# A new subgradient extragradient method for solving the split modified system of variational inequality problems and fixed point problem

## Abstract

We introduce a new subgradient extragradient algorithm utilizing the concept of the set of solutions of the split modified system of variational inequality problems (SMSVIP). Our main theorem is weak convergence theorem for such an algorithm for approximating the fixed point problem in a real Hilbert space. We also apply these results to approximate the split minimization problem. In the last section, we provide an example to illustrate the potential of our main theorem.

## Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H. The mapping $$T:C\rightarrow C$$ is called nonexpansive if $$\|Tx-Ty\|\leq \|x-y\|$$ for all $$x,y\in C$$. An element $$x\in C$$ is said to be a fixed point of T if $$Tx=x$$ and $$F(T)=\{x\in C: Tx=x\}$$ denotes the set of fixed points of T. Fixed point problem has been widely studied and developed in the literature; see [5, 11, 26, 27, 29] and the references therein.

We now recall some well-known concepts and results in a real Hilbert space H.

The variational inequality problem (VIP) is to find a point $$x^{*}\in C$$ such that

\begin{aligned} \bigl\langle Ax^{*},y-x^{*} \bigr\rangle \geq 0 \end{aligned}

for all $$y\in C$$. The set of all solutions of the variational inequality is denoted by $$VI(C,A)$$. Since its inception by Stampacchia  in 1964, the variational inequality problem has become interesting in several topics arising in structural analysis, physic, economics, optimization, and applied sciences; see [1, 3, 6, 8, 1113, 15, 18, 20, 30, 32] and the references therein.

Several algorithms for solving the VIP are projection algorithms that employ projections onto the feasible set C of the VIP, or onto some related set, in order to iteratively reach a solution. In 1976, Korpelevich  proposed an algorithm for solving the VIP in a Euclidean space, known as the extragradient method. In each iteration of her algorithm, in order to get the next iterate $$x^{k+1}$$, two orthogonal projections onto C are calculated, according to the following iterative step. Given the current iterate $$x^{k}$$, calculate

\begin{aligned} & y^{k}=P_{C} \bigl(x^{k}-\tau f \bigl(x^{k} \bigr) \bigr), \end{aligned}
(1)
\begin{aligned} &x^{k+1}=P_{C} \bigl(x^{k}-\tau f \bigl(y^{k} \bigr) \bigr) \end{aligned}
(2)

for all $$k\in \mathbb{N}$$, where τ is some positive number and $$P_{C}$$ denotes the Euclidean least distance projection onto C.

The convergence was proved in  under the assumptions of Lipschitz continuity and pseudo-monotonicity. However, there is still the need to calculate two projections onto C. If the set C is simple enough so that projections onto it can be easily computed, but if C is a general closed and convex set, a minimal distance problem has to be solved twice in order to obtain the next iterate. This might seriously affect the efficiency of the extragradient method. Korpelevich’s extragradient method has been widely studied in the literature; see [2, 4, 7, 9, 14, 16, 17, 22, 28, 31] and the references therein.

In the past decade years, Censor et al.  developed the subgradient extragradient algorithm in a Euclidean space, in which they replaced the (second) projection (2) onto C by a projection onto a specific constructible half-space as follows:

### Algorithm 1

$$\mathbf{Step\ 0:}$$ Select a starting point $$x^{0}\in H$$ and $$\tau >0$$, and set $$k=0$$.

$$\mathbf{Step\ 1:}$$ Given the current iterate $$x^{k}$$, compute

\begin{aligned} y^{k}=P_{C} \bigl(x^{k}-\tau f \bigl(x^{k} \bigr) \bigr), \end{aligned}

construct the half-space $$T_{k}$$ the bounding hyperplane of which supports C at $$y^{k}$$,

\begin{aligned} T_{k}:= \bigl\{ w\in H| \bigl\langle \bigl(x^{k}- \tau f \bigl(x^{k} \bigr) \bigr)-y^{k},w-y^{k} \bigr\rangle \leq 0 \bigr\} , \end{aligned}
(3)

and calculate the next iterate

\begin{aligned} x^{k+1}=P_{T_{k}} \bigl(x^{k}-\tau f \bigl(y^{k} \bigr) \bigr). \end{aligned}

$$\mathbf{Step\ 2:}$$ If $$x^{k}=y^{k}$$ then stop. Otherwise, set $$k\leftarrow (k+1)$$ and return to step 1.

Furthermore, under some control conditions, they proved weak convergence theorems for their algorithms.

Very recently, Sripattanet and Kangtunyakarn  introduced the following split modified system of variational inequality problems (SMSVIP), which involves finding $$(x^{*},y^{*},z^{*})\in C\times C\times C$$ such that

\begin{aligned} \textstyle\begin{cases} \langle x^{*}-(I-\zeta D_{1})(ax^{*}+(1-a)y^{*}),x-x^{*}\rangle \geq 0,\quad \forall x\in C, \\ \langle y^{*}-(I-\zeta D_{2})(ax^{*}+(1-a)z^{*}),x-y^{*}\rangle \geq 0,\quad \forall x\in C, \\ \langle z^{*}-(I-\zeta D_{3})x^{*},x-z^{*}\rangle \geq 0,\quad \forall x \in C, \end{cases}\displaystyle \end{aligned}
(4)

and finding $$(\bar{x^{*}}=Ax^{*}, \bar{y^{*}}=Ay^{*}, \bar{z^{*}}=Az^{*})\in Q\times Q\times Q$$ such that

\begin{aligned} \textstyle\begin{cases} \langle \bar{x^{*}}-(I-\bar{\zeta }\bar{D_{1}})(a\bar{x^{*}}+(1-a) \bar{y^{*}}),\bar{x}-\bar{x^{*}}\rangle \geq 0, \quad\forall \bar{x}\in Q, \\ \langle \bar{y^{*}}-(I-\bar{\zeta }\bar{D_{2}})(a\bar{x^{*}}+(1-a) \bar{z^{*}}),\bar{x}-\bar{y^{*}}\rangle \geq 0,\quad \forall \bar{x}\in Q, \\ \langle \bar{z^{*}}-(I-\bar{\zeta }\bar{D_{3}})\bar{x^{*}},\bar{x}- \bar{z^{*}}\rangle \geq 0,\quad\forall \bar{x}\in Q, \end{cases}\displaystyle \end{aligned}
(5)

where $$D_{1},D_{2},D_{3}:C\rightarrow H_{1}$$, $$\bar{D_{1}},\bar{D_{2}},\bar{D_{3}}:Q\rightarrow H_{2}$$ are six different mappings, $${\zeta,\bar{\zeta } >0,}$$ and $$a\in [0,1]$$. The sets of all solutions of (4) and (5) are denoted by $$\Psi _{D_{1},D_{2},D_{3}}$$ and $$\Psi _{\bar{D_{1}},\bar{D_{2}},\bar{D_{3}}}$$, respectively. The set of all solutions of the SMSVIP is denoted by $$\Psi ^{D_{1},D_{2},D_{3}}_{\bar{D_{1}},\bar{D_{2}},\bar{D_{3}}}$$, that is,

\begin{aligned} \Psi ^{D_{1},D_{2},D_{3}}_{\bar{D_{1}},\bar{D_{2}},\bar{D_{3}}}= \bigl\{ \bigl(x^{*},y^{*},z^{*} \bigr) \in \Psi _{D_{1},D_{2},D_{3}}: \bigl(\bar{x^{*}},\bar{y^{*}}, \bar{z^{*}} \bigr) \in \Psi _{\bar{D_{1}},\bar{D_{2}},\bar{D_{3}}} \bigr\} . \end{aligned}

If we put $$a=0$$ in (4) and (5), we have

\begin{aligned} \textstyle\begin{cases} \langle x^{*}-(I-\zeta D_{1})y^{*},x-x^{*}\rangle \geq 0,\quad \forall x \in C, \\ \langle y^{*}-(I-\zeta D_{2})z^{*},x-y^{*}\rangle \geq 0, \quad\forall x \in C, \\ \langle z^{*}-(I-\zeta D_{3})x^{*},x-z^{*}\rangle \geq 0,\quad\forall x \in C, \end{cases}\displaystyle \end{aligned}

and

\begin{aligned} \textstyle\begin{cases} \langle \bar{x^{*}}-(I-\bar{\zeta }\bar{D_{1}})\bar{y^{*}},\bar{x}- \bar{x^{*}}\rangle \geq 0,\quad \forall \bar{x}\in Q, \\ \langle \bar{y^{*}}-(I-\bar{\zeta }\bar{D_{2}})\bar{z^{*}},\bar{x}- \bar{y^{*}}\rangle \geq 0,\quad \forall \bar{x}\in Q, \\ \langle \bar{z^{*}}-(I-\bar{\zeta }\bar{D_{3}})\bar{x^{*}},\bar{x}- \bar{z^{*}}\rangle \geq 0,\quad\forall \bar{x}\in Q, \end{cases}\displaystyle \end{aligned}

which is a modified the split general system of variational inequalities (SVIP) .

Based on the above works and observation of a half-space in Algorithm 1 related to the VIP, we introduce a new half-space related to the SMSVIP and prove weak convergence theorems of the sequence $$\{x_{n}\}$$ generated by our new half-space for approximating the solutions of the SMSVIP. Moreover, using our main result, we obtain the additional results involving the split minimization problem. Finally, we perform numerical examples to illustrate the computational performance of the proposed algorithms.

## Preliminaries

We denote the weak convergence and the strong convergence by $$^{\backprime \backprime }\rightharpoonup ^{\prime \prime }$$ and $$^{\backprime \backprime }\rightarrow ^{\prime \prime }$$, respectively. For every $$x\in \mathcal{H}$$, there exists a unique nearest point $$P_{C}x$$ in C such that $$\|x-P_{C}x\|\leq \|x-y\|$$ for all $$y\in C$$. $$P_{C}$$ is called the metric projection of $$\mathcal{H}$$ onto C.

The metric projection $$P_{C}$$ is characterized by the following two properties:

1. 1.

$$P_{C} x\in C$$,

2. 2.

$$\langle x-P_{C} x,P_{C} x-y\rangle \geq 0$$, $$\forall x\in \mathcal{H}$$, $$y\in C$$,

and if C is a hyperplane, it follows that

\begin{aligned} \Vert x-y \Vert ^{2}&\geq \Vert x-P_{C} x \Vert ^{2}+ \Vert y-P_{C} x \Vert ^{2}, \end{aligned}
(6)

$$\forall x\in \mathcal{H}$$, $$y\in C$$.

### Definition 2.1

A mapping $$A:C\rightarrow H$$ is called α-inversestronglymonotone if there exists a positive real number $$\alpha >0$$ such that

\begin{aligned} \langle Ax-Ay,x-y\rangle \geq \alpha \Vert Ax-Ay \Vert ^{2} \end{aligned}

for all $$x,y\in C$$.

The following lemmas are needed to prove the main theorem.

### Lemma 2.2

Let $$\mathcal{H}$$ be a real Hilbert space, and let C be a nonempty closed convex subset of $$\mathcal{H}$$. Let $$\{x^{k}\}^{\infty }_{k=0}\subset \mathcal{H}$$ be Fejer-monotone with respect to C, i.e., for every $$u\in C$$,

\begin{aligned} \bigl\Vert x^{k+1}-u \bigr\Vert \leq \bigl\Vert x^{k}-u \bigr\Vert , \quad\forall k\geq 0. \end{aligned}

Then $$\{P_{C} x^{k}\}^{\infty }_{k=0}$$ converges strongly to some $$z\in C$$.

### Lemma 2.3

Each Hilbert space $$\mathcal{H}$$ satisfies Opial’s condition, i.e., for any sequence $$\{x_{n}\}\subset \mathcal{H}$$ with $$x_{n}\rightharpoonup x$$, the inequality

\begin{aligned} \liminf_{n \to \infty } \Vert x_{n}-x \Vert < \liminf _{n \to \infty } \Vert x_{n}-y \Vert \end{aligned}

holds for every $$y\in \mathcal{H}$$ with $$y\neq x$$.

### Lemma 2.4

()

Let $$H_{1}$$ and $$H_{2}$$ be real Hilbert spaces, and let $$C,Q$$ be nonempty closed convex subsets of $$H_{1}$$ and $$H_{2}$$, respectively. Let $$D_{1},D_{2}$$, $$D_{3}:C\rightarrow H_{1}$$ be $$d_{1},d_{2},d_{3}$$-inverse strongly monotone, respectively, where $$\zeta \in (0,2d^{*})$$ with $$d^{*}=\operatorname{min} \{d_{1},d_{2},d_{3}\}$$. Let $$\bar{D_{1}},\bar{D_{2}},\bar{D_{3}}:Q\rightarrow H_{2}$$ be $$\bar{d_{1}},\bar{d_{2}},\bar{d_{3}}$$-inverse strongly monotone, respectively, where $$\bar{\zeta }\in (0,2\hat{d})$$ with $$\hat{d}=\operatorname{min} \{\bar{d_{1}}, \bar{d_{2}},\bar{d_{3}}\}$$. Let $$A:H_{1}\rightarrow H_{2}$$ be a bounded linear operator with adjoint $$A^{*}$$ and $$\eta \in (0,\frac{1}{L})$$ with L being the spectral radius of the operator $$A^{*}A$$. Define $$M_{C}:C\rightarrow C$$ by

\begin{aligned} M_{C}(x)=P_{C}(I-\zeta D_{1}) \bigl(ax+(1-a)P_{C}(I-\zeta D_{2}) \bigl(ax+(1-a)P_{C}(I- \zeta D_{3})x \bigr) \bigr), \end{aligned}

$$\forall x\in C$$, and define $$M_{Q}:Q\rightarrow Q$$ by

\begin{aligned} M_{Q}(\hat{x})=P_{Q}(I-\bar{\zeta }\bar{D_{1}}) \bigl(a\hat{x}+(1-a)P_{Q}(I- \bar{\zeta }\bar{D_{2}}) \bigl(a\hat{x}+(1-a)P_{Q}(I-\bar{\zeta }\bar{D_{3}}) \hat{x} \bigr) \bigr), \end{aligned}

$$\forall \hat{x}\in Q$$. Define $$M:C\rightarrow C$$ by $$M(x)=M_{C}(x-\eta A^{*}(I-M_{Q})Ax)$$ for all $$x\in C$$. Then M is a nonexpansive mapping for all $$x\in C$$.

### Remark 1

From the study of Lemma 2.4, we have

\begin{aligned} & \bigl\Vert \bigl(x-\eta A^{*}(I-M_{Q})Ax \bigr)- \bigl(y- \eta A^{*}(I-M_{Q})Ay \bigr) \bigr\Vert ^{2} \\ &\quad\leq \Vert x-y \Vert ^{2}-\eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax-(I-M_{Q})Ay \bigr\Vert ^{2} \end{aligned}

for all $$x,y\in H_{1}$$.

### Lemma 2.5

()

Let $$H_{1}$$ and $$H_{2}$$ be real Hilbert spaces, and let $$C,Q$$ be nonempty closed convex subsets of $$H_{1},H_{2}$$, respectively. Define the mappings $$D_{1},D_{2},D_{3},\bar{D_{1}},\bar{D_{2}},\bar{D_{3}},M_{C}$$, and $$M_{Q}$$ as in Lemma 2.4, where $$\zeta \in (0,2d^{*})$$ with $$d^{*}=\operatorname{min} \{d_{1},d_{2},d_{3}\}$$, $$\bar{\zeta }\in (0,2\hat{d})$$ with $$\hat{d}= \operatorname{min} \{\bar{d_{1}}, \bar{d_{2}},\bar{d_{3}}\}$$. Let $$A:H_{1}\rightarrow H_{2}$$ be a bounded linear operator with adjoint $$A^{*}$$ and $$\eta \in (0,\frac{1}{L})$$ with L being the spectral radius of the operator $$A^{*}A$$.

Assume

\begin{aligned} \Psi ^{D_{1},D_{2},D_{3}}_{\bar{D_{1}},\bar{D_{2}},\bar{D_{3}}}= \bigl\{ \bigl(x^{*},y^{*},z^{*} \bigr) \in \Psi _{D_{1},D_{2},D_{3}}: \bigl(\bar{x^{*}},\bar{y^{*}}, \bar{z^{*}} \bigr) \in \Psi _{\bar{D_{1}},\bar{D_{2}},\bar{D_{3}}} \bigr\} \neq \emptyset . \end{aligned}

The following statements are equivalent:

1. (i)

$$(x^{*},y^{*},z^{*})\in \Psi ^{D_{1},D_{2},D_{3}}_{\bar{D_{1}}, \bar{D_{2}},\bar{D_{3}}}$$,

2. (ii)

$$x^{*}=M_{C}(x^{*}-\eta A^{*}(I-M_{Q})Ax^{*})$$, where $$y^{*}=P_{C}(I-\zeta D_{2})(ax^{*}+(1-a)z^{*})$$, $$z^{*}=P_{C}(I-\zeta D_{3})x^{*}$$, $$\bar{x^{*}}=Ax^{*}=P_{Q}(I-\bar{\zeta }\bar{D_{1}})(a\bar{x^{*}}+(1-a) \bar{y^{*}})$$, $$\bar{y^{*}}=Ay^{*}=P_{Q}(I-\bar{\zeta }\bar{D_{2}})(a\bar{x^{*}}+(1-a) \bar{z^{*}})$$, and $$\bar{z^{*}}=Az^{*}=P_{Q}(I-\bar{\zeta }\bar{D_{3}})\bar{x^{*}}$$.

### Lemma 2.6

()

Let $$H_{1}$$ and $$H_{2}$$ be real Hilbert spaces, and let $$C,Q$$ be nonempty closed convex subsets of $$H_{1},H_{2}$$, respectively. Define the mappings $$D_{1},D_{2},D_{3},\bar{D_{1}},\bar{D_{2}},\bar{D_{3}},M_{C}$$, and $$M_{Q}$$ as in Lemma 2.4where $$\zeta \in (0,2d^{*})$$ with $$d^{*}=\operatorname{min} \{d_{1},d_{2},d_{3}\}$$, $$\bar{\zeta }\in (0,2\hat{d})$$ with $$\hat{d}= \operatorname{min} \{\bar{d_{1}}, \bar{d_{2}},\bar{d_{3}}\}$$ and $$a\in [0,1]$$. Let $$A:H_{1}\rightarrow H_{2}$$ be a bounded linear operator with adjoint $$A^{*}$$ and $$\eta \in (0,\frac{1}{L})$$ with L being the spectral radius of the operator $$A^{*}A$$. Let $$\bigcap_{i=1}^{3}\Phi _{i}\neq \emptyset$$ and $$\Phi _{i}=\{w\in VI(C,D_{i})|Aw=\bar{w}\in VI(Q,\bar{D}_{i})\}$$ for all $$i=1,2,3$$. Then

\begin{aligned} \bigcap_{i=1}^{3}\Phi _{i}= F \bigl(M_{C} \bigl(I-\eta A^{*}(I-M_{Q})A \bigr) \bigr). \end{aligned}

In order to prove our main result, we need to prove the lemmas involving the split variational inequality problem.

### Lemma 2.7

Let $$H_{1}$$ and $$H_{2}$$ be real Hilbert spaces, and let $$C,Q$$ be nonempty closed convex subsets of $$H_{1},H_{2}$$, respectively. Define the mappings $$D_{1},D_{2},D_{3},\bar{D_{1}},\bar{D_{2}},\bar{D_{3}},M_{C}$$, and $$M_{Q}$$ as in Lemma 2.4where $$\zeta \in (0,2d^{*})$$ with $$d^{*}=\operatorname{min} \{d_{1},d_{2},d_{3}\}$$, $$\bar{\zeta }\in (0,2\hat{d})$$ with $$\hat{d}= \operatorname{min} \{\bar{d_{1}}, \bar{d_{2}},\bar{d_{3}}\}$$ and $$a\in [0,1]$$. Let $$\{x_{n}\}$$ be a sequence in $$H_{1}$$, and let $$A:H_{1}\rightarrow H_{2}$$ be a bounded linear operator with adjoint $$A^{*}$$ and $$\eta \in (0,\frac{1}{L})$$ with L being the spectral radius of the operator $$A^{*}A$$. For every $$n\in \mathbb{N}$$, let $$T_{n}=aW_{n}+(1-a)P_{C}(I-\zeta D_{2})(aW_{n}+(1-a)P_{C}(I-\zeta D_{3})W_{n}))$$ and $$W_{n}=(I-\eta A^{*}(I-M_{Q})A)x_{n}$$. If $$x^{*}\in \bigcap_{i=1}^{3}\Phi _{i}$$, then

\begin{aligned} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}&\leq \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\eta (1- \eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2} \end{aligned}

for all $$n\in \mathbb{N}$$.

### Proof

Let $$x^{*}\in \bigcap_{i=1}^{3}\Phi _{i}$$. From Lemma 2.6, we have

\begin{aligned} x^{*}\in F \bigl(M_{C} \bigl(I-\eta A^{*}(I-M_{Q})A \bigr) \bigr). \end{aligned}

It implies that $$x^{*}=M_{C}(I-\eta A^{*}(I-M_{Q})A)x^{*}$$, $$y^{*}=P_{C}(I-\zeta D_{2})(ax^{*}+(1-a)z^{*})$$, and $$z^{*}=P_{C}(I-\zeta D_{3})x^{*}$$, where $$\bar{x^{*}}=Ax^{*}=P_{Q}(I-\bar{\zeta }\bar{D_{1}})(a\bar{x^{*}}+(1-a) \bar{y^{*}})$$, $$\bar{y^{*}}=Ay^{*}=P_{Q}(I-\bar{\zeta }\bar{D_{2}})(a\bar{x^{*}}+(1-a) \bar{z^{*}})$$, and $$\bar{z^{*}}=Az^{*}=P_{Q}(I-\bar{\zeta }\bar{D_{3}})\bar{x^{*}}$$. From Lemma 2.5, we have $$(x^{*},y^{*},z^{*})\in \Omega ^{D_{1},D_{2},D_{3}}_{\bar{D_{1}}, \bar{D_{2}},\bar{D_{3}}}$$. That is, $$(x^{*},y^{*},z^{*})\in \Omega _{D_{1},D_{2},D_{3}}$$ and $$(\bar{x^{*}},\bar{y^{*}},\bar{z^{*}})\in \Omega _{\bar{D_{1}}, \bar{D_{2}},\bar{D_{3}}}$$. From $$(\bar{x^{*}},\bar{y^{*}},\bar{z^{*}})\in \Omega _{\bar{D_{1}}, \bar{D_{2}},\bar{D_{3}}}$$, we obtain that

\begin{aligned} &\bar{x^{*}}=P_{Q}(I-\bar{\zeta }\bar{D_{1}}) \bigl(a\bar{x^{*}}+(1-a) \bar{y^{*}} \bigr),\\ &\bar{y^{*}}=P_{Q}(I-\bar{\zeta }\bar{D_{2}}) \bigl(a\bar{x^{*}}+(1-a) \bar{z^{*}} \bigr),\\ &\bar{z^{*}}=P_{Q}(I-\bar{\zeta }\bar{D_{3}}) \bar{x^{*}}. \end{aligned}

It implies that

\begin{aligned} Ax^{*}&=\bar{x^{*}}=P_{Q}(I-\bar{\zeta } \bar{D_{1}}) \bigl(a\bar{x^{*}}+(1-a)P_{Q}(I- \bar{\zeta }\bar{D_{2}}) \bigl(a\bar{x^{*}}+(1-a)P_{Q}(I- \bar{\zeta }\bar{D_{3}})\bar{x^{*}} \bigr) \bigr)\\ &=M_{Q}\bar{x^{*}}=M_{Q}Ax^{*}. \end{aligned}

From the definition of $$x^{*}$$, we get $$x^{*}=P_{C}(I-\zeta D_{1})T_{x}^{*}$$, where $$T_{x}^{*}=aW_{x}^{*}+(1-a)P_{C}(I-\zeta D_{2})(aW_{x}^{*}+(1-a)P_{C}(I- \zeta D_{3})W_{x}^{*}))$$ and $$W_{x}^{*}=(I-\eta A^{*}(I-M_{Q})A)x^{*})=x^{*}$$.

From Lemma 2.6, we have that $${P_{C}}(I - {\zeta }{D_{1}}),{P_{C}}(I - {\zeta }{D_{2}})$$ and $${P_{C}}(I - {\zeta }{D_{3}})$$ are nonexpansive.

By the definition of $$T_{n}$$, Lemma 2.4, and Remark 1, we have

\begin{aligned} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}={}&\bigl\| aW_{n}+(1-a)P_{C}(I-\zeta D_{2}) \bigl(aW_{n}+(1-a) \\ &{}\times P_{C}(I- \zeta D_{3})W_{n} \bigr))- \bigl(aW_{x^{*}}+(1-a)P_{C}(I-\zeta D_{2}) \bigl(aW_{x^{*}} \\ &{} +(1-a)P_{C}(I-\zeta D_{3})W_{x^{*}} \bigr) \bigr))\bigr\| ^{2} \\ ={}& \bigl\Vert a(W_{n}-W_{x^{*}})+(1-a) \bigl[P_{C}(I- \zeta D_{2}) \bigl(aW_{n}+(1-a)P_{C}(I- \zeta D_{3})W_{n} \bigr)) \\ &{} -P_{C}(I- \zeta D_{2}) \bigl(aW_{x^{*}}+(1-a)P_{C}(I- \zeta D_{3})W_{x^{*}} \bigr)) \bigr] \bigr\Vert ^{2} \\ \leq{}& a \Vert W_{n}-W_{x^{*}} \Vert ^{2}+(1-a) \bigl\| P_{C}(I-\zeta D_{2}) \bigl(aW_{n}+(1-a)P_{C}(I- \zeta D_{3})W_{n} \bigr)) \\ &{}-P_{C}(I-\zeta D_{2}) \bigl(aW_{x^{*}}+(1-a)P_{C}(I- \zeta D_{3})W_{x^{*}} \bigr))\bigr\| ^{2} \\ \leq{}& a \Vert W_{n}-W_{x^{*}} \Vert ^{2}+(1-a) \bigl\Vert aW_{n}+(1-a)P_{C}(I-\zeta D_{3})W_{n} \\ & {}- \bigl(aW_{x^{*}}+(1-a)P_{C}(I-\zeta D_{3})W_{x^{*}} \bigr) \bigr\Vert ^{2} \\ ={}&a \Vert W_{n}-W_{x^{*}} \Vert ^{2}+(1-a) \bigl\Vert a(W_{n}-W_{x^{*}})+(1-a) \\ & {}\times \bigl[P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr] \bigr\Vert ^{2} \\ \leq{}& a \Vert W_{n}-W_{x^{*}} \Vert ^{2}+a(1-a) \Vert W_{n}-W_{x^{*}} \Vert ^{2}+(1-a)^{2} \\ & {}\times \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ ={}& \bigl(2a-a^{2} \bigr) \Vert W_{n}-W_{x^{*}} \Vert ^{2}+(1-a)^{2} \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ \leq {}& \bigl\Vert W_{n}-x^{*} \bigr\Vert ^{2} \\ ={}& \bigl\Vert x_{n}-\eta A^{*}(I-M_{Q})Ax_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2}. \end{aligned}
(7)

□

## Main results

### Theorem 3.1

Let C and Q be nonempty closed convex subsets of real Hilbert spaces $$H_{1}$$ and $$H_{2}$$, respectively, and let $$S:C\rightarrow C$$ be a nonexpansive mapping. Let $$D_{1},D_{2},D_{3}:C\rightarrow H_{1}$$ be $$d_{1},d_{2},d_{3}$$-inverse strongly monotone, respectively, with $$d^{*}=\operatorname{min} \{d_{1},d_{2},d_{3}\}$$. Let $$\bar{D_{1}},\bar{D_{2}},\bar{D_{3}}:Q\rightarrow H_{2}$$ be $$\bar{d_{1}},\bar{d_{2}},\bar{d_{3}}$$-inverse strongly monotone, respectively, with $$\hat{d}=\operatorname{min} \{\bar{d_{1}},\bar{d_{2}},\bar{d_{3}}\}$$. Let $$A:H_{1}\rightarrow H_{2}$$ be a bounded linear operator with adjoint $$A^{*}$$ and $$\eta \in (0,\frac{1}{L})$$ with L being the spectral radius of the operator $$A^{*}A$$. Define $$M_{C}:H_{1}\rightarrow C$$ by

\begin{aligned} M_{C}(x)=P_{C}(I-\zeta D_{1}) \bigl(ax+(1-a)P_{C}(I-\zeta D_{2}) \bigl(ax+(1-a)P_{C}(I- \zeta D_{3})x \bigr) \bigr), \end{aligned}

$$\forall x\in H_{1}$$, where $$a\in [0,1)$$, $$\zeta \in (0,2d^{*})$$, and define $$M_{Q}:H_{2}\rightarrow Q$$ by

\begin{aligned} M_{Q}(x)=P_{Q}(I-\bar{\zeta }\bar{D_{1}}) \bigl(a\hat{x}+(1-a)P_{Q}(I- \bar{\zeta }\bar{D_{2}}) \bigl(a \hat{x}+(1-a)P_{Q}(I-\bar{\zeta }\bar{D_{3}}) \hat{x} \bigr) \bigr), \end{aligned}

$$\forall \hat{x}\in H_{1}$$, where $$a\in [0,1)$$, $$\bar{\zeta }\in (0,2\hat{d})$$. Let the sequences $$\{x_{n}\}$$ and $$\{y_{n}\}$$ be generated by $$x_{1}\in H_{1}$$ and

\begin{aligned} y_{n}=M_{C} W_{n} =P_{C}(I-\zeta D_{1})T_{n}, \end{aligned}

where $$W_{n}=(I-\eta A^{*}(I-M_{Q})A)x_{n}$$ and $$T_{n}=aW_{n}+(1-a)P_{C}(I-\zeta D_{2})(aW_{n}+(1-a)P_{C}(I-\zeta D_{3})W_{n}))$$.

\begin{aligned} &Q_{n}= \bigl\{ z\in H: \bigl\langle (I-\zeta D_{1})T_{n}-y_{n},y_{n}-z \bigr\rangle \geq 0 \bigr\} ,\\ &x_{n+1}=\alpha _{n}T_{n}+(1-\alpha _{n})SP_{Q_{n}} \bigl(T_{n}-\zeta D_{1}(y_{n}) \bigr) \end{aligned}

for all $$n\in \mathbb{N}$$.

Assume that the following conditions hold:

1. (i)

$$\Im =F(S)\bigcap \bigcap_{i=1}^{3}\Phi _{i}\neq \emptyset$$, where $$\Phi _{i}=\{w\in VI(C,D_{i})|Aw\in VI(Q,\bar{D}_{i})\}$$ for all $$i=1,2,3$$.

2. (ii)

$$\alpha _{n}\in [c,d]\subset (0,1)$$.

Then $$\{x_{n}\}$$ converges weakly to $$x_{0}=P_{\Im }{x_{n}}$$, which $$(x_{0},y_{0},z_{0})\in \Omega ^{D_{1},D_{2},D_{3}}_{\bar{D_{1}}, \bar{D_{2}},\bar{D_{3}}}$$, $$y_{0}=P_{C}(I-\zeta D_{2})(ax_{0}+(1-a)z_{0})$$, and $$z_{0}=P_{C}(I-\zeta D_{3})x_{0}$$ with $$\bar{x_{0}}=Ax_{0}$$, $$\bar{y_{0}}=Ay_{0}$$ and $$\bar{z_{0}}=Az_{0}$$.

### Proof

Denote $$k_{n}:=P_{Q_{n}}(T_{n}-\zeta D_{1}(y_{n}))$$ for all $$n\geq 0$$. Let $$x^{*}\in \Im$$. From the definition of $$P_{Q_{n}}$$, we have $$y_{n}=P_{Q_{n}}(I-\zeta D_{1})T_{n}$$. Let $$M_{n}=T_{n}-\zeta D_{1}(y_{n})$$. From $$C\subseteq Q_{n}$$, and applying (6), we have

\begin{aligned} \bigl\Vert k_{n}-x^{*} \bigr\Vert ^{2}={}& \bigl\Vert P_{Q_{n}}M_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& \bigl\Vert M_{n}-x^{*} \bigr\Vert ^{2}- \Vert M_{n}-P_{Q_{n}}M_{n} \Vert ^{2} \\ ={}& \bigl\Vert T_{n}-\zeta D_{1}(y_{n})-x^{*} \bigr\Vert ^{2}- \bigl\Vert T_{n}-\zeta D_{1}(y_{n})-P_{Q_{n}}M_{n} \bigr\Vert ^{2} \\ ={}& \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}-2 \zeta \bigl\langle T_{n}-x^{*},D_{1}(y_{n}) \bigr\rangle + \zeta ^{2} \bigl\Vert D_{1}(y_{n}) \bigr\Vert ^{2} \\ &{} - \Vert T_{n}-P_{Q_{n}}M_{n} \Vert ^{2}+2\zeta \bigl\langle T_{n}-P_{Q_{n}}M_{n},D_{1}(y_{n}) \bigr\rangle -\zeta ^{2} \bigl\Vert D_{1}(y_{n}) \bigr\Vert ^{2} \\ ={}& \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}- \Vert T_{n}-P_{Q_{n}}M_{n} \Vert ^{2}-2 \zeta \bigl\langle P_{Q_{n}}M_{n}-x^{*},D_{1}(y_{n}) \bigr\rangle . \end{aligned}
(8)

From the monotonicity of $$D_{1}$$, we have

\begin{aligned} 0&\leq \bigl\langle D_{1}y_{n}-D_{1}x^{*},y_{n}-x^{*} \bigr\rangle \\ &= \bigl\langle D_{1}y_{n},y_{n}-x^{*} \bigr\rangle - \bigl\langle D_{1}x^{*},y_{n}-x^{*} \bigr\rangle \\ &\leq \bigl\langle D_{1}y_{n},y_{n}-x^{*} \bigr\rangle \\ &=\langle D_{1}y_{n},y_{n}-P_{Q_{n}}M_{n} \rangle - \bigl\langle D_{1}y_{n},x^{*}-P_{Q_{n}}M_{n} \bigr\rangle , \end{aligned}

which implies that

\begin{aligned} \bigl\langle D_{1}y_{n},x^{*}-P_{Q_{n}}M_{n} \bigr\rangle &\leq \langle D_{1}y_{n},y_{n}-P_{Q_{n}}M_{n} \rangle. \end{aligned}
(9)

From (8) and (9), we have

\begin{aligned} \bigl\Vert k_{n}-x^{*} \bigr\Vert ^{2}&\leq \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}- \Vert T_{n}-P_{Q_{n}}M_{n} \Vert ^{2}+2 \zeta \langle D_{1}y_{n},y_{n}-P_{Q_{n}}M_{n} \rangle. \end{aligned}
(10)

From (10) and Lemma 2.7, we have

\begin{aligned} \bigl\Vert k_{n}-x^{*} \bigr\Vert ^{2}\leq{}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2}- \Vert P_{Q_{n}}M_{n}-T_{n} \Vert ^{2} \\ & {}+2\zeta \langle D_{1}y_{n},y_{n}-P_{Q_{n}}M_{n} \rangle \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2}- \Vert P_{Q_{n}}M_{n}-y_{n} \Vert ^{2} \\ & {}- \Vert y_{n}-T_{n} \Vert ^{2} -2 \langle P_{Q_{n}}M_{n}-y_{n},y_{n}-T_{n} \rangle \\ & {}+2\zeta \langle D_{1}y_{n},y_{n}-P_{Q_{n}}M_{n} \rangle \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2}- \Vert P_{Q_{n}}M_{n}-y_{n} \Vert ^{2} \\ & {}- \Vert y_{n}-T_{n} \Vert ^{2}+2 \langle P_{Q_{n}}M_{n}-y_{n},T_{n}-y_{n}- \zeta D_{1}y_{n}\rangle \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2}- \Vert P_{Q_{n}}M_{n}-y_{n} \Vert ^{2} \\ & {}- \Vert y_{n}-T_{n} \Vert ^{2}+2 \bigl\langle (I-\zeta D_{1})T_{n}-y_{n},P_{Q_{n}}M_{n}-y_{n} \bigr\rangle \\ &{} +2\langle \zeta D_{1}T_{n}-\zeta D_{1}y_{n},P_{Q_{n}}M_{n}-y_{n} \rangle \\ \leq{}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2}- \Vert P_{Q_{n}}M_{n}-y_{n} \Vert ^{2} \\ & {}- \Vert y_{n}-T_{n} \Vert ^{2} +2 \zeta \Vert D_{1}T_{n}-D_{1}y_{n} \Vert \Vert P_{Q_{n}}M_{n}-y_{n} \Vert \\ \leq{}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2}- \Vert P_{Q_{n}}M_{n}-y_{n} \Vert ^{2} \\ &{} - \Vert y_{n}-T_{n} \Vert ^{2}+ \frac{\zeta }{d_{1}} \bigl[ \Vert T_{n}-y_{n} \Vert ^{2}+ \Vert P_{Q_{n}}M_{n}-y_{n} \Vert ^{2} \bigr] \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2} \\ &{} - \biggl(1-\frac{\zeta }{d_{1}} \biggr) \Vert P_{Q_{n}}M_{n}-y_{n} \Vert ^{2} - \biggl(1- \frac{\zeta }{d_{1}} \biggr)) \Vert T_{n}-y_{n} \Vert ^{2}. \end{aligned}
(11)

By the definition of $$x_{n+1}$$, (11), and Lemma 2.7, we have

\begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}={}& \bigl\Vert \alpha _{n} \bigl(T_{n}-x^{*} \bigr)+(1-\alpha _{n}) \bigl(Sk_{n}-x^{*} \bigr) \bigr\Vert ^{2} \\ \leq {}&\alpha _{n} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert Sk_{n}-x^{*} \bigr\Vert ^{2} \\ ={}& \alpha _{n} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert Sk_{n}-x^{*} \bigr\Vert ^{2} \\ &{} -\alpha _{n}(1-\alpha _{n}) \Vert T_{n}-Sk_{n} \Vert ^{2} \end{aligned}
(12)
\begin{aligned} ={}& \alpha _{n} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert k_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& \alpha _{n} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2} +(1-\alpha _{n}) \biggl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ &{} -\eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2} \\ & {}- \biggl(1-\frac{\zeta }{d_{1}} \biggr) \Vert P_{Q_{n}}M_{n}-y_{n} \Vert ^{2}- \biggl(1- \frac{\zeta }{d_{1}} \biggr)) \Vert T_{n}-y_{n} \Vert ^{2} \biggr] \\ \leq{}& \alpha _{n} \bigl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\alpha _{n}\eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2} \bigr] \\ & {}+(1-\alpha _{n}) \biggl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} -\eta (1-\eta L) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2} \\ & {}- \biggl(1-\frac{\zeta }{d_{1}} \biggr) \Vert P_{Q_{n}}M_{n}-y_{n} \Vert ^{2} - \biggl(1- \frac{\zeta }{d_{1}} \biggr)) \Vert T_{n}-y_{n} \Vert ^{2} \biggr] \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \eta (1-\eta L) (1+\alpha _{n}) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2} \\ & {}-(1-\alpha _{n}) \biggl(1-\frac{\zeta }{d_{1}} \biggr) \bigl[ \Vert T_{n}-y_{n} \Vert ^{2}+ \Vert y_{n}-k_{n} \Vert ^{2} \bigr]. \end{aligned}
(13)

So,

\begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}&\leq \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}. \end{aligned}

Therefore $$\lim_{n \rightarrow \infty }\|x_{n+1}-x^{*}\|$$ exists, $$\forall x^{*}\in \Im$$. So, we have $$\{x_{n}\}^{\infty }_{n=0}$$ and $$\{k_{n}\}^{\infty }_{n=0}$$ are bounded. From the last relations it follows that

\begin{aligned} \eta (1-\eta L) (1+\alpha _{n}) \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2}\leq \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2} \end{aligned}

or

\begin{aligned} \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert ^{2}& \leq \frac{ \Vert x_{n}-x^{*} \Vert ^{2}- \Vert x_{n+1}-x^{*} \Vert ^{2}}{\eta (1-\eta L)(1+\alpha _{n})}. \end{aligned}

Thus

\begin{aligned} \lim_{n \rightarrow \infty } \bigl\Vert (I-M_{Q})Ax_{n} \bigr\Vert &=0. \end{aligned}
(14)

By using the same method as above, we have

\begin{aligned} \lim_{n \rightarrow \infty } \Vert T_{n}-y_{n} \Vert &=0. \end{aligned}
(15)

From (12), we get

\begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}\leq{}& \alpha _{n} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert Sk_{n}-x^{*} \bigr\Vert ^{2} \\ & {}-\alpha _{n}(1-\alpha _{n}) \Vert T_{n}-Sk_{n} \Vert ^{2} \\ \leq{}& \alpha _{n} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ &{} -\alpha _{n}(1-\alpha _{n}) \Vert T_{n}-Sk_{n} \Vert ^{2} \\ \leq{}& \alpha _{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\alpha _{n}(1-\alpha _{n}) \Vert T_{n}-Sk_{n} \Vert ^{2}, \end{aligned}

so

\begin{aligned} \Vert T_{n}-Sk_{n} \Vert ^{2}\leq \frac{ \Vert x_{n}-x^{*} \Vert ^{2}- \Vert x_{n+1}-x^{*} \Vert ^{2}}{\alpha _{n}(1-\alpha _{n})}, \end{aligned}

which implies that

\begin{aligned} \lim_{n \rightarrow \infty } \Vert T_{n}-Sk_{n} \Vert &=0. \end{aligned}
(16)

Consider

\begin{aligned} W_{n}-x_{n}&=-\eta A^{*}(I-M_{Q})Ax_{n}, \end{aligned}

and by (14), we have

\begin{aligned} \lim_{n \rightarrow \infty } \Vert W_{n}-x_{n} \Vert &=0. \end{aligned}
(17)

From the property of $$P_{C}$$, we have

\begin{aligned} & \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad= \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-P_{C}(I- \zeta D_{3})x^{*} \bigr\Vert ^{2} \\ &\quad\leq \bigl\Vert (I-\zeta D_{3})W_{n}-(I-\zeta D_{3})x^{*} \bigr\Vert ^{2} \\ &\quad= \bigl\Vert \bigl(W_{n}-x^{*} \bigr)-\zeta \bigl(D_{3}W_{n}-D_{3}x^{*} \bigr) \bigr\Vert ^{2} \\ &\quad= \bigl\Vert W_{n}-x^{*} \bigr\Vert ^{2}-2 \zeta \bigl\langle W_{n}-x^{*}, D_{3}W_{n}-D_{3}x^{*} \bigr\rangle \\ &\qquad{} +\zeta ^{2} \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2} \\ &\quad\leq \bigl\Vert W_{n}-x^{*} \bigr\Vert ^{2}-2\zeta d_{3} \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2} \\ &\qquad{} +\zeta ^{2} \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2} \\ &\quad= \bigl\Vert W_{n}-x^{*} \bigr\Vert ^{2}- \zeta (2d_{3}-\zeta ) \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2} \\ &\quad\leq \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \zeta (2d_{3}-\zeta ) \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2}. \end{aligned}
(18)

By the definition of $$T_{n}$$, (7), Remark 1, and (18), we have

\begin{aligned} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}\leq{}& a \Vert W_{n}-W_{x^{*}} \Vert ^{2}+a(1-a) \Vert W_{n}-W_{x^{*}} \Vert ^{2} \\ &{} +(1-a)^{2} \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& a \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+a(1-a) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ & {}+(1-a)^{2} \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& \bigl(2a-a^{2} \bigr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-a)^{2} \bigl\Vert P_{C}(I- \zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& \bigl(2a-a^{2} \bigr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-a)^{2} \bigl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ &{} -\zeta (2d_{3}-\zeta ) \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2} \bigr] \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \zeta (2d_{3}-\zeta ) (1-a)^{2} \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2}. \end{aligned}
(19)

In addition, by the definition of $$x_{n+1}$$ and (19), we have

\begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}\leq{}& \alpha _{n} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert k_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& \alpha _{n} \bigl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\zeta (2d_{3}-\zeta ) (1-a)^{2} \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2} \bigr] \\ & {}+(1-\alpha _{n}) \bigl\Vert k_{n}-x^{*} \bigr\Vert ^{2} \\ ={}& \alpha _{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\alpha _{n}\zeta (2d_{3}-\zeta ) (1-a)^{2} \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2} \\ & {}+(1-\alpha _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \alpha _{n}\zeta (2d_{3}-\zeta ) (1-a)^{2} \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2}, \end{aligned}

so

\begin{aligned} \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2}\leq \frac{ \Vert x_{n}-x^{*} \Vert ^{2}- \Vert x_{n+1}-x^{*} \Vert ^{2}}{\alpha _{n}\zeta (2d_{3}-\zeta )(1-a)^{2}}, \end{aligned}

which implies that

\begin{aligned} \lim_{n \rightarrow \infty } \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert =0. \end{aligned}
(20)

From the property of $$P_{C}$$, we have

\begin{aligned} & \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad\leq \bigl\langle (I-\zeta D_{3})W_{n}-(I-\zeta D_{3})x^{*}, P_{C}(I- \zeta D_{3})W_{n}-x^{*} \bigr\rangle \\ &\quad=\frac{1}{2} \bigl[ \bigl\Vert (I-\zeta D_{3})W_{n}-(I- \zeta D_{3})x^{*} \bigr\Vert ^{2} \bigr]+ \bigl\Vert P_{C}(I- \zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ & \qquad{}- \bigl\Vert (I-\zeta D_{3})W_{n}-(I-\zeta D_{3})x^{*}- \bigl(P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr) \bigr\Vert ^{2} ] \\ &\quad\leq \frac{1}{2} \bigl[ \bigl\Vert W_{n}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ & \qquad{}- \bigl\Vert (I-\zeta D_{3})W_{n}-(I-\zeta D_{3})x^{*}- \bigl(P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr) \bigr\Vert ^{2} \bigr] \\ &\quad=\frac{1}{2} \bigl[ \bigl\Vert W_{n}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ & \qquad{}- \bigl\Vert \bigl(W_{n}-P_{C}(I-\zeta D_{3})W_{n} \bigr)-\zeta \bigl(D_{3}W_{n}-D_{3}x^{*} \bigr) \bigr\Vert ^{2} \bigr] \\ &\quad=\frac{1}{2} \bigl[ \bigl\Vert W_{n}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ &\qquad{} - \bigl\Vert W_{n}-P_{C}(I-\zeta D_{3})W_{n} \bigr\Vert ^{2} -\zeta ^{2} \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert ^{2} \\ &\qquad{} +2\zeta \bigl\langle W_{n}-P_{C}(I-\zeta D_{3})W_{n},D_{3}W_{n}-D_{3}x^{*} \bigr\rangle \bigr], \end{aligned}

so

\begin{aligned} \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2}\leq{}& \bigl\Vert W_{n}-x^{*} \bigr\Vert ^{2} - \bigl\Vert W_{n}-P_{C}(I- \zeta D_{3})W_{n} \bigr\Vert ^{2} \\ &{} +2\zeta \bigl\Vert W_{n}-P_{C}(I-\zeta D_{3})W_{n} \bigr\Vert \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert . \end{aligned}
(21)

By the definition of $$T_{n}$$, (7), Remark 1, and (21), we have

\begin{aligned} & \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad\leq a \Vert W_{n}-W_{x^{*}} \Vert ^{2}+a(1-a) \Vert W_{n}-W_{x^{*}} \Vert ^{2} \\ &\qquad{} +(1-a)^{2} \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad\leq a \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+a(1-a) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ &\qquad{} +(1-a)^{2} \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad\leq \bigl(2a-a^{2} \bigr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-a)^{2} \bigl\Vert P_{C}(I- \zeta D_{3})W_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad\leq \bigl(2a-a^{2} \bigr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-a)^{2} \bigl[ \bigl\Vert W_{n}-x^{*} \bigr\Vert ^{2} - \bigl\Vert W_{n}-P_{C} \\ &\qquad{}\times(I- \zeta D_{3})W_{n} \bigr\Vert ^{2}+2 \zeta \bigl\Vert W_{n}-P_{C}(I-\zeta D_{3})W_{n} \bigr\Vert \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert \bigr] \\ &\quad= \bigl(2a-a^{2} \bigr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-a)^{2} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ &\qquad{} -(1-a)^{2} \bigl\Vert W_{n}-P_{C}(I- \zeta D_{3})W_{n} \bigr\Vert ^{2} \\ & \qquad{}+2\zeta (1-a)^{2} \bigl\Vert W_{n}-P_{C}(I- \zeta D_{3})W_{n} \bigr\Vert \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert \\ &\quad= \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-(1-a)^{2} \bigl\Vert W_{n}-P_{C}(I- \zeta D_{3})W_{n} \bigr\Vert ^{2} \\ &\qquad{} +2\zeta (1-a)^{2} \bigl\Vert W_{n}-P_{C}(I- \zeta D_{3})W_{n} \bigr\Vert \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert . \end{aligned}
(22)

In addition, by the definition of $$x_{n+1}$$, (11), and (22), we have

\begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}\leq{}& \alpha _{n} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert k_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& \alpha _{n} \bigl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-(1-a)^{2} \bigl\Vert W_{n}-P_{C}(I- \zeta D_{3})W_{n} \bigr\Vert ^{2} \\ &{} +2\zeta (1-a)^{2} \bigl\Vert W_{n}-P_{C}(I- \zeta D_{3})W_{n} \bigr\Vert \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert \bigr] \\ &{} +(1-\alpha _{n}) \bigl\Vert k_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& \alpha _{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\alpha _{n}(1-a)^{2} \bigl\Vert W_{n}-P_{C}(I- \zeta D_{3})W_{n} \bigr\Vert ^{2} \\ &{} +2\alpha _{n}\zeta (1-a)^{2} \bigl\Vert W_{n}-P_{C}(I-\zeta D_{3})W_{n} \bigr\Vert \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert \\ & {}+(1-\alpha _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \alpha _{n}(1-a)^{2} \bigl\Vert W_{n}-P_{C}(I- \zeta D_{3})W_{n} \bigr\Vert ^{2} \\ &{} +2\alpha _{n}\zeta (1-a)^{2} \bigl\Vert W_{n}-P_{C}(I-\zeta D_{3})W_{n} \bigr\Vert \bigl\Vert D_{3}W_{n}-D_{3}x^{*} \bigr\Vert . \end{aligned}
(23)

From (20) and (23), we get

\begin{aligned} \lim_{n \rightarrow \infty } \bigl\Vert W_{n}-P_{C}(I- \zeta D_{3})W_{n} \bigr\Vert &=0. \end{aligned}
(24)

Let $$G_{n}=aW_{n}+(1-a)P_{C}(I-\lambda _{3}D_{3})W_{n}$$. From the property of $$P_{C}$$, we have

\begin{aligned} & \bigl\Vert P_{C}(I-\zeta D_{2})G_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad= \bigl\Vert P_{C}(I-\zeta D_{2})G_{n}-P_{C}(I- \zeta D_{2})x^{*} \bigr\Vert ^{2} \\ &\quad\leq \bigl\Vert (I-\zeta D_{2})G_{n}-(I-\zeta D_{2})x^{*} \bigr\Vert ^{2} \\ &\quad= \bigl\Vert \bigl(G_{n}-x^{*} \bigr)-\zeta \bigl(D_{2}G_{n}-D_{2}x^{*} \bigr) \bigr\Vert ^{2} \\ &\quad= \bigl\Vert G_{n}-x^{*} \bigr\Vert ^{2}-2 \zeta \bigl\langle G_{n}-x^{*}, D_{2}G_{n}-D_{2}x^{*} \bigr\rangle \\ &\qquad{} +\zeta ^{2} \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ^{2} \\ &\quad\leq \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-2\zeta d_{2} \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ^{2} \\ & \qquad{}+\zeta ^{2} \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ^{2} \\ &\quad= \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \zeta (2d_{2}-\zeta ) \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ^{2}. \end{aligned}
(25)

By the definition of $$T_{n}$$ and (25), we have

\begin{aligned} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}\leq {}&a \Vert W_{n}-W_{x^{*}} \Vert ^{2}+(1-a) \bigl\Vert P_{C}(I- \zeta D_{2})G_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& a \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-a) \bigl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ &{} -\zeta (2d_{2}-\zeta ) \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ^{2} \bigr] \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \zeta (1-a) (2d_{2}-\zeta ) \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ^{2}. \end{aligned}
(26)

In addition, by the definition of $$x_{n+1}$$ and (26), we have

\begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}\leq{}& \alpha _{n} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert k_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& \alpha _{n} \bigl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\zeta (1-a) (2d_{2}-\zeta ) \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ^{2} \bigr] \\ & {}+(1-\alpha _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \zeta \alpha _{n}(1-\alpha _{n}) (2d_{2}-\zeta ) \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ^{2}, \end{aligned}

so

\begin{aligned} \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ^{2}&\leq \frac{ \Vert x_{n}-x^{*} \Vert ^{2}- \Vert x_{n+1}-x^{*} \Vert ^{2}}{\zeta \alpha _{n}(1-\alpha _{n})(2d_{2}-\zeta )}. \end{aligned}

It implies that

\begin{aligned} \lim_{n \rightarrow \infty } \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert =0. \end{aligned}
(27)

From the property of $$P_{C}$$, we have

\begin{aligned} & \bigl\Vert P_{C}(I-\zeta D_{2})G_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad= \bigl\langle (I-\zeta D_{2})G_{n}-(I-\zeta D_{2})x^{*}, P_{C}(I-\zeta D_{2})G_{n}-x^{*} \bigr\rangle \\ &\quad=\frac{1}{2} \bigl[ \bigl\Vert (I-\zeta D_{2})G_{n}-(I- \zeta D_{2})x^{*} \bigr\Vert ^{2} + \bigl\Vert P_{C}(I- \zeta D_{2})G_{n}-x^{*} \bigr\Vert ^{2} \\ & \qquad{}- \bigl\Vert (I-\zeta D_{2})G_{n}-(I-\zeta D_{2})x^{*}- \bigl((I-\zeta D_{2})G_{n}-x^{*} \bigr) \bigr\Vert ^{2} \bigr] \\ &\quad\leq \frac{1}{2} \bigl[ \bigl\Vert G_{n}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert P_{C}(I-\zeta D_{2})G_{n}-x^{*} \bigr\Vert ^{2} \\ &\qquad{} - \bigl\Vert (I-\zeta D_{2})G_{n}-(I-\zeta D_{2})x^{*}- \bigl((I-\zeta D_{2})G_{n}-x^{*} \bigr) \bigr\Vert ^{2} \bigr] \\ &\quad=\frac{1}{2} \bigl[ \bigl\Vert G_{n}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert P_{C}(I-\zeta D_{2})G_{n}-x^{*} \bigr\Vert ^{2} \\ &\qquad{} - \bigl\Vert \bigl(G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr)-\zeta \bigl(D_{2}G_{n}-D_{2}x^{*} \bigr) \bigr\Vert ^{2} \bigr] \\ &\quad=\frac{1}{2} \bigl[ \bigl\Vert G_{n}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert P_{C}(I-\zeta D_{2})G_{n}-x^{*} \bigr\Vert ^{2} \\ &\qquad{} - \bigl\Vert G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr\Vert ^{2} \\ & \qquad{}+2\zeta \bigl\langle G_{n}-P_{C}(I-\zeta D_{2})G_{n}, D_{2}G_{n}-D_{2}x^{*} \bigr\rangle \\ & \qquad{}-\zeta ^{2} \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ^{2} \bigr]. \end{aligned}

It implies that

\begin{aligned} & \bigl\Vert P_{C}(I-\zeta D_{2})G_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad\leq \bigl\Vert G_{n}-x^{*} \bigr\Vert ^{2}- \bigl\Vert G_{n}-P_{C}(I- \zeta D_{2})G_{n} \bigr\Vert ^{2} \\ &\qquad{} +2\zeta \bigl\langle G_{n}-P_{C}(I-\zeta D_{2})G_{n}, D_{2}G_{n}-D_{2}x^{*} \bigr\rangle \\ &\quad\leq \bigl\Vert G_{n}-x^{*} \bigr\Vert ^{2}- \bigl\Vert G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr\Vert ^{2} \\ &\qquad{} +2\zeta \bigl\Vert G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr\Vert \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert . \end{aligned}
(28)

By the definition of $$T_{n}$$ and (28), we have

\begin{aligned} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}\leq {}&a \Vert W_{n}-W_{x^{*}} \Vert ^{2}+(1-a) \bigl\Vert P_{C}(I- \zeta D_{2})G_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& a \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-a) \bigl[ \bigl\Vert G_{n}-x^{*} \bigr\Vert ^{2}- \bigl\Vert G_{n}-P_{C} \\ &{}\times (I- \zeta D_{2})G_{n} \bigr\Vert ^{2} +2 \zeta \bigl\Vert G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr\Vert \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert \bigr] \\ \leq {}&a \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-a) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ &{} -(1-a) \bigl\Vert G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr\Vert ^{2} \\ & {}+2\zeta \bigl\Vert G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr\Vert \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ] \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-(1-a) \bigl\Vert G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr\Vert ^{2} \\ &{} +2\zeta \bigl\Vert G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr\Vert \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert ]. \end{aligned}
(29)

In addition, by the definition of $$x_{n+1}$$ and (29), we have

\begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}\leq {}&\alpha _{n} \bigl\Vert T_{n}-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert k_{n}-x^{*} \bigr\Vert ^{2} \\ \leq{}& \alpha _{n} \bigl[ \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-(1-a) \bigl\Vert G_{n}-P_{C}(I- \zeta D_{2})G_{n} \bigr\Vert ^{2} \\ &{} +2\zeta \bigl\Vert G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr\Vert \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert \bigr] \\ &{} +(1-\alpha _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ ={}& \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \alpha _{n}(1-a) \bigl\Vert G_{n}-P_{C}(I-\zeta D_{2})G_{n} \bigr\Vert ^{2} \\ &{} +2\zeta \alpha _{n}(1-a) \bigl\Vert G_{n}-P_{C}(I- \zeta D_{2})G_{n} \bigr\Vert \bigl\Vert D_{2}G_{n}-D_{2}x^{*} \bigr\Vert , \end{aligned}
(30)

by (30) and (27), we get

\begin{aligned} \lim_{n \rightarrow \infty } \bigl\Vert G_{n}-P_{C}(I- \zeta D_{2})G_{n} \bigr\Vert &=0. \end{aligned}
(31)

Since

\begin{aligned} T_{n}-W_{n}=(1-a) \bigl(P_{C}(I-\zeta D_{2}) \bigl(aW_{n}+(1-a)P_{C}(I-\zeta D_{3})W_{n} \bigr)-W_{n} \bigr). \end{aligned}

From the property of norm, we have

\begin{aligned} & \bigl\Vert P_{C}(I-\zeta D_{2}) \bigl(aW_{n}+(1-a)P_{C}(I- \zeta D_{3})W_{n} \bigr)-W_{n} \bigr\Vert \\ &\quad\leq \bigl\Vert P_{C}(I-\zeta D_{2}) \bigl(aW_{n}+(1-a)P_{C}(I-\zeta D_{3})W_{n} \bigr) \\ &\qquad{} - \bigl(aW_{n}+(1-a)P_{C}(I-\zeta D_{3})W_{n} \bigr) \bigr\Vert \\ &\qquad{} +\bigr\| (aW_{n}+(1-a)P_{C}(I-\zeta D_{3})W_{n}-W_{n} \bigr\| \\ &\quad= \bigl\Vert P_{C}(I-\zeta D_{2})G_{n}-G_{n} \bigr\Vert +(1-a) \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-W_{n} \bigr\Vert . \end{aligned}
(32)

Then we have

\begin{aligned} \Vert T_{n}-W_{n} \Vert \leq {}&(1-a) \bigl[ \bigl\Vert P_{C}(I-\zeta D_{2})G_{n}-G_{n} \bigr\Vert \\ &{} +(1-a) \bigl\Vert P_{C}(I-\zeta D_{3})W_{n}-W_{n} \bigr\Vert \bigr]. \end{aligned}

From (24) and (31), it implies that

\begin{aligned} \lim_{n \rightarrow \infty } \Vert T_{n}-W_{n} \Vert &=0. \end{aligned}
(33)

From (15), (17), (33), and

\begin{aligned} \Vert y_{n}-x_{n} \Vert &\leq \Vert y_{n}-T_{n} \Vert + \Vert T_{n}-W_{n} \Vert + \Vert W_{n}-x_{n} \Vert , \end{aligned}

we have

\begin{aligned} \lim_{n \rightarrow \infty } \Vert y_{n}-x_{n} \Vert =0. \end{aligned}
(34)

Moreover, from (16), (15), (34), and

\begin{aligned} \Vert x_{n}-Sk_{n} \Vert &\leq \Vert x_{n}-y_{n} \Vert + \Vert y_{n}-T_{n} \Vert + \Vert T_{n}-Sk_{n} \Vert , \end{aligned}

we have

\begin{aligned} \lim_{n \rightarrow \infty } \Vert x_{n}-Sk_{n} \Vert &=0. \end{aligned}
(35)

Since $$\{x_{n}\}^{\infty }_{n=0}$$ is bounded, it has a subsequence $$\{x_{n_{k}}\}^{\infty }_{k=0}$$ which weakly converges to some $$\bar{x}\in C$$.

Assume $$\bar{x} \notin F(S)$$. By the nonexpansiveness of S and Opial’s property and (35), we have

\begin{aligned} \lim_{k \rightarrow \infty }\inf \Vert x_{n_{k}}-\bar{x} \Vert &< \lim _{k \rightarrow \infty }\inf \Vert x_{n_{k}}-S\bar{x} \Vert \\ &\leq \lim_{k \rightarrow \infty }\inf \bigl[ \Vert x_{n_{k}}-Sk_{n_{k}} \Vert + \Vert Sk_{n_{k}}-S\bar{x} \Vert \bigr] \\ &\leq \lim_{k \rightarrow \infty }\inf \bigl[ \Vert x_{n_{k}}-Sk_{n_{k}} \Vert + \Vert k_{n_{k}}-\bar{x} \Vert \bigr] \\ &=\lim_{k \rightarrow \infty }\inf \Vert k_{n_{k}}-\bar{x} \Vert \\ &\leq \lim_{k \rightarrow \infty }\inf \Vert x_{n_{k}}-\bar{x} \Vert . \end{aligned}

This is a contradiction, then we have

\begin{aligned} \bar{x}\in F(S). \end{aligned}

Assume $$\bar{x}\notin \bigcap_{i=1}^{3}\Phi _{i}$$. From Lemma 2.6, we have $$\bar{x}\notin F(M_{C}(I-\eta A^{*}(I-M_{Q})A))$$. By Opial’s condition, (34), and Remark 1, we have

\begin{aligned} \lim_{k \rightarrow \infty }\inf \Vert x_{n_{k}}-\bar{x} \Vert < {}& \lim_{k \rightarrow \infty }\inf \bigl\Vert x_{n_{k}}-M_{C} \bigl(I-\eta A^{*}(I-M_{Q})A \bigr) \bar{x} \bigr\Vert \\ \leq{}& \lim_{k \rightarrow \infty }\inf \Vert x_{n_{k}}-y_{n_{k}} \Vert +\lim_{k \rightarrow \infty }\inf \bigl\Vert M_{C} \bigl(x_{n_{k}}-\eta A^{*} \\ &{}\times (I- M_{Q})Ax_{n_{k}} \bigr)-M_{C} \bigl(I-\eta A^{*}(I-M_{Q})A \bigr)\bar{x} \bigr\Vert \\ \leq{}& \lim_{k \rightarrow \infty }\inf \bigl( \Vert x_{n_{k}}-y_{n_{k}} \Vert + \Vert x_{n_{k}}- \bar{x} \Vert \bigr) \\ ={}&\lim_{k \rightarrow \infty }\inf \Vert x_{n_{k}}-\bar{x} \Vert . \end{aligned}
(36)

This is a contradiction, then we have

\begin{aligned} \bar{x}\in F \bigl(M_{C} \bigl(I-\eta A^{*}(I-M_{Q})A \bigr) \bigr). \end{aligned}

It implies that

\begin{aligned} \bar{x}\in \bigcap_{i=1}^{3}\Phi _{i}. \end{aligned}

Hence

\begin{aligned} \bar{x}\in \Im. \end{aligned}

In order to show that the entire sequence $$\{x_{n}\}$$ weakly converges to , assume $$\{x\}_{n_{k}}\rightharpoonup \hat{x}$$ as $$k \rightarrow \infty$$, with $$\bar{x}\neq \hat{x}$$ and $$\hat{x}\in \Im$$. By Opial’s condition, we have

\begin{aligned} \lim_{n \rightarrow \infty } \Vert x_{n}-\bar{x} \Vert &=\lim _{k \rightarrow \infty }\inf \Vert x_{n_{k}}-\bar{x} \Vert \\ &< \lim_{k \rightarrow \infty }\inf \Vert x_{n_{k}}-\hat{x} \Vert \\ &=\lim_{n \rightarrow \infty } \Vert x_{n}-\hat{x} \Vert \\ &=\lim_{n \rightarrow \infty }\inf \Vert x_{n_{k}}-\hat{x} \Vert \\ &< \lim_{n \rightarrow \infty }\inf \Vert x_{n_{k}}-\bar{x} \Vert \\ &=\lim_{n \rightarrow \infty } \Vert x_{n}-\bar{x} \Vert . \end{aligned}

\begin{aligned} \bar{x}\doteq \hat{x}. \end{aligned}

It implies that the sequence $$\{x_{n}\}^{\infty }_{n=0}$$ weakly converges to $$\bar{x}\in \Im$$.

From (34), we have $$\{y_{n}\}^{\infty }_{n=0}$$ weakly converges to $$\bar{x}\in \Im$$.

Finally, if we take

\begin{aligned} U_{n}&=P_{\Im }x_{n}, \end{aligned}
(37)

by Lemma 2.2, we see that $$\{P_{\Im }x_{n}\}^{\infty }_{n=0}$$ converges strongly to some $$z\in \Im$$. From (37), we get

\begin{aligned} \langle \bar{x}-U_{n},U_{n}-x_{n}\rangle \geq 0, \quad\forall \bar{x} \in \Im. \end{aligned}

Take $$n\rightarrow \infty$$, we also have

\begin{aligned} \langle \bar{x}-z,z-\bar{x}\rangle \geq 0, \end{aligned}

and hence $$\bar{x}=z$$. Therefore $$U_{n}$$ converges strongly to $$\bar{x}\in \Im$$, this completes the proof. □

## Application

Let C be a closed convex subset of H. The standard constrained convex optimization problem is to find $$x^{*}\in C$$ such that

\begin{aligned} \Im \bigl(x^{*} \bigr)=\mathop{\min } _{x\in C} \Im (x), \end{aligned}
(38)

where $$\Im:C\rightarrow \mathbb{R}$$ is a convex, Frechet differentiable function. The set of all solution of (38) is denoted by $$\Phi _{\Im }$$.

### Lemma 4.1

( Optimality condition)

A necessary condition of optimality for a point $${x^{*}} \in C$$ to be a solution of the minimization problem (38) is that $${x^{*}}$$ solves the variational inequality

\begin{aligned} \bigl\langle {\nabla \Im \bigl({x^{*}} \bigr),x - {x^{*}}} \bigr\rangle \ge 0 \end{aligned}
(39)

for all $$x \in C$$. Equivalently, $${x^{*}} \in C$$ solves the fixed point equation

\begin{aligned} {x^{*}} = {P_{C}}(I - \zeta \nabla \Im ){x^{*}} \end{aligned}

for every $$\zeta > 0$$. If, in addition, is convex, then the optimality condition (39) is also sufficient.

By using the concept of the split modified system of variational inequalities problem (SMSVIP), we consider the problem for finding $$(x^{*},y^{*},z^{*})\in C\times C\times C$$ such that

\begin{aligned} \textstyle\begin{cases} \langle x^{*}-(I-\zeta \nabla \Im _{1})(ax^{*}+(1-a)y^{*}),x-x^{*} \rangle \geq 0,\quad \forall x\in C, \\ \langle y^{*}-(I-\zeta \nabla \Im _{2})(ax^{*}+(1-a)z^{*}),x-y^{*} \rangle \geq 0,\quad \forall x\in C, \\ \langle z^{*}-(I-\zeta \nabla \Im _{3})x^{*},x-z^{*}\rangle \geq 0,\quad \forall x\in C, \end{cases}\displaystyle \end{aligned}
(40)

and finding $$(\bar{x^{*}}=Ax^{*}, \bar{y^{*}}=Ay^{*}, \bar{z^{*}}=Az^{*})\in Q\times Q\times Q$$ such that

\begin{aligned} \textstyle\begin{cases} \langle \bar{x^{*}}-(I-\bar{\zeta }\nabla \bar{\Im _{1}})(a\bar{x^{*}}+(1-a) \bar{y^{*}}),\bar{x}-\bar{x^{*}}\rangle \geq 0, \quad\forall \bar{x}\in Q, \\ \langle \bar{y^{*}}-(I-\bar{\zeta }\nabla \bar{\Im _{2}})(a\bar{x^{*}}+(1-a) \bar{z^{*}}),\bar{x}-\bar{y^{*}}\rangle \geq 0, \quad\forall \bar{x}\in Q, \\ \langle \bar{z^{*}}-(I-\bar{\zeta }\nabla \bar{\Im _{3}})\bar{x^{*}}, \bar{x}-\bar{z^{*}}\rangle \geq 0,\quad \forall \bar{x}\in Q, \end{cases}\displaystyle \end{aligned}
(41)

where $$\Im _{1},\Im _{2},\Im _{3}: C\rightarrow \mathbb{R}$$ with $$\nabla \Im _{1},\nabla \Im _{2},\nabla \Im _{3}$$ are the gradients of $$\Im _{1},\Im _{2},\Im _{3}$$, respectively, and $$\bar{\Im }_{1},\bar{\Im }_{2},\bar{\Im }_{3}:Q\rightarrow \mathbb{R}$$ with $$\nabla \bar{\Im }_{1},\nabla \bar{\Im }_{2},\nabla \bar{\Im }_{3}$$ are the gradients of $$\bar{\Im }_{1},\bar{\Im }_{2},\bar{\Im }_{3}$$, respectively, $${\zeta,\bar{\zeta } >0}$$ and $$a\in [0,1]$$. The sets of all solution of (40) and (41) are denoted by $$\Psi _{\nabla \Im _{1},\nabla \Im _{2},\nabla \Im _{3}}$$ and $$\Psi _{\nabla \bar{\Im _{1}},\nabla \bar{\Im _{2}},\nabla \bar{\Im _{3}}}$$, respectively. The set of all solutions of the split modified system of variational inequalities (SMSVIP) is denoted by $$\Psi ^{\nabla \Im _{1},\nabla \Im _{2},\nabla \Im _{3}}_{\nabla \bar{\Im _{1}},\nabla \bar{\Im _{2}},\nabla \bar{\Im _{3}}}$$, that is,

\begin{aligned} \Psi ^{\nabla \Im _{1},\nabla \Im _{2},\nabla \Im _{3}}_{\nabla \bar{\Im _{1}},\nabla \bar{\Im _{2}},\nabla \bar{\Im _{3}}}= \bigl\{ \bigl(x^{*},y^{*},z^{*} \bigr) \in \Psi _{\nabla \Im _{1},\nabla \Im _{2},\nabla \Im _{3}}: \bigl( \bar{x^{*}}, \bar{y^{*}}, \bar{z^{*}} \bigr)\in \Psi _{\nabla \bar{\Im _{1}}, \nabla \bar{\Im _{2}},\nabla \bar{\Im _{3}}} \bigr\} . \end{aligned}

### Lemma 4.2

()

Let C and Q be nonempty closed convex subsets of real Hilbert spaces $$H_{1}$$ and $$H_{2}$$, respectively. Let $$\Im _{1},\Im _{2},\Im _{3}:C \to \mathbb{R}$$ be real-valued convex functions with the gradients $$\nabla \Im _{1},\nabla \Im _{2},\nabla \Im _{3}$$ being $$\frac{1}{{{L_{\Im _{1}}}}},\frac{1}{{{L_{\Im _{2}}}}}, \frac{1}{{{L_{\Im _{3}}}}}$$-inverse strongly monotone and continuous, respectively, where $$\zeta \in (0,\frac{2}{L_{\Im }})$$ with $$\frac{1}{L_{\Im }} = \operatorname{min} \{\frac{1}{L_{\Im _{1}}}, \frac{1}{L_{\Im _{2}}},\frac{1}{L_{\Im _{3}}}\}$$. Let $$\bar{\Im }_{1},\bar{\Im }_{2},\bar{\Im }_{3}:Q \to \mathbb{R}$$ be real-valued convex functions with the gradients $$\nabla \bar{\Im _{1}}$$, $$\nabla \bar{\Im _{2}}$$, $$\nabla \bar{\Im _{3}}$$ being $$\frac{1}{{{L_{\bar{\Im _{1}}}}}}, \frac{1}{{{L_{\bar{\Im _{2}}}}}}, \frac{1}{{{L_{\bar{\Im _{3}}}}}}$$-inverse strongly monotone and continuous, respectively, where $$\bar{\zeta }\in (0,\frac{2}{L_{\bar{\Im }}})$$ with $$\frac{1}{L_{\bar{\Im }}} = \operatorname{min} \{\frac{1}{L_{\bar{\Im }_{1}}}, \frac{1}{L_{\bar{\Im }_{2}}},\frac{1}{L_{\bar{\Im }_{3}}}\}$$. Let $$A:H_{1}\rightarrow H_{2}$$ be a bounded linear operator with adjoint $$A^{*}$$ and $$\eta \in (0,\frac{1}{L})$$ with L being the spectral radius of the operator $$A^{*}A$$. Define $$M_{C}:H_{1}\rightarrow C$$ by $$M_{C}(x)=P_{C}(I-\zeta \nabla \Im _{1})(ax+(1-a)P_{C}(I-\zeta \nabla \Im _{2})(ax+(1-a)P_{C}(I-\zeta \nabla \Im _{3})x))$$, $$\forall x\in H_{1}$$, and define $$M_{Q}:H_{2}\rightarrow Q$$ by $$M_{Q}(\hat{x})=P_{Q}(I-\bar{\zeta }\nabla \bar{\Im _{1}})(a\hat{x}+(1-a)P_{Q}(I- \bar{\zeta }\nabla \bar{\Im _{2}})(a\hat{x}+(1-a)P_{Q}(I-\bar{\zeta } \nabla \bar{\Im _{3}})\hat{x}))$$, $$\forall \hat{x}\in H_{2}$$. Let $$\bigcap_{i=1}^{3}\Phi _{\Im _{i}}\neq \emptyset$$ and $$\Phi _{\Im _{i}}=\{ \Im _{i}(x)= \min_{x^{*}\in C}\Im _{i}(x^{*}) : \bar{\Im _{i}}(Ax)= \min_{Ax^{*}\in Q}\bar{\Im _{i}}(Ax^{*}) \}$$ for all $$i=1,2,3$$. Then

$$\bigcap_{i=1}^{3}\Phi _{\Im _{i}}=F \bigl(M_{C} \bigl(I-\eta A^{*}(I-M_{Q})A \bigr) \bigr).$$

### Theorem 4.3

Let C and Q be nonempty closed convex subsets of real Hilbert spaces $$H_{1}$$ and $$H_{2}$$, respectively, and let $$S:C\rightarrow C$$ be a nonexpansive mapping. Let $$\Im _{1},\Im _{2},\Im _{3}:C \to \mathbb{R}$$ be real-valued convex functions with the gradients $$\nabla \Im _{1},\nabla \Im _{2},\nabla \Im _{3}$$ being $$\frac{1}{{{L_{\Im _{1}}}}},\frac{1}{{{L_{\Im _{2}}}}}, \frac{1}{{{L_{\Im _{3}}}}}$$-inverse strongly monotone and continuous, respectively, where $$\zeta \in (0,\frac{2}{L_{\Im }})$$ with $$\frac{1}{L_{\Im }}= \operatorname{min} \{\frac{1}{L_{\Im _{1}}},\frac{1}{L_{\Im _{2}}}, \frac{1}{L_{\Im _{3}}}\}$$. Let $$\bar{\Im }_{1},\bar{\Im }_{2},\bar{\Im }_{3}:Q \to \mathbb{R}$$ be real-valued convex functions with the gradients $$\nabla \bar{\Im _{1}},\nabla \bar{\Im _{2}},\nabla \bar{\Im _{3}}$$ being $$\frac{1}{{{L_{\bar{\Im _{1}}}}}}, \frac{1}{{{L_{\bar{\Im _{2}}}}}}, \frac{1}{{{L_{\bar{\Im _{3}}}}}}$$-inverse strongly monotone and continuous, respectively, where $$\bar{\zeta }\in (0,\frac{2}{L_{\bar{\Im }}})$$ with $$\frac{1}{L_{\bar{\Im }}}= \operatorname{min} \{\frac{1}{L_{\bar{\Im }_{1}}}, \frac{1}{L_{\bar{\Im }_{2}}},\frac{1}{L_{\bar{\Im }_{3}}}\}$$. Let $$A:H_{1}\rightarrow H_{2}$$ be a bounded linear operator with adjoint $$A^{*}$$ and $$\eta \in (0,\frac{1}{L})$$ with L being the spectral radius of the operator $$A^{*}A$$. Define $$M_{C}:H_{1}\rightarrow C$$ by $$M_{C}(x)=P_{C}(I-\zeta \nabla \Im _{1})(ax+(1-a)P_{C}(I-\zeta \nabla \Im _{2})(ax+(1-a)P_{C}(I-\zeta \nabla \Im _{3})x))$$, $$\forall x\in H_{1}$$, and define $$M_{Q}:H_{2}\rightarrow Q$$ by $$M_{Q}(\hat{x})=P_{Q}(I-\bar{\zeta }\nabla \bar{\Im _{1}})(a\hat{x}+(1-a)P_{Q}(I- \bar{\zeta }\nabla \bar{\Im _{2}})(a\hat{x}+(1-a)P_{Q}(I-\bar{\zeta } \nabla \bar{\Im _{3}})\hat{x}))$$, $$\forall \hat{x}\in H_{2}$$. Let the sequences $$\{x_{n}\}$$ and $$\{y_{n}\}$$ be generated by $$x_{1}\in H_{1}$$ and

\begin{aligned} y_{n}=M_{C} W_{n} =P_{C}(I-\zeta \nabla \Im _{1})T_{n}, \end{aligned}

where $$W_{n}=(I-\eta A^{*}(I-M_{Q})A)x_{n}$$ and $$T_{n}=aW_{n}+(1-a)P_{C}(I-\zeta \nabla \Im _{2})(aW_{n}+(1-a)P_{C}(I- \zeta \nabla \Im _{3})W_{n}))$$.

\begin{aligned} &Q_{n}= \bigl\{ z\in H: \bigl\langle (I-\zeta \nabla \Im _{1})T_{n}-y_{n},y_{n}-z \bigr\rangle \geq 0 \bigr\} ,\\ &x_{n+1}=\alpha _{n}T_{n}+(1-\alpha _{n})SP_{Q_{n}} \bigl(T_{n}-\zeta \nabla \Im _{1}(y_{n}) \bigr), \quad\forall n\in \mathbb{N}. \end{aligned}

Assume that the following conditions hold:

1. (i)

$$\Im =F(S)\bigcap \bigcap_{i=1}^{3}\Phi _{\Im _{i}} \neq \emptyset$$, where $$\Phi _{\Im _{i}}=\{ \Im _{i}(x)= \min_{x^{*}\in C}\Im _{i}(x^{*}) : \bar{\Im _{i}}(Ax)= \min_{Ax^{*}\in Q}\bar{\Im _{i}}(Ax^{*}) \}$$ for all $$i=1,2,3$$.

2. (ii)

$$\alpha _{n}\in [c,d]\subset (0,1)$$.

Then $$\{x_{n}\}$$ converges weakly to $$x_{0}=P_{\Im }{x_{n}}$$, which $$(x_{0},y_{0},z_{0})\in \Omega ^{\nabla \Im _{1},\nabla \Im _{2}, \nabla \Im _{3}}_{\nabla \bar{\Im _{1}},\nabla \bar{\Im _{2}},\nabla \bar{\Im _{3}}}$$, where $$y_{0}=P_{C}(I-\zeta \nabla \Im _{2})(ax_{0}+(1-a)z_{0})$$ and $$z_{0}=P_{C}(I-\zeta \nabla \Im _{3})x_{0}$$ with $$\bar{x_{0}}=Ax_{0}$$, $$\bar{y_{0}}=Ay_{0}$$, and $$\bar{z_{0}}=Az_{0}$$.

### Proof

By using Theorem 3.1 and Lemma 4.2, we obtain the conclusion. □

## Example and numerical results

In this section, we give the following example to support our main theorem.

### Example 5.1

Let $$\mathbb{R}$$ be the set of real numbers, $$C:=\{x\in H|1\leq 2x_{1}+x_{2}\leq 7\}$$, $$Q:=\{x\in H|-10\leq 3x_{1}-x_{2}\leq 20\}$$, $$H_{1}=H_{2}=\mathbb{R}^{2}$$. Let $$D_{1},D_{2},D_{3}:C\rightarrow \mathbb{R}^{2}$$ be defined by $$D_{1}(x_{1},x_{2})=(x_{1} -2,x_{2} +1)$$, $$D_{2}(x_{1},x_{2})=(x_{1} -3,x_{2} -\frac{5}{2})$$, and $$D_{3}(x_{1},x_{2})=(x_{1} +2,x_{2} -6)$$ for all $$(x_{1},x_{2})\in C$$. Let $$\bar{D_{1}},\bar{D_{2}},\bar{D_{3}}:Q\rightarrow \mathbb{R}^{2}$$ be defined by $$\bar{D_{1}}(\bar{x_{1}},\bar{x_{2}})=(\bar{x_{1}} -4,\bar{x_{2}} +8)$$, $$\bar{D_{2}}(\bar{x_{1}},\bar{x_{2}})=(\bar{x_{1}} -12,\bar{x_{2}} -8)$$, and $$\bar{D_{3}}(\bar{x_{1}},\bar{x_{2}})=(\bar{x_{1}} +16,\bar{x_{2}} -30)$$ for all $$(\bar{x_{1}},\bar{x_{2}})\in Q$$. Let $$A:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$$ be defined by $$A(x_{1},x_{2})=(2x_{1},2x_{2})$$ and $$A^{*}:\mathbb{R}^{2}\rightarrow \mathbb{R}$$ be defined by $$A^{*}(x_{1},x_{2})=(2x_{1},2x_{2})$$. Define $$M_{C}:H_{1}\rightarrow C$$ by $$M_{C}(x)=P_{C}(I-\frac{1}{2}D_{1})(\frac{1}{2}x+\frac{1}{2}P_{C}(I- \frac{1}{2}D_{2})(\frac{1}{2}x+\frac{1}{2}P_{C}(I-\frac{1}{2}D_{3})x))$$, $$\forall x=(x_{1},x_{2})\in H_{1}$$, define $$M_{Q}:H_{2}\rightarrow Q$$ by $$M_{Q}(\hat{x})=P_{Q}(I-\frac{1}{5}\bar{D_{1}})(\frac{1}{2}\hat{x}+ \frac{1}{2}P_{Q}(I-\frac{1}{5}\bar{D_{2}})(\frac{1}{2}\hat{x}+ \frac{1}{2}P_{Q}(I-\frac{1}{5}\bar{D_{3}})\hat{x}))$$, $$\forall \hat{x}=(\hat{x_{1}},\hat{x_{2}})\in H_{2}$$, and define $$S:C\rightarrow C$$ by $$S(x_{1},x_{2})=(\frac{x_{1}}{2}+1,\frac{x_{2}}{2})$$. Let the sequences $$\{x_{n}\}$$ and $$\{y_{n}\}$$ be generated by $$x_{1}\in H_{1}$$ and

\begin{aligned} y_{n}=M_{C} W_{n} =P_{C} \biggl(I- \frac{1}{2}(x_{1} -2,x_{2} +1) \biggr)T_{n}, \end{aligned}

where $$W_{n}=(I-\frac{1}{8}A^{*}(I-M_{Q})A)x_{n}$$ and $$T_{n}=\frac{1}{2}W_{n}+\frac{1}{2}P_{C}(I-\frac{1}{2}(x_{1} -3,x_{2} - \frac{5}{2}))(\frac{1}{2}W_{n}+\frac{1}{2}P_{C}(I-\frac{1}{2}(x_{1} +2,x_{2} -6))W_{n}))$$,

\begin{aligned} Q_{n}= \biggl\{ z\in H: \biggl\langle \biggl(I-\frac{1}{2}(x_{1} -2,x_{2} +1) \biggr)T_{n}-y_{n},y_{n}-z \biggr\rangle \geq 0 \biggr\} , \end{aligned}

and

\begin{aligned} x_{n+1}=\frac{n+1}{5n}T_{n}+ \biggl(1-\frac{n+1}{5n} \biggr)SP_{Q_{n}} \biggl(T_{n}- \frac{1}{2}(x_{1} -2,x_{2} +1) (y_{n}) \biggr), \quad\forall n\in \mathbb{N}, \end{aligned}

where

\begin{aligned} P_{C}x= \textstyle\begin{cases} (x_{1},x_{2})-\frac{[2x_{1}+x_{2}-7](2,1)}{5} & \text{if } 2x_{1}+x_{2}>7, \\ (x_{1},x_{2})& \text{if } 1\leq 2x_{1}+x_{2}\leq 7, \\ (x_{1},x_{2})-\frac{[2x_{1}+x_{2}-1](2,1)}{5}& \text{if} 2x_{1}+x_{2}< 1, \end{cases}\displaystyle \end{aligned}

for every $$x=(x_{1},x_{2})\in H_{1}$$ and

\begin{aligned} P_{Q}\hat{x}= \textstyle\begin{cases} ({x_{1}},{x_{2}})-\frac{[3{x_{1}}-{x_{2}}-20](3,-1)}{10} &\text{if } 3{x_{1}}-{x_{2}}>20, \\ ({x_{1}},{x_{2}}) & \text{if } -10\leq 3{x_{1}}-{x_{2}}\leq 20, \\ ({x_{1}},{x_{2}})-\frac{[3{x_{1}}-{x_{2}}+10](3,-1)}{10}& \text{if } 3{x_{1}}-{x_{2}}< -10, \end{cases}\displaystyle \end{aligned}

for every $$\hat{x}=({x_{1}},{x_{2}})\in H_{2}$$. By the definition of $$S, D_{i}, \bar{D_{i}}, M_{C}, M_{Q}$$ for every $$i=1,2,3$$, we have $$(2,0)\in F(M_{C}(I-\frac{1}{8}A^{*}(I-M_{Q})A))$$. From Theorem 3.1, we can conclude that the sequences $$\{x_{n}\}$$ and $$\{y_{n}\}$$ converge strongly to $$(2,0)$$.

Table 1 and Fig. 1 show the numerical results of sequences $$\{x_{n}\}$$ and $$\{y_{n}\}$$ where $$x_{1}=(-5,5)$$ and $$n=N=30$$.

## Availability of data and materials

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

## References

1. Ceng, L.C., Li, X., Qin, X.: Parallel proximal point methods for systems of vector optimization problems on Hadamard manifolds without convexity. Optimization 69, 357–383 (2020)

2. Ceng, L.C., Petrusel, A., Qin, X., Yao, J.C.: A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems. Fixed Point Theory 21, 93–108 (2020)

3. Ceng, L.C., Petrusel, A., Qin, X., Yao, J.C.: Pseudomonotone variational inequalities and fixed points. Fixed Point Theory 22, 543–558 (2021)

4. Ceng, L.C., Petrusel, A., Qin, X., Yao, J.C.: Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints. Optimization 70, 1337–1358 (2021)

5. Ceng, L.C., Petrusel, A., Yao, J.C., Yao, Y.: Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces. Fixed Point Theory 19, 487–501 (2018)

6. Ceng, L.C., Petrusel, A., Yao, J.C., Yao, Y.: Systems of variational inequalities with hierarchical variational inequality constraints for Lipschitzian pseudocontractions. Fixed Point Theory 20, 113–133 (2019)

7. Ceng, L.C., Shang, M.J.: Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings. Optimization 70, 715–740 (2021)

8. Ceng, L.C., Wen, C.F.: Systems of variational inequalities with hierarchical variational inequality constraints for asymptotically nonexpansive and pseudocontractive mappings. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113, 2431–2447 (2019)

9. Ceng, L.C., Yuan, Q.: Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems. J. Inequal. Appl. 20, Article ID 274 (2019)

10. Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)

11. Cui, H.H., Zhang, H.X., Ceng, L.C.: An inertial Censor–Segal algorithm for split common fixed-point problems. Fixed Point Theory 22, 93–103 (2021)

12. Dong, Q.L., Liu, L., Yao, Y.: Self-adaptive projection and contraction methods with alternated inertial terms for solving the split feasibility problem. J. Nonlinear Convex Anal. 23(3), 591–605 (2022)

13. Guan, J.L., Ceng, L.C., Hu, B.: Strong convergence theorem for split monotone variational inclusion with constraints of variational inequalities and fixed point problems. J. Inequal. Appl. 29, Article ID 311 (2018)

14. He, L., Cui, Y.L., Ceng, L.C., et al.: Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule. J. Inequal. Appl. 37, Article ID 146 (2021)

15. He, S., Yang, C.: Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl. Anal. 2013, Article ID 942315 (2013)

16. Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997)

17. Khobotov, E.N.: Modification of the extra-gradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Math. Phys. 27, 120–127 (1989)

18. Kim, J.K., Salahuddin Lim, W.H.: General nonconvex split variational inequality problems. Korean J. Math. 25(4), 469–481 (2017)

19. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody. 12, 747–756 (1976)

20. Noor, M.A., Noor, K.I.: Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993)

21. Siriyan, K., Kangtunyakarn, A.: Algorithm method for solving the split general system of variational inequalities problem and fixed point problem of nonexpansive mapping with application. Math. Methods Appl. Sci. 41, 7766–7788 (2018)

22. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)

23. Sripattanet, A., Kangtunyakarn, A.: Convergence theorem for solving a new concept of the split variational inequality problems and application. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, Article ID 177 (2020). https://doi.org/10.1007/s13398-020-00909-0

24. Stampacchia, G.: Formes bilineaires coercivites sur les ensembles convexes. C. R. Acad. Sci. Paris, Ser. I 258, 4413–4416 (1964)

25. Su, M., Xu, H.K.: Remarks on the gradient-projection algorithm. J. Nonlinear Anal. Optim. 1(1), 35–43 (2010)

26. Taiwo, A., Mewomo, O.T., Gibali, A.: A simple strong convergent method for solving split common fixed point problems. J. Nonlinear Var. Anal. 5, 777–793 (2021)

27. Tan, B., Cho, S., Yao, J.C.: Accelerated inertial subgradient extragradient algorithms with non-monotonic step sizes for equilibrium problems and fixed point problems. J. Nonlinear Var. Anal. 6, 89–122 (2022)

28. Yao, Y., Iyiola, O.S., Shehu, Y.: Subgradient extragradient method with double inertial steps for variational inequalities. J. Sci. Comput. 90(2), Article ID 71 (2022)

29. Yao, Y., Li, H., Postolache, M.: Iterative algorithms for split equilibrium problems of monotone operators and fixed point problems of pseudo-contractions. Optimization (2020). https://doi.org/10.1080/02331934.2020.1857757

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

31. Zhao, T.Y., Wang, D.Q., Ceng, L.C., et al.: Quasi-inertial Tseng’s extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators. Numer. Funct. Anal. Optim. 42, 69–90 (2020)

32. Zhao, X., Yao, J.C., Yao, Y.: A proximal algorithm for solving split monotone variational inclusions. UPB Sci. Bull., Ser. A 82(3), 43–52 (2020)

## Acknowledgements

The authors would like to extend their sincere appreciation to the Research and Innovation Services of King Mongkut’s Institute of Technology Ladkrabang.

## Funding

This research was supported by the Royal Golden Jubilee (RGJ) Ph.D. Programme, the National Research Council of Thailand (NRCT), under Grant No. PHD/0170/2561.

## Author information

Authors

### Contributions

AK dealt with the conceptualization, formal analysis, supervision, writing—review and editing. AS writing—original draft, formal analysis, writing—review and editing. Both authors have read and approved the manuscript.

### Corresponding author

Correspondence to Atid Kangtunyakarn.

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The authors declare that they have no competing interests.

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