \Vert }{\eta \delta -\gamma \rho } \\ \leq &\max \biggl\{ \Vert x_{n}-p \Vert , \frac{ \Vert \gamma f(p)-\eta Lp \Vert }{\eta \delta -\gamma \rho } \biggr\} \\ \leq &\cdots \\ \leq &\max \biggl\{ \Vert x_{1}-p \Vert , \frac{ \Vert \gamma f(p)-\eta Lp \Vert }{\eta \delta -\gamma \rho } \biggr\} . \end{aligned}$$

That is, \(\{x_{n}\}\) is bounded, and \(\{y_{n}\}\), \(\{z_{n, j}\}\), \(\{\omega _{n, i}\}\), \(\{f(x_{n})\}\) and \(\{S_{j}\omega _{n}\}\) are also bounded.

Next, we show \(\omega _{\omega }(x_{n})\subseteq \varGamma \). To see this, we take \(q \in \omega _{\omega }(x_{n})\) and assume that \(x_{n_{l}} \rightharpoonup q\) as \(l\rightarrow \infty \) for some subsequence \(\{x_{n_{l}}\}\) of \(\{x_{n}\}\). Observe that

$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} =& \bigl\Vert \alpha _{n}\gamma f(x_{n})+(I-\alpha _{n}\eta L)y _{n}-p \bigr\Vert ^{2} \\ =& \bigl\Vert \alpha _{n} \bigl(\gamma f(x_{n})-\eta Lp \bigr)+(I-\alpha _{n}\eta L) (y_{n}-p) \bigr\Vert ^{2} \\ =&\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})- \eta Lp \bigr\Vert ^{2}+ \bigl\Vert (I-\alpha _{n} \eta L) (y_{n}-p) \bigr\Vert ^{2} \\ &{}+2\alpha _{n} \bigl\langle \gamma f(x_{n})-\eta Lp, (I- \alpha _{n}\eta L) (y _{n}-p) \bigr\rangle \\ \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n} \eta \delta )^{2} \Vert y_{n}-p \Vert ^{2} \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert y_{n}-p \Vert \\ \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n} \eta \delta )^{2} \Vert y_{n}-p \Vert ^{2} \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert . \end{aligned}$$
(3.7)

From (3.4), (3.5) and (3.6), we obtain

$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1- \alpha _{n}\eta \delta )^{2}\sum _{j=1}^{N_{2}}\nu _{n, j} \Vert z_{n, j}-p \Vert ^{2} \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert \\ \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n} \eta \delta )^{2}\sum_{j=1}^{N_{2}}\nu _{n, j} \bigl[ \Vert \omega _{n}-p \Vert ^{2} \\ &{}+(1-\beta _{n, j}) (\kappa _{j}-\beta _{n, j}) \Vert \omega _{n}-S_{j}\omega _{n} \Vert ^{2} \bigr] \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert \\ \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n} \eta \delta )^{2}\sum_{j=1}^{N_{2}}\nu _{n, j} \bigl[ \Vert x_{n}-p \Vert ^{2} \\ &{}+ \bigl(\xi _{n, i_{n}}^{2} \Vert A_{i_{n}} \Vert ^{2}-\xi _{n, i_{n}} \bigr) \bigl\Vert \bigl(T_{r_{n, i _{n}}}^{F_{i_{n}}, H_{i_{n}}}-I \bigr)A_{i_{n}}x_{n} \bigr\Vert ^{2} \\ &{}+(1-\beta _{n, j}) (\kappa _{j}-\beta _{n, j}) \Vert \omega _{n}-S_{j}\omega _{n} \Vert ^{2} \bigr] \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert \\ =&\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})- \eta Lp \bigr\Vert ^{2}+(1-\alpha _{n}\eta \delta )^{2} \Vert x_{n}-p \Vert ^{2} \\ &{}+(1-\alpha _{n}\eta \delta )^{2} \bigl(\xi _{n, i_{n}}^{2} \Vert A_{i_{n}} \Vert ^{2}-\xi _{n, i_{n}} \bigr) \bigl\Vert \bigl(T_{r_{n, i_{n}}}^{F_{i_{n}}, H_{i_{n}}}-I \bigr)A _{i_{n}}x_{n} \bigr\Vert ^{2} \\ &{}+(1-\alpha _{n}\eta \delta )^{2}\sum _{j=1}^{N_{2}}\nu _{n, j}(1- \beta _{n, j}) (\kappa _{j}-\beta _{n, j}) \Vert \omega _{n}-S_{j}\omega _{n} \Vert ^{2} \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert . \end{aligned}$$
(3.8)

Then we have

$$\begin{aligned} (1-\alpha _{n}\eta \delta )^{2} \bigl(\xi _{n, i_{n}}- \xi _{n, i_{n}}^{2} \Vert A _{i_{n}} \Vert ^{2} \bigr) \bigl\Vert \bigl(T_{r_{n, i_{n}}}^{F_{i_{n}}, H_{i_{n}}}-I \bigr)A_{i_{n}}x _{n} \bigr\Vert ^{2} \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2} \\ &{}- \Vert x_{n+1}-p \Vert ^{2} \\ &{}+(1-\alpha _{n}\eta \delta )^{2} \Vert x_{n}-p \Vert ^{2} \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \\ &{}\times \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \\ &{}\times \Vert x_{n}-p \Vert \end{aligned}$$
(3.9)

and

$$\begin{aligned} (1-\alpha _{n}\eta \delta )^{2}\sum _{j=1}^{N_{2}}\nu _{n, j}(1-\beta _{n, j}) (\beta _{n, j}-\kappa _{j}) \Vert \omega _{n}-S_{j}\omega _{n} \Vert ^{2} \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})- \eta Lp \bigr\Vert ^{2} \\ &{}- \Vert x_{n+1}-p \Vert ^{2} \\ &{}+(1-\alpha _{n}\eta \delta )^{2} \Vert x_{n}-p \Vert ^{2} \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \\ &{}\times \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \\ &{}\times \Vert x_{n}-p \Vert . \end{aligned}$$
(3.10)

Next, we analyze the inequality (3.9) and (3.10) by considering the following two cases.

Case 1. Assume that there exists \(n_{0}\) large enough such that \(\|x_{n+1}-p\|^{2}\leq \|x_{n}-p\|^{2}\) for all \(n\geq n_{0}\). Since \(\|x_{n}-p\|^{2}\) is bounded, we see that \(\lim_{n\rightarrow \infty }\|x_{n}-p\|^{2}\) exists. From the conditions (iii) and (v), we obtain

$$ \bigl\Vert \bigl(T_{r_{n, i_{n}}}^{F_{i_{n}}, H_{i_{n}}}-I \bigr)A_{i_{n}}x_{n} \bigr\Vert \rightarrow 0 \quad (n\rightarrow \infty ) $$

and

$$ \Vert \omega _{n}-S_{j}\omega _{n} \Vert \rightarrow 0 \quad (n\rightarrow \infty ), \forall j\in \{1, 2, \ldots , N_{2}\}. $$

Then we have

$$ \Vert z_{n, j}-\omega _{n} \Vert =(1-\beta _{n, j}) \Vert S_{j}\omega _{n}-\omega _{n} \Vert \rightarrow 0\quad (n\rightarrow \infty ), \forall j\in \{1, 2, \ldots , N_{2}\}. $$
(3.11)

Since \(p=T_{r_{n, i}}^{f_{i}, h_{i}}p\) and \(T_{r_{n, i}}^{f_{i}, h _{i}}\) is firmly nonexpansive, we obtain

$$\begin{aligned} \Vert \omega _{n, i}-p \Vert ^{2} =& \bigl\Vert T_{r_{n, i}}^{f_{i}, h_{i}} \bigl(x_{n}+\xi _{n, i}A_{i}^{\ast } \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr)-p \bigr\Vert ^{2} \\ =& \bigl\Vert T_{r_{n, i}}^{f_{i}, h_{i}} \bigl(x_{n}+\xi _{n, i}A_{i}^{\ast } \bigl(T_{r _{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr)-T_{r_{n, i}}^{f_{i}, h_{i}}p \bigr\Vert ^{2} \\ \leq & \bigl\langle \omega _{n, i}-p, x_{n}+\xi _{n, i}A_{i}^{\ast } \bigl(T_{r _{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n}-p \bigr\rangle \\ =&\frac{1}{2} \bigl\{ \Vert \omega _{n, i}-p \Vert ^{2}+ \bigl\Vert x_{n}+\xi _{n, i}A_{i}^{ \ast } \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n}-p \bigr\Vert ^{2} \\ &{}- \bigl\Vert \omega _{n, i}-x_{n}-\xi _{n, i}A_{i}^{\ast } \bigl(T_{r_{n, i}}^{F _{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert ^{2} \bigr\} \\ =&\frac{1}{2} \bigl\{ \Vert \omega _{n, i}-p \Vert ^{2}+ \Vert x_{n}-p \Vert ^{2}+\xi _{n, i} ^{2} \bigl\Vert A_{i}^{\ast } \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert ^{2} \\ &{}+2\xi _{n, i} \bigl\langle x_{n}-p, A_{i}^{\ast } \bigl(T_{r_{n, i}}^{F_{i}, H _{i}}-I \bigr)A_{i}x_{n} \bigr\rangle \\ &{}- \bigl[ \Vert \omega _{n, i}-x_{n} \Vert ^{2}+\xi _{n, i}^{2} \bigl\Vert A_{i}^{\ast } \bigl(T_{r _{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert ^{2} \\ &{}-2\xi _{n, i} \bigl\langle \omega _{n, i}-x_{n}, A_{i}^{\ast } \bigl(T_{r_{n, i}} ^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\rangle \bigr] \bigr\} \\ =&\frac{1}{2} \bigl\{ \Vert \omega _{n, i}-p \Vert ^{2}+ \Vert x_{n}-p \Vert ^{2}- \Vert \omega _{n, i}-x _{n} \Vert ^{2} \\ &{}+2\xi _{n, i} \bigl\langle A_{i}\omega _{n, i}-A_{i}p, \bigl(T_{r_{n, i}}^{F _{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\rangle \bigr\} \\ \leq &\frac{1}{2} \bigl\{ \Vert \omega _{n, i}-p \Vert ^{2}+ \Vert x_{n}-p \Vert ^{2}- \Vert \omega _{n, i}-x_{n} \Vert ^{2} \\ &{}+2\xi _{n, i} \Vert A_{i}\omega _{n, i}-A_{i}p \Vert \bigl\Vert \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert \bigr\} . \end{aligned}$$

Then we get

$$ \Vert \omega _{n, i}-p \Vert ^{2}\leq \Vert x_{n}-p \Vert ^{2}- \Vert \omega _{n, i}-x_{n} \Vert ^{2}+2\xi _{n, i} \Vert A_{i}\omega _{n, i}-A_{i}p \Vert \bigl\Vert \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert . $$

From (3.8), we have

$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1- \alpha _{n}\eta \delta )^{2}\sum _{j=1}^{N_{2}}\nu _{n, j} \bigl[ \Vert \omega _{n}-p \Vert ^{2} \\ &{}+(1-\beta _{n, j}) (\kappa _{j}-\beta _{n, j}) \Vert \omega _{n}-S_{j}\omega _{n} \Vert ^{2} \bigr] \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert \\ \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n} \eta \delta )^{2} \Vert \omega _{n}-p \Vert ^{2} \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert \\ \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n} \eta \delta )^{2} \bigl[ \Vert x_{n}-p \Vert ^{2}- \Vert \omega _{n}-x_{n} \Vert ^{2} \\ &{}+2\xi _{n, i_{n}} \Vert A_{i_{n}}\omega _{n}-A_{i_{n}}p \Vert \bigl\Vert \bigl(T_{r_{n, i _{n}}}^{F_{i_{n}}, H_{i_{n}}}-I \bigr)A_{i_{n}}x_{n} \bigr\Vert \bigr] \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert . \end{aligned}$$

Then

$$\begin{aligned} (1-\alpha _{n}\eta \delta )^{2} \Vert \omega _{n}-x_{n} \Vert ^{2} \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n}\eta \delta )^{2} \Vert x_{n}-p \Vert ^{2} \\ &{}- \Vert x_{n+1}-p \Vert ^{2}+2(1-\alpha _{n}\eta \delta )^{2}\xi _{n, i_{n}} \\ &{}\times \Vert A_{i_{n}}\omega _{n}-A_{i_{n}}p \Vert \bigl\Vert \bigl(T_{r_{n, i_{n}}}^{F_{i_{n}}, H_{i_{n}}}-I \bigr)A_{i_{n}}x_{n} \bigr\Vert \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert . \end{aligned}$$

Since \(\alpha _{n}\rightarrow 0\), \(\|(T_{r_{n, i_{n}}}^{F_{i_{n}}, H _{i_{n}}}-I)A_{i_{n}}x_{n}\|\rightarrow 0\), as \(n\rightarrow \infty \), and \(\lim_{n\rightarrow \infty }\|x_{n}-p\|\) exists, we obtain

$$ \Vert \omega _{n}-x_{n} \Vert \rightarrow 0 \quad (n \rightarrow \infty ). $$
(3.12)

From (3.11), we have

$$ \Vert z_{n, j}-x_{n} \Vert \rightarrow 0\quad (n \rightarrow \infty ). $$

Let \(z_{n}=\sum_{j=1}^{N_{2}}\nu _{n, j}z_{n, j}\), then \(y_{n}=P_{C}(I- \lambda _{n}B)z_{n}\), and we have

$$\begin{aligned} \Vert z_{n}-x_{n} \Vert \leq & \Biggl\Vert \sum _{j=1}^{N_{2}}\nu _{n, j}z_{n, j}-x_{n} \Biggr\Vert \\ \leq &\sum_{j=1}^{N_{2}}\nu _{n, j} \Vert z_{n, j}-x_{n} \Vert \\ \rightarrow &0\quad (n\rightarrow \infty ). \end{aligned}$$
(3.13)

Since B is σ-ism, we obtain

$$\begin{aligned} \Vert y_{n}-p \Vert ^{2} =& \bigl\Vert P_{C}(I-\lambda _{n}B)z_{n}-p \bigr\Vert ^{2} \\ =& \bigl\Vert P_{C}(I-\lambda _{n}B)z_{n}-P_{C}(I- \lambda _{n}B)p \bigr\Vert ^{2} \\ \leq & \bigl\Vert (I-\lambda _{n}B)z_{n}-(I-\lambda _{n}B)p \bigr\Vert ^{2} \\ =& \Vert z_{n}-p \Vert ^{2}+\lambda _{n}^{2} \Vert Bz_{n}-Bp \Vert ^{2}-2 \lambda _{n} \langle z_{n}-p, Bz_{n}-Bp\rangle \\ \leq & \Vert z_{n}-p \Vert ^{2}+ \bigl(\lambda _{n}^{2}-2\lambda _{n}\sigma \bigr) \Vert Bz_{n}-Bp \Vert ^{2}. \end{aligned}$$

From (3.7), we have

$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1- \alpha _{n}\eta \delta )^{2} \Vert y_{n}-p \Vert ^{2} \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert \\ \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n} \eta \delta )^{2} \bigl[ \Vert z_{n}-p \Vert ^{2} \\ &{}+ \bigl(\lambda _{n}^{2}-2\lambda _{n}\sigma \bigr) \Vert Bz_{n}-Bp \Vert ^{2} \bigr] \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert . \end{aligned}$$

Then

$$\begin{aligned} (1-\alpha _{n}\eta \delta )^{2} \bigl(2\lambda _{n} \sigma -\lambda _{n}^{2} \bigr) \Vert Bz _{n}-Bp \Vert ^{2} \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2} \\ &{}+(1-\alpha _{n}\eta \delta )^{2} \Vert z_{n}-p \Vert ^{2} \\ &{}- \Vert x_{n+1}-p \Vert ^{2}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \\ &{}\times \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert \\ \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2} \\ &{}+(1-\alpha _{n}\eta \delta )^{2} \Vert x_{n}-p \Vert ^{2} \\ &{}- \Vert x_{n+1}-p \Vert ^{2}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \\ &{}\times \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert . \end{aligned}$$

Since \(0<\liminf_{n\rightarrow \infty }\lambda _{n}\leq \limsup_{n\rightarrow \infty }\lambda _{n}<2\sigma \), \(\alpha _{n} \rightarrow 0\), as \(n\rightarrow \infty \), and \(\lim_{n\rightarrow \infty }\|x_{n}-p\|\) exists, we obtain

$$ \Vert Bz_{n}-Bp \Vert \rightarrow 0\quad (n\rightarrow \infty ). $$

Since \(P_{C}\) is firmly nonexpansive, we obtain

$$\begin{aligned} \Vert y_{n}-p \Vert ^{2} =& \bigl\Vert P_{C}(I-\lambda _{n}B)z_{n}-p \bigr\Vert ^{2} \\ =& \bigl\Vert P_{C}(I-\lambda _{n}B)z_{n}-P_{C}(I- \lambda _{n}B)p \bigr\Vert ^{2} \\ \leq & \bigl\langle y_{n}-p, (I-\lambda _{n}B)z_{n}-(I- \lambda _{n}B)p \bigr\rangle \\ =&\frac{1}{2} \bigl\{ \Vert y_{n}-p \Vert ^{2}+ \bigl\Vert (I-\lambda _{n}B)z_{n}-(I-\lambda _{n}B)p \bigr\Vert ^{2} \\ &{}- \bigl\Vert y_{n}-z_{n}+\lambda _{n}(Bz_{n}-Bp) \bigr\Vert ^{2} \bigr\} \\ =&\frac{1}{2} \bigl\{ \Vert y_{n}-p \Vert ^{2}+ \Vert z_{n}-p \Vert ^{2}+\lambda _{n}^{2} \Vert Bz _{n}-Bp \Vert ^{2} \\ &{}-2\lambda _{n}\langle z_{n}-p, Bz_{n}-Bp \rangle - \bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \lambda _{n}^{2} \Vert Bz_{n}-Bp \Vert ^{2} \\ &{}+2\lambda _{n}\langle y_{n}-z_{n}, Bz_{n}-Bp\rangle \bigr] \bigr\} \\ =&\frac{1}{2} \bigl\{ \Vert y_{n}-p \Vert ^{2}+ \Vert z_{n}-p \Vert ^{2}- \Vert y_{n}-z_{n} \Vert ^{2} \\ &{}-2\lambda _{n}\langle y_{n}-p, Bz_{n}-Bp \rangle \bigr\} \\ \leq &\frac{1}{2} \bigl\{ \Vert y_{n}-p \Vert ^{2}+ \Vert z_{n}-p \Vert ^{2}- \Vert y_{n}-z_{n} \Vert ^{2} \\ &{}+2\lambda _{n} \Vert y_{n}-p \Vert \Vert Bz_{n}-Bp \Vert \bigr\} . \end{aligned}$$

Then

$$ \Vert y_{n}-p \Vert ^{2}\leq \Vert z_{n}-p \Vert ^{2}- \Vert y_{n}-z_{n} \Vert ^{2}+2\lambda _{n} \Vert y_{n}-p \Vert \Vert Bz_{n}-Bp \Vert . $$

From (3.7), we have

$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1- \alpha _{n}\eta \delta )^{2} \Vert y_{n}-p \Vert ^{2} \\ &{}+2\alpha _{n}(1-\alpha _{n}\eta \delta ) \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert \\ \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n} \eta \delta )^{2} \bigl[ \Vert z_{n}-p \Vert ^{2}- \Vert y_{n}-z_{n} \Vert ^{2} \\ &{}+2\lambda _{n} \Vert y_{n}-p \Vert \Vert Bz_{n}-Bp \Vert \bigr]+2\alpha _{n}(1-\alpha _{n} \eta \delta ) \\ &{}\times \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert \\ \leq &\alpha _{n}^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n} \eta \delta )^{2} \bigl[ \Vert x_{n}-p \Vert ^{2}- \Vert y_{n}-z_{n} \Vert ^{2} \\ &{}+2\lambda _{n} \Vert x_{n}-p \Vert \Vert Bz_{n}-Bp \Vert \bigr]+2\alpha _{n}(1-\alpha _{n} \eta \delta ) \\ &{}\times \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert . \end{aligned}$$

Then we have

$$\begin{aligned} (1-\alpha _{n}\eta \delta )^{2} \Vert y_{n}-z_{n} \Vert ^{2} \leq &\alpha _{n} ^{2} \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert ^{2}+(1-\alpha _{n}\eta \delta )^{2} \Vert x _{n}-p \Vert ^{2} \\ &{}- \Vert x_{n+1}-p \Vert ^{2}+2\lambda _{n}(1-\alpha _{n}\eta \delta )^{2} \\ &{}\times \Vert x_{n}-p \Vert \Vert Bz_{n}-Bp \Vert +2\alpha _{n}(1-\alpha _{n}\eta \delta ) \\ &{}\times \bigl\Vert \gamma f(x_{n})-\eta Lp \bigr\Vert \Vert x_{n}-p \Vert . \end{aligned}$$

Since \(\alpha _{n}\rightarrow 0\), \(\|Bz_{n}-Bp\|\rightarrow 0\), as \(n\rightarrow \infty \), and \(\lim_{n\rightarrow \infty }\|x_{n}-p\|\) exists, we obtain

$$ \Vert y_{n}-z_{n} \Vert \rightarrow 0\quad (n\rightarrow \infty ). $$

Since \(x_{n_{l}}\rightharpoonup q\) as \(l\rightarrow \infty \) for some subsequence \(\{x_{n_{l}}\}\) of \(\{x_{n}\}\). from (3.12), we have \(\omega _{n_{l}}\rightharpoonup q\) as \(l\rightarrow \infty \) for some subsequence \(\{\omega _{n_{l}}\}\) of \(\{\omega _{n}\}\). Again since \(\lim_{n\rightarrow \infty }\|\omega _{n_{l}}-S_{j}\omega _{n_{l}}\|= 0\) and \(S_{j}-I\) are demiclosed at 0, for each \(j\in \{1, 2, \ldots , N _{2}\}\), it follows from Definition 2.1 that \(q\in \bigcap_{i=1}^{N_{2}}F(S_{j})\).

Since \(y_{n}=P_{C}(I-\lambda _{n}B)z_{n}\) and \(\lambda _{n}>0\) is bounded, with no loss of generality, we may assume that

$$ \lambda _{n_{l}}\rightarrow \lambda \quad ( l\rightarrow \infty ). $$

Then we have

$$\begin{aligned} \bigl\Vert z_{n_{l}}-P_{C}(I-\lambda B)z_{n_{l}} \bigr\Vert =& \bigl\Vert z_{n_{l}}-P_{C}(I- \lambda _{n_{l}}B)z_{n_{l}} \bigr\Vert \\ &+ \bigl\Vert P_{C}(I-\lambda _{n_{l}}B)z_{n_{l}}-P_{C}(I- \lambda B)z_{n_{l}} \bigr\Vert \\ \leq & \Vert z_{n_{l}}-y_{n_{l}} \Vert + \bigl\Vert (I- \lambda _{n_{l}}B)z_{n_{l}}-(I- \lambda B)z_{n_{l}} \bigr\Vert \\ =& \Vert z_{n_{l}}-y_{n_{l}} \Vert + \vert \lambda _{n_{l}}- \lambda \vert \Vert Bz_{n_{l}} \Vert \\ \rightarrow &0 \quad (n\rightarrow \infty ). \end{aligned}$$

From (3.13), we have \(z_{n_{l}}\rightharpoonup q\) as \(l\rightarrow \infty \) for some subsequence \(\{z_{n_{l}}\}\) of \(\{z_{n}\}\). Again since \(P_{C}(I-\lambda B)\) is nonexpansive and we have Lemma 2.2, we obtain \(q\in \operatorname{VI}(C, B)\). It follows from Lemma 2.8 that \(q\in \bigcap_{l=1}^{N_{3}}\operatorname{VI}(C, B_{l})\).

From (3.1) and (3.12), we have

$$ \Vert \omega _{n, i}-x_{n} \Vert \rightarrow 0 \quad \mbox{as }n \rightarrow \infty , 1 \leq i\leq N_{1}, $$

and from (3.3), we obtain

$$ \bigl\Vert \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert \rightarrow 0 \quad \mbox{as }n \rightarrow \infty , 1\leq i\leq N_{1}. $$
(3.14)

Let \(\omega _{n, i}=T_{r_{n, i}}^{f_{i}, h_{i}}\upsilon _{n, i}\), where \(\upsilon _{n, i}=x_{n}+\xi _{n, i}A_{i}^{\ast }(T_{r_{n, i}}^{F_{i}, H _{i}}-I)A_{i}x_{n}\), and we have

$$\begin{aligned} \Vert \upsilon _{n, i}-x_{n} \Vert =& \bigl\Vert \xi _{n, i}A_{i}^{\ast } \bigl(T_{r_{n, i}} ^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert \\ \leq &\xi _{n, i} \Vert A_{i} \Vert \bigl\Vert \bigl(T_{r_{n, i}}^{F_{i}, H_{i}}-I \bigr)A_{i}x_{n} \bigr\Vert \\ \rightarrow &0\quad (n\rightarrow \infty ). \end{aligned}$$

Then we have \(\|\omega _{n, i}-\upsilon _{n, i}\|\rightarrow 0\) as \(n \rightarrow \infty \), \(1\leq i\leq N_{1}\).

Since \(\omega _{n, i}=T_{r_{n, i}}^{f_{i}, h_{i}}\upsilon _{n, i}\), we have

$$ f_{i}(\omega _{n, i}, \omega )+h_{i}(\omega _{n, i}, \omega )+\frac{1}{r _{n, i}}\langle \omega -\omega _{n, i}, \omega _{n, i}-\upsilon _{n, i} \rangle \geq 0, \quad \forall \omega \in C, $$

then

$$ h_{i}(\omega _{n, i}, \omega )+\frac{1}{r_{n, i}}\langle \omega - \omega _{n, i}, \omega _{n, i}-\upsilon _{n, i} \rangle \geq -f_{i}( \omega _{n, i}, \omega )\geq f_{i}(\omega , \omega _{n, i}),\quad \forall \omega \in C. $$

Since \(\|\omega _{n, i}-\nu _{n, i}\|\rightarrow 0\), \(\omega _{n, i} \rightharpoonup q\), \(f_{i}\) is lower semicontinuous in the second argument and \(h_{i}\) is upper semicontinuous in the first argument, we obtain

$$ h_{i}(q, \omega )\geq f_{i}(\omega , q),\quad \forall \omega \in C. $$

Then we have

$$ f_{i}(\omega , q)+h_{i}(\omega , q)\leq f_{i}( \omega , q)-h_{i}(q, \omega )\leq 0,\quad \forall \omega \in C. $$

Let \(\hat{\omega }=t\omega +(1-t)q\in C\), we have \(\hat{\omega } \in C\) and \(f_{i}(\hat{\omega }, q)+h_{i}(\hat{\omega }, q)\leq 0\). Observe that

$$\begin{aligned} 0 =&f_{i}(\hat{\omega }, \hat{\omega })+h_{i}(\hat{\omega }, \hat{\omega }) \\ =&t \bigl[f_{i}(\hat{\omega }, \omega )+h_{i}(\hat{\omega }, \omega ) \bigr]+(1-t) \bigl[f _{i}(\hat{\omega }, q)+h_{i}(\hat{\omega }, q) \bigr] \\ \leq & t \bigl[f_{i}(\hat{\omega }, \omega )+h_{i}(\hat{ \omega }, \omega ) \bigr]. \end{aligned}$$

Hence

$$ f_{i}(\hat{\omega }, \omega )+h_{i}(\hat{\omega }, \omega )\geq 0, \quad \forall \omega \in C. $$

Since \(f_{i}\) is upper hemicontinuous and \(h_{i}\) is upper semicontinuous in the first argument, we have

$$ f_{i}(q, \omega )+h_{i}(q, \omega )\geq 0,\quad \forall \omega \in C. $$

That is, \(q\in \operatorname{GEP}(f_{i}, h_{i})\), for \(\forall i\in \{1, \ldots , N _{1}\}\).

Next, we show that \(A_{i}q\in \operatorname{GEP}(F_{i}, H_{i})\). Since \(x_{n_{l}} \rightharpoonup q\) and continuity of \(A_{i}\), we have \(A_{i}x_{n_{l}} \rightharpoonup A_{i}q\). Let \(\vartheta _{n, i}=A_{i}x_{n}-T_{r_{n, i}} ^{F_{i}, H_{i}}A_{i}x_{n}\), from (3.14), we have \(\lim_{n\rightarrow \infty }\vartheta _{n, i}=0\), for \(\forall i\in \{1, \ldots , N_{1}\}\). And since \(T_{r_{n, i}}^{F_{i}, H_{i}}A_{i}x_{n}=A _{i}x_{n}-\tau _{n, i}\), for \(\forall \varepsilon \in Q\), we have

$$ F_{i}(A_{i}x_{n}-\vartheta _{n, i}, \varepsilon )+H_{i}(A_{i}x_{n}- \vartheta _{n, i}, \varepsilon )+\frac{1}{r_{n, i}} \bigl\langle \varepsilon -(A_{i}x_{n}-\vartheta _{n, i}), (A_{i}x_{n}-\vartheta _{n, i})-A_{i}x _{n} \bigr\rangle \geq 0. $$

Since \(F_{i}\) and \(H_{i}\) are upper semicontinuous in the first argument, we have

$$ F_{i}(A_{i}q, \varepsilon )+H_{i}(A_{i}q, \varepsilon )\geq 0,\quad \forall \varepsilon \in Q. $$

Then we obtain \(A_{i}q\in \operatorname{GEP}(F_{i}, H_{i})\), for \(\forall i\in \{1, \ldots , N_{1}\}\). Therefore we conclude that \(q\in \varGamma \).

Next, we show that

$$ \limsup_{n\rightarrow \infty } \bigl\langle (\eta L-\gamma f)x^{\ast }, x ^{\ast }-x_{n} \bigr\rangle \leq 0. $$
(3.15)

Indeed, take a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) such that

$$ \limsup_{n\rightarrow \infty } \bigl\langle (\eta L-\gamma f)x^{\ast }, x ^{\ast }-x_{n} \bigr\rangle =\lim_{j\rightarrow \infty } \bigl\langle (\eta L- \gamma f)x^{\ast }, x^{\ast }-x_{n_{j}} \bigr\rangle . $$

Since \(\{x_{n}\}\) is bounded, without loss of generality, we may assume that \(x_{n_{j}}\rightharpoonup \bar{x}\in \varGamma \). Then we obtain

$$ \limsup_{n\rightarrow \infty } \bigl\langle (\eta L-\gamma f)x^{\ast }, x ^{\ast }-x_{n} \bigr\rangle = \bigl\langle (\eta L-\gamma f)x^{\ast }, x^{\ast }- \bar{x} \bigr\rangle \leq 0. $$

Finally, we show that \(x_{n}\rightarrow x^{\ast } (n\rightarrow \infty )\). From Lemma 2.9, Lemma 2.11 and (3.6), we have

$$\begin{aligned} \bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2} =& \bigl\Vert \alpha _{n}\gamma f(x_{n})+(I-\alpha _{n} \eta L)y_{n}-x^{\ast } \bigr\Vert ^{2} \\ =& \bigl\Vert \alpha _{n} \bigl(\gamma f(x_{n})-\eta Lx^{\ast } \bigr)+(I-\alpha _{n}\eta L) \bigl(y _{n}-x^{\ast } \bigr) \bigr\Vert ^{2} \\ \leq &(1-\alpha _{n}\eta \delta )^{2} \bigl\Vert y_{n}-x^{\ast } \bigr\Vert ^{2}+2\alpha _{n} \bigl\langle \gamma f(x_{n})-\eta Lx^{\ast }, x_{n+1}-x^{\ast } \bigr\rangle \\ =&(1-\alpha _{n}\eta \delta )^{2} \bigl\Vert y_{n}-x^{\ast } \bigr\Vert ^{2}+2\alpha _{n} \bigl\langle \gamma f(x_{n})-\gamma f \bigl(x^{\ast } \bigr), x_{n+1}-x^{\ast } \bigr\rangle \\ &{}+2\alpha _{n} \bigl\langle \gamma f \bigl(x^{\ast } \bigr)- \eta Lx^{\ast }, x_{n+1}-x ^{\ast } \bigr\rangle \\ \leq &(1-\alpha _{n}\eta \delta )^{2} \bigl\Vert y_{n}-x^{\ast } \bigr\Vert ^{2}+2\alpha _{n}\gamma \rho \bigl\Vert x_{n}-x^{\ast } \bigr\Vert \bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert \\ &{}+2\alpha _{n} \bigl\langle \gamma f \bigl(x^{\ast } \bigr)- \eta Lx^{\ast }, x_{n+1}-x ^{\ast } \bigr\rangle \\ \leq &(1-\alpha _{n}\eta \delta )^{2} \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+\alpha _{n} \gamma \rho \bigl( \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+ \bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2} \bigr) \\ &{}+2\alpha _{n} \bigl\langle \gamma f \bigl(x^{\ast } \bigr)- \eta Lx^{\ast }, x_{n+1}-x ^{\ast } \bigr\rangle . \end{aligned}$$

Thus

$$\begin{aligned} (1-\alpha _{n}\gamma \rho ) \bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2} \leq & \bigl[(1-\alpha _{n}\eta \delta )^{2}+\alpha _{n}\gamma \rho \bigr] \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2} \\ &{}+2\alpha _{n} \bigl\langle \gamma f \bigl(x^{\ast } \bigr)- \eta Lx^{\ast }, x_{n+1}-x ^{\ast } \bigr\rangle . \end{aligned}$$

Since \(\eta \delta >\gamma \rho \) and \(0<\alpha _{n}\leq \frac{1}{ \eta \|L\|}\leq \frac{1}{\eta \delta }\), we have \(1-\alpha _{n}\gamma \rho >1-\alpha _{n}\eta \delta \geq 0\). Hence

$$\begin{aligned} \bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2} \leq &\frac{(1-\alpha _{n}\eta \delta )^{2}+ \alpha _{n}\gamma \rho }{1-\alpha _{n}\gamma \rho } \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+\frac{2\alpha _{n}}{1-\alpha _{n}\gamma \rho } \\ &{}\times \bigl\langle \gamma f \bigl(x^{\ast } \bigr)-\eta Lx^{\ast }, x_{n+1}-x^{ \ast } \bigr\rangle \\ =& \biggl[1-\frac{2\alpha _{n}(\eta \delta -\gamma \rho )}{1-\alpha _{n} \gamma \rho } \biggr] \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+\frac{2\alpha _{n}}{1-\alpha _{n} \gamma \rho } \\ &{}\times \bigl\langle \gamma f \bigl(x^{\ast } \bigr)-\eta Lx^{\ast }, x_{n+1}-x^{ \ast } \bigr\rangle + \frac{\alpha _{n}^{2}\eta ^{2}\delta ^{2}}{1-\alpha _{n} \gamma \rho } \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2} \\ \leq & \biggl[1-\frac{2\alpha _{n}(\eta \delta -\gamma \rho )}{1-\alpha _{n} \gamma \rho } \biggr] \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2} \\ &{}+\frac{2\alpha _{n}(\eta \delta -\gamma \rho )}{1-\alpha _{n}\gamma \rho } \biggl(\frac{\langle \gamma f(x^{\ast })-\eta Lx^{\ast }, x_{n+1}-x ^{\ast }\rangle }{\eta \delta -\gamma \rho }+\alpha _{n}M \biggr), \end{aligned}$$

where M is the constant satisfying

$$ M=\sup_{n\geq 0} \biggl\{ \frac{\eta ^{2}\delta ^{2}}{2(\eta \delta -\gamma \rho )} \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2} \biggr\} . $$

From the condition \(\sum_{n=0}^{\infty }\alpha _{n}=\infty \), \(\lim_{n\rightarrow \infty }\alpha _{n}=0\) and (3.15), we have

$$ \sum_{n=0}^{\infty }\frac{2\alpha _{n}(\eta \delta -\gamma \rho )}{1- \alpha _{n}\gamma \rho }>\sum _{n=0}^{\infty }2\alpha _{n}(\eta \delta - \gamma \rho )=\infty $$

and

$$ \limsup_{n\rightarrow \infty } \biggl(\frac{\langle \gamma f(x^{\ast })- \eta Lx^{\ast }, x_{n+1}-x^{\ast }\rangle }{\eta \delta -\gamma \rho }+\alpha _{n}M \biggr)\leq 0. $$

By Lemma 2.14, we obtain \(\|x_{n}-x^{\ast }\|\rightarrow 0\) as \(n\rightarrow \infty \).

Case 2. Assume that there exists a subsequence \(\{\|x_{n_{j}}-p\|^{2} \}\) of \(\{\|x_{n}-p\|^{2}\}\) such that \(\|x_{n_{j}}-p\|^{2}<\|x_{n _{j}+1}-p\|^{2}\) for all \(j\in N\). Then it follows from Lemma 2.13 that there exists a nondecreasing sequence \(\{m_{k}\}\) of N, such that \(\lim_{k\rightarrow \infty }m_{k}=\infty \), and the following inequalities hold for all \(k\in N\):

$$ \Vert x_{m_{k}}-p \Vert ^{2}\leq \Vert x_{m_{k}+1}-p \Vert ^{2}\quad \mbox{and}\quad \Vert x_{k}-p \Vert ^{2} \leq \Vert x_{m_{k}+1}-p \Vert ^{2}. $$
(3.16)

Similarly, we get

$$\begin{aligned}& \Vert \omega _{m_{k}, i}-x_{n} \Vert \rightarrow 0 \quad (k \rightarrow \infty ), \forall i\in \{1, 2, \ldots , N_{1}\}, \\& \Vert \omega _{m_{k}}-S_{j}\omega _{m_{k}} \Vert \rightarrow 0 \quad (k\rightarrow \infty ), \forall j\in \{1, 2, \ldots , N_{2}\}, \\& \Vert z_{m_{k}, j}-\omega _{m_{k}} \Vert \rightarrow 0 \quad (k \rightarrow \infty ), \forall j\in \{1, 2, \ldots , N_{2}\}, \\& \Vert z_{m_{k}}-x_{m_{k}} \Vert \rightarrow 0\quad (k \rightarrow \infty ), \\& \Vert y_{m_{k}}-z_{m_{k}} \Vert \rightarrow 0\quad (k \rightarrow \infty ), \\& \Vert \omega _{m_{k}, i}-\upsilon _{m_{k}, i} \Vert \rightarrow 0 \quad (k \rightarrow \infty ), \forall i\in \{1, 2, \ldots , N_{1}\}, \\& \bigl\Vert A_{i}x_{m_{k}}-T_{r_{m_{k}, i}}^{F_{i}, H_{i}}A_{i}x_{m_{k}} \bigr\Vert \rightarrow 0\quad (k\rightarrow \infty ), \forall i\in \{1, 2, \ldots , N_{1}\}. \end{aligned}$$

By a similar argument to that in Case 1, we have \(\omega _{\omega }(x _{m_{k}})\subset \varGamma \).

Also, we obtain

$$ \limsup_{k\rightarrow \infty } \bigl\langle (\eta L-\gamma f)x^{\ast }, x ^{\ast }-x_{m_{k}} \bigr\rangle \leq 0 $$

and

$$\begin{aligned} \bigl\Vert x_{m_{k}+1}-x^{\ast } \bigr\Vert ^{2} \leq & \biggl[1-\frac{2\alpha _{m_{k}}(\eta \delta -\gamma \rho )}{1-\alpha _{m_{k}}\gamma \rho } \biggr] \bigl\Vert x_{m_{k}}-x ^{\ast } \bigr\Vert ^{2} \\ &{}+\frac{2\alpha _{m_{k}}(\eta \delta -\gamma \rho )}{1-\alpha _{m _{k}}\gamma \rho } \biggl(\frac{\langle \gamma f(x^{\ast })-\eta Lx^{\ast }, x_{m_{k}+1}-x^{\ast }\rangle }{\eta \delta -\gamma \rho }+\alpha _{m _{k}}M \biggr), \end{aligned}$$

where M is the constant satisfying

$$ M=\sup_{k\geq 0} \biggl\{ \frac{\eta ^{2}\delta ^{2}}{2(\eta \delta -\gamma \rho )} \bigl\Vert x_{m_{k}}-x^{\ast } \bigr\Vert ^{2} \biggr\} . $$

By a similar argument to that in Case 1, we obtain \(\|x_{m_{k}}-x^{ \ast }\|\rightarrow 0\) as \(k\rightarrow \infty \). By (3.16), we get \(\|x_{k}-x^{*}\|\leq \|x_{m_{k}}-x^{*}\|\), \(\forall k\in N\). Therefore, \(x_{k}\rightarrow x^{*}\) as \(k\rightarrow \infty \). □

Remark 3.2

We present several corollaries of Theorem 3.1 and we can consider the following cases:

  1. (i)

    \(A_{i}=A\), for \(1\leq i\leq N_{1}\);

  2. (ii)

    \(h_{i}=\varphi _{i}(\cdot , \cdot )\) and \(H_{i}=\phi _{i}( \cdot , \cdot )\), where \(\varphi _{i} : C\rightarrow R\cup \{+ \infty \}\) and \(\phi _{i} : Q\rightarrow R\cup \{+\infty \}\) is proper lower semicontinuous and convex functions, for \(1\leq i\leq N_{1}\);

  3. (iii)

    \(h_{i}=0\) and \(H_{i}=0\), for \(1\leq i\leq N_{1}\);

  4. (iv)

    \(H_{i}=0\) and \(F_{i}=0\), for \(1\leq i\leq N_{1}\);

  5. (v)

    \(N_{1}=N_{2}=N_{3}=1\).

  6. (vi)

    \(S_{j}\) is \(\kappa _{j}\)-strict pseudo-contractive for any j. From Lemma 2.2, we can remove the condition that \(S_{j}-I\) is demiclosed at 0 of Theorem 3.1;

  7. (vii)

    \(S_{j}\) is a nonspreading mapping or an N-generalized hybrid mapping. We note that nonspreading mappings with nonempty fixed point set and N-generalized hybrid mappings with nonempty fixed point set are quasi-nonexpansive operators.

Numerical examples

In this section, we give two numerical examples to demonstrate the convergence of our algorithm. All codes were written in Matlab 2010b and run on Dell i-5 Dual-Core laptop.

Example 4.1

We consider the case that \(N_{1}=N_{2}=N_{3}=1\).

Let \(\mathcal{H}_{1}=\mathcal{H}_{2}=R\), \(C=Q=[-20, 10]\) and let \(f_{1}, h_{1} : C\times C \rightarrow R\) and \(F_{1}, H_{1} : Q\times Q\rightarrow R\) be defined by \(f_{1}(z, y)=y^{2}+3zy-4z^{2}\), \(h_{1}(z, y)=y^{2}-z^{2}\) and \(F_{1}(z, y)=3y^{2}+2zy-5z^{2}\), \(H_{1}(z, y)=0\). Define \(S_{1}\), \(A_{1}\), \(B_{1}\), f, L as follows:

$$ S_{1}(x)= \textstyle\begin{cases} x, &x\in (-\infty , 0), \\ -2x, & x\in [0, \infty ). \end{cases} $$

\(A_{1}x=-x\), \(B_{1}x=\frac{1}{2}x\) and \(f(x)=L(x)=2x\). Put \(\alpha _{n}=\frac{1}{n+1}\), \(\beta _{n, 1}=\frac{1}{2}+\frac{1}{n+2}\), \(\xi _{n, 1}=1-\frac{1}{n+1}\), \(\lambda _{n}=\frac{1}{5}\) and \(\gamma =\frac{1}{2}\). It is easy to verify that \(f_{1}\), \(h_{1}\), \(F _{1}\), \(H_{1}\), \(S_{1}\), \(A_{1}\), \(B_{1}\), f, L, \(\alpha _{n}\), \(\beta _{n, 1}\), \(\xi _{n, 1}\), \(\lambda _{n}\), γ satisfy all the conditions of Theorem 3.1. Then, by Lemma 2.7, we see that \(T_{r}^{f _{1}, h_{1}}\) and \(T_{r}^{F_{1}, H_{1}}\) are single-value mappings on \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\), respectively. Hence, for \(r_{n}=r > 0\), \(x\in \mathcal{H}_{1}\) and \(x\in \mathcal{H}_{2}\), there exist \(z_{1}\in C\) and \(z_{2}\in Q\) such that

$$ f_{1}(z_{1},y)+h_{1}(z_{1},y)+ \frac{1}{r}\langle y-z_{1},z_{1}-x \rangle \geq 0, \quad \forall y\in C, $$

and

$$ F_{1}(z_{2},y)+H_{1}(z_{2},y)+ \frac{1}{r}\langle y-z_{2},z_{2}-x \rangle \geq 0, \quad \forall y\in Q. $$

We can reform the above inequalities to standard quadratic form in the variable y as follows:

$$ P_{1}(y)=2ry^{2}+(2rz_{1}+z_{1}-x)y+ \bigl(xz_{1}-5rz_{1}^{2}-z_{1}^{2} \bigr) \geq 0, \quad \forall y\in C, $$

and

$$ P_{2}(y)=3ry^{2}+(2rz_{2}+z_{2}-x)y+ \bigl(xz_{2}-5rz_{2}^{2}-z_{2}^{2} \bigr) \geq 0, \quad \forall y\in Q. $$

It is easy to check that the discriminants of the above two quadratic inequalities are nonnegative. And since \(P_{1}(y)\geq 0\) for all \(y\in C\) and \(P_{2}(y)\geq 0\) for all \(y\in Q\), we see that the discriminant must be zero. Then we obtain \(z_{1}=T_{r}^{f_{1}, h_{1}}(x)= \frac{x}{1+7r}\) and \(z_{2}=T_{r}^{F_{1}, H_{1}}(x)=\frac{x}{1+8r}\). It is clear that \(\varGamma =\{0\}\).

In what follows, we observe the convergence of the Algorithm 3.1 by considering the following three cases:

Case 1. Taking the different initial point \(x_{1}=-5, 1, 5\) with \(\eta =1\), \(r=4\), Fig. 1 presents the convergence behaviors of \(\{x_{n}\}\) for Algorithm 3.1.

Figure 1
figure1

Behaviors of \(\{x_{n}\}\) with initial points \(x_{1}=-5, 1, 5\), \(\eta =1\) and \(r=4\)

Case 2. Taking the different \(r=4, 0.4, 0.04\) with \(x_{1}=1\), \(\eta =1\), Fig. 2 presents the convergence behaviors of \(\{x_{n}\}\) for Algorithm 3.1.

Figure 2
figure2

Behaviors of \(\{x_{n}\}\) with different \(r=4, 0.4, 0.04\), \(x_{1}=1\) and \(\eta =1\)

Case 3. Taking the different \(\eta =1, 10, 50, 100, 200\) with \(x_{1}=1\), \(r=4\), Fig. 3 presents the convergence behaviors of \(\{x_{n}\}\) for Algorithm 3.1.

Figure 3
figure3

Behaviors of \(\{x_{n}\}\) with different \(\eta =1, 10, 50, 100, 200\), \(x_{1}=1\) and \(r=4\)

Example 4.2

We consider the case that \(N_{1}=N_{2}=N_{3}=2\).

Let \(H_{1}=H_{2}=R\), \(C=Q=[-20, 10]\) and let \(f_{i}, h_{i} : C\times C \rightarrow R\) and \(F_{i}, H_{i} : Q\times Q\rightarrow R\) be defined by \(f_{1}(z, y)=y^{2}+3zy-4z^{2}\), \(h_{1}(z, y)=y^{2}-z^{2}\), \(F_{1}(z, y)=3y^{2}+2zy-5z^{2}\), \(H_{1}(z, y)=0\) and \(f_{2}(z, y)=y ^{2}-z^{2}\), \(h_{2}(z, y)=0\), \(F_{2}(z, y)=y^{2}+3zy-4z^{2}\), \(H_{2}(z, y)=\frac{1}{2}(y^{2}-z^{2})\). Let \(S_{1}\), \(A_{1}\), \(B_{1}\), f, L, \(\alpha _{n}\), \(\beta _{n, 1}\), \(\xi _{n, 1}\), \(\lambda _{n}\), γ be the same as that of Example 4.1, and define \(S_{2}\), \(A_{2}\), \(B_{2}\) by \(S_{2}x=\frac{4}{5}x\), \(A_{2}x=2x\) and \(B_{2}x=2x\). Put \(\beta _{n, 2}=\frac{1}{2}-\frac{1}{n+2}\), \(\xi _{n, 2}=\frac{1}{2}(1- \frac{1}{n+1})\), \(\nu _{n, 1}=\frac{1}{2}-\frac{1}{n+2}\), \(\nu _{n, 2}= \frac{1}{2}+\frac{1}{n+2}\), \(r_{n, 1}=r_{n, 2}=r=4\), \(\mu _{1}=\mu _{2}= \frac{1}{2}\) and \(\eta =1\). It is easy to verify that they satisfy all the conditions of Theorem 3.1.

Following an argument similar to that of Example 4.1, we obtain \(T_{r}^{f_{1}, h_{1}}(x)=\frac{x}{1+7r}\), \(T_{r}^{F_{1}, H_{1}}(x)= \frac{x}{1+8r}\), \(T_{r}^{f_{2}, h_{2}}(x)=\frac{x}{1+2r}\) and \(T_{r}^{F_{2}, H_{2}}(x)=\frac{x}{1+6r}\) and \(\varGamma =\{0\}\). Figure 4 presents the convergence behaviors of \(\{x_{n}\}\) for Algorithm 3.1.

Figure 4
figure4

Behaviors of \(\{x_{n}\}\) with \(x_{1}=1\)

Conclusion

In this paper, we first propose a new parallel hybrid viscosity method for finding a common element of the set of solutions of a finite family of split generalized equilibrium problems, variational inequality problems and the set of common fixed points of a finite family of demicontractive operators in Hilbert spaces. And then we establish the corresponding strong convergence theorem under suitable conditions. We study more general split equilibrium problems and fixed point problems of operators than those in [18]. Our results in this paper improve and extend many recent results in the literature. Finally, we present numerical examples to demonstrate the effectiveness of our algorithm.

References

  1. 1.

    Takahashi, W., Wen, C.-F., Yao, J.-C.: The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space. Fixed Point Theory 19(1), 407–419 (2018)

  2. 2.

    Dehaish, B.B., Qin, X., Latif, A., Bakodah, H.: Weak and strong convergence of algorithms for the sum of two accretive operators with applications. J. Nonlinear Convex Anal. 16, 1321–1336 (2015)

  3. 3.

    Qin, X., Cho, S.Y.: Convergence analysis of a monotone projection algorithm in reflexive Banach spaces. Acta Math. Sci. 37(2), 488–502 (2017)

  4. 4.

    Chang, S.-S., Wen, C.-F., Yao, J.-C.: Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces. Optimization 67(8), 1183–1196 (2018)

  5. 5.

    Zhao, X., Ng, K.F., Li, C., Yao, J.-C.: Linear regularity and linear convergence of projection-based methods for solving convex feasibility problems. Appl. Math. Optim. 78(3), 613–641 (2018)

  6. 6.

    Ansari, Q.H., Babu, F., Yao, J.-C.: Regularization of proximal point algorithms in Hadamard manifolds. J. Fixed Point Theory Appl. 21(1), 25 (2019)

  7. 7.

    Halpern, B.: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73(6), 957–961 (1967)

  8. 8.

    Cho, S.Y., Dehaish, B.B., Qin, X.: Weak convergence of a splitting algorithm in Hilbert spaces. J. Appl. Anal. Comput. 7(2), 427–438 (2017)

  9. 9.

    Cho, S.Y., Qin, X., Yao, J.-C., Yao, Y.: Viscosity approximation splitting methods for monotone and nonexpansive operators in Hilbert spaces. J. Nonlinear Convex Anal. 19, 251–264 (2018)

  10. 10.

    Cho, S.Y., Qin, X., Wang, L.: Strong convergence of a splitting algorithm for treating monotone operators. Fixed Point Theory Appl. 2014(1), 94 (2014)

  11. 11.

    Zhang, L., Zhao, H., Lv, Y.: A modified inertial projection and contraction algorithms for quasi-variational inequalities. Appl. Set-Valued Anal. Optim. 2019(1), 63–76 (2019)

  12. 12.

    Yao, Y., Liou, Y.-C., Yao, J.-C.: Iterative algorithms for the split variational inequality and fixed point problems under nonlinear transformations. J. Nonlinear Sci. Appl. 10(2), 843–854 (2017)

  13. 13.

    Yao, Y., Liou, Y.-C., Yao, J.-C.: Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithm construction. Fixed Point Theory Appl. 2015(1), 127 (2015)

  14. 14.

    Yao, Y., Liou, Y.-C., Kang, S.M.: Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method. Comput. Math. Appl. 59(11), 3472–3480 (2010)

  15. 15.

    Kazmi, K.R., Rizvi, S.H.: Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup. Math. Sci. 7(1), 1 (2013)

  16. 16.

    Cianciaruso, F., Marino, G., Muglia, L., Yao, Y.: A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory Appl. 2010(1), 383740 (2009)

  17. 17.

    Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150(2), 275–283 (2011)

  18. 18.

    Majee, P., Nahak, C.: A hybrid viscosity iterative method with averaged mappings for split equilibrium problems and fixed point problems. Numer. Algorithms 74(2), 609–635 (2017)

  19. 19.

    Onjai-uea, N., Phuengrattana, W.: On solving split mixed equilibrium problems and fixed point problems of hybrid-type multivalued mappings in Hilbert spaces. J. Inequal. Appl. 2017(1), 137 (2017)

  20. 20.

    Sitthithakerngkiet, K., Deepho, J., Martínez-Moreno, J., Kumam, P.: An iterative approximation scheme for solving a split generalized equilibrium, variational inequalities and fixed point problems. Int. J. Comput. Math. 94(12), 2373–2395 (2017)

  21. 21.

    Qin, X., Yao, J.-C.: Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators. J. Inequal. Appl. 2016(1), 232 (2016)

  22. 22.

    Ram, T., Lal, P., Kim, J.K.: Operator solutions of generalized equilibrium problems in Hausdorff topological vector spaces. Nonlinear Funct. Anal. Appl. 24, 61–71 (2019)

  23. 23.

    Qin, X., Yao, J.-C.: Projection splitting algorithms for nonself operators. J. Nonlinear Convex Anal. 18(5), 925–935 (2017)

  24. 24.

    Kim, J.K., Salahuddin, S.: Extragradient methods for generalized mixed equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Funct. Anal. Appl. 22, 693–709 (2017)

  25. 25.

    Qin, X., Petrusel, A., Yao, J.-C.: CQ iterative algorithms for fixed points of nonexpansive mappings and split feasibility problems in Hilbert spaces. J. Nonlinear Convex Anal. 19, 157–165 (2018)

  26. 26.

    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984)

  27. 27.

    Marino, G., Xu, H.K.: Weak and strong convergence theorems for strictly pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–349 (2007)

  28. 28.

    Chidume, C.E., Maruster, S.: Iterative methods for the computation of fixed points of demicontractive mappings. J. Comput. Appl. Math. 234, 861–882 (2010)

  29. 29.

    Takahashi, W.: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000)

  30. 30.

    Mahdioui, H., Chadli, O.: On a system of generalized mixed equilibrium problems involving variational-like inequalities in Banach spaces: existence and algorithmic aspects. Adv. Oper. Res. 2012, Article ID 843486 (2012)

  31. 31.

    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in product space. Numer. Algorithms 8, 221–239 (1994)

  32. 32.

    Katchang, P., Kumam, P.: A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space. J. Appl. Math. Comput. 32(1), 19–38 (2010)

  33. 33.

    Wang, S., Gong, X., Abdou, A.A., Cho, Y.J.: Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications. Fixed Point Theory Appl. 2016(1), 4 (2016)

  34. 34.

    Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)

  35. 35.

    Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009)

  36. 36.

    Zegeye, H., Shahzad, N.: Convergence of manns type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62, 4007–4014 (2011)

  37. 37.

    Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)

  38. 38.

    Xu, H.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66(2), 240–256 (2002)

Download references

Acknowledgements

The author is most grateful to the referees and the editor for their helpful comments and advice which helped to improve the contents of this paper.

Availability of data and materials

All data generated or analysed during this study are included in this published article.

Funding

Not applicable.

Author information

The author completed the work alone, and read and approved the final manuscript.

Correspondence to Qingqing Cheng.

Ethics declarations

Competing interests

The author declares to have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Keywords

  • Split generalized equilibrium problems
  • Variational inequality problems
  • Fixed point problems
  • Parallel hybrid viscosity method
  • Lipschitzian mappings