On Mann implicit composite subgradient extragradient methods for general systems of variational inequalities with hierarchical variational inequality constraints

Abstract

In a real Hilbert space, let the VIP, GSVI, HVI, and CFPP denote a variational inequality problem, a general system of variational inequalities, a hierarchical variational inequality, and a common fixed-point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping, respectively. We design two Mann implicit composite subgradient extragradient algorithms with line-search process for finding a common solution of the CFPP, GSVI, and VIP. The suggested algorithms are based on the Mann implicit iteration method, subgradient extragradient method with line-search process, and viscosity approximation method. Under mild assumptions, we prove the strong convergence of the suggested algorithms to a common solution of the CFPP, GSVI, and VIP, which solves a certain HVI defined on their common solutions set.

1 Introduction

Let C be a nonempty, closed, and convex subset of a real Hilbert space $(H,\langle \cdot ,\cdot \rangle )$ with the induced norm $\|\cdot \|$. Let $P_{C}$ be the nearest point projection from H onto C. Given a nonlinear operator $T:C\to H$, let $\mathrm {Fix}(T)$ and R indicate the fixed-points set of T and the set of real numbers, respectively. Let → and ⇀ represent the strong and weak convergence in H, respectively. An operator $T:C\to C$ is called asymptotically nonexpansive if there exists $\{\theta _{l}\}^{\infty }_{l=1}\subset [0,+\infty )$ such that $\lim_{l\to \infty }\theta _{l}=0$ and

$$ \bigl\Vert T^{l}u-T^{l}v \bigr\Vert \leq (1+\theta _{l}) \Vert u-v \Vert \quad \forall l\geq 1, u,v \in C. $$

(1.1)

In particular, whenever $\theta _{l}=0$ $\forall l\geq 1$, T is called nonexpansive. Given a self-mapping A on H, the classical variational inequality problem (VIP) is finding $u\in C$ such that $\langle Au,v-u\rangle \geq 0$ $\forall v\in C$. We denote the solutions set of VIP by $\mathrm {VI}(C,A)$. To the best of our knowledge, one of the most popular approaches for solving the VIP is the extragradient method put forward by Korpelevich [1] in 1976, i.e., for any initial point $u_{0}\in C$, let $\{u_{l}\}$ be the sequence constructed below

$$ \textstyle\begin{cases} v_{l}=P_{C}(u_{l}-\ell Au_{l}), \\ u_{l+1}=P_{C}(u_{l}-\ell Av_{l})\quad \forall l\geq 0,\end{cases} $$

(1.2)

where $\ell \in (0,\frac{1}{L})$ and L is Lipschitz constant of A. Whenever $\mathrm {VI}(C,A)\neq \emptyset$, the sequence $\{u_{l}\}$ converges weakly to a point in $\mathrm {VI}(C,A)$. At present, the vast literature on Korpelevich’s extragradient approach shows that many authors have paid great attention to it and enhanced it in various ways; see, e.g., [2–26] and the references therein.

Suppose that $B_{1},B_{2}:C\to H$ are two nonlinear operators. Consider the following problem of finding $(u^{*},v^{*})\in C\times C$ such that

$$ \textstyle\begin{cases} \langle \mu _{1}B_{1}v^{*}+u^{*}-v^{*},w-u^{*}\rangle \geq 0\quad \forall w\in C, \\ \langle \mu _{2}B_{2}u^{*}+v^{*}-u^{*},w-v^{*}\rangle \geq 0\quad \forall w\in C,\end{cases} $$

(1.3)

with constants $\mu _{1},\mu _{2}>0$. Problem (1.3) is called a general system of variational inequalities (GSVI). Note that GSVI (1.3) can be transformed into the fixed-point problem below.

Lemma 1.1

([6])

For given $x^{*},y^{*}\in C, (x^{*},y^{*})$ is a solution of GSVI (1.3) if and only if $x^{*}\in \mathrm {Fix}(G)$, where $\mathrm {Fix}(G)$ is the fixed point set of the mapping $G:=P_{C}(I-\mu _{1}B_{1})P_{C}(I-\mu _{2}B_{2})$, and $y^{*}=P_{C}(I-\mu _{2}B_{2})x^{*}$.

Suppose that the mappings $B_{1}$, $B_{2}$ are α-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let $f:C\to C$ be a contraction with coefficient $\delta \in [0,1)$ and $F:C\to H$ be κ-Lipschitzian and η-strongly monotone with constants $\kappa ,\eta >0$ such that $\delta <\zeta :=1-\sqrt{1-\rho (2\eta -\rho \kappa ^{2})}\in (0,1]$ for $\rho \in (0,\frac{2\eta }{\kappa ^{2}})$. Let $S:C\to C$ be an asymptotically nonexpansive mapping with a sequence $\{\theta _{n}\}$. Let $\{S_{l}\}^{\infty }_{l=1}$ be a countable family of ς-uniformly Lipschitzian pseudocontractive self-mappings on C such that $\varOmega :=\bigcap^{\infty }_{l=0}\mathrm {Fix}(S_{l})\cap \mathrm {Fix}(G) \neq \emptyset$ where $S_{0}:=S$ and $\mathrm {Fix}(G)$ is the fixed-point set of the mapping $G:=P_{C}(I-\mu _{1}B_{1})P_{C}(I-\mu _{2}B_{2})$ for $\mu _{1}\in (0,2\alpha )$ and $\mu _{2}\in (0,2\beta )$. Recently, Ceng and Wen [21] proposed the hybrid extragradient-like implicit method for finding an element of Ω, that is, for any initial point $x_{1}\in C$, let $\{x_{l}\}$ be the sequence constructed below

$$ \textstyle\begin{cases} u_{l}=\beta _{l}x_{l}+(1-\beta _{l})S_{l}u_{l}, \\ v_{l}=P_{C}(u_{l}-\mu _{2}B_{2}u_{l}), \\ y_{l}=P_{C}(v_{l}-\mu _{1}B_{1}v_{l}), \\ x_{l+1}=P_{C}[\alpha _{l}f(x_{l})+(I-\alpha _{l}\rho F)S^{l}y_{l}] \quad \forall l\geq 1,\end{cases} $$

(1.4)

where $\{\alpha _{l}\}$ and $\{\beta _{l}\}$ are sequences in $(0,1]$ such that

• (i) $\sum^{\infty }_{l=1}|\alpha _{l+1}-\alpha _{l}|<\infty$ and $\sum^{\infty }_{l=1}\alpha _{l}<\infty$;

• (ii) $\lim_{l\to \infty }\alpha _{l}=0$ and $\lim_{l\to \infty }\frac{\theta _{l}}{\alpha _{l}}=0$;

• (iii) $\sum^{\infty }_{l=1}|\beta _{l+1}-\beta _{l}|<\infty$ and $0<\liminf_{l\to \infty }\beta _{l}\leq \limsup_{l\to \infty }\beta _{l}<1$;

• (iv) $\sum^{\infty }_{l=1}\|S^{l+1}y_{l}-S^{l}y_{l}\|<\infty$.

Under appropriate assumptions imposed on $\{S_{l}\}^{\infty }_{l=1}$, it was proved in [21] that the sequence $\{x_{l}\}$ converges strongly to an element $x^{*}\in \varOmega$. In 2019, Thong and Hieu [14] proposed the inertial subgradient extragradient method with line-search process for solving the monotone VIP with Lipschitz continuous A and the fixed-point problem (FPP) of a quasinonexpansive mapping S with a demiclosedness property. Assume that $\varOmega :=\mathrm {Fix}(S)\cap \mathrm {VI}(C,A)\neq \emptyset$. Let the sequences $\{\alpha _{l}\}\subset [0,1]$ and $\{\beta _{l}\}\subset (0,1)$ be given.

Algorithm 1.1

([14])

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

Iterative Steps: Compute $x_{l+1}$ below:

Step 1. Set $w_{l}=x_{l}+\alpha _{l}(x_{l}-x_{l-1})$ and calculate $v_{l}=P_{C}(w_{l}-\tau _{l}Aw_{l})$, where $\tau _{l}$ is chosen to be the largest $\tau \in \{\gamma ,\gamma \ell ,\gamma \ell ^{2},\dots \}$ satisfying $\tau \|Aw_{l}-Av_{l}\|\leq \mu \|w_{l}-v_{l}\|$.

Step 2. Calculate $z_{l}=P_{C_{l}}(w_{l}-\tau _{l}Av_{l})$ with $C_{l}:=\{v\in H:\langle w_{l}-\tau _{l}Aw_{l}-v_{l},v-v_{l}\rangle \leq 0\}$.

Step 3. Calculate $x_{l+1}=(1-\beta _{l})w_{l}+\beta _{l}Sz_{l}$. If $w_{l}=z_{l}=x_{l+1}$ then $w_{l}\in \varOmega$.

Again set $l:=l+1$ and go to Step 1.

Under suitable assumptions, it was proven in [14] that $\{x_{l}\}$ converges weakly to an element of Ω. Very recently, Ceng and Shang [22] introduced the hybrid inertial subgradient extragradient method with line-search process for solving the pseudomonotone VIP with Lipschitz continuous A and the common fixed-point problem (CFPP) of finitely many nonexpansive mappings $\{S_{l}\}^{N}_{l=1}$ and an asymptotically nonexpansive mapping S in a real Hilbert space H. Assume that $\varOmega :=\bigcap^{N}_{l=0}\mathrm {Fix}(S_{l})\cap \mathrm {VI}(C,A) \neq \emptyset$ with $S_{0}:=S$. Given a contraction $f:H\to H$ with constant $\delta \in [0,1)$, and an η-strongly monotone and κ-Lipschitzian mapping $F:H\to H$ with $\delta <\zeta :=1-\sqrt{1-\rho (2\eta -\rho \kappa ^{2})}$ for $\rho \in (0,2\eta /\kappa ^{2})$, let $\{\alpha _{l}\}\subset [0,1]$ and $\{\beta _{l}\},\{\gamma _{l}\}\subset (0,1)$ with $\beta _{l}+\gamma _{l}<1$ $\forall l\geq 1$. Besides, one writes $S_{l}:=S_{l\mathrm {mod}N}$ for integer $l\geq 1$ with the mod function taking values in the set $\{1,2,\dots ,N\}$, i.e., whenever $l=jN+q$ for some integers $j\geq 0$ and $0\leq q< N$, one has that $S_{l}=S_{N}$ if $q=0$ and $S_{l}=S_{q}$ if $0< q< N$.

Algorithm 1.2

([22])

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

Iterative Steps: Calculate $x_{l+1}$ below:

Step 1. Set $w_{l}=S_{l}x_{l}+\alpha _{l}(S_{l}x_{l}-S_{l}x_{l-1})$ and calculate $v_{l}=P_{C}(w_{l}-\tau _{l}Aw_{l})$, where $\tau _{l}$ is chosen to be the largest $\tau \in \{\gamma ,\gamma \ell ,\gamma \ell ^{2},\dots \}$ satisfying $\tau \|Aw_{l}-Av_{l}\|\leq \mu \|w_{l}-v_{l}\|$.

Step 2. Calculate $z_{l}=P_{C_{l}}(w_{l}-\tau _{l}Av_{l})$ with $C_{l}:=\{v\in H:\langle w_{l}-\tau _{l}Aw_{l}-v_{l},v-v_{l}\rangle \leq 0\}$.

Step 3. Calculate $x_{l+1}=\beta _{l}f(x_{l})+\gamma _{l}x_{l}+((1-\gamma _{l})I-\beta _{l} \rho F)S^{l}z_{l}$.

Again set $l:=l+1$ and go to Step 1.

Under appropriate assumptions, it was proven in [22] that if $S^{l}z_{l}-S^{l+1}z_{l}\to 0$, then $\{x_{l}\}$ converges strongly to $x^{*}\in \varOmega$ if and only if $x_{l}-x_{l+1}\to 0$ and $x_{l}-v_{l}\to 0$ as $l\to \infty$. In a real Hilbert space H, we always assume that the CFPP and HVI denote a common fixed-point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings $\{S_{l}\}^{\infty }_{l=1}$ and an asymptotically nonexpansive mapping $S_{0}:=S$ and a hierarchical variational inequality, respectively. Inspired by the above research works, we design two Mann implicit composite subgradient extragradient algorithms with line-search process for finding a common solution of the CFPP of $\{S_{l}\}^{\infty }_{l=0}$, the pseudomonotone VIP with Lipschitz continuous A and the GSVI for two inverse-strongly monotone $B_{1}$, $B_{2}$. The suggested algorithms are based on the viscosity approximation method, subgradient extragradient method with line-search process, and Mann implicit iteration method. Under mild assumptions, we prove the strong convergence of the suggested algorithms to a common solution of the CFPP, GSVI, and VIP, which solves a certain HVI defined on their common solution set. Finally, using the main results, we deal with the CFPP, GSVI, and VIP in an illustrated example.

2 Preliminaries

Let the nonempty set C be convex and closed in a real Hilbert space H. Given a sequence $\{\upsilon _{i}\}\subset H$, let $\upsilon _{i}\to \upsilon$ (resp., $\upsilon _{i}\rightharpoonup \upsilon$) indicate the strong (resp., weak) convergence of $\{\upsilon _{i}\}$ to υ. An operator $S:C\to H$ is called

• (a) L-Lipschitz continuous (or L-Lipschitzian) if $\exists L>0$ such that $\|Su-S\upsilon \|\leq L\|u-\upsilon \|$ ∀u, $\upsilon \in C$;

• (b) pseudocontractive if $\langle Su-S\upsilon ,u-\upsilon \rangle \leq \|u-\upsilon \|^{2}$ ∀u, $\upsilon \in C$;

• (c) pseudomonotone if $\langle Su,\upsilon -u\rangle \geq 0\Rightarrow \langle S\upsilon , \upsilon -u\rangle \geq 0$ ∀u, $\upsilon \in C$;

• (d) α-strongly monotone if $\exists \alpha >0$ such that $\langle Su-S\upsilon ,u-\upsilon \rangle \geq \alpha \|u-\upsilon \|^{2}$ ∀u, $\upsilon \in C$;

• (e) β-inverse-strongly monotone if $\exists \beta >0$ such that $\langle Su-S\upsilon ,u-\upsilon \rangle \geq \beta \|Su-S\upsilon \|^{2}$ ∀u, $\upsilon \in C$;

• (f) sequentially weakly continuous if $\forall \{\upsilon _{i}\}\subset C$, the following relation holds: $\upsilon _{i}\rightharpoonup \upsilon \Rightarrow S\upsilon _{i} \rightharpoonup S\upsilon$.

It is clear that each monotone mapping is pseudomonotone, but the converse is not true. It is known that $\forall u\in H$, ∃! (nearest point) $P_{C}u\in C$ such that $\|u-P_{C}u\|\leq \|u-\upsilon \|$ $\forall \upsilon \in C$; $P_{C}$ is refereed to as a metric (or nearest point) projection of H onto C. Recall that the following conclusions hold (see [27]):

• (a) $\langle u-\upsilon ,P_{C}u-P_{C}\upsilon \rangle \geq \|P_{C}u-P_{C} \upsilon \|^{2}$ ∀u, $\upsilon \in H$;

• (b) $w=P_{C}u\Leftrightarrow \langle u-w,\upsilon -w\rangle \leq 0$ $\forall u\in H$, $\upsilon \in C$;

• (c) $\|u-\upsilon \|^{2}\geq \|u-P_{C}u\|^{2}+\|\upsilon -P_{C}u\|^{2}$ $\forall u\in H$, $v\in C$;

• (d) $\|u-\upsilon \|^{2}=\|u\|^{2}-\|\upsilon \|^{2}-2\langle u-\upsilon , \upsilon \rangle$ ∀u, $\upsilon \in H$;

• (e) $\|su+(1-s)\upsilon \|^{2}=s\|u\|^{2}+(1-s)\|\upsilon \|^{2}-s(1-s)\|u- \upsilon \|^{2}$ ∀u, $\upsilon \in H$, $s\in [0,1]$.

The following concept will be used in the convergence analysis of the proposed algorithms.

Definition 2.1

([21])

Let $\{S_{l}\}^{\infty }_{l=1}$ be a sequence of continuous pseudocontractive self-mappings on C. Then $\{S_{l}\}^{\infty }_{l=1}$ is called a countable family of ς-uniformly Lipschitzian pseudocontractive self-mappings on C if there exists a constant $\varsigma >0$ such that each $S_{l}$ is ς-Lipschitz continuous.

The following propositions and lemmas will be needed for demonstrating our main results.

Proposition 2.1

([28])

Let C be a nonempty, closed, convex subset of a Banach space X. Suppose that $\{S_{l}\}^{\infty }_{l=1}$ is a countable family of self-mappings on C such that $\sum^{\infty }_{l=1}\sup \{\|S_{l}x -S_{l+1}x\|:x\in C\}<\infty$. Then for each $y\in C$, $\{S_{l}y\}$ converges strongly to some point of C. Moreover, let Ŝ be a self-mapping on C, defined by $\hat{S}y=\lim_{l\to \infty }S_{l}y$ for all $y\in C$. Then $\lim_{l\to \infty }\sup \{\|Sx-S_{l}x\|:x\in C\}=0$.

Proposition 2.2

([29])

Let C be a nonempty, closed, convex subset of a Banach space X and $T:C\to C$ be a continuous and strong pseudocontraction mapping. Then, T has a unique fixed point in C.

The following inequality is an immediate consequence of the subdifferential inequality of the function $\frac{1}{2}\|\cdot \|^{2}$:

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

Lemma 2.1

Let the mapping $B:C\to H$ be β-inverse-strongly monotone. Then, for a given $\lambda \geq 0$,

$$ \bigl\Vert (I-\lambda B)u-(I-\lambda B)\upsilon \bigr\Vert ^{2} \leq \Vert u-\upsilon \Vert ^{2}- \lambda (2\alpha -\lambda ) \Vert Bu-B\upsilon \Vert ^{2}. $$

In particular, if $0\leq \lambda \leq 2\alpha$, then $I-\lambda B$ is nonexpansive.

Using Lemma 2.1, we immediately derive the following lemma.

Lemma 2.2

Let the mappings $B_{1},B_{2}:C\to H$ be α-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let the mapping $G:C\to C$ be defined as $G:=P_{C}(I-\mu _{1}B_{1})P_{C}(I-\mu _{2}B_{2})$. If $0\leq \mu _{1}\leq 2\alpha$ and $0\leq \mu _{2}\leq 2\beta$, then $G:C\to C$ is nonexpansive.

Lemma 2.3

([6, Lemma 2.1])

Let $A:C\to H$ be pseudomonotone and continuous. Then $u\in C$ is a solution to the VIP $\langle Au,\upsilon -u\rangle \geq 0$ $\forall \upsilon \in C$ if and only if $\langle A\upsilon ,\upsilon -u\rangle \geq 0$ $\forall \upsilon \in C$.

Lemma 2.4

([30])

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

Lemma 2.5

([31])

Let X be a Banach space which admits a weakly continuous duality mapping, C be a nonempty, closed, convex subset of X, and $T:C\to C$ be an asymptotically nonexpansive mapping with $\mathrm {Fix} (T)\neq \emptyset$. Then $I-T$ is demiclosed at zero, i.e., if $\{u_{k}\}$ is a sequence in C such that $u_{k} \rightharpoonup u\in C$ and $(I-T)u_{k}\to 0$, then $(I-T)u=0$, where I is the identity mapping of X.

The following lemmas are crucial to the convergence analysis of the proposed algorithms.

Lemma 2.6

([25])

Let $\{\Gamma _{m}\}$ be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence $\{\Gamma _{m_{k}}\}$ of $\{\Gamma _{m}\}$ which satisfies $\Gamma _{m_{k}}<\Gamma _{m_{k}+ 1}$ for each integer $k\geq 1$. Define the sequence $\{\tau (m)\}_{m\geq m_{0}}$ of integers by

$$ \tau (m)=\max \{k\leq m:\Gamma _{k}< \Gamma _{k+1}\}, $$

where integer $m_{0}\geq 1$ is such that $\{k\leq m_{0}:\Gamma _{k}<\Gamma _{k+1}\}\neq \emptyset$. Then the following hold:

• (i) $\tau (m_{0})\leq \tau (m_{0}+1)\leq \cdots$ and $\tau (m)\to \infty$;

• (ii) $\Gamma _{\tau (m)}\leq \Gamma _{\tau (m)+1}$ and $\Gamma _{m}\leq \Gamma _{\tau (m)+1}$ $\forall m\geq m_{0}$.

3 Main results

In this section, let the feasible set C be a nonempty, closed, convex subset of a real Hilbert space H, and assume always that the following conditions hold:

• A is pseudomonotone and L-Lipschitzian self-mapping on H such that $\|Au\|\leq \liminf_{n\to \infty }\|A\upsilon _{n}\|$ for each $\{\upsilon _{n}\}\subset C$ with $\upsilon _{n}\rightharpoonup u$.

• $B_{1},B_{2}:C\to H$ are α-inverse-strongly monotone and β-inverse-strongly monotone, respectively, and $f:C\to C$ is a δ-contraction with constant $\delta \in [0,1)$.

• $\{S_{n}\}^{\infty }_{n=1}$ is a countable family of ς-uniformly Lipschitzian pseudocontractive self-mappings on C and $S:H\to C$ is an asymptotically nonexpansive mapping with a sequence $\{\theta _{n}\}$.

• $\varOmega =\bigcap^{\infty }_{n=0}\mathrm {Fix}(S_{n})\cap \mathrm {Fix}(G) \cap \mathrm {VI}(C,A)\neq \emptyset$ with $S_{0}:=S$, and $\mathrm {Fix}(G)$ is the fixed point set of mapping $G=P_{C}(I-\mu _{1}B_{1})P_{C}(I-\mu _{2}B_{2})$ for $0<\mu _{1}<2\alpha$ and $0<\mu _{2}<2\beta$.

• $\sum^{\infty }_{n=1}\sup_{x\in D}\|S_{n}x-S_{n+1}x\|<\infty$ for any bounded subset D of C and $\mathrm {Fix}(\hat{S})= \bigcap^{\infty }_{n=1}\mathrm {Fix}(S_{n})$ where $\hat{S}:C\to C$ is defined as $\hat{S}x=\lim_{n\to \infty }S_{n}x$ $\forall x\in C$.

• $\{\sigma _{n}\}\subset (0,1]$ and $\{\alpha _{n}\},\{\beta _{n}\},\{\gamma _{n}\}\subset (0,1)$ with $\alpha _{n}+\beta _{n}+\gamma _{n} =1$ $\forall n\geq 1$ such that:

  • (i) $\sum^{\infty }_{n=1}\alpha _{n}=\infty$, $\lim_{n\to \infty }\alpha _{n}=0$ and $\lim_{n\to \infty }\frac{\theta _{n}}{\alpha _{n}}=0$;

  • (ii) $0<\liminf_{n\to \infty }\sigma _{n}\leq \limsup_{n\to \infty } \sigma _{n}<1$;

  • (iii) $0<\liminf_{n\to \infty }\beta _{n}\leq \limsup_{n\to \infty }\beta _{n}<1$.

Algorithm 3.1

Initialization: Given $\gamma >0$, $\mu \in (0,1)$, $\ell \in (0,1)$, pick an initial $x_{1}\in C$ arbitrarily.

Iterative steps: Compute $x_{n+1}$ below:

Step 1. Calculate $u_{n}=\sigma _{n}x_{n}+(1-\sigma _{n})S_{n}u_{n}$ and $w_{n}=Gu_{n}$, and set $y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n})$, where $\tau _{n}$ is chosen to be the largest $\tau \in \{\gamma ,\gamma \ell ,\gamma \ell ^{2},\dots \}$ satisfying

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

(3.1)

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

Step 3. Calculate

$$ x_{n+1}=\alpha _{n}f(x_{n})+\beta _{n}x_{n}+\gamma _{n}S_{n}z_{n}. $$

(3.2)

Again put $n:=n+1$ and return to Step 1.

Lemma 3.1

The Armijo-like search rule (3.1) is well defined, and the following inequality holds: $\min \{\gamma ,\mu \ell /L\}\leq \tau _{n}\leq \gamma$.

Proof

Thanks to $\|Aw_{n}-AP_{C}(w_{n}-\gamma \ell ^{m}Aw_{n})\|\leq L\|w_{n}-P_{C}(w_{n}- \gamma \ell ^{m}Aw_{n})\|$, we know that (3.1) holds for each $\gamma \ell ^{m}\leq \frac{\mu }{L}$ and so $\tau _{n}$ is well defined. Obviously, $\tau _{n}\leq \gamma$. In the case of $\tau _{n}=\gamma$, the conclusion is true. In the case of $\tau _{n}<\gamma$, from (3.1) one gets $\|Aw_{n}-AP_{C}(w_{n}-\frac{\tau _{n}}{\ell }Aw_{n})\|> \frac{\mu }{(\tau _{n}/\ell )}\|w_{n}-P_{C}(w_{n}- \frac{\tau _{n}}{\ell }Aw_{n})\|$, which hence leads to $\tau _{n}>\mu \ell /L$. □

Lemma 3.2

Let the sequences $\{u_{n}\}$, $\{w_{n}\}$, $\{y_{n}\}$, $\{z_{n}\}$ be constructed by Algorithm 3.1. Then for each $p\in \varOmega$, one has

$$ \begin{aligned} \Vert z_{n}-p \Vert ^{2}&\leq \Vert u_{n}-p \Vert ^{2}-(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr] \\ & \quad {}-\mu _{2}(2\beta -\mu _{2}) \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}-\mu _{1}(2 \alpha -\mu _{1}) \Vert B_{1}\upsilon _{n}-B_{1}q \Vert ^{2},\end{aligned} $$

(3.3)

where $q=P_{C}(p-\mu _{2}B_{2}p)$ and $\upsilon _{n}=P_{C}(u_{n}-\mu _{2}B_{2}u_{n})$.

Proof

Define $T_{n}x:=\beta _{n}x_{n}+(1-\beta _{n})S_{n}x$, $x \in C$, for each $n\geq 0$. Then $T_{n}$ is continuous by the continuity of $S_{n}$ and

$$\begin{aligned} \langle T_{n}x-T_{n}y, x-y \rangle =&(1-\beta _{n})\langle S_{n}x-S_{n}y, x-y \rangle \\ \leq & (1-\beta _{n}) \Vert x-y \Vert ^{2} \\ \leq & \bar{\beta }_{n} \Vert x-y \Vert ^{2}, \end{aligned}$$

where $\bar{\beta }_{n}:=1-\beta _{n}\in (0, 1)$ and this implies that $T_{n}$ is a strong pseudocontractive mapping. Hence, by Proposition 2.2, there exists a unique element $u_{n}\in C$ such that for each $n\geq 0$,

$$ u_{n}=\beta _{n}x_{n}+(1-\beta _{n})S_{n}u_{n}. $$

Observe that for each $p\in \varOmega \subset C\subset C_{n}$,

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

which hence yields

$$ \Vert z_{n}-p \Vert ^{2}\leq \Vert w_{n}-p \Vert ^{2}- \Vert z_{n}-w_{n} \Vert ^{2}-2\tau _{n} \langle z_{n}-p,Ay_{n} \rangle . $$

Owing to $z_{n}=P_{C_{n}}(w_{n}-\tau _{n}Ay_{n})$ with $C_{n}:=\{y\in H:\langle w_{n}-\tau _{n}Aw_{n}-y_{n},y-y_{n}\rangle \leq 0\}$, one gets $\langle w_{n}-\tau _{n}Aw_{n}-y_{n},z_{n}-y_{n}\rangle \leq 0$. Combining (3.1) and the pseudomonotonicity of A guarantees that

$$ \begin{aligned} \Vert z_{n}-p \Vert ^{2}&\leq \Vert w_{n}-p \Vert ^{2}- \Vert z_{n}-w_{n} \Vert ^{2}-2\tau _{n} \langle Ay_{n},y_{n}-p+z_{n}-y_{n} \rangle \\ &\leq \Vert w_{n}-p \Vert ^{2}- \Vert z_{n}-w_{n} \Vert ^{2}-2\tau _{n} \langle Ay_{n},z_{n}-y_{n} \rangle \\ &= \Vert w_{n}-p \Vert ^{2}- \Vert z_{n}-y_{n} \Vert ^{2}- \Vert y_{n}-w_{n} \Vert ^{2}+2\langle w_{n}- \tau _{n}Ay_{n}-y_{n},z_{n}-y_{n} \rangle \\ &= \Vert w_{n}-p \Vert ^{2}- \Vert z_{n}-y_{n} \Vert ^{2}- \Vert y_{n}-w_{n} \Vert ^{2}+2\langle w_{n}- \tau _{n}Aw_{n}-y_{n},z_{n}-y_{n} \rangle \\ & \quad {}+2\tau _{n}\langle Aw_{n}-Ay_{n},z_{n}-y_{n} \rangle \\ &\leq \Vert w_{n}-p \Vert ^{2}- \Vert z_{n}-y_{n} \Vert ^{2}- \Vert y_{n}-w_{n} \Vert ^{2}+2\mu \Vert w_{n}-y_{n} \Vert \Vert z_{n}-y_{n} \Vert \\ &\leq \Vert w_{n}-p \Vert ^{2}- \Vert z_{n}-y_{n} \Vert ^{2}- \Vert y_{n}-w_{n} \Vert ^{2}+\mu \bigl( \Vert w_{n}-y_{n} \Vert ^{2}+ \Vert z_{n}-y_{n} \Vert ^{2}\bigr) \\ &= \Vert w_{n}-p \Vert ^{2}-(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr].\end{aligned} $$

(3.4)

Note that $q=P_{C}(p-\mu _{2}B_{2}p)$, $\upsilon _{n}=P_{C}(u_{n}-\mu _{2}B_{2}u_{n})$, and $w_{n}=P_{C}(\upsilon _{n}-\mu _{1}B_{1}\upsilon _{n})$. Then $w_{n}=Gu_{n}$. By Lemma 2.1, one has

$$ \Vert \upsilon _{n}-q \Vert ^{2}\leq \Vert u_{n}-p \Vert ^{2}-\mu _{2}(2\beta -\mu _{2}) \Vert B_{2}u_{n}-B_{2}p \Vert ^{2} $$

and

$$ \Vert w_{n}-p \Vert ^{2}\leq \Vert \upsilon _{n}-q \Vert ^{2}-\mu _{1}(2\alpha -\mu _{1}) \Vert B_{1}\upsilon _{n}-B_{1}q \Vert ^{2}. $$

Combining the last two inequalities, one gets

$$ \Vert w_{n}-p \Vert ^{2}\leq \Vert u_{n}-p \Vert ^{2}-\mu _{2}(2\beta -\mu _{2}) \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}-\mu _{1}(2\alpha -\mu _{1}) \Vert B_{1}\upsilon _{n}-B_{1}q \Vert ^{2}. $$

This, together with (3.4), implies that inequality (3.3) holds. □

Lemma 3.3

Suppose that $\{u_{n}\}$, $\{x_{n}\}$ are bounded sequences constructed by Algorithm 3.1. Assume that $x_{n}- x_{n+1}\to 0$, $u_{n}-Gu_{n}\to 0$, and $S^{n}x_{n}-S^{n+1}x_{n}\to 0$, and suppose there exists a subsequence $\{x_{n_{k}}\}\subset \{x_{n}\}$ such that $x_{n_{k}}\rightharpoonup z\in C$. Then $z\in \varOmega$.

Proof

From Algorithm 3.1, we obtain that for each $p\in \varOmega$,

$$ \begin{aligned} \Vert u_{n}-p \Vert ^{2}&= \sigma _{n}\langle x_{n}-p,u_{n}-p\rangle +(1- \sigma _{n})\langle S_{n}u_{n}-p,u_{n}-p \rangle \\ &\leq \sigma _{n}\langle x_{n}-p,u_{n}-p\rangle +(1-\sigma _{n}) \Vert u_{n}-p \Vert ^{2},\end{aligned} $$

which hence yields

$$ \begin{aligned} \Vert u_{n}-p \Vert ^{2}&\leq \langle x_{n}-p,u_{n}-p\rangle \\ &=\frac{1}{2}\bigl[ \Vert x_{n}-p \Vert ^{2}+ \Vert u_{n}-p \Vert ^{2}- \Vert x_{n}-u_{n} \Vert ^{2}\bigr].\end{aligned} $$

This immediately implies that

$$ \Vert u_{n}-p \Vert ^{2}\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n}-u_{n} \Vert ^{2}. $$

(3.5)

So it follows from (3.3) and the last inequality that

$$ \begin{aligned} \Vert z_{n}-p \Vert ^{2}&\leq \Vert u_{n}-p \Vert ^{2}-(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr] \\ &\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n}-u_{n} \Vert ^{2}-(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr],\end{aligned} $$

which, together with Algorithm 3.1, leads to

$$ \begin{aligned} &\Vert x_{n+1}-p \Vert ^{2}\\ &\quad= \bigl\Vert \alpha _{n}\bigl(f(x_{n})-p\bigr)+\beta _{n}(x_{n}-p)+ \gamma _{n}\bigl(S^{n}z_{n}-p \bigr) \bigr\Vert ^{2} \\ &\quad\leq \alpha _{n} \bigl\Vert f(x_{n})-p \bigr\Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+\gamma _{n} \bigl\Vert S^{n}z_{n}-p \bigr\Vert ^{2}-\beta _{n}\gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2} \\ &\quad\leq \alpha _{n} \bigl\Vert f(x_{n})-p \bigr\Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+\gamma _{n}(1+ \theta _{n})^{2} \Vert z_{n}-p \Vert ^{2}-\beta _{n}\gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2} \\ &\quad\leq \alpha _{n} \bigl\Vert f(x_{n})-p \bigr\Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+\gamma _{n}(1+ \theta _{n})^{2}\bigl\{ \Vert x_{n}-p \Vert ^{2}- \Vert x_{n}-u_{n} \Vert ^{2} \\ &\quad\quad {}-(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2} \bigr]\bigr\} -\beta _{n} \gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2} \\ &\quad\leq \alpha _{n} \bigl\Vert f(x_{n})-p \bigr\Vert ^{2}+ \Vert x_{n}-p \Vert ^{2}+\theta _{n}(2+ \theta _{n}) \Vert x_{n}-p \Vert ^{2}-\gamma _{n}(1+\theta _{n})^{2}\bigl\{ \Vert x_{n}-u_{n} \Vert ^{2} \\ &\quad\quad {}+(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2} \bigr]\bigr\} -\beta _{n} \gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2}.\end{aligned} $$

This immediately ensures that

$$ \begin{aligned} &\gamma _{n}(1+\theta _{n})^{2} \bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu ) \bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]\bigr\} +\beta _{n}\gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2} \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+\alpha _{n} \bigl\Vert f(x_{n})-p \bigr\Vert ^{2}+ \theta _{n}(2+\theta _{n}) \Vert x_{n}-p \Vert ^{2} \\ &\quad\leq \Vert x_{n}-x_{n+1} \Vert \bigl( \Vert x_{n}-p \Vert + \Vert x_{n+1}-p \Vert \bigr)+\alpha _{n} \bigl\Vert f(x_{n})-p \bigr\Vert ^{2}+ \theta _{n}(2+\theta _{n}) \Vert x_{n}-p \Vert ^{2}.\end{aligned} $$

Note that $\lim_{n\to \infty }\alpha _{n}=0$ and $0<\liminf_{n\to \infty }\beta _{n}\leq \limsup_{n\to \infty }\beta _{n}<1$. Thus we know that $\liminf_{n\to \infty }\gamma _{n}=\liminf_{n\to \infty }(1-\alpha _{n}- \beta _{n})=1-\limsup_{n\to \infty }\beta _{n}>0$. Since $\theta _{n}\to 0$, $x_{n}-x_{n+1}\to 0$ and $\mu \in (0,1)$, by the boundedness of $\{x_{n}\}$, we get

$$ \lim_{n\to \infty } \Vert x_{n}-u_{n} \Vert = \lim_{n\to \infty } \Vert y_{n}-z_{n} \Vert = \lim_{n\to \infty } \Vert y_{n}-w_{n} \Vert =\lim _{n\to \infty } \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert =0. $$

(3.6)

So it follows that $\|w_{n}-x_{n}\|\leq \|Gu_{n}-u_{n}\|+\|u_{n}-x_{n}\|\to 0$ ($n \to \infty$),

$$ \begin{aligned} \Vert z_{n}-x_{n} \Vert &\leq \Vert z_{n}-w_{n} \Vert + \Vert w_{n}-x_{n} \Vert \\ &\leq \Vert z_{n}-y_{n} \Vert + \Vert y_{n}-w_{n} \Vert + \Vert w_{n}-x_{n} \Vert \to 0\quad (n \to \infty ),\end{aligned} $$

and $\|x_{n}-y_{n}\|\leq \|x_{n}-z_{n}\|+\|z_{n}-y_{n}\|\to 0$ ($n\to \infty$).

We show that $\lim_{n\to \infty }\|x_{n}-Sx_{n}\|=0$. In fact, using the asymptotical nonexpansivity of S, one obtains that

$$ \begin{aligned} \Vert x_{n}-Sx_{n} \Vert & \leq \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert + \bigl\Vert S^{n}z_{n}-S^{n}x_{n} \bigr\Vert + \bigl\Vert S^{n}x_{n}-S^{n+1}x_{n} \bigr\Vert \\ &\quad {}+ \bigl\Vert S^{n+1}x_{n}-S^{n+1}z_{n} \bigr\Vert + \bigl\Vert S^{n+1}z_{n}-Sx_{n} \bigr\Vert \\ &\leq \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert +(1+\theta _{n}) \Vert z_{n}-x_{n} \Vert + \bigl\Vert S^{n}x_{n}-S^{n+1}x_{n} \bigr\Vert \\ &\quad {}+(1+\theta _{n+1}) \Vert x_{n}-z_{n} \Vert +(1+\theta _{1}) \bigl\Vert S^{n}z_{n}-x_{n} \bigr\Vert \\ &=(2+\theta _{1}) \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert +(2+\theta _{n}+\theta _{n+1}) \Vert z_{n}-x_{n} \Vert + \bigl\Vert S^{n}x_{n}-S^{n+1}x_{n} \bigr\Vert .\end{aligned} $$

Since $x_{n}-S^{n}z_{n}\to 0$, $x_{n}-z_{n}\to 0$ and $S^{n}x_{n}-S^{n+1}x_{n}\to 0$, we obtain

$$ \lim_{n\to \infty } \Vert x_{n}-Sx_{n} \Vert =0. $$

(3.7)

We show that $\lim_{n\to \infty }\|x_{n}-\bar{S}x_{n}\|=0$ where $\bar{S}:=(2I-\hat{S})^{-1}$. In fact, noticing $u_{n}=\sigma _{n} x_{n}+(1-\sigma _{n})S_{n}u_{n}$ and $x_{n}-u_{n}\to 0$, we get

$$ (1-\sigma _{n}) \Vert S_{n}u_{n}-u_{n} \Vert =\sigma _{n} \Vert x_{n}-u_{n} \Vert \leq \Vert x_{n}-u_{n} \Vert \to 0\quad (n\to \infty ), $$

which, together with $0<\liminf_{n\to \infty }(1-\sigma _{n})$, yields

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

Since $\{S_{n}\}^{\infty }_{n=1}$ is ς-uniformly Lipschitzian on C, we deduce from $x_{n}-u_{n}\to 0$ and $S_{n}u_{n}- u_{n}\to 0$ that

$$ \begin{aligned} \Vert S_{n}x_{n}-x_{n} \Vert &\leq \Vert S_{n}x_{n}-S_{n}u_{n} \Vert + \Vert S_{n}u_{n}-u_{n} \Vert + \Vert u_{n}-x_{n} \Vert \\ &\leq (\varsigma +1) \Vert u_{n}-x_{n} \Vert + \Vert S_{n}u_{n}-u_{n} \Vert \to 0\quad (n \to \infty ).\end{aligned} $$

It is clear that $\hat{S}:C\to C$ is pseudocontractive and ς-Lipschitzian where $\hat{S}x=\lim_{n\to \infty }S_{n}x$ $\forall x\in C$. We claim that $\lim_{n\to \infty }\|\hat{S}x_{n}-x_{n}\|=0$. Using the boundedness of $\{x_{n}\}$ and putting $D=\overline{\mathrm {conv}}\{x_{n}:n\geq 1\}$ (the closed convex hull of the set $\{x_{n}:n\geq 1\}$), by the hypothesis, we get $\sum^{\infty }_{n=1} \sup_{x\in D}\|S_{n}x-S_{n+1}x\|<\infty$. So, by Proposition 2.1, we have $\lim_{n\to \infty }\sup_{x\in D}\|S_{n}x-\hat{S}x\| =0$, which immediately arrives at

$$ \lim_{n\to \infty } \Vert S_{n}x_{n}- \hat{S}x_{n} \Vert =0. $$

Consequently,

$$ \Vert x_{n}-\hat{S}x_{n} \Vert \leq \Vert x_{n}-S_{n}x_{n} \Vert + \Vert S_{n}x_{n}-\hat{S}x_{n} \Vert \to 0\quad (n\to \infty ). $$

Now, let us show that if we define $\bar{S}:=(2I-\hat{S})^{-1}$, then $\bar{S}:C\to C$ is nonexpansive, $\mathrm {Fix}(\bar{S})=\mathrm {Fix}(\hat{S})=\bigcap^{\infty }_{n=1}\mathrm {Fix}(S_{n})$, and $\lim_{n\to \infty }\|x_{n}-\bar{S}x_{n}\|=0$. As a matter of fact, it is known that S̄ is nonexpansive and $\mathrm {Fix}(\bar{S})=\mathrm {Fix}(\hat{S})=\bigcap^{\infty }_{n=1}\mathrm {Fix}(S_{n})$ as a consequence of [32, Theorem 6]. From $x_{n}-\hat{S}x_{n}\to 0$, it follows that

$$ \begin{aligned} \Vert x_{n}-\bar{S} x_{n} \Vert &= \bigl\Vert \bar{S}\bar{S}^{-1}x_{n}-\bar{S} x_{n} \bigr\Vert \\ &\leq \bigl\Vert \bar{S}^{-1}x_{n}-x_{n} \bigr\Vert = \bigl\Vert (2I-\hat{S})x_{n}-x_{n} \bigr\Vert = \Vert x_{n}- \hat{S} x_{n} \Vert \to 0\quad (n\to \infty ).\end{aligned} $$

(3.8)

Next, let us show $z\in \mathrm {VI}(C,A)$. Indeed, noticing $w_{n}-x_{n}\to 0$ and $x_{n_{k}}\rightharpoonup z$, we have $w_{n_{k}} \rightharpoonup z$. We consider two cases below.

If $Az=0$, then it is clear that $z\in \mathrm {VI}(C,A)$ because $\langle Az,x-z\rangle \geq 0$ $\forall x\in C$.

Assume that $Az\neq 0$. Since $w_{n_{k}}\rightharpoonup z$ as $k\to \infty$, utilizing the assumption on A, instead of the sequentially weak continuity of A, we get $0<\|Az\|\leq \liminf_{k\to \infty }\|Aw_{n_{k}}\|$. So, we could suppose that $\|Aw_{n_{k}}\|\neq 0$ $\forall k\geq 1$. Moreover, from $y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n})$, we have $\langle w_{n}-\tau _{n} Aw_{n}-y_{n},x-y_{n}\rangle \leq 0$ $\forall x \in C$, and hence

$$ \frac{1}{\tau _{n}}\langle w_{n}-y_{n},x-y_{n} \rangle +\langle Aw_{n},y_{n}-w_{n} \rangle \leq \langle Aw_{n},x-w_{n}\rangle \quad \forall x\in C. $$

(3.9)

According to the Lipschitz continuity of A, one knows that $\{Aw_{n}\}$ is bounded. Note that $\{y_{n}\}$ is bounded as well. Using Lemma 3.1, from (3.9) we get $\liminf_{k\to \infty }\langle Aw_{n_{k}}$, $x-w_{n_{k}}\rangle \geq 0$ $\forall x\in C$.

To show that $z\in \mathrm {VI}(C,A)$, we now choose a sequence $\{\varepsilon _{k}\}\subset (0,1)$ satisfying $\varepsilon _{k}\downarrow 0$ as $k\to \infty$. For each $k\geq 1$, we denote by $m_{k}$ the smallest positive integer such that

$$ \langle Aw_{n_{j}},x-w_{n_{j}}\rangle +\varepsilon _{k} \geq 0\quad \forall j\geq m_{k}. $$

(3.10)

Since $\{\varepsilon _{k}\}$ is decreasing, it can be readily seen that $\{m_{k}\}$ is increasing. Noticing that $Aw_{m_{k}} \neq 0$ $\forall k\geq 1$ (due to $\{Aw_{m_{k}}\}\subset \{Aw_{n_{k}}\}$), we set $\varrho _{m_{k}}=\frac{Aw_{m_{k}}}{\|Aw_{m_{k}}\|^{2}}$, we get $\langle Aw_{m_{k}},\varrho _{m_{k}}\rangle =1$ $\forall k\geq 1$. So, from (3.10) we get $\langle Aw_{m_{k}},x+\varepsilon _{k}\varrho _{m_{k}}-w_{m_{k}} \rangle \geq 0$ $\forall k\geq 1$. Again from the pseudomonotonicity of A, we have $\langle A(x+\varepsilon _{k}\varrho _{m_{k}}),x+\varepsilon _{k} \varrho _{m_{k}}-w_{m_{k}}\rangle \geq 0$ $\forall k\geq 1$. This immediately leads to

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

(3.11)

We claim that $\lim_{k\to \infty }\varepsilon _{k}\varrho _{m_{k}}=0$. Note that $\{w_{m_{k}}\}\subset \{w_{n_{k}}\}$ and $\varepsilon _{k}\downarrow 0$ as $k\to \infty$. So it follows that $0\leq \limsup_{k\to \infty }\|\varepsilon _{k} \varrho _{m_{k}}\|= \limsup_{k\to \infty }\frac{\varepsilon _{k}}{\|Aw_{m_{k}}\|} \leq \frac{\limsup_{k\to \infty }\varepsilon _{k}}{\liminf_{k\to \infty }\|Aw_{n_{k}}\|}=0$. Hence we get $\varepsilon _{k} \varrho _{m_{k}}\to 0$ as $k\to \infty$. Thus, letting $k\to \infty$, we deduce that the right-hand side of (3.11) tends to zero by the Lipschitz continuity of A, the boundedness of $\{w_{m_{k}}\}$, $\{\varrho _{m_{k}}\}$ and the limit $\lim_{k\to \infty }\varepsilon _{k}\varrho _{m_{k}}=0$. Therefore, we get $\langle Ax,x-z\rangle =\liminf_{k\to \infty }\langle Ax,x-w_{m_{k}} \rangle \geq 0$ $\forall x\in C$. By Lemma 2.3, we have $z\in \mathrm {VI}(C,A)$.

Next we show that $z\in \varOmega$. In fact, from $x_{n}-u_{n}\to 0$ and $x_{n_{k}}\rightharpoonup z$, we get $u_{n_{k}} \rightharpoonup z$. Note that the condition $u_{n}-Gu_{n}\to 0$ guarantees $u_{n_{k}}-Gu_{n_{k}}\to 0$. From Lemma 2.5, it follows that $I-G$ is demiclosed at zero. Hence we get $(I-G)z=0$, i.e., $z\in \mathrm {Fix}(G)$. In the meantime, let us show that $z\in \bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})$. Again from Lemma 2.5, we know that $I-S$ and $I-\bar{S}$ are demiclosed at zero. Noticing $x_{n_{k}}-Sx_{n_{k}}\to 0$ (due to (3.7)) and $x_{n_{k}}-\bar{S}x_{n_{k}}\to 0$ (due to (3.8)), we deduce from $x_{n_{k}}\rightharpoonup z$ that $z\in \mathrm {Fix}(S)$ and $z\in \mathrm {Fix}(\bar{S})=\bigcap^{\infty }_{i=1}\mathrm {Fix}(S_{i})$. Consequently, $z\in \bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G)\cap \mathrm {VI}(C,A)=\varOmega$ with $S_{0}:=S$. This completes the proof. □

Theorem 3.1

Let $\{x_{n}\}$ be the sequence constructed in Algorithm 3.1. Then $x_{n}\to x^{*}\in \varOmega$, provided $S^{n}x_{n}-S^{n+1}x_{n}\to 0$, where $x^{*}\in \varOmega$ is the unique solution to the HVI, $\langle (I-f)x^{*}, p-x^{*}\rangle \geq 0$ $\forall p\in \varOmega$.

Proof

First of all, since $0<\liminf_{n\to \infty }\sigma _{n}\leq \limsup_{n\to \infty } \sigma _{n}<1$ and $\lim_{n\to \infty }\frac{\theta _{n}}{\alpha _{n}}=0$, we may assume, without loss of generality, that $\{\sigma _{n}\}\subset [a,b]\subset (0,1)$ and $\theta _{n}\leq \frac{\alpha _{n}(1-\delta )}{2}$ $\forall n\geq 1$. We claim that $P_{\varOmega}\circ f:C\to C$ is a contraction. In fact, it is clear that $P_{\varOmega}\circ f$ is a contraction. Banach’s contraction mapping principle guarantees that $P_{\varOmega}\circ f$ has a unique fixed point, say $x^{*}\in C$, i.e., $x^{*}=P_{\varOmega}f(x^{*})$. Thus, there exists a unique solution $x^{*}\in \varOmega =\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G) \cap \mathrm {VI}(C,A)$ of the HVI

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

(3.12)

Next we divide the rest of the proof into several steps.

Step 1. We show that $\{x_{n}\}$ is bounded. In fact, take an arbitrary $p\in \varOmega =\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G) \cap \mathrm {VI}(C,A)$. Then $Sp=p$, $S_{n}p=p$ $\forall n\geq 1$, $Gp=p$ and (3.3) holds, i.e.,

$$ \begin{aligned} \Vert z_{n}-p \Vert ^{2}&\leq \Vert u_{n}-p \Vert ^{2}-(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr] \\ &\quad {}-\mu _{2}(2\beta -\mu _{2}) \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}-\mu _{1}(2 \alpha -\mu _{1}) \Vert B_{1}\upsilon _{n}-B_{1}q \Vert ^{2},\end{aligned} $$

(3.13)

where $q=P_{C}(p-\mu _{2}B_{2}p)$ and $\upsilon _{n}=P_{C}(u_{n}-\mu _{2}B_{2}u_{n})$. Again from (3.4) and (3.5), we deduce that

$$ \Vert z_{n}-p \Vert \leq \Vert w_{n}-p \Vert = \Vert Gu_{n}-p \Vert \leq \Vert u_{n}-p \Vert \leq \Vert x_{n}-p \Vert \quad \forall n\geq 1. $$

(3.14)

Thus, using (3.14) and $\alpha _{n}+\beta _{n}+\gamma _{n}=1$ $\forall n\geq 1$, from the asymptotical nonexpansivity of S, we obtain

$$\begin{aligned} \Vert x_{n+1}-p \Vert &\leq \alpha _{n} \bigl\Vert f(x_{n})-p \bigr\Vert +\beta _{n} \Vert x_{n}-p \Vert + \gamma _{n} \bigl\Vert S^{n}z_{n}-p \bigr\Vert \\ &\leq \alpha _{n}\bigl( \bigl\Vert f(x_{n})-f(p) \bigr\Vert + \bigl\Vert f(p)-p \bigr\Vert \bigr)+\beta _{n} \Vert x_{n}-p \Vert + \gamma _{n}(1+\theta _{n}) \Vert z_{n}-p \Vert \\ &\leq \alpha _{n}\delta \Vert x_{n}-p \Vert +\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert +\beta _{n} \Vert x_{n}-p \Vert +(\gamma _{n}+\theta _{n}) \Vert x_{n}-p \Vert \\ &\leq \alpha _{n}\delta \Vert x_{n}-p \Vert +\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert +(1-\alpha _{n}) \Vert x_{n}-p \Vert +\frac{\alpha _{n}(1-\delta )}{2} \Vert x_{n}-p \Vert \\ &=\biggl[1-\frac{\alpha _{n}(1-\delta )}{2}\biggr] \Vert x_{n}-p \Vert +\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \\ &=\biggl[1-\frac{\alpha _{n}(1-\delta )}{2}\biggr] \Vert x_{n}-p \Vert + \frac{\alpha _{n}(1-\delta )}{2}\frac{2 \Vert f(p)-p \Vert }{1-\delta } \\ &\leq \max \biggl\{ \Vert x_{n}-p \Vert ,\frac{2 \Vert f(p)-p \Vert }{1-\delta }\biggr\} . \end{aligned}$$

By induction, we obtain $\|x_{n}-p\|\leq \max \{\|x_{1}-p\|,\frac{2\|f(p)-p\|}{1-\delta }\}$ $\forall n\geq 1$. Therefore, $\{x_{n}\}$ is bounded, and so are the sequences $\{u_{n}\}$, $\{w_{n}\}$, $\{y_{n}\}$, $\{z_{n}\}$, $\{f(x_{n})\}$, $\{Ay_{n}\}$, $\{S_{n}u_{n}\}$, $\{S^{n}z_{n}\}$.

Step 2. We show that

$$ \begin{aligned} &\gamma _{n}\bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]+\mu _{2}(2 \beta -\mu _{2}) \\ &\quad\quad {}\times \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}+\mu _{1}(2\alpha -\mu _{1}) \Vert B_{1} \upsilon _{n}-B_{1}q \Vert ^{2} \bigr\} \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+\theta _{n}(2+\theta _{n})M_{0}+2 \alpha _{n}M_{0}\end{aligned} $$

(3.15)

and

$$ \begin{aligned} &\gamma _{n}\bigl[ \Vert u_{n}- \upsilon _{n}+q-p \Vert ^{2}+ \Vert \upsilon _{n}-w_{n}+p-q \Vert ^{2}\bigr] \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+2\mu _{2} \Vert B_{2}p-B_{2}u_{n} \Vert \Vert \upsilon _{n}-q \Vert \\ &\quad\quad {}+2\mu _{1} \Vert B_{1}q-B_{1} \upsilon _{n} \Vert \Vert w_{n}-p \Vert +\theta _{n}(2+ \theta _{n})M_{0}+2\alpha _{n}M_{0},\end{aligned} $$

(3.16)

for some $M_{0}>0$. In fact, using (3.5), (3.13), (3.14), and the convexity of the function $\phi (s)=s^{2}$ $\forall s\in \mathbf{R}$, we get

$$\begin{aligned} & \Vert x_{n+1}-p \Vert ^{2} \\ &\quad = \bigl\Vert \alpha _{n}\bigl(f(x_{n})-f(p)\bigr)+\beta _{n}(x_{n}-p)+\gamma _{n}\bigl(S^{n}z_{n}-p \bigr)+ \alpha _{n}\bigl(f(p)-p\bigr) \bigr\Vert ^{2} \\ &\quad\leq \bigl\Vert \alpha _{n}\bigl(f(x_{n})-f(p)\bigr)+ \beta _{n}(x_{n}-p)+\gamma _{n}\bigl(S^{n}z_{n}-p \bigr) \bigr\Vert ^{2}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\quad\leq \alpha _{n} \bigl\Vert f(x_{n})-f(p) \bigr\Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+ \gamma _{n} \bigl\Vert S^{n}z_{n}-p \bigr\Vert ^{2}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p \bigr\rangle \\ &\quad\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+ \gamma _{n}(1+\theta _{n})^{2} \Vert z_{n}-p \Vert ^{2}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p \bigr\rangle \\ &\quad\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+\bigl[ \gamma _{n}+\theta _{n}(2+\theta _{n})\bigr] \Vert z_{n}-p \Vert ^{2}+2\alpha _{n} \bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\quad\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+ \gamma _{n}\bigl\{ \Vert u_{n}-p \Vert ^{2}-(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2} \bigr] \\ &\quad\quad {}-\mu _{2}(2\beta -\mu _{2}) \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}-\mu _{1}(2 \alpha -\mu _{1}) \Vert B_{1}\upsilon _{n}-B_{1}q \Vert ^{2}\bigr\} \\ &\quad\quad {}+\theta _{n}(2+ \theta _{n}) \Vert x_{n}-p \Vert ^{2}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\quad\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+ \gamma _{n}\bigl\{ \Vert x_{n}-p \Vert ^{2}- \Vert x_{n}-u_{n} \Vert ^{2}-(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2} \\ &\quad\quad {}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]- \mu _{2}(2\beta -\mu _{2}) \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}-\mu _{1}(2\alpha -\mu _{1}) \Vert B_{1}\upsilon _{n}-B_{1}q \Vert ^{2} \bigr\} \\ &\quad\quad {}+\theta _{n}(2+\theta _{n}) \Vert x_{n}-p \Vert ^{2}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p \bigr\rangle \\ &\quad=\bigl[1-\alpha _{n}(1-\delta )\bigr] \Vert x_{n}-p \Vert ^{2}-\gamma _{n}\bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr] \\ &\quad\quad {}+\mu _{2}(2\beta -\mu _{2}) \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}+\mu _{1}(2 \alpha -\mu _{1}) \Vert B_{1}\upsilon _{n}-B_{1}q \Vert ^{2}\bigr\} \\ &\quad\quad {}+\theta _{n}(2+\theta _{n}) \Vert x_{n}-p \Vert ^{2}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p \bigr\rangle \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}-\gamma _{n} \bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu ) \bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr] \\ &\quad\quad {}+\mu _{2}(2\beta -\mu _{2}) \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}+\mu _{1}(2 \alpha -\mu _{1}) \Vert B_{1}\upsilon _{n}-B_{1}q \Vert ^{2}\bigr\} \\ &\quad\quad {}+\theta _{n}(2+\theta _{n})M_{0}+2\alpha _{n}M_{0}, \end{aligned}$$

(3.17)

where $\sup_{n\geq 1}\{\|x_{n}-p\|^{2}+\|f(p)-p\|\|x_{n}-p\|\}\leq M_{0}$ for some $M_{0}>0$. This ensures that (3.15) holds.

On the other hand, by the firm nonexpansivity of $P_{C}$ we obtain that

$$ \begin{aligned} \Vert w_{n}-p \Vert ^{2}&\leq \langle \upsilon _{n}-q,w_{n}-p\rangle +\mu _{1} \langle B_{1}q-B_{1}\upsilon _{n},w_{n}-p \rangle \\ &\leq \frac{1}{2}\bigl[ \Vert \upsilon _{n}-q \Vert ^{2}+ \Vert w_{n}-p \Vert ^{2}- \Vert \upsilon _{n}-w_{n}+p-q \Vert ^{2}\bigr] \\ &\quad {}+\mu _{1} \Vert B_{1}q-B_{1}\upsilon _{n} \Vert \Vert w_{n}-p \Vert ,\end{aligned} $$

which hence gives

$$ \Vert w_{n}-p \Vert ^{2}\leq \Vert \upsilon _{n}-q \Vert ^{2}- \Vert \upsilon _{n}-w_{n}+p-q \Vert ^{2}+2\mu _{1} \Vert B_{1}q-B_{1} \upsilon _{n} \Vert \Vert w_{n}-p \Vert . $$

(3.18)

In a similar way, we have

$$ \Vert \upsilon _{n}-q \Vert ^{2}\leq \Vert u_{n}-p \Vert ^{2}- \Vert u_{n}-\upsilon _{n}+q-p \Vert ^{2}+2\mu _{2} \Vert B_{2}p-B_{2}u_{n} \Vert \Vert \upsilon _{n}-q \Vert . $$

(3.19)

Substituting (3.19) for (3.18), from (3.14) we deduce that

$$ \begin{aligned} \Vert w_{n}-p \Vert ^{2}&\leq \Vert x_{n}-p \Vert ^{2}- \Vert u_{n}- \upsilon _{n}+q-p \Vert ^{2}- \Vert \upsilon _{n}-w_{n}+p-q \Vert ^{2} \\ &\quad {}+2\mu _{2} \Vert B_{2}p-B_{2}u_{n} \Vert \Vert \upsilon _{n}-q \Vert +2\mu _{1} \Vert B_{1}q-B_{1} \upsilon _{n} \Vert \Vert w_{n}-p \Vert ,\end{aligned} $$

which, together with (3.14) and (3.17), leads to

$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} &\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+ \beta _{n} \Vert x_{n}-p \Vert ^{2}+\bigl[\gamma _{n}+\theta _{n}(2+\theta _{n})\bigr] \Vert z_{n}-p \Vert ^{2} \\ &\quad {}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+ \gamma _{n} \Vert w_{n}-p \Vert ^{2}+\theta _{n}(2+\theta _{n}) \Vert x_{n}-p \Vert ^{2} \\ &\quad {}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2} \\ &\quad {}+ \gamma _{n}\bigl\{ \Vert x_{n}-p \Vert ^{2}- \Vert u_{n}-\upsilon _{n}+q-p \Vert ^{2}- \Vert \upsilon _{n}-w_{n}+p-q \Vert ^{2} \\ &\quad {}+2\mu _{2} \Vert B_{2}p-B_{2}u_{n} \Vert \Vert \upsilon _{n}-q \Vert +2\mu _{1} \Vert B_{1}q-B_{1} \upsilon _{n} \Vert \Vert w_{n}-p \Vert \bigr\} \\ &\quad {}+\theta _{n}(2+\theta _{n}) \Vert x_{n}-p \Vert ^{2}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\leq \bigl[1-\alpha _{n}(1-\delta )\bigr] \Vert x_{n}-p \Vert ^{2}-\gamma _{n}\bigl[ \Vert u_{n}- \upsilon _{n}+q-p \Vert ^{2}+ \Vert \upsilon _{n}-w_{n}+p-q \Vert ^{2}\bigr] \\ &\quad {}+2\mu _{2} \Vert B_{2}p-B_{2}u_{n} \Vert \Vert \upsilon _{n}-q \Vert +2\mu _{1} \Vert B_{1}q-B_{1} \upsilon _{n} \Vert \Vert w_{n}-p \Vert \\ &\quad {}+\theta _{n}(2+\theta _{n}) \Vert x_{n}-p \Vert ^{2}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\leq \Vert x_{n}-p \Vert ^{2}-\gamma _{n} \bigl[ \Vert u_{n}-\upsilon _{n}+q-p \Vert ^{2}+ \Vert \upsilon _{n}-w_{n}+p-q \Vert ^{2} \bigr] \\ &\quad {}+2\mu _{2} \Vert B_{2}p-B_{2}u_{n} \Vert \Vert \upsilon _{n}-q \Vert \\ &\quad {}+2\mu _{1} \Vert B_{1}q-B_{1} \upsilon _{n} \Vert \Vert w_{n}-p \Vert +\theta _{n}(2+ \theta _{n})M_{0}+2\alpha _{n}M_{0}. \end{aligned}$$

(3.20)

This ensures that (3.16) holds.

Step 3. We show that

$$ \begin{aligned} \Vert x_{n+1}-p \Vert ^{2}&\leq \bigl[1-\alpha _{n}(1-\delta )\bigr] \Vert x_{n}-p \Vert ^{2}\\ &\quad {}+ \alpha _{n}(1-\delta )\biggl\{ \frac{2\langle (f-I)p,x_{n+1}-p\rangle }{1-\delta }+ \frac{\theta _{n}}{\alpha _{n}} \cdot \frac{(2+\theta _{n})M_{0}}{1-\delta }\biggr\} . \end{aligned} $$

In fact, from (3.14) and (3.17), we have

$$ \begin{aligned} & \Vert x_{n+1}-p \Vert ^{2} \\ &\quad\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+\bigl[ \gamma _{n}+\theta _{n}(2+\theta _{n})\bigr] \Vert z_{n}-p \Vert ^{2}\\ &\qquad {}+2\alpha _{n} \bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\quad\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+ \gamma _{n} \Vert x_{n}-p \Vert ^{2}+\theta _{n}(2+\theta _{n})M_{0}\\ &\qquad {}+2 \alpha _{n} \bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\quad=\bigl[1-\alpha _{n}(1-\delta )\bigr] \Vert x_{n}-p \Vert ^{2}+\theta _{n}(2+\theta _{n})M_{0}+2 \alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\quad=\bigl[1-\alpha _{n}(1-\delta )\bigr] \Vert x_{n}-p \Vert ^{2}\\ &\qquad {}+\alpha _{n}(1-\delta )\biggl\{ \frac{2\langle (f-I)p,x_{n+1}-p\rangle }{1-\delta }+ \frac{\theta _{n}}{\alpha _{n}}\cdot \frac{(2+\theta _{n})M_{0}}{1-\delta }\biggr\} . \end{aligned} $$

(3.21)

Step 4. We show that $\{x_{n}\}$ converges strongly to the unique solution $x^{*}\in \varOmega$ of the HVI (3.12). In fact, putting $p=x^{*}$, we deduce from (3.21) that

$$ \begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}&\leq \bigl[1-\alpha _{n}(1-\delta )\bigr] \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+ \alpha _{n}(1-\delta ) \biggl[ \frac{2\langle (f-I)x^{*},x_{n+1}-x^{*}\rangle }{1-\delta } \\ &\quad {}+\frac{\theta _{n}}{\alpha _{n}}\cdot \frac{(2+\theta _{n})M_{0}}{1-\delta }\biggr].\end{aligned} $$

(3.22)

Putting $\Gamma _{n}=\|x_{n}-x^{*}\|^{2}$, we show the convergence of $\{\Gamma _{n}\}$ to zero by the following two cases.

Case 1. Suppose that there exists an integer $n_{0}\geq 1$ such that $\{\Gamma _{n}\}$ is nonincreasing. Then the limit $\lim_{n\to \infty }\Gamma _{n}=\hbar <+\infty$ and $\lim_{n\to \infty }(\Gamma _{n}-\Gamma _{n+1})=0$. Putting $p=x^{*}$ and $q=y^{*}$, from (3.15) and (3.16) we obtain

$$ \begin{aligned} &\gamma _{n}\bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]+\mu _{2}(2 \beta -\mu _{2}) \\ &\quad\quad {}\times \bigl\Vert B_{2}u_{n}-B_{2}x^{*} \bigr\Vert ^{2}+\mu _{1}(2\alpha -\mu _{1}) \bigl\Vert B_{1}\upsilon _{n}-B_{1}y^{*} \bigr\Vert ^{2}\bigr\} \\ &\quad\leq \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}+\theta _{n}(2+\theta _{n})M_{0}+2 \alpha _{n}M_{0} \\ &\quad=\Gamma _{n}-\Gamma _{n+1}+\theta _{n}(2+\theta _{n})M_{0}+2\alpha _{n}M_{0}\end{aligned} $$

(3.23)

and

$$ \begin{aligned} &\gamma _{n}\bigl[ \bigl\Vert u_{n}-\upsilon _{n}+y^{*}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert \upsilon _{n}-w_{n}+x^{*}-y^{*} \bigr\Vert ^{2}\bigr] \\ &\quad\leq \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}+2\mu _{2} \bigl\Vert B_{2}x^{*}-B_{2}u_{n} \bigr\Vert \bigl\Vert \upsilon _{n}-y^{*} \bigr\Vert \\ &\quad\quad {}+2\mu _{1} \bigl\Vert B_{1}y^{*}-B_{1} \upsilon _{n} \bigr\Vert \bigl\Vert w_{n}-x^{*} \bigr\Vert + \theta _{n}(2+\theta _{n})M_{0}+2 \alpha _{n}M_{0} \\ &\quad=\Gamma _{n}-\Gamma _{n+1}+2\mu _{2} \bigl\Vert B_{2}x^{*}-B_{2}u_{n} \bigr\Vert \bigl\Vert \upsilon _{n}-y^{*} \bigr\Vert \\ &\quad\quad {}+2\mu _{1} \bigl\Vert B_{1}y^{*}-B_{1} \upsilon _{n} \bigr\Vert \bigl\Vert w_{n}-x^{*} \bigr\Vert + \theta _{n}(2+\theta _{n})M_{0}+2 \alpha _{n}M_{0}.\end{aligned} $$

(3.24)

Noticing $0<\liminf_{n\to \infty }(1-\alpha _{n}-\beta _{n})=\liminf_{n\to \infty }\gamma _{n}$, $\alpha _{n}\to 0$, $\theta _{n}\to 0$ and $\Gamma _{n}-\Gamma _{n+1}\to 0$, one has from (3.23) that

$$ \lim_{n\to \infty } \Vert x_{n}-u_{n} \Vert = \lim_{n\to \infty } \Vert y_{n}-z_{n} \Vert = \lim_{n\to \infty } \Vert y_{n}-w_{n} \Vert =0, $$

(3.25)

and

$$ \lim_{n\to \infty } \bigl\Vert B_{2}u_{n}-B_{2}x^{*} \bigr\Vert =\lim_{n\to \infty } \bigl\Vert B_{1} \upsilon _{n}-B_{1}y^{*} \bigr\Vert =0. $$

(3.26)

Since $0<\liminf_{n\to \infty }\gamma _{n}$, $\alpha _{n}\to 0$, $\theta _{n}\to 0$ and $\Gamma _{n}-\Gamma _{n+1}\to 0$, from (3.24), (3.26), and the boundedness of $\{\upsilon _{n}\}$, $\{w_{n}\}$, we deduce that

$$ \lim_{n\to \infty } \bigl\Vert u_{n}-\upsilon _{n}+y^{*}-x^{*} \bigr\Vert =\lim _{n\to \infty } \bigl\Vert \upsilon _{n}-w_{n}+x^{*}-y^{*} \bigr\Vert =0. $$

(3.27)

Therefore,

$$ \begin{aligned} \Vert u_{n}-Gu_{n} \Vert &= \Vert u_{n}-w_{n} \Vert \\ &\leq \bigl\Vert u_{n}- \upsilon _{n}+y^{*}-x^{*} \bigr\Vert + \bigl\Vert \upsilon _{n}-w_{n}+x^{*}-y^{*} \bigr\Vert \\ &\to 0\quad (n\to \infty ).\end{aligned} $$

(3.28)

Furthermore, using (3.14), gives

$$ \begin{aligned} & \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2} \\ &\quad\leq \bigl\Vert \alpha _{n}\bigl(f(x_{n})-x^{*} \bigr)+\beta _{n}\bigl(x_{n}-x^{*}\bigr)+\gamma _{n}\bigl(S^{n}z_{n}-x^{*}\bigr) \bigr\Vert ^{2} \\ &\quad\leq \alpha _{n} \bigl\Vert f(x_{n})-x^{*} \bigr\Vert ^{2}+\beta _{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert + \gamma _{n} \bigl\Vert S^{n}z_{n}-x^{*} \bigr\Vert ^{2}-\beta _{n}\gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2} \\ &\quad\leq \alpha _{n} \bigl\Vert f(x_{n})-x^{*} \bigr\Vert ^{2}+\beta _{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert + \gamma _{n}(1+\theta _{n})^{2} \bigl\Vert z_{n}-x^{*} \bigr\Vert ^{2}-\beta _{n}\gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2} \\ &\quad\leq \alpha _{n} \bigl\Vert f(x_{n})-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+\theta _{n}(2+\theta _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\beta _{n} \gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2} \\ &\quad\leq \alpha _{n} \bigl\Vert f(x_{n})-x^{*} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+\theta _{n}(2+\theta _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}-\beta _{n} \gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2} \\ &\quad\leq \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+ \alpha _{n}M_{1}+\theta _{n}(2+\theta _{n})M_{1}- \beta _{n}\gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2},\end{aligned} $$

where $\sup_{n\geq 1}\{\|f(x_{n})-x^{*}\|^{2}+\|x_{n}-x^{*}\|^{2}\}\leq M_{1}$ for some $M_{1}>0$. This immediately implies

$$ \begin{aligned} \beta _{n}\gamma _{n} \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert ^{2}&\leq \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}- \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}+\alpha _{n}M_{1} +\theta _{n}(2+ \theta _{n})M_{1} \\ &=\Gamma _{n}-\Gamma _{n+1}+\alpha _{n}M_{1}+ \theta _{n}(2+\theta _{n})M_{1}.\end{aligned} $$

(3.29)

Since $0<\liminf_{n\to \infty }\beta _{n}$, $0<\liminf_{n\to \infty }\gamma _{n}$, $\alpha _{n}\to 0$, $\theta _{n}\to 0$, and $\Gamma _{n}-\Gamma _{n+1}\to 0$, we infer from (3.29) that

$$ \lim_{n\to \infty } \bigl\Vert x_{n}-S^{n}z_{n} \bigr\Vert =0, $$

which, together with the boundedness of $\{x_{n}\}$, implies that

$$ \begin{aligned} \Vert x_{n+1}-x_{n} \Vert &= \bigl\Vert \alpha _{n}\bigl(f(x_{n})-x_{n}\bigr)+ \gamma _{n}\bigl(S^{n}z_{n}-x_{n}\bigr) \bigr\Vert \\ &\leq \alpha _{n} \bigl\Vert f(x_{n})-x_{n} \bigr\Vert +\gamma _{n} \bigl\Vert S^{n}z_{n}-x_{n} \bigr\Vert \\ &\leq \alpha _{n} \bigl\Vert f(x_{n})-x_{n} \bigr\Vert + \bigl\Vert S^{n}z_{n}-x_{n} \bigr\Vert \to 0\quad (n \to \infty ).\end{aligned} $$

(3.30)

From the boundedness of $\{x_{n}\}$, it follows that there exists a subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that

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

(3.31)

Since H is reflexive and $\{x_{n}\}$ is bounded, we may assume, without loss of generality, that $x_{n_{k}}\rightharpoonup \widetilde{x}$. Thus, from (3.31) one gets

$$ \begin{aligned} { \limsup_{n\to \infty }}\bigl\langle (f-I)x^{*},x_{n}-x^{*} \bigr\rangle &={ \lim _{k\to \infty }}\bigl\langle (f-I)x^{*},x_{n_{k}}-x^{*} \bigr\rangle \\ &=\bigl\langle (f-I)x^{*},\widetilde{x}-x^{*}\bigr\rangle .\end{aligned} $$

(3.32)

Since $S^{n}x_{n}-S^{n+1}x_{n}\to 0$ (due to the assumption), $u_{n}-Gu_{n}\to 0$ (due to (3.28)), $x_{n}-x_{n+1}\to 0$ (due to (3.30)), and $x_{n_{k}}\rightharpoonup \widetilde{x}$ for $\{x_{n_{k}}\}\subset \{x_{n}\}$, by Lemma 3.3, we obtain that $\widetilde{x}\in \varOmega$. Hence from (3.12) and (3.32), one gets

$$ \limsup_{n\to \infty }\bigl\langle (f-I)x^{*},x_{n}-x^{*} \bigr\rangle =\bigl\langle (f-I)x^{*}, \widetilde{x}-x^{*} \bigr\rangle \leq 0, $$

(3.33)

which, together with (3.30), leads to

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

(3.34)

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

$$ \limsup_{n\to \infty }\biggl[ \frac{2\langle (f-I)x^{*},x_{n+1}-x^{*}\rangle }{1-\delta } + \frac{\theta _{n}}{\alpha _{n}} \cdot \frac{(2+\theta _{n})M_{0}}{1-\delta }\biggr]\leq 0. $$

Consequently, applying Lemma 2.4 to (3.22), one has $\lim_{n\to \infty }\|x_{n}-x^{*}\|^{2}=0$.

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

$$ \tau (n):=\max \{k\leq n:\Gamma _{k}< \Gamma _{k+1}\}. $$

By Lemma 2.6, we get

$$ \Gamma _{\tau (n)}\leq \Gamma _{\tau (n)+1}\quad \textrm{and}\quad \Gamma _{n}\leq \Gamma _{\tau (n)+1}. $$

Putting $p=x^{*}$ and $q=y^{*}$, from (3.15) and (3.16), we obtain

$$ \begin{aligned} &\gamma _{\tau (n)}\bigl\{ \Vert x_{\tau (n)}-u_{\tau (n)} \Vert ^{2}+(1-\mu )\bigl[ \Vert y_{ \tau (n)}-z_{\tau (n)} \Vert ^{2} + \Vert y_{\tau (n)}-w_{\tau (n)} \Vert ^{2}\bigr]+\mu _{2}(2 \beta -\mu _{2}) \\ &\quad\quad {}\times \bigl\Vert B_{2}u_{\tau (n)}-B_{2}x^{*} \bigr\Vert ^{2}+\mu _{1}(2\alpha - \mu _{1}) \bigl\Vert B_{1}\upsilon _{\tau (n)}-B_{1}y^{*} \bigr\Vert ^{2}\bigr\} \\ &\quad\leq \Gamma _{\tau (n)}-\Gamma _{{\tau (n)}+1}+\theta _{\tau (n)}(2+ \theta _{\tau (n)})M_{0}+2\alpha _{\tau (n)}M_{0}\end{aligned} $$

(3.35)

and

$$ \begin{aligned} &\gamma _{\tau (n)}\bigl[ \bigl\Vert u_{\tau (n)}-\upsilon _{\tau (n)}+y^{*}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert \upsilon _{\tau (n)}-w_{\tau (n)}+x^{*}-y^{*} \bigr\Vert ^{2}\bigr] \\ &\quad\leq \Gamma _{\tau (n)}-\Gamma _{{\tau (n)}+1}+2\mu _{2} \bigl\Vert B_{2}x^{*}-B_{2}u_{ \tau (n)} \bigr\Vert \bigl\Vert \upsilon _{\tau (n)}-y^{*} \bigr\Vert \\ &\quad\quad {}+2\mu _{1} \bigl\Vert B_{1}y^{*}-B_{1} \upsilon _{\tau (n)} \bigr\Vert \bigl\Vert w_{\tau (n)}-x^{*} \bigr\Vert +\theta _{\tau (n)}(2+\theta _{\tau (n)})M_{0} +2 \alpha _{\tau (n)}M_{0}.\end{aligned} $$

(3.36)

So it follows from (3.35) that

$$ \lim_{n\to \infty } \Vert x_{\tau (n)}-u_{\tau (n)} \Vert = \lim_{n\to \infty } \Vert y_{\tau (n)}-z_{\tau (n)} \Vert = \lim_{n\to \infty } \Vert y_{\tau (n)}-w_{ \tau (n)} \Vert =0, $$

(3.37)

and

$$ \lim_{n\to \infty } \bigl\Vert B_{2}u_{\tau (n)}-B_{2}x^{*} \bigr\Vert =\lim_{n\to \infty } \bigl\Vert B_{1}\upsilon _{\tau (n)}-B_{1}y^{*} \bigr\Vert =0. $$

(3.38)

Further, from (3.36), (3.38), and the boundedness of $\{\upsilon _{\tau (n)}\}$, $\{w_{\tau (n)}\}$, we deduce that

$$ \lim_{n\to \infty } \bigl\Vert u_{\tau (n)}-\upsilon _{\tau (n)}+y^{*}-x^{*} \bigr\Vert = \lim _{n\to \infty } \bigl\Vert \upsilon _{\tau (n)}-w_{\tau (n)}+x^{*}-y^{*} \bigr\Vert =0. $$

Therefore,

$$ \begin{aligned} \Vert u_{\tau (n)}-Gu_{\tau (n)} \Vert &= \Vert u_{\tau (n)}-w_{\tau (n)} \Vert \\ &\leq \bigl\Vert u_{ \tau (n)}-\upsilon _{\tau (n)}+y^{*}-x^{*} \bigr\Vert + \bigl\Vert \upsilon _{\tau (n)}-w_{\tau (n)}+x^{*}-y^{*} \bigr\Vert \\ & \to 0\quad (n \to \infty ).\end{aligned} $$

(3.39)

Utilizing the same inferences as in the proof of Case 1, we deduce that

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

(3.40)

and

$$ \limsup_{n\to \infty }\bigl\langle (f-I)x^{*},x_{\tau (n)+1}-x^{*} \bigr\rangle \leq 0. $$

(3.41)

On the other hand, from (3.22) we obtain

$$ \begin{aligned} \alpha _{\tau (n)}(1-\delta )\Gamma _{\tau (n)} &\leq \Gamma _{\tau (n)}- \Gamma _{\tau (n)+1}+\alpha _{\tau (n)}(1-\delta ) \biggl[ \frac{2\langle (f-I)x^{*},x_{\tau (n)+1}-x^{*}\rangle }{1-\delta } \\ &\quad {}+\frac{\theta _{\tau (n)}}{\alpha _{\tau (n)}}\cdot \frac{(2+\theta _{\tau (n)})M_{0}}{1-\delta }\biggr] \\ &\leq \alpha _{\tau (n)}(1-\delta )\biggl[ \frac{2\langle (f-I)x^{*},x_{\tau (n)+1}-x^{*}\rangle }{1-\delta } + \frac{\theta _{\tau (n)}}{\alpha _{\tau (n)}}\cdot \frac{(2+\theta _{\tau (n)})M_{0}}{1-\delta }\biggr],\end{aligned} $$

which hence yields

$$ \limsup_{n\to \infty }\Gamma _{\tau (n)}\leq \limsup _{n\to \infty }\biggl[ \frac{2\langle (f-I)x^{*},x_{\tau (n)+1}-x^{*}\rangle }{1-\delta } + \frac{\theta _{\tau (n)}}{\alpha _{\tau (n)}}\cdot \frac{(2+\theta _{\tau (n)})M_{0}}{1-\delta }\biggr]\leq 0. $$

Thus, $\lim_{n\to \infty }\|x_{\tau (n)}-x^{*}\|^{2}=0$. Also, note that

$$ \begin{aligned} & \bigl\Vert x_{\tau (n)+1}-x^{*} \bigr\Vert ^{2}- \bigl\Vert x_{\tau (n)}-x^{*} \bigr\Vert ^{2} \\ &\quad=2\bigl\langle x_{\tau (n)+1}-x_{\tau (n)},x_{\tau (n)}-x^{*} \bigr\rangle + \Vert x_{ \tau (n)+1}-x_{\tau (n)} \Vert ^{2} \\ &\quad\leq 2 \Vert x_{\tau (n)+1}-x_{\tau (n)} \Vert \bigl\Vert x_{\tau (n)}-x^{*} \bigr\Vert + \Vert x_{ \tau (n)+1}-x_{\tau (n)} \Vert ^{2}.\end{aligned} $$

(3.42)

Owing to $\Gamma _{n}\leq \Gamma _{\tau (n)+1}$, we get

$$ \begin{aligned} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}&\leq \bigl\Vert x_{\tau (n)+1}-x^{*} \bigr\Vert ^{2} \\ &\leq \bigl\Vert x_{\tau (n)}-x^{*} \bigr\Vert ^{2}+2 \Vert x_{\tau (n)+1}-x_{\tau (n)} \Vert \bigl\Vert x_{ \tau (n)}-x^{*} \bigr\Vert + \Vert x_{\tau (n)+1}-x_{\tau (n)} \Vert ^{2}\\ &\to 0\quad (n \to \infty ).\end{aligned} $$

That is, $x_{n}\to x^{*}$ as $n\to \infty$. This completes the proof. □

Theorem 3.2

Let $S:H\to C$ be nonexpansive and the sequence $\{x_{n}\}$ be constructed by the modified version of Algorithm 3.1, that is, for any initial $x_{1}\in C$,

$$ \textstyle\begin{cases} u_{n}=\sigma _{n}x_{n}+(1-\sigma _{n})S_{n}u_{n}, \\ w_{n}=Gu_{n}, \\ y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n}), \\ z_{n}=P_{C_{n}}(w_{n}-\tau _{n}Ay_{n}), \\ x_{n+1}=\alpha _{n}f(x_{n})+\beta _{n}x_{n}+\gamma _{n}Sz_{n}\quad \forall n\geq 1,\end{cases} $$

(3.43)

where for each $n\geq 1$, $C_{n}$ and $\tau _{n}$ are chosen as in Algorithm 3.1. Then $x_{n}\to x^{*}\in \varOmega$, where $x^{*}\in \varOmega$ is the unique solution to the HVI, $\langle (I-f)x^{*},p-x^{*} \rangle \geq 0$ $\forall p\in \varOmega$.

Proof

We divide the proof into several steps.

Step 1. We show that $\{x_{n}\}$ is bounded. Indeed, using the same arguments as in Step 1 of the proof of Theorem 3.1, we obtain the desired assertion.

Step 2. We show that

$$ \begin{aligned} &\gamma _{n}\bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]+\mu _{2}(2 \beta -\mu _{2}) \\ &\quad\quad {}\times \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}+\mu _{1}(2\alpha -\mu _{1}) \Vert B_{1} \upsilon _{n}-B_{1}q \Vert ^{2} \bigr\} \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+2\alpha _{n}M_{0}\end{aligned} $$

and

$$ \begin{aligned} &\gamma _{n}\bigl[ \Vert u_{n}- \upsilon _{n}+q-p \Vert ^{2}+ \Vert \upsilon _{n}-w_{n}+p-q \Vert ^{2}\bigr] \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+2\mu _{2} \Vert B_{2}p-B_{2}u_{n} \Vert \Vert \upsilon _{n}-q \Vert \\ &\quad\quad {}+2\mu _{1} \Vert B_{1}q-B_{1} \upsilon _{n} \Vert \Vert w_{n}-p \Vert +2\alpha _{n}M_{0},\end{aligned} $$

where $\sup_{n\geq 1}\{\|x_{n}-p\|^{2}+\|f(p)-p\|\|x_{n}-p\|\}\leq M_{0}$ for some $M_{0}>0$. In fact, using the same arguments as in Step 2 of the proof of Theorem 3.1, we obtain the desired assertion.

Step 3. We show that

$$ \Vert x_{n+1}-p \Vert ^{2}\leq \bigl[1-\alpha _{n}(1-\delta )\bigr] \Vert x_{n}-p \Vert ^{2} + \alpha _{n}(1-\delta ) \frac{2\langle (f-I)p,x_{n+1}-p\rangle }{1-\delta }. $$

In fact, using the same arguments as in Step 3 of the proof of Theorem 3.1, we obtain the desired assertion.

Step 4. We show that $\{x_{n}\}$ converges strongly to the unique solution $x^{*}\in \varOmega$ to the HVI (3.12), with $S_{0}=S$ a nonexpansive mapping. In fact, putting $p=x^{*}$, we deduce from Step 3 that

$$ \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}\leq \bigl[1-\alpha _{n}(1-\delta )\bigr] \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+ \alpha _{n}(1-\delta ) \frac{2\langle (f-I)x^{*},x_{n+1}-x^{*}\rangle }{1-\delta }. $$

(3.44)

Putting $\Gamma _{n}=\|x_{n}-x^{*}\|^{2}$, we show the convergence of $\{\Gamma _{n}\}$ to zero by the following two cases.

Case 1. Suppose that there exists an integer $n_{0}\geq 1$ such that $\{\Gamma _{n}\}$ is nonincreasing. Then the limit $\lim_{n\to \infty }\Gamma _{n}=\hbar <+\infty$ and $\lim_{n\to \infty }(\Gamma _{n}-\Gamma _{n+1})=0$. Putting $p=x^{*}$ and $q=y^{*}$, from Step 2 we obtain

$$ \begin{aligned} &\gamma _{n}\bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]+\mu _{2}(2 \beta -\mu _{2}) \\ &\quad\quad {}\times \bigl\Vert B_{2}u_{n}-B_{2}x^{*} \bigr\Vert ^{2}+\mu _{1}(2\alpha -\mu _{1}) \bigl\Vert B_{1}\upsilon _{n}-B_{1}y^{*} \bigr\Vert ^{2}\bigr\} \\ &\quad\leq \Gamma _{n}-\Gamma _{n+1}+2\alpha _{n}M_{0}\end{aligned} $$

and

$$ \begin{aligned} &\gamma _{n}\bigl[ \bigl\Vert u_{n}-\upsilon _{n}+y^{*}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert \upsilon _{n}-w_{n}+x^{*}-y^{*} \bigr\Vert ^{2}\bigr] \\ &\quad \leq \Gamma _{n}-\Gamma _{n+1}+2\mu _{2} \bigl\Vert B_{2}x^{*}-B_{2}u_{n} \bigr\Vert \bigl\Vert \upsilon _{n}-y^{*} \bigr\Vert \\ &\qquad {} +2\mu _{1} \bigl\Vert B_{1}y^{*}-B_{1} \upsilon _{n} \bigr\Vert \bigl\Vert w_{n}-x^{*} \bigr\Vert +2\alpha _{n}M_{0}.\end{aligned} $$

By the same inferences as in Case 1 of the proof of Theorem 3.1, we deduce that

$$\begin{aligned}& \lim_{n\to \infty } \Vert u_{n}-Gu_{n} \Vert =0, \end{aligned}$$

(3.45)

$$\begin{aligned}& \lim_{n\to \infty } \Vert x_{n}-x_{n+1} \Vert =0 \quad \text{and}\quad \limsup_{n \to \infty }\bigl\langle (f-I)x^{*},x_{n+1}-x^{*}\bigr\rangle \leq 0. \end{aligned}$$

(3.46)

Consequently, applying Lemma 2.4 to (3.44), we obtain $\lim_{n\to \infty }\|x_{n}-x^{*}\|^{2}=0$.

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

$$ \tau (n):=\max \{k\leq n:\Gamma _{k}< \Gamma _{k+1}\}. $$

By Lemma 2.6, we get

$$ \Gamma _{\tau (n)}\leq \Gamma _{\tau (n)+1}\quad \text{and}\quad \Gamma _{n}\leq \Gamma _{\tau (n)+1}. $$

The conclusion follows using the same arguments as in Case 2 of the proof of Theorem 3.1. □

Next, we introduce another composite subgradient extragradient algorithm.

Algorithm 3.2

Initialization: Given $\gamma >0$, $\mu \in (0,1)$, $\ell \in (0,1)$, pick an initial $x_{1}\in C$ arbitrarily.

Iterative steps: Compute $x_{n+1}$ below:

Step 1. Calculate $u_{n}=\sigma _{n}x_{n}+(1-\sigma _{n})S_{n}u_{n}$ and $w_{n}=Gu_{n}$, and set $y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n})$, where $\tau _{n}$ is chosen to be the largest $\tau \in \{\gamma ,\gamma \ell ,\gamma \ell ^{2},\dots \}$ satisfying

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

(3.47)

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

Step 3. Calculate

$$ x_{n+1}=\alpha _{n}f(x_{n})+\beta _{n}u_{n}+\gamma _{n}S^{n}z_{n}. $$

(3.48)

Again put $n:=n+1$ and return to Step 1.

It is worth pointing out that inequality (3.5) and Lemmas 3.1–3.3 are still valid for Algorithm 3.2.

Theorem 3.3

Let $\{x_{n}\}$ be the sequence constructed in Algorithm 3.2. Then $x_{n}\to x^{*}\in \varOmega$, provided $S^{n}x_{n}-S^{n+1}x_{n}\to 0$, where $x^{*}\in \varOmega$ is the unique solution to the HVI, $\langle (I-f)x^{*}, p-x^{*}\rangle \geq 0$ $\forall p\in \varOmega$.

Proof

Using the same arguments as in the proof of Theorem 3.1, we deduce that there exists the unique solution $x^{*}\in \varOmega =\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G) \cap \mathrm {VI}(C,A)$ to the HVI (3.12). We divide the rest of the proof into several steps.

Step 1. We show that $\{x_{n}\}$ is bounded. In fact, using the same arguments as in Step 1 of the proof of Theorem 3.1, we obtain that inequalities (3.13) and (3.14) hold. Thus, from (3.14) it follows that

$$ \begin{aligned} \Vert x_{n+1}-p \Vert &\leq \alpha _{n} \bigl\Vert f(x_{n})-p \bigr\Vert +\beta _{n} \Vert u_{n}-p \Vert + \gamma _{n} \bigl\Vert S^{n}z_{n}-p \bigr\Vert \\ &\leq \alpha _{n}\bigl( \bigl\Vert f(x_{n})-f(p) \bigr\Vert + \bigl\Vert f(p)-p \bigr\Vert \bigr)+\beta _{n} \Vert u_{n}-p \Vert + \gamma _{n}(1+\theta _{n}) \Vert z_{n}-p \Vert \\ &\leq \alpha _{n}\bigl(\delta \Vert x_{n}-p \Vert + \bigl\Vert f(p)-p \bigr\Vert \bigr)+\beta _{n} \Vert x_{n}-p \Vert +(\gamma _{n}+\theta _{n}) \Vert x_{n}-p \Vert \\ &\leq \biggl[1-\frac{\alpha _{n}(1-\delta )}{2}\biggr] \Vert x_{n}-p \Vert + \alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \\ &=\biggl[1-\frac{\alpha _{n}(1-\delta )}{2}\biggr] \Vert x_{n}-p \Vert + \frac{\alpha _{n}(1-\delta )}{2}\frac{2 \Vert f(p)-p \Vert }{1-\delta } \\ &\leq \max \biggl\{ \Vert x_{n}-p \Vert ,\frac{2 \Vert f(p)-p \Vert }{1-\delta }\biggr\} .\end{aligned} $$

By induction, we obtain $\|x_{n}-p\|\leq \max \{\|x_{1}-p\|,\frac{2\|f(p)-p\|}{1-\delta }\}$ $\forall n\geq 1$. Therefore, $\{x_{n}\}$ is bounded, and so are the sequences $\{u_{n}\}$, $\{w_{n}\}$, $\{y_{n}\}$, $\{z_{n}\}$, $\{f(x_{n})\}$, $\{Ay_{n}\}$, $\{S_{n}u_{n}\}$, $\{S^{n}z_{n}\}$.

Step 2. We show that

$$ \begin{aligned} &\gamma _{n}\bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]+\mu _{2}(2 \beta -\mu _{2}) \\ &\quad \quad {}\times \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}+\mu _{1}(2\alpha -\mu _{1}) \Vert B_{1} \upsilon _{n}-B_{1}q \Vert ^{2} \bigr\} \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+\theta _{n}(2+\theta _{n})M_{0}+2 \alpha _{n}M_{0}\end{aligned} $$

(3.49)

and

$$ \begin{aligned} &\gamma _{n}\bigl[ \Vert u_{n}- \upsilon _{n}+q-p \Vert ^{2}+ \Vert \upsilon _{n}-w_{n}+p-q \Vert ^{2}\bigr] \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+2\mu _{2} \Vert B_{2}p-B_{2}u_{n} \Vert \Vert \upsilon _{n}-q \Vert \\ &\quad\quad {}+2\mu _{1} \Vert B_{1}q-B_{1} \upsilon _{n} \Vert \Vert w_{n}-p \Vert +\theta _{n}(2+ \theta _{n})M_{0}+2\alpha _{n}M_{0},\end{aligned} $$

(3.50)

for some $M_{0}>0$. In fact, using (3.5), (3.13), (3.14), and the convexity of the function $\phi (s)=s^{2}$ $\forall s\in \mathbf{R}$, we get

$$ \begin{aligned} & \Vert x_{n+1}-p \Vert ^{2} \\ &\quad\leq \alpha _{n} \bigl\Vert f(x_{n})-f(p) \bigr\Vert ^{2}+\beta _{n} \Vert u_{n}-p \Vert ^{2}+ \gamma _{n} \bigl\Vert S^{n}z_{n}-p \bigr\Vert ^{2}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p \bigr\rangle \\ &\quad\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert u_{n}-p \Vert ^{2}+\bigl[ \gamma _{n}+\theta _{n}(2+\theta _{n})\bigr] \Vert z_{n}-p \Vert ^{2}+2\alpha _{n} \bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\quad\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+ \gamma _{n}\bigl\{ \Vert x_{n}-p \Vert ^{2}- \Vert x_{n}-u_{n} \Vert ^{2}-(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2} \\ &\quad\quad {}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]- \mu _{2}(2\beta -\mu _{2}) \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}-\mu _{1}(2\alpha -\mu _{1}) \Vert B_{1}\upsilon _{n}-B_{1}q \Vert ^{2} \bigr\} \\ &\quad\quad {}+\theta _{n}(2+\theta _{n}) \Vert x_{n}-p \Vert ^{2}+2\alpha _{n}\bigl\langle f(p)-p,x_{n+1}-p \bigr\rangle \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}-\gamma _{n} \bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu ) \bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr] \\ &\quad\quad {}+\mu _{2}(2\beta -\mu _{2}) \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}+\mu _{1}(2 \alpha -\mu _{1}) \Vert B_{1}\upsilon _{n}-B_{1}q \Vert ^{2}\bigr\} \\ &\quad\quad {}+\theta _{n}(2+\theta _{n})M_{0}+2\alpha _{n}M_{0}\end{aligned} $$

(3.51)

where $\sup_{n\geq 1}\{\|x_{n}-p\|^{2}+\|f(p)-p\|\|x_{n}-p\|\}\leq M_{0}$ for some $M_{0}>0$. This ensures that (3.49) holds. Further, using similar arguments to those of (3.16), we obtain that (3.50) holds.

Step 3. We show that

$$ \begin{aligned} \Vert x_{n+1}-p \Vert ^{2}&\leq \bigl[1-\alpha _{n}(1-\delta )\bigr] \Vert x_{n}-p \Vert ^{2}\\ &\quad {}+ \alpha _{n}(1-\delta )\biggl\{ \frac{2\langle (f-I)p,x_{n+1}-p\rangle }{1-\delta }+ \frac{\theta _{n}}{\alpha _{n}} \cdot \frac{(2+\theta _{n})M_{0}}{1-\delta }\biggr\} . \end{aligned} $$

In fact, from (3.14) and (3.51), we have

$$ \begin{aligned} & \Vert x_{n+1}-p \Vert ^{2} \\ &\quad\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert u_{n}-p \Vert ^{2}+\bigl[ \gamma _{n}+\theta _{n}(2+\theta _{n})\bigr] \Vert z_{n}-p \Vert ^{2}+2\alpha _{n} \bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\quad\leq \alpha _{n}\delta \Vert x_{n}-p \Vert ^{2}+\beta _{n} \Vert x_{n}-p \Vert ^{2}+ \gamma _{n} \Vert x_{n}-p \Vert ^{2}+\theta _{n}(2+\theta _{n})M_{0}\\ &\qquad {}+2 \alpha _{n} \bigl\langle f(p)-p,x_{n+1}-p\bigr\rangle \\ &\quad=\bigl[1-\alpha _{n}(1-\delta )\bigr] \Vert x_{n}-p \Vert ^{2}+\alpha _{n}(1-\delta )\biggl\{ \frac{2\langle (f-I)p,x_{n+1}-p\rangle }{1-\delta }+ \frac{\theta _{n}}{\alpha _{n}}\cdot \frac{(2+\theta _{n})M_{0}}{1-\delta }\biggr\} . \end{aligned} $$

Step 4. We show that $\{x_{n}\}$ converges strongly to the unique solution $x^{*}\in \varOmega$ of the HVI (3.12). In fact, putting $p=x^{*}$, we deduce from Step 3 that

$$ \begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}&\leq \bigl[1-\alpha _{n}(1-\delta )\bigr] \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}\\ &\quad {}+ \alpha _{n}(1-\delta )\biggl\{ \frac{2\langle (f-I)x^{*},x_{n+1}-x^{*}\rangle }{1-\delta }+ \frac{\theta _{n}}{\alpha _{n}}\cdot \frac{(2+\theta _{n})M_{0}}{1-\delta }\biggr\} . \end{aligned} $$

(3.52)

Putting $\Gamma _{n}=\|x_{n}-x^{*}\|^{2}$, we show the convergence of $\{\Gamma _{n}\}$ to zero by the following two cases.

Case 1. Suppose that there exists an integer $n_{0}\geq 1$ such that $\{\Gamma _{n}\}$ is nonincreasing. Then the limit $\lim_{n\to \infty }\Gamma _{n}=\hbar <+\infty$ and $\lim_{n\to \infty }(\Gamma _{n}-\Gamma _{n+1})=0$. Putting $p=x^{*}$ and $q=y^{*}$, from (3.49) and (3.50), we obtain that

$$ \begin{aligned} &\gamma _{n}\bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]+\mu _{2}(2 \beta -\mu _{2}) \\ &\quad\quad {}\times \bigl\Vert B_{2}u_{n}-B_{2}x^{*} \bigr\Vert ^{2}+\mu _{1}(2\alpha -\mu _{1}) \bigl\Vert B_{1}\upsilon _{n}-B_{1}y^{*} \bigr\Vert ^{2}\bigr\} \\ &\quad\leq \Gamma _{n}-\Gamma _{n+1}+\theta _{n}(2+ \theta _{n})M_{0}+2 \alpha _{n}M_{0}\end{aligned} $$

and

$$ \begin{aligned} &\gamma _{n}\bigl[ \bigl\Vert u_{n}-\upsilon _{n}+y^{*}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert \upsilon _{n}-w_{n}+x^{*}-y^{*} \bigr\Vert ^{2}\bigr] \\ &\quad\leq \Gamma _{n}-\Gamma _{n+1}+2\mu _{2} \bigl\Vert B_{2}x^{*}-B_{2}u_{n} \bigr\Vert \bigl\Vert \upsilon _{n}-y^{*} \bigr\Vert \\ &\quad\quad {}+2\mu _{1} \bigl\Vert B_{1}y^{*}-B_{1} \upsilon _{n} \bigr\Vert \bigl\Vert w_{n}-x^{*} \bigr\Vert + \theta _{n}(2+\theta _{n})M_{0}+2 \alpha _{n}M_{0}.\end{aligned} $$

By the same inferences as in Case 1 of the proof of Theorem 3.1, we deduce that $u_{n}-Gu_{n}\to 0$, $x_{n}-x_{n+1}\to 0$ and

$$ \limsup_{n\to \infty }\bigl\langle (f-I)x^{*},x_{n+1}-x^{*} \bigr\rangle \leq 0. $$

Consequently, applying Lemma 2.4 to (3.52), we obtain $\lim_{n\to \infty }\|x_{n}-x^{*}\|^{2}=0$.

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

$$ \tau (n):=\max \{k\leq n:\Gamma _{k}< \Gamma _{k+1}\}. $$

By Lemma 2.6, we get

$$ \Gamma _{\tau (n)}\leq \Gamma _{\tau (n)+1}\quad \text{and}\quad \Gamma _{n}\leq \Gamma _{\tau (n)+1}. $$

In the remainder of the proof, using the same arguments as in Case 2 of Step 4 in the proof of Theorem 3.1, we obtain the desired conclusion. □

Theorem 3.4

Let $S:H\to C$ be nonexpansive and the sequence $\{x_{n}\}$ be constructed by the modified version of Algorithm 3.1, that is, for any initial $x_{1}\in C$,

$$ \textstyle\begin{cases} u_{n}=\sigma _{n}x_{n}+(1-\sigma _{n})S_{n}u_{n}, \\ w_{n}=Gu_{n}, \\ y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n}), \\ z_{n}=P_{C_{n}}(w_{n}-\tau _{n}Ay_{n}), \\ x_{n+1}=\alpha _{n}f(x_{n})+\beta _{n}u_{n}+\gamma _{n}Sz_{n}\quad \forall n\geq 1,\end{cases} $$

(3.53)

where for each $n\geq 1$, $C_{n}$ and $\tau _{n}$ are chosen as in Algorithm 3.2. Then $x_{n}\to x^{*}\in \varOmega$, where $x^{*}\in \varOmega$ is the unique solution to the HVI, $\langle (I-f)x^{*},p-x^{*} \rangle \geq 0$ $\forall p\in \varOmega$.

Proof

We divide the proof into several steps.

Step 1. We show that $\{x_{n}\}$ is bounded. Indeed, using the same arguments as in Step 1 of the proof of Theorem 3.3, we obtain the desired assertion.

Step 2. We show that

$$ \begin{aligned} &\gamma _{n}\bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]+\mu _{2}(2 \beta -\mu _{2}) \\ &\quad\quad {}\times \Vert B_{2}u_{n}-B_{2}p \Vert ^{2}+\mu _{1}(2\alpha -\mu _{1}) \Vert B_{1} \upsilon _{n}-B_{1}q \Vert ^{2} \bigr\} \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+2\alpha _{n}M_{0}\end{aligned} $$

and

$$ \begin{aligned} &\gamma _{n}\bigl[ \Vert u_{n}- \upsilon _{n}+q-p \Vert ^{2}+ \Vert \upsilon _{n}-w_{n}+p-q \Vert ^{2}\bigr] \\ &\quad\leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+2\mu _{2} \Vert B_{2}p-B_{2}u_{n} \Vert \Vert \upsilon _{n}-q \Vert \\ &\quad\quad {}+2\mu _{1} \Vert B_{1}q-B_{1} \upsilon _{n} \Vert \Vert w_{n}-p \Vert +2\alpha _{n}M_{0},\end{aligned} $$

where $\sup_{n\geq 1}\{\|x_{n}-p\|^{2}+\|f(p)-p\|\|x_{n}-p\|\}\leq M_{0}$ for some $M_{0}>0$. In fact, using the same arguments as in Step 2 of the proof of Theorem 3.3, we obtain the desired assertion.

Step 3. We show that

$$ \Vert x_{n+1}-p \Vert ^{2}\leq \bigl[1-\alpha _{n}(1-\delta )\bigr] \Vert x_{n}-p \Vert ^{2} + \alpha _{n}(1-\delta ) \frac{2\langle (f-I)p,x_{n+1}-p\rangle }{1-\delta }. $$

In fact, using the same arguments as in Step 3 of the proof of Theorem 3.3, we obtain the desired assertion.

Step 4. We show that $\{x_{n}\}$ converges strongly to the unique solution $x^{*}\in \varOmega$ to the HVI (3.12), with $S_{0}=S$ a nonexpansive mapping. In fact, putting $p=x^{*}$, we deduce from Step 3 that

$$ \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}\leq \bigl[1-\alpha _{n}(1-\delta )\bigr] \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+ \alpha _{n}(1-\delta ) \frac{2\langle (f-I)x^{*},x_{n+1}-x^{*}\rangle }{1-\delta }. $$

(3.54)

Putting $\Gamma _{n}=\|x_{n}-x^{*}\|^{2}$, we show the convergence of $\{\Gamma _{n}\}$ to zero by the following two cases.

Case 1. Suppose that there exists an integer $n_{0}\geq 1$ such that $\{\Gamma _{n}\}$ is nonincreasing. Then the limit $\lim_{n\to \infty }\Gamma _{n}=\hbar <+\infty$ and $\lim_{n\to \infty }(\Gamma _{n}-\Gamma _{n+1})=0$. Putting $p=x^{*}$ and $q=y^{*}$, from Step 2 we obtain

$$ \begin{aligned} &\gamma _{n}\bigl\{ \Vert x_{n}-u_{n} \Vert ^{2}+(1-\mu )\bigl[ \Vert y_{n}-z_{n} \Vert ^{2}+ \Vert y_{n}-w_{n} \Vert ^{2}\bigr]+\mu _{2}(2 \beta -\mu _{2}) \\ &\quad\quad {}\times \bigl\Vert B_{2}u_{n}-B_{2}x^{*} \bigr\Vert ^{2}+\mu _{1}(2\alpha -\mu _{1}) \bigl\Vert B_{1}\upsilon _{n}-B_{1}y^{*} \bigr\Vert ^{2}\bigr\} \\ &\quad\leq \Gamma _{n}-\Gamma _{n+1}+2\alpha _{n}M_{0}\end{aligned} $$

and

$$ \begin{aligned} &\gamma _{n}\bigl[ \bigl\Vert u_{n}-\upsilon _{n}+y^{*}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert \upsilon _{n}-w_{n}+x^{*}-y^{*} \bigr\Vert ^{2}\bigr] \\ &\quad\leq \Gamma _{n}-\Gamma _{n+1}+2\mu _{2} \bigl\Vert B_{2}x^{*}-B_{2}u_{n} \bigr\Vert \bigl\Vert \upsilon _{n}-y^{*} \bigr\Vert \\ &\qquad {} +2\mu _{1} \bigl\Vert B_{1}y^{*}-B_{1} \upsilon _{n} \bigr\Vert \bigl\Vert w_{n}-x^{*} \bigr\Vert +2\alpha _{n}M_{0}.\end{aligned} $$

By the same arguments as in Case 1 of the proof of Theorem 3.3, we deduce that $u_{n}-Gu_{n}\to 0$, $x_{n}-x_{n+1}\to 0$ and

$$ \limsup_{n\to \infty }\bigl\langle (f-I)x^{*},x_{n+1}-x^{*} \bigr\rangle \leq 0. $$

Consequently, applying Lemma 2.4 to (3.54), we obtain $\lim_{n\to \infty }\|x_{n}-x^{*}\|^{2}=0$.

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

$$ \tau (n):=\max \{k\leq n:\Gamma _{k}< \Gamma _{k+1}\}. $$

By Lemma 2.6, we get

$$ \Gamma _{\tau (n)}\leq \Gamma _{\tau (n)+1}\quad \text{and}\quad \Gamma _{n}\leq \Gamma _{\tau (n)+1}. $$

The conclusion follows using the same arguments as in Case 2 of the proof of Theorem 3.3. □

Remark 3.1

Compared with the corresponding results in Ceng and Wen [21], Ceng and Shang [22], and Thong and Hieu [14], our results improve and extend them in the following aspects:

(i) The problem of finding an element of $\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G)$ in [21] is extended to develop our problem of finding an element of $\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G)\cap \mathrm {VI}(C,A)$ where $\{S_{i}\}^{\infty }_{i=1}$ is a countable family of ς-uniformly Lipschitzian pseudocontractive mappings and $S_{0}=S$ is asymptotically nonexpansive. The hybrid extragradient-like implicit method for finding an element of $\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G)$ in [21] is extended to develop our Mann implicit composite subgradient extragradient method with line-search process for finding an element of $\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G)\cap \mathrm {VI}(C,A)$, which is based on the Mann implicit iteration method, subgradient extragradient method with line-search process, and viscosity approximation method.

(ii) The problem of finding an element of $\mathrm {Fix}(S)\cap \mathrm {VI}(C,A)$ with quasinonexpansive mapping S in [14] is extended to develop our problem of finding an element of $\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G) \cap \mathrm {VI}(C,A)$ where $\{S_{i}\}^{\infty }_{i=1}$ is a countable family of ς-uniformly Lipschitzian pseudocontractive mappings and $S_{0}=S$ is asymptotically nonexpansive. The inertial subgradient extragradient method with linear-search process for finding an element of $\mathrm {Fix}(S)\cap \mathrm {VI}(C,A)$ in [14] is extended to develop our Mann implicit composite subgradient extragradient method with line-search process for finding an element of $\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G)\cap \mathrm {VI}(C,A)$, which is based on the Mann implicit iteration method, subgradient extragradient method with line-search process, and viscosity approximation method.

(iii) The problem of finding an element of $\varOmega =\bigcap^{N}_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {VI}(C,A)$ with finitely many nonexpansive mappings $\{S_{i}\}^{N}_{i=1}$ is extended to develop our problem of finding an element of $\varOmega =\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G) \cap \mathrm {VI}(C,A)$ with a countable family of ς-uniformly Lipschitzian pseudocontractive mappings $\{S_{i}\}^{\infty }_{i=1}$. The hybrid inertial subgradient extragradient method with line-search process in [22] is extended to develop our Mann implicit composite subgradient extragradient method with line-search process, e.g., the original inertial approach $w_{n}=S_{n}x_{n}+\alpha _{n}(S_{n}x_{n} -S_{n}x_{n-1})$ is replaced by Mann implicit composite iteration method $u_{n}=\sigma _{n}x_{n}+(1-\sigma _{n})Su_{n}$ and $w_{n}=Gu_{n}$. In addition, it was shown in [22] that, under condition $S^{n}z_{n}-S^{n+1}z_{n}\to 0$, the conclusion holds:

$$ x_{n}\to x^{*}\in \varOmega \quad \Leftrightarrow\quad \Vert x_{n}-y_{n} \Vert + \Vert x_{n}-x_{n+1} \Vert \to 0\quad \text{with }x^{*}=P_{\varOmega}(I-\rho F+f)x^{*}. $$

In this paper, using Lemma 2.6, we show that, under condition $S^{n}x_{n}-S^{n+1}x_{n}\to 0$, the following conclusion holds:

$$ x_{n}\to x^{*}\in \varOmega \quad \text{with }x^{*}=P_{\varOmega}f\bigl(x^{*}\bigr). $$

4 Applications

In this section, applying our main results, we deal with the GSVI, VIP, and CFPP in an illustrated example. Put $\mu _{1}=\mu _{2}=\frac{1}{3}$, $\gamma =1$, $\mu =\ell =\frac{1}{2}$, $\sigma _{n}=\frac{2}{3}$, $\alpha _{n}=\frac{1}{3(n+1)}$, $\beta _{n} = \frac{n}{3(n+1)}$, and $\gamma _{n}=\frac{2}{3}$.

We first provide an example of two inverse-strongly monotone mappings $B_{1},B_{2}:C\to H$, Lipschitz continuous and pseudomonotone mapping A, asymptotically nonexpansive mapping S, and countably many ς-uniformly Lipschitzian pseudocontractive mappings $\{S_{i}\}^{\infty }_{i=1}$ with $\varOmega =\bigcap^{\infty }_{i=0}\mathrm {Fix}(S_{i})\cap \mathrm {Fix}(G) \cap \mathrm {VI}(C,A)\neq \emptyset$ with $S_{0}:=S$. Let $C=[-3,3]$ and $H=\mathbf{R}$ with the inner product $\langle a,b\rangle =ab$ and induced norm $\|\cdot \|=|\cdot |$. The initial point $x_{1}$ is randomly chosen in C. Take $f(x)=\frac{1}{2}x$ $\forall x\in C$ with $\delta =\frac{1}{2}$, and put $B_{1}x=B_{2}x:=Bx=x-\frac{1}{2}\sin x$ $\forall x\in C$. Let $A:H\to H$ and $S,S_{i}:C\to C$ be defined as $Au:=\frac{1}{1+|\sin u|}-\frac{1}{1+|u|}$, $Su:=\frac{5}{6}\sin u$, and $S_{i}u=Tu=\sin u$ $\forall u\in H$, $i\geq 1$. We now claim that B is $\frac{2}{9}$-inverse-strongly monotone. In fact, since B is $\frac{1}{2}$-strongly monotone and $\frac{3}{2}$-Lipschitz continuous, we know that B is $\frac{2}{9}$-inverse-strongly monotone with $\alpha =\beta =\frac{2}{9}$. Let us show that A is pseudomonotone and Lipschitz continuous. In fact, for all $u,v\in H$, we have

$$ \begin{aligned} \Vert Au-Av \Vert &\leq \biggl\vert \frac{ \Vert v \Vert - \Vert u \Vert }{(1+ \Vert u \Vert )(1+ \Vert v \Vert )} \biggr\vert + \biggl\vert \frac{ \Vert \sin v \Vert - \Vert \sin u \Vert }{(1+ \Vert \sin u \Vert )(1+ \Vert \sin v \Vert )} \biggr\vert \\ &\leq \frac{ \Vert v-u \Vert }{(1+ \Vert u \Vert )(1+ \Vert v \Vert )}+ \frac{ \Vert \sin v-\sin u \Vert }{(1+ \Vert \sin u \Vert )(1+ \Vert \sin v \Vert )} \\ &\leq \Vert u-v \Vert + \Vert \sin u-\sin v \Vert \leq 2 \Vert u-v \Vert .\end{aligned} $$

This implies that A is Lipschitz continuous with $L=2$. Next, we show that A is pseudomonotone. For each $u,v\in H$, it is easy to see that

$$ \begin{aligned} &\langle Au,v-u\rangle =\biggl(\frac{1}{1+ \vert \sin u \vert }- \frac{1}{1+ \vert u \vert }\biggr) (v-u) \geq 0 \\ &\quad \Rightarrow \quad \langle Av,v-u\rangle =\biggl(\frac{1}{1+ \vert \sin v \vert }- \frac{1}{1+ \vert v \vert }\biggr) (v-u)\geq 0.\end{aligned} $$

Besides, it is easy to verify that S is asymptotically nonexpansive with $\theta _{n}=(\frac{5}{6})^{n}$ $\forall n\geq 1$, such that $\|S^{n+1}x_{n}-S^{n}x_{n}\|\to 0$ as $n\to \infty$. Indeed, we observe that

$$ \bigl\Vert S^{n}u-S^{n}v \bigr\Vert \leq { \frac{5}{6}} \bigl\Vert S^{n-1}u-S^{n-1}v \bigr\Vert \leq \cdots \leq \biggl(\frac{5}{6}\biggr)^{n} \Vert u-v \Vert \leq (1+\theta _{n}) \Vert u-v \Vert $$

and

$$ \begin{aligned} \bigl\Vert S^{n+1}x_{n}-S^{n}x_{n} \bigr\Vert &\leq \biggl(\frac{5}{6}\biggr)^{n-1} \bigl\Vert S^{2}x_{n}-Sx_{n} \bigr\Vert =\biggl( \frac{5}{6}\biggr)^{n-1} \biggl\Vert \frac{5}{6}\sin (Sx_{n}) -\frac{5}{6}\sin x_{n} \biggr\Vert \\ &\leq 2 \biggl(\frac{5}{6}\biggr)^{n}\to 0. \end{aligned} $$

It is clear that $\mathrm {Fix}(S)=\{0\}$ and

$$ \lim_{n\to \infty }\frac{\theta _{n}}{\alpha _{n}}=\lim_{n\to \infty } \frac{(5/6)^{n}}{1/3(n+1)}=0. $$

In addition, it is clear that $S_{i}=T$ is nonexpansive and $\mathrm {Fix}(T)=\{0\}$. Therefore, $\varOmega =\mathrm {Fix}(T )\cap \mathrm {Fix}(S)\cap \mathrm {Fix}(G)\cap \mathrm {VI}(C,A)= \{0\}\neq \emptyset$. In this case, noticing $S_{n}=T$ and $G= P_{C}(I-\mu _{1}B_{1})P_{C}(I-\mu _{2}B_{2})=[P_{C}(I-\frac{1}{3}B)]^{2}$, we rewrite Algorithm 3.1 as follows:

$$ \textstyle\begin{cases} u_{n}=\frac{2}{3}x_{n}+\frac{1}{3}Tu_{n}, \\ w_{n}=[P_{C}(I-\frac{1}{3}B)]^{2}u_{n}, \\ y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n}), \\ z_{n}=P_{C_{n}}(w_{n}-\tau _{n}Ay_{n}), \\ x_{n+1}=\frac{1}{3(n+1)}\cdot \frac{1}{2}x_{n}+\frac{n}{3(n+1)}x_{n}+ \frac{2}{3}S^{n}z_{n}\quad \forall n\geq 1, \end{cases} $$

(4.1)

where for each $n\geq 1$, $C_{n}$ and $\tau _{n}$ are chosen as in Algorithm 3.1. Then, by Theorem 3.1, we know that $\{x_{n}\}$ converges to $0\in \varOmega =\mathrm {Fix}(T)\cap \mathrm {Fix}(S)\cap \mathrm {Fix}(G)\cap \mathrm { VI}(C,A)$.

In particular, since $Su:=\frac{5}{6}\sin u$ is also nonexpansive, we consider the modified version of Algorithm 3.1, that is,

$$ \textstyle\begin{cases} u_{n}=\frac{2}{3}x_{n}+\frac{1}{3}Tu_{n}, \\ w_{n}=[P_{C}(I-\frac{1}{3}B)]^{2}u_{n}, \\ y_{n}=P_{C}(w_{n}-\tau _{n}Aw_{n}), \\ z_{n}=P_{C_{n}}(w_{n}-\tau _{n}Ay_{n}), \\ x_{n+1}=\frac{1}{3(n+1)}\cdot \frac{1}{2}x_{n}+\frac{n}{3(n+1)}x_{n}+ \frac{2}{3}Sz_{n}\quad \forall n\geq 1, \end{cases} $$

(4.2)

where for each $n\geq 1$, $C_{n}$ and $\tau _{n}$ are chosen as above. Then, by Theorem 3.2, we know that $\{x_{n}\}$ converges to $0\in \varOmega =\mathrm {Fix}(T)\cap \mathrm {Fix}(S)\cap \mathrm {Fix}(G)\cap \mathrm { VI}(C,A)$.