Novel Accelerated Cyclic Iterative Approximation for Hierarchical Variational Inequalities Constrained by Multiple-Set Split Common Fixed-Point Problems

Abstract

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed-point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, with special cases being able to be found in many important engineering practical applications, such as image recognizing, signal processing, and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results.

1. Introduction

Let $\mathfrak{F}:\mathcal{H}\to \mathcal{H}$ be a mapping that is Lipschitz continuous and strongly monotone, and let $Fix\left(\mathfrak{F}\right)$ denote the fixed-point set of $\mathfrak{F}$. For $i=\{1,2,\cdots ,p\}$, suppose that ${\mathfrak{S}}_i:\mathcal{H}\to \mathcal{H}$ represents nonlinear mappings, such that $\Gamma ={\bigcap }_{i=1}^{p}Fix\left({\mathfrak{S}}_i\right)\ne \varnothing$. The variational inequality problem defined on the solution set of a common fixed-point problem, also known as the hierarchical variational inequality problem (HVIP; see [1]), is defined as follows:

$$\begin{array}{c}\hfill \mathrm{Find}{\vartheta }^{*}\in \bigcap _{i=1}^{p}Fix\left({\mathfrak{S}}_i\right)\mathrm{such}\mathrm{that}〈\mathfrak{F}{\vartheta }^{*},\varsigma -{\vartheta }^{*}〉\ge 0,\forall \varsigma \in \Gamma ,\end{array}$$

where $\langle ·,·\rangle$ denotes the inner product in Hilbert space $\mathcal{H}$ and $\Gamma$ represents the common fixed-point set. This type of HVIP is pivotal in various practical applications in real-world applications, such as power supervision [2], optimal regulating [3], network positioning [4], machine learning [5], signal restructuring, beamforming, bandwidth distribution, and so on [1,6,7]. The broad applicability of the HVIP has sparked significant research interest in recent years, leading to numerous studies exploring its various facets. For more detail, one can refer to [1,3,6] and the references cited within.

As known to all, an inverse problem is a fundamental challenge in computational mathematics, with a wide range of applications. Over the recent few decades, the study of inverse problems has seen rapid development, finding applications in diverse fields, such as computer vision, tomography, machine learning, physics, medical imaging, remote sensing, statistics, ocean acoustics, aviation, and geography. In 1994, Censor et al. [8] introduced the split feasibility problem (SFP) as a model to address certain types of inverse problems. Specifically, let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be two real Hilbert spaces. Then, the SFP can be formulated as

$$\mathrm{Find}{\vartheta }^{*}\in C\mathrm{such}\mathrm{that}\mathfrak{A}{\vartheta }^{*}\in Q,$$

where $C\subseteq {\mathcal{H}}_1$ and $Q\subseteq {\mathcal{H}}_2$ are nonempty closed convex subsets, and $\mathfrak{A}:{\mathcal{H}}_1\to {\mathcal{H}}_2$ represents a bounded linear operator. The SFP has become a crucial tool in a range of applications, such as computed tomography, image restoration, signal processing, and intensity-modulated radiation therapy (IMRT) [9,10,11,12,13,14,15,16,17,18,19,20,21]. And then, motivated by challenges in inverse problems related to IMRT, Censor et al. [10] generalized the SFP on the solution set of a common fixed-point problem and proposed the following multiple-set split feasibility problem (MSFP): Find a point ${\vartheta }^{*}$, such that

$$\begin{array}{c}\hfill {\vartheta }^{*}\in \bigcap _{i=1}^p{C}_i\mathrm{such}\mathrm{that}\mathfrak{A}{\vartheta }^{*}\in \bigcap _{j=1}^r{Q}_j,\end{array}$$

(1)

where p and r are integers with $p,r\ge 1$, and ${\left\{C_i\right\}}_{i=1}^p$ and ${\left\{Q_j\right\}}_{j=1}^r$ represent nonempty, closed, convex subsets of Hilbert spaces ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, respectively. In this case, when $p=r=1$, the MSFP (1) further simplifies to the SFP. As an extension of the convex feasibility problem, SFP, and MSFP, split common fixed-point problems (SCFPs) were later introduced by Censor and Segal [12] in 2009. SCFPs seek to find ${\vartheta }^{*}\in {\mathcal{H}}_1$, such that

$$\begin{array}{c}\hfill {\vartheta }^{*}\in Fix\left(\mathfrak{U}\right)\mathrm{and}\mathfrak{B}{\vartheta }^{*}\in Fix\left(\mathfrak{T}\right),\end{array}$$

where $\mathfrak{B}:{\mathcal{H}}_1\to {\mathcal{H}}_2$ is a bounded linear operator, and $\mathfrak{U}:{\mathcal{H}}_1\to {\mathcal{H}}_1$ and $\mathfrak{T}:{\mathcal{H}}_2\to {\mathcal{H}}_2$ are two nonlinear mappings. The SCFP has attracted significant research interest owing to its wide array of applications, such as signal processing, image reconstruction, IMRT, inverse problem modeling, and electron microscopy [12,19].

Next, we turn our attention to a class of multiple-set split common fixed-point problems (MSCFPs), which generalize SCFPs, and are recognized for their extensive applications in fields such as image reconstruction, computed tomography, and radiotherapy treatment planning [8,9,10,11]. In recent years, they have attracted considerable interest from researchers because of their broad applicability (see [1,22,23]). Formally, MSCFPs try to find ${\vartheta }^{*}\in {\mathcal{H}}_1$, such that

$$\begin{array}{c}\hfill {\vartheta }^{*}\in \bigcap _{i=1}^{p}Fix\left({\mathfrak{S}}_i\right),\mathfrak{A}{\vartheta }^{*}\in \bigcap _{j=1}^{r}Fix\left({\mathfrak{T}}_j\right),\end{array}$$

(2)

where $\mathfrak{A}:{\mathcal{H}}_1\to {\mathcal{H}}_2$ is a bounded linear operator, and for $1\le i\le p$ and $1\le j\le r$, $Fix\left({\mathfrak{S}}_i\right)$ and $Fix\left({\mathfrak{T}}_j\right)$ denote the fixed-point sets of the nonlinear mappings ${\mathfrak{S}}_i:{\mathcal{H}}_1\to {\mathcal{H}}_1$ and ${\mathfrak{T}}_j:{\mathcal{H}}_2\to {\mathcal{H}}_2$, respectively. Notably, when $p=r=1$, the MSCFP reduces to an SCFP. Furthermore, if ${\mathfrak{S}}_i$ and ${\mathfrak{T}}_j$ are severally projection operators onto nonempty closed convex subsets $C_i$ and $Q_j$, the MSCFP (2) simplifies to the MSFP (1).

On the other hand, in order to address the MSCFP (2) and circumvent the aforementioned challenges, Wang and Xu [24] introduced a cyclic iterative algorithm designed to solve MSCFPs for directed operators, as follows:

$${\vartheta }_{n+1}={\mathfrak{U}}_{{\left[n\right]}_1}\left({\vartheta }_n+\gamma {\mathfrak{A}}^{*}\left({\mathfrak{T}}_{{\left[n\right]}_2}-I\right)\mathfrak{A}{\vartheta }_n\right),$$

where $0<\gamma <{\displaystyle \frac{2}{\rho \left({\mathfrak{A}}^{*}\mathfrak{A}\right)}}$, ${\left[n\right]}_1:=n(modp)$, and ${\left[n\right]}_2:=n(modr)$. They established the weak convergence of the sequence $\left\{{\vartheta }_n\right\}$.

It is important to note that many existing algorithms rely on the calculation of operator norms to determine the appropriate step size. However, in practical applications, obtaining the operator norm is often challenging due to the complexity and high computational costs. While selecting such a step size may be theoretically optimal, it poses significant difficulties in practice, complicating the implementation of these algorithms. To overcome these difficulty, Gupta et al. [22] and Zhao et al. [23] introduced some cyclic iterative algorithms, whose step sizes do not depend on the operator norm. Especially, we note that the algorithm proposed by Zhao et al. [23] is defined as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varsigma }_n={\vartheta }_n+a_n({\vartheta }_n-{\vartheta }_{n-1}),\hfill \\ v_n={\varsigma }_n-{\gamma }_n{\mathfrak{A}}^{*}(I-{\mathfrak{T}}_n)\mathfrak{A}{\varsigma }_n,\hfill \\ {\omega }_{n+1}=(I-{\mathfrak{U}}_n)\left(v_n+(1-\lambda ){\omega }_n\right),\hfill \\ {\vartheta }_{n+1}=v_n-\lambda {\omega }_{n+1},\hfill \end{array}\right.\end{array}$$

(3)

where the step size ${\gamma }_n$ is determined by

$${\gamma }_n:=\left\{\begin{array}{cc}{\displaystyle \frac{{\rho }_n{∥(I-{\mathfrak{T}}_n)\mathfrak{A}{\varsigma }_n∥}^2}{{∥{\mathfrak{A}}^{*}(I-{\mathfrak{T}}_n)\mathfrak{A}{\varsigma }_n∥}^2}},\hfill & (I-{\mathfrak{T}}_n)\mathfrak{A}{\varsigma }_n\ne 0,\hfill \\ \gamma ,\hfill & (I-{\mathfrak{T}}_n)\mathfrak{A}{\varsigma }_n=0\hfill \end{array}\right.$$

here $\gamma >0$, and with $\mathfrak{U}$ and $\mathfrak{T}$ being quasi-nonexpansive operators. Under suitable conditions, the sequence $\left\{{\vartheta }_n\right\}$ produced by this algorithm converges weakly to a solution of the MSCFP, offering a practical advantage by eliminating the need to compute the operator norm.

Moreover, over the recent few years, there has been a significant increase in activity and significant progress in the field of MSCFPs, driven by the need to broaden their applicability to more generalized operators. Researchers have extended the framework to include various types of operators, such as quasi-nonexpansive mappings, firmly quasi-nonexpansive mappings [23], demicontractive mappings [22], and directed mappings [25]. These advancements have allowed for more robust and versatile solutions to MSCFPs across different application domains. In this paper, we focus on quasi-pseudocontractive operators, which further expand the scope of MSCFs to encompass a broader class of nonlinear mappings, including demicontractive mappings, directed mappings, and firmly quasi-nonexpansive mappings (see [26]). By examining quasi-pseudocontractive operators, we aim to provide new insights and extend the applicability of MSCFPs to a wider range of practical problems.

Recently, there has been an increasing focus on constructing iterative schemes that achieve faster convergence rates, particularly in the context of solving fixed-point and optimization problems. Inertial techniques, which discretely simulate second-order dissipative dynamical systems, have gained recognition for their ability to accelerate the convergence behavior of iterative methods. The most prevalent method in this category is single-step inertial extrapolation, given by ${\vartheta }_n+{\Theta }_n({\vartheta }_n-{\vartheta }_{n-1})$. Since the introduction of inertial-type algorithms, many researchers have incorporated the inertial term $\left[{\Theta }_n({\vartheta }_n-{\vartheta }_{n-1})\right]$ into various iterative schemes, such as Mann, Krasnoselski, Halpern, and Viscosity methods, to approximate solutions for fixed-point and optimization problems. While most studies have established weak convergence results, achieving strong convergence remains challenging and relatively rare. Recently, Kim [27] and Maingé [28] explored the inertial extrapolation technique, incorporating an extra correction term, in the context of optimization problems. This correction term plays a crucial role in enhancing the acceleration rate of the algorithm, and their studies have demonstrated some promising weak convergence results, paving the way for further exploration in this direction.

Inspired by recent advancements in iterative methods and motivated by the need for more efficient algorithms, the purpose of this paper is to explore a class of novel self-adaptive cyclic iterative algorithms for approximating solutions of HVIPs for quasi-pseudocontractive mappings. Our iterative approximation integrates the inertial approach combined with correction terms and the primal-dual approach, enhancing the rate of convergence of the iterative process. A key feature of the iterative approximation algorithms presented in this paper is their use of a self-adaptive step size strategy, which can be implemented without requiring prior knowledge of the operator norm, thus making it more practical for real-world applications. By imposing appropriate control conditions on the relevant parameters, we establish that the iterates converge strongly to the unique solution of the hierarchical variational inequality problem under consideration. We further apply our theoretical results to the multiple set split feasibility problem, demonstrating the broad applicability of our approach. To confirm the effectiveness of the proposed algorithm, we provide numerical examples that illustrate its superior performance in comparison to existing methods.

2. Preliminaries

In this paper, the inner product is denoted by $\langle ·,·\rangle$ and the norm by $\parallel ·\parallel$. The identity operator on the Hilbert space $\mathcal{H}$ is denoted by I. We represent the fixed-point set of an operator $\mathfrak{T}$ as $Fix\left(\mathfrak{T}\right)$. Strong convergence is indicated by →, while weak convergence is represented by ⇀. The weak $\omega$-limit set of the sequence $\left\{{\vartheta }_n\right\}$ is denoted by ${\omega }_w\left({\vartheta }_n\right)$.

Definition 1. 

Let $\mathcal{H}$ be a real Hilbert space. Then, for all $\vartheta ,\varsigma \in \mathcal{H}$ and $\alpha \in (0,1)$, one has

(i)

$2\langle \vartheta ,\varsigma \rangle ={\parallel \vartheta \parallel }^2+{\parallel \varsigma \parallel }^2{-\parallel \vartheta -\varsigma \parallel }^2={\parallel \vartheta +\varsigma \parallel }^2-{\parallel \vartheta \parallel }^2-{\parallel \varsigma \parallel }^2$,

(ii)

${\parallel \alpha \vartheta +(1-\alpha )\varsigma \parallel }^2={\alpha \parallel \vartheta \parallel }^2+{(1-\alpha )\parallel \varsigma \parallel }^2-\alpha (1-\alpha ){\parallel \vartheta -\varsigma \parallel }^2$,

(iii)

${\parallel \vartheta +\varsigma \parallel }^2\le {\parallel \vartheta \parallel }^2+2\langle \varsigma ,\vartheta +\varsigma \rangle .$

Definition 2. 

A mapping $\mathfrak{F}:\mathcal{H}\to \mathcal{H}$ is termed l-Lipschitz continuous, provided that there exists a constant $l>0$, such that

$$\parallel \mathfrak{F}\vartheta -\mathfrak{F}\varsigma \parallel \le l\parallel \vartheta -\varsigma \parallel ,\forall \vartheta ,\varsigma \in \mathcal{H}.$$

The mapping $\mathfrak{F}$ is referred to as τ-strongly monotone if a constant $\tau >0$ can be found, such that

$$\langle \mathfrak{F}\vartheta -\mathfrak{F}\varsigma ,\vartheta -\varsigma \rangle \ge {\tau \parallel \vartheta -\varsigma \parallel }^2,\forall \vartheta ,\varsigma \in \mathcal{H}.$$

Lemma 1 

([29]). Let $\mathfrak{T}:\mathcal{H}\to \mathcal{H}$ be a l-Lipschitizian mapping with $l\ge 1$. Denote

$$\mathfrak{K}:=(1-\xi )I+\xi \mathfrak{T}\left(\right(1-\eta )I+\eta \mathfrak{T}).$$

If $0<\xi <\eta <{\displaystyle \frac{1}{1+\sqrt{1+l^2}}}$, then the following conclusions hold:

(i)

$Fix\left(\mathfrak{T}\right)=Fix\left(\mathfrak{T}\right((1-\eta )I+\eta \mathfrak{T}\left)\right)=Fix\left(\mathfrak{K}\right)$;

(ii)

If $\mathfrak{T}$ is demiclosed at 0, then $\mathfrak{K}$ is also demiclosed at 0;

(iii)

In addition, if $\mathfrak{T}:\mathcal{H}\to \mathcal{H}$ is quasi-pseudocontractive, then the mapping $\mathfrak{K}$ is quasi-nonexpansive, that is,

$$∥\mathfrak{K}\vartheta -{\vartheta }^{*}∥⩽∥\vartheta -{\vartheta }^{*}∥,\forall \vartheta \in \mathcal{H},{\vartheta }^{*}\in Fix\left(\mathfrak{T}\right)=Fix\left(\mathfrak{K}\right).$$

Lemma 2. 

Let $\mathcal{H}$ be a real Hilbert space, and let $\mathfrak{T}:\mathcal{H}\to \mathcal{H}$ be both l-Lipschitzian and quasi-pseudocontractive with coefficient $l\ge 1$, $\mathfrak{K}:=(1-\xi )I+\xi \mathfrak{T}\left(\right(1-\eta )I+\eta \mathfrak{T})$, where $0<\xi <\eta <{\displaystyle \frac{1}{1+\sqrt{1+l^2}}}$. Set ${\mathfrak{K}}_{\mu }=(1-\mu )I+\mu \mathfrak{K}$ for $\mu \in (0,1)$; then, for all $\vartheta \in \mathcal{H}$, ${\vartheta }^{*}\in Fix\left(\mathfrak{T}\right)$, we have the following results:

(i)

$\langle \vartheta -\mathfrak{K}\vartheta ,\vartheta -{\vartheta }^{*}\rangle \ge {\displaystyle \frac{1}{2}}{\parallel \vartheta -\mathfrak{K}\vartheta \parallel }^2$ and $\langle \vartheta -\mathfrak{K}\vartheta ,{\vartheta }^{*}-\mathfrak{K}\vartheta \rangle \le {\displaystyle \frac{1}{2}}{\parallel \vartheta -\mathfrak{K}\vartheta \parallel }^2$,

(ii)

${∥{\mathfrak{K}}_{\mu }\vartheta -{\vartheta }^{*}∥}^2\le \parallel \vartheta -{\vartheta }^{*}{\parallel }^2-\mu (1-\mu ){\parallel \mathfrak{K}\vartheta -\vartheta \parallel }^2$,

(iii)

$〈\vartheta -{\mathfrak{K}}_{\mu }\vartheta ,\vartheta -{\vartheta }^{*}〉\ge {\displaystyle \frac{\mu }{2}}{\parallel \mathfrak{K}\vartheta -\vartheta \parallel }^2$.

Proof. 

By Lemma 1, it is known that $\mathfrak{K}$ is quasi-nonexpansive and satisfies $∥\mathfrak{K}\vartheta -{\vartheta }^{*}∥⩽∥\vartheta -{\vartheta }^{*}∥$, for all $\vartheta \in \mathcal{H}$ and ${\vartheta }^{*}\in Fix\left(\mathfrak{T}\right)=Fix\left(\mathfrak{K}\right)$.

(i) Combining the classic $\langle \vartheta ,\varsigma \rangle =-{\displaystyle \frac{1}{2}}{\parallel \vartheta -\varsigma \parallel }^2+{\displaystyle \frac{1}{2}}{\parallel \vartheta \parallel }^2+{\displaystyle \frac{1}{2}}{\parallel \varsigma \parallel }^2$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \langle \vartheta -\mathfrak{K}\vartheta ,\vartheta -{\vartheta }^{*}\rangle =& -{\displaystyle \frac{1}{2}}\parallel \mathfrak{K}\vartheta -{\vartheta }^{*}{\parallel }^2+{\displaystyle \frac{1}{2}}{\parallel \vartheta -\mathfrak{K}\vartheta \parallel }^2+{\displaystyle \frac{1}{2}}{\parallel \vartheta -{\vartheta }^{*}\parallel }^2\hfill \\ \hfill \ge & -{\displaystyle \frac{1}{2}}\parallel \vartheta -{\vartheta }^{*}{\parallel }^2+{\displaystyle \frac{1}{2}}{\parallel \vartheta -\mathfrak{K}\vartheta \parallel }^2+{\displaystyle \frac{1}{2}}{\parallel \vartheta -{\vartheta }^{*}\parallel }^2\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\parallel \vartheta -\mathfrak{K}\vartheta \parallel }^2.\hfill \end{array}\end{array}$$

Similarly, we can obtain $\langle \vartheta -\mathfrak{K}\vartheta ,{\vartheta }^{*}-\mathfrak{K}\vartheta \rangle \le {\displaystyle \frac{1}{2}}{\parallel \vartheta -\mathfrak{K}\vartheta \parallel }^2$.

(ii) From (i), we know that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {∥{\mathfrak{K}}_{\mu }\vartheta -{\vartheta }^{*}∥}^2=& \parallel [(1-\mu )I+\mu \mathfrak{K}]\vartheta -{\vartheta }^{*}{\parallel }^2\hfill \\ \hfill =& \parallel \vartheta -{\vartheta }^{*}{\parallel }^2-2\mu \langle \vartheta -{\vartheta }^{*},\vartheta -\mathfrak{K}\vartheta \rangle +{\mu }^2{\parallel \mathfrak{K}\vartheta -\vartheta \parallel }^2\hfill \\ \hfill \le & \parallel \vartheta -{\vartheta }^{*}{\parallel }^2-\mu (1-\mu ){\parallel \mathfrak{K}\vartheta -\vartheta \parallel }^2.\hfill \end{array}\end{array}$$

(iii) With (i), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 〈\vartheta -{\mathfrak{K}}_{\mu }\vartheta ,\vartheta -{\vartheta }^{*}〉=& \mu \langle \vartheta -\mathfrak{K}\vartheta ,\vartheta -{\vartheta }^{*}\rangle \hfill \\ \hfill \ge & {\displaystyle \frac{\mu }{2}}{\parallel \mathfrak{K}\vartheta -\vartheta \parallel }^2.\hfill \end{array}\end{array}$$

   □

Remark 1. 

Let ${\mathfrak{K}}_{\mu }=(1-\mu )I+\mu \mathfrak{K}$ for $\mu \in (0,1)$, where $\mathfrak{K}:=(1-\xi )I+\xi \mathfrak{T}\left(\right(1-\eta )I+\eta \mathfrak{T})$ with $0<\xi <\eta <{\displaystyle \frac{1}{1+\sqrt{1+l^2}}}$, $\mathfrak{T}:\mathcal{H}\to \mathcal{H}$ be both l-Lipschitzian and quasi-pseudocontractive with $l\ge 1$. We have $Fix\left({\mathfrak{K}}_{\mu }\right)=Fix\left(\mathfrak{K}\right)$ and ${∥{\mathfrak{K}}_{\mu }\vartheta -\vartheta ∥}^2={\mu }^2{\parallel \mathfrak{K}\vartheta -\vartheta \parallel }^2$. It follows form (ii) of Lemma 2 that ${∥{\mathfrak{K}}_{\mu }\vartheta -{\vartheta }^{*}∥}^2\le {\parallel \vartheta -{\vartheta }^{*}\parallel }^2-{\displaystyle \frac{1-\mu }{\mu }}{∥{\mathfrak{K}}_{\mu }\vartheta -\vartheta ∥}^2$, which implies that ${\mathfrak{K}}_{\mu }$ is firmly quasi-nonexpansive when $\mu ={\displaystyle \frac{1}{2}}$. On the other hand, if $\widehat{\mathfrak{K}}$ is a firmly quasi-nonexpansive operator, we can easily obtain $\widehat{\mathfrak{K}}={\displaystyle \frac{1}{2}}I+{\displaystyle \frac{1}{2}}\mathfrak{K}$, where $\mathfrak{K}$ is a quasi-nonexpansive operator.

Lemma 3. 

Let $\mathfrak{T}:\mathcal{H}\to \mathcal{H}$ be both l-Lipschitzian and quasi-pseudocontractive with $l\ge 1$ and $\mu \in (0,1)$. If $\mathfrak{K}:=(1-\xi )I+\xi \mathfrak{T}\left(\right(1-\eta )I+\eta \mathfrak{T})$ with $0<\xi <\eta <{\displaystyle \frac{1}{1+\sqrt{1+l^2}}}$, ${\mathfrak{K}}_{\mu }=(1-\mu )I+\mu \mathfrak{K}$; then, ${∥\left(I-{\mathfrak{K}}_{\mu }\right)\vartheta ∥}^2\le 2\mu 〈\vartheta -{\vartheta }^{*},\left(I-{\mathfrak{K}}_{\mu }\right)\vartheta 〉$ for all $\vartheta \in \mathcal{H},{\vartheta }^{*}\in Fix\left(\mathfrak{T}\right)$.

Proof. 

By Lemma 1, it is known that $\mathfrak{K}$ is quasi-nonexpansive and satisfies ${\vartheta }^{*}\in Fix\left(\mathfrak{T}\right)=Fix\left(\mathfrak{K}\right)$. It follows easily from (iii) of Lemma 2.    □

Lemma 4 

([30]). Let the operator $\mathfrak{F}:\mathcal{H}\to \mathcal{H}$ be l-Lipschitz continuous and δ-strongly monotone with constants $l>0,\delta >0$. Assume that $ϵ\in \left(0,{\displaystyle \frac{2\delta }{l^2}}\right)$. Define ${\mathfrak{G}}_{\mu }=I-\mu ϵ\mathfrak{F}$ for $\mu \in (0,1)$. Then, for all $\vartheta ,\varsigma \in \mathcal{H}$,

$$∥{\mathfrak{G}}_{\mu }\vartheta -{\mathfrak{G}}_{\mu }\varsigma ∥\le (1-\mu \chi )\parallel \vartheta -\varsigma \parallel$$

holds, where $\chi =1-\sqrt{1-ϵ\left(2\delta -ϵl^2\right)}\in (0,1)$.

Lemma 5 

([1]). Suppose that $\left\{{\vartheta }_n\right\}$ is a sequence of non-negative real numbers satisfying the condition that

$$\left\{\begin{array}{c}{\vartheta }_{n+1}\le \left(1-{\tau }_n\right){\vartheta }_n+{\tau }_n{\psi }_n,n\ge 0,\hfill \\ {\vartheta }_{n+1}\le {\vartheta }_n-{\varrho }_n+{\theta }_n,n\ge 0,\hfill \end{array}\right.$$

where $\left\{{\tau }_n\right\}\subseteq (0,1)$, ${\varrho }_n\ge 0$, and $\left\{{\psi }_n\right\}$ and $\left\{{\theta }_n\right\}$ are two sequences in $\mathbb{R}$, such that

(i)${\sum }_{n=1}^{\infty }{\tau }_n=\infty$,(ii)${lim}_{n\to \infty }{\theta }_n=0$,

(iii)${lim}_{l\to \infty }{\varrho }_{n_l}=0$ implies ${lim\; sup}_{l\to \infty }{\psi }_{n_l}\le 0$ for any subsequence $\left\{n_l\right\}\subset \left\{n\right\}$.

Then, ${lim}_{n\to \infty }{\vartheta }_n=0$.

3. Main Results

In this section, we present our novel accelerated cyclic iterative approximation algorithm and convergence analysis. We begin by outlining the assumptions necessary for achieving strong convergence.

Assumption 1. 

Let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be real Hilbert spaces. Additionally, we assume that the following conditions are satisfied:

(i) $\mathfrak{A}:{\mathcal{H}}_1\to {\mathcal{H}}_2$ is a nonzero bounded linear operator with an adjoint operator denoted by ${\mathfrak{A}}^{*}$, ${\mathfrak{S}}_i:{\mathcal{H}}_1\to {\mathcal{H}}_1$ is an $l_{1i}$-Lipschitzian quasi-pseudocontractive operator with $1<l_{1i}\le l_1$, and ${\mathfrak{T}}_j:{\mathcal{H}}_2\to {\mathcal{H}}_2$ is an $l_{2j}$-Lipschitzian quasi-pseudocontractive operator with $1<l_{2j}\le l_2$, such that $I_1-{\mathfrak{S}}_i$ and $I_2-{\mathfrak{T}}_j$ are demiclosed at 0 for all $1\le i\le p$ and $1\le j\le r$. Here, $I_1$ and $I_2$ are separate identity operators on ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, and $\mathfrak{F}:{\mathcal{H}}_1\to {\mathcal{H}}_1$ is l-Lipschitz continuous and δ-strongly monotone.

(ii) $\Omega :=\left\{\upsilon \in {\mathcal{H}}_1\upsilon \in {\bigcap }_{i=1}^{p}Fix\left({\mathfrak{S}}_i\right)and\mathfrak{A}\upsilon \in {\bigcap }_{j=1}^{r}Fix\left({\mathfrak{T}}_j\right)\right\}\ne \varnothing$.

(iii) For all $n\ge 1$, ${\mathfrak{J}}_n:=(1-{\kappa }_n)I_1+{\kappa }_n{\mathfrak{J}}_{n,{\left[n\right]}_1}$ and ${\mathfrak{D}}_n:=(1-{\iota }_n)I_2+{\iota }_n{\mathfrak{D}}_{n,{\left[n\right]}_2}$, where $\left\{{\kappa }_n\right\}\subset (0,{\displaystyle \frac{1}{2}}]$ with $\kappa ={sup}_{n\ge 1}\left\{{\kappa }_n\right\}$ and $\left\{{\iota }_n\right\}\subset (0,1)$ with $\iota ={sup}_{n\ge 1}\left\{{\iota }_n\right\}$, ${\left[n\right]}_1:=n(modp)+1$, ${\left[n\right]}_2:=n(mod)r+1$, and for $i\in \{1,2,\cdots ,p\}$ and $j\in \{1,2,\cdots ,r\}$, ${\mathfrak{J}}_{n,i}:=(1-{\xi }_n)I_1+{\xi }_n{\mathfrak{S}}_i((1-{\eta }_n)I_1+{\eta }_n{\mathfrak{S}}_i)$ and ${\mathfrak{D}}_{n,j}:=(1-{\xi }_n)I_2+{\xi }_n{\mathfrak{T}}_j((1-{\eta }_n)I_2+{\eta }_n{\mathfrak{T}}_j)$ with

$$0<{\xi }_n\le {\eta }_n<{\displaystyle \frac{1}{1+\sqrt{1+l_3^2}}}$$

for $l_3=max\{l_1,l_2\}$.

(iv) $\left\{{\epsilon }_n\right\}$ and $\left\{{\rho }_n\right\}$ are two positive sequences such that ${lim}_{n\to \infty }{\displaystyle \frac{{\varphi }_n}{{\sigma }_n}}=0$, ${lim}_{n\to \infty }{\displaystyle \frac{{\epsilon }_n}{{\sigma }_n}}=0$, ${lim}_{n\to \infty }{\displaystyle \frac{{\rho }_n}{{\sigma }_n}}=0$, where $\left\{{\varphi }_n\right\}\subset [0,1]$, $\left\{{\sigma }_n\right\}\subset (0,1)$ satisfies ${lim}_{n\to \infty }{\sigma }_n=0$ and ${\sum }_{n=0}^{\infty }{\sigma }_n=\infty$, and $0\le lim{inf}_{n\to \infty }{\rho }_n\le lim{sup}_{n\to \infty }{\rho }_n<{\displaystyle \frac{1}{\iota }}$.

Next, we shall propose the following novel iterative approximation (i.e., Algorithm 1) to solve a HVIP controlled by the MSCFP (2).

Remark 2. 

Since Ω in Assumption 1 (ii) is a nonempty closed convex set, the variational inequality

$$\begin{array}{c}\hfill 〈\mathfrak{F}\left({\vartheta }^{*}\right),z-{\vartheta }^{*}〉\ge 0,\forall z\in \Omega ,\end{array}$$

(4)

 has a unique solution by Assumption 1 (i).

Remark 3. 

It can be seen that the newly proposed adaptive cyclic iterative algorithm (i.e., Algorithm 1), distinct from Zhao et al. [23], combines the inertial approach with a correction term and the primal-dual ideal, without requiring prior knowledge of the operator norm. This makes the algorithm easier to implement in practice and helps to improve the convergence speed of the iterative process. In addition, we have conducted our research using a more generalized quasi-pseudocontractive operator, which expands the scope to various categories of nonlinear mappings, including demicontractive, directed, and firmly quasi-nonexpansive mappings [26].

Theorem 1. 

Suppose that $\left\{{\vartheta }_n\right\}$ is a sequence generated by Algorithm 1 under Assumption 1. Then, the sequence $\left\{{\vartheta }_n\right\}$ converges strongly to ${\vartheta }^{*}\in \Omega$, which is the unique solution to the HVIP, as follows:

$$〈\mathfrak{F}\left({\vartheta }^{*}\right),z-{\vartheta }^{*}〉\ge 0,\forall z\in \Omega ,$$

where Ω is defined by Assumption 1 (ii). This implies that ${\vartheta }^{*}$ is also a solution of the MSCFP (2).

Algorithm 1

$\mathbf{Initialization}$ Let $\alpha ,\beta ,\gamma >0$. Pick out sequences $\left\{{\epsilon }_n\right\}$, $\left\{{\rho }_n\right\}$, $\left\{{\varphi }_n\right\}$ and $\left\{{\sigma }_n\right\}$ such that Assumption 1 is satisfied, and give the initial points ${\vartheta }_0,{\vartheta }_1,{\nu }_0,{\omega }_0\in {\mathcal{H}}_1$. $\mathbf{IterativeSteps}:$ For $(n\ge 1)$, based on the iterates ${\vartheta }_{n-1}$ and ${\vartheta }_n$, make the following computation for ${\vartheta }_{n+1}$: $\mathbf{Step}\mathbf{1}$: Calculate $${\nu }_n={\vartheta }_n+{\alpha }_n\left({\vartheta }_n-{\vartheta }_{n-1}\right)+{\beta }_n\left({\nu }_{n-1}-{\vartheta }_{n-1}\right),$$ where ${\alpha }_n\in (0,{\overline{\alpha }}_n]$ and ${\beta }_n\in \left[0,{\overline{\beta }}_n\right]$ with the condition that $$\begin{array}{c}\hfill {\overline{\alpha }}_n=\left\{\begin{array}{c}min\left\{{\displaystyle \frac{{\epsilon }_n}{∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel }},\alpha \right\},\mathrm{if}{\vartheta }_n\ne {\vartheta }_{n-1}\mathrm{or}{\omega }_n\ne 0,\hfill \\ \alpha ,\mathrm{otherwise},\hfill \end{array}\right.\end{array}$$ (5) and $$\begin{array}{c}\hfill {\overline{\beta }}_n=\left\{\begin{array}{c}min\left\{{\displaystyle \frac{{\epsilon }_n}{∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel }},\beta \right\},\mathrm{if}{\nu }_{n-1}\ne {\vartheta }_{n-1}\mathrm{or}{\omega }_n\ne 0,\hfill \\ \beta ,\mathrm{otherwise},\hfill \end{array}\right.\end{array}$$ (6) $\mathbf{Step}\mathbf{2}:$ Evaluate $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill z_n=& {\nu }_n-{\gamma }_n{\mathfrak{A}}^{*}(I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_n,\hfill \\ \hfill {\omega }_{n+1}& =(I_1-{\mathfrak{J}}_n)(z_n+{\lambda }_n{\omega }_n),\hfill \end{array}\right.\end{array}$$ where ${\lambda }_n={\alpha }_n+{\beta }_n$ and the step size ${\gamma }_n$ is selected in such a manner that $$\begin{array}{c}\hfill {\gamma }_n:=\left\{\begin{array}{c}{\displaystyle \frac{{\rho }_n{\parallel (I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_n\parallel }^2}{\parallel {\mathfrak{A}}^{*}(I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_n{\parallel }^2}},\mathrm{if}(I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_n\ne 0\hfill \\ \gamma ,\mathrm{otherwise},\hfill \end{array}\right.\end{array}$$ (7) $\mathbf{Step}\mathbf{3}:$ Ascertain $$\begin{array}{c}\hfill {\varsigma }_n=(1-{\varphi }_n)z_n+{\varphi }_n{\omega }_{n+1}.\end{array}$$ $\mathbf{Step}\mathbf{4}:$ Compute $${\vartheta }_{n+1}=\left(I_1-{\sigma }_n\mathfrak{F}\right){\varsigma }_n.$$ Set $n:=n+1$ and go to step 1.

Proof. 

Firstly, we show that the sequences $\left\{{\vartheta }_n\right\}$ is bounded.

Taking ${\vartheta }^{*}\in \Omega$, we have ${\vartheta }^{*}\in {\bigcap }_{i=1}^{p}Fix\left({\mathfrak{S}}_i\right)$ such that $\mathfrak{A}{\vartheta }^{*}\in {\bigcap }_{j=1}^{r}Fix\left({\mathfrak{T}}_j\right)$. For each $i\in \{1,2,\cdots ,p\}$, $j\in \{1,2,\cdots ,r\}$, it follows from the definitions of ${\mathfrak{J}}_n$ and ${\mathfrak{D}}_n$, Assumption 1, Lemma 1, and Remark 1 that ${\vartheta }^{*}\in {\bigcap }_{n=1}^{\infty }Fix\left({\mathfrak{J}}_n\right)$ and $\mathfrak{A}{\vartheta }^{*}\in$${\bigcap }_{n=1}^{\infty }Fix\left({\mathfrak{D}}_n\right)$.

According to Lemmas 1 and 3, we have

$$\begin{array}{cc}\hfill 〈{\nu }_n-{\vartheta }^{*},{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n〉& =〈\mathfrak{A}{\nu }_n-\mathfrak{A}{\vartheta }^{*},\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n〉\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2{\iota }_n}}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2.\hfill \end{array}$$

(8)

From Algorithm 1 and Equation (8), it follows that

$$\begin{array}{cc}\hfill {∥z_n-{\vartheta }^{*}∥}^2=& {∥{\nu }_n-{\gamma }_n{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n-{\vartheta }^{*}∥}^2\hfill \\ \hfill =& {∥{\nu }_n-{\vartheta }^{*}∥}^2-2{\gamma }_n〈{\nu }_n-{\vartheta }^{*},{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n〉\hfill \\ \hfill & +{\gamma }_n^2{∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2\hfill \\ \hfill \le & {∥{\nu }_n-{\vartheta }^{*}∥}^2-{\displaystyle \frac{{\gamma }_n}{\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2+{\gamma }_n^2{∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2.\hfill \end{array}$$

(9)

For the case $(I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_n=0$, one obtains

$$\begin{array}{cc}\hfill {∥z_n-{\vartheta }^{*}∥}^2\le & {∥{\nu }_n-{\vartheta }^{*}∥}^2-{\gamma }_n\left({\displaystyle \frac{1}{\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2-{\gamma }_n{∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2\right)\hfill \\ \hfill =& {∥{\nu }_n-{\vartheta }^{*}∥}^2.\hfill \end{array}$$

(10)

Otherwise, we deduce from (7) and (9) that

$$\begin{array}{c}\hfill {∥z_n-{\vartheta }^{*}∥}^2\le {∥{\nu }_n-{\vartheta }^{*}∥}^2-{\rho }_n\left({\displaystyle \frac{1}{\iota }}-{\rho }_n\right){\displaystyle \frac{\parallel (I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_n{\parallel }^4}{\parallel {\mathfrak{A}}^{*}(I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_n{\parallel }^2}}.\end{array}$$

(11)

By Assumption 1 (iv), (10), and (11), we see that

$$\begin{array}{c}\hfill ∥z_n-{\vartheta }^{*}∥\le ∥{\nu }_n-{\vartheta }^{*}∥.\end{array}$$

(12)

By Lemma 3, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {∥{\omega }_{n+1}∥}^2=& {∥(I_1-{\mathfrak{J}}_n)(z_n+{\lambda }_n{\omega }_n)-(I_1-{\mathfrak{J}}_n){\vartheta }^{*}∥}^2\hfill \\ \hfill \le & 2{\kappa }_n\langle {\omega }_{n+1},z_n-{\vartheta }^{*}+{\lambda }_n{\omega }_n\rangle \hfill \\ \hfill \le & \langle {\omega }_{n+1},z_n-{\vartheta }^{*}+{\lambda }_n{\omega }_n\rangle \hfill \\ \hfill \le & \parallel {\omega }_{n+1}\parallel \parallel z_n-{\vartheta }^{*}+{\lambda }_n{\omega }_n\parallel ,\hfill \end{array}\end{array}$$

Then, we obtain

$$\begin{array}{c}\hfill ∥{\omega }_{n+1}∥\le \parallel z_n-{\vartheta }^{*}+{\lambda }_n{\omega }_n\parallel \le \parallel z_n-{\vartheta }^{*}\parallel +{\lambda }_n\parallel {\omega }_n\parallel .\end{array}$$

(13)

It follows from Algorithm 1 and the triangle inequality that

$$\begin{array}{cc}\hfill ∥{\nu }_n-{\vartheta }^{*}∥& =∥{\vartheta }_n+{\alpha }_n\left({\vartheta }_{n-1}-{\vartheta }_n\right)+{\beta }_n\left({\nu }_{n-1}-{\vartheta }_{n-1}\right)-{\vartheta }^{*}∥\hfill \\ \hfill & \le ∥{\vartheta }_n-{\vartheta }^{*}∥+{\alpha }_n∥{\vartheta }_n-{\vartheta }_{n-1}∥+{\beta }_n∥{\nu }_{n-1}-{\vartheta }_{n-1}∥.\hfill \end{array}$$

(14)

From (5), we have ${\alpha }_n(∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )\le {\epsilon }_n$ for all n. This, combined with Assumption 1 (iv), leads to

$$\underset{n\to \infty }{lim}{\displaystyle \frac{{\alpha }_n}{{\sigma }_n}}(∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )=0.$$

Following a similar reasoning from (6), we determine that

$$\underset{n\to \infty }{lim}{\displaystyle \frac{{\beta }_n}{{\sigma }_n}}(∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )=0.$$

By (12)–(14), one knows that

$$\begin{array}{cc}\hfill ∥{\varsigma }_n-{\vartheta }^{*}∥=& \parallel (1-{\varphi }_n)z_n+{\varphi }_n{\omega }_{n+1}-{\vartheta }^{*}\parallel \hfill \\ \hfill \le & ∥(1-{\varphi }_n)z_n-(1-{\varphi }_n){\vartheta }^{*}∥+\parallel {\varphi }_n{\vartheta }^{*}\parallel +{\varphi }_n\parallel {\omega }_{n+1}\parallel \hfill \\ \hfill \le & (1-{\varphi }_n)∥z_n-{\vartheta }^{*}∥+{\varphi }_n\parallel z_n-{\vartheta }^{*}\parallel +{\lambda }_n\parallel {\omega }_n\parallel +\parallel {\varphi }_n{\vartheta }^{*}\parallel \hfill \\ \hfill \le & ∥{\nu }_n-{\vartheta }^{*}∥+{\lambda }_n\parallel {\omega }_n\parallel +\parallel {\varphi }_n{\vartheta }^{*}\parallel \hfill \\ \hfill \le & ∥{\vartheta }_n-{\vartheta }^{*}∥+{\alpha }_n∥{\vartheta }_n-{\vartheta }_{n-1}∥+{\beta }_n∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+{\lambda }_n\parallel {\omega }_n\parallel +\parallel {\varphi }_n{\vartheta }^{*}\parallel \hfill \\ \hfill =& ∥{\vartheta }_n-{\vartheta }^{*}∥+{\sigma }_n{\displaystyle \frac{{\alpha }_n}{{\sigma }_n}}(∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )\hfill \\ \hfill & +{\sigma }_n{\displaystyle \frac{{\beta }_n}{{\sigma }_n}}(∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )+\parallel {\varphi }_n{\vartheta }^{*}\parallel \hfill \\ \hfill \le & ∥{\vartheta }_n-{\vartheta }^{*}∥+2{\sigma }_n{M}_1,\hfill \end{array}$$

(15)

where $M_1={sup}_{n\in \mathbb{N}}\left\{{\displaystyle \frac{{\alpha }_n}{{\sigma }_n}}(∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel ),{\displaystyle \frac{{\beta }_n}{{\sigma }_n}}(∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel ),\parallel {\varphi }_n{\vartheta }^{*}\parallel \right\}<\infty$.

Consider $ϵ\in \left(0,{\displaystyle \frac{2\delta }{l^2}}\right)$. Given that ${lim}_{n\to \infty }{\sigma }_n=0$, there exists $n_0\in \mathbb{N}$, such that for all $n>n_0$, ${\sigma }_n<ϵ$. Consequently, ${\displaystyle \frac{{\sigma }_n}{ϵ}}\in (0,1)$. According to Lemma 4 for all $n>n_0$, one can deduce that

$$\begin{array}{cc}\hfill ∥\left(I_1-{\sigma }_n\mathfrak{F}\right){\varsigma }_n-\left(I_1-{\sigma }_n\mathfrak{F}\right){\vartheta }^{*}∥& =∥\left(I_1-{\displaystyle \frac{{\sigma }_n}{ϵ}}·ϵ\mathfrak{F}\right){\varsigma }_n-\left(I_1-{\displaystyle \frac{{\sigma }_n}{ϵ}}·ϵ\mathfrak{F}\right){\vartheta }^{*}∥\hfill \\ \hfill & \le \left(1-{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \right)∥{\varsigma }_n-{\vartheta }^{*}∥,\hfill \end{array}$$

(16)

where $\chi =1-\sqrt{1-ϵ\left(2\delta -ϵl^2\right)}\in (0,1)$. By applying Inequalities (15) and (16), we know that

$$\begin{array}{cc}\hfill ∥{\vartheta }_{n+1}-{\vartheta }^{*}∥& =∥\left(I_1-{\sigma }_n\mathfrak{F}\right){\varsigma }_n-{\vartheta }^{*}∥\hfill \\ & =∥\left(I_1-{\sigma }_n\mathfrak{F}\right){\varsigma }_n-\left(I_1-{\sigma }_n\mathfrak{F}\right){\vartheta }^{*}-{\sigma }_n\mathfrak{F}{\vartheta }^{*}∥\hfill \\ & \le ∥\left(I_1-{\sigma }_n\mathfrak{F}\right){\varsigma }_n-\left(I_1-{\sigma }_n\mathfrak{F}\right){\vartheta }^{*}∥+{\sigma }_n∥\mathfrak{F}{\vartheta }^{*}∥\hfill \\ & \le \left(1-{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \right)∥{\varsigma }_n-{\vartheta }^{*}∥+{\sigma }_n∥\mathfrak{F}{\vartheta }^{*}∥\hfill \\ & \le \left(1-{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \right)∥{\vartheta }_n-{\vartheta }^{*}∥+2{\sigma }_n{M}_1+{\sigma }_n∥\mathfrak{F}{\vartheta }^{*}∥\hfill \\ & \le \left(1-{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \right)∥{\vartheta }_n-{\vartheta }^{*}∥+{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \left[{\displaystyle \frac{ϵ\left(2M_1+∥\mathfrak{F}{\vartheta }^{*}∥\right)}{\chi }}\right]\hfill \\ & \le max\left\{∥{\vartheta }_n-{\vartheta }^{*}∥,{\displaystyle \frac{ϵ\left(2M_1+∥\mathfrak{F}{\vartheta }^{*}∥\right)}{\chi }}\right\}\hfill \\ & \le \cdots \le max\left\{∥{\vartheta }_{n_0}-{\vartheta }^{*}∥,{\displaystyle \frac{ϵ\left(2M_1+∥\mathfrak{F}{\vartheta }^{*}∥\right)}{\chi }}\right\}.\hfill \end{array}$$

Thus, the sequence $\left\{{\vartheta }_n\right\}$ is bounded. As a result, the sequences $\left\{{\varsigma }_n\right\}$, $\left\{w_{n+1}\right\}$ and $\left\{{\nu }_n\right\}$ are also bounded.

According to the definition of ${\nu }_n$, along with applying the Cauchy–Schwarz inequality, we obtain

$$\begin{array}{cc}\hfill {∥{\nu }_n-{\vartheta }^{*}∥}^2=& \parallel {\vartheta }_n+{\alpha }_n\left({\vartheta }_n-{\vartheta }_{n-1}\right)+{\beta }_n\left({\nu }_{n-1}-{\vartheta }_{n-1}\right)-{\vartheta }^{*}{\parallel }^2\hfill \\ \hfill =& {∥{\vartheta }_n-{\vartheta }^{*}∥}^2+{\alpha }_n^2{∥{\vartheta }_{n-1}-{\vartheta }_n∥}^2+{\beta }_n^2{∥{\nu }_{n-1}-{\vartheta }_{n-1}∥}^2\hfill \\ & +2{\alpha }_n{\beta }_n〈{\vartheta }_{n-1}-{\vartheta }_n,{\nu }_{n-1}-{\vartheta }_{n-1}〉\hfill \\ & +2{\alpha }_n〈{\vartheta }_n-{\vartheta }^{*},{\vartheta }_{n-1}-{\vartheta }_n〉+2{\beta }_n〈{\vartheta }_n-{\vartheta }^{*},{\nu }_{n-1}-{\vartheta }_{n-1}〉\hfill \\ \hfill \le & {∥{\vartheta }_n-{\vartheta }^{*}∥}^2+{\alpha }_n^2{∥{\vartheta }_n-{\vartheta }_{n-1}∥}^2+{\beta }_n^2{∥{\nu }_{n-1}-{\vartheta }_{n-1}∥}^2\hfill \\ & +2{\alpha }_n{\beta }_n∥{\vartheta }_n-{\vartheta }_{n-1}∥∥{\nu }_{n-1}-{\vartheta }_{n-1}∥\hfill \\ & +2{\alpha }_n∥{\vartheta }_n-{\vartheta }^{*}∥∥{\vartheta }_n-{\vartheta }_{n-1}∥+2{\beta }_n∥{\vartheta }_n-{\vartheta }^{*}∥∥{\nu }_{n-1}-{\vartheta }_{n-1}∥\hfill \\ \hfill =& {∥{\vartheta }_n-{\vartheta }^{*}∥}^2+{\beta }_n∥{\nu }_{n-1}-{\vartheta }_{n-1}∥\left(2∥{\vartheta }_n-{\vartheta }^{*}∥+{\beta }_n∥{\nu }_{n-1}-{\vartheta }_{n-1}∥\right)\hfill \\ & +{\alpha }_n∥{\vartheta }_n-{\vartheta }_{n-1}∥\left({\alpha }_n∥{\vartheta }_n-{\vartheta }_{n-1}∥+2{\beta }_n∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+2∥{\vartheta }_n-{\vartheta }^{*}∥\right).\hfill \end{array}$$

Since $\left\{{\vartheta }_n\right\},\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}$ and $\left\{{\nu }_n\right\}$ are bounded, there exists a constant $M_2>0$ such that

$$\begin{array}{c}\hfill {∥{\nu }_n-{\vartheta }^{*}∥}^2\le {∥{\vartheta }_n-{\vartheta }^{*}∥}^2+M_2{\alpha }_n∥{\vartheta }_n-{\vartheta }_{n-1}∥+M_2{\beta }_n∥{\nu }_{n-1}-{\vartheta }_{n-1}∥.\end{array}$$

(17)

By Algorithm 1 and (12), we have

$$\begin{array}{cc}\hfill {∥{\varsigma }_n-{\vartheta }^{*}∥}^2=& {∥(1-{\varphi }_n)z_n+{\varphi }_n{\omega }_{n+1}-{\vartheta }^{*}∥}^2\hfill \\ \hfill \le & {\varphi }_n\parallel {\omega }_{n+1}-{\vartheta }^{*}{\parallel }^2+(1-{\varphi }_n){\parallel z_n-{\vartheta }^{*}\parallel }^2\hfill \\ \hfill \le & {∥z_n-{\vartheta }^{*}∥}^2+{\varphi }_n{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2\hfill \\ \hfill \le & {∥{\nu }_n-{\vartheta }^{*}∥}^2+{\varphi }_n{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2.\hfill \end{array}$$

(18)

Utilizing (16) and (17), one arrives at the following for all $n>n_0$:

$$\begin{array}{cc}\hfill {∥{\vartheta }_{n+1}-{\vartheta }^{*}∥}^2=& {∥\left(I_1-{\sigma }_n\mathfrak{F}\right){\varsigma }_n-\left(I_1-{\sigma }_n\mathfrak{F}\right){\vartheta }^{*}-{\sigma }_n\mathfrak{F}{\vartheta }^{*}∥}^2\hfill \\ \hfill \le & {∥\left(I_1-{\sigma }_n\mathfrak{F}\right){\varsigma }_n-\left(I_1-{\sigma }_n\mathfrak{F}\right){\vartheta }^{*}∥}^2-2{\sigma }_n〈\mathfrak{F}{\vartheta }^{*},{\vartheta }_{n+1}-{\vartheta }^{*}〉\hfill \\ \hfill \le & {\left(1-{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \right)}^2{∥{\varsigma }_n-{\vartheta }^{*}∥}^2+2{\sigma }_n〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n+1}〉\hfill \\ \hfill \le & \left(1-{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \right){∥{\nu }_n-{\vartheta }^{*}∥}^2+2{\sigma }_n〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n+1}〉\hfill \\ \hfill & +{\varphi }_n{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2\hfill \\ \hfill \le & \left(1-{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \right){∥{\vartheta }_n-{\vartheta }^{*}∥}^2+2{\sigma }_n〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n+1}〉\hfill \\ \hfill & +M_2{\alpha }_n∥{\vartheta }_n-{\vartheta }_{n-1}∥+M_2{\beta }_n∥{\nu }_{n-1}-{\vartheta }_{n-1}∥\hfill \\ \hfill & +{\varphi }_n{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2\hfill \\ \hfill \le & \left(1-{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \right){∥{\vartheta }_n-{\vartheta }^{*}∥}^2+{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \left[{\displaystyle \frac{2ϵ}{\chi }}〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n+1}〉\right.\hfill \\ \hfill & +{\displaystyle \frac{ϵ{\alpha }_n}{{\sigma }_n}}{\displaystyle \frac{M_2}{\chi }}∥{\vartheta }_n-{\vartheta }_{n-1}∥+{\displaystyle \frac{ϵ{\beta }_n}{{\sigma }_n}}{\displaystyle \frac{M_2}{\chi }}∥{\nu }_{n-1}-{\vartheta }_{n-1}∥\hfill \\ \hfill & \left.+{\displaystyle \frac{ϵ{\varphi }_n}{{\sigma }_n\chi }}{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2\right]\hfill \\ \hfill =& \left(1-{\tau }_n\right){∥{\vartheta }_n-{\vartheta }^{*}∥}^2+{\tau }_n{\psi }_n,\hfill \end{array}$$

(19)

where

$$\begin{array}{cc}\hfill {\tau }_n=& {\displaystyle \frac{{\sigma }_n}{ϵ}}\chi ,\hfill \\ \hfill {\psi }_n=& {\displaystyle \frac{2ϵ}{\chi }}〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n+1}〉+{\displaystyle \frac{ϵ{\alpha }_n}{{\sigma }_n}}{\displaystyle \frac{M_2}{\chi }}∥{\vartheta }_n-{\vartheta }_{n-1}∥+{\displaystyle \frac{ϵ{\beta }_n}{{\sigma }_n}}{\displaystyle \frac{M_2}{\chi }}∥{\nu }_{n-1}-{\vartheta }_{n-1}∥\hfill \\ & +{\displaystyle \frac{ϵ{\varphi }_n}{{\sigma }_n\chi }}{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2.\hfill \end{array}$$

It is evident that ${\tau }_n\to 0,{\sum }_{n=1}^{\infty }{\tau }_n=\infty$. Given that the sequence $\left\{{\vartheta }_n\right\}$ is bounded, this implies the existence of a constant $M_3>0$, such that

$$2〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n+1}〉\le M_3.$$

Hence, by (16), we obtain

$$\begin{array}{cc}\hfill {∥{\vartheta }_{n+1}-{\vartheta }^{*}∥}^2& ={∥\left(I_1-{\sigma }_n\mathfrak{F}\right){\varsigma }_n-{\vartheta }^{*}∥}^2\hfill \\ & ={∥\left(I_1-{\sigma }_n\mathfrak{F}\right){\varsigma }_n-\left(I_1-{\sigma }_n\mathfrak{F}\right){\vartheta }^{*}-{\sigma }_n\mathfrak{F}{\vartheta }^{*}∥}^2\hfill \\ & \le {∥\left(I_1-{\sigma }_n\mathfrak{F}\right){\varsigma }_n-\left(I_1-{\sigma }_n\mathfrak{F}\right){\vartheta }^{*}∥}^2-2{\sigma }_n〈\mathfrak{F}{\vartheta }^{*},{\vartheta }_{n+1}-{\vartheta }^{*}〉\hfill \\ & \le {\left(1-{\displaystyle \frac{{\sigma }_n}{ϵ}}\chi \right)}^2{∥{\varsigma }_n-{\vartheta }^{*}∥}^2+2{\sigma }_n〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n+1}〉\hfill \\ & \le {∥{\varsigma }_n-{\vartheta }^{*}∥}^2+{\sigma }_n{M}_3\forall n>n_0,\hfill \end{array}$$

from the preceding inequality, as well as (9) and (18), we have the following for all $n>n_0$:

$$\begin{array}{cc}\hfill {∥{\vartheta }_{n+1}-{\vartheta }^{*}∥}^2\le & {∥z_n-{\vartheta }^{*}∥}^2+{\varphi }_n{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2+{\sigma }_n{M}_3\hfill \\ \hfill \le & {∥{\nu }_n-{\vartheta }^{*}∥}^2-{\displaystyle \frac{{\gamma }_n}{\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2+{\gamma }_n^2{∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2\hfill \\ \hfill & +{\varphi }_n{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2+{\sigma }_n{M}_3.\hfill \end{array}$$

(20)

We know from (14) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ∥{\nu }_n-{\vartheta }^{*}∥\le & ∥{\vartheta }_n-{\vartheta }^{*}∥+{\alpha }_n∥{\vartheta }_n-{\vartheta }_{n-1}∥+{\beta }_n∥{\nu }_{n-1}-{\vartheta }_{n-1}∥\hfill \\ \hfill =& ∥{\vartheta }_n-{\vartheta }^{*}∥+{\alpha }_n(∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )\hfill \\ & +{\beta }_n(∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )-{\lambda }_n\parallel {\omega }_n\parallel \hfill \\ \hfill \le & ∥{\vartheta }_n-{\vartheta }^{*}∥+{\sigma }_n{\displaystyle \frac{{\alpha }_n}{{\sigma }_n}}(∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )\hfill \\ & +{\sigma }_n{\displaystyle \frac{{\beta }_n}{{\sigma }_n}}(∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )\hfill \\ \hfill \le & ∥{\vartheta }_n-{\vartheta }^{*}∥+2{\sigma }_n{M}_1.\hfill \end{array}\end{array}$$

Then, for some constant $M_4>0$,

$$\begin{array}{cc}\hfill {∥{\nu }_n-{\vartheta }^{*}∥}^2& \le {\left(∥{\vartheta }_n-{\vartheta }^{*}∥+2{\sigma }_n{M}_1\right)}^2\hfill \\ \hfill & ={∥{\vartheta }_n-{\vartheta }^{*}∥}^2+{\sigma }_n\left(4M_1∥{\vartheta }_n-{\vartheta }^{*}∥+4{\sigma }_n{M}_1^2\right)\hfill \\ \hfill & \le {∥{\vartheta }_n-{\vartheta }^{*}∥}^2+{\sigma }_n{M}_4.\hfill \end{array}$$

(21)

Combining (20) and (21), we get the following for all $n>n_0$:

$$\begin{array}{cc}\hfill {∥{\vartheta }_{n+1}-{\vartheta }^{*}∥}^2\le & {∥{\vartheta }_n-{\vartheta }^{*}∥}^2+{\sigma }_n{M}_4+{\gamma }_n^2{∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2+{\sigma }_n{M}_3\hfill \\ \hfill & +{\varphi }_n{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2-{\displaystyle \frac{{\gamma }_n}{\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2\hfill \\ \hfill \le & {∥{\vartheta }_n-{\vartheta }^{*}∥}^2+{\sigma }_n{M}_4+{\sigma }_n{\displaystyle \frac{{\gamma }_n^2}{{\sigma }_n}}{∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2+{\sigma }_n{M}_3\hfill \\ \hfill & +{\sigma }_n{\displaystyle \frac{{\varphi }_n}{{\sigma }_n}}{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2-{\displaystyle \frac{{\gamma }_n}{\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2\hfill \\ \hfill \le & {∥{\vartheta }_n-{\vartheta }^{*}∥}^2+{\sigma }_n(M_3+M_4+M_5)-{\displaystyle \frac{{\gamma }_n}{\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2,\hfill \end{array}$$

(22)

where $M_5={sup}_{n\in \mathbb{N}}\left\{{\displaystyle \frac{{\gamma }_n^2}{{\sigma }_n}}{∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2,{\displaystyle \frac{{\varphi }_n}{{\sigma }_n}}{\parallel {\omega }_{n+1}-{\vartheta }^{*}\parallel }^2\right\}<\infty$.

Now we set

$${\varrho }_n={\displaystyle \frac{{\gamma }_n}{\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥}^2,$$

and

$${\theta }_n={\sigma }_n\left(M_3+M_4+M_5\right),{\Gamma }_n={∥{\vartheta }_n-{\vartheta }^{*}∥}^2.$$

Thus, (22) can be reformulated as follows:

$$\begin{array}{c}\hfill {\Gamma }_{n+1}\le {\Gamma }_n-{\varrho }_n+{\theta }_n.\end{array}$$

(23)

It is easy to see that ${lim}_{n\to \infty }{\theta }_n=0$. To establish that ${\Gamma }_n\to 0$, according to Lemma 5 (and taking into account (19) and (23)), it is enough to demonstrate that for any subsequence $\left\{n_k\right\}\subset \left\{n\right\}$, if ${lim}_{k\to \infty }{\varrho }_{n_k}=0$, we can obtain

$$\underset{k\to \infty }{lim\; sup}{\psi }_{n_k}\le 0.$$

We suppose that ${lim}_{k\to \infty }{\varrho }_{n_k}=0$. When $(I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_{n_k}=0$, it is clear that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\displaystyle \frac{\gamma }{\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}^2=0.\end{array}$$

(24)

Otherwise, as indicated by (7), it follows that

$$\underset{k\to \infty }{lim}{\displaystyle \frac{{\gamma }_{n_k}}{\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}^2=0.$$

By our assumption, we get

$$\underset{k\to \infty }{lim}{\displaystyle \frac{{\gamma }_{n_k}}{\iota }}{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}^2=\underset{k\to \infty }{lim}{\displaystyle \frac{{\rho }_{n_k}{\parallel (I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_{n_k}\parallel }^4}{\iota \parallel {\mathfrak{A}}^{*}(I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_{n_k}{\parallel }^2}},$$

which implies that

$$\underset{k\to \infty }{lim}{\displaystyle \frac{\parallel (I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_{n_k}{\parallel }^4}{\parallel {\mathfrak{A}}^{*}(I_2-{\mathfrak{D}}_n)\mathfrak{A}{\nu }_{n_k}{\parallel }^2}}=0.$$

Further, we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\displaystyle \frac{{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}^2}{∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}}=0.\end{array}$$

(25)

Since $\mathfrak{A}$ is a bounded linear operator, we can conclude that

$$∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥\le \parallel \mathfrak{A}\parallel ∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥.$$

Hence, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\parallel \mathfrak{A}\parallel }}∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥={\displaystyle \frac{{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}^2}{\parallel \mathfrak{A}\parallel ∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}}\le {\displaystyle \frac{{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}^2}{∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}},\end{array}$$

(26)

by considering that $\mathfrak{A}\ne 0$. Using (24)–(26), it follows that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥=0.\end{array}$$

Since ${lim}_{n\to \infty }{\sigma }_n=0$ and the sequence $\left\{\mathfrak{F}\left({\varsigma }_n\right)\right\}$ is bounded, it follows that

$$\begin{array}{c}\hfill ∥{\vartheta }_{n_k+1}-{\varsigma }_{n_k}∥={\sigma }_{n_k}∥\mathfrak{F}\left({\varsigma }_{n_k}\right)∥\to 0,(k\to \infty ).\end{array}$$

It is evident that as $n\to \infty$, we have

$$\begin{array}{cc}\hfill ∥{\nu }_n-{\vartheta }_n∥\le & {\alpha }_n(∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )+{\beta }_n(∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )\hfill \\ \hfill =& {\sigma }_n{\displaystyle \frac{{\alpha }_n}{{\sigma }_n}}\left(∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel \right)\hfill \\ \hfill & +{\sigma }_n{\displaystyle \frac{{\beta }_n}{{\sigma }_n}}\left(∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel \right)\to 0.\hfill \end{array}$$

(27)

Further, according to (25) and (7), we get

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}∥{\nu }_{n_k}-z_{n_k}∥& =\underset{k\to \infty }{lim}∥{\gamma }_{n_k}{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥\hfill \\ \hfill & =\underset{k\to \infty }{lim}{\rho }_{n_k}{\displaystyle \frac{{∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}^2}{∥{\mathfrak{A}}^{*}\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_{n_k}∥}}=0.\hfill \end{array}$$

(28)

From the above inequalities, we arrive at

$$\begin{array}{cc}\hfill ∥{\vartheta }_{n_k+1}-{\vartheta }_{n_k}∥\le & ∥{\vartheta }_{n_k+1}-{\varsigma }_{n_k}∥+∥{\varsigma }_{n_k}-{\nu }_{n_k}∥\hfill \\ \hfill & +∥{\nu }_{n_k}-{\vartheta }_{n_k}∥\to 0,(k\to \infty ).\hfill \end{array}$$

(29)

According to the arbitrariness of $n_k$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill ∥\left(I_2-{\mathfrak{D}}_n\right)\mathfrak{A}{\nu }_n∥\to 0;∥{\vartheta }_{n+1}-{\vartheta }_n∥\to 0,(n\to \infty ).\end{array}\end{array}$$

By Algorithm 1, we show that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ∥{\nu }_n-{\vartheta }_n∥\le & {\alpha }_n∥{\vartheta }_n-{\vartheta }_{n-1}∥+{\beta }_n∥{\nu }_{n-1}-{\vartheta }_{n-1}∥\hfill \\ \hfill =& {\alpha }_n(∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )+{\beta }_n(∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )-{\lambda }_n\parallel {\omega }_n\parallel \hfill \\ \hfill \le & {\sigma }_n{\displaystyle \frac{{\alpha }_n}{{\sigma }_n}}(∥{\vartheta }_n-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )+{\sigma }_n{\displaystyle \frac{{\beta }_n}{{\sigma }_n}}(∥{\nu }_{n-1}-{\vartheta }_{n-1}∥+\parallel {\omega }_n\parallel )-{\lambda }_n\parallel {\omega }_n\parallel \hfill \\ \hfill \le & 2{\sigma }_n{M}_1-{\lambda }_n\parallel {\omega }_n\parallel .\hfill \end{array}\end{array}$$

Then, we get

$$\begin{array}{c}\hfill {\lambda }_n\parallel {\omega }_n\parallel \le 2{\sigma }_n{M}_1-∥{\nu }_n-{\vartheta }_n∥,\end{array}$$

which means that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\lambda }_n\parallel {\omega }_n\parallel =\underset{n\to \infty }{lim}\left(2{\sigma }_n{M}_1-∥{\nu }_n-{\vartheta }_n∥\right)\to 0,\end{array}$$

and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel {\omega }_n\parallel \to 0.\end{array}$$

(30)

Given that $\left\{{\vartheta }_{n_k}\right\}$ is bounded, there exists a subsequence $\left\{{\vartheta }_{n_{k_l}}\right\}$ that converges weakly to $\widehat{\vartheta }$. Without a loss of generality, we assume that the entire sequence $\left\{{\vartheta }_{n_k}\right\}$ converges weakly to $\widehat{\vartheta }$.

At the same time, it follows from (27) that ${\nu }_{n_k}\rightharpoonup \widehat{\vartheta }$ and $\mathfrak{A}{\nu }_{n_k}\rightharpoonup \mathfrak{A}\widehat{\vartheta }$ as $k\to \infty$. By (27)–(30), we have $z_{n_k}+{\lambda }_{n_k}{\omega }_{n_k}\rightharpoonup \widehat{\vartheta }$ as $k\to \infty$. Noting that the pool of indexes is finite and $\left\{{\vartheta }_n\right\}$ is asymptotically regular, for any $1\le i\le p$, we can select a subsequence $\left\{n_{i_m}\right\}\subset \left\{n\right\}$ such that ${\vartheta }_{n_{i_m}}\rightharpoonup \widehat{\vartheta }$, $z_{n_{i_m}}+{\lambda }_{n_{i_m}}{\omega }_{n_{i_m}}\rightharpoonup \widehat{\vartheta }$ as $m\to \infty$, with ${\left[n_{i_m}\right]}_1=i$ for all m. Consequently, we find that

$$\begin{array}{cc}\hfill & \underset{m\to \infty }{lim}∥\left(I_1-{\mathfrak{J}}_{n_{i_m},i}\right)\left(z_{n_{i_m}}+{\lambda }_{n_{i_m}}{\omega }_{n_{i_m}}\right)∥\hfill \\ \hfill & =\underset{m\to \infty }{lim}∥\left(I_1-{\mathfrak{J}}_{n_{i_m},{\left[n_{i_m}\right]}_1}\right)\left(z_{n_{i_m}}+{\lambda }_{n_{i_m}}{\omega }_{n_{i_m}}\right)∥\hfill \\ \hfill & =\underset{m\to \infty }{lim}{\displaystyle \frac{1}{{\kappa }_{n_{i_m}}}}∥\left(I_1-{\mathfrak{J}}_{n_{i_m},n_{i_m}}\right)\left(z_{n_{i_m}}+{\lambda }_{n_{i_m}}{\omega }_{n_{i_m}}\right)∥\hfill \\ \hfill & =\underset{m\to \infty }{lim}{\displaystyle \frac{1}{{\kappa }_{n_{i_m}}}}∥{\omega }_{n_{i_m}+1}∥\hfill \\ \hfill & =0.\hfill \end{array}$$

(31)

Similarly, for any $1\le j\le r$, we can select a subsequence $\left\{n_{j_s}\right\}\subset \left\{n\right\}$ such that $\mathfrak{A}{\nu }_{n_{j_s}}\rightharpoonup \mathfrak{A}\widehat{\vartheta }$ as $s\to \infty$, and ${\left[n_{j_s}\right]}_2=j$ for all s. It turns out that

$$\begin{array}{cc}\hfill \underset{s\to \infty }{lim}∥\left(I_2-{\mathfrak{D}}_{n_{j_s},j}\right)\mathfrak{A}{\nu }_{n_{j_s}}∥& =\underset{s\to \infty }{lim}∥\left(I_2-{\mathfrak{D}}_{n_{j_s},{\left[n_{j_s}\right]}_2}\right)\mathfrak{A}{\nu }_{n_{j_s}}∥\hfill \\ \hfill & =\underset{s\to \infty }{lim}{\displaystyle \frac{1}{{\iota }_{n_{j_s}}}}∥\left(I_2-{\mathfrak{D}}_{n_{j_s},n_{j_s}}\right)\mathfrak{A}{\nu }_{n_{j_s}}∥\hfill \\ \hfill & =0.\hfill \end{array}$$

(32)

Since $I_1-{\mathfrak{S}}_i(1\le i\le p)$ and $I_2-{\mathfrak{T}}_j(1\le j\le r)$ are demiclosed at 0, by employing Lemma 1, we have $I_1-{\mathfrak{J}}_{n,i}(1\le i\le p)$ and $I_2-{\mathfrak{D}}_{n,j}(1\le j\le r)$, which are demiclosed at 0; moreover, it follows from (31) and (32) that $\widehat{\vartheta }\in {\cap }_{i=1}^{p}Fix\left({\mathfrak{J}}_{n,i}\right)$, $\mathfrak{A}\widehat{\vartheta }\in {\cap }_{j=1}^{r}Fix\left({\mathfrak{D}}_{n,j}\right)$. Additionally, with Lemma 1, we know that $\widehat{\vartheta }\in {\cap }_{i=1}^{p}Fix\left({\mathfrak{S}}_i\right)$, $\mathfrak{A}\widehat{\vartheta }\in {\cap }_{j=1}^{r}Fix\left({\mathfrak{T}}_j\right)$.

Subsequently, we demonstrate that

$$\underset{k\to \infty }{lim\; sup}〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n_k}〉\le 0.$$

To demonstrate this inequality, we select a subsequence $\left\{{\vartheta }_{n_{k_l}}\right\}$ from $\left\{{\vartheta }_{n_k}\right\}$ such that

$$\underset{l\to \infty }{lim}〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n_{k_l}}〉=\underset{k\to \infty }{lim\; sup}〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n_k}〉.$$

Given that ${\vartheta }^{*}$ is the unique solution to the variational inequality (4) and $\left\{{\vartheta }_{n_{k_l}}\right\}$ converges weakly to $\widehat{\vartheta }\in \Omega$, we can deduce that

$$\underset{k\to \infty }{lim\; sup}〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n_k}〉=\underset{l\to \infty }{lim}〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n_{k_l}}〉=〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-\widehat{\vartheta }〉\le 0.$$

From (29), we get

$$\underset{k\to \infty }{lim\; sup}〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n_k+1}〉=\underset{k\to \infty }{lim\; sup}〈\mathfrak{F}{\vartheta }^{*},{\vartheta }^{*}-{\vartheta }_{n_k}〉\le 0$$

and

$$\underset{k\to \infty }{lim\; sup}{\psi }_{n_k}\le 0.$$

Therefore, all the conditions specified in Lemma 5 are met. As a result, we can directly conclude that ${lim}_{n\to \infty }{\Gamma }_n={lim}_{n\to \infty }{∥{\vartheta }_n-{\vartheta }^{*}∥}^2=0$. This implies that the sequence $\left\{{\vartheta }_n\right\}$ converges strongly to ${\vartheta }^{*}$, which serves as the unique solution to the variational inequality (4). □

4. Application to the MSFP and a Numerical Example

It is widely recognized that the projection operator $P_Q$ on a nonempty closed convex subset Q is firmly nonexpansive, implying that it is demiclosed at 0. Now, as an application, we will solve the MSFP (1).

Theorem 2. 

Let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be real Hilbert spaces, $C_i\subset {\mathcal{H}}_1$ for all $i=1,2,\cdots ,p$ and $Q_j\subset {\mathcal{H}}_2$ for each $j=1,2,\cdots ,$ and r be nonempty closed convex subsets. Suppose that $\mathfrak{A}$, $\mathfrak{F}$, ${\left[n\right]}_1$, ${\left[n\right]}_2$, ${\mathfrak{J}}_n$, ${\mathfrak{D}}_n$, $\left\{{\varphi }_n\right\}$, $\left\{{\epsilon }_n\right\}$ and $\left\{{\sigma }_n\right\}$ are the same as in Assumption 1. Additionally, if $\overline{\Omega }=\left\{\overline{\upsilon }\in {\mathcal{H}}_1\overline{\upsilon }\in {\bigcap }_{i=1}^p{C}_{i}and\mathfrak{A}\overline{\upsilon }\in {\bigcap }_{j=1}^r{Q}_j\right\}\ne \varnothing$, $0\le lim{inf}_{n\to \infty }{\rho }_n\le lim{sup}_{n\to \infty }{\rho }_n<2$, ${lim}_{n\to \infty }{\displaystyle \frac{{\rho }_n}{{\sigma }_n}}=0$, and for $i\in \{1,2,\cdots ,p\}$ and $j\in \{1,2,\cdots ,r\}$, ${\mathfrak{J}}_{n,i}=(1-{\xi }_n)I_1+{\xi }_n{P}_{C_i}((1-{\eta }_n)I_1+{\eta }_n{P}_{C_i})$ and ${\mathfrak{D}}_{n,j}=(1-{\xi }_n)I_2+{\xi }_n{P}_{Q_j}((1-{\eta }_n)I_2+{\eta }_n{P}_{Q_j})$ with $0<{\xi }_n<{\eta }_n<{\displaystyle \frac{1}{\sqrt{2}+1}}$, then the sequence $\left\{{\vartheta }_n\right\}$ constructed by the following Algorithm 1 converges strongly to a point $\overline{\upsilon }\in \overline{\Omega }$, that is to say, $\overline{\upsilon }$ is a solution of the MSFP (1), which is also the sole solution of the following HVIP:

$$〈\mathfrak{F}\left(\overline{\upsilon }\right),z-\overline{\upsilon }〉\ge 0,\forall z\in \overline{\Omega }.$$

Proof. 

Let $l_1=l_2=1$, ${\mathfrak{S}}_i=P_{C_i}$ for all $i=1,2,\cdots ,p$ and ${\mathfrak{T}}_j=P_{Q_j}$ for each $j=1,2,\cdots ,r$ in Assumption 1 (i). Consequently, it follows from Theorem 1 that conclusions can be easily drawn. □

In this section, we showcase the effectiveness of the proposed Algorithm 1 by using it to solve the MSFP (1). The algorithm was implemented using MATLAB R2020a and executed on a laptop with an Intel(R) Core(TM) i5-8300H CPU @ 2.30GHz and 8.00GB of RAM.

Example 1. 

For $1\le i\le p$ and $1\le j\le r$, we select subsets $C_i\subset {\mathbb{R}}^M$ and $Q_j\subset {\mathbb{R}}^N$ in the MSFP (1), defined by $C_i:=\left\{\vartheta \in {\mathbb{R}}^M\mid 〈a_i^C,\vartheta 〉\le b_i^C\right\}$ and $Q_j:=\left\{\vartheta \in {\mathbb{R}}^N\mid 〈a_j^Q,\vartheta 〉\le b_j^Q\right\}$, respectively. Here, $a_i^C\in {\mathbb{R}}^M$, $a_j^Q\in {\mathbb{R}}^N$, and $b_i^C$, $b_j^Q\in \mathbb{R}$. For these ranges, the components of $a_i^C$ and $a_j^Q$ are randomly selected from the closed interval $[1,3]$, and $b_i^C$ and $b_j^Q$ are randomly chosen from the closed interval $[2,4]$. Additionally, we define $A={\left({\widehat{a}}_{kl}\right)}_{N\times M}$ as a bounded linear operator with entries ${\widehat{a}}_{kl}$ being randomly generated within the closed interval $[20,120]$. Further, we define the function $\Phi :{\mathbb{R}}^M\to \mathbb{R}$ by

$$\Phi \left(\vartheta \right)=\sum _{i=1}^p{\displaystyle \frac{1}{p}}{∥\vartheta -P_{C_i}\left(\vartheta \right)∥}^2+\sum _{j=1}^r{\displaystyle \frac{1}{r}}{∥A\vartheta -P_{Q_j}\left(A\vartheta \right)∥}^2,$$

and use the stopping rule $\Phi \left(\vartheta \right)<\epsilon ={10}^{-20}$. Set $p=r=10$, $\alpha =0.9$, $\beta =0$, $\mathfrak{F}=I$, and ${\kappa }_n={\iota }_n\equiv {\displaystyle \frac{1}{2}}$, ${\epsilon }_n={\displaystyle \frac{1}{n^2}}$, ${\rho }_n={\displaystyle \frac{1}{{(n+1)}^{0.7}}}$, ${\sigma }_n={\displaystyle \frac{1}{log(n+2)}}$, ${\varphi }_n={\displaystyle \frac{1}{log{(n+2)}^{1.1}}}$ and ${\rho }_n=1.95$ for each $n\ge 1$.

By applying Algorithm 1 to solve Example 1,we can select the inertial extrapolation factor ${\alpha }_n$ within the interval $(0,{\overline{\alpha }}_n]$. Specifically, by setting ${\alpha }_n=\varpi {\overline{\alpha }}_n$, we can vary the inertial extrapolation factors by adjusting the parameter $\varpi$ within the range $(0,1]$. Setting ${\vartheta }_0=-5e_1,{\vartheta }_1=10e_1$, and ${\omega }_0=10e_1$, where $e_1={(1,1,\cdots ,1)}^T$, we compare different inertial extrapolation factors of Algorithm 1 across various dimensional spaces.

Table 1. Numerical results with different ${\alpha }_n$, where ${\alpha }_n=\varpi {\overline{\alpha }}_n$, $N=10$, $M=15$.

		$\mathit{\varpi }=0.1$	$\mathit{\varpi }=0.2$	$\mathit{\varpi }=0.3$	$\mathit{\varpi }=0.4$	$\mathit{\varpi }=1$
$\lambda =0.93$	Iter	292	252	212	171	16
Equation (3)	CPU(s)	0.0116	0.0109	0.0179	0.0069	0.0010
	Iter	6	6	6	6	5
Algorithm 1	CPU(s)	0.0012	0.0012	0.0008	0.0006	0.0005

and

Table 2. Numerical results with different ${\alpha }_n$, where ${\alpha }_n=\varpi {\overline{\alpha }}_n$, $N=50$, $M=50$.

		$\mathit{\varpi }=0.1$	$\mathit{\varpi }=0.2$	$\mathit{\varpi }=0.3$	$\mathit{\varpi }=0.4$	$\mathit{\varpi }=1$
$\lambda =0.98$	Iter	267	227	187	139	8
Equation (3)	CPU(s)	0.0226	0.0131	0.0115	0.0070	0.0007
	Iter	8	7	6	6	6
Algorithm 1	CPU(s)	0.0023	0.0013	0.0012	0.0004	0.0004

present the iteration numbers and CPU time of Algorithm 1 for $\varpi =0.1,0.2,0.3,0.4,0.5$, and 1 when the dimensions are $(N,M)=(10,15)$ and $(N,M)=(50,50)$, respectively.

Based on the data presented in Table 1 and Table 2, it is evident that Algorithm 1 outperforms Equation (3) in terms of convergence speed across various dimensions. The computational results indicate that Algorithm 1, with the adjustment parameter $\varpi =1$, demonstrates superior performance across different dimensions.

5. Conclusions

In this paper, we introduced a novel algorithm to approximate solutions for strongly monotone variational inequality problems within the solution set of the split common fixed-point problem with quasi-pseudocontractive mappings. Our method integrates a self-adaptive step size, which negates the need for prior operator norm knowledge, and incorporates an inertial method with a correction term to accelerate convergence. We proved a strong convergence theorem under reasonable conditions and applied our results to solve a multiple-set split feasibility problem. Numerical experiments further validated the effectiveness of our proposed algorithm. The issue addressed in this paper holds significant potential for practical applications in real-world scenarios, including image recognition, signal processing, and machine learning. Our findings contribute to the development of more efficient and practical iterative methods, potentially influencing future research and applications in these fields.