Convergence Properties and Numerical Illustration of a Resolvent-Based Inertial Extrapolation Method for Variational Inclusions in Banach Space

Abstract

This paper examines $\mathcal{H}(·,·)$-accretive mappings in Banach spaces and proves that the resolvent operator related to these mappings is Lipschitz continuous. Using the resolvent operator technique, we formulate iterative algorithms to solve a class of variational inclusions in Banach spaces. We also concentrate on examining the convergence of the problem by employing the inertial extrapolation scheme and proving the convergence of the iterative scheme produced by the algorithm. The theoretical analysis is corroborated with a numerical result, which highlights the effectiveness and practical relevance of the proposed approaches.

1. Introduction

The theory of variational inequalities has undergone extensive development and generalization, largely due to its wide-ranging applications in mechanics, physics, optimization, economics, and engineering. A significant extension of this theory is the concept of variational inclusion, originally introduced by Hassouni and Moudafi [1]. Over the years, various approaches have been suggested to address variational inclusion problems, with the resolvent operator technique emerging as one of the most prominent and effective tools.

The idea of generalized m-accretive mappings, together with the formulation of the resolvent operator for such mappings in Banach spaces, was first explored by Huang and Fang [2]. They further investigated several properties of the resolvent operator associated with generalized $\mathcal{H}(·,·)$-accretive mappings in Banach spaces (see also [2,3,4,5,6,7,8,9] for related contributions).

Noor [10,11] later introduced resolvent equations, which generalize Wiener–Hopf-type equations. Several researchers have shown the equivalence between variational inclusion problems and resolvent equations, thereby reinforcing the importance of the resolvent operator method. Unlike projection methods, which may fail in some cases, the resolvent approach has proven to be highly effective in tackling inclusion problems. In fact, many generalized forms of resolvent operators involving multiple monotone operators have been developed in the literature.

Several iterative schemes based on generalized resolvent operators have been developed in the literature. To enhance computational performance, methods that achieve faster convergence are particularly desirable. One effective acceleration strategy involves the use of inertial extrapolation, in which an extrapolation term $v(x_n-x_{n-1})$, governed by a scalar parameter v is incorporated to speed up the convergence process. The idea of inertial-type iterations was originally introduced by Polyak [12] through the well-known heavy-ball method. Such two-step inertial algorithms utilize information from the two most recent iterates to produce the next approximation, thereby improving both stability and convergence rate (see, e.g., [13,14,15,16,17,18,19,20,21]). More recently, Rajpoot et al. [22,23] studied the Yosida resolvent equation related to Yosida variational inclusions. They examined the existence of solutions and the convergence of iterative algorithms based on the Yosida resolvent operator, which further extended some earlier results on variational inclusion problems.

Building upon these advancements, the present study focuses on the investigation of the Lipschitz continuity of the resolvent operator corresponding to $\mathcal{H}(·,·)$-accretive mappings under appropriate conditions. Within this theoretical framework, an iterative algorithm was developed to address a class of variational inclusion problems in Banach spaces. The existence of solutions is ensured under sufficient assumptions, and the convergence analysis of the proposed inertial-type scheme is rigorously established. In addition, a numerical experiment implemented in MATLAB R2024a is presented to illustrate the validity of the theoretical findings through convergence graphs and tabulated results. Furthermore, the efficiency and stability of the proposed algorithm are demonstrated through a comparative study with the Mann-type and Ishikawa-type algorithms.

This paper is organized as follows: Section 1 presents the Introduction and outlines the motivation, background, and related work in the field. Section 2 provides preliminaries, including the basic definitions, lemmas, and mathematical tools used throughout the paper. Section 3 introduces the concept of $\mathcal{H}(·,·)$-accretive operators and discusses their fundamental properties. Section 4 formulates the generalized variational inclusion problem within the Banach space framework. Section 5 is devoted to the fixed-point formulation and iterative algorithm, where the proposed inertial-type scheme is constructed. Section 6 presents the Convergence Analysis of the proposed algorithm under suitable assumptions. Section 7 provides the Numerical Results and a comparative study with existing methods, followed by the Conclusion, which summarizes the main contributions and highlights possible directions for future research.

2. Basic Concepts

Let $\stackrel{\sim }{\mathcal{Q}}$ be a real Banach space, with its dual denoted by ${\stackrel{\sim }{\mathcal{Q}}}^{*}$. The duality pairing between elements of $\stackrel{\sim }{\mathcal{Q}}$ and ${\stackrel{\sim }{\mathcal{Q}}}^{*}$ is represented by $⟨·,·⟩$. We write $2^{\stackrel{\sim }{\mathcal{Q}}}$ for the collection of all nonempty subsets of $\stackrel{\sim }{\mathcal{Q}}$. The generalized duality mapping ${\mathcal{D}}_p:\stackrel{\sim }{\mathcal{Q}}\to 2^{{\stackrel{\sim }{\mathcal{Q}}}^{*}}$ is defined by

$${\mathcal{D}}_p\left(z\right)=\{g^{*}\in {\stackrel{\sim }{\mathcal{Q}}}^{*}:⟨z,g^{*}⟩={\parallel z\parallel }^p,\parallel g^{*}{\parallel =\parallel z\parallel }^{p-1}\},\forall z\in \stackrel{\sim }{\mathcal{Q}},$$

where $p>1$. For $p=2$, ${\mathcal{D}}_p$ becomes the normalized duality mapping. More generally, for any $z\ne 0$, one can write ${\mathcal{D}}_p\left(z\right)={\parallel z\parallel }^{p-1}{\mathcal{D}}_2\left(z\right)$. If the dual space ${\stackrel{\sim }{\mathcal{Q}}}^{*}$ is strictly convex, then the operator ${\mathcal{D}}_p$ is single-valued. In this work, we assume that $\stackrel{\sim }{\mathcal{Q}}$ is a real Banach space. The mapping ${\mathcal{D}}_p$ is single-valued if $\stackrel{\sim }{\mathcal{Q}}$ is uniformly smooth (see [24]). The modulus of smoothness of $\stackrel{\sim }{\mathcal{Q}}$ is introduced through the function ${\psi }_z:[0,\infty )\to [0,\infty )$ defined by

$${\psi }_z\left(t\right)=sup\left\{{\displaystyle \frac{1}{2}}\left(\parallel z_1+z_2\parallel +\parallel z_1-z_2\parallel \right)-1:\parallel z_1\parallel \le 1,\parallel z_2\parallel \le t\right\}.$$

A Banach space $\stackrel{\sim }{\mathcal{Q}}$ is said to be uniformly smooth whenever

$$\underset{t\to 0}{lim}{\displaystyle \frac{{\psi }_z\left(t\right)}{t}}=0.$$

Furthermore, $\stackrel{\sim }{\mathcal{Q}}$ is called p-uniformly smooth if there exists a constant $r>0$ such that

$${\psi }_z\left(t\right)\le r{t}^p,p>1.$$

In a uniformly smooth Banach space, the duality mapping ${\mathcal{D}}_p$ is single-valued. The following result, proved by Xu [25], deals with characteristic inequalities in p-uniformly smooth Banach spaces.

Lemma 1

([25]). Let $\stackrel{\sim }{\mathcal{Q}}$ be a uniformly smooth Banach space. Then, $\stackrel{\sim }{\mathcal{Q}}$ is p-uniformly smooth if and only if there exists a constant $r_p>0$ such that, for all $z_1,z_2\in \stackrel{\sim }{\mathcal{Q}}$,

$${\parallel z_1+z_2\parallel }^p\le {\parallel z_1\parallel }^p+p⟨z_1,{\mathcal{D}}_p\left(z_1\right)⟩+r_p{\parallel z_1\parallel }^p.$$

For completeness, we now recall some standard notions in the case where $\stackrel{\sim }{\mathcal{Q}}=H$ is a Hilbert space.

Definition 1. 

Let $A:\stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ be a single-valued mapping. Then,

(i) 

A is monotone if

$$⟨A\left(z_1\right)-A\left(z_2\right),z_1-z_2⟩\ge 0,\forall z_1,z_2\in \stackrel{\sim }{\mathcal{Q}}.$$

(ii) 

A is strongly monotone if there exists ${\delta }_{\stackrel{\sim }{\mathcal{A}}}>0$ such that

$$⟨A\left(z_1\right)-A\left(z_2\right),z_1-z_2⟩\ge {\delta }_{\stackrel{\sim }{\mathcal{A}}}{\parallel z_1-z_2\parallel }^2,\forall z_1,z_2\in \stackrel{\sim }{\mathcal{Q}}.$$

Definition 2.

A set-valued operator $\Delta :\stackrel{\sim }{\mathcal{Q}}\to 2^{\stackrel{\sim }{\mathcal{Q}}}$ is said to be monotone if

$$⟨w_1-w_2,z_1-z_2⟩\ge 0,\forall z_1,z_2\in \stackrel{\sim }{\mathcal{Q}},w_1\in \Delta \left(z_1\right),w_2\in \Delta \left(z_2\right).$$

Definition 3.

Let $\mathcal{H}:\stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ be a mapping. A set-valued operator $\Delta :\stackrel{\sim }{\mathcal{Q}}\to 2^{\stackrel{\sim }{\mathcal{Q}}}$ is called $\mathcal{H}$-monotone if Δ is monotone and

$$[\mathcal{H}+\lambda \Delta ]\left(\stackrel{\sim }{\mathcal{Q}}\right)=\stackrel{\sim }{\mathcal{Q}},\forall \lambda >0.$$

The following extends Definitions 1–3 to the framework of p-uniformly smooth Banach spaces.

Definition 4

([24]). Let $f,g,u,v:\stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ and $\mathcal{H}:\stackrel{\sim }{\mathcal{Q}}\times \stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ be single-valued mappings. Then,

(i) 

f is accretive if

$$⟨f\left(z_1\right)-f\left(z_2\right),{\mathcal{D}}_p(z_1-z_2)⟩\ge 0,\forall z_1,z_2\in \stackrel{\sim }{\mathcal{Q}}.$$

(ii) 

f is strictly accretive if f is accretive and

$$⟨f\left(z_1\right)-f\left(z_2\right),{\mathcal{D}}_p(z_1-z_2)⟩=0,\iff z_1=z_2\forall z_1,z_2\in \stackrel{\sim }{\mathcal{Q}}.$$

(iii) 

$\mathcal{H}(f,·)$ is called α-strongly accretive with respect to f if there exists $\alpha >0$ such that

$$⟨\mathcal{H}(f\left(z_1\right),w)-\mathcal{H}(f\left(z_2\right),w),{\mathcal{D}}_p(z_1-z_2)⟩\ge \alpha {\parallel z_1-z_2\parallel }^p,\forall z_1,z_2,w\in \stackrel{\sim }{\mathcal{Q}}.$$

(iv) 

g is τ-Lipschitz continuous if there exists $\tau >0$ such that

$$\parallel g\left(z_1\right)-g\left(z_2\right)\parallel \le \tau \parallel z_1-z_2\parallel ,\forall z_1,z_2\in \stackrel{\sim }{\mathcal{Q}}.$$

(v) 

$\mathcal{H}(f,·)$ is μ-co-coercive with respect to f if there exists $\mu >0$ such that

$$⟨\mathcal{H}(f\left(z_1\right),w)-\mathcal{H}(f\left(z_2\right),w),{\mathcal{D}}_p(z_1-z_2)⟩\ge \mu {\parallel f\left(z_1\right)-f\left(z_2\right)\parallel }^p,\forall z_1,z_2,w\in \stackrel{\sim }{\mathcal{Q}}.$$

(vi) 

$\mathcal{H}(·,g)$ is β-relaxed co-coercive with respect to g if there exists $\beta >0$ such that

$$⟨\mathcal{H}(w,g\left(z_1\right))-\mathcal{H}(w,g\left(z_2\right)),{\mathcal{D}}_p(z_1-z_2)⟩\ge (-\beta ){\parallel g\left(z_1\right)-g\left(z_2\right)\parallel }^p,\forall z_1,z_2,w\in \stackrel{\sim }{\mathcal{Q}}.$$

(vii) 

$\mathcal{H}(f,g)$ is called mixed Lipschitz continuous with respect to f and g if there exists $t_{\mathcal{H}}>0$ such that

$$\parallel \mathcal{H}(f\left(z_1\right),g\left(z_1\right))-\mathcal{H}(f\left(z_2\right),g\left(z_2\right))\parallel \le t_{\mathcal{H}}\parallel z_1-z_2\parallel ,\forall z_1,z_2\in \stackrel{\sim }{\mathcal{Q}}.$$

(viii) 

$\mathsf{\Psi }(u,v)$ is called mixed Lipschitz continuous with respect to u and v if there exists $t_{\mathsf{\Psi }}>0$ such that

$$\parallel \mathsf{\Psi }(u\left(z_1\right),v\left(z_1\right))-\mathsf{\Psi }(u\left(z_2\right),v\left(z_2\right))\parallel \le t_{\mathsf{\Psi }}\parallel z_1-z_2\parallel ,\forall z_1,z_2\in \stackrel{\sim }{\mathcal{Q}}.$$

(ix) 

A set-valued operator $\Delta :\stackrel{\sim }{\mathcal{Q}}\to 2^{\stackrel{\sim }{\mathcal{Q}}}$ is called accretive if

$$⟨w_1-w_2,{\mathcal{D}}_p(z_1-z_2)⟩\ge 0,\forall z_1,z_2\in \stackrel{\sim }{\mathcal{Q}},w_1\in \Delta \left(z_1\right),w_2\in \Delta \left(z_2\right).$$

Lemma 2

([22]). Let $\left\{{\mathsf{\Phi }}_n\right\}$ be a sequence of nonnegative real numbers satisfying

$${\mathsf{\Phi }}_{n+1}\le (1-{\zeta }_n){\mathsf{\Phi }}_n+{\zeta }_n{\sigma }_n+{\delta }_n,\forall n\ge 0,$$

where

(i) 

$\left\{{\zeta }_n\right\}\subset [0,1]$ with ${\sum }_{n=1}^{\infty }{\zeta }_n=\infty$;

(ii) 

$lim\; sup{\sigma }_n\le 0$;

(iii) 

${\delta }_n\ge 0$ and ${\sum }_{n=1}^{\infty }{\delta }_n<\infty$.

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

3. $\mathcal{H}(·,·)$-Accretive Operator

This section introduces the concept of an $\mathcal{H}$-accretive operator and highlights some of its essential properties.

Definition 5

([24]). Let $\mathcal{H}:\stackrel{\sim }{\mathcal{Q}}\times \stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ and $f,g,h:\stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ be single-valued mappings. Let $\Delta :\stackrel{\sim }{\mathcal{Q}}\to 2^{\stackrel{\sim }{\mathcal{Q}}}$ be a set-valued operator. Then, Δ is called $\mathcal{H}(·,·)$-accretive with respect to f and g (or simply $\mathcal{H}(·,·)$-accretive) if the following hold:

(i) 

Δ is accretive;

(ii) 

For each $\lambda >0$,

$$\left(\mathcal{H}(f,g)+\lambda \Delta \left(h\right)\right)(\stackrel{\sim }{\mathcal{Q}})=\stackrel{\sim }{\mathcal{Q}}.$$

Theorem 1.

Assume that $\mathcal{H}(f,g)$ is α-strongly accretive with respect to f and β-relaxed co-coercive with respect to g, where g is τ-Lipschitz continuous. If $\alpha >\beta {\tau }^p$ and Δ is an $\mathcal{H}(·,·)$-accretive operator with respect to f and g, then the mapping

$${\left(\mathcal{H}(f,g)+\lambda \Delta \left(h\right)\right)}^{-1}$$

is single-valued for every $\lambda >0$.

Proof. 

Take any $w\in \stackrel{\sim }{\mathcal{Q}}$ and let $z_1,z_2\in {(\mathcal{H}(f,g)+\lambda \Delta \left(h\right))}^{-1}\left(w\right)$. Then,

$${\displaystyle \frac{1}{\lambda }}\left(-\mathcal{H}(f\left(z_1\right),g\left(z_1\right))+w\right)\in \Delta \left(h\left(z_1\right)\right),{\displaystyle \frac{1}{\lambda }}\left(-\mathcal{H}(f\left(z_2\right),g\left(z_2\right))+w\right)\in \Delta \left(h\left(z_2\right)\right).$$

Since $\Delta$ is accretive, we obtain

$$\begin{array}{cc}\hfill 0& \le {\displaystyle \frac{1}{\lambda }}⟨-\mathcal{H}(f\left(z_1\right),g\left(z_1\right))+w-(-\mathcal{H}(f\left(z_2\right),g\left(z_2\right))+w),{\mathcal{D}}_p(z_1-z_2)⟩\hfill \\ \hfill & =-{\displaystyle \frac{1}{\lambda }}⟨\mathcal{H}(f\left(z_1\right),g\left(z_1\right))-\mathcal{H}(f\left(z_2\right),g\left(z_2\right)),{\mathcal{D}}_p(z_1-z_2)⟩\hfill \\ \hfill & =-{\displaystyle \frac{1}{\lambda }}(⟨\mathcal{H}(f\left(z_1\right),g\left(z_1\right))-\mathcal{H}(f\left(z_2\right),g\left(z_1\right)),{\mathcal{D}}_p(z_1-z_2)⟩\hfill \\ \hfill & +⟨\mathcal{H}(f\left(z_2\right),g\left(z_1\right))-\mathcal{H}(f\left(z_2\right),g\left(z_2\right)),{\mathcal{D}}_p(z_1-z_2)⟩).\hfill \end{array}$$

Applying the $\alpha$-strong accretivity of $\mathcal{H}(f,g)$ with respect to f and its $\beta$-relaxed co-coercivity with respect to g, we deduce

$$0\le {\displaystyle \frac{-1}{\lambda }}\left(\alpha {\parallel z_1-z_2\parallel }^p-\beta {\parallel g\left(z_1\right)-g\left(z_2\right)\parallel }^p\right).$$

As g is $\tau$-Lipschitz continuous, this becomes

$$0\le -{\displaystyle \frac{1}{\lambda }}\left((\alpha -\beta {\tau }^p){\parallel z_1-z_2\parallel }^p\right).$$

Since $\alpha >\beta {\tau }^p$ and $\lambda >0$, the above inequality yields $z_1=z_2$. Thus, ${\left(\mathcal{H}(f,g)+\lambda \Delta \left(h\right)\right)}^{-1}$ is single-valued.    □

Definition 6.

Let $\mathcal{H}(f,g)$ be α-strongly accretive with respect to f and β-relaxed co-coercive with respect to g, and let g be τ-Lipschitz continuous, with $\alpha >\beta {\tau }^p$. If Δ is an $\mathcal{H}(·,·)$-accretive operator with respect to f and g, the resolvent operator ${\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}:\stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ is defined as

$${\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(w\right)={\left(\mathcal{H}(f,g)+\lambda \Delta \left(h\right)\right)}^{-1}\left(w\right),\forall w\in \stackrel{\sim }{\mathcal{Q}}.$$

The next theorem shows that this resolvent operator shares properties similar to those studied in [6].

Theorem 2.

Suppose $\mathcal{H}(f,g)$ is α-strongly accretive with respect to f and β-relaxed co-coercive with respect to g and g is τ-Lipschitz continuous. If $\alpha >\beta {\tau }^p$, then the resolvent operator

$${\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}:\stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$$

is ${\displaystyle \frac{1}{\alpha -\beta {\tau }^p}}$-Lipschitz continuous. Specifically, for any $v,w\in \stackrel{\sim }{\mathcal{Q}}$,

$$\parallel {\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(v\right)-{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(w\right)\parallel \le {\displaystyle \frac{1}{\alpha -\beta {\tau }^p}}\parallel v-w\parallel .$$

Proof. 

Let $v,w\in \stackrel{\sim }{\mathcal{Q}}$. By definition,

$${\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(v\right)={(\mathcal{H}(f,g)+\lambda \Delta \left(h\right))}^{-1}\left(v\right),{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(w\right)={(\mathcal{H}(f,g)+\lambda \Delta \left(h\right))}^{-1}\left(w\right).$$

Thus,

$${\displaystyle \frac{1}{\lambda }}\left(v-\mathcal{H}(f\left({\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(v\right)\right),g\left({\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(v\right)\right))\right)\in \Delta \left(h\left({\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(v\right)\right)\right),$$

$${\displaystyle \frac{1}{\lambda }}\left(w-\mathcal{H}(f\left({\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(w\right)\right),g\left({\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(w\right)\right))\right)\in \Delta \left(h\left({\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(w\right)\right)\right).$$

For brevity, set

$$\overline{J}\left(v\right)={\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(v\right),\overline{J}\left(w\right)={\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(w\right).$$

Since $\Delta$ is accretive, it follows that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\lambda }}⟨v-\mathcal{H}(f\left(\overline{J}\left(v\right)\right),g\left(\overline{J}\left(v\right)\right))-\left(w-\mathcal{H}(f\left(\overline{J}\left(w\right)\right),g\left(\overline{J}\left(w\right)\right))\right),{\mathcal{D}}_p(\overline{J}\left(v\right)-\overline{J}\left(w\right))⟩\ge 0.\hfill \end{array}$$

This gives

$$\begin{array}{cc}\hfill ⟨v-w,{\mathcal{D}}_p(\overline{J}\left(v\right)-\overline{J}\left(w\right))⟩\ge & ⟨\mathcal{H}(f\left(\overline{J}\left(v\right)\right),g\left(\overline{J}\left(v\right)\right))\hfill \\ \hfill & -\mathcal{H}(f\left(\overline{J}\left(w\right)\right),g\left(\overline{J}\left(w\right)\right)),{\mathcal{D}}_p(\overline{J}\left(v\right)-\overline{J}\left(w\right))⟩\hfill \end{array}$$

(1)

Using the property of duality mapping, we obtain

$$\parallel v-w\parallel ·\parallel \overline{J}\left(v\right)-\overline{J}{\left(w\right)\parallel }^{p-1}=⟨v-w,{\mathcal{D}}_p(\overline{J}\left(v\right)-\overline{J}\left(w\right))⟩.$$

(2)

Combining (1) and (2) and using the $\alpha$-accretivity and $\beta$-relaxed co-coercivity of $\mathcal{H}$, we have

$$\parallel v-w\parallel ·\parallel \overline{J}\left(v\right)-\overline{J}{\left(w\right)\parallel }^{p-1}\ge \alpha \parallel \overline{J}\left(v\right)-\overline{J}{\left(w\right)\parallel }^p-\beta {\parallel g\left(\overline{J}\left(v\right)\right)-g\left(\overline{J}\left(w\right)\right)\parallel }^p.$$

Since g is $\tau$-Lipschitz continuous, it follows that

$$\parallel v-w\parallel ·\parallel \overline{J}\left(v\right)-\overline{J}{\left(w\right)\parallel }^{p-1}\ge (\alpha -\beta {\tau }^p){\parallel \overline{J}\left(v\right)-\overline{J}\left(w\right)\parallel }^p.$$

Therefore,

$$\parallel {\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(v\right)-{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(w\right)\parallel \le {\displaystyle \frac{1}{\alpha -\beta {\tau }^p}}\parallel v-w\parallel .$$

   □

4. Generalized Variational Inclusion Problem

In this section, we demonstrate that, under suitable assumptions, the generalized $\mathcal{H}(·,·)$-accretive operator introduced earlier can be effectively applied to solve variational inclusion problems in Banach spaces. Let $\stackrel{\sim }{\mathcal{Q}}$ denote a p-uniformly smooth Banach space. Consider single-valued mappings $f,g,h,u,v:\stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ and $\mathsf{\Psi }:\stackrel{\sim }{\mathcal{Q}}\times \stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$, together with a set-valued operator $\Delta :\stackrel{\sim }{\mathcal{Q}}\to 2^{\stackrel{\sim }{\mathcal{Q}}}$. We study the following generalized variational inclusion problem: find $z\in \stackrel{\sim }{\mathcal{Q}}$ such that

$$0\in \mathsf{\Psi }\left(u\right(z),v(z\left)\right)+\Delta \left(h\right(z\left)\right).$$

(3)

When $\mathsf{\Psi }(·,·)=0$ and h is the identity mapping, problem (3) reduces to the simpler task of finding $z\in \stackrel{\sim }{\mathcal{Q}}$ such that

$$0\in \Delta \left(z\right).$$

(4)

The formulation in (4) represents a standard inclusion problem that forms the basis for a wide range of practical applications in the applied sciences.

5. Fixed-Point Formulation and Iterative Algorithm

The next lemma establishes a fixed-point characterization of problem (3), utilizing the resolvent operator introduced in Definition 6.

Lemma 3.

The generalized variational inclusion problem (3) admits a solution $a\in \stackrel{\sim }{\mathcal{Q}}$ if and only if

$$a={\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}[\mathcal{H}(f\left(a\right),g\left(a\right))-\lambda \mathsf{\Psi }(u\left(a\right),v\left(a\right))].$$

(5)

Proof. 

Suppose $a\in \stackrel{\sim }{\mathcal{Q}}$ satisfies (5). Then,

$$a={\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}[\mathcal{H}(f\left(a\right),g\left(a\right))-\lambda \mathsf{\Psi }(u\left(a\right),v\left(a\right))].$$

This is equivalent to

$$\begin{array}{cc}\hfill a& ={(\mathcal{H}(f,g)+\lambda \Delta \left(h\right))}^{-1}[\mathcal{H}(f\left(a\right),g\left(a\right))-\lambda \mathsf{\Psi }(u\left(a\right),v\left(a\right))]\hfill \\ \hfill \iff & \mathcal{H}\left(f\right(a),g(a\left)\right)+\lambda \Delta \left(h\right(a\left)\right)=\mathcal{H}\left(f\right(a),g(a\left)\right)-\lambda \mathsf{\Psi }\left(u\right(a),v(a\left)\right)\hfill \\ \hfill \iff & 0\in \mathsf{\Psi }\left(u\right(a),v(a\left)\right)+\Delta \left(h\right(a\left)\right).\hfill \end{array}$$

   □

Based on the Lemma 5, we now design an iterative algorithm to compute a solution to the problem (3).

6. Convergence Analysis

In this section, we study the convergence of the proposed scheme for the generalized variational inclusion problem in a real Banach space.

Algorithm 1: Inertial Extrapolation Method

Step 1 (Initialisation): For an initial point $a_0\in \stackrel{\sim }{\mathcal{Q}}$, and ${\alpha }_n,{\nu }_n\in [0,1]$, where ${\nu }_n$ is the extrapolation parameter, and $\lambda >0$ is a fixed constant.

Step 2: Given $a_0,a_1\in \stackrel{\sim }{\mathcal{Q}},$ compute $w_n$ as follows: $$w_n=a_n+{\nu }_n(a_n-a_{n-1}),n\ge 1.$$

Step 3: Compute $a_{n+1}$ as follows: $$a_{n+1}=(1-{\alpha }_n)a_n+{\alpha }_n{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left[{\displaystyle \frac{\mathcal{H}(f\left(a_n\right),g\left(a_n\right))+\mathcal{H}(f\left(w_n\right),g\left(w_n\right))}{2}}-\lambda \mathsf{\Psi }(u\left(w_n\right),v\left(w_n\right))\right].$$

Theorem 3.

Let $\stackrel{\sim }{\mathcal{Q}}$ be a p-uniformly smooth Banach space. Let $f,g,u,v:\stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ and $\mathcal{H},\mathsf{\Psi }:\stackrel{\sim }{\mathcal{Q}}\times \stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ be single-valued mappings such that $\mathcal{H}(f,g)$ is α strongly accretive with respect to f and β-relaxed co-coercive with respect to g, where g is τ-Lipschitz continuous. Also, assume that $\alpha >\beta {\tau }^p$, $\mathcal{H}$, and Ψ are mixed Lipschitz continuous with respect to mappings $f,g$ and $u,v$ with constant $t_{\mathcal{H}}$ and $t_{\mathsf{\Psi }}$, respectively. Let $\Delta :\stackrel{\sim }{\mathcal{Q}}\to 2^{\stackrel{\sim }{\mathcal{Q}}}$ be an $\mathcal{H}$-accretive set-valued operator. Suppose that ${\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}:\stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ is a resolvent operator such that ${\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}$ is ${\displaystyle \frac{1}{\kappa }}$-Lipschitz continuous, where $\kappa =\alpha -\beta {\tau }^p$. Suppose that the following conditions are satisfied:

$$\left|\kappa -{\displaystyle \frac{3t_{\mathcal{H}}}{4}}\right|>{\displaystyle \frac{\sqrt{9t_{\mathcal{H}}^2+16}}{4}},$$

(6)

$$t_{\mathcal{H}}<\kappa -\lambda t_{\mathsf{\Psi }},$$

(7)

where

$$t_{\mathsf{\Psi }}={\displaystyle \frac{t_{\mathcal{H}}\kappa +1}{\lambda \kappa }},\kappa \ne 0,\lambda \kappa \ne 0.$$

Let ${\alpha }_n,{\nu }_n\in [0,1]$ for all $n\ge 1$ such that

$$\sum _{n=1}^{\infty }{\alpha }_n=\infty \mathit{as}\mathit{well}\mathit{as}\sum _{n=1}^{\infty }{\nu }_n(a_n-a_{n-1})<\infty ,$$

(8)

where all constants are positive, and ${\nu }_n$ is an extrapolating term Then, the sequence $\left\{a_n\right\}$ generated by Algorithm 1 strongly converges to the unique solution $a^{*}\in \stackrel{\sim }{\mathcal{Q}}$ of generalized variational inclusion problem (3).

Proof. 

Let $a^{*}\in \stackrel{\sim }{\mathcal{Q}}$ be the solution of generalized variational inclusion problem (3). Then, by the algorithm, we get

$$a^{*}=(1-{\alpha }_n)a^{*}+{\alpha }_n{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left[{\displaystyle \frac{\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))+\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))}{2}}-\lambda \mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))\right],$$

(9)

where ${\alpha }_n\in [0,1],$ for all $n\ge 1.$

Using iterative sequence $a_n$ and (9), we find that

$$\begin{array}{cc}\hfill & \parallel a_{n+1}-a^{*}\parallel \hfill \\ \hfill & =\parallel (1-{\alpha }_n)a_n+{\alpha }_n{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left[{\displaystyle \frac{\mathcal{H}(f\left(a_n\right),g\left(a_n\right))+\mathcal{H}(f\left(w_n\right),g\left(w_n\right))}{2}}-\lambda \mathsf{\Psi }(u\left(w_n\right),v\left(w_n\right))\right]\hfill \\ \hfill & -(1-{\alpha }_n)a^{*}-{\alpha }_n{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left[{\displaystyle \frac{\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))+\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))}{2}}-\lambda \mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))\right]\parallel \hfill \\ & =\parallel (1-{\alpha }_n)(a_n-a^{*})+{\alpha }_n\left\{{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\right[{\displaystyle \frac{\mathcal{H}(f\left(a_n\right),g\left(a_n\right))+\mathcal{H}(f\left(w_n\right),g\left(w_n\right))}{2}}-\hfill \\ & \lambda \mathsf{\Psi }(u\left(w_n\right),v\left(w_n\right))]-{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left[{\displaystyle \frac{\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))+\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))}{2}}-\lambda \mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))\right]\}\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel a_n-a^{*}\parallel +{\alpha }_n\parallel {\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}[{\displaystyle \frac{\mathcal{H}(f\left(a_n\right),g\left(a_n\right))+\mathcal{H}(f\left(w_n\right),g\left(w_n\right))}{2}}-\hfill \\ & \lambda \mathsf{\Psi }(u\left(w_n\right),v\left(w_n\right))]-{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left[{\displaystyle \frac{\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))+\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))}{2}}-\lambda \mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))\right]\parallel .\hfill \end{array}$$

Applying the ${\displaystyle \frac{1}{\kappa }}$-Lipschitz continuity of resolvent operator ${\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}$, where $\kappa =\alpha -\beta {\tau }^p$, we get

$$\begin{array}{cc}\hfill & \parallel a_{n+1}-a^{*}\parallel \hfill \\ \hfill & \le (1-{\alpha }_n)\parallel a_n-a^{*}\parallel +{\displaystyle \frac{{\alpha }_n}{\kappa }}\parallel \left[{\displaystyle \frac{\mathcal{H}(f\left(a_n\right),g\left(a_n\right))+\mathcal{H}(f\left(w_n\right),g\left(w_n\right))}{2}}\right]\hfill \\ & -\left[{\displaystyle \frac{\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))+\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))}{2}}\right]-\lambda [\mathsf{\Psi }(u\left(w_n\right),v\left(w_n\right))-\mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))]\parallel \hfill \\ & \le (1-{\alpha }_n)\parallel a_n-a^{*}\parallel +{\displaystyle \frac{{\alpha }_n}{2\kappa }}\parallel \mathcal{H}(f\left(a_n\right),g\left(a_n\right))-\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))\parallel +{\displaystyle \frac{{\alpha }_n}{2\kappa }}\parallel \mathcal{H}(f\left(w_n\right),g\left(w_n\right))\hfill \\ & -\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))\parallel +{\displaystyle \frac{{\alpha }_n\lambda }{\kappa }}\parallel \mathsf{\Psi }(u\left(w_n\right),v\left(w_n\right))-\mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))\parallel .\hfill \end{array}$$

Using the mixed Lipschitz continuity of $\mathcal{H}$ and $\mathsf{\Psi }$ with respect to $f,g$ and $u,v$, respectively, we obtain

$$\begin{array}{cc}\hfill & \parallel a_{n+1}-a^{*}\parallel \hfill \\ \hfill & \le (1-{\alpha }_n)\parallel a_n-a^{*}\parallel +{\displaystyle \frac{{\alpha }_n}{2\kappa }}{t}_{\mathcal{H}}\parallel a_n-a^{*}\parallel +{\displaystyle \frac{{\alpha }_n}{2\kappa }}{t}_{\mathcal{H}}\parallel w_n-a^{*}\parallel +{\displaystyle \frac{{\alpha }_n\lambda }{\kappa }}{t}_{\mathsf{\Psi }}\parallel w_n-a^{*}\parallel \hfill \\ \hfill & =\left[(1-{\alpha }_n)+{\displaystyle \frac{{\alpha }_n}{2\kappa }}{t}_{\mathcal{H}}\right]\parallel a_n-a^{*}\parallel +\left[{\displaystyle \frac{{\alpha }_n}{2\kappa }}{t}_{\mathcal{H}}+{\displaystyle \frac{{\alpha }_n\lambda }{\kappa }}{t}_{\mathsf{\Psi }}\right]\parallel w_n-a^{*}\parallel .\hfill \end{array}$$

(10)

From Algorithm 1, we have

$$\begin{array}{ccc}\hfill \parallel w_n-a^{*}\parallel & =& \parallel a_n-a^{*}+{\nu }_n(a_n-a_{n-1})\parallel \hfill \\ & \le & \parallel a_n-a^{*}\parallel +{\nu }_n\parallel a_n-a_{n-1}\parallel .\hfill \end{array}$$

(11)

Combining (10) and (11), we obtain

$$\begin{array}{c}\parallel a_{n+1}-a^{*}\parallel \le \left[(1-{\alpha }_n)+{\displaystyle \frac{{\alpha }_n}{2\kappa }}{t}_{\mathcal{H}}\right]\parallel a_n-a^{*}\parallel +{\displaystyle \frac{{\alpha }_n(t_{\mathcal{H}}+2\lambda t_{\mathsf{\Psi }})}{2\kappa }}\parallel a_n-a^{*}\parallel \\ +{\displaystyle \frac{{\alpha }_n(t_{\mathcal{H}}+2\lambda t_{\mathsf{\Psi }})}{2\kappa }}{\nu }_n\parallel a_n-a_{n-1}\parallel .\end{array}$$

Thus, we have

$$\begin{array}{cc}\hfill & \parallel a_{n+1}-a^{*}\parallel \hfill \\ \hfill & \le \left[(1-{\alpha }_n)+{\displaystyle \frac{{\alpha }_n}{2\kappa }}{t}_{\mathcal{H}}+{\displaystyle \frac{{\alpha }_n(t_{\mathcal{H}}+2\lambda t_{\mathsf{\Psi }})}{2\kappa }}\right]\parallel a_n-a^{*}\parallel +{\displaystyle \frac{{\alpha }_n(t_{\mathcal{H}}+2\lambda t_{\mathsf{\Psi }})}{2\kappa }}{\nu }_n\parallel a_n-a_{n-1}\parallel \hfill \\ \hfill & =[(1-{\alpha }_n)+{\alpha }_n{S}_1+{\alpha }_n{S}_2]\parallel a_n-a^{*}\parallel +{\alpha }_n{S}_2{\nu }_n\parallel a_n-a_{n-1}\parallel ,\hfill \end{array}$$

or

$$\parallel a_{n+1}-a^{*}\parallel \le [1-{\alpha }_n(1-(S_1+S_2))]\parallel a_n-a^{*}\parallel +{\alpha }_n{S}_2{\nu }_n\parallel a_n-a_{n-1}\parallel ,$$

(12)

where

$$\begin{array}{c}{S}_1={\displaystyle \frac{t_{\mathcal{H}}}{2\kappa }}\\ S_2={\displaystyle \frac{t_{\mathcal{H}}+2\lambda t_{\mathsf{\Psi }}}{2\kappa }}<1,\hspace{1em}\hspace{1em}(\mathrm{By}\mathrm{condition} (6))\\ S_1+S_2={\displaystyle \frac{t_{\mathcal{H}}}{\kappa }}+{\displaystyle \frac{\lambda t_{\mathsf{\Psi }}}{\kappa }}.\end{array}$$

letting $S=S_1+S_2,S<1$ from condition (7). By condition (8), we have

$$\sum _{n=1}^{\infty }{\alpha }_n=\infty \mathrm{and}\sum _{n=1}^{\infty }{\nu }_n\parallel a_n-a_{n-1}\parallel <\infty .$$

Setting ${\sigma }_n=0$ and ${\delta }_n={\sum }_{n=1}^{\infty }{\nu }_n\parallel a_n-a_{n-1}\parallel <\infty$, it follows from Lemma 2 and Equation (12) that $a_n\to a^{*}$ as $n\to \infty$. Therefore, the iterative Algorithm 1 strongly converges to the unique element $a^{*}\in \stackrel{\sim }{\mathcal{Q}}$ solving the generalized variational inclusion problem (3). Furthermore, we establish the uniqueness of the solution to the problem (3). Let $a,a^{*}\in \stackrel{\sim }{\mathcal{Q}}$ be two distinct solutions of the generalized variational inclusion problem (3). Then, by Lemma 3, we have

$$a={\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left[\mathcal{H}\left(f\right(a),g(a\left)\right)-\lambda \mathsf{\Psi }\left(u\right(a),v(a\left)\right)\right]$$

and

$$a^{*}={\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left[\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))-\lambda \mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))\right].$$

It follows that

$$\begin{array}{cc}\hfill \parallel a-a^{*}\parallel =\parallel {\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}[\mathcal{H}(f\left(a\right),g\left(a\right))-\lambda \mathsf{\Psi }(u\left(a\right),v\left(a\right))]-& {\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}[\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))\hfill \\ \hfill & -\lambda \mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))]\parallel .\hfill \end{array}$$

Using the Lipschitz continuity of the resolvent operator ${\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}$,

$$\begin{array}{cc}\hfill & \parallel a-a^{*}\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\kappa }}\parallel [\mathcal{H}(f\left(a\right),g\left(a\right))-\lambda \mathsf{\Psi }(u\left(a\right),v\left(a\right))]-[\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))-\lambda \mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))]\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\kappa }}\parallel [\mathcal{H}(f\left(a\right),g\left(a\right))-\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))-\lambda [\mathsf{\Psi }(u\left(a\right),v\left(a\right)-\mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))]\hfill \\ \hfill & \le {\displaystyle \frac{1}{\kappa }}\parallel [\mathcal{H}(f\left(a\right),g\left(a\right))-\mathcal{H}(f\left(a^{*}\right),g\left(a^{*}\right))]\parallel +{\displaystyle \frac{\lambda }{\kappa }}\parallel [\mathsf{\Psi }(u\left(a\right),v\left(a\right)-\mathsf{\Psi }(u\left(a^{*}\right),v\left(a^{*}\right))]\parallel .\hfill \end{array}$$

Using the mixed Lipschitz continuity of $\mathcal{H}$ and $\mathsf{\Psi }$ with respect to $f,g$ and $u,v$, respectively, we obtain

$$\begin{array}{ccc}\hfill \parallel a-a^{*}\parallel & \le & {\displaystyle \frac{1}{\kappa }}{t}_{\mathcal{H}}\parallel a-a^{*}\parallel +{\displaystyle \frac{\lambda }{\kappa }}{t}_{\mathsf{\Psi }}\parallel a-a^{*}\parallel \hfill \\ & =& \left({\displaystyle \frac{t_{\mathcal{H}}}{\kappa }}+{\displaystyle \frac{\lambda t_{\mathsf{\Psi }}}{\kappa }}\right)\parallel a-a^{*}\parallel \hfill \\ & =& P\left(\theta \right)\parallel a-a^{*}\parallel ,\hfill \end{array}$$

(13)

where $P\left(\theta \right)=\left({\displaystyle \frac{t_{\mathcal{H}}}{\kappa }}+{\displaystyle \frac{\lambda t_{\mathsf{\Psi }}}{\kappa }}\right),t_{\mathsf{\Psi }}=\left({\displaystyle \frac{t_{\mathcal{H}}r+1}{\lambda \kappa }}\right).$ From condition (7), we obtain $0<P\left(\theta \right)<1$. Hence, in applying (13), the generalized variational inclusion problem (3) admits a unique solution $a=a^{*}$.    □

When ${\alpha }_n=1$ in Algorithm 1, we obtain the following algorithm and its convergence result.

Algorithm 2: Inertial Extrapolation Method with ${\alpha }_n=1$

Step 1 (Initialisation): For an initial point $a_0\in \stackrel{\sim }{\mathcal{Q}}$, and ${\nu }_n\in [0,1]$, where ${\nu }_n$ is the extrapolation parameter, and $\lambda >0$ is a fixed constant.

Step 2: Given $a_0,a_1\in \stackrel{\sim }{\mathcal{Q}},$ compute $w_n$ as follows: $$w_n=a_n+{\nu }_n(a_n-a_{n-1}),n\ge 1.$$

Step 3: Compute $a_{n+1}$ as follows: $$a_{n+1}={\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left[{\displaystyle \frac{\mathcal{H}(f\left(a_n\right),g\left(a_n\right))+\mathcal{H}(f\left(w_n\right),g\left(w_n\right))}{2}}-\lambda \mathsf{\Psi }(u\left(w_n\right),v\left(w_n\right))\right].$$

Corollary 1.

Suppose that all the assumptions of Theorem 1 hold. Then, the sequence generated by Algorithm 2 strongly converges to the unique solution $a^{*}\in \stackrel{\sim }{\mathcal{Q}}$ of the generalized variational inclusion problem (3).

When ${\nu }_n=0$ in Algorithm 1, we obtain the following algorithm and its convergence result:

Algorithm 3: Without Inertial Parameter

Step 1 (Initialisation): For an initial point $a_0\in \stackrel{\sim }{\mathcal{Q}}$, and ${\alpha }_n\in [0,1]$, and $\lambda >0$ is a fixed constant.

Step 2: Compute $a_{n+1}$ as follows: $$a_{n+1}=(1-{\alpha }_n)a_n+{\alpha }_n{\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left[\mathcal{H}(f\left(a_n\right),g\left(a_n\right))-\lambda \mathsf{\Psi }(u\left(a_n\right),v\left(a_n\right))\right].$$

Corollary 2.

Suppose that all conditions of Theorem 1 hold except condition (6). Then, the sequence generated by Algorithm 3 strongly converges to the unique solution $a^{*}\in \stackrel{\sim }{\mathcal{Q}}$ of the generalized variational inclusion problem (3).

7. Numerical Results

To verify Theorem 3, we provide a numerical illustration implemented in MATLAB R2024a. The example includes a step-by-step computation table and a convergence plot to illustrate the results.

Example 1.

Consider $\stackrel{\sim }{\mathcal{Q}}=\mathbb{R}$ with the standard inner product and norm, and let $p=2$. Let $f,g,h,u,v:\stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ and $\mathcal{H},\mathsf{\Psi }:\stackrel{\sim }{\mathcal{Q}}\times \stackrel{\sim }{\mathcal{Q}}\to \stackrel{\sim }{\mathcal{Q}}$ be single-valued mappings and let $\Delta :\stackrel{\sim }{\mathcal{Q}}\to 2^{\stackrel{\sim }{\mathcal{Q}}}$ be a set-valued operator. Suppose

$$\mathcal{H}(z_1,z_2)={\displaystyle \frac{6}{5}}{z}_1-{\displaystyle \frac{2}{3}}{z}_2,\mathit{for}\mathit{all}{z}_1,z_2\in \stackrel{\sim }{\mathcal{Q}},$$

and

$$f\left(z\right)=g\left(z\right)=2z,\mathit{for}\mathit{all}z\in \stackrel{\sim }{\mathcal{Q}}.$$

Then,

(i) 

$\mathcal{H}(f,·)$ is α-strongly accretive with respect to f. Computation yields

$$\begin{array}{ccc}\hfill ⟨\mathcal{H}(f\left(z_1\right),w)-\mathcal{H}(f\left(z_2\right),w),{\mathcal{D}}_p(z_1-z_2)⟩& =& {\displaystyle \frac{12}{5}}{\parallel z_1-z_2\parallel }^2,\hfill \end{array}$$

so $\mathcal{H}(f,·)$ is ${\displaystyle \frac{12}{5}}$-strongly accretive with respect to f.

(ii) 

$\mathcal{H}(·,g)$ is β-relaxed co-coercive with respect to g:

$$\begin{array}{ccc}\hfill ⟨\mathcal{H}(w,g\left(z_1\right))-\mathcal{H}(w,g\left(z_2\right)),{\mathcal{D}}_p(z_1-z_2)⟩& =& -{\displaystyle \frac{1}{3}}{\parallel g\left(z_1\right)-g\left(z_2\right)\parallel }^2.\hfill \end{array}$$

Hence, $\mathcal{H}(·,g)$ is ${\displaystyle \frac{1}{3}}$-relaxed co-coercive with respect to g.

(iii) 

$\mathcal{H}(f,g)$ is ${\displaystyle \frac{16}{15}}$-mixed Lipschitz continuous with respect to f and g:

$$\parallel \mathcal{H}(f\left(z_1\right),g\left(z_1\right))-\mathcal{H}(f\left(z_2\right),g\left(z_2\right))\parallel ={\displaystyle \frac{16}{15}}\parallel z_1-z_2\parallel .$$

(iv) 

$\mathsf{\Psi }(u,v)$ is ${\displaystyle \frac{5}{2}}$-mixed Lipschitz continuous with respect to u and v, with

$$\mathsf{\Psi }(z_1,z_2)={\displaystyle \frac{z_1}{2}}+{\displaystyle \frac{z_2}{3}},u\left(z\right)=v\left(z\right)=3z.$$

(v) 

g is 2-Lipschitz continuous:

$$\parallel g\left(z_1\right)-g\left(z_2\right)\parallel =2\parallel z_1-z_2\parallel .$$

(vi) 

Δ is $\mathcal{H}(·,·)$-accretive. Let

$$\Delta \left(x\right)=\left\{{\displaystyle \frac{1}{5}}z\right\},h\left(z\right)={\displaystyle \frac{1}{2}}z.$$

Then,

$$\parallel \Delta \left(h\left(z_1\right)\right)-\Delta \left(h\left(z_2\right)\right)\parallel ={\displaystyle \frac{1}{10}}\parallel z_1-z_2\parallel ,$$

and

$$[\mathcal{H}(f,g)+\lambda \Delta \left(h\right)]\left(\stackrel{\sim }{\mathcal{Q}}\right)=\stackrel{\sim }{\mathcal{Q}},\mathit{for}\lambda =1.$$

(vii) 

From the above, the constants are

$$\alpha ={\displaystyle \frac{12}{5}},\beta ={\displaystyle \frac{1}{3}},\tau =2,p=2,r=\alpha -\beta {\tau }^p={\displaystyle \frac{16}{15}}.$$

(viii) 

For $\lambda =1$, the resolvent operator is

$${\mathfrak{R}}_{\Delta ,\lambda }^{\mathcal{H}(·,·)}\left(z\right)={[\mathcal{H}(f,g)+\lambda \Delta ]}^{-1}\left(z\right)={\displaystyle \frac{30}{35}}z,$$

which is ${\displaystyle \frac{1}{\kappa }}={\displaystyle \frac{15}{16}}$-Lipschitz continuous.

(ix) 

Using these constants, all conditions (6) and (7) of Theorem 3 are satisfied.

(x) 

By choosing ${\alpha }_n={\displaystyle \frac{1}{n+1}}$ and ${\nu }_n={\displaystyle \frac{1}{n}}$, the inertial iterative Algorithm 1 becomes

$$\begin{array}{ccc}\hfill w_n& =& a_n+{\nu }_n(a_n-a_{n-1}),\hfill \\ \hfill a_{n+1}& =& (1-{\alpha }_n)a_n+{\alpha }_n\left[{\displaystyle \frac{16}{35}}{a}_n-{\displaystyle \frac{59}{35}}{w}_n\right].\hfill \end{array}$$

Table 1 and Figure 1 shows that the convergence behavior of sequence $\left\{a_n\right\}$ converges to $a^{*}=0\in \stackrel{\sim }{\mathcal{Q}}$ for the initial values $a_0=1,a_1=0.5$; $a_0=4,a_1=2$; and $a_0=-4,a_1=-4$. The generalized variational inclusion problem (3) has the solution given by the limit $a^{*}=0.$ Next, we analyze the performance of Algorithm 1 for this example and compare it with Algorithms 4.1 and 4.2 from [26]. All computations and the convergence graph were produced using MATLAB_R2024a. In the MATLAB implementation, the computational complexity was evaluated based on two stopping criteria: either when the sequence satisfies the condition $|a_{n+1}-a^{*}|<{10}^{-6}$, indicating convergence, or when the maximum number of iterations reaches 20. The corresponding numerical results are presented in Figure 1.

Table 1. Results of the computations with different initial values $1,4\mathrm{and}$ −4.

No. of Iterations	Initial Value ${\mathit{a}}_0=1$	Initial Value ${\mathit{a}}_0=4$	Initial Value ${\mathit{a}}_0=-4$
(n)	${\mathit{a}}_{\mathit{n}}$	${\mathit{a}}_{\mathit{n}}$	${\mathit{a}}_{\mathit{n}}$
1	0.50000	2.00000	−2.00000
2	0.26904	1.07619	−1.07619
3	0.15159	0.60637	−0.60637
4	0.09392	0.37570	−0.37570
5	0.06227	0.24911	−0.24911
6	0.04372	0.17488	−0.17488
7	0.03210	0.12840	−0.12840
8	0.02442	0.09769	−0.09769
9	0.01912	0.07649	−0.07649
10	0.01533	0.06132	−0.06132
11	0.01253	0.05013	−0.05013
12	0.01041	0.04165	−0.04165
13	0.00877	0.03510	−0.03510
14	0.00748	0.02994	−0.02994
15	0.00645	0.02580	−0.02580
16	0.00561	0.02245	−0.02245
17	0.00492	0.01968	−0.01968
18	0.00434	0.01739	−0.01739
19	0.00386	0.01546	−0.01546
50	0.0005	0.0019	−0.0019
100	0.0001	0.0004	−0.0004
150	0.0000	0.0002	−0.0002
200	0.0000	0.0001	−0.0001
256	0.0000	0.0000	0.0000

Figure 1. Convergence of sequence $a_n$ with different starting points represented graphically.

The numerical experiments were conducted on a MacBook Air equipped with an Apple M1 chip (8-core CPU) running macOS Sequoia 15.6.1. The comparison behaviour of Algorithm 1 is shown in Table 2, Figure 2 and Figure 3. We observe that Algorithm 1 is faster than Algorithms 4.1 and 4.2 in [26].

Table 2. Comparison rate analysis of Algorithm 1 with Algorithms 4.1 and 4.2 in [26] with initial value $x_0=3.$

No. of Iterations	Algorithm 1	Algorithm 4.1 in [26]	Algorithm 4.2 in [26]
(n)	${\mathit{a}}_{\mathit{n}}$	${\mathit{a}}_{\mathit{n}}$	${\mathit{a}}_{\mathit{n}}$
0	3	3	3
1	2.0000	2.0513	2.1750
2	0.7952	1.6329	1.7762
3	0.5214	1.3882	1.5320
4	0.3121	1.2240	1.3635
5	0.2079	1.1046	1.2385
6	0.1459	1.0129	1.1412
7	0.1071	0.9398	1.0627
8	0.0815	0.8798	0.9978
9	0.0638	0.8294	0.9429
10	0.0512	0786.4	0.8958
20	0.0115	0.5486	0.6311
30	0.0048	0.4421	0.5104
40	0.0030	0.3787	0.4380
50	0.0015	0.3357	0.3887
60	0.0010	0.3041	0.3524
70	0.0008	0.2797	0.3243
80	0.0006	0.2601	0.3017
90	0.0004	0.2439	0.2830
100	0.0003	0.2303	0.2673
120	0.0002	0.2085	0.2420
140	0.0002	0.1916	0.2225
160	0.0001	0.1781	0.2069
180	0.0001	0.1670	0.1940
200	0.0001	0.1576	0.1832
220	0.0001	0.1496	0.1739
240	0.0001	0.1426	0.1658
250	0.0000	0.1395	0.1621

Figure 2. Comparison of Algorithm 1 and Ishikawa-type Algorithm 4.1 in [26] with initial point $x_0=3.$

Figure 3. Comparison of Algorithm 1 and Mann-type Algorithm 4.2 in [26] with initial point $x_0=3.$

By setting ${\alpha }_n=1$ and ${\nu }_n=0$, new algorithmic variants were developed. Four comparative graphs Figure 4 and Figure 5 were then plotted to evaluate their convergence behavior with respect to the proposed Algorithm 1, Ishikawa-type Algorithm 4.1 and Mann-type Algorithm 4.2 in [26].

Figure 4. (a) Comparison of Algorithm 1, Algorithm 2, and Ishikawa-type Algorithm 4.1 in [26]; (b) comparison of Algorithm 1, Algorithm 3, and Ishikawa-type Algorithm 4.1 in [26].

Figure 5. (a) Comparison of Algorithm 1, Algorithm 2, and Mann-type Algorithm 4.2 in [26]; (b) comparison of Algorithm 1, Algorithm 3, and Mann-type Algorithm 4.2 in [26].

8. Conclusions

In this paper, we investigated the concept of $\mathcal{H}(·,·)$-accretive mappings in Banach spaces and proposed iterative algorithms based on the resolvent operator technique for solving a specific class of variational inclusions. The convergence of the algorithm was established using an inertial extrapolation scheme, confirming that the iterative sequence generated by the algorithm converges reliably and efficiently to the solution.

The results obtained are also applicable to higher-dimensional Banach spaces, making them relevant for engineers, physicists, and other practitioners in a variety of applied contexts. Furthermore, the performance of the proposed scheme was validated through numerical result, demonstrating its practical effectiveness.