Some variants of the hybrid extragradient algorithm in Hilbert spaces

Abstract

This paper provides convergence analysis of some variants of the hybrid extragradient algorithm (HEA) in Hilbert spaces. We employ the HEA to compute the common solution of the equilibrium problem and split fixed-point problem associated with the finite families of $\Bbbk$-demicontractive mappings. We also incorporate appropriate numerical results concerning the viability of the proposed variants with respect to various real-world applications.

1 Introduction

The class of split inverse problems (SIP) plays a prominent role in medical image reconstruction and in signal processing. One of the important generalizations of the SIP is the split common fixed-point problem (SCFPP). The class of SCFPP associated with a variety of nonlinear mappings has been analyzed in the framework of Hilbert as well as Banach spaces. In this paper, we are interested in solving the SCFPP for finite families of $\Bbbk$-demicontractive mappings in Hilbert spaces.

In 1994, Blum and Oettli [14] proposed the (monotone-) equilibrium problem (EP) theory in Hilbert spaces. Since then, several iterative algorithms have been employed to compute the optimal solution of the (monotone-) EP as well as EP together with the fixed-point problem (FPP). In 2006, Tada and Takahashi [32] suggested a hybrid framework for the analysis of monotone EP and FPP in Hilbert spaces. However, the iterative algorithm proposed in [32] fails for the case of pseudomonotone EP. To overcome this drawback, Anh [2] employed the hybrid extragradient method, based on the seminal work of Korpelevich [27], to compute the optimal common solution of the pseudomonotone EP and the FPP.

Inspired and motivated by the ongoing research, it is natural to study the pseudomonotone EP together with the SCFPP associated with the class of $\Bbbk$-demicontractive mappings in Hilbert spaces. We propose some accelerated variants, based on the inertial extrapolation technique [29] (see also [1, 3–12, 15–17, 19, 21–24]), of the hybrid extragradient algorithm in Hilbert spaces.

The rest of the paper is organized as follows. We present some relevant preliminary concepts and useful results regarding the pseudomonotone EP and SCFPP in Sect. 2. Section 3 comprises strong convergence results of the proposed variants of the hybrid inertial extragradient algorithm under a suitable set of constraints. In Sect. 4, we provide numerical results for the demonstration of the main results in Sect. 3 as well as the viability of the proposed variants with respect to various real-world applications.

2 Preliminaries

Let $\mathcal{K}$ be a nonempty, closed, convex subset of a real Hilbert space $\mathcal{H}_{1}$. The metric projector $\mathcal{P_{K}}$ from $\mathcal{H}_{1}$ onto $\mathcal{K}$ is defined for each $\mu \in \mathcal{H}_{1}$ there exists a unique nearest point $\mathcal{P_{K}}\mu$ in $\mathcal{K}$ such that $\Vert \mu - \mathcal{P_{K}}\mu \Vert \leq \Vert \mu - \nu \Vert$, $\forall \nu \in \mathcal{K}$. A subset $\mathcal{X}$ of $\mathcal{K}$ is said to be proximal if for each $\mu \in \mathcal{K}$, there exists $\nu \in \mathcal{X}$ such that $d(\mu ,\mathcal{X})=\Vert \mu -\nu \Vert$. Throughout the rest of the paper, we denote by $\mathcal{CB({H}}_{1})$, $\mathcal{CC({H}}_{1})$, and $\mathcal{PB({H}}_{1})$ the families of nonempty, closed, bounded subsets, nonempty, closed, convex subsets, and nonempty, proximal, bounded subsets of $\mathcal{H}_{1}$. The Hausdorff metric on $\mathcal{CB({H}}_{1})$ is defined as:

$$ \mathcal{D}(A,B):=\max \Bigl\{ \sup_{\mu \in A}d(\mu ,B),\sup _{\nu \in B}d( \nu ,A) \Bigr\} ,\quad \forall A,B \in \mathcal{CB({H}}_{1}), $$

where $d(\mu ,B)=\inf_{b \in B}\Vert \mu -b\Vert$.

Let $\mathcal{T}:\mathcal{H}_{1} \rightarrow 2^{\mathcal{H}_{1}}$ be a multivalued mapping with a nonempty, closed, and convex fixed-point set denoted by $\operatorname{Fix}(\mathcal{T}):= \{\nu \in \mathcal{H}_{1}; \nu \in \mathcal{T} \nu \}$. Recall that the multivalued mapping $\mathcal{T}$ is said to be a (i) contraction if there exists $\Bbbk \in (0,1)$ such that $\mathcal{D}(\mathcal{T}\mu ,\mathcal{T}\nu )\leq \Bbbk \Vert \mu - \nu \Vert$ for all $\mu ,\nu \in \mathcal{H}_{1}$; (ii) nonexpansive if $\mathcal{D}(\mathcal{T}\mu ,\mathcal{T}\nu ) \leq \Vert \mu -\nu \Vert$ for all $\mu ,\nu \in \mathcal{H}_{1}$; (iii) quasinonexpansive if $\operatorname{Fix}(\mathcal{T}) \neq \emptyset$ and $\mathcal{D}(\mathcal{T}\mu ,\nu ) \leq \Vert \mu -\nu \Vert$ for all $\mu \in \mathcal{H}_{1}$, $\nu \in \operatorname{Fix}(\mathcal{T})$; (iv) demicontractive [20] if $\operatorname{Fix}(\mathcal{T}) \neq \emptyset$, and there exists $\Bbbk \in [0,1)$ such that $\mathcal{D}(\mathcal{T}\mu ,\mathcal{T}\nu )^{2} \leq \Vert \mu - \nu \Vert ^{2}+\Bbbk d(\mu ,\mathcal{T}\nu )^{2}$ for all $\mu \in \mathcal{H}_{1}$, $\nu \in \operatorname{Fix}(\mathcal{T})$. It is worth mentioning that every multivalued quasinonexpansive mapping $\mathcal{T}$ with $\operatorname{Fix}(\mathcal{T}) \neq \emptyset$ is demicontractive, but not all multivalued demicontractive mappings are quasinonexpansive (see Example 2.2 in Ref. [25] for the proper inclusion).

Recall that the best approximation operator $\mathcal{P_{T}}$ of a multivalued mapping $\mathcal{T}:\mathcal{H}_{1}\rightarrow \mathcal{PB({H}}_{1})$ is defined as $\mathcal{P_{T}}(\mu ) := \{\nu \in \mathcal{T}\mu : d(\mu , \mathcal{T}\mu )=\Vert \mu -\nu \Vert \}$. Observe that $\operatorname{Fix}(\mathcal{T}) = \operatorname{Fix}(\mathcal{P_{T}})$ and $\mathcal{P_{T}}$ satisfies the endpoint condition, i.e., $\mathcal{T}\mu = \{\mu \}$, for all $\mu \in \operatorname{Fix}(\mathcal{T})$. Meanwhile, there is an example for the best approximation operator $\mathcal{P_{T}}$ that is nonexpansive, but $\mathcal{T}$ is not necessarily nonexpansive [31]. Recall also that the multivalued mapping $\mathcal{T}:\mathcal{H}_{1}\rightarrow \mathcal{CB({H}}_{1})$ satisfies the demiclosedness principle at 0 if for any sequence $(x_{k})$ in $\mathcal{H}_{1}$ that converges weakly to $\mu \in \mathcal{H}_{1}$ and the sequence $(\Vert x_{k} - y_{k}\Vert )$ converges strongly to 0, where $y_{k} \in \mathcal{T}x_{k}$, then $\mu \in \operatorname{Fix}(\mathcal{T})$. The Hilbert space $\mathcal{H}_{1}$ satisfies Opial’s condition if for a sequence $(\nu _{k})\subset \mathcal{H}_{1}$ with $\nu _{k}\rightharpoonup \nu$ then the inequality $\liminf_{k \rightarrow \infty}\Vert \nu _{k}-\nu \Vert < \liminf_{k \rightarrow \infty}\Vert \nu _{k}-\mu \Vert$ holds for all $\mu \in \mathcal{H}_{1}$ with $\nu \neq \mu$. Moreover, $\mathcal{H}_{1}$ satisfies the Kadec–Klee property, i.e., if $\nu _{k}\rightharpoonup \nu$ and $\Vert \nu _{k}\Vert \rightarrow \Vert \nu \Vert$ as $k \rightarrow \infty$, then $\Vert \nu _{k}-\nu \Vert \rightarrow 0$ as $k \rightarrow \infty$.

Let $g:\mathcal{H}_{1}\times \mathcal{H}_{1}\rightarrow \mathbb{R}\cup \{+ \infty \}$ be a monotone bifunction, i.e., $g(\mu ,\nu )+g(\nu ,\mu )\leq 0$, for all $\mu ,\nu \in \mathcal{H}_{1}$, then the equilibrium problem associated with the bifunction g is to find $\mu \in \mathcal{H}_{1}$ such that $g(\mu ,\nu )\geq 0$ for all $\nu \in \mathcal{H}_{1}$. The set of solutions of the equilibrium problem is denoted by $\mathcal{EP}(g)$.

Assumption 2.1

([13, 14])

Let $g:\mathcal{H}_{1}\times \mathcal{H}_{1}\rightarrow \mathbb{R}\cup \{+ \infty \}$ be the bifunction satisfying the following assumptions:

(A1): $g(\mu ,\nu )\geq 0\Rightarrow g(\mu ,\nu )\leq 0$, for all $\mu ,\nu \in \mathcal{H}_{1}$ (pseudomonotonicity);

(A2): There exist constants $d_{1}>0$ and $d_{2}>0$ for Lipschitz-type continuity such that

$$ g(\mu ,\nu )+g(\nu ,\xi )\geq g(\mu ,\xi )-d_{1} \Vert \mu -\nu \Vert ^{2}-d_{2} \Vert \nu -\xi \Vert ^{2}, \quad \text{for all } \mu ,\nu ,\xi \in \mathcal{H}_{1}; $$

(A3): If $\mu ,\nu \in \mathcal{H}_{1}$ and two sequences $(\mu _{k})$, $(\nu _{k})$ such that $\mu _{k}\rightharpoonup \mu$ and $\nu _{k}\rightharpoonup \nu$, respectively, then $f(\mu _{k},\nu _{k}) \rightarrow f(\mu ,\nu )$ (weakly continuity property of g);

(A4): For a fixed $\mu \in \mathcal{H}_{1}$, the function $g(\mu ,\cdot )$ is convex and subdifferentiable on $\mathcal{H}_{1}$.

The set $\mathcal{EP}(g)$ is weakly closed and convex provided that the bifunction g satisfies Assumption 2.1. For a finite family of bifunctions $g_{i}$ satisfying Assumption 2.1, we can compute the same Lipschitz coefficients $(d_{1},d_{2})$ for the family of bifunctions $g_{i}$ by employing the condition (A2) as

$$\begin{aligned} g_{i}(\mu ,\xi )-g_{i}(\mu ,\nu )-g_{i}(\nu , \xi )\leq d_{1,i} \Vert \mu -\nu \Vert ^{2}+d_{2,i} \Vert \nu -\xi \Vert ^{2} \leq d_{1} \Vert \mu -\nu \Vert ^{2}+d_{2} \Vert \nu -\xi \Vert ^{2}, \end{aligned}$$

where $(d_{1},d_{2})=\max_{1\leq i \leq M} (d_{1,i},d_{2,i})$. Therefore, $g_{i}(\mu ,\nu )+g_{i}(\nu ,\xi )\geq g_{i}(\mu ,\xi )-d_{1}\Vert \mu -\nu \Vert ^{2}-d_{2}\Vert \nu -\xi \Vert ^{2}$. In addition, for all $j =1,2,\ldots N$, let $\mathcal{T}_{j}:\mathcal{H}_{1} \rightarrow \mathcal{CB}(\mathcal{H}_{1})$ and $\mathcal{S}_{j}:\mathcal{H}_{2}\rightarrow \mathcal{CB}(\mathcal{H}_{2})$ be finite families of multivalued demicontractive mappings with constants $\Bbbk _{j}$ and $\tilde{\Bbbk}_{j}$, respectively, such that $\mathcal{T}_{j}- \mathrm{Id}$ and $\mathcal{S}_{j}- \mathrm{Id}$ are demiclosed at zero. If we assume $\Theta : \mathcal{H}_{1} \rightarrow \mathcal{H}_{2}$ to be a bounded linear operator then the solution set of the SCFPP for two finite families of multivalued mappings $(\mathcal{T}_{j})^{N}_{j=1}$ and $(\mathcal{S}_{j})^{N}_{j=1}$, is denoted as

$$ \Omega =: \Biggl\{ \nu \in \bigcap^{N}_{j=1} \operatorname{Fix}(\mathcal{T}_{j}): \Theta \nu \in \bigcap ^{N}_{j=1}\operatorname{Fix}(\mathcal{S}_{j}) \Biggr\} . $$

In [34], it was shown that the fixed-point set of a multivalued demicontractive mapping is closed and convex provided it satisfies the endpoint condition. In a similar fashion, we can choose $(\Bbbk ,\tilde{\Bbbk})=\sup_{1\leq j \leq N}(\Bbbk _{j}, \tilde{\Bbbk}_{j})$. Suppose that $\Gamma := (\bigcap^{M}_{i=1}\mathcal{EP}(g_{i}) )\cap \Omega \neq \emptyset$. Then, we are interested in the following problem:

$$ \nu ^{\ast} \in \Gamma . $$

(2.1)

Lemma 2.2

Let $\mu , \nu \in \mathcal{H}_{1}$ and $\theta \in \mathbb{R}$, then

$$ \Vert \mu +\nu \Vert ^{2} \leq \Vert \mu \Vert ^{2}+2 \langle \nu , \mu +\nu \rangle . $$

Lemma 2.3

([26])

Let $\mathcal{T}:\mathcal{H}_{1}\rightarrow \mathcal{CB}(\mathcal{H}_{1})$ be a $\Bbbk$-demicontractive multivalued mapping. If $\mu \in \operatorname{Fix}(\mathcal{T})$ such that $\mathcal{T}\mu =(\mu )$, then the following inequalities hold: for all $\tilde{\mu} \in \mathcal{H}_{1}$, $\tilde{\nu} \in \mathcal{T}\tilde{\mu}$,

• (I) $\langle \tilde{\mu}-\tilde{\nu},\mu -\tilde{\nu}\rangle \leq \frac{1+\Bbbk}{2}\Vert \tilde{\mu}-\tilde{\nu}\Vert ^{2}$;

• (II) $\langle \tilde{\mu}-\tilde{\nu},\tilde{\mu}-\mu \rangle \geq \frac{1-\Bbbk}{2}\Vert \tilde{\mu}-\tilde{\nu}\Vert ^{2}$.

Lemma 2.4

([18])

Let $\mathcal{H}_{1}$ be a Hilbert space and $(\nu _{k})$ be a sequence in $\mathcal{H}_{1}$. Then, for any given $(\alpha _{k})^{\infty}_{k=1} \subset (0, 1)$ with $\sum^{\infty}_{k=1}\alpha _{k}=1$ and for any positive integer i, j with $i \leq j$,

$$ \Biggl\Vert \sum^{\infty}_{k=1}\alpha _{k}\nu _{k} \Biggr\Vert ^{2}\leq \sum ^{\infty}_{k=1}\alpha _{k} \Vert \nu _{k} \Vert ^{2}-\alpha _{i} \alpha _{j} \Vert \nu _{i}-\nu _{j} \Vert ^{2}. $$

Lemma 2.5

([33])

Assume a convex and subdifferentiable function $h:\mathcal{K}\rightarrow \mathbb{R}$ defined on a nonempty, closed, convex subset $\mathcal{K}$ of a real Hilbert space $\mathcal{H}_{1}$. A point $\nu _{\ast}$ solves the convex problem $\min \{h(\nu ):\nu \in \mathcal{K}\}$ if and only if $0\in \partial h(\nu _{\ast})+N_{\mathcal{K}}(\nu _{\ast})$, where $\partial h(\cdot )$ indicates the subdifferential of h and $N_{\mathcal{K}}(\bar{\nu})$ is the normal cone of $\mathcal{K}$ at ν̄.

Lemma 2.6

([28])

Let $\mathcal{K}$ be a nonempty, closed, convex subset of a real Hilbert space $\mathcal{H}_{1}$. For every $p,q,r\in \mathcal{H}_{1}$ and $\gamma \in \mathbb{R}$, the following set is closed and convex:

$$ D=\bigl\{ v\in \mathcal{K}: \Vert q-v \Vert ^{2}\leq \Vert p-v \Vert ^{2}+ \langle r,v\rangle +\gamma \bigr\} . $$

3 Algorithm and convergence analysis

Our main iterative algorithm of this section has the following architecture (Algorithm 1).

Algorithm 1

Hybrid Inertial Extragradient Algorithm (Alg.1)

Theorem 3.1

Let the following conditions:

• (C1) $\sum^{\infty}_{k=1}\xi _{k}\|\nu _{k}-\nu _{k-1}\|<\infty$;

• (C2) $\sum^{N}_{j=1}\tilde{\alpha}_{k,j}=1$ and $\liminf_{k \rightarrow \infty} \tilde{\alpha}_{k,j}>0$, for all $j =1,2, \ldots N$;

• (C3) $\sum^{N}_{j=0}\tilde{\beta}_{k,j}=1$ and $\liminf_{k \rightarrow \infty} \tilde{\beta}_{k,j}>0$, for all $j =1,2, \ldots N$;

hold. Then, the sequence $(\nu _{k})$ generated by Algorithm 1 converges strongly to an element in Γ.

The following result is crucial for the strong convergence result of Algorithm 1.

Lemma 3.2

([2, 30])

Suppose that $\nu ^{\ast} \in \mathcal{EP}(g_{i})$, then

$$ \bigl\Vert w_{k}-\nu ^{\ast} \bigr\Vert ^{2}\leq \bigl\Vert b_{k}-\nu ^{\ast} \bigr\Vert ^{2}-(1-2 \vartheta d_{1}) \Vert u_{k}-b_{k} \Vert ^{2}-(1-2\vartheta d_{2}) \Vert u_{k}-w_{k} \Vert ^{2}, $$

where $\nu _{k}$, $b_{k}$, $u_{k}$, and $v_{k}$ are defined in Algorithm 1.

Proof of Theorem 3.1

Step 1. Algorithm 1 is well defined.

It is obvious by recalling Lemma 2.6 that the set $\mathcal{C}_{k}$ is closed and convex. Moreover, the set Ω is closed and convex. Therefore, Γ is nonempty, closed, and convex. For any $\nu ^{\ast}\in \Gamma$, it follows from Algorithm 1 that

$$\begin{aligned} \bigl\Vert b_{k}-\nu ^{\ast} \bigr\Vert ^{2} =& \bigl\Vert \nu _{k}-\nu ^{\ast}+\xi _{k}( \nu _{k}-\nu _{k-1}) \bigr\Vert ^{2} \\ \leq & \bigl\Vert \nu _{k}-\nu ^{\ast} \bigr\Vert ^{2}+\xi ^{2}_{k} \Vert \nu _{k}- \nu _{k-1} \Vert ^{2}+2\xi _{k}\bigl\langle \nu _{k}-\nu ^{\ast},\nu _{k}- \nu _{k-1}\bigr\rangle . \end{aligned}$$

(3.1)

Recalling the estimate (3.1), Lemmas 2.3 and 2.4, and Lemma 3.2, we obtain

$$\begin{aligned} \bigl\Vert y_{k}-\nu ^{\ast} \bigr\Vert ^{2} =& \Biggl\Vert w_{k}+\sum^{N}_{j=1} \tilde{\alpha}_{k,j}\lambda \Theta ^{\ast}\bigl( \mathcal{S}_{j}(\Theta w_{k})- \Theta w_{k} \bigr)-\nu ^{\ast} \Biggr\Vert ^{2} \\ =& \Biggl\Vert \sum^{N}_{j=1}\tilde{ \alpha}_{k,j}\bigl(w_{k}-\nu ^{\ast}+ \lambda \Theta ^{\ast}\bigl(\mathcal{S}_{j}(\Theta w_{k})-\Theta w_{k}\bigr)\bigr) \Biggr\Vert ^{2} \\ \leq & \sum^{N}_{j=1}\tilde{ \alpha}_{k,j} \bigl\Vert w_{k}-\nu ^{\ast}+ \lambda \Theta ^{\ast}\bigl(\mathcal{S}_{j}(\Theta w_{k})- \Theta w_{k}\bigr) \bigr\Vert ^{2} \\ = & \sum^{N}_{j=1}\tilde{ \alpha}_{k,j} \bigl( \bigl\Vert w_{k}-\nu ^{\ast} \bigr\Vert ^{2}+\lambda ^{2} \bigl\Vert \Theta ^{\ast}\bigl(\mathcal{S}_{j}(\Theta w_{k})- \Theta w_{k}\bigr) \bigr\Vert ^{2} \\ & {}+2\lambda \bigl\langle w_{k}-\nu ^{\ast},\Theta ^{\ast}\bigl(\mathcal{S}_{j}( \Theta w_{k})-\Theta w_{k}\bigr)\bigr\rangle \bigr) \\ \leq & \sum^{N}_{j=1}\tilde{ \alpha}_{k,j} \bigl( \bigl\Vert w_{k}-\nu ^{ \ast} \bigr\Vert ^{2}+\lambda ^{2} \Vert \Theta \Vert ^{2} \bigl\Vert \mathcal{S}_{j}( \Theta w_{k})- \Theta w_{k} \bigr\Vert ^{2} \\ &{} +2\lambda \bigl\langle w_{k}-\nu ^{\ast},\Theta ^{\ast}\bigl(\mathcal{S}_{j}( \Theta w_{k})-\Theta w_{k}\bigr)\bigr\rangle \bigr). \end{aligned}$$

(3.2)

Putting $M_{k}=2\lambda \langle w_{k}-\nu ^{\ast},\Theta ^{\ast}(\mathcal{S}_{j}( \Theta w_{k})-\Theta w_{k})\rangle$, since $\mathcal{S}_{j}$ is $\tilde{\Bbbk}_{j}$-demicontractive, then by Lemma 2.3, we have

$$\begin{aligned} M_{k} = & 2\lambda \bigl\langle \Theta \bigl(w_{k}-\nu ^{\ast}\bigr), \mathcal{S}_{j}( \Theta w_{k})-\Theta w_{k}\bigr\rangle \\ = & 2\lambda \bigl\langle \Theta \bigl(w_{k}-\nu ^{\ast} \bigr)+\bigl(\mathcal{S}_{j}( \Theta w_{k})-\Theta w_{k}\bigr)-\bigl(\mathcal{S}_{j}(\Theta w_{k})- \Theta w_{k}\bigr), \mathcal{S}_{j}(\Theta w_{k})-\Theta w_{k}\bigr\rangle \\ = & 2\lambda \bigl\langle \mathcal{S}_{j}(\Theta w_{k})- \Theta \nu ^{\ast}, \mathcal{S}_{j}(\Theta w_{k})- \Theta w_{k}\bigr\rangle - \bigl\Vert \mathcal{S}_{j}( \Theta w_{k})-\Theta w_{k} \bigr\Vert ^{2} \\ \leq & 2\lambda \biggl( \frac{1+\tilde{\Bbbk}_{j}}{2} \biggr) \bigl\Vert \mathcal{S}_{j}(\Theta w_{k})-\Theta w_{k} \bigr\Vert ^{2}- \bigl\Vert \mathcal{S}_{j}(\Theta w_{k})-\Theta w_{k} \bigr\Vert ^{2} \\ = & -(1-\tilde{\Bbbk}_{j})\lambda \bigl\Vert \mathcal{S}_{j}( \Theta w_{k})- \Theta w_{k} \bigr\Vert ^{2} \\ \leq & -(1-\tilde{\Bbbk})\lambda \bigl\Vert \mathcal{S}_{j}(\Theta w_{k})- \Theta w_{k} \bigr\Vert ^{2}. \end{aligned}$$

(3.3)

Utilizing (3.2) and (3.3), we obtain

$$ \bigl\Vert y_{k}-\nu ^{\ast} \bigr\Vert ^{2} \leq \bigl\Vert w_{k}-\nu ^{\ast} \bigr\Vert ^{2}- \sum^{N}_{j=1}\tilde{\alpha}_{k,j} \lambda \bigl(1-\tilde{\Bbbk}-\lambda \Vert \Theta \Vert ^{2}\bigr) \bigl\Vert \mathcal{S}_{j}(\Theta w_{k})-\Theta w_{k} \bigr\Vert ^{2}. $$

(3.4)

Since $\mathcal{T}_{j}$ is $\Bbbk _{j}$-demicontractive and by using Lemma 2.4, we have

$$\begin{aligned} \bigl\Vert z_{k}-\nu ^{\ast} \bigr\Vert ^{2} = & \Biggl\Vert \tilde{\beta}_{k,0}y_{k}+ \sum ^{N}_{j=1}\tilde{\beta}_{k,j} \mathcal{T}_{j}y_{k}-\nu ^{\ast} \Biggr\Vert ^{2} \\ \leq &\tilde{\beta}_{k,0} \bigl\Vert y_{k}-\nu ^{\ast} \bigr\Vert ^{2}+\sum^{N}_{j=1} \tilde{\beta}_{k,j} \bigl\Vert \mathcal{T}_{j}y_{k}- \nu ^{\ast} \bigr\Vert ^{2}- \tilde{\beta}_{k,0} \tilde{\beta}_{k,j} \Vert y_{k}-\mathcal{T}_{j}y_{k} \Vert ^{2} \\ = &\tilde{\beta}_{k,0} \bigl\Vert y_{k}-\nu ^{\ast} \bigr\Vert ^{2}+\sum^{N}_{j=1} \tilde{\beta}_{k,j}d\bigl(\mathcal{T}_{j}y_{k}- \mathcal{T}_{j}\nu ^{\ast}\bigr)- \tilde{\beta}_{k,0} \tilde{\beta}_{k,j} \Vert y_{k}-\mathcal{T}_{j}y_{k} \Vert ^{2} \\ \leq &\tilde{\beta}_{k,0} \bigl\Vert y_{k}-\nu ^{\ast} \bigr\Vert ^{2}+\sum^{N}_{j=1} \tilde{\beta}_{k,j}\mathcal{H}\bigl(\mathcal{T}_{j}y_{k}- \mathcal{T}_{j} \nu ^{\ast}\bigr)-\tilde{\beta}_{k,0} \tilde{\beta}_{k,j} \Vert y_{k}- \mathcal{T}_{j}y_{k} \Vert ^{2} \\ \leq &\tilde{\beta}_{k,0} \bigl\Vert y_{k}-\nu ^{\ast} \bigr\Vert ^{2}+\sum^{N}_{j=1} \tilde{\beta}_{k,j} \bigl( \bigl\Vert y_{k}-\nu ^{\ast} \bigr\Vert ^{2}+\Bbbk _{j}d(y_{k}, \mathcal{T}_{j}y_{k}) \bigr) \\ &{}-\tilde{\beta}_{k,0}\tilde{\beta}_{k,j} \Vert y_{k}-\mathcal{T}_{j}y_{k} \Vert ^{2} \\ \leq &\tilde{\beta}_{k,0} \bigl\Vert y_{k}-\nu ^{\ast} \bigr\Vert ^{2}+\sum^{N}_{j=1} \tilde{\beta}_{k,j} \bigl\Vert y_{k}-\nu ^{\ast} \bigr\Vert ^{2} \Vert ^{2}+\sum ^{N}_{j=1} \tilde{\beta}_{k,j}\Bbbk \Vert y_{k}-\mathcal{T}_{j}y_{k}) \bigl\Vert ^{2} \\ &{}-\tilde{\beta}_{k,0}\tilde{\beta}_{k,j} \Vert y_{k}-\mathcal{T}_{j}y_{k} \Vert ^{2} \\ = & \bigl\Vert y_{k}-\nu ^{\ast} \bigr\Vert ^{2} \bigr\Vert ^{2}-(\tilde{\beta}_{k,0}- \Bbbk ) \tilde{\beta}_{k,j} \Vert y_{k}-\mathcal{T}_{j}y_{k} \Vert ^{2} \\ \leq & \bigl\Vert w_{k}-\nu ^{\ast} \bigr\Vert ^{2}-\sum^{N}_{j=1} \tilde{ \alpha}_{k,j}\lambda \bigl(1-\tilde{\Bbbk}-\lambda \Vert \Theta \Vert ^{2}\bigr) \bigl\Vert \mathcal{S}_{j}(\Theta w_{k})-\Theta w_{k} \bigr\Vert ^{2} \\ &{}-(\tilde{\beta}_{k,0}-\Bbbk )\tilde{\beta}_{k,j} \Vert y_{k}- \mathcal{T}_{j}y_{k} \Vert ^{2}. \end{aligned}$$

(3.5)

This shows that Γ is contained in $\mathcal{C}_{k}$, for all $k \geq 0$. Recalling the definition of the set $\mathcal{C}_{k}$ the above estimate infers that Algorithm 1 is well defined.

Step 2. The limit $\lim_{k\rightarrow \infty }\Vert \nu _{k}-\nu _{1}\Vert$ exists.

From $\nu _{k+1}=\mathcal{P}^{\mathcal{H}_{1}}_{\mathcal{C}_{k+1}}\nu _{1}$, we have $\langle \nu _{k+1}-\nu _{1}, \nu _{k+1}-p \rangle \leq 0$ for each $p\in \mathcal{C}_{k+1}$. In particular, we have $\langle \nu _{k+1}-\nu _{1}, \nu _{k+1}-\nu ^{\ast} \rangle \leq 0$ for each $\nu ^{\ast}\in \Gamma$. This establishes that the sequence $(\Vert \nu _{k}-\nu _{1}\Vert )$ is bounded. Nevertheless, from $\nu _{k}=\mathcal{P}^{\mathcal{H}_{1}}_{\mathcal{C}_{k}}\nu _{1}$ and $\nu _{k+1}=\mathcal{P}^{\mathcal{H}_{1}}_{\mathcal{C}_{k+1}}\nu _{1}\in \mathcal{C}_{k+1}$, we have that

$$ \Vert \nu _{k}-\nu _{1} \Vert \leq \Vert \nu _{k+1}-\nu _{1} \Vert . $$

This infers that $(\Vert \nu _{k}-\nu _{1}\Vert )$ is nondecreasing and consequently

$$ \lim_{k\rightarrow \infty } \Vert \nu _{k}-\nu _{1} \Vert \quad \text{exists}. $$

(3.6)

Step 3. $\tilde{\nu}\in \Gamma$.

We first compute

$$\begin{aligned} \Vert \nu _{k+1}-\nu _{k} \Vert ^{2} =& \Vert \nu _{k+1}- \nu _{1}+\nu _{1}-\nu _{k} \Vert ^{2} \\ =& \Vert \nu _{k+1}-\nu _{1} \Vert ^{2}+ \Vert \nu _{k}- \nu _{1} \Vert ^{2}-2 \langle \nu _{k}-\nu _{1},\nu _{k+1}- \nu _{1} \rangle \\ =& \Vert \nu _{k+1}-\nu _{1} \Vert ^{2}+ \Vert \nu _{k}- \nu _{1} \Vert ^{2}-2 \langle \nu _{k}-\nu _{1},\nu _{k+1}- \nu _{k}+ \nu _{k}-\nu _{1} \rangle \\ =& \Vert \nu _{k+1}-\nu _{1} \Vert ^{2}- \Vert \nu _{k}- \nu _{1} \Vert ^{2}-2 \langle \nu _{k}-\nu _{1},\nu _{k+1}- \nu _{k} \rangle \\ \leq & \Vert \nu _{k+1}-\nu _{1} \Vert ^{2}- \Vert \nu _{k}-\nu _{1} \Vert ^{2}. \end{aligned}$$

Employing the limsup and recalling (3.6), we have

$$ \lim_{k\rightarrow \infty } \Vert \nu _{k+1}-\nu _{k} \Vert =0. $$

(3.7)

By recalling $(b_{k})$ from Algorithm 1 and the condition (C1), we have

$$ \lim_{k\rightarrow \infty } \Vert b_{k}-\nu _{k} \Vert =0. $$

(3.8)

By recalling the estimates (3.7), (3.8), and the following triangle inequality:

$$ \Vert b_{k}-\nu _{k+1} \Vert \leq \Vert b_{k}-\nu _{k} \Vert + \Vert \nu _{k}-\nu _{k+1} \Vert , $$

we have

$$ \lim_{k \rightarrow \infty} \Vert b_{k}-\nu _{k+1} \Vert =0. $$

(3.9)

Recall that $\nu _{k+1}\in \mathcal{C}_{k+1}$, therefore we have

$$ \Vert z_{k}-\nu _{k+1} \Vert \leq \Vert \nu _{k}-\nu _{k+1} \Vert +2 \xi _{k} \Vert \nu _{k}-\nu _{k-1} \Vert +2\xi _{k}\langle \nu _{k}- \nu _{k+1},\nu _{k}-\nu _{k-1} \rangle . $$

Recalling the estimate (3.7) and the condition (C1), the above estimate infers that

$$ \lim_{k\rightarrow \infty } \Vert z_{k}-\nu _{k+1} \Vert =0. $$

(3.10)

By employing the estimates (3.7), (3.10), and the following triangular inequality:

$$ \Vert z_{k}-\nu _{k} \Vert \leq \Vert z_{k}-\nu _{k+1} \Vert + \Vert \nu _{k+1}-\nu _{k} \Vert , $$

we obtain

$$ \lim_{k\rightarrow \infty } \Vert z_{k}-\nu _{k} \Vert =0. $$

(3.11)

In view of Lemma 3.2, it is easy to obtain the following variant of the estimate (3.5):

$$\begin{aligned} &(1-2\vartheta d_{1}) \Vert u_{k}-b_{k} \Vert ^{2}-(1-2\vartheta d_{2}) \Vert u_{k}-w_{k} \Vert ^{2} \\ & \leq \bigl( \bigl\Vert \nu _{k}-\nu ^{\ast} \bigr\Vert + \bigl\Vert z_{k}-\nu ^{\ast} \bigr\Vert \bigr) \Vert \nu _{k}-z_{k} \Vert +\xi ^{2}_{k} \Vert \nu _{k}-\nu _{k-1} \Vert ^{2}+2 \xi _{k} \bigl\Vert \nu _{k}-\nu ^{\ast} \bigr\Vert \Vert \nu _{k}-\nu _{k-1} \Vert . \end{aligned}$$

Recalling the estimate (3.11) and the condition (C1), we have

$$ (1-2\vartheta d_{1})\lim_{k \rightarrow \infty} \Vert u_{k}-b_{k} \Vert ^{2}-(1-2\vartheta d_{2})\lim_{k \rightarrow \infty} \Vert u_{k}-w_{k} \Vert ^{2} = 0. $$

(3.12)

The estimate (3.12) implies that

$$ \lim_{k \rightarrow \infty} \Vert u_{k}-b_{k} \Vert ^{2} = \lim_{k \rightarrow \infty} \Vert u_{k}-w_{k} \Vert ^{2} =0. $$

(3.13)

We can also extract the following two inequalities from the estimate (3.5):

$$\begin{aligned} & \sum^{N}_{j=1}\tilde{ \alpha}_{k,j}\lambda \bigl(1-\tilde{\Bbbk}- \lambda \Vert \Theta \Vert ^{2}\bigr) \bigl\Vert \mathcal{S}_{j}(\Theta w_{k})- \Theta w_{k} \bigr\Vert ^{2} \\ &\quad \leq \bigl\Vert w_{k}-\nu ^{\ast} \bigr\Vert ^{2}- \bigl\Vert z_{k}-\nu ^{\ast} \bigr\Vert ^{2} \\ &\quad \leq \bigl( \bigl\Vert \nu _{k}-\nu ^{\ast} \bigr\Vert + \bigl\Vert z_{k}-\nu ^{\ast} \bigr\Vert \bigr) \Vert \nu _{k}-z_{k} \Vert +\xi ^{2}_{k} \Vert \nu _{k}-\nu _{k-1} \Vert ^{2} \\ &\qquad {}+2\xi _{k}\bigl\langle \nu _{k}-\nu ^{\ast},\nu _{k}-\nu _{k-1} \bigr\rangle \end{aligned}$$

(3.14)

and

$$\begin{aligned} &(\tilde{\beta}_{k,0}-\Bbbk )\tilde{\beta}_{k,j} \Vert y_{k}- \mathcal{T}_{j}y_{k} \Vert ^{2} \\ &\quad \leq \bigl\Vert w_{k}-\nu ^{\ast} \bigr\Vert ^{2}- \bigl\Vert z_{k}-\nu ^{\ast} \bigr\Vert ^{2} \\ &\quad \leq \bigl( \bigl\Vert \nu _{k}-\nu ^{\ast} \bigr\Vert + \bigl\Vert z_{k}-\nu ^{\ast} \bigr\Vert \bigr) \Vert \nu _{k}-z_{k} \Vert +\xi ^{2}_{k} \Vert \nu _{k}-\nu _{k-1} \Vert ^{2} \\ &\qquad {}+2\xi _{k}\bigl\langle \nu _{k}-\nu ^{\ast},\nu _{k}-\nu _{k-1} \bigr\rangle . \end{aligned}$$

(3.15)

Utilizing (3.14) and the conditions (C1) and (C2), we have

$$ \lim_{k \rightarrow \infty} \bigl\Vert \mathcal{S}_{j}(\Theta w_{k})- \Theta w_{k} \bigr\Vert = 0, \quad \text{for all }j =1,2,\ldots N. $$

(3.16)

Since $\liminf_{k \rightarrow \infty}(\tilde{\beta}_{k,0}-\Bbbk ) \tilde{\beta}_{k,j} > 0$, we obtain from (3.15) that

$$ \lim_{k \rightarrow \infty} \bigl\Vert (\mathrm{Id}-\mathcal{T}_{j})y_{k} \bigr\Vert = 0\quad \text{for all }j =1,2,\ldots N. $$

(3.17)

In order to establish the claim of this section, we first show that $\nu ^{\ast} \in \bigcap^{M}_{i=1}\mathcal{EP}(g_{i})$.

Observe that

$$ u_{k}=\arg \min \biggl\{ \vartheta g_{i}(b_{k}, \tilde{y})+\frac{1}{2} \Vert b_{k}- \tilde{y} \Vert ^{2}:\tilde{y} \in \mathcal{K}\biggr\} . $$

Recalling Lemma 2.5, we obtain

$$ 0 \in \partial _{2}\biggl\{ \vartheta g_{i}(b_{k}, \tilde{y})+\frac{1}{2} \Vert b_{k}-\tilde{y} \Vert ^{2}\biggr\} (u_{k})+N_{\mathcal{K}}(u_{k}). $$

This implies the existence of $p \in \partial _{2} g_{i}(b_{k},u_{k})$ and $\bar{p} \in N_{\mathcal{K}}(u_{k})$ such that

$$ \vartheta p+b_{k}-u_{k}+\bar{p}. $$

(3.18)

Since $\bar{p} \in N_{\mathcal{K}}(u_{k})$ and $\langle \bar{p},\tilde{y}-u_{k} \rangle \leq 0$ for all $\tilde{y} \in \mathcal{K}$. Hence, recalling the estimate (3.18), we have

$$ \vartheta \langle p, \tilde{y}-u_{k}\rangle \geq \langle u_{k}-b_{k}, \tilde{y}-u_{k} \rangle , \quad \forall \tilde{y} \in \mathcal{K}. $$

(3.19)

Since $p \in \partial _{2}g_{i}(b_{k},u_{k})$,

$$ g_{i}(b_{k},\tilde{y})-g_{i}(b_{k},u_{k}) \geq \langle p,\tilde{y}-u_{k} \rangle ,\quad \forall \tilde{y} \in \mathcal{K}. $$

(3.20)

Therefore, recalling the estimates (3.19) and (3.20), we obtain

$$ \vartheta \bigl(g_{i}(b_{k},\tilde{y})-g_{i}(b_{k},u_{k}) \bigr)\geq \langle u_{k}-b_{k}, \tilde{y}-u_{k} \rangle , \quad \forall \tilde{y} \in \mathcal{K}. $$

(3.21)

Since $(\nu _{k})$ is bounded, there exists a subsequence $(\nu _{k_{t}})$ of $(\nu _{k})$ such that $\nu _{k_{t}} \rightharpoonup \tilde{\nu} \in \mathcal{H}_{1}$ as $t \rightarrow \infty$. This also implies that $z_{k_{t}} \rightharpoonup \tilde{\nu}$ and $w_{k_{t}} \rightharpoonup \tilde{\nu}$ as $t \rightarrow \infty$. Since $u_{k} \rightharpoonup \tilde{\nu}$, therefore, recalling the assumption (A3) and the estimate (3.21), we deduce that $0\leq g_{i}(\tilde{\nu},y)$ for all $y \in \mathcal{K}$ and $i \in \{1,2,3,\ldots ,M\}$. This infers that $\tilde{\nu} \in \bigcap^{M}_{i=1}\mathcal{EP}(g_{i})$.

Finally, we show that $\tilde{\nu} \in \Omega$. Since $x_{k_{t}} \rightharpoonup \tilde{\nu}$ as $t \rightarrow \infty$, recalling the demiclosed principal along with the estimate (3.16) and (3.17), we have $\tilde{\nu} \in \Omega$. Hence, $\tilde{\nu} \in \Gamma$.

Step 4. $\nu _{k}\rightarrow \nu ^{*}=\mathcal{P}^{\mathcal{H}_{1}}_{\Gamma } \nu _{1}$.

Since $\nu ^{*}=\mathcal{P}^{\mathcal{H}_{1}}_{\Gamma}\nu _{1}$ and $\tilde{\nu} \in \Gamma$, we have

$$\begin{aligned} \bigl\Vert \nu _{1}-\nu ^{*} \bigr\Vert \leq \Vert \nu _{1}-\tilde{\nu} \Vert \leq &\liminf_{t \rightarrow \infty} \Vert \nu _{1}-\nu _{k_{t}} \Vert \leq \limsup _{t \rightarrow \infty} \Vert \nu _{1}-\nu _{k_{t}} \Vert \leq \Vert \nu _{1}-\tilde{\nu} \Vert . \end{aligned}$$

Recalling the uniqueness of the metric projection operator yields that $\tilde{\nu}=\nu ^{*}=\mathcal{P}^{\mathcal{H}_{1}}_{\Gamma }\nu _{1}$. This completes the proof. □

If for each $j=1,2,\ldots , N$, let $S_{j}=\mathrm{Id}$, then we have the following result:

Corollary 3.3

Let $\mathcal{K} \subseteq \mathcal{H}_{1}$ be a nonempty, closed, convex subset of a real Hilbert space $\mathcal{H}_{1}$ and let $g_{i}:\mathcal{K}\times \mathcal{K}\rightarrow \mathbb{R}\cup \{+ \infty \}$ be a finite family of bifunctions satisfying Assumptions 2.1for all $i =1,2,\ldots ,M$. For all $j =1,2,\ldots ,N$, let $\mathcal{T}_{j}:\mathcal{H}_{1} \rightarrow \mathcal{CB}(\mathcal{H}_{1})$ be a finite family of multivalued $\Bbbk _{j}$-demicontractive mappings such that $\mathcal{T}_{j}- \mathrm{Id}$ are demiclosed at zero. Assume that $\Gamma := \bigcap^{M}_{i=1}\mathcal{EP}(g_{i})\cap \bigcap^{N}_{j=1}\operatorname{Fix}( \mathcal{T}_{j}) \neq \emptyset$ and calculate

$$\begin{aligned} \textstyle\begin{cases} b_{k}=\nu _{k}+\xi _{k}(\nu _{k}-\nu _{k-1}); \\ u_{k}= \operatorname{argmin}\{\vartheta g_{i}(b_{k},\tilde{y})+\frac{1}{2} \Vert b_{k}-\tilde{y} \Vert ^{2}:\tilde{y} \in \mathcal{K}\}, & i=1,2, \ldots ,M; \\ w_{k}=\operatorname{argmin}\{\vartheta g_{i}(u_{k},\tilde{y})+\frac{1}{2} \Vert b_{k}-\tilde{y} \Vert ^{2}:\tilde{y} \in \mathcal{K}\}, & i=1,2, \ldots ,M; \\ z_{k}=\tilde{\beta}_{k,0}w_{k}+\sum^{N}_{j=1}\tilde{\beta}_{k,j} \mathcal{T}_{j}w_{k}, &j=1,2,\ldots ,N; \\ \mathcal{C}_{k+1}=\{z^{\ast} \in \mathcal{C}_{k}: \Vert z_{k}-z^{ \ast} \Vert ^{2} \\ \hphantom{\mathcal{C}_{k+1}}\leq \Vert \nu _{k}-z^{\ast} \Vert ^{2}+\xi ^{2}_{k} \Vert \nu _{k}-\nu _{k-1} \Vert ^{2}+2\xi _{k}\langle \nu _{k}-z^{ \ast},\nu _{k}-\nu _{k-1}\rangle \}; \\ \nu _{k+1}=\mathcal{P}^{\mathcal{H}_{1}}_{\mathcal{C}_{k+1}} \nu _{1}, &\forall k \geq 1. \end{cases}\displaystyle \end{aligned}$$

(3.22)

Assume that the conditions (C1) and (C3) hold, then the sequence $(\nu _{k})$ generated by (3.22) converges strongly to an element in Γ.

As a direct application of Theorem 3.1, we have the following result for the variational inequality problem (i.e., find $\bar{\nu} \in \mathcal{K}$ for which $\langle A\bar{\nu},\bar{\mu}-\bar{\nu} \rangle \geq 0$ $\forall \bar{\mu} \in \mathcal{K}$, where $A:\mathcal{K} \rightarrow \mathcal{H}_{1}$ is a nonlinear, monotone mapping defined on a nonempty, closed, convex subset $\mathcal{K} \subseteq \mathcal{H}_{1}$):

Theorem 3.4

Let $\mathcal{K} \subseteq \mathcal{H}_{1}$ be a nonempty, closed, convex subset of a real Hilbert space $\mathcal{H}_{1}$ and for each $i = 1,2,\ldots ,M$ let $A_{i}: \mathcal{K} \rightarrow \mathcal{H}_{1}$ be a finite family of pseudomonotone and L-Lipschitz continuous mappings. For all $j = 1,2,\ldots ,N$, let $\mathcal{T}_{j}:\mathcal{H}_{1} \rightarrow \mathcal{CB}(\mathcal{H}_{1})$ and $\mathcal{S}_{j}:\mathcal{H}_{2}\rightarrow \mathcal{CB}(\mathcal{H}_{2})$ be two finite families of multivalued, demicontractive mappings with constants $\Bbbk _{j}$ and $\tilde{\Bbbk}_{j}$, respectively, such that $\mathcal{T}_{j}- \mathrm{Id}$ and $\mathcal{S}_{j}- \mathrm{Id}$ are demiclosed at zero. Let $\Gamma =\bigcap^{M}_{i=1}VI(\mathcal{K}, A_{i})\cap \Omega \neq \emptyset$. Let $\xi _{k}$ be a bounded real sequence and $0 < \lambda < \frac{1-\tilde{\Bbbk}}{\Vert \Theta \Vert ^{2}}$. Given $\nu _{k} \in \mathcal{H}_{1}$, calculate

$$\begin{aligned} \textstyle\begin{cases} b_{k}=\nu _{k}+\xi _{k}(\nu _{k}-\nu _{k-1}); \\ u_{k}=P_{\mathcal{C}}(b_{k}-\vartheta A_{i}(b_{k})),& i=1,2, \ldots ,M; \\ w_{k}=P_{\mathcal{C}}(b_{k}-\vartheta A_{i}(u_{k})), & i=1,2, \ldots ,M; \\ y_{k}=w_{k}+\sum^{N}_{j=1}\tilde{\alpha}_{k,j}\lambda \Theta ^{\ast}(\mathcal{S}_{j}(\Theta w_{k})-\Theta w_{k}),& j=1,2, \ldots ,N; \\ z_{k}=\tilde{\beta}_{k,0}y_{k}+\sum^{N}_{j=1}\tilde{\beta}_{k,j} \mathcal{T}_{j}y_{k}, & j=1,2,\ldots ,N; \\ \mathcal{C}_{k+1}=\{z^{\ast} \in \mathcal{C}_{k}: \Vert z_{k}-z^{ \ast} \Vert ^{2} \\ \hphantom{\mathcal{C}_{k+1}}\leq \Vert \nu _{k}-z^{\ast} \Vert ^{2}+\xi ^{2}_{k} \Vert \nu _{k}-\nu _{k-1} \Vert ^{2}+2\xi _{k}\langle \nu _{k}-z^{ \ast},\nu _{k}-\nu _{k-1}\rangle \}, \\ \nu _{k+1}=\mathcal{P}^{\mathcal{H}_{1}}_{\mathcal{C}_{k+1}} \nu _{1}, &\forall k \geq 1. \end{cases}\displaystyle \end{aligned}$$

(3.23)

Assume that the conditions (C1)–(C3) hold, then the sequence $(\nu _{k})$ generated by (3.23) converges strongly to an element in Γ.

Proof

Let $g_{i}(\bar{\nu},\bar{\mu})=\langle A_{i}(\bar{\nu}), \bar{\mu}- \bar{\nu} \rangle$ for all $\nu ,y \in \mathcal{K}$ and $i = 1,2,\ldots ,M$. Since $A_{i}$ is L-Lipschitz continuous, we observe that for all $\bar{\nu},\bar{\mu},\bar{\xi} \in \mathcal{C}$

$$\begin{aligned} g_{i}(\bar{\nu},\bar{\mu})+g_{i}(\bar{\mu},\bar{ \xi})-g_{i}(\bar{\nu}, \bar{\xi}) =& \bigl\langle A_{i}( \bar{\nu}), \bar{\mu}-\bar{\nu}\bigr\rangle + \bigl\langle A_{i}(\bar{ \mu}), \bar{\xi}-\bar{\mu}\bigr\rangle -\bigl\langle A_{i}( \bar{\nu}), \bar{\xi}-\bar{\nu} \bigr\rangle \\ =& - \bigl\langle A_{i}(\bar{\mu})-A_{i}(\bar{\nu}), \bar{\mu}-\bar{\xi} \bigr\rangle \\ \geq & - \bigl\Vert A_{i}(\bar{\mu})-A_{i}(\bar{\nu}) \bigr\Vert \Vert \bar{\mu}-\bar{\xi} \Vert \\ \geq & - L \Vert \bar{\mu}-\bar{\nu} \Vert \Vert \bar{\mu}-\bar{\xi} \Vert \\ \geq & - \frac{L}{2} \Vert \bar{\mu}-\bar{\nu} \Vert ^{2} - \frac{L}{2} \Vert \bar{\mu}-\bar{\xi} \Vert ^{2}. \end{aligned}$$

This infers that $g_{i}$ is Lipschitz-type continuous with $d_{1}=d_{2}=\frac{L}{2}$. Moreover, the pseudomonotonicity of $A_{i}$ ensures the pseudomonotonicity of $g_{i}$. From Algorithm 1, we have

$$\begin{aligned}& u_{k} = \arg \min \biggl\{ \vartheta \bigl\langle A_{i}( \nu _{k}), \mu - \nu _{k} \bigr\rangle +\frac{1}{2} \Vert \nu _{k}-\mu \Vert ^{2}: \mu \in \mathcal{K}\biggr\} , \\& w_{k} = \arg \min \biggl\{ \vartheta \bigl\langle A_{i}(u_{k}),\mu - u_{k} \bigr\rangle + \frac{1}{2} \Vert \nu _{k}-\mu \Vert ^{2}: \mu \in \mathcal{K}\biggr\} . \end{aligned}$$

Equivalently, we have

$$\begin{aligned}& u_{k} = \arg \min \biggl\{ \frac{1}{2}\Vert \tilde{y}-(\nu _{k}-\vartheta A_{i}( \nu _{k})\Vert ^{2}: \tilde{y} \in \mathcal{K}\biggr\} =P_{\mathcal{K}}\bigl(\nu _{k}- \vartheta A_{i}(\nu _{k})\bigr), \\& w_{k} = \arg \min \biggl\{ \frac{1}{2}\Vert \tilde{y}-(\nu _{k}-\vartheta A_{i}(u_{k}) \Vert ^{2}: \tilde{y} \in \mathcal{K}\biggr\} =P_{\mathcal{K}}\bigl(\nu _{k}- \vartheta A_{i}(u_{k})\bigr). \end{aligned}$$

Recalling the proof of Theorem 3.1 with the above-mentioned $g_{i}(\nu ,\mu )$ for all $i \in \{1,2,\ldots , M\}$ leads to the desired result. □

Setting a terminating criterion by fixing $k > k_{\max}$ for an appropriately chosen large number $k_{\max}$, we now propose a Halpern-type variant of Algorithm 1.

Theorem 3.5

If $\Gamma \neq \emptyset$ such that the conditions (C1)–(C3) with $C^{\ast}=\lim_{k \rightarrow \infty}\tilde{\gamma}_{k}=0$ hold. Then, the sequence $(\nu _{k})$ generated by Algorithm 2 converges strongly to an element in Γ.

Algorithm 2

Hybrid Inertial Halpern-Extragradient Algorithm (Alg.2)

Proof

Observe that the set $\mathcal{C}_{k}$ can be expressed as:

$$\begin{aligned} \mathcal{C}_{k} =&\bigl\{ \nu ^{\ast} \in \mathcal{C}_{k}: \bigl\Vert h_{k}-\nu ^{ \ast} \bigr\Vert ^{2}\leq \tilde{\gamma}_{k} \bigl\Vert q-\nu ^{\ast} \bigr\Vert ^{2}+(1- \tilde{\gamma}_{k}) \bigl( \Vert \nu _{k}-\nu _{k+1} \Vert ^{2}+\xi ^{2}_{k} \Vert \nu _{k}-\nu _{k-1} \Vert ^{2} \\ &{}+2\xi _{k}\langle \nu _{k}-\nu _{k+1},\nu _{k}-\nu _{k-1}\rangle \bigr) \bigr\} . \end{aligned}$$

Recalling the proof of Theorem 3.1, it infers that (i) the sets $\Gamma \text{and} \mathcal{C}_{k}$ are closed and convex, satisfying $\Gamma \subset \mathcal{C}_{k+1}$ for all $k \geq 0$; (ii) $(\nu _{k})$ is bounded such that

$$ \lim_{k \rightarrow \infty} \Vert \nu _{k+1} - \nu _{k} \Vert = 0. $$

(3.24)

Since $\nu _{k+1}=\mathcal{P}^{\mathcal{H}_{1}}_{\mathcal{C}_{k}}(q) \in \mathcal{C}_{k}$ and by the definition of $\mathcal{C}_{k}$, we have

$$\begin{aligned} \Vert h_{k}-\nu _{k+1} \Vert ^{2} \leq & \tilde{\gamma}_{k} \Vert q- \nu _{k+1} \Vert ^{2}+(1-\tilde{\gamma}_{k}) \\ &{}\times \bigl( \Vert \nu _{k}-\nu _{k+1} \Vert ^{2} +\xi ^{2}_{k} \Vert \nu _{k}-\nu _{k-1} \Vert ^{2} \\ &{}+2\xi _{k}\langle \nu _{k}-\nu _{k+1},\nu _{k}-\nu _{k-1}\rangle \bigr). \end{aligned}$$

Recalling the estimate (3.24), the conditions (C1)–(C3) and the boundedness of $(\nu _{k})$, we obtain

$$ \lim_{k \rightarrow \infty} \Vert h_{k}-\nu _{k+1} \Vert =0, $$

implying that

$$ \lim_{k \rightarrow \infty} \Vert h_{k}-\nu _{k} \Vert =0. $$

(3.25)

Also, observe that

$$\begin{aligned} &(1-\tilde{\gamma}_{k}) (1-2\vartheta d_{1}) \Vert u_{k}-\nu _{k} \Vert ^{2}+(1-2 \vartheta d_{2}) \Vert u_{k}-w_{k} \Vert ^{2} \\ & \quad \leq \tilde{\gamma}_{k}\bigl( \bigl\Vert q-\nu ^{\ast} \bigr\Vert ^{2}- \bigl\Vert \nu _{k}-\nu ^{\ast} \bigr\Vert ^{2}\bigr)+ \Vert \nu _{k}-z_{k} \Vert \bigl( \bigl\Vert \nu _{k}- \nu ^{\ast} \bigr\Vert + \bigl\Vert z_{k}-\nu ^{\ast} \bigr\Vert \bigr), \end{aligned}$$

for each $\nu ^{\ast} \in \Gamma$. Recalling the estimate (3.24), the conditions (C1)–(C3), and the boundedness of $(\nu _{k})$, we obtain

$$ \lim_{k \rightarrow \infty} \Vert u_{k}-\nu _{k} \Vert =0= \lim_{k \rightarrow \infty} \Vert w_{k}-\nu _{k} \Vert , \quad i \in \{1,2,\ldots ,M \}. $$

Recalling $h_{k}=\tilde{\gamma}_{k}q+(1-\tilde{\gamma}_{k})z_{k}$ and the conditions (C2) and (C3) with $C^{\ast}$, we obtain

$$ \Vert z_{k}-\nu _{k} \Vert \leq \frac{1}{(1-\tilde{\gamma}_{k})} \Vert h_{k}- \nu _{k} \Vert +\frac{\tilde{\gamma}_{k}}{(1-\tilde{\gamma}_{k})} \Vert q-\nu _{k} \Vert . $$

Recalling the estimate (3.25) again, the above estimate implies that

$$ \lim_{k \rightarrow \infty} \Vert z_{k}-\nu _{k} \Vert =0. $$

The rest of the proof of Theorem 3.5 is similar to the proof of Theorem 3.1 and is therefore omitted here. □

Remark 3.6

From the numerical standpoint, the condition (C1) can easily be aligned in an algorithm as $\|\nu _{k}-\nu _{k-1}\|$ is a priorly known before selecting $\xi _{k}$ satisfying $0 \leq \xi _{k} \leq \widehat{\xi _{k}}$, where

$$ \widehat{\xi _{k}} = \textstyle\begin{cases} \min \{\frac{\sigma _{k}}{ \Vert \nu _{k}-\nu _{k-1} \Vert }, \xi \}& \text{if } \nu _{k} \neq \nu _{k-1}; \\ \xi & otherwise,\end{cases} $$

where $\{ \sigma _{k}\}$ is a positive sequence such that $\sum^{\infty}_{k = 1}\sigma _{k} < \infty$ and $\xi _{k} \in [0,1)$.

4 Numerical experiment and results

This section provides the effective viability of our algorithm supported by a suitable example.

Example 4.1

Let $\mathcal{H}_{1} = \mathbb{R}=\mathcal{H}_{2}$ be the set of all real numbers with the inner product defined by $\langle \mu , \nu \rangle = \mu \nu$, for all $\mu , \nu \in \mathbb{R}$ and the induced usual norm $|\cdot |$. For each $i=\{1,2,3,\ldots ,M\}$ and $\mathcal{K} = [0,1] \subset \mathcal{H}_{1}$, let the bifunctions $g_{i}(\mu ,\nu ): \mathcal{K}\times \mathcal{K} \rightarrow \mathbb{R}$ be defined by $g_{i}(\mu ,\nu )= h_{i}(\mu )(\nu -\mu )$ with

$$ h_{i}(\mu ) = \textstyle\begin{cases} 0, & \text{if } 0 \leq \mu \leq \varrho _{i}, \\ \sin (\mu -\varrho _{i})+\exp ( \mu -\varrho _{i})-1,& \text{if } \varrho _{i} \leq \mu \leq 1, \end{cases} $$

where $0 < \varrho _{1} < \varrho _{2} < \cdots <\varrho _{m} < 1$. It is easy to prove that $g_{i}(\mu ,\nu )$ is pseudomonotone satisfying the Assumptions 2.1 with $h_{i}(\mu )$ being 4-Lipschitz continuous. Observe that $\mathcal{EP}(g_{i})=[0,\varrho _{i}]$ if and only if $0 \leq \mu \leq \varrho _{i}$ for all $\nu \in [0,1]$. Hence, $\bigcap^{M}_{i=1}\mathcal{EP}(g_{i})=[0,\varrho _{1}]$. For each $j =1,2,\ldots , N$, let $\mathcal{T}_{j}$ and $\mathcal{S}_{j}$ be defined as:

$$ \mathcal{T}_{j}(\mu )= \textstyle\begin{cases} 0, & \text{if } \mu < 0; \\ {[}\frac{\mu}{j+1}, \mu ], & \text{if } \mu \geq 0 \end{cases} $$

and

$$ \mathcal{S}_{j}(\mu )= \textstyle\begin{cases} [ 0,\frac{ \vert \mu \vert }{j+2} ], &\text{if } \mu < j+2; \\ [1, j+1 ], & \text{if } \mu \geq j+2. \end{cases} $$

It is not difficult to show that $\mathcal{T}_{j}$ and $\mathcal{S}_{j}$ are 0-demicontractive, and $\mathrm{Id} - \mathcal{T}_{j}$ and $\mathrm{Id} - \mathcal{S}_{j}$ are demiclosed at zero for all $j =1,2,\ldots , N$. We also define a bounded linear operator $\Theta : \mathbb{R} \rightarrow \mathbb{R}$ by $\Theta \mu = 3\mu$. Thus, $\Theta ^{\ast}\mu = 3\mu$ and $\Theta = 3$. It is clear that $0 \in \Omega$, where $\Omega = \{\mu \in \bigcap^{N}_{j=1}\operatorname{Fix}(\mathcal{T}_{j}): \Theta \mu \in \bigcap^{N}_{j=1}\operatorname{Fix}(\mathcal{S}_{j}) \}$. Hence, $\Gamma = \bigcap^{M}_{i=1}\mathcal{EP}(g_{i}) \cap \Omega = 0$.

Set $\xi _{k}=\xi = 0.5$, $\tilde{\alpha}_{k,j} =\tilde{\alpha}_{k}= \frac{1}{2^{k+1}}$, $\tilde{\beta}_{k,j} =\tilde{\beta}_{k}= \frac{1}{2^{k}}$, $\vartheta = \frac{1}{7}$, $\varrho _{i}=\frac{i}{(M+1)}$, $M=2 \times 10^{5}$ and $N=3 \times 10^{5}$. Since

$$ \textstyle\begin{cases} \min \{\frac{1}{k^{2} \Vert \nu _{k}-\nu _{k-1} \Vert },0.5\} & \text{if } \nu _{k}\neq \nu _{k-1}; \\ 0.5& \text{otherwise}. \end{cases} $$

The terminating criteria of Algorithm 1 is set as $\mathrm{Error}= E_{k}= \|\nu _{k}-\nu _{k-1}\|<10^{-6}$. Table 1 summarizes the computation of Algorithm 1 and its variant.

Table 1 Summary of the Numerical Computation for Algorithm 1

The terminating criterion $E_{k}$ and $(\nu _{k})$ summarized in Table 1 for Algorithm 1 are depicted in Fig. 1.

Figure 1

Comparison of Alg.1, $\xi _{k} \neq 0$, $\xi _{k} = 0$

We can see from Table 1 and Fig. 1 that Alg.1 with $\xi _{k}\neq 0$ outperforms Alg.1 with $\xi _{k} = 0$ with respect to the reduction in the error, time consumption, and the number of iterations required for the convergence towards the common solution.

5 Conclusions

In this paper, we have investigated an inertial-based, parallel, hybrid, extragradient algorithm for constructing iteratively a common solution of the pseudomonotone EP and the SCFPP associated with the finite families demicontractive mappings in Hilbert spaces. The abstract formalism of the problem has been strengthened with the computer-assisted simulation for the algorithm via an appropriate numerical example. We emphasize that our proposed abstract formalism together with the computer-assisted iterative algorithm arise naturally in various forms of real-world applications and would be an important topic of future research.