Triple-adaptive subgradient extragradient with extrapolation procedure for bilevel split variational inequality

Abstract

This paper introduces a triple-adaptive subgradient extragradient process with extrapolation to solve a bilevel split pseudomonotone variational inequality problem (BSPVIP) with the common fixed point problem constraint of finitely many nonexpansive mappings. The problem under consideration is in real Hilbert spaces, where the BSPVIP involves a fixed point problem of demimetric mapping. The proposed rule exploits the strong monotonicity of one operator at the upper level and the pseudomonotonicity of another mapping at the lower level. The strong convergence result for the proposed algorithm is established under some suitable assumptions. In addition, a numerical example is given to demonstrate the viability of the proposed rule. Our results improve and extend some recent developments to a great extent.

1 Introduction

Suppose that $\emptyset \neq C\subset{\mathcal {H}}$ with C being a closed convex set in a real Hilbert space $\mathcal {H}$, and $\langle \cdot ,\cdot \rangle$ and $\|\cdot \|$ are the inner product and the induced norm in $\mathcal {H}$, respectively. Let $P_{C}$ be the metric projection of $\mathcal {H}$ onto C, and for a given mapping $S:C\to{\mathcal {H}}$, let its set of fixed points be denoted by $\operatorname{Fix}(S)$.

Let $A:{\mathcal {H}}\to{\mathcal {H}}$ be a Lipschitz continuous mapping with Lipschitz constant L, and consider the classical variational inequality problem (VIP) of finding $x^{*}\in C$ such that $\langle Ax^{*},x-x^{*}\rangle \geq 0 \ \forall x\in C$. We denote the solution set of the VIP by VI(C, A). One of the most popular approaches for settling the VIP is the extragradient method invented by Korpelevich [1] in 1976. For any given initial point $p_{0}\in C$, the method of Korpelevich [1] generates a sequence $\{p_{t}\}$ as fabricated below:

$$ \textstyle\begin{cases} q_{t} = P_{C}(p_{t}-\ell Ap_{t}), \\ p_{t+1} =P_{C}(p_{t}-\ell Aq_{t}), & t = 0, 1, 2, \ldots , \end{cases} $$

where the constant ℓ lies in $(0,\frac{1}{L} )$. The literature on the VIP is numerous, and Korpelevich’s extragradient method has received extensive attention of many scholars, who intensely enhanced it in various aspects; for example, please see [2–26] and the references therein, to name but a few.

Thong and Hieu [26] put forward subgradient extragradient process with extrapolation, which generates a sequence $\{p_{t}\}$ for any given $p_{1},p_{0}\in{\mathcal {H}}$ as follows:

$$ \textstyle\begin{cases} w_{t} = p_{t}+\alpha _{t}(p_{t}-p_{t-1}), \\ y_{t} = P_{C}(w_{t}-\zeta Aw_{t}), \\ C_{t} = \{p\in{\mathcal {H}}:\langle w_{t}-\zeta Aw_{t}-y_{t},y_{t}-p \rangle \geq 0 \}, \\ p_{t+1} = P_{C_{t}}(w_{t}-\ell Ay_{t}),\quad t = 1, 2, 3, \ldots , \end{cases} $$

where $\zeta \in (0,\frac{1}{L} )$ and weak convergence is obtained. Given nonexpansive mappings $S_{i}:{\mathcal {H}}\rightarrow {\mathcal {H}}$, $i=1,2,\ldots , N$, Ceng and Shang [16] presented a subgradient extragradient-type process for computing a common element of the common fixed point set and $\operatorname{VI}(C,A)$ when

$$ \Omega :=\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i}) \cap\operatorname{VI}(C,A) \neq \emptyset . $$

Furthermore, the following strongly convergent algorithm was studied in [21] when $\Omega :=\bigcap^{N}_{i=1} \operatorname{Fix}(S_{i})\cap\operatorname{VI}(C,A)$ is nonempty.

Algorithm 1.1

(See [21, Algorithm 3.1])

Modified inertial subgradient extragradient method.

Initialization

Let $\lambda _{1}>0$, $\alpha >0$, $\mu \in (0,1)$, and $x_{1},x_{0}\in{\mathcal {H}}$ be arbitrary.

Iterative steps

Calculate $x_{t+1}$ as follows:

Step 1. Given the iterates $x_{t}$ and $x_{t-1}$ ($t\geq 1$), choose $\alpha _{t}$ such that $0\leq \alpha _{t}\leq \bar{\alpha}_{t}$, where

$$ \bar{\alpha}_{t} = \textstyle\begin{cases} \min \{\alpha ,\frac {\varepsilon _{t}}{ \Vert x_{t}-x_{t-1} \Vert } \} & \text{if } x_{t} \neq x_{t-1}, \\ \alpha & \text{otherwise}. \end{cases} $$

Step 2. Compute $w_{t}=S_{t}x_{t}+\alpha _{t}(S_{t}x_{t}-S_{t}x_{t-1})$ and $y_{t}=P_{C}(w_{t}-\lambda _{t}Aw_{t})$.

Step 3. Identify $C_{t} = \{y\in{\mathcal {H}}:\langle w_{t}-\lambda _{t}Aw_{t}-y_{t},y_{t}-y \rangle \geq 0\}$, then calculate

$$ z_{t}=P_{C_{t}}(w_{t}-\lambda _{t}Ay_{t}). $$

Step 4. Update $x_{t+1}=\beta _{t}f(x_{t})+\gamma _{t}x_{t}+((1-\gamma _{t})I-\beta _{t} \rho F)z_{t}$, where $\rho \in (0, \frac{2\eta}{\kappa ^{2}} )$ and update

$$ \lambda _{t+1} = \textstyle\begin{cases} \min \{\mu \frac { \Vert w_{t}-y_{t} \Vert ^{2}+ \Vert z_{t}-y_{t} \Vert ^{2}}{2\langle Aw_{t}-Ay_{t},z_{t}-y_{t}\rangle}, \lambda _{t} \} & \text{if } \langle Aw_{t}-Ay_{t},z_{t}-y_{t} \rangle >0, \\ \lambda _{t} & \text{otherwise.} \end{cases} $$

Set $t:=t+1$ and return to Step 1, where f is a contraction ($f:{\mathcal {H}}\rightarrow { \mathcal {H}}$ is a contraction if there exists $\nu \in [0,1)$ such that $\|f(x)-f(y)\| \leq \nu \|x-y\|$, $\forall x,y \in {\mathcal {H}}$), F is η-strongly monotone and κ-Lipschitz continuous (kindly see Sect. 2 for its definition) with $\{\beta _{t}\},\{\gamma _{t}\}, \{\varepsilon _{t}\} \subset (0,1)$ fulfilling some conditions.

Next, suppose that C and Q are nonempty, closed, and convex subsets of Hilbert spaces ${\mathcal {H}}_{1}$ and ${\mathcal {H}}_{2}$, respectively. Let $T:{\mathcal {H}}_{1}\to{\mathcal {H}}_{2}$ denote a bounded linear operator and $A,F:{\mathcal {H}}_{1}\to{\mathcal {H}}_{1}$ and $B:{\mathcal {H}}_{2}\to{\mathcal {H}}_{2}$ be nonlinear mappings. Then, the bilevel split variational inequality problem (BSVIP) (see [27]) is as specified below:

$$ \text{Seek } q^{*}\in\Omega \text{ such that}\quad \bigl\langle Fq^{*},z-q^{*} \bigr\rangle \geq 0 \quad \forall z\in\Lambda , $$

(1.1)

where $\Lambda :=\{z\in\operatorname{VI}(C,A):Tz\in\operatorname{VI}(Q,B)\}$ is the solution set of the split variational inequality problem (SVIP), which was introduced by Censor et al. [28] and formulated as follows:

$$ \text{Find }x^{*}\in C \text{ such that}\quad \bigl\langle Ax^{*},x-x^{*}\bigr\rangle \geq 0 \quad \forall x\in C $$

(1.2)

and

$$ y^{*}=Tx^{*}\in Q \text{ such that}\quad \bigl\langle By^{*},y-y^{*}\bigr\rangle \geq 0 \quad \forall y\in Q $$

(1.3)

with $\operatorname{VI}(C,A)$ and $\operatorname{VI}(Q,B)$ representing the solution sets of variational inequalities (1.2) and (1.3), respectively. Note that the SVIP involves finding $x^{*}\in\operatorname{VI}(C,A)$ such that $Tx^{*}\in\operatorname{VI}(Q,B)$. Censor et al. [28] proposed a weakly convergent method for approximating the solution of (1.2)–(1.3): for any given initial $x_{1}\in{\mathcal {H}}_{1}$, identify the sequence $\{x_{t}\}$ generated by

$$ x_{t+1}=P_{C}(I-\lambda A) \bigl(x_{t}+\gamma T^{*}\bigl(P_{Q}(I-\lambda B)-I \bigr)Tx_{t}\bigr),\quad t = 1, 2, 3, \ldots , $$

(1.4)

where A and B both are inverse-strongly monotone and T is a bounded linear operator. Under appropriate assumptions, it was proven in [28] that the sequence $\{x_{t}\}$ converges weakly to a solution of (1.2)–(1.3).

We note that the VIP can be expressed as the FPP: $Sz=P_{Q}(z-\mu Bz)$, $\mu >0$, with $\operatorname{VI}(Q,B)=\operatorname{Fix}(S)$. Consequently, we can reformulate the BSVIP in (1.1) as follows: Let $A:{\mathcal {H}}_{1}\to{\mathcal {H}}_{1}$ be quasimonotone and L-Lipschitz continuous, $F:{\mathcal {H}}_{1}\to{\mathcal {H}}_{1}$ be κ-Lipschitzian and η-strongly monotone, $T:{\mathcal {H}}_{1}\to{\mathcal {H}}_{2}$ be a nonzero bounded linear operator, and $S:{\mathcal {H}}_{2}\to{\mathcal {H}}_{2}$ be a τ-demimetric mapping with $\tau \in (-\infty ,1)$; then,

$$ \text{Find } q^{*}\in\Omega \text{ such that}\quad \bigl\langle Fq^{*},z-q^{*} \bigr\rangle \geq 0 \quad \forall z\in\Omega , $$

(1.5)

where $\Omega :=\{z\in\operatorname{VI}(C,A): Tz\in\operatorname{Fix}(S)\}$. In this case, such a problem is referred to as a bilevel split quasimonotone variational inequality problem (BSQVIP) and its strong convergence results are obtained in [18].

Assume that $f:{\mathcal {H}}_{1}\to {\mathcal {H}}_{1}$ is a contractive mapping with $\nu \in [0,1)$ with $\nu <\zeta :=1-\sqrt{1-\rho (2\eta -\rho \kappa ^{2})}$ for $\rho \in (0,\frac{2\eta}{\kappa ^{2}} )$, $A:{\mathcal {H}}_{1}\to {\mathcal {H}}_{1}$ is pseudomonotone and L-Lipschitz continuous with $\|Au\|\leq \liminf_{t\to \infty}\|Au_{t}\|$ for each $\{u_{t}\}\subset C$ with $u_{t}\rightharpoonup u$, $\{S_{i}\}^{N}_{i=1}$ is finitely many nonexpansive mappings on ${\mathcal {H}}_{1}$ and $\Xi :=\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\Omega \neq \emptyset$. Then, the bilevel split pseudomonotone variational inequality problem (BSPVIP) with the common fixed point problem (CFPP) constraint is formulated as follows:

$$ \text{Seek }q^{*}\in\Xi \text{ such that} \quad \bigl\langle (\rho F-f)q^{*},p-q^{*} \bigr\rangle \geq 0 \quad \forall p\in\Xi . $$

(1.6)

We propose triple-adaptive subgradient extragradient-type rule with inertial extrapolation to solve (1.6) in real Hilbert spaces, where the BSPVIP involves the FPP of demimetric mapping S. The rule exploits the strong monotonicity of the operator F at the upper-level problem and the pseudomonotonicity of the mapping A at the lower level. Consequently, we obtain strong convergence result. In addition, a numerical test is provided to show the viability of the suggested rule.

The article is organized as follows: In Sect. 2, we provide some concepts and basic tools for further use. Section 3 gives the convergence analysis of the suggested algorithm. Lastly, Sect. 4 gives a numerical illustration. Our results improve and extend the corresponding ones in [21, 29], and the relevant explanatory argument is given after the main proof of convergence result in Sect. 3.

2 Preliminaries

A mapping $S:C\to{\mathcal {H}}$ is (see [30]):

• (i) L-Lipschitz continuous or L-Lipschitzian if $\exists L>0$ such that $\|S\tilde{u}-S\bar{y}\|\leq L\|\tilde{u}-\bar{y}\| \ \forall \tilde{u}, \bar{y}\in C$. If $L=1$, then S is nonexpansive;

• (ii) ς-strongly monotone if $\exists \varsigma >0$ such that $\langle S\tilde{u}-S\bar{y},\tilde{u}-\bar{y} \rangle \geq \varsigma \|\tilde{u}-\bar{y}\|^{2} \ \forall \tilde{u}$, $\bar{y}\in C$;

• (iii) monotone if $\langle S\tilde{u}-S\bar{y},\tilde{u}-\bar{y}\rangle \geq 0 \ \forall \tilde{u}$, $\bar{y}\in C$;

• (iv) pseudomonotone if $\langle S\tilde{u},\bar{y}-\tilde{u}\rangle \geq 0\Longrightarrow \langle S\bar{y},\bar{y}-\tilde{u}\rangle \geq 0 \ \forall \tilde{u}, \bar{y}\in C$;

• (v) quasimonotone if $\langle S\tilde{u},\bar{y}-\tilde{u}\rangle >0\Longrightarrow \langle S\bar{y},\bar{y}-\tilde{u}\rangle \geq 0 \ \forall \tilde{u}, \bar{y}\in C$;

• (vi) τ-demicontractive if $\exists \tau \in (0,1)$ such that

  $$ \Vert S\tilde{u}-p \Vert ^{2}\leq \Vert \tilde{u}-p \Vert ^{2}+\tau \Vert \tilde{u}-S \tilde{u} \Vert ^{2} \quad \forall \tilde{u}\in C, p\in\operatorname{Fix}(S)\neq \emptyset ; $$

• (vii) τ-demimetric if $\exists \tau \in (-\infty ,1)$ such that

  $$ \langle \tilde{u}-S\tilde{u},\tilde{u}-p\rangle \geq \frac {1-\tau}{2} \Vert \tilde{u}-S\tilde{u} \Vert ^{2} \quad \forall \tilde{u}\in C, p \in\operatorname{Fix}(S)\neq \emptyset ; $$

• (viii) sequentially weakly continuous if $\forall \{x_{t}\}\subset C$, $x_{t}\rightharpoonup x\Longrightarrow Sx_{t}\rightharpoonup Sx$.

Given $\grave{u}\in{\mathcal {H}}$, there exists unique $P_{C}\grave{u} \in C$ with the following properties.

Lemma 2.1

(See [31])

The following hold:

• (i) $\langle \check{u}-\grave{v},P_{C} \check{u} - P_{C} \grave{v} \rangle \geq \|P_{C}\check{u}-P_{C}\grave{v}\|^{2} \ \forall \check{u}, \grave{v}\in{\mathcal {H}}$;

• (ii) $w=P_{C}\check{u} \Longleftrightarrow \langle \check{u}-w,\grave{v}-w \rangle \leq 0 \ \forall \check{u}\in{\mathcal {H}},\grave{v}\in C$;

• (iii) $\|\check{u}-\grave{v}\|^{2}\geq \|\check{u}-P_{C}\check{u}\|^{2}+\| \grave{v}-P_{C}\check{u}\|^{2} \ \forall \check{u}\in{\mathcal {H}}, \grave{v}\in C$;

• (iv) $\|\check{u}-\grave{v}\|^{2}=\|\check{u}\|^{2}-\|\grave{v}\|^{2}-2 \langle \check{u}-\grave{v},\grave{v}\rangle \ \forall \check{u}, \grave{v}\in{\mathcal {H}}$;

• (v) $\|\vartheta \check{u}+(1-\vartheta )\grave{v}\|^{2}=\vartheta \| \check{u}\|^{2}+(1-\vartheta )\|\grave{v}\|^{2}-\vartheta (1- \vartheta )\|\check{u}-\grave{v}\|^{2} \ \forall \check{u},\grave{v} \in{\mathcal {H}}, \vartheta \in {\mathbb{R}}$.

Clearly, (ii) ⟹ (iii) ⟹ (iv) ⟹ (v). However, the converse is not generally true.

Lemma 2.2

(See [32])

Let $\varpi \in (0,1]$, $S:C\to{\mathcal {H}}$ be nonexpansive and $S^{\varpi}: C\to{\mathcal {H}}$ be defined by $S^{\varpi }\acute{x}:=S\acute{x}-\varpi \rho F(S\acute{x}) \ \forall \acute{x}\in C$, where F is ϱ-Lipschitz continuous and ς-strongly monotone. Then $S^{\varpi}$ is a contraction provided $0<\rho <\frac{2\varsigma}{\varrho ^{2}}$, i.e., $\|S^{\varpi }\acute{x}-S^{\varpi }\acute{y}\|\leq (1-\varpi \zeta ) \|\acute{x}-\acute{y}\| \ \forall \acute{x},\acute{y}\in C$, where $\zeta =1-\sqrt{1-\rho (2\varsigma -\rho \varrho ^{2})}\in (0,1]$.

Lemma 2.3

If $A:C\to{\mathcal {H}}$ is pseudomonotone and continuous, then $u^{*}\in C$ solves VIP ⇔ $\langle Av,v-u^{*}\rangle \geq 0 \ \forall v\in C$.

Proof

The proof is straightforward and thus we skip it. □

Lemma 2.4

(See [32])

Let $\{a_{t}\} \subset (0,\infty )$ satisfying the condition $a_{t+1} \leq (1-\lambda _{t})a_{t}+\lambda _{t}\gamma _{t} \ \forall t \geq 1$, where $\{\lambda _{t}\}, \{\gamma _{t}\} \subset \mathbb{R}$ and (i) $\{\lambda _{t}\}\subset [0,1]$ and $\sum^{\infty}_{t=1}\lambda _{t}=\infty$, and (ii) $\limsup_{t\to \infty}\gamma _{t}\leq 0$ or $\sum^{\infty}_{t=1}|\lambda _{t}\gamma _{t}|<\infty$. Then $\lim_{t\to \infty}a_{t}=0$.

Lemma 2.5

(See [31, demiclosedness principle])

If S is nonexpansive with $\operatorname{Fix} (S)\neq \emptyset$, then $I-S$ is demiclosed at zero, i.e., if $\{x_{t}\}$ is a sequence in C such that $x_{t} \rightharpoonup x\in C$ and $(I-S)x_{t}\to 0$, then $(I-S)x=0$, where I is the identity mapping of $\mathcal {H}$.

Lemma 2.6

(See [6])

Let $\{{\boldsymbol{\Gamma}}_{s}\} \subset \mathbb{R}$ with $\exists \{{\boldsymbol{\Gamma}}_{s_{k}}\}\subset \{{\boldsymbol{\Gamma}}_{s}\}$ such that ${\boldsymbol{\Gamma}}_{s_{k}}<{\boldsymbol{\Gamma}}_{s_{k}+1} \ \forall k\geq 1$. Let $\{\phi (s)\}_{s\geq s_{0}}$ be formulated as

$$ \phi (s)=\max \{k\leq s:{\boldsymbol{\Gamma}}_{k}< {\boldsymbol{\Gamma}}_{k+1}\} $$

with $s_{0}\geq 1$ satisfying $\{k\leq s_{0}:{\boldsymbol{\Gamma}}_{k}<{\boldsymbol{\Gamma}}_{k+1}\}\neq \emptyset$. Then:

• (i) $\phi (s_{0})\leq \phi (s_{0}+1)\leq \cdots$ and $\phi (s)\to \infty$;

• (ii) ${\boldsymbol{\Gamma}}_{\phi (s)}\leq{\boldsymbol{\Gamma}}_{\phi (s)+1}$ and ${\boldsymbol{\Gamma}}_{s}\leq{\boldsymbol{\Gamma}}_{\phi (s)+1} \ \forall s\geq s_{0}$.

3 Convergence analysis

For the convergence analysis of our proposed rule for treating BSPVIP (1.6) with the CFPP constraint, we assume throughout that

• $T:{\mathcal {H}}_{1}\to{\mathcal {H}}_{2}$ is a nonzero bounded linear operator with the adjoint $T^{*}$, and $S:{\mathcal {H}}_{2}\to{\mathcal {H}}_{2}$ is τ-demimetric with $I-S$ being demiclosed at zero, where $\tau \in (-\infty , 1)$.

• $A:{\mathcal {H}}_{1}\to{\mathcal {H}}_{1}$ is a pseudomonotone and L-Lipschitz continuous mapping satisfying the condition: $\|Au\|\leq \liminf_{t\to \infty}\|Au_{t}\|$ for each $\{u_{t}\}\subset C$ with $u_{t}\rightharpoonup u$.

• $\{S_{i}\}^{N}_{i=1}$ is finitely many nonexpansive self-mappings on ${\mathcal {H}}_{1}$ such that $\Xi :=\bigcap^{N}_{i=1} \operatorname{Fix}(S_{i})\cap\Omega \neq \emptyset$ with $\Omega :=\{z\in\operatorname{VI}(C,A):Tz\in\operatorname{Fix}(S)\}$. In addition, when required, we write $S_{t}:=S_{t \operatorname{mod} N}$, $t = 1, 2, 3, \ldots$ .

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

• $\{\beta _{t}\},\{\gamma _{t}\},\{\varepsilon _{t}\} \subset (0, \infty )$ such that $\beta _{t}+\gamma _{t}<1$, $\sum^{\infty}_{t=1} \beta _{t}=\infty$, $\lim_{t\to \infty}\beta _{t}=0$, $0<\liminf_{t\to \infty}\gamma _{t} \leq \limsup_{t\to \infty}\gamma _{t}<1$ and $\varepsilon _{t}=o(\beta _{t})$.

Algorithm 3.1

(Triple-adaptive inertial subgradient extragradient rule)

Initialization: Let $\lambda _{1}>0$, $\epsilon >0$, $\sigma \geq 0$, $\mu \in (0,1)$, $\alpha \in [0,1)$, and $x_{0},x_{1}\in{\mathcal {H}}_{1}$ be arbitrary.

Iterative steps: Calculate $x_{t+1}$ as follows:

Step 1. Given the iterates $x_{t-1}$ and $x_{t}$ ($t\geq 1$), choose $\alpha _{t}$ such that $0\leq \alpha _{t}\leq \bar{\alpha}_{t}$, where

$$ \bar{\alpha}_{t}=\textstyle\begin{cases} \min \{\alpha ,\frac{\varepsilon _{t}}{ \Vert x_{t}-x_{t-1} \Vert }\} & \text{if }x_{t} \neq x_{t-1}, \\ \alpha & \text{otherwise}.\end{cases} $$

(3.1)

Step 2. Compute $w_{t}=S_{t}x_{t}+\alpha _{t}(S_{t}x_{t}-S_{t}x_{t-1})$ and $y_{t}=P_{C}(w_{t}-\lambda _{t}Aw_{t})$.

Step 3. Construct $C_{t}:=\{y\in{\mathcal {H}}_{1}:\langle w_{t}-\lambda _{t}Aw_{t}-y_{t},y_{t}-y \rangle \geq 0\}$, and compute $v_{t}=P_{C_{t}}(w_{t}-\lambda _{t}Ay_{t})$ and $z_{t}=v_{t}-\sigma _{t}T^{*}(I-S)Tv_{t}$.

Step 4. Calculate $x_{t+1}=\beta _{t}f(x_{t})+\gamma _{t}x_{t}+((1-\gamma _{t})I-\beta _{t} \rho F)z_{t}$ and update

$$ \lambda _{t+1}=\textstyle\begin{cases} \min \{\mu \frac{ \Vert w_{t}-y_{t} \Vert ^{2}+ \Vert v_{t}-y_{t} \Vert ^{2}}{2\langle Aw_{t}-Ay_{t},v_{t}-y_{t}\rangle}, \lambda _{t}\} &\text{if }\langle Aw_{t}-Ay_{t},v_{t}-y_{t}\rangle >0, \\ \lambda _{t} & \text{otherwise},\end{cases} $$

(3.2)

and for any fixed $\epsilon >0$, $\sigma _{t}$ is chosen to be the bounded sequence satisfying

$$ 0< \epsilon \leq \sigma _{t}\leq \frac{(1-\tau ) \Vert Tv_{t}-STv_{t} \Vert ^{2}}{ \Vert T^{*}(Tv_{t}-STv_{t}) \Vert ^{2}}- \epsilon \quad \text{if }Tv_{t}\neq STv_{t}, $$

(3.3)

otherwise set $\sigma _{t}=\sigma \geq 0$.

Set $t:=t+1$ and go to Step 1.

Remark 3.1

We have from (3.1) that $\lim_{t\to \infty}\frac{\alpha _{t}}{\beta _{t}}\|x_{t}-x_{t-1}\| =0$. Indeed, we have $\alpha _{t}\|x_{t}-x_{t-1}\|\leq \varepsilon _{t} \ \forall t\geq 1$, which together with $\lim_{t\to \infty} \frac{\varepsilon _{t}}{\beta _{t}}=0$ implies that $\frac{\alpha _{t}}{\beta _{t}}\|x_{t}-x_{t-1}\|\leq \frac{\varepsilon _{t}}{\beta _{t}}\to 0$. It is easy to see that $C_{t}$ is closed and convex. Furthermore, $C_{t} \neq \emptyset$ since $C \subset C_{t}$ and $C\neq \emptyset$. Hence, $\{v_{t}\}$ is well defined.

Lemma 3.1

The step size $\{\lambda _{t}\}$ is nonincreasing with $\lambda _{t}\geq \lambda :=\min \{\lambda _{1},\frac {\mu}{L}\} \ \forall t\geq 1$, and $\lim_{t\to \infty}\lambda _{t}\geq \lambda :=\min \{\lambda _{1}, \frac {\mu}{L}\}$.

Proof

By (3.2), we get $\lambda _{t}\geq \lambda _{t+1} \ \forall t\geq 1$. Now, observe that

$$ \left . \textstyle\begin{array}{rl} &\frac{1}{2}( \Vert w_{t}-y_{t} \Vert ^{2}+ \Vert v_{t}-y_{t} \Vert ^{2})\geq \Vert w_{t}-y_{t} \Vert \Vert v_{t}-y_{t} \Vert \\ & \langle Aw_{t}-Ay_{t},v_{t}-y_{t}\rangle \leq L \Vert w_{t}-y_{t} \Vert \Vert v_{t}-y_{t} \Vert \end{array}\displaystyle \right \}\Longrightarrow \lambda _{t+1}\geq \min \biggl\{ \lambda _{t}, \frac {\mu}{L} \biggr\} . $$

 □

We prove the following lemmas.

Lemma 3.2

The step size $\sigma _{t}$ formulated in (3.3) is well defined.

Proof

It suffices to show that $\|T^{*}(Tv_{t}-STv_{t})\|^{2}\neq 0$. Take $p\in\Xi$ arbitrarily. Since S is a τ-demimetric mapping, we obtain

$$ \begin{aligned} \Vert v_{t}-p \Vert \bigl\Vert T^{*}(Tv_{t}-STv_{t}) \bigr\Vert &\geq \bigl\langle v_{t}-p,T^{*}(Tv_{t}-STv_{t}) \bigr\rangle \\ &=\langle Tv_{t}-Tp,Tv_{t}-STv_{t}\rangle \\ &\geq \frac{1-\tau}{2} \Vert Tv_{t}-STv_{t} \Vert ^{2}.\end{aligned} $$

(3.4)

If $Tv_{t}\neq STv_{t}$, then $\|Tv_{t}-STv_{t}\|^{2}>0$. Thus, $\|T^{*}(Tv_{t}-STv_{t})\|^{2}>0$. □

Lemma 3.3

The sequences $\{w_{t}\}$, $\{y_{t}\}$, $\{v_{t}\}$ satisfy

$$ \Vert v_{t}-p \Vert ^{2}\leq \Vert w_{t}-p \Vert ^{2}- \biggl(1-\mu \frac {\lambda _{t}}{\lambda _{t+1}} \biggr) \Vert w_{t}-y_{t} \Vert ^{2}- \biggl(1-\mu \frac {\lambda _{t}}{\lambda _{t+1}} \biggr) \Vert v_{t}-y_{t} \Vert ^{2} \quad \forall p\in\Xi . $$

Proof

Observe that

$$ 2\langle Aw_{t}-Ay_{t},v_{t}-y_{t} \rangle \leq \frac{\mu}{\lambda _{t+1}} \Vert w_{t}-y_{t} \Vert ^{2}+ \frac {\mu}{\lambda _{t+1}} \Vert v_{t}-y_{t} \Vert ^{2} \quad \forall t\geq 1. $$

(3.5)

Note that (3.5) holds when $\langle Aw_{t}-Ay_{t},v_{t}-y_{t}\rangle \leq 0$. Conversely, we have (3.5) by (3.2). Also, $\forall \hat{p}\in\Xi \subset C\subset C_{t}$,

$$ \begin{aligned} \Vert v_{t}-\hat{p} \Vert ^{2}&= \bigl\Vert P_{C_{t}}(w_{t}-\lambda _{t}Ay_{t})-P_{C_{t}} \hat{p} \bigr\Vert ^{2} \\ &\leq \langle v_{t}-\hat{p},w_{t}-\lambda _{t}Ay_{t}- \hat{p}\rangle \\ &=\frac{1}{2} \Vert v_{t}-\hat{p} \Vert ^{2}+ \frac{1}{2} \Vert w_{t}-\hat{p} \Vert ^{2}- \frac{1}{2} \Vert v_{t}-w_{t} \Vert ^{2}-\langle v_{t}-\hat{p},\lambda _{t}Ay_{t} \rangle ,\end{aligned} $$

which hence yields

$$ \Vert v_{t}-\hat{p} \Vert ^{2}\leq \Vert w_{t}-\hat{p} \Vert ^{2}- \Vert v_{t}-w_{t} \Vert ^{2}-2 \langle v_{t}-\hat{p},\lambda _{t}Ay_{t}\rangle . $$

(3.6)

Since $\hat{p}\in\operatorname{VI}(C,A)$, we get $\langle A\hat{p},\breve{x}-\hat{p}\rangle \geq 0 \ \forall \breve{x} \in C$. Pseudomonotonicity of A implies $\langle Au,u-\hat{p}\rangle \geq 0 \ \forall u\in C$. Letting $u:=y_{t}\in C$ gives $\langle Ay_{t},\hat{p}-y_{t}\rangle \leq 0$. Thus,

$$ \langle Ay_{t},\hat{p}-v_{t}\rangle = \langle Ay_{t},\hat{p}-y_{t} \rangle +\langle Ay_{t},y_{t}-v_{t}\rangle \leq \langle Ay_{t},y_{t}-v_{t} \rangle . $$

(3.7)

Substituting (3.7) for (3.6), we obtain

$$ \Vert v_{t}-\hat{p} \Vert ^{2}\leq \Vert w_{t}-\hat{p} \Vert ^{2}- \Vert v_{t}-y_{t} \Vert ^{2}- \Vert y_{t}-w_{t} \Vert ^{2}+2\langle w_{t}-\lambda _{t}Ay_{t}-y_{t},v_{t}-y_{t} \rangle . $$

(3.8)

Since $v_{t}=P_{C_{t}}(w_{t}-\lambda _{t}Ay_{t})$, we have that $v_{t}\in C_{t}$, and hence

$$ \begin{aligned} 2\langle w_{t}-\lambda _{t}Ay_{t}-y_{t},v_{t}-y_{t} \rangle ={}&2 \langle w_{t}-\lambda _{t}Aw_{t}-y_{t},v_{t}-y_{t} \rangle \\ &{} +2\lambda _{t}\langle Aw_{t}-Ay_{t},v_{t}-y_{t} \rangle \\ \leq{}& 2\lambda _{t}\langle Aw_{t}-Ay_{t},v_{t}-y_{t} \rangle ,\end{aligned} $$

which together with (3.5) implies that

$$ 2\langle w_{t}-\lambda _{t}Ay_{t}-y_{t},v_{t}-y_{t} \rangle \leq \mu \frac {\lambda _{t}}{\lambda _{t+1}} \Vert w_{t}-y_{t} \Vert ^{2}+\mu \frac {\lambda _{t}}{\lambda _{t+1}} \Vert v_{t}-y_{t} \Vert ^{2}. $$

(3.9)

Therefore, substituting (3.9) for (3.8), the result follows. □

Lemma 3.4

$\{x_{t}\}$ is bounded.

Proof

First of all, we show that $P_{\Xi}(f+I-\rho F)$ is a contraction. Indeed, for any $x,y\in{\mathcal {H}}_{1}$, by Lemma 2.2, we have

$$\begin{aligned}& \bigl\Vert P_{\Xi}(f+I-\rho F)x-P_{\Xi}(f+I-\rho F)y \bigr\Vert \\& \quad \leq \bigl\Vert f(x)-f(y) \bigr\Vert + \bigl\Vert (I-\rho F)x-(I-\rho F)y \bigr\Vert \\& \quad \leq \nu \Vert x-y \Vert +(1-\zeta ) \Vert x-y \Vert =\bigl[1-(\zeta - \nu )\bigr] \Vert x-y \Vert , \end{aligned}$$

which implies that $P_{\Xi}(f+I-\rho F)$ is a contraction. Banach’s contraction mapping principle guarantees that $P_{\Xi}(f+I-\rho F)$ has a unique fixed point. Say $q^{*}\in{\mathcal {H}}_{1}$, i.e., $q^{*}=P_{\Xi}(f+I-\rho F) q^{*}$. Hence, there exists unique $q^{*}\in\Xi$ that solves

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

(3.10)

This also means that there exists a unique solution $q^{*}\in\Xi$ to BSPVIP (1.6) with the CFPP constraint.

Now, by the definition of $w_{t}$ in Algorithm 3.1, we have

$$ \begin{aligned} \bigl\Vert w_{t}-q^{*} \bigr\Vert & = \bigl\Vert S_{t}x_{t}+\alpha _{t}(S_{t}x_{t}-S_{t}x_{t-1})-q^{*} \bigr\Vert \\ &\leq \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t} \frac{\alpha _{t}}{\beta _{t}} \Vert x_{t}-x_{t-1} \Vert .\end{aligned} $$

From Remark 3.1, we know that $\lim_{t\to \infty}\frac{\alpha _{t}}{\beta _{t}}\|x_{t}-x_{t-1}\|=0$. This means that $\{\frac{\alpha _{t}}{\beta _{t}}\|x_{t}-x_{t-1}\|\}$ is bounded. Thus, $\exists M_{1}>0$ such that $\frac{\alpha _{t}}{\beta _{t}}\|x_{t}-x_{t-1}\|\leq M_{1} \ \forall t \geq 1$. Hence,

$$ \bigl\Vert w_{t}-q^{*} \bigr\Vert \leq \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}M_{1} \quad \forall t \geq 1. $$

(3.11)

From Step 3 of Algorithm 3.1, using the definition of $z_{t}$, we get

$$ \begin{aligned} \bigl\Vert z_{t}-q^{*} \bigr\Vert ^{2}={}& \bigl\Vert v_{t}-\sigma _{t}T^{*}(I-S)Tv_{t}-q^{*} \bigr\Vert ^{2} \\ ={}& \bigl\Vert v_{t}-q^{*} \bigr\Vert ^{2}-2 \sigma _{t}\bigl\langle v_{t}-q^{*},T^{*}(I-S)Tv_{t} \bigr\rangle \\ &{}+\sigma ^{2}_{t} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2} \\ ={}& \bigl\Vert v_{t}-q^{*} \bigr\Vert ^{2}-2 \sigma _{t}\bigl\langle T\bigl(v_{t}-q^{*} \bigr),(I-S)Tv_{t} \bigr\rangle \\ &{}+\sigma ^{2}_{t} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}.\end{aligned} $$

(3.12)

Since the operator S is τ-demimetric, from (3.12), we get

$$ \begin{aligned} \bigl\Vert z_{t}-q^{*} \bigr\Vert ^{2}&\leq \bigl\Vert v_{t}-q^{*} \bigr\Vert ^{2}-\sigma _{t}(1-\tau ) \bigl\Vert (I-S)Tv_{t} \bigr\Vert ^{2}+\sigma ^{2}_{t} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2} \\ &= \bigl\Vert v_{t}-q^{*} \bigr\Vert ^{2}+ \sigma _{t}\bigl[\sigma _{t} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}-(1- \tau ) \bigl\Vert (I-S)Tv_{t} \bigr\Vert ^{2}\bigr].\end{aligned} $$

(3.13)

However, from the step size $\sigma _{t}$ in (3.3), we get

$$ \sigma _{t}+\epsilon \leq \frac{(1-\tau ) \Vert Tv_{t}-STv_{t} \Vert ^{2}}{ \Vert T^{*}(I-S)Tv_{t} \Vert ^{2}} $$

if and only if

$$ \sigma _{t}\bigl(\sigma _{t} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}-(1-\tau ) \Vert Tv_{t}-STv_{t} \Vert ^{2}\bigr)\leq - \sigma _{t}\epsilon \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}. $$

(3.14)

Using $0<\epsilon \leq \sigma _{t}$ in (3.3), we have that $-\epsilon ^{2}\geq -\sigma _{t}\epsilon$, and hence

$$ -\sigma _{t}\epsilon \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}\leq -\epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}. $$

(3.15)

Combining (3.13), (3.14), and (3.15), we obtain

$$ \begin{aligned} \bigl\Vert z_{t}-q^{*} \bigr\Vert ^{2}&\leq \bigl\Vert v_{t}-q^{*} \bigr\Vert ^{2}-\sigma _{t}\epsilon \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2} \\ &\leq \bigl\Vert v_{t}-q^{*} \bigr\Vert ^{2}- \epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2} \\ &\leq \bigl\Vert v_{t}-q^{*} \bigr\Vert ^{2}.\end{aligned} $$

(3.16)

In addition, by Lemma 3.1, we have $\lim_{t\to \infty}\lambda _{t}\geq \lambda :=\min \{\lambda _{1}, \frac{\mu}{L}\}$, which leads to $\lim_{t\to \infty}(1-\mu \frac{\lambda _{t}}{\lambda _{t+1}})=1- \mu >0$. Without loss of generality, we may assume that $1-\mu \frac{\lambda _{t}}{\lambda _{t+1}}>0 \ \forall t\geq 1$. Thus, by Lemma 3.3, we get

$$ \begin{aligned} \bigl\Vert v_{t}-q^{*} \bigr\Vert ^{2}\leq {}& \bigl\Vert w_{t}-q^{*} \bigr\Vert ^{2}-\biggl(1-\mu \frac{\lambda _{t}}{\lambda _{t+1}}\biggr) \Vert w_{t}-y_{t} \Vert ^{2} \\ &{} -\biggl(1-\mu \frac{\lambda _{t}}{\lambda _{t+1}}\biggr) \Vert v_{t}-y_{t} \Vert ^{2} \\ \leq {}& \bigl\Vert w_{t}-q^{*} \bigr\Vert ^{2}.\end{aligned} $$

(3.17)

Combining (3.11), (3.16), and (3.17), we obtain

$$ \bigl\Vert z_{t}-q^{*} \bigr\Vert \leq \bigl\Vert v_{t}-q^{*} \bigr\Vert \leq \bigl\Vert w_{t}-q^{*} \bigr\Vert \leq \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}M_{1} \quad \forall t\geq 1. $$

(3.18)

Since $\beta _{t}+\gamma _{t}<1 \ \forall t\geq 1$, we get $\frac{\beta _{t}}{1-\gamma _{t}}<1 \ \forall t\geq 1$. So, from Lemma 2.2 and (3.18) it follows that

$$\begin{aligned} \bigl\Vert x_{t+1}-q^{*} \bigr\Vert = {}& \bigl\Vert \beta _{t}f(x_{t})+\gamma _{t}x_{t}+ \bigl((1- \gamma _{t})I-\beta _{t}\rho F \bigr)z_{t}-q^{*} \bigr\Vert \\ \leq {}& \beta _{t} \bigl\Vert f(x_{t})-q^{*} \bigr\Vert +\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert \\ & {}+(1-\beta _{t}-\gamma _{t}) \biggl\Vert \biggl( \frac{1-\gamma _{t}}{1-\beta _{t}-\gamma _{t}}I - \frac{\beta _{t}}{1-\beta _{t}-\gamma _{t}}\rho F\biggr)z_{t}-q^{*} \biggr\Vert \\ \leq {}& \beta _{t}\bigl( \bigl\Vert f(x_{t})-f \bigl(q^{*}\bigr) \bigr\Vert + \bigl\Vert f\bigl(q^{*} \bigr)-q^{*} \bigr\Vert \bigr)+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert \\ & {}+(1-\beta _{t}-\gamma _{t}) \biggl\Vert \biggl( \frac{1-\gamma _{t}}{1-\beta _{t}-\gamma _{t}}I - \frac{\beta _{t}}{1-\beta _{t}-\gamma _{t}}\rho F\biggr)z_{t}-q^{*} \biggr\Vert \\ \leq {}& \beta _{t}\bigl(\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert + \bigl\Vert f\bigl(q^{*}\bigr)-q^{*} \bigr\Vert \bigr)+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert \\ & {}+(1-\gamma _{t}) \biggl\Vert \biggl(I-\frac{\beta _{t}}{1-\gamma _{t}}\rho F \biggr)z_{t}-\biggl(1- \frac{\beta _{t}}{1-\gamma _{t}}\biggr)q^{*} \biggr\Vert \\ = {}& \beta _{t}\bigl(\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert + \bigl\Vert f\bigl(q^{*}\bigr)-q^{*} \bigr\Vert \bigr)+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert \\ & {}+(1-\gamma _{t}) \biggl\Vert \biggl(I-\frac{\beta _{t}}{1-\gamma _{t}}\rho F \biggr)z_{t}-\biggl(I- \frac{\beta _{t}}{1-\gamma _{t}}\rho F\biggr)q^{*}+ \frac{\beta _{t}}{1-\gamma _{t}}(I-\rho F)q^{*} \biggr\Vert \\ \leq {}& \beta _{t}\bigl(\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert + \bigl\Vert f\bigl(q^{*}\bigr)-q^{*} \bigr\Vert \bigr)+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert \\ & {}+(1-\gamma _{t})\biggl[\biggl(1-\frac{\beta _{t}}{1-\gamma _{t}}\zeta \biggr) \bigl\Vert z_{t}-q^{*} \bigr\Vert +\frac{\beta _{t}}{1-\gamma _{t}} \bigl\Vert (I-\rho F)q^{*} \bigr\Vert \biggr] \\ = {}& \beta _{t}\bigl(\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert + \bigl\Vert f\bigl(q^{*}\bigr)-q^{*} \bigr\Vert \bigr)+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert \\ & {}+(1-\gamma _{t}-\beta _{t}\zeta ) \bigl\Vert z_{t}-q^{*} \bigr\Vert +\beta _{t} \bigl\Vert (I- \rho F)q^{*} \bigr\Vert \\ \leq {}& \beta _{t}\bigl(\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert + \bigl\Vert f\bigl(q^{*}\bigr)-q^{*} \bigr\Vert \bigr)+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert \\ & {}+(1-\gamma _{t}-\beta _{t}\zeta ) \bigl( \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}M_{1} \bigr)+ \beta _{t} \bigl\Vert (I-\rho F)q^{*} \bigr\Vert \\ \leq {}& \bigl[1-\beta _{t}(\zeta -\nu )\bigr] \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t} \bigl(M_{1}+ \bigl\Vert f\bigl(q^{*}\bigr)-q^{*} \bigr\Vert + \bigl\Vert (I-\rho F)q^{*} \bigr\Vert \bigr) \\ = {}& \bigl[1-\beta _{t}(\zeta -\nu )\bigr] \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}(\zeta - \nu ) \frac{M_{1}+ \Vert f(q^{*})-q^{*} \Vert + \Vert (I-\rho F)q^{*} \Vert }{\zeta -\nu} \\ \leq {}& \max \biggl\{ \bigl\Vert x_{t}-q^{*} \bigr\Vert , \frac {M_{1}+ \Vert f(q^{*})-q^{*} \Vert + \Vert (I-\rho F)q^{*} \Vert }{\zeta -\nu} \biggr\} . \end{aligned}$$

Thus, $\|x_{t}-q^{*}\|\leq \max \{\|x_{1}-q^{*}\|, \frac{M_{1}+\|f(q^{*})-q^{*}\|+\|(I-\rho F)q^{*}\|}{\zeta -\nu} \}$ for all $t\geq 1$. Thus, $\{x_{t}\}$ is bounded, and so are the sequences $\{v_{t}\}$, $\{w_{t}\}$, $\{y_{t}\}$, $\{z_{t}\}$, $\{f(x_{t})\}$, $\{Fz_{t}\}$, $\{S_{t}x_{t}\}$. □

Lemma 3.5

Let $\{v_{t}\}$, $\{w_{t}\}$, $\{x_{t}\}$, $\{y_{t}\}$, $\{z_{t}\}$ be the sequences generated by Algorithm 3.1. Suppose that $x_{t}-x_{t+1}\to 0$, $w_{t}-x_{t}\to 0$, $w_{t}-y_{t}\to 0$, and $v_{t}-z_{t}\to 0$. Then $\omega _{w}(\{x_{t}\})\subset\Xi$ with $\omega _{w}(\{x_{t}\})=\{z\in{\mathcal {H}}_{1}:x_{t_{k}} \rightharpoonup z\textit{ for some }\{x_{t_{k}}\}\subset \{x_{t}\}\}$.

Proof

Take an arbitrary fixed $z\in \omega _{w}(\{x_{t}\})$. Then $\exists \{x_{t_{k}}\}\subset \{x_{t}\}$ such that $x_{t_{k}} \rightharpoonup z\in{\mathcal {H}}_{1}$. Thanks to $w_{t}-x_{t}\to 0$, by which $\exists \{w_{t_{k}}\}\subset \{w_{t}\}$ such that $w_{t_{k}}\rightharpoonup z\in{\mathcal {H}}_{1}$. In what follows, we claim that $z\in\Xi$. In fact, from Algorithm 3.1, we get $w_{t}-x_{t}=S_{t}x_{t}-x_{t}+\alpha _{t}(S_{t}x_{t}-S_{t}x_{t-1}) \ \forall t\geq 1$, and hence

$$ \begin{aligned} \Vert S_{t}x_{t}-x_{t} \Vert &= \bigl\Vert w_{t}-x_{t}-\alpha _{t}(S_{t}x_{t}-S_{t}x_{t-1}) \bigr\Vert \\ &\leq \Vert w_{t}-x_{t} \Vert +\alpha _{t} \Vert S_{t}x_{t}-S_{t}x_{t-1} \Vert \\ &\leq \Vert w_{t}-x_{t} \Vert +\beta _{t} \frac{\alpha _{t}}{\beta _{t}} \Vert x_{t}-x_{t-1} \Vert .\end{aligned} $$

Using Remark 3.1 and the assumption $w_{t}-x_{t}\to 0$, we have

$$ \lim_{t\to \infty} \Vert x_{t}-S_{t}x_{t} \Vert =0. $$

(3.19)

Also, from $y_{t}=P_{C}(w_{t}-\lambda _{t}Aw_{t})$, we have $\langle w_{t}-\lambda _{t}Aw_{t}-y_{t},y_{t}-y\rangle \geq 0 \ \forall y\in C$, and hence

$$ \frac {1}{\lambda _{t}}\langle w_{t}-y_{t},v-y_{t} \rangle +\langle Aw_{t},y_{t}-w_{t} \rangle \leq \langle Aw_{t},v-w_{t}\rangle \quad \forall v\in C. $$

(3.20)

Observe that $\lambda _{t}\geq \min \{\lambda _{1},\frac{\mu}{L}\}$. So, from (3.20), we get $\liminf_{k\to \infty}\langle Aw_{t_{k}},y-w_{t_{k}}\rangle \geq 0 \ \forall y\in C$. In the meantime, observe that $\langle Ay_{t},y-y_{t}\rangle =\langle Ay_{t}-Aw_{t},y-w_{t}\rangle + \langle Aw_{t},y-w_{t}\rangle +\langle Ay_{t},w_{t}-y_{t}\rangle$. Since $w_{t}-y_{t}\to 0$, we obtain $Aw_{t}-Ay_{t}\to 0$, which together with (3.20) arrives at $\liminf_{k\to \infty}\langle Ay_{t_{k}},v-y_{t_{k}}\rangle \geq 0 \ \forall v\in C$.

For $i=1,2,\ldots ,N$,

$$ \begin{aligned} \Vert x_{t}-S_{t+i}x_{t} \Vert &\leq \Vert x_{t}-x_{t+i} \Vert + \Vert x_{t+i}-S_{t+i}x_{t+i} \Vert + \Vert S_{t+i}x_{t+i}-S_{t+i}x_{t} \Vert \\ &\leq 2 \Vert x_{t}-x_{t+i} \Vert + \Vert x_{t+i}-S_{t+i}x_{t+i} \Vert .\end{aligned} $$

Hence, from (3.19) and the assumption $x_{t}-x_{t+1}\to 0$, we get $\lim_{t\to \infty}\|x_{t}-S_{t+i}x_{t}\|=0$ for $i=1,2,\ldots ,N$. This immediately implies that

$$ \lim_{t\to \infty} \Vert x_{t}-S_{l}x_{t} \Vert =0\quad \text{for }l=1,2,\ldots ,N. $$

(3.21)

Pick $\{\varsigma _{k}\}\subset (0,1)$, $\varsigma _{k}\downarrow 0$. For all $k\geq 1$, let $m_{k}$ be the smallest positive integer such that

$$ \langle Ay_{t_{k}},y-y_{t_{k}}\rangle +\varsigma _{k}\geq 0 \quad \forall k\geq m_{k}. $$

(3.22)

Since $\{\varsigma _{k}\}$ is nonincreasing, it is clear that $\{m_{k}\}$ is nondecreasing.

Again from the assumption on A, we know that $\liminf_{k\to \infty}\|Ay_{t_{k}}\|\geq \|Az\|$. If $Az=0$, then z is a solution, i.e., $z\in\operatorname{VI}(C,A)$. Let $Az\neq 0$. Then we have $0<\|Az\|\leq \liminf_{k\to \infty}\|Ay_{t_{k}}\|$. Without loss of generality, we may assume that $Ay_{t_{k}}\neq 0 \ \forall k\geq 1$. Noticing $\{y_{m_{k}}\}\subset \{y_{t_{k}}\}$ and $Ay_{t_{k}}\neq 0 \ \forall k\geq 1$, set $u_{m_{k}}=\frac{Ay_{m_{k}}}{\|Ay_{m_{k}}\|^{2}}$, and then $\langle Ay_{m_{k}},u_{m_{k}}\rangle =1 \ \forall k\geq 1$. So, from (3.22), we get $\langle Ay_{m_{k}},y+\varsigma _{k}u_{m_{k}}-y_{m_{k}}\rangle \geq 0 \ \forall k\geq 1$. By the pseudomonotonicity of A, we obtain $\langle A(y+\varsigma _{k}u_{m_{k}}),y+\varsigma _{k}u_{m_{k}}-y_{m_{k}} \rangle \geq 0 \ \forall k\geq 1$. This immediately yields

$$ \langle Ay,y-y_{m_{k}}\rangle \geq \bigl\langle Ay-A(y+ \varsigma _{k}u_{m_{k}}),y+ \varsigma _{k}u_{m_{k}}-y_{m_{k}} \bigr\rangle -\varsigma _{k}\langle Ay,u_{m_{k}} \rangle \quad \forall k\geq 1. $$

(3.23)

From $x_{t_{k}}\rightharpoonup z$ and $x_{t}-y_{t}\to 0$ (due to $w_{t}-x_{t}\to 0$ and $w_{t}-y_{t}\to 0$), we obtain $y_{t_{k}}\rightharpoonup z$. So, $\{y_{t}\}\subset C$ guarantees $z\in C$. Since $\{y_{m_{k}}\}\subset \{y_{t_{k}}\}$ and $\varsigma _{k}\downarrow 0$, we have $0\leq \limsup_{k\to \infty}\|\varsigma _{k}u_{m_{k}}\|=\limsup_{k \to \infty}\frac{\varsigma _{k}}{\|Ay_{m_{k}}\|} \leq \frac{\limsup_{k\to \infty}\varsigma _{k}}{\liminf_{k\to \infty}\|Ay_{t_{k}}\|}=0$. Hence, we get $\varsigma _{k}u_{m_{k}}\to 0$.

Next, we show that $z\in\Xi$. Indeed, using (3.21), we have $x_{t_{k}}-S_{l}x_{t_{k}}\to 0$ for $l=1,2,\ldots ,N$. By Lemma 2.5, $I-S_{l}$ is demiclosed at zero for $l=1,2,\ldots ,N$. Thus, from $x_{t_{k}}\rightharpoonup z$, we get $z\in\operatorname{Fix}(S_{l})$. Since l is an arbitrary element in the finite set $\{1,2,\ldots ,N\}$, it follows that $z\in \bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})$. Also, letting $k\to \infty$, we have that the right-hand side of (3.23) tends to zero. Thus, $\langle A\vec{y},\vec{y}-z\rangle =\liminf_{k\to \infty}\langle A \vec{y},\vec{y}-y_{m_{k}}\rangle \geq 0 \ \forall \vec{y}\in C$. By Lemma 2.3 we have $z\in\operatorname{VI}(C,A)$. Furthermore, we claim $Tz\in\operatorname{Fix}(S)$. In fact, noticing $z_{t}=v_{t}-\sigma _{t}T^{*}(I-S)Tv_{t}$, from $0<\epsilon \leq \sigma _{t}$ and $v_{t}-z_{t}\to 0$, we get

$$ \epsilon \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert \leq \sigma _{t} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert = \Vert v_{t}-z_{t} \Vert \to 0 \quad (t\to \infty ), $$

which together with the τ-demimetricness of S leads to

$$ \begin{aligned} \frac{1-\tau}{2} \bigl\Vert (I-S)Tv_{t} \bigr\Vert ^{2}&\leq \bigl\langle (I-S)Tv_{t},T\bigl(v_{t}-q^{*}\bigr) \bigr\rangle \\ &\leq \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert \bigl\Vert v_{t}-q^{*} \bigr\Vert \to 0 \quad (t\to \infty ).\end{aligned} $$

(3.24)

Noticing $x_{t+1}=\beta _{t}f(x_{t})+\gamma _{t}x_{t}+((1-\gamma _{t})I-\beta _{t} \rho F)z_{t}$, we have

$$ \begin{aligned} (1-\gamma _{t}) \Vert z_{t}-x_{t} \Vert &= \bigl\Vert x_{t+1}-x_{t}-\beta _{t} \bigl(f(x_{t})- \rho Fz_{t}\bigr) \bigr\Vert \\ &\leq \Vert x_{t+1}-x_{t} \Vert +\beta _{t} \bigl( \bigl\Vert f(x_{t}) \bigr\Vert + \Vert \rho Fz_{t} \Vert \bigr).\end{aligned} $$

Since $0<\liminf_{t\to \infty}(1-\gamma _{t})$, $x_{t}-x_{t+1}\to 0$ and $\beta _{t}\to 0$, from the boundedness of $\{x_{t}\}$ and $\{z_{t}\}$, we get $\lim_{t\to \infty}\|z_{t}-x_{t}\|=0$, which hence yields

$$ \Vert v_{t}-x_{t} \Vert \leq \Vert v_{t}-z_{t} \Vert + \Vert z_{t}-x_{t} \Vert \to 0 \quad (t\to \infty ). $$

From $x_{t_{k}}\rightharpoonup z$, we get $v_{t_{k}}\rightharpoonup z$. It follows that $Tv_{t_{k}}\rightharpoonup Tz$. From (3.24) one derives $Tz\in\operatorname{Fix}(S)$. Therefore, $z\in \bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\Omega =\Xi$. This completes the proof. □

Theorem 3.1

$\{x_{t}\}$ generated by Algorithm 3.1converges strongly to the unique solution $q^{*}\in\Xi$ of BSPVIP (1.6) with the CFPP constraint.

Proof

First of all, in terms of Lemma 3.4 we obtain that $\{x_{t}\}$ is bounded. From its proof we know that there exists a unique solution $q^{*}\in\Xi$ of BSPVIP (1.6) with the CFPP constraint, i.e., VIP (3.10) has a unique solution $q^{*}\in\Xi$.

Step 1. We claim that

$$ \begin{aligned} &(1-\beta _{t}\zeta -\gamma _{t}) \biggl[\biggl(1-\mu \frac{\lambda _{t}}{\lambda _{t+1}}\biggr) \bigl( \Vert w_{t}-y_{t} \Vert ^{2} + \Vert v_{t}-y_{t} \Vert ^{2}\bigr)+\epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}\biggr] \\ &\quad \leq \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}- \bigl\Vert x_{t+1}-q^{*} \bigr\Vert ^{2}+\beta _{t}M_{4}\end{aligned} $$

for some $M_{4}>0$. Also

$$ \begin{aligned} x_{t+1}-q^{*} = {}& \beta _{t}\bigl(f(x_{t})-q^{*}\bigr)+\gamma _{t}\bigl(x_{t}-q^{*}\bigr)+(1- \beta _{t}-\gamma _{t})\biggl\{ \frac{1-\gamma _{t}}{ 1-\beta _{t}-\gamma _{t}}\biggl[ \biggl(I-\frac{\beta _{t}}{1-\gamma _{t}}\rho F\biggr)z_{t} \\ & {} -\biggl(I-\frac{\beta _{t}}{1-\gamma _{t}}\rho F\biggr)q^{*}\biggr]+ \frac{\beta _{t}}{1-\beta _{t}-\gamma _{t}}(I-\rho F)q^{*}\biggr\} \\ ={}& \beta _{t}\bigl(f(x_{t})-f\bigl(q^{*}\bigr) \bigr)+\gamma _{t}\bigl(x_{t}-q^{*}\bigr) \\ &{} +(1-\gamma _{t})\biggl[\biggl(I-\frac{\beta _{t}}{1-\gamma _{t}}\rho F \biggr)z_{t} -\biggl(I- \frac{\beta _{t}}{1-\gamma _{t}}\rho F\biggr)q^{*} \biggr] + \beta _{t}(f-\rho F)q^{*}.\end{aligned} $$

Using Lemma 2.2, we get

$$\begin{aligned} \bigl\Vert x_{t+1}-q^{*} \bigr\Vert ^{2} \leq {}& \biggl\Vert \beta _{t}\bigl(f(x_{t})-f \bigl(q^{*}\bigr)\bigr)+ \gamma _{t}\bigl(x_{t}-q^{*} \bigr) \end{aligned}$$

(3.25)

$$\begin{aligned} & {}+ (1-\gamma _{t})\biggl[\biggl(I-\frac{\beta _{t}}{1-\gamma _{t}}\rho F \biggr)z_{t}-\biggl(I- \frac{\beta _{t}}{1-\gamma _{t}}\rho F\biggr)q^{*} \biggr] \biggr\Vert ^{2} \\ & {}+2\beta _{t}\bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ \leq {}& \biggl[\beta _{t}\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert +\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert +(1- \gamma _{t}) \biggl(1-\frac{\beta _{t}}{1-\gamma _{t}}\zeta \biggr) \bigl\Vert z_{t}-q^{*} \bigr\Vert \biggr]^{2} \\ & {}+2\beta _{t}\bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ = {}& \bigl[\beta _{t}\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert +\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert +(1- \beta _{t}\zeta -\gamma _{t}) \bigl\Vert z_{t}-q^{*} \bigr\Vert \bigr]^{2} \\ & {}+ 2\beta _{t}\bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ \leq {}& \beta _{t}\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+(1- \beta _{t}\zeta -\gamma _{t}) \bigl\Vert z_{t}-q^{*} \bigr\Vert ^{2} \\ & {}+2\beta _{t}\bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ \leq {}& \beta _{t}\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+(1- \beta _{t}\zeta -\gamma _{t}) \bigl\Vert z_{t}-q^{*} \bigr\Vert ^{2} \\ & {}+\beta _{t}M_{2} \end{aligned}$$

(3.26)

(due to $\beta _{t}\nu +\gamma _{t}+(1-\beta _{t}\zeta -\gamma _{t})=1-\beta _{t}( \zeta -\nu )\leq 1$), where $\sup_{t\geq 1}2\|(f-\rho F)q^{*}\|\|x_{t}-q^{*}\|\leq M_{2}$ for some $M_{2}>0$. Substituting (3.16) for (3.25), by Lemma 3.3 we get

$$\begin{aligned} \bigl\Vert x_{t+1}-q^{*} \bigr\Vert ^{2}\leq {}&\beta _{t}\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2} +(1-\beta _{t}\zeta -\gamma _{t})\bigl[ \bigl\Vert v_{t}-q^{*} \bigr\Vert ^{2} \\ & {}-\epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}\bigr]+\beta _{t}M_{2} \\ \leq {}& \beta _{t}\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+(1- \beta _{t}\zeta -\gamma _{t})\biggl[ \bigl\Vert w_{t}-q^{*} \bigr\Vert ^{2} \\ & {}- \biggl(1-\mu \frac {\lambda _{t}}{\lambda _{t+1}} \biggr) \bigl( \Vert w_{t}-y_{t} \Vert ^{2}+ \Vert v_{t}-y_{t} \Vert ^{2}\bigr) \\ &{} -\epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}\biggr]+\beta _{t}M_{2}. \end{aligned}$$

(3.27)

Also, from (3.18) we have

$$ \begin{aligned} \bigl\Vert w_{t}-q^{*} \bigr\Vert ^{2}&\leq \bigl( \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}M_{1}\bigr)^{2} \\ &= \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+ \beta _{t}\bigl(2M_{1} \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}M^{2}_{1}\bigr) \\ &\leq \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+ \beta _{t}M_{3},\end{aligned} $$

(3.28)

where $\sup_{t\geq 1}(2M_{1}\|x_{t}-q^{*}\|+\beta _{t}M^{2}_{1})\leq M_{3}$ for some $M_{3}>0$. Combining (3.27) and (3.28), we obtain

$$\begin{aligned} \bigl\Vert x_{t+1}-q^{*} \bigr\Vert ^{2} \leq {}& \beta _{t}\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+ \gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2} \\ & {}+(1-\beta _{t}\zeta -\gamma _{t})\bigl[ \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+\beta _{t}M_{3}\bigr] \\ & {}-(1-\beta _{t}\zeta -\gamma _{t})\biggl[\biggl(1-\mu \frac {\lambda _{t}}{\lambda _{t+1}}\biggr) \bigl( \Vert w_{t}-y_{t} \Vert ^{2}+ \Vert v_{t}-y_{t} \Vert ^{2} \bigr) \\ & {}+\epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}\biggr]+\beta _{t}M_{2} \\ \leq{} & \bigl[1-\beta _{t}(\zeta -\nu )\bigr] \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}-(1-\beta _{t} \zeta -\gamma _{t}) \biggl[\biggl(1-\mu \frac {\lambda _{t}}{\lambda _{t+1}}\biggr) \bigl( \Vert w_{t}-y_{t} \Vert ^{2} \\ & {}+ \Vert v_{t}-y_{t} \Vert ^{2}\bigr) + \epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2} \biggr]+ \beta _{t}M_{4} \\ \leq{} & \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}-(1-\beta _{t}\zeta -\gamma _{t})\biggl[ \biggl(1-\mu \frac {\lambda _{t}}{\lambda _{t+1}}\biggr) \bigl( \Vert w_{t}-y_{t} \Vert ^{2}+ \Vert v_{t}-y_{t} \Vert ^{2}\bigr) \\ & {}+\epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}\biggr]+\beta _{t}M_{4}, \end{aligned}$$

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

$$ \begin{aligned} &(1-\beta _{t}\zeta -\gamma _{t})\biggl[\biggl(1-\mu \frac{\lambda _{t}}{\lambda _{t+1}}\biggr) \bigl( \Vert w_{t}-y_{t} \Vert ^{2}+ \Vert v_{t}-y_{t} \Vert ^{2}\bigr) +\epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}\biggr] \\ &\quad \leq \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}- \bigl\Vert x_{t+1}-q^{*} \bigr\Vert ^{2}+\beta _{t}M_{4}.\end{aligned} $$

(3.29)

Step 2. We claim that

$$ \begin{aligned} \bigl\Vert x_{t+1}-q^{*} \bigr\Vert ^{2}\leq{}& \bigl[1-\beta _{t}(\zeta -\nu )\bigr] \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2} + \beta _{t}(\zeta -\nu )\biggl[\frac{2}{\zeta -\nu}\bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ & {} +\frac{M}{\zeta -\nu}\cdot \frac{\alpha _{t}}{\beta _{t}}\cdot \Vert x_{t}-x_{t-1} \Vert \biggr]\end{aligned} $$

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

$$ \begin{aligned} \bigl\Vert w_{t}-q^{*} \bigr\Vert ^{2}&\leq \bigl[ \bigl\Vert x_{t}-q^{*} \bigr\Vert +\alpha _{t} \Vert x_{t}-x_{t-1} \Vert \bigr]^{2} \\ &\leq \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+ \alpha _{t} \Vert x_{t}-x_{t-1} \Vert \bigl[2 \bigl\Vert x_{t}-q^{*} \bigr\Vert +\alpha _{t} \Vert x_{t}-x_{t-1} \Vert \bigr].\end{aligned} $$

(3.30)

Combining (3.18), (3.25), and (3.30), we have

$$\begin{aligned} \bigl\Vert x_{t+1}-q^{*} \bigr\Vert ^{2} \leq &\beta _{t}\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+ \gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2} \\ & {}+(1-\beta _{t}\zeta -\gamma _{t}) \bigl\Vert z_{t}-q^{*} \bigr\Vert ^{2} +2\beta _{t} \bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ \leq &\beta _{t}\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+(1- \beta _{t}\zeta -\gamma _{t}) \bigl\Vert w_{t}-q^{*} \bigr\Vert ^{2} \\ & {}+2\beta _{t}\bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ \leq & \beta _{t}\nu \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+\gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+(1- \beta _{t}\zeta -\gamma _{t})\bigl\{ \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2} \\ & {}+\alpha _{t} \Vert x_{t}-x_{t-1} \Vert \bigl[2 \bigl\Vert x_{t}-q^{*} \bigr\Vert +\alpha _{t} \Vert x_{t}-x_{t-1} \Vert \bigr]\bigr\} \\ & {}+2\beta _{t}\bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ \leq &\bigl[1-\beta _{t}(\zeta -\nu )\bigr] \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+\alpha _{t} \Vert x_{t}-x_{t-1} \Vert \bigl[2 \bigl\Vert x_{t}-q^{*} \bigr\Vert +\alpha _{t} \Vert x_{t}-x_{t-1} \Vert \bigr] \\ & {}+2\beta _{t}\bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ \leq &\bigl[1-\beta _{t}(\zeta -\nu )\bigr] \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+\alpha _{t} \Vert x_{t}-x_{t-1} \Vert M \\ & {}+ 2\beta _{t}\bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ = &\bigl[1-\beta _{t}(\zeta -\nu )\bigr] \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+\beta _{t}(\zeta - \nu )\cdot \biggl[ \frac{2\langle (f-\rho F)q^{*},x_{t+1}-q^{*}\rangle}{\zeta -\nu} \\ & {}+\frac{M}{\zeta -\nu}\cdot \frac{\alpha _{t}}{\beta _{t}}\cdot \Vert x_{t}-x_{t-1} \Vert \biggr], \end{aligned}$$

(3.31)

where $\sup_{t\geq 1}\{2\|x_{t}-q^{*}\|+\alpha _{t}\|x_{t}-x_{t-1}\|\} \leq M$.

Step 3. We show that $\{x_{t}\}$ converges strongly to $q^{*}\in\Xi$. Put ${\boldsymbol{\Gamma}}_{t}=\|x_{t}-q^{*}\|^{2}$.

Case 1. Assume that integer $t_{0}\geq 1$ with $\{{\boldsymbol{\Gamma}}_{t}\}_{t\geq t_{0}}$ is nonincreasing. Then $\lim_{t\to \infty}{\boldsymbol{\Gamma}}_{t}=d<+\infty$, $\lim_{t\to \infty}({\boldsymbol{\Gamma}}_{t}-{\boldsymbol{\Gamma}}_{t+1})=0$. By (3.29), one obtains

$$ \begin{aligned} &(1-\beta _{t}\zeta -\gamma _{t}) \biggl[\biggl(1-\mu \frac{\lambda _{t}}{\lambda _{t+1}}\biggr) \bigl( \Vert w_{t}-y_{t} \Vert ^{2}+ \Vert v_{t}-y_{t} \Vert ^{2}\bigr) +\epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert ^{2}\biggr] \\ &\quad \leq \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}- \bigl\Vert x_{t+1}-q^{*} \bigr\Vert ^{2}+\beta _{t}M_{4}={ \boldsymbol{\Gamma}}_{t}-{\boldsymbol{\Gamma}}_{t+1}+\beta _{t}M_{4}.\end{aligned} $$

Since $\lim_{t\to \infty}(1-\mu \frac{\lambda _{t}}{\lambda _{t+1}})=1- \mu >0$, $\liminf_{t\to \infty}(1-\gamma _{t})>0$, $\beta _{t}\to 0$, and ${\boldsymbol{\Gamma}}_{t}-{\boldsymbol{\Gamma}}_{t+1}\to 0$, one has

$$ \lim_{t\to \infty} \Vert w_{t}-y_{t} \Vert =\lim_{t\to \infty} \Vert v_{t}-y_{t} \Vert = \lim_{t\to \infty} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert =0. $$

(3.32)

Noticing $z_{t}=v_{t}-\sigma _{t}T^{*}(I-S)Tv_{t}$ and the boundedness of $\{\sigma _{t}\}$, from (3.32) we get

$$ \Vert v_{t}-z_{t} \Vert =\sigma _{t} \bigl\Vert T^{*}(I-S)Tv_{t} \bigr\Vert \to 0 \quad (t\to \infty ), $$

(3.33)

and hence

$$ \Vert w_{t}-z_{t} \Vert \leq \Vert w_{t}-y_{t} \Vert + \Vert y_{t}-v_{t} \Vert + \Vert v_{t}-z_{t} \Vert \to 0 \quad (t\to \infty ). $$

(3.34)

Moreover, noticing $x_{t+1}-q^{*}=\gamma _{t}(x_{t}-q^{*})+(1-\gamma _{t})(z_{t}-q^{*})+ \beta _{t}(f(x_{t})-\rho Fz_{t})$, we obtain from (3.18) that

$$\begin{aligned} \bigl\Vert x_{t+1}-q^{*} \bigr\Vert ^{2} =& \bigl\Vert \gamma _{t}\bigl(x_{t}-q^{*} \bigr)+(1-\gamma _{t}) \bigl(z_{t}-q^{*}\bigr)+ \beta _{t}\bigl(f(x_{t})-\rho Fz_{t}\bigr) \bigr\Vert ^{2} \\ \leq& \bigl\Vert \gamma _{t}\bigl(x_{t}-q^{*} \bigr)+(1-\gamma _{t}) \bigl(z_{t}-q^{*}\bigr) \bigr\Vert ^{2} \\ &{}+2\bigl\langle \beta _{t}\bigl(f(x_{t})-\rho Fz_{t}\bigr),x_{t+1}-q^{*}\bigr\rangle \\ \leq& \gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+(1-\gamma _{t}) \bigl\Vert z_{t}-q^{*} \bigr\Vert ^{2}- \gamma _{t}(1-\gamma _{t}) \Vert x_{t}-z_{t} \Vert ^{2} \\ &{} +2 \bigl\Vert \beta _{t}\bigl(f(x_{t})-\rho Fz_{t}\bigr) \bigr\Vert \bigl\Vert x_{t+1}-q^{*} \bigr\Vert \\ \leq& \gamma _{t} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+(1-\gamma _{t}) \bigl\Vert z_{t}-q^{*} \bigr\Vert ^{2}- \gamma _{t}(1-\gamma _{t}) \Vert x_{t}-z_{t} \Vert ^{2} \\ &{} +2\beta _{t}\bigl( \bigl\Vert f(x_{t}) \bigr\Vert + \Vert \rho Fz_{t} \Vert \bigr) \bigl\Vert x_{t+1}-x^{*} \bigr\Vert \\ \leq& \gamma _{t}\bigl( \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}M_{1}\bigr)^{2}+(1-\gamma _{t}) \bigl( \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}M_{1}\bigr)^{2} \\ &{} -\gamma _{t}(1-\gamma _{t}) \Vert x_{t}-z_{t} \Vert ^{2}+2\beta _{t} \bigl( \bigl\Vert f(x_{t}) \bigr\Vert + \Vert \rho Fz_{t} \Vert \bigr) \bigl\Vert x_{t+1}-q^{*} \bigr\Vert \\ =&\bigl( \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}M_{1}\bigr)^{2}-\gamma _{t}(1- \gamma _{t}) \Vert x_{t}-z_{t} \Vert ^{2} \\ &{} +2\beta _{t}\bigl( \bigl\Vert f(x_{t}) \bigr\Vert + \Vert \rho Fz_{t} \Vert \bigr) \bigl\Vert x_{t+1}-q^{*} \bigr\Vert \\ =& \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}+ \beta _{t}M_{1}\bigl[2 \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}M_{1}\bigr] \\ &{} -\gamma _{t}(1-\gamma _{t}) \Vert x_{t}-z_{t} \Vert ^{2}+2\beta _{t} \bigl( \bigl\Vert f(x_{t}) \bigr\Vert + \Vert \rho Fz_{t} \Vert \bigr) \bigl\Vert x_{t+1}-q^{*} \bigr\Vert , \end{aligned}$$

which immediately leads to

$$ \begin{aligned} \gamma _{t}(1-\gamma _{t}) \Vert x_{t}-z_{t} \Vert ^{2}\leq{}& \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}- \bigl\Vert x_{t+1}-q^{*} \bigr\Vert ^{2} \\ & {} +\beta _{t}M_{1}\bigl[2 \bigl\Vert x_{t}-q^{*} \bigr\Vert +\beta _{t}M_{1} \bigr]+2\beta _{t}\bigl( \bigl\Vert f(x_{t}) \bigr\Vert \\ &{}+ \Vert \rho Fz_{t} \Vert \bigr) \bigl\Vert x_{t+1}-q^{*} \bigr\Vert \\ \leq{}&{\boldsymbol{\Gamma}}_{t}-{\boldsymbol{\Gamma}}_{t+1}+\beta _{t}M_{1}\bigl[2{\boldsymbol{\Gamma}}^{ \frac{1}{2}}_{t}+ \beta _{t}M_{1}\bigr] \\ & {}+2\beta _{t}\bigl( \bigl\Vert f(x_{t}) \bigr\Vert + \Vert \rho Fz_{t} \Vert \bigr){\boldsymbol{\Gamma}}^{\frac{1}{2}}_{t+1}.\end{aligned} $$

Since $0<\liminf_{t\to \infty}\gamma _{t}\leq \limsup_{t\to \infty} \gamma _{t}<1$, $\beta _{t}\to 0$, ${\boldsymbol{\Gamma}}_{t}-{\boldsymbol{\Gamma}}_{t+1}\to 0$, and $\lim_{t\to \infty}{\boldsymbol{\Gamma}}_{t}=d<+\infty$, from the boundedness of $\{x_{t}\}$, $\{z_{t}\}$, we infer that

$$ \lim_{t\to \infty} \Vert x_{t}-z_{t} \Vert =0. $$

So, it follows from (3.34) that

$$ \Vert w_{t}-x_{t} \Vert \leq \Vert w_{t}-z_{t} \Vert + \Vert z_{t}-x_{t} \Vert \to 0\quad (t\to \infty ). $$

(3.35)

Also, from Algorithm 3.1 we obtain that

$$ \begin{aligned} \Vert x_{t+1}-x_{t} \Vert &= \bigl\Vert \beta _{t}f(x_{t})+(1-\gamma _{t}) (z_{t}-x_{t})- \beta _{t}\rho Fz_{t} \bigr\Vert \\ &\leq (1-\gamma _{t}) \Vert z_{t}-x_{t} \Vert +\beta _{t} \bigl\Vert f(x_{t})-\rho Fz_{t} \bigr\Vert \\ &\leq \Vert z_{t}-x_{t} \Vert +\beta _{t} \bigl( \bigl\Vert f(x_{t}) \bigr\Vert + \Vert \rho Fz_{t} \Vert \bigr)\to 0\quad (t\to \infty ).\end{aligned} $$

(3.36)

In addition, the boundedness of $\{x_{t}\}$ means there is $\{x_{t_{k}}\}\subset \{x_{t}\}$ such that

$$ \limsup_{t\to \infty}\bigl\langle (f-\rho F)q^{*},x_{t}-q^{*}\bigr\rangle = \lim _{k\to \infty}\bigl\langle (f-\rho F)q^{*},x_{t_{k}}-q^{*} \bigr\rangle . $$

(3.37)

Since $\{x_{t}\}$ is bounded, we may assume that $x_{t_{k}}\rightharpoonup \widetilde{z}$. We get from (3.37)

$$ \begin{aligned} { \limsup_{t\to \infty}}\bigl\langle (f-\rho F)q^{*},x_{t}-q^{*} \bigr\rangle &={ \lim_{k\to \infty}}\bigl\langle (f-\rho F)q^{*},x_{t_{k}}-q^{*} \bigr\rangle \\ &=\bigl\langle (f-\rho F)q^{*},\widetilde{z}-q^{*}\bigr\rangle .\end{aligned} $$

(3.38)

Since $x_{t}-x_{t+1}\to 0$, $w_{t}-x_{t}\to 0$, $w_{t}-y_{t}\to 0$, and $v_{t}-z_{t}\to 0$, by Lemma 3.5 we deduce that $\widetilde{z}\in \omega _{w}(\{x_{t}\})\subset\Xi$. Hence, from (3.10) and (3.38), one gets

$$ \limsup_{t\to \infty}\bigl\langle (f-\rho F)q^{*},x_{t}-q^{*}\bigr\rangle = \bigl\langle (f- \rho F)q^{*},\widetilde{z}-q^{*}\bigr\rangle \leq 0, $$

(3.39)

which together with (3.36) leads to

$$ \begin{aligned} &{ \limsup_{t\to \infty}}\bigl\langle (f-\rho F)q^{*},x_{t+1}-q^{*} \bigr\rangle \\ &\quad ={ \limsup_{t\to \infty}} \bigl[\bigl\langle (f-\rho F)q^{*},x_{t+1}-x_{t} \bigr\rangle +\bigl\langle (f-\rho F)q^{*},x_{t}-q^{*}\bigr\rangle \bigr] \\ &\quad \leq{ \limsup_{t\to \infty}} \bigl[ \bigl\Vert (f-\rho F)q^{*} \bigr\Vert \Vert x_{t+1}-x_{t} \Vert +\bigl\langle (f-\rho F)q^{*},x_{t}-q^{*}\bigr\rangle \bigr]\leq 0. \end{aligned} $$

(3.40)

Note that $\{\beta _{t}(\zeta -\nu )\}\subset [0,1]$, $\sum^{\infty}_{t=1}\beta _{t}( \zeta -\nu )=\infty$, and

$$ \limsup_{t\to \infty} \biggl[ \frac{2\langle (f-\rho F)q^{*},x_{t+1}-q^{*}\rangle}{\zeta -\nu} + \frac{M}{\zeta -\nu} \cdot \frac{\alpha _{t}}{\beta _{t}}\cdot \Vert x_{t}-x_{t-1} \Vert \biggr]\leq 0. $$

By Lemma 2.4 and (3.31), $\lim_{t\to \infty}\|x_{t}-q^{*}\|^{2}=0$.

Case 2. Suppose that $\exists \{{\boldsymbol{\Gamma}}_{t_{k}}\}\subset \{{\boldsymbol{\Gamma}}_{t}\}$ such that ${\boldsymbol{\Gamma}}_{t_{k}}<{\boldsymbol{\Gamma}}_{t_{k}+1} \ \forall k\in{\mathcal {N}}$, where ${\mathcal {N}}$ is the set of all positive integers. Define the mapping $\phi :{\mathcal {N}}\to{\mathcal {N}}$ by

$$ \phi (t):=\max \{k\leq t:{\boldsymbol{\Gamma}}_{k}< {\boldsymbol{\Gamma}}_{k+1}\}. $$

By Lemma 2.6, we get

$$ {\boldsymbol{\Gamma}}_{\phi (t)}\leq{\boldsymbol{\Gamma}}_{\phi (t)+1}\quad \text{and}\quad {\boldsymbol{\Gamma}}_{t}\leq{\boldsymbol{\Gamma}}_{\phi (t)+1}. $$

From (3.29) we have

$$ \begin{aligned} & (1-\beta _{\phi (t)}\zeta -\gamma _{\phi (t)})\biggl[\biggl(1-\mu \frac{\lambda _{\phi (t)}}{\lambda _{{\phi (t)}+1}}\biggr) \bigl( \Vert w_{\phi (t)}-y_{ \phi (t)} \Vert ^{2}+ \Vert v_{\phi (t)}-y_{\phi (t)} \Vert ^{2}\bigr) \\ & \qquad {} +\epsilon ^{2} \bigl\Vert T^{*}(I-S)Tv_{\phi (t)} \bigr\Vert ^{2}\biggr] \\ &\quad \leq \bigl\Vert x_{\phi (t)}-q^{*} \bigr\Vert ^{2}- \bigl\Vert x_{{\phi (t)}+1}-q^{*} \bigr\Vert ^{2}+ \beta _{\phi (t)}M_{4} \\ &\quad ={\boldsymbol{\Gamma}}_{\phi (t)}-{\boldsymbol{\Gamma}}_{{\phi (t)}+1}+\beta _{\phi (t)}M_{4},\end{aligned} $$

(3.41)

which immediately yields

$$ \lim_{t\to \infty} \Vert w_{\phi (t)}-y_{\phi (t)} \Vert = \lim_{t\to \infty} \Vert v_{\phi (t)}-y_{\phi (t)} \Vert = \lim_{t\to \infty} \bigl\Vert T^{*}(I-S)Tv_{ \phi (t)} \bigr\Vert =0. $$

Similar to Case 1,

$$\begin{aligned}& \lim_{t\to \infty} \Vert v_{\phi (t)}-z_{\phi (t)} \Vert = \lim_{t\to \infty} \Vert w_{\phi (t)}-x_{\phi (t)} \Vert = \lim_{t\to \infty} \Vert x_{\phi (t)+1}-x_{ \phi (t)} \Vert =0, \\& \limsup_{t\to \infty}\bigl\langle (f-\rho F)q^{*},x_{\phi (t)+1}-q^{*} \bigr\rangle \leq 0. \end{aligned}$$

(3.42)

By (3.31),

$$\begin{aligned} \beta _{\phi (t)}(\zeta -\nu ){\boldsymbol{\Gamma}}_{\phi (t)}\leq{}& { \boldsymbol{\Gamma}}_{\phi (t)}-{\boldsymbol{\Gamma}}_{\phi (t)+1}+\beta _{\phi (t)}(\zeta - \nu ) \biggl[ \frac{2\langle (f-\rho F)q^{*},x_{\phi (t)+1}-q^{*}\rangle}{\zeta -\nu} \\ & {}+\frac{M}{\zeta -\nu}\cdot \frac{\alpha _{\phi (t)}}{\beta _{\phi (t)}}\cdot \Vert x_{\phi (t)}-x_{ \phi (t)-1} \Vert \biggr] \\ \leq {}& \beta _{\phi (t)}(\zeta -\nu ) \biggl[ \frac{2\langle (f-\rho F)q^{*},x_{\phi (t)+1}-q^{*}\rangle}{\zeta -\nu} \\ & {}+\frac{M}{\zeta -\nu}\cdot \frac{\alpha _{\phi (t)}}{\beta _{\phi (t)}}\cdot \Vert x_{\phi (t)}-x_{ \phi (t)-1} \Vert \biggr], \end{aligned}$$

and so

$$\begin{aligned} & { \limsup_{t\to \infty}} {\boldsymbol{\Gamma}}_{\phi (t)} \\ &\quad \leq { \limsup_{t\to \infty}} \biggl[ \frac{2\langle (f-\rho F)q^{*},x_{\phi (t)+1}-q^{*}\rangle}{\zeta -\nu} + \frac{M}{\zeta -\nu}\cdot \frac{\alpha _{\phi (t)}}{\beta _{\phi (t)}}\cdot \Vert x_{\phi (t)}-x_{ \phi (t)-1} \Vert \biggr] \\ &\quad \leq 0. \end{aligned}$$

Thus, $\lim_{t\to \infty}\|x_{\phi (t)}-q^{*}\|^{2}=0$. Also note that

$$ \begin{aligned} \bigl\Vert x_{\phi (t)+1}-q^{*} \bigr\Vert ^{2}- \bigl\Vert x_{\phi (t)}-q^{*} \bigr\Vert ^{2} ={}&2\bigl\langle x_{ \phi (t)+1}-x_{\phi (t)},x_{\phi (t)}-q^{*} \bigr\rangle \\ &{}+ \Vert x_{\phi (t)+1}-x_{\phi (t)} \Vert ^{2} \\ \leq{}& 2 \Vert x_{\phi (t)+1}-x_{\phi (t)} \Vert \bigl\Vert x_{\phi (t)}-q^{*} \bigr\Vert \\ &{}+ \Vert x_{\phi (t)+1}-x_{\phi (t)} \Vert ^{2}.\end{aligned} $$

(3.43)

Owing to ${\boldsymbol{\Gamma}}_{t}\leq{\boldsymbol{\Gamma}}_{\phi (t)+1}$, we get

$$\begin{aligned} \bigl\Vert x_{t}-q^{*} \bigr\Vert ^{2}\leq{}& \bigl\Vert x_{\phi (t)+1}-q^{*} \bigr\Vert ^{2} \\ \leq{}& \bigl\Vert x_{\phi (t)}-q^{*} \bigr\Vert ^{2}+2 \Vert x_{\phi (t)+1}-x_{\phi (t)} \Vert \bigl\Vert x_{ \phi (t)}-q^{*} \bigr\Vert \\ & {}+ \Vert x_{\phi (t)+1}-x_{\phi (t)} \Vert ^{2} \to 0, \end{aligned}$$

i.e., $x_{t}\to q^{*}$ as $t\to \infty$. □

Remark 3.2

• (i) The results in [21] are extended to develop BSPVIP (1.6) with the CFPP constraint, i.e., the problem of finding $q^{*}\in\Xi =\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\Omega$ such that $\langle (\rho F-f)q^{*},p-q^{*}\rangle \geq 0 \ \forall p\in\Xi$, where $\Omega =\{z\in\operatorname{VI}(C,A):Tz\in\operatorname{Fix}(S)\}$ with A being pseudomonotone and Lipschitzian mapping. The results in [21] are extended to develop our triple-adaptive inertial subgradient extragradient rule for settling BSPVIP (1.6) with the CFPP constraint, which is on the basis of the subgradient extragradient method with adaptive step sizes, accelerated inertial approach, hybrid deepest-descent method, and viscosity approximation technique. In [21] the following holds:

  $$ x_{t}\to q^{*}\in\Omega =\bigcap ^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\operatorname{VI}(C,A)\quad \Leftrightarrow\quad \Vert x_{t}-x_{t+1} \Vert \to 0 $$

  with $q^{*}=P_{\Omega}(I-\rho F+f)q^{*}$. In our results, Lemma 2.6 implies that

  $$ x_{t}\to q^{*}\in\Xi =\bigcap ^{N}_{i=1}\operatorname{Fix}(S_{i})\cap \bigl\{ z \in\operatorname{VI}(C,A):Tz\in\operatorname{Fix}(S)\bigr\} $$

  with $q^{*} = P_{\Xi}(I-\rho F+f)q^{*}$.

• (ii) BSQVIP (1.5) (i.e., the problem of finding $q^{*}\in\Omega$ such that $\langle Fq^{*},p-q^{*}\rangle \geq 0 \ \forall p\in\Omega$, where $\Omega =\{z\in\operatorname{VI}(C,A):Tz\in\operatorname{Fix}(S)\}$ with A being quasimonotone and Lipschitzian mapping) in [29] is extended to develop BSPVIP (1.6) with the CFPP constraint, i.e., the problem of finding $q^{*}\in\Xi =\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\Omega$ such that $\langle (\rho F-f)q^{*},p-q^{*}\rangle \geq 0 \ \forall p\in\Xi$, where $\Omega =\{z\in\operatorname{VI}(C,A):Tz\in\operatorname{Fix}(S)\}$ with A being pseudomonotone and Lipschitzian mapping.

4 Numerical implementation

In this section, we compare our proposed Algorithm 3.1 with Algorithm 1 of [27] using the example below. All codes were written in MATLAB R2017a and performed on a PC Desktop Intel(R) Core(TM) i7-8700U CPU @ 3.20GHz 3.19GHz, RAM 8.00 GB.

Suppose that $H_{1}=H_{2}=L_{2}([0,1])$ is endowed with the inner product $\langle x,y\rangle = \int _{0}^{1} x(t)y(t)\,dt$, $\forall x, y \in L_{2}([0,1])$ and the induced norm $\|x\|:= \int _{0}^{1} |x(t)|^{2}\,dt$, $\forall x, y \in L_{2}([0,1])$. Let $T: L_{2}([0,1])\rightarrow L_{2}([0,1])$ be defined by

$$ T x(s) = \int _{0}^{1}e^{-st}x(t)\,dt, \quad \forall x \in L_{2}\bigl([0,1]\bigr), \forall s,t \in [0,1]. $$

Then T is a bounded linear operator with adjoint

$$ T^{*}x(s) = \int _{0}^{1}e^{-st}x(t)\,dt, \quad \forall x \in L_{2}\bigl([0,1]\bigr), \forall s,t \in [0,1]. $$

Let $C = \{x \in L_{2}([0,1]): \langle t+1,x\rangle \leq 1\}$. Then C is a nonempty closed and convex subset. The projection $P_{C}$ is given as

$$ P_{C}(x)=\textstyle\begin{cases} \frac{1-\langle t+1,x\rangle}{ \Vert y \Vert ^{2}} (t+1)+x, & \text{if } \langle t+1,x\rangle >1, \\ x, & \text{if } \langle t+1,x\rangle \leq 1. \end{cases} $$

Also, let $Q= \{x \in L_{2}([0,1]): \|x\|\leq 2\}$. Then Q is a nonempty closed and convex subset. $P_{Q}$ is

$$ P_{Q}(x)=\textstyle\begin{cases} x & \text{if } x \in Q, \\ \frac{2x}{ \Vert x \Vert } & \text{if otherwise}. \end{cases} $$

Let $A:L_{2}([0,1])\rightarrow L_{2}([0,1])$ be defined by

$$ Ax(t):=e^{-\|x\|^{2}} \int _{0}^{t} x(s)\,ds, \quad \forall x \in L_{2}\bigl([0,1]\bigr), t \in [0,1]. $$

Then A is pseudomonotone and Lipschitz continuous but not monotone. Also define $B:L_{2}([0,1])\rightarrow L_{2}([0,1])$ by

$$ Bx(t):=\max \bigl\{ x(t),0\bigr\} , \quad \forall t \in [0,1]. $$

Take $f(x)=\frac{x}{2}$, $x \in L_{2}([0,1])$, $\beta _{t}=\frac{1}{t+1}$ and $F=I$.

To test the algorithms, we choose the following parameters for the algorithm: for our algorithm, we used $\lambda _{1} = 0.06$, $\epsilon = 10^{-4}$, $\sigma = 0.5$, $\mu = 0.06$, $\alpha = 10^{-3}$, $\epsilon _{t} = (t+1)^{-2}$, $\beta _{t} = (t+1)^{-1}$, $\gamma _{t} = 2t(5t+9)^{-1}$, $\rho = 0.07$. For Anh’s algorithm, we choose $\eta = 0.06$, $\gamma = 0.05$, $\mu = 0.07$, $\delta _{t} = 10^{-3}$, $\lambda _{t} = 2t(5t+1)^{-1}$, $\alpha _{t} = (t+1)^{-1}$. We used $Err = \|x_{t+1} - x_{t}\| < 10^{-4}$ as a stopping criterion for each algorithm. We test the algorithms using the following starting points:

Case I: $x_{0} = 2t^{2} +1$, $x_{1} = \exp (3t)$

Case II: $x_{0} = 2t^{2} -2t+1$, $x_{1} = -4(t^{3} +2t -3)$;

Case III: $x_{0} = t^{4} -1$, $x_{1} = t^{5} -9$;

Case IV: $x_{0} = \frac{1}{4}t^{2} +2t$, $x_{1} = \frac{1}{3}\cos (2t)$.

The numerical results are shown in Table 1 and Fig. 1.

Figure 1

Numerical results, Top Left: Case I; Top Right: Case II; Bottom Left: Case III; Bottom Right: Case IV

Table 1 Computational result

Algorithm 4.1

Initialization: Let $\lambda _{1}>0$, $\epsilon >0$, $\sigma \geq 0$, $\mu \in (0,1)$, $\alpha \in [0,1)$, and $x_{0},x_{1}\in{\mathcal {H}}_{1}$ be arbitrary.

Iterative steps: Calculate $x_{t+1}$ as follows:

Step 1. Given the iterates $x_{t-1}$ and $x_{t}$ ($t\geq 1$), choose $\alpha _{t}$ such that $0\leq \alpha _{t}\leq \bar{\alpha}_{t}$, where

$$ \bar{\alpha}_{t}=\textstyle\begin{cases} \min \{\alpha ,\frac{\varepsilon _{t}}{ \Vert x_{t}-x_{t-1} \Vert }\} & \text{if }x_{t} \neq x_{t-1}, \\ \alpha & \text{otherwise}.\end{cases} $$

(4.1)

Step 2. Compute $w_{t}=x_{t}+\alpha _{t}(x_{t}-x_{t-1})$ and $y_{t}=P_{C}(w_{t}-\lambda _{t}Aw_{t})$.

Step 3. Construct $C_{t}:=\{y\in{\mathcal {H}}_{1}:\langle w_{t}-\lambda _{t}Aw_{t}-y_{t},y_{t}-y \rangle \geq 0\}$, and compute $v_{t}=P_{C_{t}}(w_{t}-\lambda _{t}Ay_{t})$ and $z_{t}=v_{t}-\sigma _{t}T^{*}(I-S)Tv_{t}$, where $S=P_{Q}(I-\varphi B)-\varphi (B(P_{Q}(I-\varphi B))-B)$ and $\varphi \in (0,1)$.

Step 4. Calculate $x_{t+1}=\beta _{t}\frac{x_{t}}{2}+\gamma _{t}x_{t}+((1-\gamma _{t})I- \beta _{t}\rho )z_{t}$ and update

$$ \lambda _{t+1}=\textstyle\begin{cases} \min \{\mu \frac{ \Vert w_{t}-y_{t} \Vert ^{2}+ \Vert v_{t}-y_{t} \Vert ^{2}}{2\langle Aw_{t}-Ay_{t},v_{t}-y_{t}\rangle}, \lambda _{t}\}& \text{if }\langle Aw_{t}-Ay_{t},v_{t}-y_{t}\rangle >0, \\ \lambda _{t} & \text{otherwise},\end{cases} $$

(4.2)

and for any fixed $\epsilon >0$, $\sigma _{t}$ is chosen to be the bounded sequence satisfying

$$ 0< \epsilon \leq \sigma _{t}\leq \frac{(1-\tau ) \Vert Tv_{t}-STv_{t} \Vert ^{2}}{ \Vert T^{*}(Tv_{t}-STv_{t}) \Vert ^{2}}- \epsilon \quad \text{if } Tv_{t}\neq STv_{t}, $$

(4.3)

otherwise set $\sigma _{t}=\sigma \geq 0$.

Set $t:=t+1$ and go to Step 1.