# Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems

## Abstract

In this paper, we introduce and investigate composite inertial gradient-based algorithms with a line-search process for solving a variational inequality problem (VIP) with a pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in the framework infinite-dimensional Hilbert spaces. The proposed algorithms are based on an inertial subgradient–extragradient method with the line-search process, hybrid steepest-descent methods, viscosity approximation methods and Mann iteration methods. Under weak conditions, we prove strong convergence of the proposed algorithms to the element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

## 1 Introduction

Throughout this work, H is assumed to be a real Hilbert space with norm $$\|\cdot \|$$ and inner product $$\langle \cdot ,\cdot \rangle$$. We suppose that C is a convex and closed set in Hilbert space H and $$P_{C}$$ is the metric projection from space H onto set C. Let $$S:C\to H$$ be a nonlinear operator. We denote by $$\operatorname{Fix}(S)$$ the set of fixed points of operator S. A mapping $$T:C\to C$$ is called strictly pseudocontractive if

$$\Vert Tx-Ty \Vert ^{2}\leq \zeta \bigl\Vert (T-I)x-(T-I)y \bigr\Vert ^{2}+ \Vert x-y \Vert ^{2},\quad \forall x,y\in C,$$
(1.1)

where $$\zeta \in [0,1)$$ is some constant. With $$\zeta =0$$, T is called a nonexpansive operator. It is well known that the class of strict pseudocontractions includes the class of nonexpansive mappings and is also continuous. From their direct and indirect applications to other branches of mathematics and other engineering fields, to the the class of nonexpansive operators and the class of strictly pseudocontractive operators are now under the spotlights; see, e.g., [1,2,3,4,5,6,7,8] and the references therein.

Let $$A:H\to H$$ be a mapping defined on the whole space. The known variational inequality problem (VIP), which find lots of applications in sparse reconstruction, image reconstruction, traffic and transportation systems, is to get $$x^{*}\in C$$ with $$\langle Ax^{*},x-x ^{*}\rangle \geq 0$$ $$\forall x\in C$$. The solution set of the VIP is presented with VI(C, A). At present, one of the most effective methods for solving the VIP is the extragradient method introduced by Korpelevich  in 1976, that is, for any initial $$x_{0}\in C$$, the sequence $$\{x_{n}\}$$ is generated by

$$\textstyle\begin{cases} y_{n}=P_{C}(I-\tau A)x_{n}, \\ x_{n+1}=P_{C}(x_{n}-\tau Ay_{n})\quad \forall n\geq 0, \end{cases}$$
(1.2)

with τ is such that $$L\tau <1$$. If $$\operatorname{VI}(C,A)\neq \emptyset$$, then the sequence $$\{x_{n}\}$$ generated by process (1.2) converges weakly to a point, which lies in $$\operatorname{VI}(C,A)$$. The literature on the VIP is quite vast and Korpelevich’s extragradient method, which is projection-based, has received great attention given by many researchers working on nonlinear programming, who improved it in various ways; see, e.g., [10,11,12,13,14,15,16,17] and the references therein. For the well-known extragradient method, one needs to compute two nearest point projections for every iterative step/process. Without question, the projection onto a feasibility (closed-convex) set is closely related to a minimizer problem (min distance). If feasibility sets are general, this might require a prohibitive amount of computation time; For recent efforts on projection-based methods, see, e.g., [18,19,20,21,22]. In 2011, Censor et al.  deeply studied Korpelevich’s extragradient method and first introduced the subgradient–extragradient method, in which the second projection onto set C is replaced by a projection onto a half-space:

$$\textstyle\begin{cases} y_{n}=P_{C}(I-\tau A)x_{n}, \\ C_{n}=\{x\in H:\langle x_{n}-\tau Ax_{n}-y_{n},x-y_{n}\rangle \leq 0 \}, \\ x_{n+1}=P_{C_{n}}(x_{n}-\tau Ay_{n})\quad \forall n\geq 0, \end{cases}$$
(1.3)

with τ is a parameter such that $$L\tau <1$$. In 2014, Kraikaew and Saejung  proposed the Halpern subgradient–extragradient method for solving the VIP

$$\textstyle\begin{cases} y_{n}=P_{C}(I-\tau A)x_{n}, \\ C_{n}=\{x\in H:\langle x_{n}-\tau Ax_{n}-y_{n},x-y_{n}\rangle \leq 0 \}, \\ z_{n}=P_{C_{n}}(x_{n}-\tau Ay_{n}), \\ x_{n+1}=\alpha _{n}x_{0}+(1-\alpha _{n})z_{n}\quad \forall n\geq 0, \end{cases}$$
(1.4)

where τ is a parameter such that $$L\tau <1$$, $$\{\alpha _{n}\}$$ is a sequence in $$(0,1)$$ with $$\lim_{n\to \infty }\alpha _{n}=0$$, and $$\sum^{\infty }_{n=1}\alpha _{n}= +\infty$$. They proved the strong convergence of the sequence $$\{x_{n}\}$$ generated by (1.4) to $$P_{\operatorname{VI}(C,A)}x_{0}$$.

In 2018, by virtue of the inertial technique, Thong and Hieu  first proposed the inertial subgradient–extragradient method, that is, for any initial $$x_{0},x_{1}\in H$$, the sequence $$\{x_{n}\}$$ is generated by

$$\textstyle\begin{cases} w_{n}=x_{n}+\alpha _{n}(x_{n}-x_{n-1}), \\ y_{n}=P_{C}(w_{n}-\tau Aw_{n}), \\ C_{n}=\{x\in H:\langle w_{n}-\tau Aw_{n}-y_{n},x-y_{n}\rangle \leq 0 \}, \\ x_{n+1}=P_{C_{n}}(w_{n}-\tau Ay_{n})\quad \forall n\geq 1, \end{cases}$$
(1.5)

with τ is a parameter such that $$L\tau <1$$. Under some suitable conditions, they proved the weak convergence of $$\{x_{n}\}$$ to an element of $$\operatorname{VI}(C,A)$$. Very recently, Thong and Hieu  introduced two inertial subgradient–extragradient algorithms with linear-search process for solving the VIP with monotone and Lipschitz continuous mapping A and the fixed-point problem (FPP) of a quasi-nonexpansive mapping T with a demiclosedness property in a real Hilbert space.

### Algorithm 1.1

(see [25, Algorithm 1])

Initialization: Let $$x_{0},x_{1}\in H$$ be arbitrary. Given $$\gamma >0$$, $$l\in (0,1)$$, $$\mu \in (0,1)$$.

Iterative Steps: Compute $$x_{n+1}$$ as follows:

Step 1. Set $$w_{n}=\alpha _{n}(x_{n}-x_{n-1})+x_{n}$$ and compute $$y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n})$$, where $$\tau _{n}$$ is chosen to be the largest $$\tau \in \{\gamma ,\gamma l,\gamma l^{2},\ldots\}$$ satisfying $$\tau \|Aw_{n}-Ay_{n}\|\leq \mu \|w_{n}-y_{n}\|$$.

Step 2. Compute $$z_{n}=P_{C_{n}}(w_{n}-\tau _{n}Ay_{n})$$ with $$C_{n}:=\{x\in H:\langle w_{n}-\tau _{n}Aw_{n}-y_{n},x-y_{n}\rangle \leq 0\}$$.

Step 3. Compute $$x_{n+1}=(1-\beta _{n})w_{n}+\beta _{n}Tz_{n}$$. If $$w_{n}=z_{n}=x_{n+1}$$ then $$w_{n}\in \operatorname{Fix}(T)\cap \operatorname{VI}(C,A)$$. Set $$n:=n+1$$ and go to Step 1.

### Algorithm 1.2

(see [25, Algorithm 2])

Initialization: Let $$x_{0},x_{1}\in H$$ be arbitrary. Given $$\gamma >0$$, $$l\in (0,1)$$, $$\mu \in (0,1)$$.

Iterative Steps: Calculate $$x_{n+1}$$ as follows:

Step 1. Set $$w_{n}=\alpha _{n}(x_{n}-x_{n-1})+x_{n}$$ and compute $$y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n})$$, where $$\tau _{n}$$ is chosen to be the largest $$\tau \in \{\gamma ,\gamma l,\gamma l^{2},\ldots\}$$ satisfying $$\tau \|Aw_{n}-Ay_{n}\|\leq \mu \|w_{n}-y_{n}\|$$.

Step 2. Compute $$z_{n}=P_{C_{n}}(w_{n}-\tau _{n}Ay_{n})$$ with $$C_{n}:=\{x\in H:\langle w_{n}-\tau _{n}Aw_{n}-y_{n},x-y_{n}\rangle \leq 0\}$$.

Step 3. Compute $$x_{n+1}=(1-\beta _{n})x_{n}+\beta _{n}Tz_{n}$$. If $$w_{n}=z_{n}=x_{n}=x_{n+1}$$ then $$x_{n}\in \operatorname{Fix}(T)\cap \operatorname{VI}(C,A)$$. Set $$n:=n+1$$ and go to Step 1.

Under some appropriate mild conditions, they proved the weak convergence of the proposed algorithms to an element of $$\operatorname{Fix}(T)\cap \operatorname{VI}(C,A)$$. Inspired by the research work of [23,24,25,26,27,28], we introduce two composite inertial subgradient–extragradient algorithms with a line-search process for solving the VIP with pseudomonotone and Lipschitz continuous mapping and the CFPP of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in a real Hilbert space. The proposed algorithms are based on an inertial subgradient–extragradient method with a line-search process, hybrid steepest-descent methods, viscosity approximation methods and the Mann iteration method. We prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set. Finally, our main results are applied to solve the VIP and CFPP in an illustrative example.

This paper is organized as follows: In Sect. 2, we recall some definitions and preliminary results for further use. Section 3 deals with the convergence analysis of the proposed algorithms. Some examples are presented to solve the VIP and CFPP.

## 2 Preliminaries

Let C be a convex nonempty closed set in a real Hilbert space H. Let $$\{x_{n}\}$$ be a vector sequence in H, then we denote by $$x_{n} \to x$$ (resp., $$x_{n}\rightharpoonup x$$) the strong (resp., weak) convergence of $$\{x_{n}\}$$ to x. A single-valued mapping $$T:C\to H$$ is called

1. (i)

L-Lipschitz continuous (or L-Lipschitzian) if $$\exists L>0$$ such that $$\|Tx-Ty\|\leq L\|x-y\|$$ $$\forall x, y\in C$$;

2. (ii)

monotone if $$\langle Tx-Ty,x-y\rangle \geq 0$$ $$\forall x, y\in C$$;

3. (iii)

pseudomonotone if $$\langle Tx,y-x\rangle \geq 0\Rightarrow \langle Ty,y-x\rangle \geq 0$$ $$\forall x, y\in C$$;

4. (iv)

α-strongly monotone if $$\exists \alpha >0$$ such that $$\langle Tx-Ty,x-y\rangle \geq \alpha \|x-y\|^{2}$$ $$\forall x, y\in C$$;

5. (v)

sequentially weakly continuous if $$\forall \{x_{n}\}\subset C$$, the relation holds: $$x_{n}\rightharpoonup x\Rightarrow Tx_{n}\rightharpoonup Tx$$.

It is easy to see that every monotone operator is pseudomonotone but the converse is not true. Also, recall that the mapping $$T:C\to C$$ is a ζ-strict pseudocontraction for some $$\zeta \in [0,1)$$ if and only if the inequality holds $$\langle Tx-Ty,x-y\rangle \leq \|x-y\|^{2}-\frac{1- \zeta }{2}\|(I-T)x-(I-T)y\|^{2}$$ $$\forall x, y\in C$$. We know that if T is a ζ-strictly pseudocontractive mapping, then T satisfies Lipschitz condition $$\|Tx-Ty\|\leq \frac{1+\zeta }{1-\zeta }\|x-y\| \forall x$$, $$y\in C$$. For each point $$x\in H$$, we know that there exists a unique nearest point in C, denoted by $$P_{C}x$$, such that $$\|x-P_{C}x\|\leq \|x-y\|$$ $$\forall y\in C$$. The mapping $$P_{C}$$ is called the metric projection of H onto C. The complementary operator of the metric projection operator is monotone.

The following two lemmas are trivial.

### Lemma 2.1

The following hold:

1. (i)

$$\langle x-y,P_{C}x-P_{C}y\rangle \geq \|P_{C}x-P_{C}y\|^{2}$$ $$\forall x, y\in H$$;

2. (ii)

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

3. (iii)

$$\|x-y\|^{2}\geq \|x-P_{C}x\|^{2}+\|y-P_{C}x\|^{2}$$ $$\forall x \in H$$, $$y\in C$$;

4. (iv)

$$\|x-y\|^{2}+2\langle x-y,y\rangle =\|x\|^{2}-\|y\|^{2}$$ $$\forall x, y\in H$$;

5. (v)

$$\|\lambda x+\mu y\|^{2}+\lambda \mu \|x-y\|^{2}=\lambda \|x\| ^{2}+\mu \|y\|^{2}$$ $$\forall x, y\in H$$, $$\forall \lambda ,\mu \in [0,1]$$ with $$\lambda +\mu =1$$.

### Lemma 2.2

We have the inequality

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

### Lemma 2.3

()

Let $$A:C\to H$$ be pseudomonotone and continuous. Then $$x^{*}\in C$$ is a solution to the VIP $$\langle Ax ^{*},x-x^{*}\rangle \geq 0$$ $$\forall x\in C$$, if and only if $$\langle Ax,x-x ^{*}\rangle \geq 0$$ $$\forall x\in C$$.

### Lemma 2.4

()

Let $$\{a_{n}\}$$ be a sequence of nonnegative real numbers satisfying the conditions: $$a_{n+1} \leq \lambda _{n}\gamma _{n}+(1-\lambda _{n})a_{n}$$ $$\forall n\geq 1$$, where $$\{\lambda _{n}\}$$ and $$\{\gamma _{n}\}$$ are sequences of real numbers such that (i) $$\{\lambda _{n}\}\subset [0,1]$$ and $$\sum^{\infty }_{n=1} \lambda _{n}=\infty$$, and (ii) $$\limsup_{n\to \infty }\gamma _{n} \leq 0$$ or $$\sum^{\infty }_{n=1}|\lambda _{n}\gamma _{n}|<\infty$$. Then $$\lim_{n\to \infty }a_{n}=0$$.

### Lemma 2.5

()

Let $$T:C\to C$$ be a ζ-strict pseudocontraction. Then $$I-T$$ is demiclosed at zero, i.e., if $$\{x_{n}\}$$ is a sequence in C such that $$x_{n}\rightharpoonup x \in C$$ and $$(T-I)x_{n}\to 0$$, then $$(I-T)x=0$$, where I is the identity mapping of H.

### Lemma 2.6

()

Let $$T:C\to C$$ be a ζ-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers. Then $$\|\gamma (x-y)+\delta (Tx-Ty)\|\leq ( \gamma + \delta )\|x-y\|$$ $$\forall x, y\in C$$ provided $$(\gamma +\delta )\zeta \leq \gamma$$.

The following lemma is a direct consequence of Yamada .

### Lemma 2.7

Let $$\lambda \in (0,1]$$, $$T:C\to H$$ be a nonexpansive mapping, and the mapping $$T^{\lambda }: C\to H$$ be defined by $$T^{\lambda }x:=Tx-\lambda \mu F(Tx)$$ $$\forall x\in C$$, where $$F:H\to H$$ is κ-Lipschitzian and η-strongly monotone. Then $$T^{\lambda }$$ is a contraction with the coefficient $$1-\lambda \tau$$ provided $$0<\mu <\frac{2\eta }{\kappa ^{2}}$$.

## 3 Main results

In this section, we use C to denote the the feasible set and assume always that the following hold.

$$T_{i}:H\to H$$ is a nonexpansive mapping for $$i=1,\ldots,N$$ and $$T:H\to H$$ is a ζ-strictly pseudocontractive mapping.

$$A:H\to H$$ is Lipschitz continuous with the module L, pseudomonotone on H, and sequentially weakly continuous on C, such that $${\varOmega }=\bigcap ^{N}_{i=0}\operatorname{Fix}(T_{i})\cap \operatorname{VI}(C,A) \neq \emptyset$$ with $$T_{0}:=T$$.

$$f:H\to H$$ is a contraction with constant $$\delta \in [0,1)$$, and $$F:H\to H$$ is η-strongly monotone and κ-Lipschitzian such that $$\delta <\tau :=1-\sqrt{1-\rho (2\eta -\rho \kappa ^{2})}$$ for $$\rho \in (0,\frac{2\eta }{\kappa ^{2}})$$.

$$\{\sigma _{n}\}\subset [0,1]$$ and $$\{\alpha _{n}\},\{\beta _{n}\},\{ \gamma _{n}\},\{\delta _{n}\}\subset (0,1)$$ such that

1. (i)

$$\sup_{n\geq 1}\frac{\sigma _{n}}{\alpha _{n}}<\infty$$ and $$\beta _{n}+\gamma _{n}+\delta _{n}=1$$ $$\forall n\geq 1$$;

2. (ii)

$$\zeta (\gamma _{n}+\delta _{n})\leq \gamma _{n}$$ $$\forall n\geq 1$$;

3. (iii)

$$\sum^{\infty }_{n=1}\alpha _{n}=\infty$$ and $$\lim_{n\to \infty }\alpha _{n}=0$$;

4. (iv)

$$\limsup_{n\to \infty }\beta _{n}<1$$, $$\liminf_{n\to \infty }\beta _{n}>0$$ and $$\liminf_{n\to \infty }\delta _{n}>0$$.

In terms of Xu and Kim , we write $$T_{n}:=T_{n~\mathrm{mod}~N}$$ for integer $$n\geq 1$$ with the mod function taking values in the set $$\{1,2,\ldots,N\}$$, that is, if $$n=jN+q$$ for some integers $$j\geq 0$$ and $$0\leq q< N$$, then $$T_{n}=T_{N}$$ if $$q=0$$ and $$T_{n}=T_{q}$$ if $$0< q< N$$.

### Algorithm 3.1

Initialization: Given $$l\in (0,1)$$, $$\gamma >0$$ and $$\mu \in (0,1)$$. Let $$x_{0},x_{1}\in H$$ be arbitrary.

Iterative Steps: Calculate $$x_{n+1}$$ as follows:

Step 1. Set $$w_{n}=\sigma _{n}(T_{n}x_{n}-T_{n}x_{n-1})+T_{n}x_{n}$$ and compute $$y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n})$$, where $$\tau _{n}$$ is chosen to be the largest $$\tau \in \{\gamma ,\gamma l,\gamma l^{2},\ldots\}$$ satisfying

$$\tau \Vert Aw_{n}-Ay_{n} \Vert \leq \mu \Vert w_{n}-y_{n} \Vert .$$
(3.1)

Step 2. Compute $$z_{n}=(I-\alpha _{n}\rho F)P_{C_{n}}(w_{n}-\tau _{n}Ay _{n})+\alpha _{n}f(x_{n})$$ with $$C_{n}:=\{x\in H:\langle w_{n}-\tau _{n}Aw_{n}-y_{n},x-y_{n}\rangle \leq 0\}$$.

Step 3. Compute

$$x_{n+1}=\beta _{n}x_{n}+\gamma _{n}z_{n}+\delta _{n}Tz_{n}.$$
(3.2)

Again set $$n:=n+1$$ and go to Step 1.

### Lemma 3.1

The Armijo-like search rule (3.1) is well defined, and $$\min \{\gamma ,\frac{\mu l}{L}\}\leq \tau _{n}\leq \gamma$$.

### Proof

From the L-Lipschitz continuity of A we get $$\mu \|w_{n}-P_{C}(w_{n}-\gamma l^{m} Aw_{n})\|\geq \frac{\mu }{L}\|Aw _{n}-AP_{C}(w_{n}-\gamma l^{m} Aw_{n})\|$$. Thus, (3.1) holds for all $$\gamma l^{m}\leq \frac{\mu }{L}$$ and so $$\tau _{n}$$ is well defined. Obviously, $$\tau _{n}\geq \gamma$$. If $$\tau _{n}=\gamma$$, then the inequality is true. If $$\tau _{n}<\gamma$$, then from (3.1) we get $$\|Aw_{n}-AP_{C}(w_{n}-\frac{\tau _{n}}{l}Aw_{n})\|>\frac{\mu }{\frac{ \tau _{n}}{l}}\|w_{n}-P_{C}(w_{n} -\frac{\tau _{n}}{l}Aw_{n})\|$$. Again from the L-Lipschitz continuity of A we obtain $$\tau _{n}>\frac{ \mu l}{L}$$. Hence the inequality is valid. □

### Lemma 3.2

Let $$\{w_{n}\}$$, $$\{y_{n}\}$$, $$\{z_{n}\}$$ be the sequences generated by Algorithm 3.1. Then

\begin{aligned}[b] \Vert z_{n}-p \Vert ^{2}& \leq (1-\alpha _{n}\tau ) \Vert w_{n}-p \Vert ^{2}-(1-\alpha _{n} \tau ) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+\alpha _{n} \delta \Vert x_{n}-p \Vert ^{2} \\ & \quad {} + \Vert u_{n}-y_{n} \Vert ^{2} \bigr]+2\alpha _{n}\bigl\langle (f-\rho F)p,z_{n}-p\bigr\rangle \quad \forall p\in {\varOmega }, \end{aligned}
(3.3)

where $$u_{n}:=P_{C_{n}}(w_{n}-\tau _{n}Ay_{n})$$ $$\forall n\geq 1$$.

### Proof

First, take an arbitrary $$p\in {\varOmega }\subset C \subset C_{n}$$. We note that

\begin{aligned} \Vert u_{n}-p \Vert ^{2} &= \bigl\Vert P_{C_{n}}p-P_{C_{n}}(w_{n}-\tau _{n}Ay_{n}) \bigr\Vert ^{2} \leq \langle w_{n}-\tau _{n}Ay_{n}-p,u_{n}-p \rangle \\ &=\frac{1}{2} \Vert u_{n}-p \Vert ^{2}+ \frac{1}{2} \Vert w_{n}-p \Vert ^{2}- \frac{1}{2} \Vert u _{n}-w_{n} \Vert ^{2}-\langle u_{n}-p,\tau _{n}Ay_{n} \rangle . \end{aligned}

So, it follows that $$\|u_{n}-p\|^{2}\leq \|w_{n}-p\|^{2}-\|u_{n}-w _{n}\|^{2}-2\langle u_{n}-p,\tau _{n}Ay_{n}\rangle$$, from which, together with (3.1) and the pseudomonotonicity of A, we deduce that $$\langle Ay_{n},p-y_{n}\rangle \leq 0$$ and

\begin{aligned}[b] \Vert u_{n}-p \Vert ^{2}& \leq 2\tau _{n}\bigl(\langle Ay_{n},p-y_{n} \rangle +\langle Ay_{n},y_{n}-u_{n}\rangle \bigr)- \Vert u_{n}-w_{n} \Vert ^{2}+ \Vert w_{n}-p \Vert ^{2} \\ &\leq 2\tau _{n}\langle Ay_{n},y_{n}-u_{n} \rangle - \Vert u_{n}-w_{n} \Vert ^{2}+ \Vert w_{n}-p \Vert ^{2} \\ &=2\langle w_{n}-\tau _{n}Ay_{n}-y_{n},u_{n}-y_{n} \rangle - \Vert y_{n}-w _{n} \Vert ^{2}- \Vert u_{n}-y_{n} \Vert ^{2}+ \Vert w_{n}-p \Vert ^{2}. \end{aligned}
(3.4)

Since $$u_{n}=P_{C_{n}}(w_{n}-\tau _{n}Ay_{n})$$ with $$C_{n}:=\{x\in H: \langle w_{n}-\tau _{n}Aw_{n}-y_{n},x-y_{n}\rangle \leq 0\}$$, we have $$\langle w_{n}-\tau _{n}Aw_{n}-y_{n},u_{n}-y_{n}\rangle \leq 0$$, which, together with (3.1), implies that

\begin{aligned} 2\langle w_{n}-\tau _{n}Ay_{n}-y_{n},u_{n}-y_{n} \rangle &=2\langle w _{n}-\tau _{n}Aw_{n}-y_{n},u_{n}-y_{n} \rangle +2\tau _{n}\langle Aw_{n}-Ay _{n},u_{n}-y_{n} \rangle \\ &\leq 2\mu \Vert w_{n}-y_{n} \Vert \Vert u_{n}-y_{n} \Vert \leq \mu \bigl( \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr). \end{aligned}

Therefore, substituting the last inequality for (3.4), we infer that

$$\Vert u_{n}-p \Vert ^{2}\leq \Vert w_{n}-p \Vert ^{2}-(1-\mu ) \Vert w_{n}-y_{n} \Vert ^{2}-(1- \mu ) \Vert u_{n}-y_{n} \Vert ^{2}\quad \forall p\in {\varOmega }.$$
(3.5)

In addition, from Algorithm 3.1 we have

$$z_{n}-p =\alpha _{n}(f-\rho F)p+\alpha _{n} \bigl(f(x_{n})-f(p)\bigr)+(I-\alpha _{n}\rho F)u_{n}-(I-\alpha _{n}\rho F)p.$$

Using Lemma 2.2, Lemma 2.7, and the convexity of the function $$h(t)=t^{2}$$ $$\forall t\in {\mathbf{R}}$$, from (3.5) we get

\begin{aligned} & \Vert z_{n}-p \Vert ^{2} \\ &\quad \leq 2\alpha _{n}\bigl\langle p-z_{n},(\rho F-f)p \bigr\rangle + \bigl\Vert (I-\alpha _{n} \rho F)u_{n}-(I- \alpha _{n}\rho F)p+\alpha _{n}\bigl(f(x_{n})-f(p) \bigr) \bigr\Vert ^{2} \\ &\quad \leq 2\alpha _{n}\bigl\langle p-z_{n},(\rho F-f)p \bigr\rangle +\bigl[\alpha _{n} \delta \Vert x_{n}-p \Vert +(1-\alpha _{n}\tau ) \Vert u_{n}-p \Vert \bigr]^{2} \\ &\quad \leq 2\alpha _{n}\bigl\langle p-z_{n},(\rho F-f)p \bigr\rangle +\alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+(1-\alpha _{n}\tau ) \Vert u_{n}-p \Vert ^{2} \\ &\quad \leq (1-\alpha _{n}\tau )\bigl[ \Vert w_{n}-p \Vert ^{2}-(1-\mu ) \Vert w_{n}-y_{n} \Vert ^{2}-(1- \mu ) \Vert u_{n}-y_{n} \Vert ^{2}\bigr] \\ & \qquad {} +2\alpha _{n}\bigl\langle p-z_{n},(\rho F-f)p \bigr\rangle +\alpha _{n}\delta \Vert x _{n}-p \Vert ^{2} \\ &\quad =(1-\alpha _{n}\tau ) \Vert w_{n}-p \Vert ^{2}-(1-\alpha _{n}\tau ) (1-\mu )\bigl[ \Vert w _{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr] \\ & \qquad {} +2\alpha _{n}\bigl\langle p-z_{n},(\rho F-f)p \bigr\rangle +\alpha _{n}\delta \Vert x _{n}-p \Vert ^{2}. \end{aligned}

This completes the proof. □

### Lemma 3.3

Let $$\{w_{n}\}$$, $$\{x_{n}\}$$, $$\{y_{n}\}$$, $$\{z_{n}\}$$ be bounded sequences generated by Algorithm 3.1. If $$x_{n}-x_{n+1}\to 0$$, $$w _{n}-x_{n}\to 0$$, $$w_{n}-y_{n}\to 0$$, $$w_{n}-z_{n}\to 0$$ and $$\exists \{w _{n_{k}}\}\subset \{w_{n}\}$$ such that $$w_{n_{k}}\rightharpoonup z \in H$$, then $$z\in {\varOmega }$$.

### Proof

From Algorithm 3.1, we get $$T_{n}x_{n}-x_{n}+\alpha _{n}(T_{n}x_{n}-T_{n}x_{n-1})=w_{n}-x_{n}$$ $$\forall n\geq 1$$, and hence $$\|T_{n}x_{n}-x_{n}\|\leq \|w_{n}-x_{n}\|+\alpha _{n}\|T_{n}x_{n}-T _{n}x_{n-1}\|\leq \|w_{n}-x_{n}\|+\|x_{n}-x_{n-1}\|$$. Utilizing the assumptions $$x_{n}-x_{n+1}\to 0$$ and $$w_{n}-x_{n}\to 0$$, we obtain

$$\lim_{n\to \infty } \Vert x_{n}-T_{n}x_{n} \Vert =0.$$
(3.6)

Combining the assumptions $$w_{n}-x_{n}\to 0$$ and $$w_{n}-z_{n}\to 0$$ implies that, as $$n\to \infty$$,

$$\Vert z_{n}-x_{n} \Vert \leq \Vert w_{n}-z_{n} \Vert + \Vert w_{n}-x_{n} \Vert \to 0.$$

Moreover, by Algorithm 3.1 we get $$z_{n}-x_{n}=u_{n}-x_{n}-\alpha _{n} \rho Fu_{n}+\alpha _{n}f(x_{n})$$ with $$u_{n}:=P_{C_{n}} (w_{n}-\tau _{n}Ay_{n})$$. So it follows from the boundedness of $$\{x_{n}\}$$, $$\{u _{n}\}$$ that, as $$n\to \infty$$,

$$\Vert u_{n}-x_{n} \Vert = \bigl\Vert z_{n}-x_{n}-\alpha _{n}f(x_{n})+ \alpha _{n}\rho Fu _{n} \bigr\Vert \leq \Vert z_{n}-x_{n} \Vert +\alpha _{n}\bigl( \bigl\Vert f(x_{n}) \bigr\Vert + \Vert \rho Fu_{n} \Vert \bigr) \to 0.$$

Also, by Algorithm 3.1 we get $$\beta _{n}(x_{n}-z_{n})+\delta _{n}(Tz _{n}-z_{n})=x_{n+1}-z_{n}$$, which immediately yields

$$\delta _{n} \Vert Tz_{n}-z_{n} \Vert = \bigl\Vert x_{n+1}-z_{n}-\beta _{n}(x_{n}-z_{n}) \bigr\Vert \leq \Vert x_{n+1}-x_{n} \Vert +2 \Vert x_{n}-z_{n} \Vert .$$

Since $$x_{n}-x_{n+1}\to 0$$, $$z_{n}-x_{n}\to 0$$ and $$\liminf_{n\to \infty }\delta _{n}>0$$, we obtain

$$\lim_{n\to \infty } \Vert z_{n}-Tz_{n} \Vert =0.$$
(3.7)

Noticing $$y_{n}=P_{C}(I-\tau _{n}A)w_{n}$$, we have $$\langle w_{n}-\tau _{n}Aw_{n}-y_{n},x-y_{n}\rangle \leq 0$$ $$\forall x\in C$$, and hence

$$\langle w_{n}-y_{n},x-y_{n}\rangle +\tau _{n}\langle Aw_{n},y_{n}-w _{n} \rangle \leq \tau _{n}\langle Aw_{n},x-w_{n} \rangle \quad \forall x\in C.$$
(3.8)

Being weakly convergent, $$\{w_{n_{k}}\}$$ is bounded. Then, according to the Lipschitz continuity of A, $$\{Aw_{n_{k}}\}$$ is bounded. Since $$w_{n}-y_{n}\to 0$$, $$\{y_{n_{k}}\}$$ is bounded as well. Note that $$L\tau _{n}\geq \min \{L\gamma ,\mu l \}$$. So, from (3.8) we get $$\liminf_{k\to \infty }\langle Aw_{n_{k}},x-w_{n_{k}}\rangle \geq 0 \forall x\in C$$. Meanwhile, observe that $$\langle x-y_{n},Ay_{n} \rangle =\langle Ay_{n}-Aw_{n},x-w_{n}\rangle +\langle Aw_{n},x-w_{n} \rangle +\langle Ay_{n},w_{n}-y_{n} \rangle$$. Since $$w_{n}-y_{n} \to 0$$, from L-Lipschitz continuity of A we obtain $$Ay_{n}-Aw_{n} \to 0$$, which together with (3.8) yields $$\liminf_{k\to \infty } \langle Ay_{n_{k}},x-y_{n_{k}}\rangle \geq 0$$ $$\forall x\in C$$.

Next we show that $$\lim_{n\to \infty }\|x_{n}-T_{r}x_{n}\|=0$$ for $$r=1,\ldots,N$$. Indeed, note that, for $$i=1,\ldots,N$$,

\begin{aligned} &2 \Vert x_{n}-x_{n+i} \Vert + \Vert x_{n+i}-T_{n+i}x_{n+i} \Vert \\ &\quad \geq \Vert x_{n}-x_{n+i} \Vert + \Vert T_{n+i}x_{n+i}-T_{n+i}x_{n} \Vert + \Vert x_{n+i}-T _{n+i}x_{n+i} \Vert \\ &\quad \geq \Vert x_{n}-T_{n+i}x_{n} \Vert . \end{aligned}

Hence from (3.6) and the assumption $$x_{n}-x_{n+1}\to 0$$ we get $$\lim_{n\to \infty }\|x_{n}-T_{n+i}x_{n}\|=0$$ for $$i=1,\ldots,N$$. This immediately implies that

$$\lim_{n\to \infty } \bigl\Vert (T_{r}-I)x_{n} \bigr\Vert =0\quad \text{for }r=1,\ldots,N.$$
(3.9)

We now take a sequence $$\{\varepsilon _{k}\}\subset (0,1)$$ satisfying $$\varepsilon _{k}\downarrow 0$$ as $$k\to \infty$$. For all $$k\geq 1$$, we denote by $$m_{k}$$ the smallest positive integer such that

$$\varepsilon _{k}+\langle Ay_{n_{j}},x-y_{n_{j}}\rangle \geq 0\quad \forall j\geq m_{k}.$$
(3.10)

Since $$\{\varepsilon _{k}\}$$ is decreasing, it is clear that $$\{m_{k}\}$$ is increasing. Noticing that $$\{y_{m_{k}}\}\subset C$$ guarantees $$Ay_{m_{k}}\neq 0$$ $$\forall k\geq 1$$, we set $$u_{m_{k}}=\frac{Ay _{m_{k}}}{\|Ay_{m_{k}}\|^{2}}$$, we get $$\langle Ay_{m_{k}},u_{m_{k}} \rangle =1$$ $$\forall k\geq 1$$. So, from (3.10) we get $$\langle Ay_{m _{k}},x+\varepsilon _{k}u_{m_{k}}-y_{m_{k}} \rangle \geq 0$$ $$\forall k \geq 1$$. Again from the pseudomonotonicity of A we have $$\langle A(x+ \varepsilon _{k}u_{m_{k}}), x+\varepsilon _{k}u_{m_{k}}-y_{m_{k}}\rangle \geq 0$$ $$\forall k\geq 1$$. This immediately leads to

$$\langle Ax,x-y_{m_{k}}\rangle \geq \bigl\langle Ax-A(x+\varepsilon _{k}u _{m_{k}}),x+\varepsilon _{k}u_{m_{k}}-y_{m_{k}} \bigr\rangle -\varepsilon _{k}\langle Ax,u_{m_{k}}\rangle \quad \forall k\geq 1.$$
(3.11)

We claim $$\lim_{k\to \infty }\varepsilon _{k}u_{m_{k}}=0$$. As fact, from $$w_{n_{k}}\rightharpoonup z$$ and $$w_{n}-y_{n}\to 0$$, we obtain $$y_{n_{k}}\rightharpoonup z$$. So, $$\{y_{n}\}\subset C$$ guarantees $$z\in C$$. Again from the sequentially weak continuity of A, one knows $$Ay_{n_{k}}\rightharpoonup Az$$. Thus, one has $$Az\neq 0$$ (otherwise, z is a solution). Taking into account the sequentially weak lower semicontinuity of the norm $$\|\cdot \|$$, one gets $$0<\|Az\|\leq \liminf_{k\to \infty }\|Ay_{n_{k}}\|$$. Note that $$\{y_{m_{k}}\}\subset \{y_{n_{k}}\}$$ and $$\varepsilon _{k}\downarrow 0$$ as $$k\to \infty$$. So it follows that $$0\leq \limsup_{k\to \infty }\|\varepsilon _{k}u_{m _{k}}\| =\limsup_{k\to \infty }\frac{\varepsilon _{k}}{\|Ay_{m_{k}}\|} \leq \frac{\limsup_{k\to \infty }\varepsilon _{k}}{\liminf_{k\to \infty }\|Ay_{n_{k}}\|}=0$$. Hence $$\varepsilon _{k}u_{m_{k}}\to 0$$.

Next one claims $$z\in {\varOmega }$$. Indeed, from $$w_{n}-x_{n}\to 0$$ and $$w_{n_{k}}\rightharpoonup z$$, we get $$x_{n_{k}} \rightharpoonup z$$. From (3.9) we have $$(x_{n_{k}}-T_{r}x_{n_{k}})\to 0$$ for $$r=1,\ldots,N$$. Note that Lemma 2.5 guarantees the demiclosedness of $$I-T_{r}$$ at zero for $$r=1,\ldots,N$$. Thus $$z\in \operatorname{Fix}(T_{r})$$. Since r is an arbitrary element in the finite set $$\{1,\ldots,N\}$$, we get $$z\in \bigcap ^{N}_{r=1}\operatorname{Fix}(T_{r})$$. Meanwhile, from $$(w_{n}-z_{n}) \to 0$$ and $$w_{n_{k}}\rightharpoonup z$$, we get $$z_{n_{k}}\rightharpoonup z$$. From (3.7) we have $$(z_{n_{k}}-Tz_{n_{k}})\to 0$$. From Lemma 2.5 it follows that $$I-T$$ is demiclosed at zero, and hence we get $$(I-T)z=0$$, i.e., $$z\in \operatorname{Fix}(T)$$. On the other hand, letting $$k\to \infty$$, we deduce that the right hand side of (3.11) tends to zero by the uniform continuity of A, the boundedness of $$\{w_{m_{k}}\}$$, $$\{u _{m_{k}}\}$$ and the limit $$\lim_{k\to \infty }\varepsilon _{k}u_{m_{k}}=0$$. Thus, we get $$\langle Ax,x-z\rangle =\liminf_{k\to \infty }\langle Ax,x-y_{m_{k}} \rangle \geq 0$$ $$\forall x\in C$$. By Lemma 2.3 we have $$z\in \operatorname{VI}(C,A)$$. Therefore, $$z\in \bigcap ^{N}_{i=0}\operatorname{Fix}(T _{i})\cap \operatorname{VI}(C,A)={\varOmega }$$. This completes the proof. □

### Theorem 3.1

Let the sequence $$\{x_{n}\}$$ be generated by Algorithm 3.1. Then

$$x_{n}\to x^{*}\in {\varOmega } \quad \Leftrightarrow \quad \textstyle\begin{cases} x_{n}-x_{n+1}\to 0, \\ x_{n}-y_{n}\to 0, \end{cases}$$

where $$x^{*}\in {\varOmega }$$ is a unique solution to the VIP: $$\langle (\rho F-f)x^{*}, p-x^{*}\rangle \geq 0$$ $$\forall p\in {\varOmega }$$.

### Proof

First of all, since $$\limsup_{n\to \infty }\beta _{n}<1$$ and $$\liminf_{n\to \infty }\beta _{n}>0$$, we may assume, without loss of generality, that $$\{\beta _{n}\}\subset [a,b]\subset (0,1)$$. one asserts that $$P_{\varOmega }(f+I-\rho F)$$ is a contraction. This sends to the situation that $$P_{\varOmega }(f+I-\rho F)$$ is a contraction. Then $$P_{\varOmega }(f+I-\rho F)$$ has a unique fixed point. Say $$x^{*}\in H$$, that is, $$x^{*}=P_{\varOmega }(f+I-\rho F) x^{*}$$. Thus, there is a unique solution $$x^{*}\in {\varOmega }=\bigcap ^{N}_{i=0}\operatorname{Fix}(T_{i}) \cap \operatorname{VI}(C,A)$$ to the VIP

$$\bigl\langle (\rho F-f)x^{*},p-x^{*}\bigr\rangle \geq 0\quad \forall p\in {\varOmega }.$$
(3.12)

It is now easy to see that the necessity of the theorem is valid. Indeed, if $$x_{n}\to x^{*}\in {\varOmega }=\bigcap ^{N}_{i=0} \operatorname{Fix}(T _{i})\cap \operatorname{VI}(C,A)$$, then $$x^{*}=T_{i}x^{*}$$ for $$i=0,1,\ldots,N$$ and $$x^{*}=P_{C}(x^{*}-\tau _{n}Ax^{*})$$, which, together with Algorithm 3.1, imply that

\begin{aligned} \bigl\Vert w_{n}-x^{*} \bigr\Vert &= \bigl\Vert T_{n}x_{n}-x^{*}+\alpha _{n}(T_{n}x_{n}-T_{n}x_{n-1}) \bigr\Vert \\ &\leq \bigl\Vert x_{n}-x^{*} \bigr\Vert + \alpha _{n} \Vert x_{n}-x_{n-1} \Vert \to 0 \quad (n\to \infty ), \end{aligned}

and hence

\begin{aligned} \Vert y_{n}-x_{n} \Vert & \leq \bigl\Vert y_{n}-x^{*} \bigr\Vert + \bigl\Vert x_{n}-x^{*} \bigr\Vert \\ &= \bigl\Vert P_{C}(w_{n}-\tau _{n}Aw_{n})-P_{C} \bigl(x^{*}-\tau _{n}Ax^{*}\bigr) \bigr\Vert + \bigl\Vert x_{n}-x ^{*} \bigr\Vert \\ &\leq \bigl\Vert w_{n}-x^{*} \bigr\Vert +\tau _{n} \bigl\Vert Aw_{n}-Ax^{*} \bigr\Vert + \bigl\Vert x_{n}-x^{*} \bigr\Vert \\ &\leq (1+\gamma L) \bigl\Vert w_{n}-x^{*} \bigr\Vert + \bigl\Vert x_{n}-x^{*} \bigr\Vert \to 0 \quad (n\to \infty ). \end{aligned}

In addition, it is clear that

$$\Vert x_{n}-x_{n+1} \Vert \leq \bigl\Vert x_{n}-x^{*} \bigr\Vert + \bigl\Vert x_{n+1}-x^{*} \bigr\Vert \to 0 \quad (n\to \infty ).$$

Next we show the sufficiency of the theorem. To this aim, we assume $$\lim_{n\to \infty }(\|x_{n}-x_{n+1}\|+\|x_{n}-y_{n}\|)=0$$ and divide the proof of the sufficiency into several steps.

Step 1. We show that $$\{x_{n}\}$$ is bounded. Indeed, take an arbitrary $$p\in {\varOmega }=\bigcap ^{N}_{i=0}\operatorname{Fix}(T_{i})\cap \operatorname{VI}(C,A)$$. Then $$Tp=p$$, $$T_{n}p=p$$ $$\forall n\geq 1$$, and (3.5) holds, i.e.,

$$\Vert u_{n}-p \Vert ^{2}+(1-\mu ) \Vert w_{n}-y_{n} \Vert ^{2}+(1-\mu ) \Vert u_{n}-y_{n} \Vert ^{2}\leq \Vert w_{n}-p \Vert ^{2}.$$
(3.13)

This immediately implies that

$$\Vert u_{n}-p \Vert \leq \Vert w_{n}-p \Vert \quad \forall n\geq 1.$$
(3.14)

From the definition of $$w_{n}$$, we get

$$\Vert w_{n}-p \Vert \leq \Vert T_{n}x_{n}-p \Vert +\sigma _{n} \Vert T_{n}x_{n}-T_{n}x_{n-1} \Vert \leq \Vert x_{n}-p \Vert +\alpha _{n}\cdot \frac{\sigma _{n}}{\alpha _{n}} \Vert x _{n}-x_{n-1} \Vert .$$
(3.15)

Since $$\sup_{n\geq 1}\frac{\sigma _{n}}{\alpha _{n}}<\infty$$ and $$\sup_{n\geq 1}\|x_{n}-x_{n-1}\|<\infty$$, $$\sup_{n\geq 1}\frac{\sigma _{n}}{\alpha _{n}}\|x_{n}-x_{n-1}\|<\infty$$, which hence guarantees that there is $$M_{1}>0$$ such that

$$\frac{\sigma _{n}}{\alpha _{n}} \Vert x_{n}-x_{n-1} \Vert \leq M_{1}\quad \forall n\geq 1.$$
(3.16)

Combining (3.14), (3.15) and (3.16), we obtain

$$\Vert u_{n}-p \Vert \leq \Vert w_{n}-p \Vert \leq \alpha _{n}M_{1}+ \Vert x_{n}-p \Vert \quad \forall n\geq 1.$$
(3.17)

So, from Algorithm 3.1, Lemma 2.7 and (3.17) it follows that

\begin{aligned} \Vert z_{n}-p \Vert &\leq (1-\alpha _{n}\tau ) \Vert u_{n}-p \Vert +\alpha _{n} \bigl\Vert (f- \rho F)p \bigr\Vert +\alpha _{n}\delta \Vert x_{n}-p \Vert \\ &\leq (1-\alpha _{n}\tau ) \bigl( \Vert x_{n}-p \Vert +\alpha _{n}M_{1}\bigr)+\alpha _{n} \bigl\Vert (f- \rho F)p \bigr\Vert +\alpha _{n}\delta \Vert x_{n}-p \Vert \\ &\leq \bigl(1-\alpha _{n}(\tau -\delta )\bigr) \Vert x_{n}-p \Vert +\alpha _{n}\bigl(M_{1}+ \bigl\Vert (f- \rho F)p \bigr\Vert \bigr), \end{aligned}

which, together with Lemma 2.6 and $$(\gamma _{n}+\delta _{n})\zeta \leq \gamma _{n}$$, yields

\begin{aligned} \Vert x_{n+1}-p \Vert &\leq (1-\beta _{n}) \biggl\Vert \frac{1}{1-\beta _{n}}\bigl[\gamma _{n}(z _{n}-p)+\delta _{n}(Tz_{n}-p)\bigr] \biggr\Vert +\beta _{n} \Vert x_{n}-p \Vert \\ &\leq (1-\beta _{n}) \Vert z_{n}-p \Vert +\beta _{n} \Vert x_{n}-p \Vert \\ &\leq (1-\beta _{n})\bigl[\bigl(1-\alpha _{n}(\tau -\delta )\bigr) \Vert x_{n}-p \Vert +\alpha _{n} \bigl(M_{1}+ \bigl\Vert (f-\rho F)p \bigr\Vert \bigr)\bigr]+\beta _{n} \Vert x_{n}-p \Vert \\ &= \alpha _{n}(1-\beta _{n}) \bigl(M_{1}+ \bigl\Vert (f-\rho F)p \bigr\Vert \bigr)+\bigl[1-\alpha _{n}(1- \beta _{n}) (\tau -\delta )\bigr] \Vert x_{n}-p \Vert \\ &=\alpha _{n}(1-\beta _{n}) (\tau -\delta ) \frac{M_{1}+ \Vert (f-\rho F)p \Vert }{ \tau -\delta }+\bigl[1-\alpha _{n}(1-\beta _{n}) (\tau - \delta )\bigr] \Vert x_{n}-p \Vert \\ &\leq \max \biggl\{ \frac{M_{1}+ \Vert (f-\rho F)p \Vert }{\tau -\delta }, \Vert x_{n}-p \Vert \biggr\} . \end{aligned}

By induction, we obtain $$\|x_{n}-p\|\leq \max \{\|x_{1}-p\|,\frac{M _{1}+\|(\rho F-f)p\|}{\tau -\delta }\}$$ $$\forall n\geq 1$$. Thus, $$\{x_{n}\}$$ is a bounded point sequence, and so are the sequences $$\{u_{n}\}$$, $$\{w_{n}\}$$, $$\{y_{n}\}$$, $$\{z_{n}\}$$, $$\{f(x_{n})\}$$, $$\{Tz_{n}\}$$, $$\{Fu _{n}\}$$, $$\{T_{n}x_{n}\}$$.

Step 2. We show that

$$(1-\alpha _{n}\tau ) (1-\beta _{n}) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y _{n} \Vert ^{2}\bigr]\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+\alpha _{n}M_{4},$$

for some $$M_{4}>0$$. Indeed, utilizing Lemma 2.6, Lemma 3.2 and the convexity of $$\|\cdot \|^{2}$$, from $$(\gamma _{n}+\delta _{n}) \zeta \leq \gamma _{n}$$ we get

\begin{aligned}[b] &\Vert x_{n+1}-p \Vert ^{2}\\ &\quad \leq (1-\beta _{n}) \biggl\Vert \frac{1}{1-\beta _{n}}\bigl[\gamma _{n}(z_{n}-p)+\delta _{n}(Tz_{n}-p) \bigr] \biggr\Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2} \\ &\quad \leq (1-\beta _{n}) \Vert z_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2} \\ &\quad \leq (1-\beta _{n})\bigl\{ \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+(1-\alpha _{n} \tau ) \Vert w_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2} \\ & \qquad{} -(1-\alpha _{n}\tau ) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr]+2 \alpha _{n}\bigl\langle (f-\rho F)p,z_{n}-p\bigr\rangle \bigr\} \\ &\quad \leq (1-\beta _{n})\bigl\{ \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+(1-\alpha _{n} \tau ) \Vert w_{n}-p \Vert ^{2} \\ & \qquad {} -(1-\alpha _{n}\tau ) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr]+ \alpha _{n}M_{2}\bigr\} +\beta _{n} \Vert x_{n}-p \Vert ^{2}, \end{aligned}
(3.18)

where $$\sup_{n\geq 1}2\|(f-\rho F)p\|\|z_{n}-p\|\leq M_{2}$$ for some $$M_{2}>0$$. Also, from (3.17) we have

\begin{aligned}[b] \Vert w_{n}-p \Vert ^{2}& \leq \bigl( \Vert x_{n}-p \Vert +\alpha _{n}M_{1} \bigr)^{2} \\ &=\alpha _{n}\bigl(2M_{1} \Vert x_{n}-p \Vert +\alpha _{n}M^{2}_{1}\bigr)+ \Vert x_{n}-p \Vert ^{2} \\ &\leq \alpha _{n}M_{3}+ \Vert x_{n}-p \Vert ^{2}, \end{aligned}
(3.19)

where $$\sup_{n\geq 1}(2M_{1}\|x_{n}-p\|+\beta _{n}M^{2}_{1})\leq M_{3}$$ for some $$M_{3}>0$$. Substituting (3.19) for (3.18), we obtain

\begin{aligned}[b] & \Vert x_{n+1}-p \Vert ^{2}\\ &\quad \leq (1-\beta _{n})\bigl\{ \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+(1-\alpha _{n}\tau )\bigl[ \Vert x_{n}-p \Vert ^{2}+\alpha _{n}M_{3} \bigr]+\beta _{n} \Vert x _{n}-p \Vert ^{2} \\ & \qquad {} -(1-\alpha _{n}\tau ) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr]+ \alpha _{n}M_{2}\bigr\} \\ &\quad =(1-\beta _{n})\bigl\{ \bigl(1-\alpha _{n}(\tau -\delta ) \bigr) \Vert x_{n}-p \Vert ^{2}+(1- \alpha _{n}\tau )\alpha _{n}M_{3}+\beta _{n} \Vert x_{n}-p \Vert ^{2} \\ & \qquad {} -(1-\alpha _{n}\tau ) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr]+ \alpha _{n}M_{2}\bigr\} \\ &\quad =\bigl[1-\alpha _{n}(1-\beta _{n}) (\tau -\delta )\bigr] \Vert x_{n}-p \Vert ^{2}+(1-\beta _{n}) (1- \alpha _{n}\tau )\alpha _{n}M_{3} \\ & \qquad {} -(1-\alpha _{n}\tau ) (1-\beta _{n}) (1-\mu ) \bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y _{n} \Vert ^{2}\bigr]+(1-\beta _{n})\alpha _{n}M_{2} \\ &\quad \leq \Vert x_{n}-p \Vert ^{2}-(1-\alpha _{n}\tau ) (1-\beta _{n}) (1-\mu )\bigl[ \Vert w _{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr]+\alpha _{n}M_{4}, \end{aligned}
(3.20)

where $$M_{4}:=M_{2}+M_{3}$$. This immediately implies that

\begin{aligned}[b] &(1-\alpha _{n}\tau ) (1-\beta _{n}) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y _{n} \Vert ^{2}\bigr]\\ &\quad \leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+\alpha _{n}M_{4}. \end{aligned}
(3.21)

Step 3. We show that

\begin{aligned} \Vert x_{n+1}-p \Vert ^{2}& \leq \bigl[1-\alpha _{n}(1-\beta _{n}) (\tau -\delta )\bigr] \Vert x _{n}-p \Vert ^{2}+\alpha _{n}(1-\beta _{n}) (\tau -\delta ) \\ & \quad {} \times \biggl[\frac{2}{\tau -\delta }\bigl\langle (f-\rho F)p,z_{n}-p\bigr\rangle +\frac{M}{ \tau -\delta }\cdot \frac{\sigma _{n}}{\alpha _{n}} \cdot 3 \Vert x_{n}-x_{n-1} \Vert \biggr] \end{aligned}

for some $$M>0$$. Indeed, we have

\begin{aligned}[b] \Vert w_{n}-p \Vert ^{2}& \leq \bigl( \Vert x_{n}-p \Vert +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr)^{2} \\ &= \Vert x_{n}-p \Vert ^{2}+\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigl(2 \Vert x_{n}-p \Vert +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr). \end{aligned}
(3.22)

Combining (3.18) and (3.22), we have

\begin{aligned} & \Vert x_{n+1}-p \Vert ^{2} \\ &\quad \leq \beta _{n} \Vert x_{n}-p \Vert ^{2}+(1-\beta _{n})\bigl[\alpha _{n}\delta \Vert x_{n}-p \Vert ^{2} \\ &\qquad {}+(1-\alpha _{n}\tau ) \Vert w_{n}-p \Vert ^{2}+2\alpha _{n}\bigl\langle (f- \rho F)p,z_{n}-p\bigr\rangle \bigr] \\ &\quad \leq \beta _{n} \Vert x_{n}-p \Vert ^{2}+(1-\beta _{n})\bigl\{ \alpha _{n}\delta \Vert x _{n}-p \Vert ^{2}+(1-\alpha _{n}\tau ) \bigl[ \Vert x_{n}-p \Vert ^{2} \\ & \qquad {} +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigl(2 \Vert x_{n}-p \Vert +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr)\bigr]+2 \alpha _{n} \bigl\langle (f-\rho F)p,z_{n}-p\bigr\rangle \bigr\} \\ &\quad \leq \beta _{n} \Vert x_{n}-p \Vert ^{2}+(1-\beta _{n})\bigl\{ \bigl(1-\alpha _{n}( \tau - \delta )\bigr) \Vert x_{n}-p \Vert ^{2} \\ & \qquad {} +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigl(2 \Vert x_{n}-p \Vert +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr) +2\alpha _{n}\bigl\langle (f-\rho F)p,z_{n}-p\bigr\rangle \bigr\} \\ &\quad =\bigl[1-\alpha _{n}(1-\beta _{n}) (\tau -\delta ) \bigr] \Vert x_{n}-p \Vert ^{2} +(1-\beta _{n})\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \\ & \qquad {} \times \bigl(2 \Vert x_{n}-p \Vert +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr)+2\alpha _{n}(1-\beta _{n})\bigl\langle (f-\rho F)p,z_{n}-p \bigr\rangle \\ &\quad \leq \bigl[1-\alpha _{n}(1-\beta _{n}) (\tau - \delta )\bigr] \Vert x_{n}-p \Vert ^{2}+\alpha _{n}(1-\beta _{n}) (\tau -\delta )\biggl[\frac{2}{\tau -\delta } \bigl\langle (f- \rho F)p,z_{n}-p\bigr\rangle \\ & \qquad {} +\frac{M}{\tau -\delta }\cdot \frac{\sigma _{n}}{\alpha _{n}}\cdot 3 \Vert x _{n}-x_{n-1} \Vert \biggr], \end{aligned}
(3.23)

where $$\sup_{n\geq 1}\{\|x_{n}-p\|,\alpha _{n}\|x_{n}-x_{n-1}\|\} \leq M$$ for some $$M>0$$.

Step 4. We show that $$\{x_{n}\}$$ converges strongly to a unique solution $$x^{*}\in {\varOmega }$$ to the VIP (3.12). Indeed, putting $$p=x^{*}$$, we deduce from (3.23) that

\begin{aligned}[b] & \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}\leq \bigl[1-\alpha _{n}(1-\beta _{n}) (\tau -\delta )\bigr] \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\ &\quad{} +\alpha _{n}(1-\beta _{n}) (\tau -\delta ) \biggl[\frac{2}{\tau -\delta } \bigl\langle (f-\rho F)x^{*},z_{n}-x^{*} \bigr\rangle +\frac{M}{\tau -\delta } \cdot \frac{\sigma _{n}}{\alpha _{n}}\cdot 3 \Vert x_{n}-x_{n-1} \Vert \biggr]. \end{aligned}
(3.24)

By Lemma 2.4, it suffices to show that $$\limsup_{n\to \infty }\langle (f-\rho F)x^{*},z_{n}-x^{*}\rangle \leq 0$$. From (3.21), $$x_{n}-x_{n+1} \to 0$$, $$\alpha _{n}\to 0$$ and $$\{\beta _{n}\}\subset [a,b]\subset (0,1)$$, we obtain

\begin{aligned} &{ \limsup_{n\to \infty }}(1-\alpha _{n}\tau ) (1-b) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr] \\ &\quad \leq { \limsup_{n\to \infty }}(1-\alpha _{n}\tau ) (1-\beta _{n}) (1-\mu )\bigl[ \Vert w _{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr] \\ &\quad \leq { \limsup_{n\to \infty }}\bigl[ \Vert x_{n}-p \Vert ^{2}+\alpha _{n}M_{4}- \Vert x_{n+1}-p \Vert ^{2}\bigr] \\ &\quad \leq { \limsup_{n\to \infty }}\bigl[ \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}\bigr]+{ \limsup_{n\to \infty }}\alpha _{n}M_{4} \\ &\quad \leq { \limsup_{n\to \infty }}\bigl( \Vert x_{n}-p \Vert + \Vert x_{n+1}-p \Vert \bigr) \Vert x_{n}-x_{n+1} \Vert =0. \end{aligned}

This immediately implies that

$$\lim_{n\to \infty } \Vert w_{n}-y_{n} \Vert =0\quad \text{and}\quad \lim_{n\to \infty } \Vert u_{n}-y_{n} \Vert =0.$$
(3.25)

Since $$z_{n}=(I-\alpha _{n}\rho F)u_{n}+\alpha _{n}f(x_{n})$$ with $$u_{n}:=P_{C_{n}}(w_{n}-\tau _{n}Ay_{n})$$, from (3.25) and the boundedness of $$\{x_{n}\}$$, $$\{u_{n}\}$$, we get

\begin{aligned}[b] \Vert z_{n}-y_{n} \Vert &= \bigl\Vert \alpha _{n}f(x_{n})-\alpha _{n} \rho Fu_{n}+u_{n}-y _{n} \bigr\Vert \\ &\leq \Vert u_{n}-y_{n} \Vert +\alpha _{n} \bigl( \bigl\Vert f(x_{n}) \bigr\Vert + \Vert \rho Fu_{n} \Vert \bigr)\to 0 \quad (n\to \infty ), \end{aligned}
(3.26)

and hence

$$\Vert z_{n}-x_{n} \Vert \leq \Vert z_{n}-y_{n} \Vert + \Vert y_{n}-x_{n} \Vert \to 0\quad (n \to \infty )$$
(3.27)

(due to the assumption $$\|x_{n}-y_{n}\|\to 0$$). Obviously, the assumption $$\|x_{n}-y_{n}\|\to 0$$ together with (3.25) and (3.26), guarantees that, as $$n\to \infty$$, $$\|w_{n}-x_{n}\|\leq \|w_{n}-y _{n}\|+\|y_{n}-x_{n}\|\to 0$$ and $$\|w_{n}-z_{n}\|\leq \|w_{n}-y_{n}\|+ \|y_{n}-z_{n}\|\to 0$$. From the property of $$\{z_{n}\}$$, it follows that there is a subsequence $$\{z_{n_{k}}\}$$ of $$\{z_{n}\}$$ such that

$$\limsup_{n\to \infty }\bigl\langle (f-\rho F)x^{*},z_{n}-x^{*} \bigr\rangle = \lim_{k\to \infty }\bigl\langle (f-\rho F)x^{*},z_{n_{k}}-x^{*}\bigr\rangle .$$
(3.28)

Since H is reflexive and $$\{z_{n}\}$$ is bounded, it is guaranteed that $$z_{n_{k}}\rightharpoonup \tilde{z}$$. Hence from (3.28) one gets

$$\limsup_{n\to \infty }\bigl\langle (f-\rho F)x^{*},z_{n}-x^{*} \bigr\rangle = \lim_{k\to \infty }\bigl\langle (f-\rho F)x^{*},z_{n_{k}}-x^{*}\bigr\rangle = \bigl\langle (f- \rho F)x^{*},\tilde{z}-x^{*}\bigr\rangle .$$
(3.29)

It is easy to see from $$w_{n}-z_{n}\to 0$$ and $$z_{n_{k}}\rightharpoonup \tilde{z}$$ that $$w_{n_{k}}\rightharpoonup \tilde{z}$$. Since $$x_{n}-x_{n+1}\to 0$$, $$w_{n}-x_{n}\to 0$$, $$w_{n}-y_{n}\to 0$$, $$w_{n}-z_{n} \to 0$$ and $$w_{n_{k}}\rightharpoonup \tilde{z}$$, by Lemma 3.3 we infer that $$\tilde{z}\in {\varOmega }$$. Therefore, from (3.12) and (3.29) we conclude that

$$\limsup_{n\to \infty }\bigl\langle (f-\rho F)x^{*},z_{n}-x^{*} \bigr\rangle = \bigl\langle (f-\rho F)x^{*},\tilde{z}-x^{*} \bigr\rangle \leq 0.$$
(3.30)

Note that $$\{\beta _{n}\}\subset [a,b]\subset (0,1)$$, $$\{\alpha _{n}(1- \beta _{n})(\tau -\delta )\}\subset [0,1]$$, $$\sum^{\infty }_{n=1} \alpha _{n}(1-\beta _{n})(\tau -\delta )=\infty$$, and

$$\limsup_{n\to \infty }\biggl[\frac{2}{\tau -\delta }\bigl\langle (f-\rho F)x^{*},z _{n}-x^{*}\bigr\rangle + \frac{M}{\tau -\delta }\cdot \frac{\sigma _{n}}{ \alpha _{n}}\cdot 3 \Vert x_{n}-x_{n-1} \Vert \biggr]\leq 0.$$
(3.31)

Consequently, applying Lemma 2.4 to (3.24), we have $$\lim_{n\to 0}\|x _{n}-x^{*}\|=0$$. This completes the proof. □

Next, we introduce another composite inertial subgradient–extragradient algorithms with a line-search process.

### Algorithm 3.2

Initialization: Given $$\gamma >0$$, $$l \in (0,1)$$, $$\mu \in (0,1)$$. Let $$x_{0},x_{1}\in H$$ be arbitrary.

Iterative Steps: Calculate $$x_{n+1}$$ as follows:

Step 1. Set $$w_{n}=\sigma _{n}(T_{n}x_{n}-T_{n}x_{n-1})+T_{n}x_{n}$$ and compute $$y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n})$$, where $$\tau _{n}$$ is chosen to be the largest $$\tau \in \{\gamma ,\gamma l,\gamma l^{2},\ldots\}$$ satisfying

$$\tau \Vert Aw_{n}-Ay_{n} \Vert \leq \mu \Vert w_{n}-y_{n} \Vert .$$
(3.32)

Step 2. Compute $$z_{n}=(I-\alpha _{n}\rho F)P_{C_{n}}(w_{n}-\tau _{n}Ay _{n})+\alpha _{n}f(x_{n})$$ with $$C_{n}:=\{x\in H:\langle w_{n}-\tau _{n}Aw_{n}-y_{n},x-y_{n}\rangle \leq 0\}$$.

Step 3. Compute

$$x_{n+1}=\beta _{n}w_{n}+\gamma _{n}z_{n}+\delta _{n}Tz_{n}.$$
(3.33)

Again set $$n:=n+1$$ and go to Step 1.

It is worth pointing out that Lemmas 3.1, 3.2 and 3.3 are still valid for Algorithm 3.2.

### Theorem 3.2

Let the sequence $$\{x_{n}\}$$ be generated by Algorithm 3.2. Then

$$x_{n}\to x^{*}\in {\varOmega } \quad \Leftrightarrow \quad \textstyle\begin{cases} x_{n}-x_{n+1}\to 0, \\ x_{n}-y_{n}\to 0, \end{cases}$$

where $$x^{*}\in {\varOmega }$$ is a unique solution to the VIP: $$\langle (\rho F-f)x^{*}, p-x^{*}\rangle \geq 0$$ $$\forall p\in {\varOmega }$$.

### Proof

Utilizing the same arguments as in the proof of Theorem 3.1, we deduce that there exists a unique solution $$x^{*} \in {\varOmega }=\bigcap ^{N}_{i=0}\operatorname{Fix}(T_{i})\cap \operatorname{VI}(C,A)$$ to the VIP (3.12), and that the necessity of the theorem is valid.

Next we show the sufficiency of the theorem. To the aim, we assume $$\lim_{n\to \infty }(\|x_{n}-x_{n+1}\|+\|x_{n}-y_{n}\|)=0$$ and divide the proof of the sufficiency into several steps.

Step 1. We show that $$\{x_{n}\}$$ is bounded. Indeed, utilizing the same arguments as in Step 1 of the proof of Theorem 3.1, we obtain inequalities (3.13)–(3.17). So, from Algorithm 3.2, Lemma 2.7 and (3.17) it follows that

\begin{aligned} \Vert z_{n}-p \Vert &= \bigl\Vert (I-\alpha _{n}\rho F)u_{n}-(I-\alpha _{n}\rho F)p+\alpha _{n}(f-\rho F)p+\alpha _{n}\bigl(f(x_{n})-f(p) \bigr) \bigr\Vert \\ &\leq \alpha _{n}\bigl( \bigl\Vert (f-\rho F)p \bigr\Vert +M_{1}\bigr)+\bigl(1-\alpha _{n}(\tau -\delta )\bigr) \Vert x_{n}-p \Vert , \end{aligned}

which, together with Lemma 2.6 and $$(\gamma _{n}+\delta _{n})\zeta \leq \gamma _{n}$$, yields

\begin{aligned} \Vert x_{n+1}-p \Vert &= \bigl\Vert \gamma _{n}(z_{n}-p)+\delta _{n}(Tz_{n}-p)+ \beta _{n}(w _{n}-p) \bigr\Vert \\ &\leq (1-\beta _{n}) \biggl\Vert \frac{1}{1-\beta _{n}}\bigl[\gamma _{n}(z_{n}-p)+\delta _{n}(Tz_{n}-p) \bigr] \biggr\Vert +\beta _{n} \Vert w_{n}-p \Vert \\ &\leq (1-\beta _{n}) \Vert z_{n}-p \Vert +\beta _{n}\bigl(\alpha _{n}M_{1}+ \Vert x_{n}-p \Vert \bigr) \\ &\leq (1-\beta _{n})\bigl[\bigl(1-\alpha _{n}(\tau -\delta )\bigr) \Vert x_{n}-p \Vert +\alpha _{n} \bigl(M_{1}+ \bigl\Vert (f-\rho F)p \bigr\Vert \bigr)\bigr]\\ &\quad {}+\beta _{n}\bigl(\alpha _{n}M_{1}+ \Vert x_{n}-p \Vert \bigr) \\ &=\alpha _{n}(1-\beta _{n}) \biggl(\frac{M_{1}}{1-\beta _{n}}+ \bigl\Vert (f-\rho F)p \bigr\Vert \biggr)+\bigl[1- \alpha _{n}(1- \beta _{n}) (\tau -\delta )\bigr] \Vert x_{n}-p \Vert \\ &\leq \alpha _{n}(1-\beta _{n}) (\tau -\delta )\cdot \frac{ \frac{M_{1}}{1-b}+ \Vert (f-\rho F)p \Vert }{\tau -\delta }+\bigl[1-\alpha _{n}(1- \beta _{n}) (\tau -\delta )\bigr] \Vert x_{n}-p \Vert \\ &\leq \max \biggl\{ \frac{\frac{M_{1}}{1-b}+ \Vert (f-\rho F)p \Vert }{\tau -\delta }, \Vert x_{n}-p \Vert \biggr\} . \end{aligned}

By induction, we obtain $$\|x_{n}-p\|\leq \max \{\frac{ \frac{M_{1}}{1-b}+\|(f-\rho F)p\|}{\tau -\delta },\|x_{1}-p\|\} \forall n\geq 1$$. Thus, $$\{x_{n}\}$$ is bounded point sequence. Hence the sequences $$\{u_{n}\}$$, $$\{w_{n}\}$$, $$\{y_{n}\}$$, $$\{z_{n}\}$$, $$\{f(x_{n})\}$$, $$\{Tz _{n}\}$$, $$\{Fu_{n}\}$$, $$\{T_{n}x_{n}\}$$ are also bounded sequences.

Step 2. We show that

\begin{aligned} &(1-\alpha _{n}\tau ) (1-\beta _{n}) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y _{n} \Vert ^{2}\bigr]\\ &\quad \leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+\alpha _{n}M_{4}, \end{aligned}

for some $$M_{4}>0$$. Indeed, utilizing Lemma 2.6, Lemma 3.2 and the convexity of $$\|\cdot \|^{2}$$, from $$(\gamma _{n}+\delta _{n}) \zeta \leq \gamma _{n}$$ we get

\begin{aligned}[b] &\Vert x_{n+1}-p \Vert ^{2}\\ &\quad = \bigl\Vert \gamma _{n}(z_{n}-p)+\delta _{n}(Tz_{n}-p)+\beta _{n}(w_{n}-p) \bigr\Vert ^{2} \\ &\quad \leq (1-\beta _{n}) \biggl\Vert \frac{1}{1-\beta _{n}}\bigl[\gamma _{n}(z_{n}-p)+\delta _{n}(Tz_{n}-p) \bigr] \biggr\Vert ^{2}+\beta _{n} \Vert w_{n}-p \Vert ^{2} \\ &\quad \leq (1-\beta _{n}) \Vert z_{n}-p \Vert ^{2}+\beta _{n} \Vert w_{n}-p \Vert ^{2} \\ &\quad \leq (1-\beta _{n})\bigl\{ \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+(1-\alpha _{n} \tau ) \Vert w_{n}-p \Vert ^{2}+\beta _{n} \Vert w_{n}-p \Vert ^{2} \\ & \qquad {} -(1-\alpha _{n}\tau ) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr]+2 \alpha _{n}\bigl\langle (f-\rho F)p,z_{n}-p\bigr\rangle \bigr\} \\ &\quad \leq (1-\beta _{n})\bigl\{ \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+(1-\alpha _{n} \tau ) \Vert w_{n}-p \Vert ^{2}+\beta _{n} \Vert w_{n}-p \Vert ^{2} \\ & \qquad {} -(1-\alpha _{n}\tau ) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr]+ \alpha _{n}M_{2}\bigr\} , \end{aligned}
(3.34)

where $$\sup_{n\geq 1}2\|(f-\rho F)p\|\|z_{n}-p\|\leq M_{2}$$ for some $$M_{2}>0$$. Also, from (3.17) we have

\begin{aligned}[b] \Vert w_{n}-p \Vert ^{2}& \leq \bigl( \Vert x_{n}-p \Vert +\alpha _{n}M_{1} \bigr)^{2} \\ &=\alpha _{n}\bigl(2M_{1} \Vert x_{n}-p \Vert +\alpha _{n}M^{2}_{1}\bigr)+ \Vert x_{n}-p \Vert ^{2} \\ &\leq \alpha _{n}M_{3}+ \Vert x_{n}-p \Vert ^{2}, \end{aligned}
(3.35)

where $$\sup_{n\geq 1}(2M_{1}\|x_{n}-p\|+\beta _{n}M^{2}_{1})\leq M_{3}$$ for some $$M_{3}>0$$. Substituting (3.35) for (3.34) guarantees

\begin{aligned}[b]& \Vert x_{n+1}-p \Vert ^{2}\\ &\quad \leq \beta _{n}\bigl[ \Vert x_{n}-p \Vert ^{2}+\alpha _{n}M_{3}\bigr]+(1- \beta _{n})\bigl\{ \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+(1-\alpha _{n}\tau )\bigl[ \Vert x _{n}-p \Vert ^{2} \\ & \qquad {} +\alpha _{n}M_{3}\bigr]-(1-\alpha _{n}\tau ) (1-\mu )\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u _{n}-y_{n} \Vert ^{2}\bigr]+\alpha _{n}M_{2}\bigr\} \\ &\quad =\beta _{n} \Vert x_{n}-p \Vert ^{2}+\beta _{n}\alpha _{n}M_{3}+(1-\beta _{n}) \bigl\{ \bigl(1- \alpha _{n}(\tau -\delta )\bigr) \Vert x_{n}-p \Vert ^{2} \\ & \qquad {} +(1-\alpha _{n}\tau )\alpha _{n}M_{3}-(1- \alpha _{n}\tau ) (1-\mu )\bigl[ \Vert w _{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr]+\alpha _{n}M_{2}\bigr\} \\ &\quad =\bigl[1-\alpha _{n}(1-\beta _{n}) (\tau -\delta )\bigr] \Vert x_{n}-p \Vert ^{2}+\beta _{n} \alpha _{n}M_{3}+(1-\beta _{n}) (1-\alpha _{n}\tau )\alpha _{n}M_{3} \\ & \qquad {} -(1-\alpha _{n}\tau ) (1-\beta _{n}) (1-\mu ) \bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y _{n} \Vert ^{2}\bigr]+(1-\beta _{n})\alpha _{n}M_{2} \\ &\quad \leq \Vert x_{n}-p \Vert ^{2}-(1-\alpha _{n}\tau ) (1-\beta _{n}) (1-\mu )\bigl[ \Vert w _{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y_{n} \Vert ^{2}\bigr]+\alpha _{n}M_{4}, \end{aligned}
(3.36)

where $$M_{4}:=M_{2}+M_{3}$$. This immediately implies that

\begin{aligned}[b] &(1-\alpha _{n}\tau ) (1-\mu ) (1-\beta _{n})\bigl[ \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert u_{n}-y _{n} \Vert ^{2}\bigr]\\ &\quad \leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+\alpha _{n}M_{4}. \end{aligned}
(3.37)

Step 3. We show that

\begin{aligned} \Vert x_{n+1}-p \Vert ^{2}& \leq \bigl[1-\alpha _{n}(1-\beta _{n}) (\tau -\delta )\bigr] \Vert x _{n}-p \Vert ^{2}+\alpha _{n}(1-\beta _{n}) (\tau -\delta ) \\ & \quad {} \times \biggl[\frac{2}{\tau -\delta }\bigl\langle (f-\rho F)p,z_{n}-p\bigr\rangle +\frac{M}{ \tau -\delta }\cdot \frac{\sigma _{n}}{\alpha _{n}} \cdot \frac{3}{1-b} \Vert x_{n}-x_{n-1} \Vert \biggr] \end{aligned}

for some $$M>0$$. Indeed, we have

\begin{aligned}[b] \Vert w_{n}-p \Vert ^{2}& \leq \bigl(\sigma _{n} \Vert x_{n}-x_{n-1} \Vert + \Vert x_{n}-p \Vert \bigr)^{2} \\ &=\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigl(2 \Vert x_{n}-p \Vert +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr)+ \Vert x_{n}-p \Vert ^{2}. \end{aligned}
(3.38)

Combining (3.34) and (3.38), we have

\begin{aligned}[b] & \Vert x_{n+1}-p \Vert ^{2} \\ &\quad \leq (1-\beta _{n})\bigl[\alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+(1-\alpha _{n} \tau ) \Vert w_{n}-p \Vert ^{2}\\ &\qquad {}+2\alpha _{n}\bigl\langle (f-\rho F)p,z_{n}-p\bigr\rangle \bigr]+ \beta _{n} \Vert w_{n}-p \Vert ^{2} \\ &\quad \leq \beta _{n}\bigl[ \Vert x_{n}-p \Vert ^{2}+\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigl(2 \Vert x_{n}-p \Vert +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr)\bigr]\\ &\qquad {}+(1-\beta _{n}) \bigl\{ \alpha _{n}\delta \Vert x _{n}-p \Vert ^{2} \\ & \qquad {} +(1-\alpha _{n}\tau )\bigl[ \Vert x_{n}-p \Vert ^{2}+\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigl(2 \Vert x _{n}-p \Vert +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr)\bigr] \\ & \qquad {} +2\alpha _{n}\bigl\langle (f-\rho F)p,z_{n}-p \bigr\rangle \bigr\} \\ &\quad \leq \beta _{n}\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigl(2 \Vert x_{n}-p \Vert +\sigma _{n} \Vert x _{n}-x_{n-1} \Vert \bigr)+\beta _{n} \Vert x_{n}-p \Vert ^{2} \\ & \qquad {} +(1-\beta _{n})\bigl\{ \bigl(1-\alpha _{n}(\tau -\delta )\bigr) \Vert x_{n}-p \Vert ^{2}+\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigl(2 \Vert x_{n}-p \Vert +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr) \\ & \qquad {} +2\alpha _{n}\bigl\langle (f-\rho F)p,z_{n}-p \bigr\rangle \bigr\} \\ &\quad =\bigl[1-\alpha _{n}(1-\beta _{n}) (\tau -\delta ) \bigr] \Vert x_{n}-p \Vert ^{2}+\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \\ & \qquad {} \times \bigl(2 \Vert x_{n}-p \Vert +\sigma _{n} \Vert x_{n}-x_{n-1} \Vert \bigr)+2\alpha _{n}(1-\beta _{n})\bigl\langle (f-\rho F)p,z_{n}-p \bigr\rangle \\ &\quad \leq \bigl[1-\alpha _{n}(1-\beta _{n}) (\tau - \delta )\bigr] \Vert x_{n}-p \Vert ^{2}+\alpha _{n}(1-\beta _{n}) (\tau -\delta )\biggl[\frac{2}{\tau -\delta } \bigl\langle p-z _{n},(\rho F-f)p\bigr\rangle \\ & \qquad {} +\frac{M}{(1-\beta _{n})(\tau -\delta )}\cdot \frac{\sigma _{n}}{\alpha _{n}}\cdot 3 \Vert x_{n}-x_{n-1} \Vert \biggr] \\ &\quad \leq \bigl[1-\alpha _{n}(1-\beta _{n}) (\tau - \delta )\bigr] \Vert x_{n}-p \Vert ^{2}+\alpha _{n}(1-\beta _{n}) (\tau -\delta )\biggl[\frac{2}{\tau -\delta } \bigl\langle p-z _{n},(\rho F-f)p\bigr\rangle \\ & \qquad {} +\frac{M}{\tau -\delta }\cdot \frac{\sigma _{n}}{\alpha _{n}}\cdot \frac{3}{1-b} \Vert x_{n}-x_{n-1} \Vert \biggr], \end{aligned}
(3.39)

where $$\sup_{n\geq 1}\{\|x_{n}-p\|,\alpha _{n}\|x_{n}-x_{n-1}\|\} \leq M$$ for some $$M>0$$.

Step 4. We show that $$\{x_{n}\}$$ converges strongly to a unique solution $$x^{*}\in {\varOmega }$$ to the VIP (3.12). Indeed, utilizing the same argument as in Step 4 of the proof of Theorem 3.1, we obtain the desired assertion. This completes the proof. □

### Remark 3.1

Compared with the corresponding results in Kraikaew and Saejung , Thong and Hieu [24, 25], our results improve and extend them in the following aspects.

(i) The problem of finding an element of $$\operatorname{VI}(C,A)$$ in  is extended to develop our problem of finding an element of $$\bigcap ^{N}_{i=0}\operatorname{Fix}(T_{i})\cap \operatorname{VI}(C,A)$$ where $$T_{i}$$ is nonexpansive for $$i=1,\ldots,N$$ and $$T_{0}=T$$ is strictly pseudocontractive. The Halpern subgradient–extragradient method for solving the VIP in  is extended to develop our composite inertial subgradient–extragradient method with line-search process for solving the VIP and CFPP, which is based on an inertial subgradient–extragradient method with a line-search process, a hybrid steepest-descent method, a viscosity approximation method and a Mann iteration method.

(ii) The problem of finding an element of $$\operatorname{VI}(C,A)$$ in  is extended to develop our problem of finding an element of $$\bigcap ^{N}_{i=0}\operatorname{Fix}(T_{i})\cap \operatorname{VI}(C,A)$$ where $$T_{i}$$ is nonexpansive for $$i=1,\ldots,N$$ and $$T_{0}=T$$ is strictly pseudocontractive. The inertial subgradient–extragradient method with weak convergence for solving the VIP in  is extended to develop our composite inertial subgradient–extragradient method with line-search process (which is strongly convergent) for solving the VIP and CFPP, which is based on an inertial subgradient–extragradient method with line-search process, a hybrid steepest-descent method, a viscosity approximation method and a Mann iteration method.

(iii) The problem of finding an element of $$\operatorname{VI}(C,A)\cap \operatorname{Fix}(T)$$ (where A is monotone and T is quasi-nonexpansive) in  is extended to develop our problem of finding an element of $$\bigcap ^{N}_{i=0}\operatorname{Fix}(T_{i})\cap \operatorname{VI}(C,A)$$ where A is pseudomonotone, $$T_{i}$$ is nonexpansive for $$i=1,\ldots,N$$ and $$T_{0}=T$$ is strictly pseudocontractive. The inertial subgradient–extragradient method with line-search (which is weakly convergent) for solving the VIP and FPP in  is extended to develop our composite inertial subgradient–extragradient method with line-search process (which is strongly convergent) for solving the VIP and CFPP, which is based on an inertial subgradient–extragradient method with a line-search process, a hybrid steepest-descent method, a viscosity approximation method and a Mann iteration method. It is worth pointing out that the inertial subgradient–extragradient method with a line-search process in  combines the inertial subgradient–extragradient method  with the Mann iteration method.

## 4 An example

In this section, our main results are applied to solve the VIP and CFPP in an illustrated example. The initial point $$x_{0}=x_{1}$$ is randomly chosen in R. Take $$f(x)=F(x)=\frac{1}{2}x$$, $$\gamma =l= \mu =\frac{1}{2}$$, $$\sigma _{n}=\alpha _{n}=\frac{1}{n+1}$$, $$\beta _{n}= \frac{1}{3}$$, $$\gamma _{n}=\frac{1}{2}$$, $$\delta _{n}=\frac{1}{6}$$ and $$\rho =2$$. Then we know that $$\delta =\kappa =\eta =\frac{1}{2}$$, and

$$\tau =1-\sqrt{1-\rho \bigl(2\eta -\rho \kappa ^{2}\bigr)}=1-\sqrt{1-2 \biggl(2 \cdot \frac{1}{2}-2\biggl(\frac{1}{2}\biggr)^{2} \biggr)}=1\in (0,1].$$

We first provide an example of Lipschitz continuous and pseudomonotone mapping A, strictly pseudocontractive mapping T and nonexpansive mapping $$T_{1}$$ with $${\varOmega }=\operatorname{Fix}(T_{1})\cap \operatorname{Fix}(T) \cap \operatorname{VI}(C,A)\neq \emptyset$$. Let $$C=[-1,0]$$ and $$H=\mathbf{R}$$ with the inner product $$\langle a,b\rangle =ab$$ and induced norm $$\|\cdot \|=|\cdot |$$. Let $$A,T,T_{1}:H\to H$$ be defined as $$Ax:=\frac{1}{1+|\sin x|}-\frac{1}{1+|x|}$$, $$Tx:=\frac{1}{2}x+ \frac{3}{8}\sin x$$ and $$T_{1}x:=\sin x$$ for all $$x\in H$$. Now, we first show that A is pseudomonotone and Lipschitz continuous with $$L=2$$. Indeed, for all $$x,y\in H$$ we have

\begin{aligned} \Vert Ax-Ay \Vert &= \biggl\vert \frac{1}{1+ \Vert \sin x \Vert }-\frac{1}{1+ \Vert x \Vert }-\frac{1}{1+ \Vert \sin y \Vert }+\frac{1}{1+ \Vert y \Vert } \biggr\vert \\ &\leq \biggl\vert \frac{1}{1+ \Vert x \Vert }-\frac{1}{1+ \Vert y \Vert } \biggr\vert + \biggl\vert \frac{1}{1+ \Vert \sin x \Vert }-\frac{1}{1+ \Vert \sin y \Vert } \biggr\vert \\ &= \biggl\vert \frac{1+ \Vert y \Vert -1- \Vert x \Vert }{(1+ \Vert x \Vert )(1+ \Vert y \Vert )} \biggr\vert + \biggl\vert \frac{1+ \Vert \sin y \Vert -1- \Vert \sin x \Vert }{(1+ \Vert \sin x \Vert )(1+ \Vert \sin y \Vert )} \biggr\vert \\ &= \biggl\vert \frac{ \Vert y \Vert - \Vert x \Vert }{(1+ \Vert x \Vert )(1+ \Vert y \Vert )} \biggr\vert + \biggl\vert \frac{ \Vert \sin y \Vert - \Vert \sin x \Vert }{(1+ \Vert \sin x \Vert )(1+ \Vert \sin y \Vert )} \biggr\vert \\ &\leq \frac{ \Vert x-y \Vert }{(1+ \Vert x \Vert )(1+ \Vert y \Vert )}+\frac{ \Vert \sin x-\sin y \Vert }{(1+ \Vert \sin x \Vert )(1+ \Vert \sin y \Vert )} \\ &\leq \Vert x-y \Vert + \Vert \sin x-\sin y \Vert \\ &\leq 2 \Vert x-y \Vert . \end{aligned}

This implies that A is Lipschitz continuous with $$L=2$$. Next, we show that A is pseudomonotone. For any given $$x,y\in H$$, it is clear that

\begin{aligned} &\langle Ax,y-x\rangle =\biggl(\frac{1}{1+ \vert \sin x \vert }-\frac{1}{1+ \vert x \vert }\biggr) (y-x) \geq 0\\ &\quad \Rightarrow\quad \langle Ay,y-x\rangle =\biggl(\frac{1}{1+ \vert \sin y \vert }- \frac{1}{1+ \vert y \vert }\biggr) (y-x)\geq 0. \end{aligned}

Furthermore, it is easy to see that T is strictly pseudocontractive with constant $$\zeta =\frac{1}{2}$$. Indeed, we observe that, for all $$x,y\in H$$,

$$\Vert Tx-Ty \Vert \leq \frac{1}{2} \Vert x-y \Vert + \frac{3}{8} \Vert \sin x-\sin y \Vert \leq \Vert x-y \Vert + \frac{1}{2} \bigl\Vert (I-T)x-(I-T)y \bigr\Vert .$$

It is clear that $$(\gamma _{n}+\delta _{n})\zeta =(\frac{1}{2}+ \frac{1}{6})\cdot \frac{1}{2}\leq \frac{1}{2}=\gamma _{n}$$ for all $$n\geq 1$$. In addition, it is clear that $$T_{1}$$ is nonexpansive and $$\operatorname{Fix}(T_{1})=\{0\}$$. Therefore, $${\varOmega }=\operatorname{Fix}(T_{1}) \cap \operatorname{Fix}(T)\cap \operatorname{VI}(C,A)=\{0\}\neq \emptyset$$. In this case, Algorithm 3.1 can be rewritten as follows:

$$\textstyle\begin{cases} w_{n}=T_{1}x_{n}+\frac{1}{n+1}(T_{1}x_{n}-T_{1}x_{n-1}), \\ y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n}), \\ z_{n}=\frac{1}{n+1}\cdot \frac{1}{2}x_{n}+\frac{n}{n+1}P_{C_{n}}(w _{n}-\tau _{n}Ay_{n}), \\ x_{n+1}=\frac{1}{3}x_{n}+\frac{1}{2}z_{n}+\frac{1}{6}Tz_{n}\quad \forall n\geq 1, \end{cases}$$
(4.1)

where for each $$n\geq 1$$, $$C_{n}$$ and $$\tau _{n}$$ are chosen as in Algorithm 3.2. Then, by Theorem 3.2, we know that $$\{x_{n}\}$$ converges to $$0\in {\varOmega }=\operatorname{Fix}(T_{1})\cap \operatorname{Fix}(T)\cap \operatorname{VI}(C,A)$$ if and only if $$|x_{n}-x_{n+1}| +|x_{n}-y_{n}| \to 0$$ as $$n\to \infty$$.

On the other hand, Algorithm 3.2 can be rewritten as follows:

$$\textstyle\begin{cases} w_{n}=T_{1}x_{n}+\frac{1}{n+1}(T_{1}x_{n}-T_{1}x_{n-1}), \\ y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n}), \\ z_{n}=\frac{1}{n+1}\cdot \frac{1}{2}x_{n}+\frac{n}{n+1}P_{C_{n}}(w _{n}-\tau _{n}Ay_{n}), \\ x_{n+1}=\frac{1}{3}w_{n}+\frac{1}{2}z_{n}+\frac{1}{6}Tz_{n}\quad \forall n\geq 1, \end{cases}$$
(4.2)

where for each $$n\geq 1$$, $$C_{n}$$ and $$\tau _{n}$$ are chosen as in Algorithm 3.2. Then, by Theorem 3.2, we know that $$\{x_{n}\}$$ converges to $$0\in {\varOmega }=\operatorname{Fix}(T_{1})\cap \operatorname{Fix}(T)\cap \operatorname{VI}(C,A)$$ if and only if $$|x_{n}-x_{n+1}| +|x_{n}-y_{n}| \to 0$$ as $$n\to \infty$$.

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### Acknowledgements

We thank the two anonymous referees for useful suggestions, which improved the presentation of this paper a lot.

Not applicable.

## Funding

This research was supported by the Natural Science Foundation of Shandong Province of China (ZR2017LA001) and Youth Foundation of Linyi University (LYDX2016BS023). The first author was partially supported by the Innovation Program of Shanghai Municipal Education Commission (15ZZ068), Ph.D. Program Foundation of Ministry of Education of China (20123127110002) and Program for Outstanding Academic Leaders in Shanghai City (15XD1503100).

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The two authors contributed equally. Both authors read and approved the final version of the manuscript.

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Correspondence to Qing Yuan.

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