On the Strong Convergence of Combined Generalized Equilibrium and Fixed Point Problems in a Banach Space

Abstract

This work develops and analyzes an iterative method to solve the combined generalized equilibrium and fixed point problems involving two relatively nonexpansive mappings. We establish that the generated sequence converges strongly to a shared solution within a two-uniformly convex and uniformly smooth real Banach space. We also highlight some immediate consequences of the main result. To confirm the algorithm’s efficiency, a numerical example is provided. Furthermore, the practical utility of the proposed algorithm is illustrated using comprehensive tables and figures.

1. Introduction

Let $\mathcal{X}$ be a real Banach space, and let ${\mathcal{X}}^{*}$ denote its dual space. The duality pairing between these spaces is represented by $⟨·,·⟩$. Suppose a nonempty, closed, and convex subset $\mathcal{D}\subset \mathcal{X}$. A mapping $J:\mathcal{X}\to 2^{{\mathcal{X}}^{*}}$ with $J\left(\mathfrak{d}\right)=\{\mathfrak{t}\in {\mathcal{X}}^{*}:⟨\mathfrak{t},\mathfrak{d}⟩=\parallel \mathfrak{d}{\parallel }^2=\parallel \mathfrak{t}{\parallel }^2\},\forall \mathfrak{d}\in \mathcal{X},$ is known as a normalized duality mapping. It is fundamental in studying the geometric and functional characteristics of Banach spaces.

Let $T:\mathcal{D}\to \mathcal{D}$ be a mapping. Let $\mathrm{Fix}\left(T\right)$ denote the set of fixed points of T, i.e., $\mathrm{Fix}\left(T\right)=\{t\in \mathcal{D}:Tt=t\}$. Fixed point theory, a fundamental area of nonlinear analysis, has been extensively applied to the study of nonlinear phenomena. Over the past five decades, numerous notable fixed point existence theorems have been established (see, for example, [1,2,3,4]).

However, from a practical perspective, it is not sufficient to merely establish the existence of fixed points for nonlinear mappings; it is equally important to develop iterative methods to approximate these fixed points. The computation of fixed points plays a crucial role in solving many real-world problems, including inverse problems. For instance, both the split feasibility problem and the convex feasibility problem, which arise in signal processing and image reconstruction, can be reformulated as problems of finding fixed points of specific operators (see [5,6] and references therein for further details). Fixed point theory is a cornerstone of mathematics, with applications in solving equations, optimization, and modeling in pure and applied sciences. The study of fixed points in the moduli spaces of vector bundles over algebraic curves is fundamental in understanding the geometry of these spaces; see [7]. It is foundational to fields like topology (Brouwer’s theorem), analysis (Banach’s contraction principle), and mathematical physics (e.g., Hitchin integrable systems and mirror symmetry). Fixed points also play a vital role in economics and game theory, such as in Nash equilibria; see [8,9,10,11].

In our study, we consider the combination of generalized equilibrium problems (CGEPs), which extends EPs (3) as a particular instance. Let $H_i:\mathcal{D}\times \mathcal{D}\to \mathbb{R}$, $i=1,2,\dots ,N$, and real numbers $a_i\in (0,1)$ satisfy ${\displaystyle \sum _{i=1}^N}{a}_i=1$. Consider ${\displaystyle \sum _{i=1}^N}{a}_i{H}_i:\mathcal{D}\times \mathcal{D}\to \mathbb{R}$. Then, the CGEP is to find $\mathfrak{t}\in \mathcal{D}$ such that

$$\left(\sum _{i=1}^N{a}_i{H}_i\right)(\mathfrak{t},\mathfrak{q})+b(\mathfrak{t},\mathfrak{q})-b(\mathfrak{t},\mathfrak{t})\ge 0,\forall \mathfrak{q}\in \mathcal{D},$$

(1)

where $b:\mathcal{D}\times \mathcal{D}\to \mathbb{R}$ fulfills the assumptions defined below:

Assumption 1. 

(i)

Skew-symmetric, i.e., $b(\mathfrak{t},\mathfrak{t})-b(\mathfrak{t},\mathfrak{q})-b(\mathfrak{q},\mathfrak{t})+b(\mathfrak{q},\mathfrak{q})\ge 0\mathit{for}\mathit{all}\mathfrak{t},\mathfrak{q}\in \mathcal{D}.$

(ii)

Convex with respect to its second argument.

(iii)

Continuous.

Assume $\Omega =\mathrm{Sol}\left(\mathrm{CGEP}\right(1\left)\right)$ as a solution set of CGEP (1). Assume that $b\equiv 0$; then, the CGEP (1) simplifies to the combination of equilibrium problems (CEP), where we seek $\mathfrak{t}\in \mathcal{D}$ such that

$$\left(\sum _{i=1}^N{a}_i{H}_i\right)(\mathfrak{t},\mathfrak{q})\ge 0,\forall \mathfrak{q}\in \mathcal{D},$$

(2)

which was studied by Suwannaut et al. [12]. For further studies on the combination of equilibrium problems, see [13,14,15].

Inspired by minimax problems arising in economic equilibrium theory, the concept of equilibrium problems was first introduced by Zuhovickii et al. [16], Fan [17], and Brezis et al. [18]. Blum and Oettli [19] later adopted this notion and formulated the abstract equilibrium problem (commonly referred to as the EP), which is a special case of the CEP (2). For $i=1$ and $b=0$, the CGEP simplifies to the equilibrium problem (EP), where we seek $\mathfrak{t}\in \mathcal{D}$ such that

$$H(\mathfrak{t},\mathfrak{q})\ge 0,\forall \mathfrak{q}\in \mathcal{D}.$$

(3)

  In 2009, Marino et al. [20] analyzed the convergence properties of a Krasnoselski–Mann-type iterative scheme designed to identify a common element of $\mathrm{Fix}\left(T\right)$ and the solution set of EP (3). They established that EP (3) serves as a unifying framework encompassing various well-known mathematical problems, such as optimization, variational inequality, Nash equilibrium, complementarity, and fixed point problems. Its solution is represented as ${\Omega }_1=\mathrm{Sol}\left(\mathrm{EP}\right(3\left)\right)$ (see [19,21,22]).

  If we define $H(\mathfrak{t},\mathfrak{q})=⟨\mathfrak{q}-\mathfrak{t},\mathcal{A}\mathfrak{t}⟩$ for all $\mathfrak{t},\mathfrak{q}\in \mathcal{D}$, where $\mathcal{A}:\mathcal{D}\to \mathcal{X}$, then we find $\mathfrak{t}\in \mathcal{D}$ such that

$$⟨\mathfrak{q}-\mathfrak{t},\mathcal{A}\mathfrak{t}⟩\ge 0,\forall \mathfrak{q}\in \mathcal{D},$$

(4)

which is known as a variational inequality problem (VIP) [23]. Its solution is represented as $\mathrm{Sol}\left(\mathrm{VIP}\right(4\left)\right)$.

  In a Hilbert space $\mathcal{X}$, Korpelevich [24] proposed an extragradient method to solve VIP (4) as

$$\left.\begin{array}{c}{\mathfrak{d}}_0\in \mathcal{D}\subseteq \mathcal{X},\hfill \\ {\mathfrak{v}}_n=P_{\mathcal{D}}({\mathfrak{d}}_n-\lambda \mathcal{A}{\mathfrak{d}}_n),\hfill \\ {\mathfrak{d}}_{n+1}=P_{\mathcal{D}}({\mathfrak{d}}_n-\lambda \mathcal{A}{\mathfrak{v}}_n).\hfill \end{array}\right\}$$

(5)

Under suitable assumptions, he studied the convergence analysis.

A hybrid extragradient method studied by Nadezkhina et al. [25] in Hilbert space $\mathcal{X}$ is defined as

$$\left.\begin{array}{c}{\mathfrak{d}}_0\in \mathcal{D}\subseteq \mathcal{X},\hfill \\ {\mathfrak{v}}_n=P_{\mathcal{D}}({\mathfrak{d}}_n-r_n\mathcal{A}{\mathfrak{d}}_n),\hfill \\ {\mathfrak{q}}_n={\xi }_n{\mathfrak{d}}_n+(1-{\xi }_n)T{P}_{\mathcal{D}}({\mathfrak{d}}_n-r_n\mathcal{A}{\mathfrak{v}}_n),\hfill \\ {\mathcal{D}}_n=\{z\in \mathcal{D}:\parallel {\mathfrak{q}}_n{-z\parallel }^2\le \parallel {\mathfrak{d}}_n-z{\parallel }^2\},\hfill \\ {\mathcal{B}}_n=\{z\in \mathcal{D}:⟨{\mathfrak{d}}_n-z,x-{\mathfrak{d}}_n⟩\ge 0\},\hfill \\ {\mathfrak{d}}_{n+1}=P_{{\mathcal{D}}_n\cap {\mathcal{B}}_n}{\mathfrak{d}}_0.\hfill \end{array}\right\}$$

(6)

They established that the sequence initiated by (6) converges strongly to a shared solution of VIP (4) and the FPP for the nonexpansive mapping T on $\mathcal{D}$. Several extensions and modifications of this method have been explored in the literature (see [26,27,28,29,30] and the references therein).

A hybrid iterative algorithm studied by Matsushita et al. [31] utilizes generalized projection techniques within the broader setting of a Banach space. The iterative process is formulated as follows:

$$\left.\begin{array}{c}{\mathfrak{d}}_0\in \mathcal{D},\hfill \\ {\mathfrak{v}}_n=J^{-1}({\xi }_n{\mathfrak{d}}_n+(1-{\xi }_n){\mathfrak{d}}_n),\hfill \\ {\mathcal{D}}_n=\{z\in \mathcal{D}:\varphi (z,{\mathfrak{v}}_n)\le \varphi (z,{\mathfrak{d}}_n)\},\hfill \\ {\mathcal{B}}_n=\{z\in \mathcal{D}:⟨{\mathfrak{d}}_n-z,J{\mathfrak{d}}_0-J{\mathfrak{d}}_n⟩\}\hfill \\ {\mathfrak{d}}_{n+1}={\Pi }_{{\mathcal{D}}_n\cap {\mathcal{B}}_n}{\mathfrak{d}}_0.\hfill \end{array}\right\}$$

Takahashi et al. [32] established an iterative scheme to find a shared solution to the EP and FPP involving a relatively nonexpansive mapping T in a Banach space as

$$\left.\begin{array}{c}{\mathfrak{d}}_0\in \mathcal{D},\hfill \\ {\mathfrak{v}}_n=J^{-1}({\xi }_{n}J{\mathfrak{d}}_n+(1-{\xi }_n)JT{\mathfrak{d}}_n),\hfill \\ {\varrho }_n=T_{r_n}{\mathfrak{v}}_n\in \mathcal{D}\mathrm{such}\mathrm{that}H({\varrho }_n,y)+{\displaystyle \frac{1}{r_n}}⟨y-{\varrho }_n,J{\varrho }_n-J{\mathfrak{v}}_n⟩\ge 0,\forall y\in \mathcal{D},\hfill \\ {\mathcal{D}}_n=\{z\in \mathcal{D}:\varphi (z,{\varrho }_n)\le \varphi (z,{\mathfrak{d}}_n)\},\hfill \\ {\mathcal{B}}_n=\{z\in \mathcal{D}:⟨{\mathfrak{d}}_n-z,J{\mathfrak{d}}_0-J{\mathfrak{d}}_n⟩\ge 0\},\hfill \\ {\mathfrak{d}}_{n+1}={\Pi }_{{\mathcal{D}}_n\bigcap {\mathcal{B}}_n}{\mathfrak{d}}_0,\hfill \end{array}\right\}$$

(7)

Under suitable assumptions, they studied the convergence analysis.

Building upon the contributions in [12,14,15,20,31,32], we establish a strong convergence theorem for determining a shared solution of CGEP (3) and the FPP associated with two relatively nonexpansive mappings in a uniformly smooth and uniformly convex Banach space. The approach and findings presented extend and unify various established methods in Banach spaces; see [12,14,15,20,31,32].

2. Preliminaries

Let the symbols → and ⇀ denote strong and weak convergence, respectively. The unit sphere of a Banach space $\mathcal{X}$ is given by $S=\{\mathfrak{d}\in \mathcal{X}:\parallel \mathfrak{d}\parallel =1\}$. A Banach space $\mathcal{X}$ is said to be strictly convex if, for all distinct elements $\mathfrak{d},\mathfrak{q}\in S$, the inequality ${\displaystyle \frac{\parallel \mathfrak{d}+\mathfrak{q}\parallel }{2}}<1$ holds for all $\mathfrak{d},\mathfrak{q}\in S$ with $\mathfrak{d}\ne \mathfrak{q}$. It is called uniformly convex if for any $\epsilon \in (0,2]$, there exists a $\delta >0$ such that for all $\mathfrak{d},\mathfrak{q}\in S$,

$$\parallel \mathfrak{d}-\mathfrak{q}\parallel \ge \epsilon \mathrm{implies}{\displaystyle \frac{\parallel \mathfrak{d}+\mathfrak{q}\parallel }{2}}\le 1-\delta .$$

It is well known that every uniformly convex Banach space is both reflexive and strictly convex. A Banach space $\mathcal{X}$ is said to be smooth if the limit $\underset{t\to 0}{lim}{\displaystyle \frac{\parallel \mathfrak{d}+t\mathfrak{q}\parallel -\parallel \mathfrak{d}\parallel }{t}}$ exists for all $\mathfrak{d},\mathfrak{q}\in S$. Furthermore, if this limit is attained uniformly for all $\mathfrak{d},\mathfrak{q}\in S$, then $\mathcal{X}$ is called uniformly smooth.

A Banach space $\mathcal{X}$ is said to satisfy the Kadec–Klee property if for any sequence $\left\{{\mathfrak{d}}_n\right\}\subset \mathcal{X}$ and any $\mathfrak{d}\in \mathcal{X}$, the conditions $\left\{{\mathfrak{d}}_n\right\}\rightharpoonup \mathfrak{d}$ and $|{\mathfrak{d}}_n|\to |\mathfrak{d}|$ imply that $|{\mathfrak{d}}_n-\mathfrak{d}|\to 0$ as $n\to \infty$. It is well established that every uniformly convex Banach space possesses the Kadec–Klee property.

Additionally, if $\mathcal{X}$ is smooth, the normalized duality mapping J is single-valued. If $\mathcal{X}$ is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subsets of $\mathcal{X}$. Moreover, J is strictly monotone if and only if $\mathcal{X}$ is strictly convex. In the special case where $\mathcal{X}$ is a Hilbert space, the duality mapping reduces to the identity operator, i.e., $J=I$, where I is the identity mapping.

The Lyapunov function $\varphi :\mathcal{X}\times \mathcal{X}\to \mathbb{R}$ is defined as

$$\varphi (\mathfrak{d},\mathfrak{q})={\parallel \mathfrak{d}\parallel }^2-2⟨\mathfrak{d},J\mathfrak{q}⟩+{\parallel \mathfrak{q}\parallel }^2,\forall \mathfrak{d},\mathfrak{q}\in \mathcal{X}.$$

(8)

From the definition of $\varphi$, it follows that

$$\left(\right|\mathfrak{d}|-{\left|\mathfrak{q}\right|)}^2\le \varphi (\mathfrak{d},\mathfrak{q})\le \left(\right|\mathfrak{d}|+{\left|\mathfrak{q}\right|)}^2,\forall \mathfrak{d},\mathfrak{q}\in \mathcal{X}.$$

(9)

Additionally, the function satisfies the inequality

$$\varphi (\mathfrak{d},J^{-1}(\lambda J\mathfrak{q}+(1-\lambda )Jz))\le \lambda \varphi (\mathfrak{d},\mathfrak{q})+(1-\lambda )\varphi (\mathfrak{d},z),\forall \mathfrak{d},\mathfrak{q}\in \mathcal{X},\lambda \in [0,1],$$

(10)

and can also be expressed as

$$\varphi (\mathfrak{d},\mathfrak{q})=\left|\mathfrak{d}\right||J\mathfrak{d}-J\mathfrak{q}|+\left|\mathfrak{q}\right||\mathfrak{d}-\mathfrak{q}|,\forall \mathfrak{d},\mathfrak{q}\in \mathcal{X}.$$

(11)

Remark 1. 

If $\mathcal{X}$ is a reflexive, strictly convex, and smooth Banach space, then for any $\mathfrak{d},\mathfrak{q}\in \mathcal{X},\varphi (\mathfrak{d},\mathfrak{q})=0$ holds if and only if $\mathfrak{d}=\mathfrak{q}$.

Lemma 1 

([33]). Let $\mathcal{X}$ be a Banach space that is both smooth and uniformly convex. Consider two sequences $\left\{{\mathfrak{d}}_n\right\}$ and $\left\{{\mathfrak{q}}_n\right\}$ in $\mathcal{X}$, where at least one of them is bounded. If $\underset{n\to \infty }{lim}\varphi ({\mathfrak{d}}_n,{\mathfrak{q}}_n)=0,$ then it follows that $\underset{n\to \infty }{lim}\parallel {\mathfrak{d}}_n-{\mathfrak{q}}_n\parallel =0.$

Remark 2. 

By applying (11), it follows that the converse of Lemma 1 also holds provided that both sequences $\left\{{\mathfrak{d}}_n\right\}$ and $\left\{{\mathfrak{q}}_n\right\}$ are bounded.

Lemma 2 

([34]). In a two-uniformly convex Banach space $\mathcal{X}$, the following inequality holds for all $\mathfrak{d},\mathfrak{q}\in \mathcal{X}$:

$$\parallel \mathfrak{d}-\mathfrak{q}\parallel \le {\displaystyle \frac{2}{c}}\parallel J\mathfrak{d}-J\mathfrak{q}\parallel ,$$

where c is a constant satisfying $0<c\le 1$, which is referred to as the two-uniformly convex constant of $\mathcal{X}$.

Definition 1. 

Let $T:\mathcal{D}\to \mathcal{D}$ be a mapping. Then, we have the following:

(i)

$\mathrm{Fix}\left(T\right)$ represents the set of fixed points of T, i.e., $\mathrm{Fix}\left(T\right)=\{\mathfrak{d}\in \mathcal{D}:T\mathfrak{d}=\mathfrak{d}\};$

(ii)

A point $p\in \mathcal{D}$ is called an asymptotic fixed point of T if there exists a sequence $\left\{{\mathfrak{d}}_n\right\}\subset \mathcal{D}$ that converges weakly to p and satisfies $\underset{n\to \infty }{lim}\parallel T{\mathfrak{d}}_n-{\mathfrak{d}}_n\parallel =0$. The set of all asymptotic fixed points of T is denoted by $\widehat{\mathrm{Fix}}\left(T\right)$.

(iii)

The mapping T is called relatively nonexpansive if

$$\widehat{\mathrm{Fix}}\left(T\right)=\mathrm{Fix}\left(T\right)\ne \varnothing \mathrm{and}\varphi (p,T\mathfrak{d})\le \varphi (p,\mathfrak{d}),\forall \mathfrak{d}\in \mathcal{D},p\in \mathrm{Fix}\left(T\right).$$

(iv)

T is closed if for any sequence $\left\{{\mathfrak{d}}_n\right\}\subset \mathcal{D}$ such that $\left\{{\mathfrak{d}}_n\right\}\to \mathfrak{d}$ and $\left\{T{\mathfrak{d}}_n\right\}\to \mathfrak{q}$, it follows that $T\mathfrak{d}=\mathfrak{q}$.

Lemma 3 

([31]). Let $\mathcal{X}$ be a reflexive, strictly convex, and smooth Banach space, and let $\mathcal{D}$ be a nonempty closed convex subset of $\mathcal{X}$. Suppose $T:\mathcal{D}\to \mathcal{D}$ is a relatively nonexpansive mapping. Then, the set $\mathrm{Fix}\left(T\right)$ of fixed points of T is a closed convex subset of $\mathcal{D}$.

Lemma 4 

([35]). Let $\mathcal{D}$ be a nonempty closed convex subset of $\mathcal{X}$, and let $\mathcal{A}$ be a monotone and hemicontinuous mapping from $\mathcal{D}$ into ${\mathcal{X}}^{*}$. Then, the set $\mathrm{VIP}(\mathcal{D},\mathcal{A})$ is closed and convex.

Lemma 5 

([36]). Let $\mathcal{X}$ be a uniformly convex Banach space, $r>0$ be a positive number, and $B_r\left(0\right)$ denote the closed ball in $\mathcal{X}$. For a given set $\{{\mathfrak{d}}_1,{\mathfrak{d}}_2,\dots ,{\mathfrak{d}}_N\}\subset B_r\left(0\right)$ and a set of positive numbers $\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_N\}$ such that ${\displaystyle \sum _{i=1}^N}{\lambda }_i=1$, there exists a continuous, strictly increasing, and convex function $g:[0,2r)\to [0,\infty )$ with $g\left(0\right)=0$ such that, for any $i,j\in \{1,2,3,\dots ,N\}$ where $i<j$, the following inequality holds:

$${∥\sum _{n=1}^N{\lambda }_n{\mathfrak{d}}_n∥}^2\le \sum _{n=1}^N{\lambda }_n\parallel {\mathfrak{d}}_n{\parallel }^2-{\lambda }_i{\lambda }_{j}g(\parallel {\mathfrak{d}}_i-{\mathfrak{d}}_j\parallel ).$$

In the following, we utilize the function $V:\mathcal{X}\times {\mathcal{X}}^{*}\to \mathbb{R}$ defined by

$$V(\mathfrak{d},{\mathfrak{d}}^{*})={\parallel \mathfrak{d}\parallel }^2-⟨\mathfrak{d},{\mathfrak{d}}^{*}⟩+{\parallel {\mathfrak{d}}^{*}\parallel }^2.$$

Note that $V(\mathfrak{d},{\mathfrak{d}}^{*})=\varphi (\mathfrak{d},J^{-1}{\mathfrak{d}}^{*})$. The following lemma is well known.

Lemma 6 

([37]). Let $\mathcal{X}$ be a smooth, strictly convex, and reflexive Banach space, with ${\mathcal{X}}^{*}$ denoting its dual. Then, for all $\mathfrak{d}\in \mathcal{X}$ and all ${\mathfrak{d}}^{*},{\mathfrak{q}}^{*}\in {\mathcal{X}}^{*}$, the following inequality holds:

$$V(\mathfrak{d},{\mathfrak{d}}^{*})+2⟨J^{-1}{\mathfrak{d}}^{*}-x,{\mathfrak{q}}^{*}⟩\le V(\mathfrak{d},{\mathfrak{d}}^{*}+{\mathfrak{q}}^{*}).$$

Lemma 7 

([33]). Let $\mathcal{D}$ be a nonempty closed convex subset of a real reflexive, strictly convex, and smooth Banach space $\mathcal{X}$, and let $\mathfrak{d}\in \mathcal{X}$. Then, there exists a unique element ${\mathfrak{d}}_0\in \mathcal{D}$ such that

$$\varphi ({\mathfrak{d}}_0,\mathfrak{d})=\underset{\mathfrak{q}\in \mathcal{D}}{inf}\varphi (\mathfrak{q},\mathfrak{d}).$$

Recently, Alber [37] introduced the generalized projection operator ${\Pi }_{\mathcal{D}}$ in a Banach space $\mathcal{X}$, which serves as an analogue of the metric projection $P_{\mathcal{D}}$ in a Hilbert space H.

Definition 2. 

A generalized projection ${\Pi }_{\mathcal{D}}:\mathcal{X}\to \mathcal{D}$ is a mapping that assigns to any point $\mathfrak{d}\in \mathcal{X}$ the point $\overline{\mathfrak{d}}\in \mathcal{D}$ that minimizes the functional $\varphi (\mathfrak{d},\mathfrak{q})$, i.e., ${\Pi }_{\mathcal{D}}\mathfrak{d}=\overline{\mathfrak{d}}$, where $\overline{\mathfrak{d}}$ is the solution to the minimization problem $\varphi (\overline{\mathfrak{d}},\mathfrak{d})={inf}_{\mathfrak{q}\in \mathcal{D}}\varphi (\mathfrak{q},\mathfrak{d})$.

Lemma 8 

([37]). Let $\mathcal{X}$ be a reflexive, strictly convex, and smooth Banach space, and let $\mathcal{D}$ be a nonempty closed convex subset of $\mathcal{X}$. Then, the following inequality holds:

$$\varphi (\mathfrak{d},{\Pi }_{\mathcal{D}}\mathfrak{q})+\varphi ({\Pi }_{\mathcal{D}}\mathfrak{q},\mathfrak{q})\le \varphi (\mathfrak{d},\mathfrak{q}),\forall \mathfrak{d}\in \mathcal{D},\mathfrak{q}\in \mathcal{X}.$$

Lemma 9 

([37]). Let $\mathcal{X}$ be a reflexive, strictly convex Banach space, and let $\mathcal{D}$ be a nonempty closed convex subset of $\mathcal{X}$. For $\mathfrak{d}\in \mathcal{X}$ and $z\in \mathcal{D}$, the following holds:

$$z={\Pi }_{\mathcal{D}}\mathfrak{d}⟺⟨z-\mathfrak{q},J\mathfrak{d}-Jz⟩\ge 0,\forall \mathfrak{q}\in \mathcal{D}.$$

Next, we present the following assumption for the bifunction $H:\mathcal{D}\times \mathcal{D}\to \mathbb{R}$.

Assumption 2. 

The bifunction $H:\mathcal{D}\times \mathcal{D}⟶\mathbb{R}$ satisfies the following conditions:

(i)

$H(\mathfrak{d},\mathfrak{d})=0\forall \mathfrak{d}\in \mathcal{D};$

(ii)

H is monotone, i.e., $H(\mathfrak{d},\mathfrak{q})+H(\mathfrak{q},\mathfrak{d})\le 0\forall \mathfrak{d}\in \mathcal{D};$

(iii)

For each $\mathfrak{d},\mathfrak{q},z\in \mathcal{D}$, we have

$$\underset{t\to 0}{lim\; sup}H(tz+(1-t)\mathfrak{d},\mathfrak{q})\le H(\mathfrak{d},\mathfrak{q});$$

(iv)

For each $\mathfrak{d}\in \mathcal{D}$, the mapping $\mathfrak{q}\to H(\mathfrak{d},\mathfrak{q})$ is convex and lower semicontinuous.

Lemmas 10 and 11 can be derived from Lemma 2.7, Lemma 2.11, and Remark 2.12, as presented by S. Suwannaut and A. Kangtunyakarn [12].

Lemma 10 

([12]). For each $i=1,2,\dots ,N$, let $H_i:\mathcal{D}\times \mathcal{D}\times \mathcal{D}\to \mathbb{R}$ satisfy Assumption 2, and suppose that

$$\bigcap _{i=1}^N\mathrm{Sol}(\mathrm{CGEP}\left(H_i\right))\ne \varnothing .$$

Then, the solution set of the combined problem satisfies

$$\mathrm{Sol}(\mathrm{CGEP}(\sum _{i=1}^N{a}_i{H}_i))=\bigcap _{i=1}^N\mathrm{Sol}(\mathrm{CGEP}(H_i)),$$

where the coefficients $a_i$ belong to the interval $(0,1)$ for all $i=1,2,\dots ,N$, and they sum to 1, i.e., ${\displaystyle \sum _{i=1}^N}{a}_i=1$.

For a fixed $r\ge 0$, and for each $i\in \{1,2,\dots ,N\}$, we define the mapping $T_r^{H_i}:\mathcal{X}\to \mathcal{D}$ as follows:

$$T_r^{H_i}\left(\mathfrak{d}\right)=\{z\in \mathcal{D}:H_i(z,\mathfrak{q})+{\displaystyle \frac{1}{r}}⟨\mathfrak{q}-z,Jz-J\mathfrak{d}⟩\ge 0,\forall \mathfrak{q}\in \mathcal{D}\},\forall \mathfrak{d}\in \mathcal{X}.$$

(12)

Lemma 11 

([12]). For each $i\in \{1,2,\dots ,N\}$, consider a function $H_i:\mathcal{D}\times \mathcal{D}\times \mathcal{D}\to \mathbb{R}$ that satisfies Assumption 2. Let the operator $J_r^{H_i}:\mathcal{X}\to \mathcal{D}$ be given as defined in (12). Then, the following properties hold:

(a)

The set $T_r^{H_i}\left(\mathfrak{d}\right)$ is nonempty for every $\mathfrak{d}\in \mathcal{X}$.

(b)

The mapping $T_r^{H_i}$ is uniquely defined.

(c)

The operator $J_r^{H_i}$ is firmly nonexpansive, satisfying the inequality

$$⟨T_r\mathfrak{d}-T_r\mathfrak{q},J{T}_r\mathfrak{d}-J{T}_r\mathfrak{q}⟩\le ⟨T_r\mathfrak{d}-T_r\mathfrak{q},J\mathfrak{d}-J\mathfrak{q}⟩.$$

(d)

The set of fixed points of $T_r^{H_i}$ coincides with the solution set of the CGEP $\left(H_i\right)$, i.e.,

$$\mathrm{Fix}\left(T_r^{H_i}\right)=\mathrm{Sol}(\mathrm{CGEP}\left(H_i\right)).$$

(e)

The solution set $\mathrm{Sol}(\mathrm{CGEP}\left(H_i\right))$ is both closed and convex.

Remark 3. 

It is straightforward to verify that ${\displaystyle \sum _{i=1}^N}{a}_i{H}_i$ satisfies Assumption 2. Additionally, we observe that Lemma 11 remains valid for $H:={\displaystyle \sum _{i=1}^N}{a}_i{H}_i$. Moreover, applying Lemmas 10 and 11, we obtain the following result:

$$\mathrm{Fix}\left(J_r^H\right)=\mathrm{Sol}(\mathrm{GVLIP}\left(H\right))=\bigcap _{i=1}^N\mathrm{Sol}(\mathrm{GVLIP}\left(H_i\right)),\forall a_i\in (0,1).$$

3. Main Result

Now, we introduce an iterative scheme, referred to as Algorithm 1, designed to address the combined problem of generalized equilibrium and fixed point formulations involving a pair of relatively nonexpansive mappings. Furthermore, it is shown that the sequence produced by this algorithm converges strongly to a shared solution in a real Banach space that is both two-uniformly convex and uniformly smooth.

Algorithm 1: Iterative algorithm for combined equilibrium problem

Initialization: Let ${\mathfrak{d}}_0=\mathfrak{d}\in \mathcal{D}$, and $n=0$.

Iterative Steps: Step 1. Calculate: $$\begin{array}{c}\hfill \left.\begin{array}{c}{\mathfrak{d}}_0\in \mathcal{D},{\mathcal{D}}_0:=\mathcal{D},\hfill \\ {\mathfrak{g}}_n={\Pi }_{\mathcal{D}}{J}^{-1}(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n),\hfill \\ {\mathfrak{q}}_n=J^{-1}({\sigma }_{n}JS{\mathfrak{d}}_n+(1-{\sigma }_n)J{\mathfrak{g}}_n),\hfill \\ {\mathfrak{v}}_n=J^{-1}((1-{\xi }_n)J{\mathfrak{d}}_n+{\xi }_{n}JT{\mathfrak{q}}_n),\hfill \\ {\varrho }_n=T_{r_n}{\mathfrak{v}}_n,\mathrm{such}\mathrm{that}\hfill \\ \left(\sum _{i=1}^N{a}_i{H}_i\right)({\varrho }_n,\mathfrak{q})+{\displaystyle \frac{1}{r_n}}⟨\mathfrak{q}-{\varrho }_n,J{\varrho }_n-J{\mathfrak{v}}_n⟩+b(\mathfrak{q},{\varrho }_n)-b({\varrho }_n,{\varrho }_n)\ge 0,\forall \mathfrak{q}\in \mathcal{D},\hfill \\ {\mathcal{D}}_n=\{z\in \mathcal{D}:\varphi (z,{\varrho }_n)\le \varphi (z,{\mathfrak{d}}_n)\},\hfill \\ {\mathcal{B}}_n=\{z\in \mathcal{D}:⟨{\mathfrak{d}}_n-z,J{\mathfrak{d}}_n-J{\mathfrak{d}}_0⟩\le 0\},\hfill \\ {\mathfrak{d}}_{n+1}={\Pi }_{{\mathcal{D}}_n\cap {\mathcal{B}}_n}{\mathfrak{d}}_0,\hfill \end{array}\right\}\end{array}$$ (13) where $\left\{r_n\right\}\subseteq [a,\infty )$, $a>0$. Assume that $\left\{{\zeta }_n\right\}$, $\left\{{\xi }_n\right\}$, and $\left\{{\sigma }_n\right\}$ are positive real sequences satisfying: (i) $\left\{{\zeta }_n\right\}\subseteq (0,\infty ),0<{lim\; inf}_{n\to \infty }{\zeta }_n\le {lim\; sup}_{n\to \infty }{\zeta }_n<{\displaystyle \frac{c^2\gamma }{2}}$, c be two-uniformly constant; (ii) $\left\{{\sigma }_n\right\}\in (0,1)$, and ${lim}_{n\to \infty }{\sigma }_n=\sigma \in [c,d]$. Step 2. For $n:=n+1$ and return to Step 1.

Theorem 1. 

Let $\mathcal{D}\ne \varnothing$ be a closed convex subset of a Banach space $\mathcal{X}$, that is, both two-uniformly convex and uniformly smooth, with ${\mathcal{X}}^{*}$ denoting its dual space. Suppose that $H_i:\mathcal{D}\times \mathcal{D}\to \mathbb{R}$, $i\in \{1,2,\dots ,N\}$, and $b:\mathcal{D}\times \mathcal{D}\to \mathbb{R}$ to be maps meet Assumptions 2 and 1, respectively. Let $\mathcal{A}:\mathcal{D}\to {\mathcal{X}}^{*}$ be a γ-ism, $\gamma \in (0,1)$, and let $T,S:\mathcal{D}\to \mathcal{D}$ be two relatively nonexpansive mappings. Suppose $\Gamma =\Omega \cap \mathrm{Fix}\left(T\right)\cap \mathrm{Fix}\left(S\right)\cap {\mathcal{A}}^{-1}\left(0\right)\ne \varnothing$. Then, the sequences $\left\{{\mathfrak{d}}_n\right\}$ and $\left\{{\varrho }_n\right\}$ initiated by the iterative scheme (1) converge strongly to an element $\mathfrak{t}\in {\Pi }_{\Gamma }{\mathfrak{d}}_0$.

Proof. 

The proof is structured into multiple steps.

• Step 1. By applying Lemmas 3, 4, and 11, it follows that $\Gamma \ne \varnothing$ is a closed convex set. Consequently, the projection ${\Pi }_{\Gamma }{\mathfrak{d}}_0$ is well defined.
• Step 2. We establish that ${\mathcal{D}}_n\cap {\mathcal{B}}_n$ is a closed and convex set. According to its concept, ${\mathcal{B}}_n$ is clearly both closed and convex. For all $n\ge 0$, we verify that ${\mathcal{D}}_n$ maintains these properties. Initially, ${\mathcal{D}}_0=\mathcal{D}$ is closed and convex. The closedness of ${\mathcal{D}}_n$ follows naturally, so it remains to demonstrate its convexity. From the concept of ${\mathcal{D}}_n$,

$$\begin{array}{ccc}\hfill {\mathcal{D}}_n& =& \{z\in \mathcal{D}:\varphi (z,{\varrho }_n)\le \varphi (z,{\mathfrak{d}}_n)\}\hfill \\ & =& \{z\in \mathcal{D}:2⟨z,J{\mathfrak{d}}_n-J{\varrho }_n⟩\le \parallel {\mathfrak{d}}_n{\parallel }^2-\parallel {\varrho }_n{\parallel }^2\}.\hfill \end{array}$$

(14)

Utilizing (14), along with the continuity of $J{\mathfrak{d}}_n-J{\varrho }_n$, ${\mathcal{D}}_n$ is indeed closed and convex. Therefore, ${\mathcal{D}}_n\cap {\mathcal{B}}_n$ is closed and convex.

• Step 3. We demonstrate that $\Gamma$ is contained within ${\mathcal{D}}_n\cap {\mathcal{B}}_n$ for all $n\ge 0$. Suppose that for $p\in \Gamma$,

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\varphi (p,{\varrho }_n)& =& \varphi (p,T_{r_n}{\mathfrak{v}}_n)\hfill \\ & \le & \varphi (p,{\mathfrak{v}}_n)\hfill \\ & =& \varphi (p,J^{-1}((1-{\xi }_n)J{\mathfrak{d}}_n+{\xi }_{n}JT{\mathfrak{q}}_n))\hfill \\ & \le & (1-{\xi }_n)\varphi (p,{\mathfrak{d}}_n)+{\xi }_n\varphi (p,T{\mathfrak{q}}_n)-{\xi }_n(1-{\xi }_n)g(\parallel J{\mathfrak{d}}_n-JT{\mathfrak{q}}_n\parallel )\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} & \le & (1-{\xi }_n)\varphi (p,{\mathfrak{d}}_n)+{\xi }_n(\varphi (p,{\mathfrak{q}}_n)-{\xi }_n(1-{\xi }_n)g(\parallel J{\mathfrak{d}}_n-JT{\mathfrak{q}}_n\parallel )\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& \le & (1-{\xi }_n)\varphi (p,{\mathfrak{d}}_n)+{\xi }_n(\varphi (p,{\mathfrak{q}}_n).\hfill \end{array}$$

(17)

Furthermore, applying Lemmas 2, 5, and 6, we obtain

$$\begin{array}{ccc}\hfill \varphi (p,{\mathfrak{g}}_n)& =& \varphi (p,{\Pi }_C{J}^{-1}(J{\mathfrak{d}}_n-{\zeta }_{n}D{\mathfrak{d}}_n))\hfill \\ & \le & \varphi (p,J^{-1}(J{\mathfrak{d}}_n-{\zeta }_{n}D{\mathfrak{d}}_n))\hfill \\ & =& V(p,J{\mathfrak{d}}_n-{\zeta }_{n}D{\mathfrak{d}}_n)\hfill \\ & \le & V(p,(J{\mathfrak{d}}_n-{\zeta }_{n}D{\mathfrak{d}}_n)+{\zeta }_{n}D{\mathfrak{d}}_n)-2⟨J^{-1}(J{\mathfrak{d}}_n-{\zeta }_{n}D{\mathfrak{d}}_n)-p,{\zeta }_{n}D{\mathfrak{d}}_n⟩\hfill \\ & =& V(p,J{\mathfrak{d}}_n)-2{\zeta }_n⟨J^{-1}(J{\mathfrak{d}}_n-{\zeta }_{n}D{\mathfrak{d}}_n)-p,D{\mathfrak{d}}_n⟩\hfill \\ & =& \varphi (p,{\mathfrak{d}}_n)-2⟨{\mathfrak{d}}_n-p,D{\mathfrak{d}}_n⟩-2{\zeta }_n⟨J^{-1}(J{\mathfrak{d}}_n-{\zeta }_{n}D{\mathfrak{d}}_n)-{\mathfrak{d}}_n,D{\mathfrak{d}}_n⟩\hfill \\ & =& \varphi (p,{\mathfrak{d}}_n)-2⟨{\mathfrak{d}}_n-p,D{\mathfrak{d}}_n-Dp⟩-2{\zeta }_n⟨J^{-1}(J{\mathfrak{d}}_n-{\zeta }_{n}D{\mathfrak{d}}_n)-{\mathfrak{d}}_n,D{\mathfrak{d}}_n⟩\hfill \\ & \le & \varphi (p,{\mathfrak{d}}_n)-2{\zeta }_n\gamma \parallel D{\mathfrak{d}}_n{\parallel }^2+2{\zeta }_n\parallel J^{-1}(J{\mathfrak{d}}_n-D{\mathfrak{d}}_n)-J^{-1}J{\mathfrak{d}}_n\parallel \parallel D{\mathfrak{d}}_n{\parallel }^2\hfill \\ & \le & \varphi (p,{\mathfrak{d}}_n)-2{\zeta }_n\gamma \parallel D{\mathfrak{d}}_n{\parallel }^2+{\displaystyle \frac{4{\zeta }_n^2}{c^2}}{\parallel D{\mathfrak{d}}_n\parallel }^2\hfill \\ & =& \varphi (p,{\mathfrak{d}}_n)+2{\zeta }_n({\displaystyle \frac{2{\zeta }_n}{c^2}}-\gamma ){\parallel D{\mathfrak{d}}_n\parallel }^2\hfill \end{array}$$

(18)

which, together with the condition ${\zeta }_n<{\displaystyle \frac{c^2\gamma }{2}}$, yields that

$$\varphi (p,{\mathfrak{g}}_n)\le \varphi (p,{\mathfrak{d}}_n).$$

(19)

Subsequently, applying (19), we derive the following estimate:

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\varphi (p,{\mathfrak{q}}_n)& =& \varphi (p,J^{-1}({\sigma }_{n}JS{\mathfrak{d}}_n+(1-{\sigma }_n)J{\mathfrak{g}}_n))\hfill \\ & \le & {\sigma }_n\varphi (p,S{\mathfrak{d}}_n)+(1-{\sigma }_n)\varphi (p,{\mathfrak{g}}_n)-{\sigma }_n(1-{\sigma }_n)g(\parallel JS{\mathfrak{d}}_n-J{\mathfrak{g}}_n\parallel )\hfill \\ & \le & {\sigma }_n\varphi (p,{\mathfrak{d}}_n)+(1-{\sigma }_n)\varphi (p,{\mathfrak{g}}_n)-{\sigma }_n(1-{\sigma }_n)g(\parallel JS{\mathfrak{d}}_n-J{\mathfrak{g}}_n\parallel )\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& \le & \varphi (p,{\mathfrak{d}}_n)-{\sigma }_n(1-{\sigma }_n)g(\parallel JS{\mathfrak{d}}_n-J{\mathfrak{g}}_n\parallel )\hfill \end{array}$$

(21)

$$\begin{array}{ccc}& \le & \varphi (p,{\mathfrak{d}}_n).\hfill \end{array}$$

(22)

As a consequence of (17) and (22), we obtain

$$\varphi (p,{\varrho }_n)\le \varphi (p,{\mathfrak{d}}_n)$$

(23)

This indicates that $p\in {\mathcal{D}}_n$, and hence, $\Gamma \subset {\mathcal{D}}_n$. We now proceed by induction to show that $\Gamma \subset {\mathcal{D}}_n\cap {\mathcal{B}}_n$ for all $n\ge 0$.

As ${\mathcal{B}}_0=\mathcal{D}$, therefore $\Gamma \subset {\mathcal{D}}_0\cap {\mathcal{B}}_0$. For some $k>0$, consider $\Gamma \subset {\mathcal{D}}_k\cap {\mathcal{B}}_k$. Then, there exists ${\mathfrak{d}}_{k+1}\in {\mathcal{D}}_k\cap {\mathcal{B}}_k$ such that ${\mathfrak{d}}_{k+1}={\Pi }_{{\mathcal{D}}_k\cap {\mathcal{B}}_k}{\mathfrak{d}}_0$. By the concept of ${\mathfrak{d}}_{k+1}$, we obtain, for $z\in {\mathcal{D}}_k\cap {\mathcal{B}}_k$,

$$⟨{\mathfrak{d}}_{k+1}-z,J{\mathfrak{d}}_0-J{\mathfrak{d}}_{k+1}⟩\ge 0.$$

(24)

As $\Gamma \subset {\mathcal{D}}_k\cap {\mathcal{B}}_k$, it follows that

$$⟨{\mathfrak{d}}_{k+1}-p,J{\mathfrak{d}}_0-J{\mathfrak{d}}_{k+1}⟩\ge 0,\forall p\in \Gamma ,$$

implies that $p\in {\mathcal{B}}_{k+1}$. Therefore, $\Gamma \subset {\mathcal{D}}_{k+1}\cap {\mathcal{B}}_{k+1}$, as $\Gamma \subset {\mathcal{D}}_n$ for all n. Hence, $\Gamma \subset {\mathcal{D}}_n\cap {\mathcal{B}}_n$ for all $n\ge 0$, and thus, ${\mathfrak{d}}_{n+1}={\Pi }_{{\mathcal{D}}_n\cap {\mathcal{B}}_n}{\mathfrak{d}}_0$ is well defined. Therefore, $\left\{{\mathfrak{d}}_n\right\}$ is well defined.

• Step 4. Next, prove that $\left\{{\mathfrak{d}}_n\right\}$, $\left\{{\mathfrak{g}}_n\right\}$, $\left\{{\mathfrak{q}}_n\right\}$, $\left\{{\mathfrak{v}}_n\right\}$, and $\left\{{\varrho }_n\right\}$ are bounded. Additionally, we prove that $\underset{n\to \infty }{lim}\varphi ({\mathfrak{d}}_n,{\mathfrak{d}}_0)$ exists, and $\underset{n\to \infty }{lim}\varphi ({\mathfrak{d}}_{n+1},{\mathfrak{d}}_n)=0$.

By the definition of ${\mathcal{B}}_n$, ${\mathfrak{d}}_n={\Pi }_{{\mathcal{B}}_n}{\mathfrak{d}}_0$. Using this and applying Lemma 8, we obtain

$$\varphi ({\mathfrak{d}}_n,{\mathfrak{d}}_0)=\varphi ({\Pi }_{{\mathcal{B}}_n}{\mathfrak{d}}_0,{\mathfrak{d}}_0)\le \varphi (u,{\mathfrak{d}}_0)-\varphi (u,{\Pi }_{{\mathcal{B}}_n}{\mathfrak{d}}_0)\le \varphi (u,{\mathfrak{d}}_0),\forall u\in \Gamma \subset {\mathcal{B}}_n.$$

This indicates that the sequence $\left\{\varphi ({\mathfrak{d}}_n,{\mathfrak{d}}_0)\right\}$ is bounded, which implies, by (9), that $\left\{{\mathfrak{d}}_n\right\}$ is bounded.

Next, from Equations (15), (19), (22), and (23), we conclude that the sequences $\left\{{\mathfrak{g}}_n\right\}$, $\left\{{\mathfrak{q}}_n\right\}$, $\left\{{\varrho }_n\right\}$, and $\left\{{\mathfrak{v}}_n\right\}$ are bounded. Therefore, the sequences $\left\{D{\mathfrak{d}}_n\right\}$, $\left\{S{\mathfrak{d}}_n\right\}$, and $\left\{T{\mathfrak{q}}_n\right\}$ are also bounded.

Since ${\mathfrak{d}}_{n+1}={\Pi }_{{\mathcal{D}}_n\cap {\mathcal{B}}_n}{\mathfrak{d}}_0\in {\mathcal{B}}_n$ and ${\mathfrak{d}}_n\in {\Pi }_{{\mathcal{B}}_n}{\mathfrak{d}}_0$, we have

$$\varphi ({\mathfrak{d}}_n,{\mathfrak{d}}_0)\le \varphi ({\mathfrak{d}}_{n+1},{\mathfrak{d}}_0),\forall n\ge 0,$$

which yields that $\left\{\varphi ({\mathfrak{d}}_n,{\mathfrak{d}}_0)\right\}$ is nondecreasing. As the sequence is bounded, therefore $\underset{n\to \infty }{lim}\varphi ({\mathfrak{d}}_n,{\mathfrak{d}}_0)$ exists.

Using Lemma 8 and $m>n$,

$$\varphi ({\mathfrak{d}}_m,{\mathfrak{d}}_n)=\varphi ({\mathfrak{d}}_m,{\Pi }_{{\mathcal{B}}_n}{\mathfrak{d}}_0)\le \varphi ({\mathfrak{d}}_m,{\mathfrak{d}}_0)-\varphi ({\Pi }_{{\mathcal{B}}_n}{\mathfrak{d}}_0,{\mathfrak{d}}_0)=\varphi ({\mathfrak{d}}_m,{\mathfrak{d}}_0)-\varphi ({\mathfrak{d}}_n,{\mathfrak{d}}_0)\to 0,\mathrm{as}m,n\to \infty .$$

(25)

Thus, by Lemma 1, $\parallel {\mathfrak{d}}_m-{\mathfrak{d}}_n\parallel \to 0$ as $m,n\to \infty$. This shows that $\left\{{\mathfrak{d}}_n\right\}$ is a Cauchy sequence in $\mathcal{D}$. As $\mathcal{D}$ is a nonempty closed subset of $\mathcal{X}$, it is complete. Hence, ∃$\mathfrak{t}\in \mathcal{D}$, with

$$\underset{n\to \infty }{lim}{\mathfrak{d}}_n=\mathfrak{t}.$$

(26)

Furthermore, from (25), we obtain

$$\varphi ({\mathfrak{d}}_{n+1},{\mathfrak{d}}_n)\le \varphi ({\mathfrak{d}}_{n+1},{\mathfrak{d}}_0)-\varphi ({\mathfrak{d}}_n,{\mathfrak{d}}_0),$$

which implies that

$$\underset{n\to \infty }{lim}\varphi ({\mathfrak{d}}_{n+1},{\mathfrak{d}}_n)=0.$$

By Lemma 1, we obtain

$$\underset{n\to \infty }{lim}\parallel {\mathfrak{d}}_{n+1}-{\mathfrak{d}}_n\parallel =0.$$

(27)

• Step 5. We now demonstrate that as $n\to \infty$, the sequences $\left\{{\mathfrak{d}}_n\right\}\to \mathfrak{t}$, $\left\{{\mathfrak{q}}_n\right\}\to \mathfrak{t}$, $\left\{{\mathfrak{v}}_n\right\}\to \mathfrak{t}$, and $\left\{{\varrho }_n\right\}\to \mathfrak{t}$, where $\mathfrak{t}$ is a point in $\mathcal{D}$.

Since ${\mathfrak{d}}_{n+1}={\Pi }_{{\mathcal{D}}_n\cap {\mathcal{B}}_n}{\mathfrak{d}}_0\in {\mathcal{D}}_n$, we can write the following inequality:

$$\varphi ({\mathfrak{d}}_{n+1},{\varrho }_n)\le \varphi ({\mathfrak{d}}_{n+1},{\mathfrak{d}}_n).$$

(28)

From (27), we obtain

$$\varphi ({\mathfrak{d}}_{n+1},{\varrho }_n)=0,$$

and therefore, applying Lemma 1, we have

$$\underset{n\to \infty }{lim}\parallel {\mathfrak{d}}_{n+1}-{\varrho }_n\parallel =0.$$

(29)

Next,

$$\parallel {\mathfrak{d}}_n-{\varrho }_n\parallel \le \parallel {\mathfrak{d}}_n-{\mathfrak{d}}_{n+1}\parallel +\parallel {\mathfrak{d}}_{n+1}-{\varrho }_n\parallel .$$

(30)

From (27), (29), and (30),

$$\underset{n\to \infty }{lim}\parallel {\mathfrak{d}}_n-{\varrho }_n\parallel =0,$$

(31)

which implies that

$$\underset{n\to \infty }{lim}{\varrho }_n=\mathfrak{t}.$$

(32)

On a bounded subset of $\mathcal{X}$, J is norm-to-norm uniformly continuous; therefore,

$$\underset{n\to \infty }{lim}\parallel J{\mathfrak{d}}_n-J{\varrho }_n\parallel =0.$$

(33)

By (16) and (21), we obtain

$$\begin{array}{ccc}\hfill {\xi }_n(1-{\xi }_n)g(\parallel J{\mathfrak{d}}_n-JT{\mathfrak{q}}_n\parallel )& \le & \varphi (p,{\varrho }_n)-\varphi (p,{\mathfrak{d}}_n)\hfill \\ & \le & \parallel {\mathfrak{d}}_n-{\varrho }_n\parallel (\parallel {\mathfrak{d}}_n\parallel +\parallel {\varrho }_n\parallel )+2\parallel p\parallel \parallel J{\mathfrak{d}}_n-J{\varrho }_n\parallel ,\hfill \end{array}$$

(34)

which yields that from (31), (33), and (34),

$$\underset{n\to \infty }{lim}g(\parallel J{\mathfrak{d}}_n-JT{\mathfrak{q}}_n\parallel )=0.$$

(35)

Using the property of g and J, we obtain

$$\underset{n\to \infty }{lim}\parallel J{\mathfrak{d}}_n-JT{\mathfrak{q}}_n\parallel =0.$$

(36)

Similarly, from (17) and (21), we obtain

$$\begin{array}{ccc}\hfill {\sigma }_n(1-{\sigma }_n)g(\parallel JS{\mathfrak{d}}_n-J{\mathfrak{g}}_n\parallel )& \le & \varphi (p,{\mathfrak{d}}_n)-\varphi (p,{\varrho }_n)\hfill \\ & \le & \parallel {\mathfrak{d}}_n-{\varrho }_n\parallel (\parallel {\mathfrak{d}}_n+{\varrho }_n\parallel )+2\parallel p\parallel \parallel J{\mathfrak{d}}_n-J{\varrho }_n\parallel ,\hfill \end{array}$$

(37)

Applying (31), (33), (37), and condition (ii),

$$\underset{n\to \infty }{lim}g(\parallel JS{\mathfrak{d}}_n-J{\mathfrak{g}}_n\parallel )=0.$$

(38)

Again, applying the property of g and J,

$$\underset{n\to \infty }{lim}\parallel S{\mathfrak{d}}_n-{\mathfrak{g}}_n\parallel =0.$$

(39)

From (17), (18), and (20), we deduce that

$$\begin{array}{ccc}\hfill \varphi (p,{\varrho }_n)& \le & (1-{\xi }_n)\varphi (p,{\mathfrak{d}}_n)+{\xi }_n\varphi (p,{\mathfrak{q}}_n)\hfill \\ & \le & (1-{\xi }_n)\varphi (p,{\mathfrak{d}}_n)+{\xi }_n[{\sigma }_n\varphi (p,{\mathfrak{d}}_n)+(1-{\sigma }_n)\varphi (p,{\mathfrak{g}}_n)]\hfill \\ & \le & (1-{\xi }_n)\varphi (p,{\mathfrak{d}}_n)+{\xi }_n\left[{\sigma }_n\varphi (p,{\mathfrak{d}}_n)+(1-{\sigma }_n)\varphi (p,{\mathfrak{d}}_n)+2(1-{\sigma }_n){\zeta }_n\left({\displaystyle \frac{2{\zeta }_n}{c^2}}-\gamma \right)\right]{\parallel \mathcal{A}{\mathfrak{d}}_n\parallel }^2,\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {\xi }_n(1-{\sigma }_n)2{\zeta }_n\left(\gamma -{\displaystyle \frac{2{\zeta }_n}{c^2}}\right){\parallel \mathcal{A}{\mathfrak{d}}_n\parallel }^2& \le & \varphi (p,{\mathfrak{d}}_n)-\varphi (p,{\varrho }_n)\hfill \\ & \le & \parallel {\mathfrak{d}}_n-{\varrho }_n\parallel (\parallel {\mathfrak{d}}_n+{\varrho }_n\parallel )+2\parallel p\parallel \parallel J{\mathfrak{d}}_n-J{\varrho }_n\parallel ,\hfill \end{array}$$

(40)

Using (31), (33), and (40), we have the given conditions

$$\underset{n\to \infty }{lim}\parallel \mathcal{A}{\mathfrak{d}}_n\parallel =0.$$

(41)

Using the concept of $\mathcal{A}$, (26), and (41), we immediately have $\mathfrak{t}\in {\mathcal{A}}^{-1}\left(0\right)$.

Furthermore, combining (13) with (41), we obtain

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}\parallel {\mathfrak{g}}_n-\mathfrak{t}\parallel & =& \underset{n\to \infty }{lim}\parallel {\Pi }_{\mathcal{D}}{J}^{-1}(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n)-{\Pi }_{\mathcal{D}}\mathfrak{t}\parallel \hfill \\ & \le & \underset{n\to \infty }{lim}\parallel J^{-1}(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n)-\mathfrak{t}\parallel \hfill \\ & =& 0\hfill \end{array}$$

(42)

From (13), we have

$$\parallel J{\mathfrak{q}}_n-J{\mathfrak{g}}_n\parallel ={\sigma }_n\parallel JS{\mathfrak{d}}_n-J{\mathfrak{g}}_n\parallel ,$$

(43)

which, from (33) and (38), implies that

$$\underset{n\to \infty }{lim}\parallel J{\mathfrak{q}}_n-J{\mathfrak{g}}_n\parallel =0.$$

(44)

This leads to

$$\underset{n\to \infty }{lim}\parallel {\mathfrak{q}}_n-{\mathfrak{g}}_n\parallel =0.$$

(45)

By Lemmas 6 and 3,

$$\begin{array}{ccc}\hfill \varphi ({\mathfrak{d}}_n,{\mathfrak{g}}_n)& =& \varphi ({\mathfrak{d}}_n,{\Pi }_{\mathcal{D}}{J}^{-1}(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n))\hfill \\ & \le & \varphi ({\mathfrak{d}}_n,J^{-1}(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n))\hfill \\ & \le & V({\mathfrak{d}}_n,(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n))\hfill \\ & \le & V({\mathfrak{d}}_n,(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n)+{\zeta }_n\mathcal{A}{\mathfrak{d}}_n)-2⟨J^{-1}(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n)-{\mathfrak{d}}_n,{\zeta }_n\mathcal{A}{\mathfrak{d}}_n⟩\hfill \\ & =& \varphi ({\mathfrak{d}}_n,{\mathfrak{d}}_n)+2⟨J^{-1}(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n)-{\mathfrak{d}}_n,-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n⟩\hfill \\ & =& 2{\zeta }_n⟨J^{-1}(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n)-{\mathfrak{d}}_n,-\mathcal{A}{\mathfrak{d}}_n⟩\hfill \\ & \le & \parallel J^{-1}(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n)-J^{-1}J{\mathfrak{d}}_n\parallel \hfill \\ & \le & {\displaystyle \frac{4}{c^2}}{\zeta }_n^2{\parallel \mathcal{A}{\mathfrak{d}}_n\parallel }^2,\hfill \end{array}$$

(46)

Thus, from (41), we conclude that

$$\underset{n\to \infty }{lim}\varphi ({\mathfrak{d}}_n,{\mathfrak{g}}_n)=0,$$

(47)

and with Lemma 1, it yields that

$$\underset{n\to \infty }{lim}\parallel {\mathfrak{d}}_n-{\mathfrak{g}}_n\parallel =0.$$

(48)

Since

$$\parallel T{\mathfrak{q}}_n-{\mathfrak{q}}_n\parallel \le \parallel {\mathfrak{q}}_n-{\mathfrak{g}}_n\parallel +\parallel {\mathfrak{g}}_n-{\mathfrak{d}}_n\parallel +\parallel {\mathfrak{d}}_n-T{\mathfrak{q}}_n\parallel ,$$

(49)

this implies with (36), (45), (48), and (49) that

$$\underset{n\to \infty }{lim}\parallel T{\mathfrak{q}}_n-{\mathfrak{q}}_n\parallel =0,$$

(50)

which implies that $\mathfrak{t}\in \mathrm{Fix}\left(T\right)$.

Moreover, since

$$\parallel T{\mathfrak{d}}_n-{\mathfrak{d}}_n\parallel \le \parallel S{\mathfrak{d}}_n-{\mathfrak{g}}_n\parallel +\parallel {\mathfrak{g}}_n-{\mathfrak{d}}_n\parallel ,$$

(51)

which implies with (39), (48), and (51) that

$$\underset{n\to \infty }{lim}\parallel S{\mathfrak{d}}_n-{\mathfrak{d}}_n\parallel =0,$$

(52)

this implies that $\mathfrak{t}\in \mathrm{Fix}\left(S\right)$. Therefore, $\mathfrak{t}\in \mathrm{Fix}\left(T\right)\cap \mathrm{Fix}\left(S\right)$.

• Step 6. We now demonstrate that $\mathfrak{t}\in \Gamma$. It remains to be shown that $\mathfrak{t}$ also belongs to $\mathrm{Sol}(\mathrm{CGEP}\left({\sum }_{i=1}^N{a}_i{H}_i\right))$.

Therefore, let $p\in \Gamma \subset {\mathcal{D}}_n$, and by (16),

$$\begin{array}{ccc}\hfill \varphi (p,{\mathfrak{v}}_n)& \le & \varphi (p,J^{-1}((1-{\xi }_n)J{\mathfrak{d}}_n+{\xi }_{n}JT{\mathfrak{q}}_n))\hfill \\ & \le & \varphi (p,{\mathfrak{d}}_n).\hfill \end{array}$$

(53)

Applying Lemma 11 and ${\varrho }_n=T_{r_n}{\mathfrak{v}}_n$, we have

$$\begin{array}{ccc}\hfill \varphi ({\varrho }_n,{\mathfrak{v}}_n)& =& \varphi (T_{r_n}{\mathfrak{v}}_n,{\mathfrak{v}}_n)\hfill \\ & \le & \varphi (p,{\mathfrak{v}}_n)-\varphi (p,T_{r_n}{\mathfrak{v}}_n)\hfill \\ & \le & \varphi (p,{\mathfrak{d}}_n)-\varphi (p,{\varrho }_n).\hfill \\ & \le & \parallel {\mathfrak{d}}_n-{\varrho }_n\parallel (\parallel {\mathfrak{d}}_n\parallel +\parallel {\varrho }_n\parallel )+2\parallel p\parallel \parallel J{\mathfrak{d}}_n-J{\varrho }_n\parallel \hfill \end{array}$$

(54)

By (31), (33), and (54)

$$\underset{n\to \infty }{lim}\varphi ({\varrho }_n,{\mathfrak{v}}_n)=0.$$

Therefore, by Lemma 1,

$$\underset{n\to \infty }{lim}\parallel {\varrho }_n-{\mathfrak{v}}_n\parallel =0.$$

Using the property of J,

$$\underset{n\to \infty }{lim}\parallel J{\varrho }_n-J{\mathfrak{v}}_n\parallel =0.$$

(55)

Given $r_n\ge a$ and (55), therefore

$$\underset{n\to \infty }{lim}{\displaystyle \frac{\parallel J{\varrho }_n-J{\mathfrak{v}}_n\parallel }{r_n}}=0.$$

(56)

Let ${\varrho }_n=T_{r_n}^H{\mathfrak{v}}_n$, where $H={\displaystyle \sum _{i=1}^N}{a}_i{H}_i$; then, we have

$$\left(\sum _{i=1}^N{a}_i{H}_i\right)({\varrho }_n,\mathfrak{q})+{\displaystyle \frac{1}{r_n}}⟨\mathfrak{q}-{\varrho }_n,J{\varrho }_n-J{\mathfrak{v}}_n⟩+b(\mathfrak{q},{\varrho }_n)-b({\varrho }_n,{\varrho }_n)\ge 0,\forall \mathfrak{q}\in \mathcal{D}.$$

Utilizing Assumption 2(ii), we obtain

$${\displaystyle \frac{1}{r_n}}⟨\mathfrak{q}-{\varrho }_n,J{\varrho }_n-J{\mathfrak{v}}_n⟩\ge -b(\mathfrak{q},{\varrho }_n)+b({\varrho }_n,{\varrho }_n)-\left(\sum _{i=1}^N{a}_i{H}_i\right)({\varrho }_n,\mathfrak{q})\ge \left(\sum _{i=1}^N{a}_i{H}_i\right)(\mathfrak{q},{\varrho }_n)-b(\mathfrak{q},{\varrho }_n)+b({\varrho }_n,{\varrho }_n).$$

Taking the limit as $n\to \infty$, by (56) and the lower semicontinuity of $\mathfrak{q}\to \left({\displaystyle \sum _{i=1}^N}{a}_i{H}_i\right)(\mathfrak{q},·)$, we obtain

$$\left(\sum _{i=1}^N{a}_i{H}_i\right)(\mathfrak{q},\mathfrak{t})-b(y,\mathfrak{t})+b(\mathfrak{t},\mathfrak{t})\le 0,\forall \mathfrak{q}\in \mathcal{D}.$$

Let ${\mathfrak{q}}_{\kappa }:=\kappa \mathfrak{q}+(1-\kappa )\mathfrak{t}$ for all $\kappa \in (0,1]$ and $\mathfrak{q}\in \mathcal{D}$. Then, it follows that ${\mathfrak{q}}_{\kappa }\in \mathcal{D}$, and hence, we have

$$\left(\sum _{i=1}^N{a}_i{H}_i\right)({\mathfrak{q}}_{\kappa },\mathfrak{t})-b({\mathfrak{q}}_{\kappa },\mathfrak{t})+b(\mathfrak{t},\mathfrak{t})\le 0.$$

Therefore, by Assumption 2(i)–(iv), we deduce

$$\begin{array}{ccc}\hfill 0=\left(\sum _{i=1}^N{a}_i{H}_i\right)({\mathfrak{q}}_{\kappa },{\mathfrak{q}}_{\kappa })& \le & \kappa \left(\sum _{i=1}^N{a}_i{H}_i\right)({\mathfrak{q}}_{\kappa },\mathfrak{q})+(1-\kappa )\left(\sum _{i=1}^N{a}_i{H}_i\right)({\mathfrak{q}}_{\kappa },\mathfrak{t})\hfill \\ & \le & \kappa \left(\sum _{i=1}^N{a}_i{H}_i\right)({\mathfrak{q}}_{\kappa },\mathfrak{q})+(1-\kappa )[b({\mathfrak{q}}_{\kappa },\mathfrak{t})-b(\mathfrak{t},\mathfrak{t})]\hfill \\ & \le & t\left(\sum _{i=1}^N{a}_i{H}_i\right)({\mathfrak{q}}_{\kappa },\mathfrak{q})+\kappa (1-\kappa )[b(\mathfrak{q},\mathfrak{t})-b(\mathfrak{t},\mathfrak{t})]\hfill \end{array}$$

(57)

Since $\kappa >0$, we obtain

$$\left(\sum _{i=1}^N{a}_i{H}_i\right)({\mathfrak{q}}_{\kappa },\mathfrak{q})+(1-\kappa )[b(\mathfrak{q},\mathfrak{t})-b(\mathfrak{t},\mathfrak{t})]\ge 0,\forall \mathfrak{q}\in \mathcal{D}.$$

Letting $\kappa \downarrow 0_{+}$, we obtain from Assumption 2(iii) that

$$\left(\sum _{i=1}^N{a}_i{H}_i\right)(\mathfrak{t},\mathfrak{q})+b(\mathfrak{q},\mathfrak{t})-b(\mathfrak{t},\mathfrak{t})\ge 0,\forall \mathfrak{q}\in \mathcal{D}.$$

Thus, $\mathfrak{t}\in \mathrm{Sol}(\mathrm{CGEP}({\displaystyle \sum _{i=1}^N}{a}_i{H}_i)).$ Therefore, $\mathfrak{t}\in \mathrm{Sol}(\mathrm{CGEP}({\displaystyle \sum _{i=1}^N}{a}_i{H}_i))={\displaystyle \bigcap _{i=1}^N}\mathrm{Sol}(\mathrm{CGEP}(H_i)).$ Hence, $\mathfrak{t}\in \Gamma =\Omega \bigcap \mathrm{Fix}\left(T\right)\bigcap \mathrm{Fix}\left(S\right)\bigcap {\mathcal{A}}^{-1}\left(0\right).$

• Step 7. Finally, we demonstrate that $\mathfrak{t}={\Pi }_{\Gamma }{\mathfrak{d}}_0$. Let $k\to \infty$ in (24), and we have

$$⟨\mathfrak{t}-p,J{\mathfrak{d}}_0-J\mathfrak{t}⟩\ge 0,\forall p\in \Gamma .$$

Now, by Lemma 9, it follows that $\mathfrak{t}={\Pi }_{\Gamma }{\mathfrak{d}}_0$. □

We now present a consequence of Algorithm 1 by formulating a second iterative Algorithm 2, which focuses specifically on the equilibrium problem.

Algorithm 2: Iterative algorithm for equilibrium problem

Initialization: Choose $\left\{{\xi }_n\right\},\left\{{\beta }_n\right\}\subseteq (0,1)$. Let ${\mathfrak{d}}_0=\mathfrak{d}\in \mathcal{D}$ and $n=0$.

Iterative Steps: Step 1. Calculate $$\begin{array}{c}\hfill \left.\begin{array}{c}{\mathfrak{d}}_0\in \mathcal{D},{\mathcal{D}}_0:=\mathcal{D},\hfill \\ {\mathfrak{g}}_n={\Pi }_{\mathcal{D}}{J}^{-1}(J{\mathfrak{d}}_n-{\zeta }_n\mathcal{A}{\mathfrak{d}}_n),\hfill \\ {\mathfrak{q}}_n=J^{-1}({\sigma }_{n}JS{\mathfrak{d}}_n+(1-{\sigma }_n)J{\mathfrak{g}}_n),\hfill \\ {\mathfrak{v}}_n=J^{-1}((1-{\xi }_n)J{\mathfrak{d}}_n+{\xi }_{n}JT{\mathfrak{q}}_n),\hfill \\ {\varrho }_n=T_{r_n}{\mathfrak{v}}_n,\mathrm{such}\mathrm{that}\hfill \\ H({\varrho }_n,\mathfrak{q})+{\displaystyle \frac{1}{r_n}}⟨\mathfrak{q}-{\varrho }_n,J{\varrho }_n-J{\mathfrak{v}}_n⟩+b(\mathfrak{q},{\varrho }_n)-b({\varrho }_n,{\varrho }_n)\ge 0,\forall \mathfrak{q}\in \mathcal{D},\hfill \\ {\mathcal{D}}_n=\{z\in \mathcal{D}:\varphi (z,{\varrho }_n)\le \varphi (z,{\mathfrak{d}}_n)\},\hfill \\ {\mathcal{B}}_n=\{z\in \mathcal{D}:⟨{\mathfrak{d}}_n-z,J{\mathfrak{d}}_n-J{\mathfrak{d}}_0⟩\le 0\},\hfill \\ {\mathfrak{d}}_{n+1}={\Pi }_{{\mathcal{D}}_n\cap {\mathcal{B}}_n}{\mathfrak{d}}_0,\hfill \end{array}\right\}\end{array}$$ (58) where $\left\{r_n\right\}\subset [a,\infty )$, $a>0$. Assume that $\left\{{\zeta }_n\right\}$, $\left\{{\xi }_n\right\}$, and $\left\{{\sigma }_n\right\}$ are satisfying the below: (i) $\left\{{\zeta }_n\right\}\subseteq (0,\infty ),0<lim{inf}_{n\to \infty }{\zeta }_n\le lim{sup}_{n\to \infty }{\zeta }_n<{\displaystyle \frac{c^2\gamma }{2}}$, where c is the two-uniformly constant; (ii) $\left\{{\sigma }_n\right\}\in (0,1)$, ${lim}_{n\to \infty }{\sigma }_n=\sigma \in [c,d]$. Step 2. Let $n:=n+1$ and return Step 1.

We establish a strong convergence result for determining the common solution of the CGEP as presented in (1). To do so, we focus on the specific case where $N=1$.

Corollary 1. 

Let $\mathcal{D}\ne \varnothing$ be closed convex subset of a Banach space $\mathcal{X}$, that is, both two-uniformly convex and uniformly smooth, with ${\mathcal{X}}^{*}$ denoting its dual space. Suppose that $H:\mathcal{D}\times \mathcal{D}\to \mathbb{R}$ and $b:\mathcal{D}\times \mathcal{D}\to \mathbb{R}$ are maps that meet Assumptions 2 and 1, respectively. Let $\mathcal{A}:\mathcal{D}\to {\mathcal{X}}^{*}$ be a γ-ism, $\gamma \in (0,1)$, and let $T,S:\mathcal{D}\to \mathcal{D}$ be two relatively nonexpansive mappings. Suppose that ${\Gamma }_1:=\Omega \cap \mathrm{Fix}\left(T\right)\cap \mathrm{Fix}\left(S\right)\cap {\mathcal{A}}^{-1}\left(0\right)\ne \varnothing$. Then, the sequences $\left\{{\mathfrak{d}}_n\right\}$ and $\left\{{\varrho }_n\right\}$ initiated by the iterative scheme (2) converge strongly to an element $\mathfrak{t}\in {\Pi }_{{\Gamma }_1}{\mathfrak{d}}_0$.

Numerical Example

We now provide examples to illustrate the main theorem.

Example 1. 

Let $\mathcal{X}=\mathbb{R}$ and $\mathcal{D}=[0,+\infty )$. For $i=1,2,3,\dots$, consider $H_i(\mathfrak{d},\mathfrak{q})=(1+i)\mathfrak{d}(\mathfrak{q}-\mathfrak{d}),\forall \mathfrak{d},\mathfrak{q}\in \mathcal{D}$ and $b(\mathfrak{d},\mathfrak{q})=\mathfrak{d}\mathfrak{q},\forall \mathfrak{d},\mathfrak{q}\in \mathcal{D}$. It is straightforward to verify that the functions $H_i$, for each $i=1,2,3,\dots$, satisfy Assumption 2, and $b:\mathcal{D}\times \mathcal{D}\to \mathbb{R}$ satisfies Assumption 1. Let $\mathcal{A}\mathfrak{d}=2\mathfrak{d},\forall \mathfrak{d}\in \mathcal{D}$ and $T\mathfrak{d}=S\mathfrak{d}={\displaystyle \frac{1}{3}}\mathfrak{d},\forall \mathfrak{d}\in \mathcal{D}$. These mappings can also be easily checked to satisfy the requirements of Theorem 1. The execution of the algorithms involves specific parameter settings. Let $r_n=1$, $a_i={\displaystyle \frac{1}{3}},\mathrm{for}\mathrm{each}\mathrm{i}=1,2,3$. Under these configurations, the sequence produced by Algorithm 1 converges to $\left\{0\right\}\in \Gamma$.

The computations and graphical visualizations for this algorithm were carried out using MATLAB R2015a. The stopping criterion was set as $\parallel {\mathfrak{d}}_{n+1}-{\mathfrak{d}}_n\parallel <{10}^{-10}$. Various initial points ${\mathfrak{d}}_0$ were tested, and the results are summarized in Table 1 and Table 2, where we also compare our findings with those in [31,32]. Additionally, the convergence behavior is illustrated in Figure 1 and Figure 2. We observe that by taking distinct initial points, we found our result more quickly than others.

Table 1. Comparison of our main results for initial point ${\mathfrak{d}}_0=0.7$.

No. of Iterations	Main Theorem	Takahashi et al. [32]	Matsushita [31]
	cpu Time (in seconds)	cpu Time (in seconds)	cpu Time (in seconds)
1	0.225082	0.303333	0.606667
2	0.075319	0.141556	0.566222
3	0.025535	0.067632	0.541057
4	0.008713	0.032689	0.523022
5	0.002984	0.015909	0.509074
6	0.001025	0.007778	0.497762
7	0.000353	0.003815	0.488280
8	0.000121	0.001876	0.480142
9	0.000042	0.000924	0.473029
10	0.000014	0.000456	0.466722
11	0.000005	0.000225	0.461065
12	0.000002	0.000111	0.455942
13	0.000001	0.000055	0.451266
14	0.000000	0.000027	0.446968
15	0.000000	0.000014	0.442995

Table 2. Comparison of our main results for initial point ${\mathfrak{d}}_0=0.001$.

No. of Iterations	Main Theorem	Takahashi et al. [32]	Matsushita [31]
	cpu Time (in seconds)	cpu Time (in seconds)	cpu Time (in seconds)
1	0.000322	0.000433	0.000867
2	0.000108	0.000202	0.000809
3	0.000036	0.000097	0.000773
4	0.000012	0.000047	0.000747
5	0.000004	0.000023	0.000727
6	0.000001	0.000011	0.000711
7	0.000001	0.000005	0.000698
8	0.000000	0.000003	0.000686
9	0.000000	0.000001	0.000676
10	0.000000	0.000001	0.000667

Figure 1. Convergence of $\left\{{\mathfrak{d}}_n\right\}\mathrm{at}{\mathfrak{d}}_0=0.7$ [31,32].

Figure 2. Convergence of $\left\{{\mathfrak{d}}_n\right\}\mathrm{at}{\mathfrak{d}}_0=0.001$ [31,32].

Example 2. 

Let $\mathcal{X}=l_2$, where $l_2$ consists of square-summable infinite sequences of real numbers. Define $\mathcal{D}=\{\mathfrak{q}\in l_2:\parallel \mathfrak{q}\parallel \le 3\}$. For $i=1,2,3,\dots$, consider $H_i(\mathfrak{d},\mathfrak{q})=(1+i)(4\mathfrak{q}+5\mathfrak{d})(\mathfrak{q}-\mathfrak{d}),\forall \mathfrak{d},\mathfrak{q}\in \mathcal{D}$ and $b(\mathfrak{d},\mathfrak{q})=\mathfrak{d}(5\mathfrak{q}-4\mathfrak{d}),\forall \mathfrak{d},\mathfrak{q}\in \mathcal{D}$, where $\mathfrak{d}=\{{\mathfrak{d}}_1,{\mathfrak{d}}_2,\dots ,{\mathfrak{d}}_n,\dots \},\mathfrak{q}=\{{\mathfrak{q}}_1,{\mathfrak{q}}_2,\dots ,{\mathfrak{q}}_n,\dots \}$. The norm and inner product on $l_2$ are defined by $\parallel \mathfrak{d}\parallel =({\displaystyle \sum _{j=1}^{\infty }}|{\mathfrak{d}}_j{{|}^2)}^{{\displaystyle \frac{1}{2}}}$, $⟨\mathfrak{d},\mathfrak{q}⟩={\displaystyle {\displaystyle \sum _{j=1}^{\infty }}}{\mathfrak{d}}_j{\mathfrak{q}}_j$. It is straightforward to verify that the functions $H_i$, for each $i=1,2,3,\dots$, satisfy Assumption 2, and $b:\mathcal{D}\times \mathcal{D}\to \mathbb{R}$ satisfies Assumption 1. Let $\mathcal{A}\mathfrak{d}=10\mathfrak{d},\forall \mathfrak{d}\in \mathcal{D}$ and $T\mathfrak{d}=S\mathfrak{d}={\displaystyle \frac{1}{100}}\mathfrak{d},\forall \mathfrak{d}\in \mathcal{D}$. These mappings can also be easily checked to satisfy the requirements of Theorem 1. The execution of the algorithms involves specific parameter settings. Let $r_n=1$, $a_i={\displaystyle \frac{1}{3}},\mathrm{for}\mathrm{each}\mathrm{i}=1,2,3$. Under these configurations, the sequence produced by Algorithm 1 converges to $\left\{0\right\}\in \Gamma$.

The computations and graphical visualizations for this algorithm were carried out using MATLAB R2015a. The stopping criterion was set to $\parallel {\mathfrak{d}}_{n+1}-{\mathfrak{d}}_n\parallel <{10}^{-10}$. Several initial points ${\mathfrak{d}}_0$ were tested, and the convergence behavior is illustrated in Figure 3 and Figure 4.

Figure 3. Convergence of $\left\{{\mathfrak{d}}_n\right\}\mathrm{at}\mathrm{initial}\mathrm{point}{\mathfrak{d}}_0=\{0.01,0.01,\dots ,0.01,\dots \}$ [31,32].

Figure 4. Convergence of $\left\{{\mathfrak{d}}_n\right\}\mathrm{at}\mathrm{initial}\mathrm{point}{\mathfrak{d}}_0=\{0.6,0.6,\dots ,0.6,\dots \}$ [31,32].

Application in optimization problem: We explore the application of our algorithms to optimization problems. Let $\mathcal{G}:\mathcal{D}\to \mathbb{R}$ be a mapping. Define $H_i(\mathfrak{d},\mathfrak{q})=\mathcal{G}\mathfrak{q}-\mathcal{G}\mathfrak{d},\forall \mathfrak{d},\mathfrak{q}\in \mathcal{D}$. The objective is to determine $\mathfrak{v}\in \mathcal{D}$ such that

$$\mathcal{G}\mathfrak{v}\le \mathcal{G}{\mathfrak{v}}^{*},\forall {\mathfrak{v}}^{*}\in \mathcal{D}.$$

(59)

Assume that $\Omega$ is the solution set of (59) and $\Omega \ne \varnothing$. It is straightforward to verify that Assumption 2 holds. Consequently, we have $\Gamma =\Omega$.

4. Conclusions

In this work, we establish an iterative method to solve the combined generalized equilibrium and fixed point problems involving two relatively nonexpansive mappings. We prove that the generated sequence converges strongly to a shared solution within a two-uniformly convex and uniformly smooth real Banach space. We also highlight some immediate consequences of the main result. To confirm the algorithm’s efficiency, a numerical example is provided. Furthermore, the practical utility of the proposed algorithm is illustrated using comprehensive tables and figures. The practical applicability of the method is further demonstrated through two detailed tables and four illustrative figures, which provide comparisons with existing approaches and an in-depth analysis of convergence behavior. These findings emphasize the effectiveness and computational advantages of our approach. Moreover, this study generalizes and integrates several well-known results from the literature, providing a unified framework for solving various problems in optimization and computational mathematics. Theorem 3.1 presents an enhancement over the findings of [25,31,32] in the following ways:

(i)

In [25], a strong convergence theorem was established for nonexpansive mapping in a real Hilbert spaces. In contrast, our theorem extends this result to a broader context by proving convergence in two uniformly convex and uniformly smooth real Banach space and for two relatively nonexpansive mappings.

(ii)

Iterative Algorithm 1 is a generalization of the method introduced in Theorem 1 of [31,32], offering a more comprehensive approach to solving fixed point problems in the specified Banach spaces.

5. Future Direction

(i)

Our work is useful from a theoretical and applied perspective, as the result of this paper enables the further development of fixed point theory and its application by using the concept of a specific notion of Galois bundles, as has been introduced to describe fixed points [7].

(ii)

A direction for future research is also to extend the method proposed in this paper to an extragradient iterative approach for the considered problems, assuming that $\mathcal{A}$ is monotone, within the framework of a uniformly smooth and strictly convex Banach space $\mathcal{X}$, which generalizes the setting of a uniformly smooth and uniformly convex Banach space.