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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.

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., [226] 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

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (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.

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

  1. (a)

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

  2. (b)

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

  3. (c)

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

  4. (d)

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

  5. (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, \(\upsilon \in C\);

  6. (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]):

  1. (a)

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

  2. (b)

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

  3. (c)

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

  4. (d)

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

  5. (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:

  1. (i)

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

  2. (ii)

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

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:

    1. (i)

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

    2. (ii)

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

    3. (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 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 3.13.3 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:

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

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 \(0\in \varOmega =\mathrm {Fix}(T)\cap \mathrm {Fix}(S)\cap \mathrm {Fix}(G)\cap \mathrm { VI}(C,A)\).

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Acknowledgements

The research of Lu-Chuan Ceng was supported by the 2020 Shanghai Leading Talents Program of the Shanghai Municipal Human Resources and Social Security Bureau (20LJ2006100), the Innovation Program of Shanghai Municipal Education Commission (15ZZ068) and the Program for Outstanding Academic Leaders in Shanghai City (15XD1503100). The research of Jen-Chih Yao was partially supported by the grant MOST 108-2115-M039-005-MY3.

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This research was partially supported by the grant MOST 108-2115-M039-005-MY3.

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Conceptualization and Formal analysis are done by L-CC, YS and J-CY. Funding acquisition, Project administration and Supervision are done by J-CY. Investigation and Methodology are done by L-CC, YS and J-CY. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Yekini Shehu.

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Ceng, LC., Yao, JC. & Shehu, Y. On Mann implicit composite subgradient extragradient methods for general systems of variational inequalities with hierarchical variational inequality constraints. J Inequal Appl 2022, 78 (2022). https://doi.org/10.1186/s13660-022-02813-0

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MSC

  • 47H09
  • 47H10
  • 47J20
  • 47J25

Keywords

  • Mann implicit composite subgradient extragradient method
  • Variational inequality problem
  • General system of variational inequalities
  • Asymptotically nonexpansive mapping
  • Lipschitzian pseudocontractive mapping