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 ${\mathfrak{S}}_i:\mathcal{H}\to \mathcal{H}$ is 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 also [1]), is defined as follows: 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 fields, such as signal restructuring, power supervision, bandwidth distribution, optimal regulating, network positioning, beamforming, and machine learning [2,3]. The broad applicability of HVIP has sparked significant research interest in recent years, leading to numerous studies exploring its various facets. For more detail, one can refer to [1,2,3] and the references cited within.

As known to all, inverse problem is a fundamental challenge in computational mathematics, with a wide range of applications. Over the past 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. [4] introduced 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 SFP can be formulated as 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. 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) [5,6,7,8,9,10,11,12,13,14,15,16,17]. And then, motivated by challenges in inverse problems related to IMRT, Censor et al. [6] generalized SFP on the solution set of a common fixed point problem and proposed the following multiple-sets split feasibility problem (MSFP): Find a point ${\vartheta }^{*}$ such that 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 SFP. As an extension of convex feasibility problem, SFP, and MSFP, split common fixed point problems (SCFPs) were later introduced by Censor and Segal [8] in 2009. SCFP seeks to find ${\vartheta }^{*}\in {\mathcal{H}}_1$ such that 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. 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 [8,15].

Next, we turn our attention to a class of multiple-sets split common fixed point problems (MSCFPs), which generalize SCFPs , and are recognized for its extensive applications in fields such as image reconstruction, computed tomography, and radiotherapy treatment planning [4,5,6,7]. In recent years, it has attracted considerable interest from researchers because of its broad applicability (see [1,18,19]). Formally, MSCFP tries to find ${\vartheta }^{*}\in {\mathcal{H}}_1$ such that: 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 $p=r=1$, MSCFP reduces to 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 [20] introduced a cyclic iterative algorithm designed to solve MSCFP for directed operators 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 cost. 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. [18] and Zhao et al. [19] introduced some cyclic iterative algorithms, whose step size do not depend on the operator norm. Especially, we note that the algorithm proposed by Zhao et al. [19] is defined as follows: where the step size ${\gamma }_n$ is determined by with $\gamma >0$, and $\mathfrak{U}$ 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 MSCFP, offering a practical advantage by eliminating the need to compute the operator norm.

Moreover, over the past few years, there has been a significant increase in activity and significant progress in the field of MSCFP, driven by the need to broaden its 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 [19], demicontractive mappings [18], and directed mappings [21]. These advancements have allowed for more robust and versatile solutions to MSCFP across different application domains. In this paper, we focus on quasi-pseudocontractive operators, which further expand the scope of MSCFP to encompass a broader class of nonlinear mappings, including demicontractive mappings, directed mappings, and firmly quasi-nonexpansive mappings (see [22]). By examining quasi-pseudocontractive operators, we aim to provide new insights and extend the applicability of MSCFP 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 the 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 [23] and Maing$\acute{\mathrm{e}}$ [24] 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 its 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.

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 ${\omega }_w\left({\vartheta }_n\right)$.

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.

Next, we shall propose the following novel iterative approximation (i.e., Algorithm 3.1) to solve HVIP controlled the MSCFP (2).