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An iterative scheme for split monotone variational inclusion, variational inequality and fixed point problems

Abstract

We propose and analyze a new type iterative algorithm to find a common solution of split monotone variational inclusion, variational inequality, and fixed point problems for an infinite family of nonexpansive mappings in the framework of Hilbert spaces. Further, we show that a sequence generated by the algorithm converges strongly to common solution. Furthermore, we list some consequences of our established theorem. Finally, we provide a numerical example to demonstrate the applicability of the algorithm. We emphasize that the result accounted in manuscript unifies and extends various results in this field of study.

Introduction

Throughout the paper, let \(C_{1}\) be a nonempty subset of a real Hilbert space \(H_{1}\).

A mapping \(S_{1}:C_{1}\to C_{1}\) is said to be nonexpansive if

$$ \Vert S_{1}x_{1}-S_{1}x_{2} \Vert \leq \Vert x_{1}-x_{2} \Vert , \quad \forall x_{1},x_{2} \in C_{1}. $$

Let \(\operatorname{Fix}(S_{1})\) denote the fixed point of \(S_{1}\), that is, \(\operatorname{Fix}(S_{1})=\{x_{1}\in C_{1}: S_{1}x_{1}=x_{1}\}\).

The classical scalar nonlinear variational inequality problem (in brief, VIP) is: Find \(x_{1}\in C_{1}\) such that

$$ \langle Dx_{1},x_{2}-x_{1} \rangle \geq 0,\quad \forall x_{2}\in C_{1}, $$
(1.1)

where \(D:C_{1}\to H_{1}\) is a nonlinear mapping. It was introduced by Hartman and Stampacchia [1].

A mapping \(S:H_{1}\to H_{1}\) is said to be

  1. (i)

    monotone, if

    $$ \langle Sx_{1}-Sx_{2},x_{1}-x_{2} \rangle \geq 0, \quad \forall x_{1}, x_{2} \in H_{1}; $$
  2. (ii)

    γ-inverse strongly monotone (in brief, ism), if

    $$ \langle Sx_{1}-Sx_{2} , x_{1}-x_{2} \rangle \geq \gamma \Vert Sx_{1}-Sx_{2} \Vert ^{2}, \quad \forall x_{1},x_{2} \in H_{1} \text{ and } \gamma >0; $$
  3. (iii)

    firmly nonexpansive, if

    $$ \langle Sx_{1}-Sx_{2} , x_{1}-x_{2} \rangle \geq \Vert Sx_{1}-Sx_{2} \Vert ^{2}, \quad \forall x_{1},x_{2} \in H_{1}; $$
  4. (iv)

    L-Lipschitz continuous, if

    $$ \Vert Sx_{1}-Sx_{2} \Vert \leq L \Vert x_{1}-x_{2} \Vert , \quad \forall x_{1}, x_{2}\in H_{1} \text{ and } L >0. $$

A multi-valued mapping \(M_{1}:D(M_{1})\subseteq H_{1}\to 2^{H_{1}}\) is called monotone if, for all \(x_{1}, x_{2}\in D(M_{1})\), \(u_{1}\in M_{1}x_{1}\) and \(u_{2}\in M_{1}x_{2}\) such that

$$ \langle x_{1}-x_{2}, u_{1}-u_{2} \rangle \geq 0. $$

And it is maximal if \(G(M_{1})\), the graph of \(M_{1}\) defined as

$$ G(M_{1})=\bigl\{ (x_{1},u_{1}): u_{1}\in M_{1}x_{1} \bigr\} , $$

is not contained properly in the graph of other. It is well known that a monotone mapping \(M_{1}\) is maximal iff for \(x_{1}\in D(M_{1})\), \(u_{1}\in H_{1}\), \(\langle x_{1}-x_{2}, u_{1}-u_{2}\rangle \geq 0\) for each \((x_{2},u_{2})\in G(M_{1})\) implies that \(u_{1}\in M_{1}x_{1}\).

Let \(M_{1}:D(M_{1})\subseteq H_{1}\to 2^{H_{1}}\) be a multi-valued maximal monotone mapping. Then the resolvent operator \(J_{\rho _{1}}^{M_{1}}:H_{1}\to D(M_{1})\) is defined by

$$ J_{\rho _{1}}^{M_{1}}x_{1}:=(1+\rho _{1} M_{1})^{-1}(x_{1}),\quad \forall x_{1}\in H_{1} $$

for \(\rho _{1} >0\), where I stands for the identity operator on \(H_{1}\). We observe that \(J_{\rho _{1}}^{M_{1}}\) is single-valued nonexpansive and firmly nonexpansive.

Moudafi [2] was first to introduce the split monotone variational inclusion problem: Find \(\tilde{x}\in H_{1}\) such that

$$ 0\in f_{1}(\tilde{x})+M_{1}( \tilde{x}), $$
(1.2)

and

$$ \tilde{y}=B\tilde{x}\in H_{2} \quad {\text{solves }} 0\in f_{2}(\tilde{y})+M_{2}( \tilde{y}), $$
(1.3)

where \(f_{1}:H_{1}\to H_{1}\), \(f_{2}:H_{2}\to H_{2}\) are inverse strongly monotone mappings, \(B: H_{1}\to H_{2}\) is a bounded linear mapping, and \(M_{1}:H_{1} \to 2^{H_{1}}\), \(M_{2}:H_{2} \to 2^{H_{2}}\) are multi-valued maximal monotone mappings.

The split feasibility, split zero, and split fixed point problems are included as special cases. They have been studied broadly by various authors and solve real life problems essentially in modeling of inverse problems, sensor networks in computerized tomography and radiation therapy; for details, see [35].

If \(f_{1}\equiv 0\) and \(f_{2}\equiv 0\), then we find a split null point problem (in brief, \({\mathrm{S_{P}NPP}}\)): Find \(\tilde{x} \in H_{1}\) such that

$$ 0\in M_{1}(\tilde{x}), $$
(1.4)

and

$$ \tilde{y}=B\tilde{x}\in H_{2} \quad \text{solves } 0\in M_{2}(\tilde{y}). $$
(1.5)

In this paper, we consider the split monotone variational inclusion problem (in brief, \(\mathrm{S}_{\mathrm{P}}\mathrm{MVIP}\)): Find \(\tilde{x} \in H_{1}\) such that

$$ 0\in M_{1}(\tilde{x}), $$
(1.6)

and

$$ \tilde{y}=B\tilde{x}\in H_{2} \quad \text{solves } 0\in f(\tilde{y})+M_{2}( \tilde{y}). $$
(1.7)

Let \(\Lambda =\{\tilde{x} \in H_{1}:\tilde{x} \in \operatorname{Sol}(\mathrm{MVIP}( \text{1.6})) \text{ and } B\tilde{x} \in\operatorname{Sol}(\mathrm{MVIP}\text{(1.7)})\}\) denote the solution of \({\mathrm{S_{P}}}\mathrm{MVIP}\text{(1.6)--(1.7)}\).

The iterative algorithm for SPMVIP(1.2)–(1.3) was introduced and studied by Moudafi [2]:

$$ x_{0}\in H_{1}, x_{n+1}=P \bigl(x_{n}+\eta A^{*}(Q-I)Ax_{n}\bigr) \quad {\text{for }} \rho >0, $$
(1.8)

where \(P:=J_{\rho }^{M_{1}}(I-\lambda f_{1})\), \(Q:=J_{\rho }^{M_{2}}(I-\rho f_{2})\), \(A^{*}\) is the adjoint operator of A and \(0<\eta <\frac{1}{\varsigma }\), ς is the spectral radius of \(A^{*}A\).

The convergence analysis was studied by Byrne et al. [6] of some iterative algorithm for SPNPP(1.4)–(1.5). Moreover, Kazmi et al. [7] established an iterative method to find a common solution of SPNPP(1.4)–(1.5) and fixed point problem. For instance, see [8, 9].

Recently, Qin et al. [10] proposed an algorithm for an infinite family of nonexpansive mappings as follows:

$$ x_{0}\in C_{1},\quad x_{n+1}=\mu _{n}\theta g(x_{n})+\eta _{n}x_{n}+\bigl((1- \eta _{n})I-\mu _{n}A\bigr)\mathbb{W}_{n}u_{n}, $$
(1.9)

where g is a contraction mapping on \(H_{1}\), A is a strongly positive bounded linear operator, \(W_{n}\) is generated by \(S_{1},S_{2},\ldots \) as follows:

$$\begin{aligned}& \mathbb{V}_{n,n+1}:=I, \\& \mathbb{V}_{n,n}:=\lambda _{n}S_{n} \mathbb{V}_{n,n+1}+(1-\lambda _{n})I, \\& \mathbb{V}_{n,n-1}:=\lambda _{n-1}S_{n-1} \mathbb{V}_{n,n}+(1- \lambda _{n-1})I, \\& \vdots \\& \mathbb{V}_{n,m}:=\lambda _{m}S_{m} \mathbb{V}_{n,m+1}+(1-\lambda _{m})I, \\& \mathbb{V}_{n,m-1}:=\lambda _{m-1}S_{m-1} \mathbb{V}_{n,m}+(1- \lambda _{m-1})I, \\& \vdots \\& \mathbb{V}_{n,2}:=\lambda _{2}S_{2} \mathbb{V}_{n,3}+(1-\lambda _{3})I, \\& \mathbb{W}_{n}\equiv \mathbb{V}_{n,1}:=\lambda _{1}S_{1}\mathbb{V}_{n,2}+(1- \lambda _{1})I, \end{aligned}$$
(1.10)

where \(S_{1},S_{2},\ldots , W_{n}\) are nonexpansive mappings, \(\{\lambda _{n}\}\subset (0,1]\) for \(n\geq 1\); for further work, see [11, 12].

Inspirited by Moudafi [2], Byrne et al. [6], Kazmi et al. [7, 8], Qin et al. [10] and by continuing work, we propose and analyze a new type iterative algorithm to find a common solution of split monotone variational inclusion, variational inequality, and fixed point problems for an infinite family of nonexpansive mappings in the framework of Hilbert spaces. Further, we show that the sequence generated by the algorithm converges strongly to common solution. Furthermore, we list some consequences of our established theorem. Finally, we provide a numerical example to demonstrate the applicability of the algorithm. We emphasize that the result accounted in the manuscript unifies and extends various results in this field of study.

Preliminaries

This section is devoted to recalling few definitions, entailing mathematical tools, and helpful results that are required in the sequel.

To each \(x_{1}\in H_{1}\), there exists a unique nearest point \(P_{C_{1}}x_{1}\) to \(x_{1}\) in \(C_{1}\) such that

$$ \Vert x_{1}-P_{C_{1}}x_{1} \Vert \leq \Vert x_{1}-x_{2} \Vert , \quad \forall x_{2}\in C_{1}, $$
(2.1)

where \(P_{C_{1}}\) is a metric projection of \(H_{1}\) onto \(C_{1}\). Also, \(P_{C_{1}}\) is nonexpansive and satisfies

$$ \langle x_{1}-x_{2},P_{C_{1}}x_{1}-P_{C_{1}}x_{2} \rangle \geq \Vert P_{C_{1}}x_{1}-P_{C_{1}}x_{2} \Vert ^{2}, \quad \forall x_{1},x_{2}\in H_{1}. $$
(2.2)

Moreover, \(P_{C_{1}}x_{1}\) is characterized by the fact that \(P_{C_{1}}x_{1}\in C_{1}\) and

$$ \langle x_{1}-P_{C_{1}}x_{1},x_{2}-P_{C_{1}}x_{1} \rangle \leq 0,\quad \forall x_{2}\in C_{1}. $$
(2.3)

This implies that

$$ \Vert x_{1}-x_{2} \Vert ^{2}\geq \Vert x_{1}-P_{C_{1}}x_{1} \Vert ^{2}+ \Vert x_{2}-P_{C_{1}}x_{1} \Vert ^{2}, \quad \forall x_{1}\in H_{1}, x_{2}\in C_{1}, $$
(2.4)

and

$$ \bigl\Vert \mu x_{1}+(1-\mu )x_{2} \bigr\Vert ^{2}=\mu \Vert x_{1} \Vert ^{2}+(1-\mu ) \Vert x_{2} \Vert ^{2}- \mu (1-\mu ) \Vert x_{1}-x_{2} \Vert ^{2} $$
(2.5)

for all \(x_{1} ,x_{2}\in H_{1}\) and \(\mu \in [0,1]\).

Also, on \(H_{1}\) the following inequalities hold:

  1. 1.

    Opial’s condition [13], that is, for any \(\{x_{n}\}\) with \(x_{n}\rightharpoonup x_{1}\) and

    $$ \liminf_{n\to \infty } \Vert x_{n}-x_{1} \Vert < \liminf_{n\to \infty } \Vert x_{n}-x_{2} \Vert $$
    (2.6)

    holds, \(\forall x_{2}\in H_{1}\) with \(x_{2}\neq x_{1}\);

  2. 2.
    $$ \Vert x_{1}+x_{2} \Vert ^{2} \leq \Vert x_{1} \Vert ^{2}+2 \langle x_{2},x_{1}+x_{2} \rangle , \quad \forall x_{1},x_{2} \in H_{1}. $$
    (2.7)

Definition 2.1

([14])

A mapping \(T_{1}:H_{1}\to H_{1}\) is called averaged iff

$$ T_{1}=(1-\lambda )I+\lambda S_{1}, $$

where \(\lambda \in (0,1)\), I is the identity mapping on \(H_{1}\), and \(S_{1}:H_{1}\to H_{1}\) is a nonexpansive mapping.

Lemma 2.1

([2])

  1. (i)

    If \(T_{2}=(1-\lambda )T_{1}+\lambda S_{1}\), where \(T_{1}:H_{1}\to H_{1}\)is averaged, \(S_{1}:H_{1}\to H_{1}\)is nonexpansive, and \(0<\lambda <1\), then \(T_{2}\)is averaged;

  2. (ii)

    If \(T_{1}\)is γ-ism, then \(\beta T_{1}\)is \(\frac{\gamma }{\beta }\)-ism for \(\beta >0\);

  3. (iii)

    \(T_{1}\)is averaged iff \(I-T_{1}\)is γ-ism for some \(\gamma >\frac{1}{2}\).

Lemma 2.2

([2])

Let \(\rho > 0\), f be a γ-ism, and M be a maximal monotone mapping. If \(\rho \in (0, 2\gamma )\), then \(J_{\rho }^{M}(I-\rho f)\)is averaged.

Lemma 2.3

([2])

Let \(\rho _{1},\rho _{2} > 0\)and \(M_{1}\), \(M_{2}\)be maximal monotone mappings. Then

$$ \tilde{x} \quad \textit{solves }((\text{1.2})\mbox{--}(\mbox{1.3})) \quad \Leftrightarrow\quad \tilde{x}=J_{\rho _{1}}^{M_{1}}(I-\rho _{1}f_{1}) \tilde{x} \quad {\textit{and}}\quad B \tilde{x}=J_{\rho _{2}}^{M_{2}}(I- \rho _{2}f_{2})B\tilde{x}. $$

Lemma 2.4

([15])

Let \(\{u_{n}\}\)and \(\{v_{n}\}\)be bounded sequences in E, a Banach space, and let \(0<\mu _{n}<1\)with \(0<\liminf_{n\to \infty }\mu _{n}\leq \limsup_{n\to \infty }\mu _{n}<1\). Consider \(v_{n+1}=(1-\mu _{n})v_{n}+\mu _{n}u_{n}\), \(n\geq 0\)and \(\limsup_{n\to \infty }(\|v_{n+1}-v_{n}\|-\|u_{n+1}-u_{n}\|) \leq 0\). Then

$$ \lim_{n\to \infty } \Vert v_{n}-u_{n} \Vert =0. $$

Lemma 2.5

([16])

Assume that B is a strongly positive self-adjoint bounded linear operator on \(H_{1}\)with coefficient \(\overline{\gamma }>0\)and \(0<\rho \leq \|B\|^{-1}\). Then \(\|I-\rho B\|\leq 1-\rho \overline{\gamma }\).

Lemma 2.6

([17])

Let \(\{a_{n}\}\)be a sequence of nonnegative real numbers with

$$ a_{n+1}\leq (1-\lambda _{n})a_{n}+\alpha _{n}, \quad n\geq 0, $$

where \(\lambda _{n}\in (0,1)\)and \(\{\alpha _{n}\}\)in \(\mathbb{R}\)with

  1. (i)

    \(\sum_{n=1}^{\infty }\lambda _{n}=\infty \);

  2. (ii)

    \(\limsup_{n\to \infty }\frac{\alpha _{n}}{\lambda _{n}}\leq 0\)or \(\sum_{n=1}^{\infty }|\alpha _{n}|< +\infty \).

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

Lemma 2.7

([18])

Let \(S_{1}:C_{1}\to H_{1}\)be a nonexpansive mapping. If \(S_{1}\)has a fixed point, then \((I-S_{1})\), where I is the identity mapping, it is demiclosed, that is, if \(x_{n}\rightharpoonup x_{1}\in H_{1}\)and \(x_{n}-S_{1}x_{n}\rightarrow x_{2}\), then \((I-S_{1})x_{1}=x_{2}\).

Lemma 2.8

([19])

Let \(C_{1}\neq \emptyset \)be a closed convex subset of a strictly convex Banach space E. Let \(S_{1}, S_{2},\ldots \)be nonexpansive mappings of \(C_{1}\)to \(C_{1}\)such that \(\bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})\neq \emptyset \), and let \(\lambda _{1},\lambda _{2},\ldots \)be real numbers satisfying \(0<\lambda _{i}<1\), \(\forall i\geq 1\). Then \(\lim_{i\to \infty }\mathbb{V}_{i,j}\tilde{x}\)exists, \(\forall \tilde{x}\in C_{1}\)and \(j\in \mathbb{N}\).

Remark 2.9

By Lemma 2.8, define a mapping \(\mathbb{W}:C_{1}\to C_{1}\) such that \(\mathbb{W}\tilde{x}=\lim_{i\to \infty }\mathbb{W}_{i} \tilde{x}=\lim_{i\to \infty }\mathbb{V}_{i,1}\tilde{x}\), \(\forall \tilde{x}\in C_{1}\), which is called the \(\mathbb{W}\)-mapping generated by \(S_{1}, S_{2},\ldots \) and \(\lambda _{1},\lambda _{2},\ldots \) . In the whole paper, we consider \(0<\lambda _{i}<1\), \(\forall i\geq 1\).

Lemma 2.10

([19])

Let \(C_{1}\neq \emptyset \)be a closed convex subset of a strictly convex Banach space E. Let \(S_{1}, S_{2},\ldots \)be nonexpansive mappings of \(C_{1}\)to \(C_{1}\)such that \(\bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})\neq \emptyset \), and let \(\lambda _{1},\lambda _{2},\ldots \)be real numbers satisfying \(0<\lambda _{i}<1\), \(\forall i\geq 1\). Then \(\operatorname{Fix}(\mathbb{W})=\bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})\).

Lemma 2.11

([20])

Let \(C_{1}\neq \emptyset \)be a closed convex subset of \(H_{1}\). Let \(S_{1}, S_{2},\ldots \)be nonexpansive mappings of \(C_{1}\)to \(C_{1}\)such that \(\bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})\neq \emptyset \), and let \(\lambda _{1},\lambda _{2},\ldots \)be real numbers satisfying \(0<\lambda _{i}<1\), \(\forall i\geq 1\). For any bounded subset K of \(C_{1}\), \(\lim_{i\to \infty }\sup_{\tilde{x}\in K}\| \mathbb{W}_{i}\tilde{x}-\mathbb{W}\tilde{x}\|=0 \).

Main result

We study the following convergence result for a new type iterative method to find a common solution of SPMVIP(1.6)–(1.7), VIP(1.1), and fixed point problem.

Theorem 3.1

Let \(H_{1}\)and \(H_{2}\)denote the Hilbert spaces and \(C_{1}\subset H_{1}\)be a nonempty closed convex subset of Hilbert space \(H_{1}\). Let \(D:C_{1}\to H_{1}\)be a γinverse strongly monotone mapping, \(B: H_{1}\to H_{2}\)be a bounded linear operator with its adjoint operator \(B^{*}\), \(M_{1}:C_{1}\to 2^{H_{1}}\), and \(M_{2}:H_{2}\to 2^{H_{2}}\)be multi-valued maximal monotone operators and \(f:H_{2}\to H_{2}\)be an α-inverse strongly monotone mapping. Let \(g:C_{1}\to C_{1}\)be a contraction mapping with constant \(\tau \in (0,1)\), A be a strongly positive bounded linear self-adjoint operator on \(C_{1}\)with constant \(\bar{\theta }>0\)such that \(0<\theta <\frac{\bar{\theta }}{\tau }<\theta +\frac{1}{\tau }\), and \(\{S_{i}\}_{i=1}^{\infty }: C_{1}\to C_{1}\)be an infinite family of nonexpansive mappings such that \(\Gamma :=\Lambda \cap\operatorname{Sol}(\mathrm{VIP}(\mbox{1.1}))\cap (\bigcap_{i=1}^{ \infty }\operatorname{Fix}(S_{i})) \neq \emptyset \). Let \(\{x_{n}\}\)be a sequence generated as follows:

$$ \left . \textstyle\begin{array}{lll} x_{1}\in C_{1}, \\ v_{n}=J^{M_{1}}_{\rho _{1}}[x_{n}+\eta B^{*}(Q-I)Bx_{n}], \\ u_{n}=P_{C_{1}}(v_{n}-\sigma _{n}Dv_{n}), \\ x_{n+1}=\mu _{n}\theta g(\mathbb{W}_{n}x_{n})+\delta _{n}x_{n}+((1- \delta _{n})I-\mu _{n}A)\mathbb{W}_{n}u_{n}, \quad n\geq 1, \end{array}\displaystyle \right \} $$
(3.1)

where \(\mathbb{W}_{n}\)is defined in (1.10), \(Q=J^{f,M_{2}}_{\rho _{2}}(I-\rho _{2}f)\), \(\{\mu _{n}\}, \{\delta _{n}\}\subset (0,1)\)and \(\eta \in (0,\frac{1}{\epsilon })\), ϵ is the spectral radius of \(B^{*}B\). Let the control sequences satisfy the following conditions:

  1. (i)

    \(\lim_{n\to \infty }\mu _{n}=0\), \(\sum_{n=1}^{\infty } \mu _{n}=\infty \);

  2. (ii)

    \(\rho _{1}> 0\), \(0<\rho _{2}<2\alpha \);

  3. (iii)

    \(0<\liminf_{n\to \infty }\delta _{n}\leq \limsup_{n \to \infty }\delta _{n}<1\);

  4. (iv)

    \(0<\liminf_{n\to \infty }\sigma _{n}\leq \limsup_{n \to \infty }\sigma _{n}<2\gamma \); \(\sum_{n=1}^{\infty }| \sigma _{n+1}-\sigma _{n}|<\infty \).

Then the sequence \(\{x_{n}\}\)converges strongly to some \(\tilde{x}\in \Gamma \), where \(\tilde{x}=P_{\Gamma }(\theta g+(I-A))\tilde{x}\)which solves

$$ \bigl\langle (A-\theta g)\tilde{x},v-\tilde{x}\bigr\rangle \geq 0 \quad \textit{for all } v \in \Gamma . $$
(3.2)

Proof

For the sake of simplicity, we divide the proof into several steps.

Step 1. We prove that \(\{x_{n}\}\) is bounded.

Let \(\tilde{x}\in \Gamma \), then \(\tilde{x}\in \Lambda \) and thus \(J^{M_{1}}_{\rho _{1}}\tilde{x}=\tilde{x}\), \(J^{f,M_{2}}_{\rho _{2}}(I-\rho _{2}f)B\tilde{x}=B\tilde{x}\) and \((I+\eta B^{*}(Q-I)B)\tilde{x}=\tilde{x}\). By Lemma 2.2 and firm nonexpansiveness, \(J^{M_{1}}_{\rho _{1}}\) and \(J^{f,M_{2}}_{\rho _{2}}(I-\rho _{2}f)\) are averaged. Also, \((I+\eta B^{*}(Q-I)B)\) is averaged since it is \(\frac{\nu }{\epsilon }\)-ism for some \(\nu >\frac{1}{2}\). From Lemma 2.1(iii), \(I-Q\) is ν-ism. Thus, we obtain

$$\begin{aligned} \bigl\langle B^{*}(I-Q)Bx_{1}-B^{*}(I-Q)Bx_{2}, x_{1}-x_{2}\bigr\rangle =&\bigl\langle (I-Q)Bx_{1}-(I-Q)Bx_{2}, \\ &{}Bx_{1}-Bx_{2}\bigr\rangle \\ \geq &\nu \bigl\Vert (I-Q)Bx_{1}-(I-Q)Bx_{2} \bigr\Vert ^{2} \\ \geq &\frac{\nu }{\epsilon } \bigl\Vert B^{*}(I-Q)Bx_{1} \\ &{}-B^{*}(I-Q)Bx_{2} \bigr\Vert ^{2}. \end{aligned}$$
(3.3)

This implies that \(\eta B^{*}(I-Q)B\) is \(\frac{\nu }{\eta \epsilon }\)-ism. Since \(0<\eta <\frac{1}{\epsilon }\), its complement \((I-\eta B^{*}(I-Q)B)\) is averaged and hence \(J^{M_{1}}_{\rho _{1}}[I+\eta B^{*}(Q-I)B]=\mathbb{R}({\mathrm{say}})\). Thus, \(I+\eta B^{*}(Q-I)B\), \(J^{M_{1}}_{\rho _{1}}\), Q and \(\mathbb{R}\) are nonexpansive mappings.

Next, we calculate

$$\begin{aligned} \Vert v_{n}-\tilde{x} \Vert ^{2} =& \bigl\Vert J^{M_{1}}_{\rho _{1}} \bigl(x_{n}+\eta B^{*}(Q-I)Bx_{n} \bigr)-J^{M_{1}}_{ \rho _{1}}\tilde{x} \bigr\Vert ^{2} \\ \leq & \bigl\Vert x_{n}+\eta B^{*}(Q-I)Bx_{n}- \tilde{x} \bigr\Vert ^{2} \\ =& \Vert x_{n}-\tilde{x} \Vert ^{2}+\eta ^{2} \bigl\Vert B^{*}(Q-I)Bx_{n} \bigr\Vert ^{2} \\ &{}+2\eta \bigl\langle x_{n}-\tilde{x}, B^{*}(Q-I)Bx_{n} \bigr\rangle , \end{aligned}$$
(3.4)

and hence

$$\begin{aligned} \Vert v_{n}-\tilde{x} \Vert ^{2} \leq & \Vert x_{n}-\tilde{x} \Vert ^{2}+\eta ^{2} \bigl\langle (Q-I)Bx_{n}, BB^{*}(Q-I)Bx_{n} \bigr\rangle \\ &{}+2\eta \bigl\langle x_{n}-\tilde{x}, B^{*}(Q-I)Bx_{n} \bigr\rangle . \end{aligned}$$
(3.5)

Consider \(\Upsilon _{1}:=\eta ^{2}\langle (Q-I)Bx_{n}, BB^{*}(Q-I)Bx_{n} \rangle \), and we have

$$\begin{aligned} \Upsilon _{1} =&\eta ^{2}\bigl\langle (Q-I)Bx_{n}, BB^{*}(Q-I)Bx_{n} \bigr\rangle \\ \leq &\epsilon \eta ^{2}\bigl\langle (Q-I)Bx_{n}, (Q-I)Bx_{n} \bigr\rangle \\ =&\epsilon \eta ^{2} \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2}. \end{aligned}$$
(3.6)

Also, let \(\Upsilon _{2}:=2\eta \langle x_{n}-\tilde{x}, B^{*}(Q-I)Bx_{n} \rangle \), and we calculate

$$\begin{aligned} \Upsilon _{2} =&2\eta \bigl\langle x_{n}-\tilde{x}, B^{*}(Q-I)Bx_{n} \bigr\rangle \\ =&2\eta \bigl\langle B(x_{n}-\tilde{x}), (Q-I)Bx_{n} \bigr\rangle \\ =&2\eta \bigl\langle B(x_{n}-\tilde{x})+(Q-I)Bx_{n}-(Q-I)Bx_{n}, (Q-I)Bx_{n} \bigr\rangle \\ =&2\eta \bigl(\bigl\langle QB(x_{n}-B\tilde{x}),(Q-I)Bx_{n} \bigr\rangle - \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2} \bigr) \\ \leq &2\eta (\frac{1}{2} \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2}- \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2} \\ \leq &-\eta \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2}. \end{aligned}$$
(3.7)

By (3.6) and (3.7) in (3.5), we get

$$ \Vert v_{n}-\tilde{x} \Vert ^{2} \leq \Vert x_{n}-\tilde{x} \Vert ^{2}+\eta (\epsilon \eta -1) \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2}. $$
(3.8)

Since \(0<\eta <\frac{1}{\epsilon }\), therefore

$$ \Vert v_{n}-\tilde{x} \Vert \leq \Vert x_{n}-\tilde{x} \Vert . $$
(3.9)

Using γ-ism and \(0<\sigma _{n}<2\gamma \), we have

$$\begin{aligned} \Vert u_{n}-\tilde{x} \Vert ^{2} =&\|P_{C_{1}}(v_{n}-\sigma _{n}Dv_{n})-P_{C_{1}}(v_{n}- \sigma _{n}D\tilde{x}\|^{2} \\ \leq &\|v_{n}-\sigma _{n}Dv_{n}-(v_{n}- \sigma _{n}D\tilde{x}\|^{2} \\ =& \bigl\Vert (v_{n}-\tilde{x})-\sigma _{n}(Dv_{n}-D \tilde{x}) \bigr\Vert ^{2} \\ =& \Vert v_{n}-\tilde{x} \Vert ^{2}-2\sigma _{n}\langle Dv_{n}-D\tilde{x}, v_{n}- \tilde{x} \rangle +\sigma _{n}^{2} \Vert Dv_{n}-D\tilde{x} \Vert ^{2} \\ \leq & \Vert v_{n}-\tilde{x} \Vert ^{2}-2\sigma _{n}\gamma \Vert Dv_{n}-D\tilde{x} \Vert ^{2}+\sigma _{n}^{2} \Vert Dv_{n}-D\tilde{x} \Vert ^{2} \\ =& \Vert v_{n}-\tilde{x} \Vert ^{2}+\sigma _{n}(\sigma _{n}-2\gamma ) \Vert Dv_{n}-D \tilde{x} \Vert ^{2} \\ \leq & \Vert v_{n}-\tilde{x} \Vert ^{2}, \end{aligned}$$
(3.10)

this implies

$$ \Vert u_{n}-\tilde{x} \Vert \leq \Vert v_{n}-\tilde{x} \Vert . $$
(3.11)

By using (3.9) and (3.11), we calculate

$$\begin{aligned} \Vert x_{n+1}-\tilde{x} \Vert =& \bigl\Vert \mu _{n}\theta g(\mathbb{W}_{n} x_{n})+ \delta _{n} x_{n}+\bigl((1-\delta _{n})I- \mu _{n}A\bigr)\mathbb{W}_{n}u_{n}- \tilde{x} \bigr\Vert \\ =& \bigl\Vert \mu _{n}\bigl(\theta g(\mathbb{W}_{n}x_{n})-A \tilde{x}\bigr)+\delta _{n}(x_{n}- \tilde{x})+\bigl((1- \delta _{n})I-\mu _{n}A\bigr) (\mathbb{W}_{n}u_{n}- \tilde{x}) \bigr\Vert \\ \leq &\mu _{n} \bigl\Vert \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x} \bigr\Vert +\delta _{n} \Vert x_{n}- \tilde{x} \Vert +\bigl((1-\delta _{n})I-\mu _{n} \overline{\theta }\bigr) \Vert \mathbb{W}_{n}u_{n}- \tilde{x} \Vert \\ \leq &\mu _{n} \bigl\Vert \theta g(\mathbb{W}_{n}x_{n})- \theta g(\tilde{x})+ \theta g(\tilde{x})-A\tilde{x} \bigr\Vert \\ &{}+\delta _{n} \Vert x_{n}-\tilde{x} \Vert + \bigl((1-\delta _{n})I-\mu _{n} \overline{\theta }\bigr) \Vert u_{n}-\tilde{x} \Vert \\ \leq &\mu _{n}\theta \bigl\Vert g(\mathbb{W}_{n}x_{n})- g(\tilde{x}) \bigr\Vert +\mu _{n} \bigl\Vert \theta g( \tilde{x})-A\tilde{x} \bigr\Vert \\ &{}+\delta _{n} \Vert x_{n}-\tilde{x} \Vert + \bigl((1-\delta _{n})I-\mu _{n} \overline{\theta }\bigr) \Vert x_{n}-\tilde{x} \Vert \\ \leq &\mu _{n}\theta \tau \Vert x_{n}- \tilde{x} \Vert +\mu _{n} \bigl\Vert \theta g( \tilde{x})-A\tilde{x} \bigr\Vert +(1-\mu _{n}\overline{\theta }) \Vert x_{n}- \tilde{x} \Vert \\ \leq &\bigl(1-\mu _{n}(\overline{\theta }-\theta \tau )\bigr) \Vert x_{n}-\tilde{x} \Vert +\mu _{n} \bigl\Vert \theta g(\tilde{x})-A\tilde{x} \bigr\Vert \\ \leq &\max \biggl\{ \Vert x_{n}-\tilde{x} \Vert , \frac{ \Vert \theta g(\tilde{x})-A\tilde{x} \Vert }{\overline{\theta }-\theta \tau } \biggr\} , \quad n\geq 1. \end{aligned}$$
(3.12)

Using induction, we get

$$ \Vert x_{n+1}-\tilde{x} \Vert \leq \max \biggl\{ \Vert x_{1}-\tilde{x} \Vert , \frac{ \Vert \theta g(\tilde{x})-A\tilde{x} \Vert }{\overline{\theta }-\theta \tau } \biggr\} . $$

Thus, \(\{x_{n}\}\) is bounded and hence \(\{u_{n}\}\), \(\{\mathbb{W}u_{n}\}\), and \(\{g(\mathbb{W}_{n}x_{n})\}\).

Step 2. We show that \(\lim_{n\to \infty }\|x_{n+1}-x_{n}\|=0\), \(\lim_{n\to \infty }\|x_{n}-\mathbb{W}_{n}u_{n}\|=0\), \(\lim_{n\to \infty }\|v_{n}-x_{n}\|=0\), and \(\lim_{n\to \infty }\|v_{n}-u_{n}\|=0\).

Since \(J^{M_{1}}_{\rho _{1}}[I+\eta B^{*}(Q-I)B]\) is nonexpansive, therefore

$$\begin{aligned} \Vert v_{n+1}-v_{n} \Vert =& \bigl\Vert J^{M_{1}}_{\rho _{1}}\bigl[I+\eta B^{*}(Q-I)B \bigr]x_{n+1}-J^{M_{1}}_{ \rho _{1}}\bigl[I+\eta B^{*}(Q-I)B\bigr]x_{n} \bigr\Vert \\ \leq & \Vert x_{n+1}-x_{n} \Vert . \end{aligned}$$
(3.13)

Using (3.13), we estimate

$$\begin{aligned} \Vert u_{n+1}-u_{n} \Vert =& \bigl\Vert P_{C}(I-\sigma _{n+1}A)v_{n+1}-P_{C}(I- \sigma _{n}S)v_{n} \bigr\Vert \\ \leq & \bigl\Vert (I-\sigma _{n+1}A)v_{n+1}-(I-\sigma _{n}A)v_{n} \bigr\Vert \\ =& \bigl\Vert (I-\sigma _{n+1}A)v_{n+1}-(I-\sigma _{n+1}A)v_{n}+(\sigma _{n}- \sigma _{n+1})Av_{n} \bigr\Vert \\ \leq & \Vert v_{n+1}-v_{n} \Vert + \vert \sigma _{n}-\sigma _{n+1} \vert \Vert Av_{n} \Vert \\ \leq & \Vert x_{n+1}-x_{n} \Vert + \vert \sigma _{n}-\sigma _{n+1} \vert \Vert Av_{n} \Vert \\ \leq & \Vert x_{n+1}-x_{n} \Vert + \mathbb{N}_{1} \vert \sigma _{n}-\sigma _{n+1} \vert , \end{aligned}$$
(3.14)

where \(\mathbb{N}_{1}=\sup_{n\geq 1}\|Av_{n}\|\).

For \(i\in 1,2,\ldots,n\), \(S_{i}\) and \(\mathbb{V}_{n,i}\) are nonexpansive, therefore from (1.10) we obtain

$$\begin{aligned} \Vert \mathbb{W}_{n+1}u_{n}- \mathbb{W}_{n}u_{n} \Vert =& \Vert \lambda _{1}S_{1} \mathbb{V}_{n+1,2}u_{n}- \lambda _{1}S_{1}\mathbb{V}_{n,2}u_{n} \Vert \\ \leq &\lambda _{1} \Vert \mathbb{V}_{n+1,2}u_{n}- \mathbb{V}_{n,2}u_{n} \Vert \\ \leq &\lambda _{1} \Vert \lambda _{2}S_{2} \mathbb{V}_{n+1,3}u_{n}- \lambda _{2}S_{2} \mathbb{V}_{n,3}u_{n} \Vert \\ \leq &\lambda _{1}\lambda _{2} \Vert \mathbb{V}_{n+1,3}u_{n}-\mathbb{V}_{n,3}u_{n} \Vert \\ \vdots & \\ \leq &\lambda _{1}\lambda _{2}...\lambda _{n} \Vert \mathbb{V}_{n+1,n+1}u_{n}- \mathbb{V}_{n,n+1}u_{n} \Vert \\ \leq &\mathbb{N}_{2}\prod_{i=1}^{n} \lambda _{i}, \end{aligned}$$
(3.15)

where \(\mathbb{N}_{2}\geq 0\) with \(\|\mathbb{V}_{n+1,n+1}u_{n}-\mathbb{V}_{n,n+1}u_{n}\|\leq \mathbb{N}_{2}\), \(\forall n\geq 1\).

Setting \(x_{n+1}=(1-\delta _{n})s_{n}+\delta _{n}x_{n}\), then we have \(s_{n}=\frac{x_{n+1}-\delta _{n}x_{n}}{1-\delta _{n}}\) and

$$\begin{aligned} s_{n+1}-s_{n} =&\frac{\mu _{n+1}\theta g(\mathbb{W}_{n+1}x_{n+1})+((1-\delta _{n+1})I-\mu _{n+1}A)\mathbb{W}_{n+1}u_{n+1}}{1-\delta _{n+1}} \\ &{}- \frac{\mu _{n}\theta g(\mathbb{W}_{n}x_{n})+((1-\delta _{n})I-\mu _{n}A)\mathbb{W}_{n}u_{n}}{1-\delta _{n}} \\ =&\biggl(\frac{\mu _{n+1}}{1-\delta _{n+1}}\biggr) \bigl(\theta g(\mathbb{W}_{n+1}x_{n+1})-A \mathbb{W}_{n+1}u_{n+1}\bigr) \\ &{}+\biggl(\frac{\mu _{n}}{1-\delta _{n}}\biggr) \bigl(A\mathbb{W}_{n}u_{n}- \theta g( \mathbb{W}_{n}x_{n})\bigr)+ \mathbb{W}_{n+1}u_{n+1}-\mathbb{W}_{n}y_{n} \\ =&\biggl(\frac{\mu _{n+1}}{1-\delta _{n+1}}\biggr) \bigl(\theta g(\mathbb{W}_{n+1}x_{n+1})-A \mathbb{W}_{n+1}u_{n+1}\bigr) \\ &{}+\biggl(\frac{\mu _{n}}{1-\delta _{n}}\biggr) \bigl(A\mathbb{W}_{n}u_{n}- \theta g( \mathbb{W}_{n}x_{n})\bigr) \\ &{}+\mathbb{W}_{n+1}u_{n+1}-\mathbb{W}_{n+1}u_{n}+ \mathbb{W}_{n+1}u_{n}- \mathbb{W}_{n}u_{n}. \end{aligned}$$
(3.16)

Hence,

$$\begin{aligned} \Vert s_{n+1}-s_{n} \Vert \leq & \frac{\mu _{n+1}}{1-\delta _{n+1}}\bigl( \bigl\Vert \theta g( \mathbb{W}_{n+1}x_{n+1}) \bigr\Vert + \Vert A\mathbb{W}_{n+1}u_{n+1} \Vert \bigr) \\ &{}+\frac{\mu _{n}}{1-\delta _{n}}\bigl( \Vert A\mathbb{W}_{n}u_{n} \Vert + \bigl\Vert \theta g( \mathbb{W}_{n}x_{n}) \bigr\Vert \bigr) \\ &{}+ \Vert \mathbb{W}_{n+1}u_{n+1}- \mathbb{W}_{n+1}u_{n} \Vert + \Vert \mathbb{W}_{n+1}u_{n}- \mathbb{W}_{n}u_{n} \Vert \\ \leq &\frac{\mu _{n+1}}{1-\delta _{n+1}}\mathbb{N}_{3}+ \frac{\mu _{n}}{1-\delta _{n}} \mathbb{N}_{4} \\ &{}+ \Vert u_{n+1}-u_{n} \Vert + \Vert \mathbb{W}_{n+1}u_{n}-\mathbb{W}_{n}u_{n} \Vert , \end{aligned}$$
(3.17)

where \(\mathbb{N}_{3}=\sup_{n\geq 1}(\|\theta g(\mathbb{W}_{n+1}x_{n+1})\|+ \|A\mathbb{W}_{n+1}u_{n+1}\|)\) and \(\mathbb{N}_{4}=\sup_{n\geq 1}(\|A\mathbb{W}_{n}u_{n}\|+ \|\theta g( \mathbb{W}_{n}x_{n})\|)\).

Using (3.14) and (3.15) in the above inequality

$$\begin{aligned} \Vert s_{n+1}-s_{n} \Vert \leq & \frac{\mu _{n+1}}{1-\delta _{n+1}}\mathbb{N}_{3}+ \frac{\mu _{n}}{1-\delta _{n}} \mathbb{N}_{4}+ \Vert x_{n+1}-x_{n} \Vert \\ &{}+\mathbb{N}_{1} \vert \sigma _{n}-\sigma _{n+1} \vert +\mathbb{N}_{2}\prod _{i=1}^{n} \lambda _{i}, \end{aligned}$$
(3.18)

and thus

$$\begin{aligned} \Vert s_{n+1}-s_{n} \Vert - \Vert x_{n+1}-x_{n} \Vert \leq &\frac{\mu _{n+1}}{1-\delta _{n+1}} \mathbb{N}_{3}+\frac{\mu _{n}}{1-\delta _{n}}\mathbb{N}_{4} \\ &{}+\mathbb{N}_{1} \vert \sigma _{n}-\sigma _{n+1} \vert +\mathbb{N}_{2}\prod _{i=1}^{n} \lambda _{i}. \end{aligned}$$
(3.19)

Using the given conditions in the above inequality, we have

$$ \limsup_{n\to \infty }\bigl( \Vert s_{n+1}-s_{n} \Vert - \Vert x_{n+1}-x_{n} \Vert \bigr)\leq 0. $$

By Lemma 2.4, we get

$$ \lim_{n\to \infty } \Vert s_{n}-x_{n} \Vert = 0. $$
(3.20)

As \(x_{n+1}=(1-\delta _{n})s_{n}+\delta _{n}x_{n}\) therefore

$$ \Vert x_{n+1}-x_{n} \Vert = \bigl\Vert (1-\delta _{n}) (s_{n}-x_{n}) \bigr\Vert , $$

which yields

$$ \lim_{n\to \infty } \Vert x_{n+1}-x_{n} \Vert = 0. $$
(3.21)

Now,

$$\begin{aligned} \Vert x_{n}-\mathbb{W}_{n}u_{n} \Vert =& \Vert x_{n}-x_{n+1}+x_{n+1}- \mathbb{W}_{n}u_{n} \Vert \\ \leq & \Vert x_{n+1}-x_{n} \Vert + \bigl\Vert \mu _{n}\theta g(\mathbb{W}_{n}x_{n})+ \delta _{n}x_{n} \\ &{}+\bigl((1-\delta _{n})I-\mu _{n}A\bigr) \mathbb{W}_{n}u_{n}-\mathbb{W}_{n}u_{n} \bigr\Vert \\ =& \Vert x_{n+1}-x_{n} \Vert + \bigl\Vert \mu _{n}\bigl(\theta g(\mathbb{W}_{n}x_{n})-A \mathbb{W}_{n}u_{n}\bigr) \bigr\Vert \\ &{}+\bigl((1-\delta _{n})I-\mu _{n}A\bigr) ( \mathbb{W}_{n}u_{n}-\mathbb{W}_{n}u_{n})+ \delta _{n}(x_{n}-\mathbb{W}_{n}u_{n}) \\ \leq & \Vert x_{n+1}-x_{n} \Vert +\mu _{n} \bigl\Vert \theta g(\mathbb{W}_{n}x_{n})-A \mathbb{W}_{n}u_{n} \bigr\Vert \\ &{}+\beta _{n} \Vert x_{n}-\mathbb{W}_{n}u_{n} \Vert . \end{aligned}$$
(3.22)

Hence,

$$ (1-\delta _{n}) \Vert x_{n}-\mathbb{W}_{n}u_{n} \Vert \leq \Vert x_{n+1}-x_{n} \Vert + \mu _{n} \bigl\Vert \theta g(\mathbb{W}_{n}x_{n})-A \mathbb{W}_{n}u_{n} \bigr\Vert . $$

Using the given conditions and (3.21) in (3.22), we get

$$ \lim_{n\to \infty } \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert =0. $$
(3.23)

By (3.8) and (3.11), we compute

$$\begin{aligned} \Vert x_{n+1}-\tilde{x} \Vert ^{2} =& \bigl\Vert \mu _{n}\bigl(\theta g( \mathbb{W}_{n}x_{n})-A \tilde{x}\bigr)+\delta _{n}(x_{n}-\mathbb{W}_{n}u_{n})+(1- \mu _{n} A) ( \mathbb{W}_{n}u_{n}- \tilde{x}) \bigr\Vert ^{2} \\ \leq & \bigl\Vert (1-\mu _{n} A) (\mathbb{W}_{n}u_{n}- \tilde{x})+\delta _{n}(x_{n}- \mathbb{W}_{n}u_{n}) \bigr\Vert ^{2} \\ &{}+2\bigl\langle \mu _{n}\theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ \leq &\bigl[ \bigl\Vert (1-\mu _{n} A) (\mathbb{W}_{n}u_{n}- \tilde{x}) \bigr\Vert +\delta _{n} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \bigr]^{2} \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ \leq &\bigl[(1-\mu _{n} \bar{\theta }) \Vert u_{n}- \tilde{x} \Vert +\delta _{n} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \bigr]^{2} \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ =&(1-\mu _{n} \bar{\theta })^{2} \Vert u_{n}-\tilde{x} \Vert ^{2}+\delta _{n}^{2} \Vert x_{n}-\mathbb{W}_{n}u_{n} \Vert ^{2} \\ &{}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ \leq &(1-\mu _{n} \bar{\theta })^{2}\bigl[ \Vert x_{n}-\tilde{x} \Vert ^{2}+\eta ( \epsilon \eta -1) \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2}\bigr] \\ &{}+\delta _{n}^{2} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert ^{2}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ =&\bigl(1-2\mu _{n}\bar{\theta }+(\mu _{n}\bar{\theta })^{2}\bigr) \Vert x_{n}- \tilde{x} \Vert ^{2}+(1-\mu _{n} \bar{\theta })^{2}\eta ( \epsilon \eta -1) \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2} \\ &{}+\delta _{n}^{2} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert ^{2}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ \leq & \Vert x_{n}-\tilde{x} \Vert ^{2}+(\mu _{n}\bar{\theta })^{2}) \Vert x_{n}- \tilde{x} \Vert ^{2}+(1-\mu _{n} \bar{\theta })^{2}\eta (\epsilon \eta -1) \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2} \\ &{}+\delta _{n}^{2} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert ^{2}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle . \end{aligned}$$
(3.24)

Therefore,

$$\begin{aligned} (1-\mu _{n} \bar{\theta })^{2}\eta (1- \epsilon \eta ) \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2} \leq & \Vert x_{n}-\tilde{x} \Vert ^{2}- \Vert x_{n+1}-\tilde{x} \Vert ^{2} \\ &{}+\delta _{n}^{2} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert ^{2}+\mu _{n} \bar{\theta }^{2} \Vert x_{n}- \tilde{x} \Vert ^{2} \\ &{}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ \leq &\bigl( \Vert x_{n}-\tilde{x} \Vert + \Vert x_{n+1}-\tilde{x} \Vert \bigr) \Vert x_{n}-x_{n+1} \Vert \\ &{}+\delta _{n}^{2} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert ^{2}+\mu _{n} \bar{\theta }^{2} \Vert x_{n}- \tilde{x} \Vert ^{2} \\ &{}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl(\theta \bigl\Vert g(\mathbb{W}_{n}x_{n}) \bigr\Vert \\ &{}+ \Vert A\tilde{x} \Vert \bigr) \Vert x_{n+1}-\tilde{x} \Vert . \end{aligned}$$
(3.25)

Since \(\eta (1-\epsilon \eta )>0\), \(\lim_{n \to \infty }\mu _{n}=0\) and \(\{x_{n}\}\), \(\{u_{n}\}\) are bounded, and using (3.21) and (3.23), we have

$$ \lim_{n \to \infty } \bigl\Vert (Q-I)Bx_{n} \bigr\Vert ^{2}=0. $$
(3.26)

Next, we calculate

$$\begin{aligned} \Vert v_{n}-\tilde{x} \Vert ^{2} =& \bigl\Vert J^{M_{1}}_{\rho _{1}}\bigl(x_{n}+\eta B^{*}(Q-I)Bx_{n}\bigr)- \tilde{x} \bigr\Vert ^{2} \\ \leq & \bigl\Vert J^{M_{1}}_{\rho _{1}}\bigl(x_{n}+ \eta B^{*}(Q-I)Bx_{n}\bigr)-J^{M_{1}}_{ \rho _{1}} \tilde{x} \bigr\Vert ^{2} \\ \leq &\bigl\langle v_{n}-\tilde{x},x_{n}+\eta B^{*}(Q-I)Bx_{n}-\tilde{x} \bigr\rangle \\ =&\frac{1}{2} \bigl\{ \Vert v_{n}-\tilde{x} \Vert ^{2}+ \bigl\Vert x_{n}+\eta B^{*}(Q-I)Bx_{n}- \tilde{x} \bigr\Vert ^{2}- \bigl\Vert (v_{n}- \tilde{x}) \\ &{}-\bigl[x_{n}+\eta B^{*}(Q-I)Bx_{n}- \tilde{x}\bigr] \bigr\Vert ^{2} \bigr\} \\ =&\frac{1}{2} \bigl\{ \Vert v_{n}-\tilde{x} \Vert ^{2}+ \Vert x_{n}-\tilde{x} \Vert ^{2}- \bigl\Vert v_{n}-x_{n}-\eta B^{*}(Q-I)Bx_{n} \bigr\Vert ^{2} \bigr\} \\ =&\frac{1}{2} \bigl\{ \Vert v_{n}-\tilde{x} \Vert ^{2}+ \Vert x_{n}-\tilde{x} \Vert ^{2}- \bigl[ \Vert v_{n}-x_{n} \Vert ^{2}+\eta ^{2} \bigl\Vert B^{*}(Q-I)Bx_{n} \bigr\Vert ^{2} \\ &{}-2\eta \bigl\langle v_{n}-x_{n},B^{*}(Q-I)Bx_{n} \bigr\rangle \bigr] \bigr\} . \end{aligned}$$

Hence, we obtain

$$ \Vert v_{n}-\tilde{x} \Vert ^{2} \leq \Vert x_{n}-\tilde{x} \Vert ^{2}- \Vert v_{n}-x_{n} \Vert ^{2}+2 \eta \bigl\Vert B(v_{n}-x_{n}) \bigr\Vert \bigl\Vert (Q-I)Bx_{n} \bigr\Vert . $$
(3.27)

By (3.11) and (3.24), we obtain

$$\begin{aligned} \Vert x_{n+1}-\tilde{x} \Vert ^{2} \leq &(1-\mu _{n} \bar{\theta })^{2} \Vert u_{n}- \tilde{x} \Vert ^{2}+\delta _{n}^{2} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert ^{2} \\ &{}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ \leq &(1-\mu _{n} \bar{\theta })^{2}\bigl[ \Vert x_{n}-\tilde{x} \Vert ^{2}- \Vert v_{n}-x_{n} \Vert ^{2} \\ &{}+2\eta \bigl\Vert A(u_{n}-x_{n}) \bigr\Vert \bigl\Vert (Q-I)Bx_{n} \bigr\Vert \bigr] \\ &{}+\delta _{n}^{2} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert ^{2}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ \leq & \Vert x_{n}-\tilde{x} \Vert ^{2}+(\mu _{n} \bar{\theta })^{2} \Vert x_{n}- \tilde{x} \Vert ^{2} \\ &{}-2\mu _{n}\bar{\theta } \Vert x_{n}-\tilde{x} \Vert ^{2}-(1-\mu _{n} \bar{\theta })^{2} \Vert v_{n}-x_{n} \Vert ^{2} \\ &{}+2(1-\mu _{n} \bar{\theta })^{2}\eta \bigl\Vert A(u_{n}-x_{n}) \bigr\Vert \bigl\Vert (Q-I)Bx_{n} \bigr\Vert +\delta _{n}^{2} \Vert x_{n}-\mathbb{W}_{n}u_{n} \Vert ^{2} \\ &{}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle . \end{aligned}$$
(3.28)

Hence,

$$\begin{aligned} (1-\mu _{n} \bar{\theta })^{2} \Vert v_{n}-x_{n} \Vert ^{2} \leq & \Vert x_{n}- \tilde{x} \Vert ^{2}- \Vert x_{n+1}-\tilde{x} \Vert ^{2} \\ &{}+(\mu _{n} \bar{\theta })^{2} \Vert x_{n}-\tilde{x} \Vert ^{2}-2\mu _{n} \bar{\theta } \Vert x_{n}-\tilde{x} \Vert ^{2} \\ &{}+2(1-\mu _{n} \bar{\theta })^{2}\eta \bigl\Vert A(u_{n}-x_{n}) \bigr\Vert \bigl\Vert (Q-I)Bx_{n} \bigr\Vert \\ &{}+\delta _{n}^{2} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert ^{2} \\ &{}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ \leq &\bigl( \Vert x_{n}-\tilde{x} \Vert + \Vert x_{n+1}-\tilde{x} \Vert \bigr) \Vert x_{n}-x_{n+1} \Vert \\ &{}+(\mu _{n} \bar{\theta })^{2} \Vert x_{n}-\tilde{x} \Vert ^{2}-2\mu _{n} \bar{\theta } \Vert x_{n}-\tilde{x} \Vert ^{2} \\ &{}+2(1-\mu _{n} \bar{\theta })^{2}\eta \bigl\Vert A(u_{n}-x_{n}) \bigr\Vert \bigl\Vert (Q-I)Bx_{n} \bigr\Vert \\ &{}+\delta _{n}^{2} \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert ^{2} \\ &{}+2(1-\mu _{n} \bar{\theta })\delta _{n} \Vert u_{n}-\tilde{x} \Vert \Vert x_{n}- \mathbb{W}_{n}u_{n} \Vert \\ &{}+2\mu _{n}\bigl(\theta \bigl\Vert g(\mathbb{W}_{n}x_{n}) \bigr\Vert + \Vert A\tilde{x} \Vert \bigr) \Vert x_{n+1}- \tilde{x} \Vert . \end{aligned}$$
(3.29)

As \(\{x_{n}\}\), \(\{u_{n}\}\) are bounded, and using (3.21), (3.23), (3.26) and the given conditions, we have

$$ \lim_{n \to \infty } \Vert v_{n}-x_{n} \Vert =0. $$
(3.30)

Next, we prove that \(\lim_{n\to \infty }\|v_{n}-u_{n}\|=0\).

We estimate

$$\begin{aligned} \Vert x_{n+1}-\tilde{x} \Vert ^{2} =& \bigl\Vert \mu _{n}\theta g( \mathbb{W}_{n}x_{n})+ \delta _{n}x_{n}+ \bigl((1-\delta _{n})I-\mu _{n}A\bigr) \mathbb{W}_{n}u_{n}- \tilde{x} \bigr\Vert ^{2} \\ =& \bigl\Vert (1-\delta _{n}) (\mathbb{W}_{n}u_{n}- \tilde{x})+\delta _{n}(x_{n}- \tilde{x})+\mu _{n}\bigl(\theta g(\mathbb{W}_{n}x_{n})-A \mathbb{W}_{n}u_{n}\bigr) \bigr\Vert ^{2} \\ \leq &(1-\delta _{n}) \Vert \mathbb{W}_{n}u_{n}- \tilde{x} \Vert ^{2}+\delta _{n} \Vert x_{n}-\tilde{x} \Vert ^{2}+2\mu _{n} \langle \kappa _{n},x_{n+1}- \tilde{x} \rangle \\ \leq &(1-\delta _{n}) \Vert \mathbb{W}_{n}u_{n}- \tilde{x} \Vert ^{2}+\delta _{n} \Vert x_{n}-\tilde{x} \Vert ^{2}+2\omega ^{2}\mu _{n} \\ \leq &(1-\delta _{n}) \Vert u_{n}-\tilde{x} \Vert ^{2}+\delta _{n} \Vert x_{n}- \tilde{x} \Vert ^{2}+2\omega ^{2}\mu _{n}. \end{aligned}$$
(3.31)

In the above inequality we set \(\kappa _{n}=\theta g(\mathbb{W}_{n}x_{n})-A\mathbb{W}_{n}u_{n}\), and let \(\omega >0\) be a suitable constant with \(\omega \geq \sup_{n}\{\|\kappa _{n}\|,\|x_{n}-\tilde{x}\|\}\). Thus,

$$\begin{aligned} \Vert x_{n+1}-\tilde{x} \Vert ^{2} \leq &(1-\delta _{n}) \Vert u_{n}-\tilde{x} \Vert ^{2}+ \delta _{n} \Vert x_{n}-\tilde{x} \Vert ^{2}+2\omega ^{2}\mu _{n} \\ \leq &(1-\delta _{n})\bigl\{ \bigl\Vert P_{C_{1}}(v_{n}- \sigma _{n}Dv_{n})-P_{C_{1}}( \tilde{x}-\sigma _{n}D\tilde{x}) \bigr\Vert ^{2}\bigr\} \\ &{}+\delta _{n} \Vert x_{n}-\tilde{x} \Vert ^{2}+2\omega ^{2}\mu _{n} \\ \leq &(1-\delta _{n})\bigl\{ \Vert v_{n}-\tilde{x} \Vert ^{2}+\sigma _{n}(\sigma _{n}-2 \gamma ) \Vert Dv_{n}-D\tilde{x} \Vert ^{2}\bigr\} \\ &{}+\delta _{n} \Vert x_{n}-\tilde{x} \Vert ^{2}+2\omega ^{2}\mu _{n} \\ \leq &(1-\delta _{n})\bigl\{ \Vert x_{n}-\tilde{x} \Vert ^{2}+\sigma _{n}(\sigma _{n}-2 \gamma ) \Vert Dv_{n}-D\tilde{x} \Vert ^{2}\bigr\} \\ &{}+\delta _{n} \Vert x_{n}-\tilde{x} \Vert ^{2}+2\omega ^{2}\mu _{n} \\ \leq &(1-\delta _{n})\sigma _{n}(\sigma _{n}-2\omega ) \Vert Dv_{n}-D \tilde{x} \Vert ^{2} \\ &{}+ \Vert x_{n}-\tilde{x} \Vert ^{2}+2\omega ^{2}\mu _{n}, \end{aligned}$$
(3.32)

which implies

$$ \begin{aligned} (1-\delta _{n})\sigma _{n}(2 \omega -\sigma _{n}) \Vert Dv_{n}-D\tilde{x} \Vert ^{2}& \leq \Vert x_{n}-\tilde{x} \Vert ^{2}- \Vert x_{n+1}-\tilde{x} \Vert ^{2}+2 \omega ^{2} \mu _{n} \\ &\leq \bigl( \Vert x_{n}-\tilde{x} \Vert + \Vert x_{n+1}-\tilde{x} \Vert \bigr) \Vert x_{n}-x_{n+1} \Vert +2 \omega ^{2}\mu _{n}. \end{aligned} $$

By (3.21) and the given conditions, we get

$$ \lim_{n\to \infty } \Vert Dv_{n}-D \tilde{x} \Vert =0. $$
(3.33)

From (2.7), we compute

$$\begin{aligned} \Vert u_{n}-\tilde{x} \Vert ^{2} =& \bigl\Vert P_{C_{1}}(v_{n}-\sigma _{n}Dv_{n})-P_{C_{1}}( \tilde{x}-\sigma _{n}D\tilde{x}) \bigr\Vert ^{2} \\ \leq &\bigl\langle u_{n}-\tilde{x},(v_{n}-\sigma _{n}Dv_{n})-(\tilde{x}- \sigma _{n}D \tilde{x}) \bigr\rangle \\ \leq &\frac{1}{2}\bigl\{ \Vert u_{n}-\tilde{x} \Vert ^{2}+ \bigl\Vert (v_{n}-\sigma _{n}Dv_{n}) \\ &{}-(\tilde{x}-\sigma _{n}D\tilde{x}) \bigr\Vert ^{2}- \bigl\Vert (u_{n}-v_{n})+\sigma _{n}(Dv_{n}-D \tilde{x}) \bigr\Vert ^{2} \bigr\} \\ \leq &\frac{1}{2}\bigl\{ \Vert u_{n}-\tilde{x} \Vert ^{2}+ \Vert v_{n}-\tilde{x} \Vert ^{2}- \bigl\Vert (u_{n}-v_{n})+\sigma _{n}(Dv_{n}-D \tilde{x}) \bigr\Vert ^{2}\bigr\} \\ \leq & \Vert v_{n}-\tilde{x} \Vert ^{2}- \Vert u_{n}-v_{n} \Vert ^{2}-\sigma _{n}^{2} \Vert Dv_{n}-D \tilde{x} \Vert ^{2} \\ &{}+2\sigma _{n}\langle u_{n}-v_{n}, Du_{n}-D\tilde{x} \rangle \\ \leq & \Vert v_{n}-\tilde{x} \Vert ^{2}- \Vert u_{n}-v_{n} \Vert ^{2}+2\sigma _{n} \Vert u_{n}-v_{n} \Vert \Vert Dv_{n}-D\tilde{x} \Vert \\ \leq & \Vert x_{n}-\tilde{x} \Vert ^{2}- \Vert u_{n}-v_{n} \Vert ^{2}+2\sigma _{n} \Vert u_{n}-v_{n} \Vert \Vert Dv_{n}-D\tilde{x} \Vert . \end{aligned}$$

By (3.32), we obtain

$$\begin{aligned} \Vert x_{n+1}-\tilde{x} \Vert ^{2} \leq &(1-\delta _{n}) \Vert u_{n}-\tilde{x} \Vert ^{2}+ \delta _{n} \Vert x_{n}-\tilde{x} \Vert ^{2}+2\omega ^{2}\mu _{n} \\ \leq &(1-\delta _{n})\bigl\{ \Vert x_{n}-\tilde{x} \Vert ^{2}- \Vert u_{n}-v_{n} \Vert ^{2} \\ &{}+2\sigma _{n} \Vert u_{n}-v_{n} \Vert \Vert Dv_{n}-D\tilde{x} \Vert \bigr\} +\delta _{n} \Vert x_{n}- \tilde{x} \Vert ^{2}+2\omega ^{2}\mu _{n}, \end{aligned}$$

which implies

$$\begin{aligned} (1-\delta _{n}) \Vert u_{n}-v_{n} \Vert ^{2} \leq & \Vert x_{n}-\tilde{x} \Vert ^{2}- \Vert x_{n+1}- \tilde{x} \Vert ^{2} \\ &{}+2(1-\delta _{n})\sigma _{n} \Vert u_{n}-v_{n} \Vert \Vert Dv_{n}-D \tilde{x} \Vert +2 \omega ^{2}\mu _{n} \\ \leq &\bigl( \Vert x_{n}-\tilde{x} \Vert + \Vert x_{n+1}-\tilde{x} \Vert \bigr) \Vert x_{n}-x_{n+1} \Vert \\ &{}+2(1-\delta _{n})\sigma _{n} \Vert u_{n}-v_{n} \Vert \Vert Dv_{n}-D \tilde{x} \Vert +2 \omega ^{2}\mu _{n}. \end{aligned}$$

Using (3.21), (3.33) and the given conditions, we get

$$ \lim_{n\to \infty } \Vert u_{n}-v_{n} \Vert =0. $$
(3.34)

By (3.23), (3.30), and (3.34), we get

$$ \lim_{n\to \infty } \Vert \mathbb{W}_{n}u_{n}-u_{n} \Vert =0. $$
(3.35)

By Lemma 2.11, we have \(\lim_{n\to \infty }\|\mathbb{W}u_{n}-\mathbb{W}_{n}u_{n}\|=0\). Thus,

$$ \lim_{n\to \infty } \Vert \mathbb{W}u_{n}-u_{n} \Vert =0. $$
(3.36)

Step 3. We claim that \(\tilde{x}\in \Gamma \).

Since \(\{x_{n}\}\) is bounded, therefore consider \(\tilde{x}\in H_{1}\) to be any weak cluster point of \(\{x_{n}\}\). Hence, there exists a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) with \(x_{n}\rightharpoonup \tilde{x}\). By Lemma 2.7 and (3.35), we have \(\tilde{x}\in \bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})\). And \(v_{n_{j}}=J^{M_{1}}_{\rho _{1}}[x_{n_{j}}+\eta B^{*}(Q-I)Bx_{n_{j}}]\) can be rewritten as

$$ \frac{(x_{n_{j}}-v_{n_{j}})+B^{*}(Q-I)Bx_{n_{j}}}{\rho _{1}}\in M_{1}v_{n_{j}}. $$
(3.37)

Taking \(j\to \infty \) in (3.37) and by (3.26), (3.30) and the concept of the graph of a maximal monotone mapping, we get \(0\in M_{1}\tilde{x}\), that is, \(\tilde{x}\in \operatorname{Sol}(\mathrm{MVIP}(\mbox{1.6}))\). Furthermore, since \(\{x_{n}\}\) and \(\{v_{n}\}\) have the same asymptotical behavior, \(Bx_{n_{j}}\rightharpoonup B\tilde{x}\). As Q is nonexpansive, by (3.26)and Lemma 2.7, we get \((I-Q)B\tilde{x}=0\). Hence, by Lemma 2.3, \(0\in f(B\tilde{x})+M_{1}B\tilde{x}\), that is, \(B\tilde{x}\in \operatorname{Sol}(\mathrm{MVIP}(\mbox{1.7}))\). Thus, \(\tilde{x}\in \Lambda \).

Next, we prove \(\tilde{x}\in \operatorname{Sol}(\mathrm{VIP}(\mbox{1.1}))\). Since \(\lim_{n\to \infty }\|v_{n}-u_{n}\|=0\) and \(\lim_{n\to \infty }\|v_{n}-x_{n}\|=0\), there exist subsequences \(\{v_{n_{i}}\}\) and \(\{u_{n_{i}}\}\) of \(\{v_{n}\}\) and \(\{u_{n}\}\), respectively, such that \(v_{n_{i}}\rightharpoonup \tilde{x}\) and \(u_{n_{i}}\rightharpoonup \tilde{x}\).

Define the mapping \(\mathbb{M}\) as

$$\begin{aligned} \mathbb{M}(z_{1}) =\textstyle\begin{cases} D(z_{1})+\mathbb{N}_{C_{1}}(z_{1}), & \text{if } z_{1} \in C_{1} , \\ \emptyset , & \text{if } z_{1} \notin C_{1}, \end{cases}\displaystyle \end{aligned}$$
(3.38)

where \(\mathbb{N}_{C_{1}}(z_{1}):=\{z_{2}\in H_{1}:\langle z_{1}-y,z_{2} \rangle \geq 0, \forall y\in C_{1}\}\) is the normal cone to \(C_{1}\) at \(z_{1} \in H_{1}\). Thus, \(\mathbb{M}\) is a maximal monotone mapping, and hence \(0\in \mathbb{M}z_{1}\) if and only if \(z_{1}\in \operatorname{Sol}(\mathrm{VIP}(\mbox{1.1}))\). Let \((z_{1},z_{2})\in \operatorname{graph}(\mathbb{M})\). Then we have \(z_{2}\in \mathbb{M}z_{1}=Dz_{1}+\mathbb{N}_{C_{1}}(z_{1})\), and hence \(z_{2}-Dz_{1}\in \mathbb{N}_{C_{1}}(z_{1})\). So, we have \(\langle z_{1}-y,z_{2}-Dz_{1}\rangle \geq 0\) for all \(y\in C_{1}\). On the other hand, from \(u_{n}=P_{C_{1}}(v_{n}-\sigma _{n}Dv_{n})\) and \(z_{1}\in C_{1}\), we have

$$ \bigl\langle (v_{n}-\sigma _{n}Dv_{n})-u_{n},u_{n}-z_{1} \bigr\rangle \geq 0. $$

This implies that

$$ \biggl\langle z_{1}-u_{n}, \frac{u_{n}-v_{n}}{\sigma _{n}}+Dv_{n} \biggr\rangle \geq 0. $$

Since \(\langle z_{1}-y,z_{2}-Dz_{1}\rangle \geq 0\), for all \(y\in C_{1}\) and \(u_{n_{i}}\in C_{1}\), using the monotonicity of D, we have

$$\begin{aligned} \langle z_{1}-u_{n_{i}},z_{2}\rangle \geq & \langle z_{1}-u_{n_{i}}, Dz_{1} \rangle \\ \geq &\langle z_{1}-u_{n_{i}}, Dz_{1} \rangle - \biggl\langle z_{1}-u_{n_{i}}, \frac{u_{n_{i}}-v_{n_{i}}}{\sigma _{n}}+Du_{n_{i}} \biggr\rangle \\ =&\langle z_{1}-u_{n_{i}}, Dz_{1}-Du_{n_{i}} \rangle +\langle z_{1}-u_{n_{i}}, Du_{n_{i}}-Dv_{n_{i}} \rangle \\ &{}-\biggl\langle z_{1}-u_{n_{i}}, \frac{u_{n_{i}}-v_{n_{i}}}{\sigma _{n}} \biggr\rangle \\ \geq &\langle z_{1}-u_{n_{i}}, Du_{n_{i}}-Dv_{n_{i}} \rangle - \biggl\langle z_{1}-u_{n_{i}}, \frac{u_{n_{i}}-v_{n_{i}}}{\sigma _{n}} \biggr\rangle . \end{aligned}$$

Since D is continuous, on taking limit \(i\to \infty \), we have \(\langle z_{1}-\tilde{x}, z_{2} \rangle \geq 0\). Since \(\mathbb{M}\) is maximal monotone, we have \(\tilde{x}\in \mathbb{M}^{-1}(0)\) and hence \(\tilde{x}\in \operatorname{Sol}(\mathrm{VIP}(\mbox{1.1}))\). Thus, \(\tilde{x}\in \Gamma \).

Step 4. Finally, we prove that \(\lim \sup_{n \rightarrow \infty } \langle (\theta g-A)z,x_{n}-z \rangle \leq 0\), where \(z=P_{\Gamma }(I-A+\theta g)z\) and \(x_{n}\to \tilde{x}\).

By (2.3) and (3.23), we obtain

$$\begin{aligned} \lim \sup_{n \rightarrow \infty } \bigl\langle (\theta g-A)z,x_{n}-z \bigr\rangle =&\lim \sup_{n \rightarrow \infty } \bigl\langle (\theta g-A)z, \mathbb{W}_{n}u_{n}-z\bigr\rangle \\ \leq &\lim \sup_{i \rightarrow \infty } \bigl\langle (\theta g-A)z, \mathbb{W}_{n}u_{n_{i}}-z\bigr\rangle \\ =&\bigl\langle (\theta g-A)z,\tilde{x}-z\bigr\rangle \\ \leq & 0. \end{aligned}$$
(3.39)

Using (3.9) and (3.11), we calculate

$$\begin{aligned} \Vert x_{n+1}-\tilde{x} \Vert ^{2} =&\bigl\langle \mu _{n} \bigl(\theta g(\mathbb{W}_{n}x_{n})-A \tilde{x}\bigr) \\ &{}+\delta _{n} (x_{n}-\tilde{x})+\bigl((1-\delta _{n})I-\mu _{n} A\bigr) ( \mathbb{W}_{n}u_{n}- \tilde{x}),x_{n+1}-\tilde{x}\bigr\rangle \\ =&\mu _{n} \bigl\langle \theta g(\mathbb{W}_{n}x_{n})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle +\delta _{n} \langle x_{n}-\tilde{x},x_{n+1}- \tilde{x}\rangle \\ &{}+ \bigl\langle \bigl((1-\delta _{n})I-\mu _{n} A \bigr) (\mathbb{W}_{n}u_{n}- \tilde{x}),x_{n+1}- \tilde{x}\bigr\rangle \\ \leq &\mu _{n} \bigl(\theta \bigl\langle g(\mathbb{W}_{n}x_{n})-g( \tilde{x}),x_{n+1}-\tilde{x}\bigr\rangle + \bigl\langle \theta g( \tilde{x})-A \tilde{x},x_{n+1}-\tilde{x}\bigr\rangle \bigr) \\ &{}+\delta _{n} \Vert x_{n}-\tilde{x} \Vert \Vert x_{n+1}-\tilde{x} \Vert \\ &{}+ \bigl\Vert (1-\delta _{n})I-\mu _{n} A \bigr\Vert \Vert \mathbb{W}_{n}u_{n}-\tilde{x} \Vert \Vert x_{n+1}- \tilde{x} \Vert \\ \leq &\mu _{n}\tau \theta \Vert x_{n}-\tilde{x} \Vert \Vert x_{n+1}-\tilde{x} \Vert + \mu _{n}\bigl\langle \theta g(\tilde{x})-A\tilde{x},x_{n+1}-\tilde{x} \bigr\rangle \\ &{}+\delta _{n} \Vert x_{n}-\tilde{x} \Vert \Vert x_{n+1}-\tilde{x} \Vert +(1-\delta _{n}- \mu _{n} \bar{\theta }) \Vert u_{n}-\tilde{x} \Vert \Vert x_{n+1}-\tilde{x} \Vert \\ \leq &\mu _{n}\tau \theta \Vert x_{n}-\tilde{x} \Vert \Vert x_{n+1}-\tilde{x} \Vert + \mu _{n}\bigl\langle \theta g(\tilde{x})-A\tilde{x},x_{n+1}-\tilde{x} \bigr\rangle \\ &{}+\delta _{n} \Vert x_{n}-\tilde{x} \Vert \Vert x_{n+1}-\tilde{x} \Vert +(1-\delta _{n}- \mu _{n} \bar{\theta }) \Vert x_{n}-\tilde{x} \Vert \Vert x_{n+1}-\tilde{x} \Vert \\ =&\bigl[1-\mu _{n}(\bar{\theta }-\theta \tau )\bigr] \Vert x_{n}-\tilde{x} \Vert \Vert x_{n+1}- \tilde{x} \Vert +\mu _{n}\bigl\langle \theta g(\tilde{x})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ \leq &\frac{1-\mu _{n}(\bar{\theta }-\theta \tau )}{2} \bigl( \Vert x_{n}- \tilde{x} \Vert ^{2}+ \Vert x_{n+1}-\tilde{x} \Vert ^{2} \bigr) \\ &{}+\mu _{n}\bigl\langle \theta g(\tilde{x})-A \tilde{x},x_{n+1}-\tilde{x} \bigr\rangle \\ \leq &\frac{1-\mu _{n}(\bar{\theta }-\theta \tau )}{2} \Vert x_{n}- \tilde{x} \Vert ^{2}+\frac{1}{2} \Vert x_{n+1}-\tilde{x} \Vert ^{2} \\ &{}+\mu _{n}\bigl\langle \theta g(\tilde{x})-A \tilde{x},x_{n+1}-\tilde{x} \bigr\rangle , \end{aligned}$$
(3.40)

which yields that

$$\begin{aligned} \Vert x_{n+1}-\tilde{x} \Vert ^{2} \leq & \bigl[1-\mu _{n}(\bar{\theta }-\theta \tau )\bigr] \Vert x_{n}-\tilde{x} \Vert ^{2} \\ &{}+2\mu _{n} (\bigl\langle \theta g(\tilde{x})-A \tilde{x},x_{n+1}- \tilde{x}\bigr\rangle \\ =& \bigl[1-\mu _{n}(\bar{\theta }-\theta \tau )\bigr] \Vert x_{n}-\tilde{x} \Vert ^{2}+2 \mu _{n} \bigl\langle \theta g(\tilde{x})-A\tilde{x},x_{n+1}- \tilde{x} \bigr\rangle . \end{aligned}$$
(3.41)

Thus, by (3.39), (3.41), Lemma 2.6 and using \(\lim_{n \rightarrow \infty } \mu _{n}=0\), we get \(x_{n} \to \tilde{x}\), where \(\tilde{x}=P_{\Gamma }(I+\theta g-A)\). □

Now, we list the following consequences from Theorem 3.1.

Corollary 3.1

Let \(H_{1}\)and \(H_{2}\)denote the Hilbert spaces and \(C_{1}\subset H_{1}\)be a nonempty closed convex subset of Hilbert space \(H_{1}\). Let \(D:C_{1}\to H_{1}\)be a γinverse strongly monotone mapping, \(B: H_{1}\to H_{2}\)be a bounded linear operator with its adjoint operator \(B^{*}\), \(M_{1}:C_{1}\to 2^{H_{1}}\), and \(M_{2}:H_{2}\to 2^{H_{2}}\)be multi-valued maximal monotone operators and \(f:H_{2}\to H_{2}\)be an α-inverse strongly monotone mapping. Let \(g:C_{1}\to C_{1}\)be a contraction mapping with constant \(\tau \in (0,1)\), A be a strongly positive bounded linear self-adjoint operator on \(C_{1}\)with constant \(\bar{\theta }>0\)such that \(0<\theta <\frac{\bar{\theta }}{\tau }<\theta +\frac{1}{\tau }\), and \(S: C_{1}\to C_{1}\)be a nonexpansive mapping such that \(\Gamma :=\Lambda \cap \operatorname{Sol}(\mathrm{VIP}(\mbox{1.1}))\cap \operatorname{Fix}(S)) \neq \emptyset \). Let \(\{x_{n}\}\)be a sequence generated as follows:

$$ \left . \textstyle\begin{array}{lll} x_{1}\in C_{1}, \\ v_{n}=J^{M_{1}}_{\rho _{1}}[x_{n}+\eta B^{*}(Q-I)Bx_{n}], \\ u_{n}=P_{C_{1}}(v_{n}-\sigma _{n}Dv_{n}), \\ x_{n+1}=\mu _{n}\theta g(Sx_{n})+\delta _{n}x_{n}+((1-\delta _{n})I- \mu _{n}A)Su_{n}, \end{array}\displaystyle \right \} $$
(3.42)

where \(Q=J^{f,M_{2}}_{\rho _{2}}(I-\rho _{2}f)\), \(\{\mu _{n}\}, \{\delta _{n}\}\subset (0,1)\), and \(\eta \in (0,\frac{1}{\epsilon })\), ϵ is the spectral radius of \(B^{*}B\). Let the control sequences satisfy the following conditions:

  1. (i)

    \(\lim_{n\to \infty }\mu _{n}=0\), \(\sum_{n=1}^{\infty } \mu _{n}=\infty \);

  2. (ii)

    \(\rho _{1}> 0\), \(0<\rho _{2}<2\alpha \);

  3. (iii)

    \(0<\liminf_{n\to \infty }\delta _{n}\leq \limsup_{n \to \infty }\delta _{n}<1\);

  4. (iv)

    \(0<\liminf_{n\to \infty }\sigma _{n}\leq \limsup_{n \to \infty }\sigma _{n}<2\gamma \); \(\sum_{n=1}^{\infty }| \sigma _{n+1}-\sigma _{n}|<\infty \).

Then the sequence \(\{x_{n}\}\)converges strongly to some \(\tilde{x}\in \Gamma \), where \(\tilde{x}=P_{\Gamma }(\theta g+(I-A))\tilde{x}\), which solves

$$ \bigl\langle (A-\theta g)\tilde{x},v-\tilde{x}\bigr\rangle \geq 0,\quad \forall v \in \Gamma . $$
(3.43)

Corollary 3.2

Let \(H_{1}\)and \(H_{2}\)denote the Hilbert spaces and \(C_{1}\subset H_{1}\)be a nonempty closed convex subset of Hilbert space \(H_{1}\). Let \(D:C_{1}\to H_{1}\)be a γinverse strongly monotone mapping, \(B: H_{1}\to H_{2}\)be a bounded linear operator with its adjoint operator \(B^{*}\), \(M_{1}:C_{1}\to 2^{H_{1}}\), and \(M_{2}:H_{2}\to 2^{H_{2}}\)be multi-valued maximal monotone operators. Let \(g:C_{1}\to C_{1}\)be a contraction mapping with constant \(\tau \in (0,1)\), A be a strongly positive bounded linear self-adjoint operator on \(C_{1}\)with constant \(\bar{\theta }>0\)such that \(0<\theta <\frac{\bar{\theta }}{\tau }<\theta +\frac{1}{\tau }\), and \(S: C_{1}\to C_{1}\)be a nonexpansive mapping such that \(\Gamma :=\operatorname{Sol}(\mathrm{S}_{\mathrm{P}}\mathrm{NPP}(\mbox{1.4})-(\mbox{1.5})) \cap \operatorname{Sol}(\mathrm{VIP}( \mbox{1.1}))\cap \operatorname{Fix}(S)) \neq \emptyset \). Let \(\{x_{n}\}\)be a sequence generated as follows:

$$ \left . \textstyle\begin{array}{lll} x_{1}\in C_{1}, \\ v_{n}=J^{M_{1}}_{\rho _{1}}[x_{n}+\eta B^{*}(J^{M_{2}}_{\rho _{2}}-I)Bx_{n}], \\ u_{n}=P_{C_{1}}(v_{n}-\sigma _{n}Dv_{n}), \\ x_{n+1}=\mu _{n}\theta g(Sx_{n})+\delta _{n}x_{n}+((1-\delta _{n})I- \mu _{n}A)Su_{n}, \end{array}\displaystyle \right \} $$
(3.44)

where \(\{\mu _{n}\}, \{\delta _{n}\}\subset (0,1)\)and \(\eta \in (0,\frac{1}{\epsilon })\), ϵ is the spectral radius of \(B^{*}B\). Let the control sequences satisfy the following conditions:

  1. (i)

    \(\lim_{n\to \infty }\mu _{n}=0\), \(\sum_{n=1}^{\infty } \mu _{n}=\infty \);

  2. (ii)

    \(0<\liminf_{n\to \infty }\delta _{n}\leq \limsup_{n \to \infty }\delta _{n}<1\);

  3. (iii)

    \(0<\liminf_{n\to \infty }\sigma _{n}\leq \limsup_{n \to \infty }\sigma _{n}<2\gamma ; \sum_{n=1}^{\infty }| \sigma _{n+1}-\sigma _{n}|<\infty \).

Then the sequence \(\{x_{n}\}\)converges strongly to some \(\tilde{x}\in \Gamma \), where \(\tilde{x}=P_{\Gamma }(\theta g+(I-A))\tilde{x}\), which solves

$$ \bigl\langle (A-\theta g)\tilde{x},v-\tilde{x}\bigr\rangle \geq 0, \quad \forall v \in \Gamma . $$
(3.45)

Numerical example

Example 4.1

Let \(H_{1}=H_{2}=\mathbb{R}\), the set of all real numbers, with the inner product defined by \(\langle x,y \rangle =xy\), \(\forall x,y\in \mathbb{R}\), and the induced usual norm \(|\cdot |\). Let \(C_{1}= [0, \infty )\); let the mapping \(f:\mathbb{R}\to \mathbb{R}\) be defined by \(f(y)=y+6\), \(\forall y\in H_{2}\); let \(M_{1}: C_{1}\to 2^{\mathbb{R}}\), \(M_{2}: \mathbb{R}\to 2^{\mathbb{R}}\) be defined by \(M_{1}(x)=\{3x-2\}\), \(\forall x\in C_{1}\) and \(M_{2}(y)=\{3y\}\), \(\forall y\in \mathbb{R}\); let the mapping \(B:\mathbb{R} \to \mathbb{R}\) be defined by \(B(x)=-\frac{9}{4}x\), \(\forall x\in \mathbb{R}\); let the mappings \(\{S_{i}\}_{i=1}^{\infty }: C_{1}\to C_{1}\) be defined by \(S_{i}x=\frac{x+2i}{1+3i}\) for each \(i\in \mathbb{N}\), let the mapping \(D: C_{1}\to \mathbb{R}\) be defined by \(Dx=3x-2\), \(\forall x \in C_{1}\); let the mapping \(g: C_{1}\to C_{1}\) be defined by \(g(x)=\frac{x}{5}\), \(\forall x \in C_{1}\) and \(Ax=\frac{x}{2}\) with \(\theta =\frac{1}{10}\). Setting \(\{\mu _{n}\}=\{\frac{1}{10n}\}\), \(\{\delta _{n}\}=\{\frac{1}{2n^{2}}\}\), \(\{\sigma _{n}\}=\frac{1}{4}\), and \(\{\lambda _{n}\}=\{\frac{1}{3n^{2}}\}\), \(\forall n\geq 1\). Let \(\mathbb{W}_{n}\) be the \(\mathbb{W}\)-mapping generated by \(S_{1},S_{2},\ldots\) , and \(\lambda _{1},\lambda _{2},\ldots\) which is defined by (1.10). Then there are sequences \(\{x_{n}\}\), \(\{u_{n}\}\), and \(\{v_{n}\}\) as follows: Given \(x_{1}\),

$$ \left . \textstyle\begin{array}{lll} t_{n}=QBx_{n}=J^{f,M_{2}}_{\rho _{2}}(I-\rho _{2}f)Bx_{n} \\ y_{n}= x_{n}+\eta B^{*}(t_{n}-Bx_{n}) \\ v_{n}=J^{M_{1}}_{\rho _{1}}y_{n} \\ u_{n}=P_{C_{1}}(v_{n}-\sigma _{n}Dv_{n}), \\ x_{n+1}=\mu _{n}\theta g(\mathbb{W}_{n}x_{n})+\delta _{n}x_{n}+((1- \delta _{n})I-\mu _{n}A)\mathbb{W}_{n}u_{n}. \end{array}\displaystyle \right \} $$
(4.1)

Then \(\{x_{n}\}\) converges to \(\tilde{x}=\{\frac{2}{3}\}\in \Gamma \).

Proof

Obviously, B is a bounded linear operator on \(\mathbb{R}\) with adjoint \(B^{*}\) and \(\|B\|=\|B^{*}\|=\frac{9}{4}\), and hence \(\eta \in (0, \frac{16}{81})\). Therefore, we choose \(\eta =0.1\). Further, f is 1-ism, \(\rho _{1}=\frac{1}{4}>0\) and thus \(\rho _{2} \subset (0, 2)\), so we take \(\rho _{2}=\frac{1}{4}\). For each i, \(S_{i}\) is nonexpansive with \(\operatorname{Fix}(S_{i})= \{ \frac{2}{3} \} \). Further, D is 3-ism and \(\operatorname{Sol}(\mathrm{VIP}(\mbox{1.1}))=\{\frac{2}{3}\}\). Furthermore, \(\operatorname{Sol}(\mathrm{MVIP}(\mbox{1.6}))= \{ \frac{2}{3} \} \) and \(\operatorname{Sol}(\mathrm{MVIP}(\mbox{1.7}))= \{ -\frac{3}{2} \} \), and thus \(\Lambda =\{\frac{2}{3}\in C_{1}:\frac{2}{3}\in \operatorname{Sol}(\mathrm{MVIP}( \mbox{1.6})): B(\frac{2}{3})\in \operatorname{Sol}(\mathrm{MVIP}(\mbox{1.7})) \}= \{ \frac{2}{3} \} \). Therefore, \(\Gamma :=\Lambda \cap \operatorname{Sol}(\mathrm{VIP}(\mbox{1.1}))\cap (\bigcap_{i=1}^{ \infty }\operatorname{Fix}(S_{i})) \neq \emptyset \). Simplify(4.1)as follows: Given \(x_{1}\),

$$\begin{aligned}& t_{n} =\frac{-27x_{n}-24}{28}; \qquad y_{n} = \frac{79x_{n}-36t_{n}}{160}; \\& v_{n}=\frac{4}{7}y_{n}+\frac{2}{7}; \\& u_{n}=P_{C_{1}}(v_{n}-\sigma _{n}Dv_{n}); \\& \hphantom{u_{n}}=\textstyle\begin{cases} 0, & \text{if } x< 0, \\ 1, & \text{if } x>1, \\ \frac{v_{n}+2}{4} & \text{otherwise}; \end{cases}\displaystyle \\& \mathbb{W}_{n}=x_{n}; \\& \textit{Step }1: \\& i=1; \\& \mathbb{W}_{n}=\biggl(\frac{1}{3n^{2}}\biggr) \frac{(\mathbb{W}_{n}+2i)}{1+3i}+ \biggl(1- \frac{1}{3n^{2}}\biggr)x_{n}; \\& i=i+1; \\& \mbox{if } (i \leq N) \mbox{ go to Step 1}; \\& \mathbb{W}^{\prime }_{n}=u_{n}; \\& \textit{Step } 1^{\prime }: \\& i=1; \\& \mathbb{W}^{\prime }_{n}=\biggl(\frac{1}{3n^{2}}\biggr) \frac{(\mathbb{W}^{\prime }_{n}+2i)}{1+3i}+ \biggl(1-\frac{1}{3n^{2}}\biggr)u_{n}; \\& i=i+1; \\& \mbox{if } (i \leq \mathbb{N})\quad \mbox{go to Step } 1^{\prime }; \\& x_{n+1}=\mu _{n}\theta \frac{\mathbb{W}_{n}x_{n}}{5}+\delta _{n}x_{n}+\bigl((1- \delta _{n})I-\mu _{n}A\bigr)\mathbb{W}^{\prime }_{n}u_{n}, \end{aligned}$$

Finally, by the software Matlab 7.8.0, we obtain the Figures 1 and 2 which show that \(\{x_{n}\}\) converges to \(\tilde{x}=\frac{2}{3}\) as \(n \to +\infty \), and \(\lim_{n\to \infty }\|\mathbb{W}_{n} x - \mathbb{W}x\|=0\) for each \(x \in C_{1}\).

Figure 1
figure1

Convergence of \(\{x_{n}\}\)

Figure 2
figure2

Convergence of \(\| W_{n}x - Wx \|\)

 □

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Acknowledgements

This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (D-642-363-1441). The authors, therefore, gratefully acknowledge the DSR technical and financial support.

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This research was funded by King Abdulaziz University, Jeddah, KSA.

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Alansari, M., Farid, M. & Ali, R. An iterative scheme for split monotone variational inclusion, variational inequality and fixed point problems. Adv Differ Equ 2020, 485 (2020). https://doi.org/10.1186/s13662-020-02942-0

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MSC

  • 47J25
  • 49J40
  • 47H09
  • 90C99

Keywords

  • Iterative method
  • Strong convergence
  • Fixed point problem
  • Split monotone variational inclusion problem
  • Nonexpansive mapping
  • Variational inequality problem