A viscosity-based iterative method for solving split generalized equilibrium and fixed point problems of strict pseudo-contractions

1.   Introduction

Let $Y_{1}$ and $Y_{2}$ represent real Hilbert spaces with the inner product $\langle \cdot, \cdot \rangle$ and induced norm $\| \cdot \|$. Define $Q_{1}$ and $Q_{2}$ as nonempty, closed, convex subsets of $Y_{1}$ and $Y_{2}$, respectively. This paper addresses the task of finding a unified solution for the split generalized equilibrium problem, variational inequality problem, and fixed point problem, focusing on a finite family of $\epsilon$-strict pseudo-contractive and nonexpansive mappings within real Hilbert spaces. These problems are commonly encountered in a wide range of mathematical models, where equilibrium and fixed-point conditions play a crucial role. Notable examples include applications in game theory, as seen in Nash's foundational work ^[24], image reconstruction ^[12,17], network optimization in telecommunications, public infrastructure planning ^[27], and the analysis of Nash equilibria in strategic decision-making ^[25]. The variational inequality problem (VIP) involves finding an element $s^*\in Q_{1}$ such that

where $A:Q_{1}\to Y_{1}$ is a nonlinear mapping, as introduced by Hartman and Stampacchia ^[13].

In 1994, Blum and Oettli ^[3] introduced and studied the following equilibrium problem (EP): finding $s^*\in Q_{1}$ that satisfies

where $f_{1}:Q_{1}\times Q_{1}\to \mathbb {R}$ is a bifunction, with the solution set denoted by Sol(EP(1.2)). EP(1.2) has been widely studied and extended in multiple directions over the past two decades due to its importance. For details on existence and iterative solution approximations, see ^[9,10,29,31] and the references therein. Censor et al. ^[7] introduced the split feasibility problem (SFP) for finite-dimensional Hilbert spaces, primarily for applications in phase retrieval and medical imaging, defined as:

where $B: Y_{1} \to Y_{2}$ is a bounded linear operator.

We introduce the split generalized equilibrium problem (SGEP) as follows. Let $f_{j}, \phi_{j}: Q_{j}\times Q_{j}\to \mathbb{R}, \; j = 1, 2,$ be non-linear bifunctions, with $B:Y_{1}\to Y_{2}$ as a bounded linear operator. SGEP aims to find $s^* \in Q_{1}$ that satisfies

such that

If $\phi_{1}, \phi_{2} \equiv 0$, SGEP becomes split equilibrium problem (SEP) as:

such that

We note that SGEP generalizes the multiple-set split feasibility problem and includes split variational inequalities as a special case, which further extends split zero problems and split feasibility problems for the existence and iterative approaches (see, e.g., ^{[5,8,11,15,20]}).

The fixed point problem (in short, FPP) for a map $T:Q_{1}\to Q_{1}$ is to find $v\in Q_{1}$ such that $Tv = v$. The fixed point set of $T$ is denoted by Fix(T) and ${\rm Fix}(T) = \{v\in Q_{1}:v = Tv\}$. 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 ^[1]. 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 ^[2,14,24,34].

Korpelevich ^[16] introduced the extragradient iterative method in Hilbert space $Y_{1}$ for solving VIP (1.1):

where $\alpha > 0$, $A:Q_{1}\to Y_{1}$ is a monotone and Lipschitz continuous mapping, and $P_{Q_{1}}$ denotes the metric projection onto $Q_{1}$. Under certain conditions, this sequence converges to a solution of VIP (1.1).

In 2006, Nadezkhina and Takahashi ^[21] introduced a modified form of (1.7) as follows::

By setting appropriate conditions on control sequences, they examined the weak convergence of the generated sequence toward a common solution of ${\rm Fix}(T)$ and VIP (1.1).

In the same year, Nadezhkina and Takahashi ^[22] proposed an alternative extragradient approach. This method combined the hybrid method ^[23] with the extragradient iterative approach ^[16] and was formulated as:

Using specific control sequences, they demonstrated the strong convergence of this iterative sequence to a common solution of ${\rm Fix}(T)$ and VIP (1.1). For additional generalizations of the iterative method (1.9), refer to ^[6].

Notably, only a few strong convergence theorems exist for extragradient iterative methods, other than the hybrid extragradient approach. Therefore, our primary objective is to develop a novel extragradient method distinct from the hybrid type.

A mapping $S:Q_{1}\to Y_{1}$ is defined as an $\epsilon$-strict pseudo-contractive if there exists $\epsilon\in [0, 1)$ such that:

When $\epsilon = 0$, the mapping $S$ is termed nonexpansive, and when $\epsilon = 1$, it is termed pseudo-contractive. $S$ is said to be strongly pseudo-contractive if $\exists\; \eta \in (0, 1)$ with $\langle Sv_{1} \; - \; Sv_{2}, v_{1} \; - \; v_{2} \rangle \leq \eta\|v_{1} \; - \; v_{2}\|^{2}, \; \; \forall v_{1}, v_{2}\in Q_{1}$. Thus, the $\epsilon$-strict pseudo-contractive class lies between the nonexpansive and pseudo-contractive mappings. Note that the class of strongly pseudo-contractive mappings is independent from $\epsilon$-strict pseudo-contractive (for more, see ^[4]). It is obvious that for real Hilbert space $Y_{1}$, (1.10) is equivalent to

Moreover, iterative approaches for strict pseudo-contractive are less advanced than those for nonexpansive mappings, despite Browder and Petryshyn's early work in 1967 ^[4]. This gap may be due to the additional term on the right-hand side of (1.10), which complicates the convergence analysis for algorithms that locate a fixed point of the strict pseudo-contractive $S$. However, strict pseudo-contractive offer stronger applications than nonexpansive mappings for solving inverse problems (see ^[30]). This motivates the development of iterative methods to find a common solution to SGEP((1.3) and (1.4)) and fixed-point problems for nonexpansive mappings, as well as for a finite family of $\epsilon$-strict pseudo-contractive mappings. For further reading, refer to ^[19,33] and the references therein.

Inspired by previous contributions (e.g., ^{[11,15,22,33]}), we propose a viscosity-based extragradient iterative approach for approximating solutions to split generalized equilibrium, variational inequality, and fixed point problems involving nonexpansive and $\epsilon$-strict pseudo-contractive mappings in Hilbert space. We discuss strong convergence and highlight specific results derived from our theorems, along with numerical analysis to demonstrate the significance of our findings.

This paper is structured as follows: Section 2 covers foundational concepts, lemmas, and assumptions. In Section 3, we present main results, numerical analyses, and graphical illustrations. Section 4 provides an interpretation of our findings.

2.   Preliminaries

In this section, we compile key concepts and results needed for the presentation of this work. We denote strong and weak convergence by $\rightarrow$ and $\rightharpoonup$, respectively.

For any $v_{1}\in Y_{1}\; \exists$ a unique nearest point to $v_{1}$ in $Q_{1}$ denoted by $P_{Q_{1}}v_{1}$ such that

The operator $P_{Q_{1}}$ is called the metric projection of $Y_1$ onto $Q_{1}$. This projection is nonexpansive and satisfies

Additionally, $P_{Q_{1}}v_{1}$ is characterized by $P_{Q_{1}}v_{1}\in Q_{1}$ and

This implies that

In a real Hilbert space $Y_1$, it is known that

and

Lemma 2.1. ^[18] Let $\{b_{n}\}$ be a sequence of nonnegative real numbers with a subsequence $\{b_{n_{i}}\}$ such that $b_{n_{i}} < b_{n_{i+1}}$ for all $i\in \mathbb{N}$. Then, there exists a non-decreasing sequence $\{m_{j}\}\subset \mathbb{N}$ such that $\lim_{j\to \infty}m_{j} = \infty$ and, for all sufficiently large $j\in \mathbb{N}$, the following hold:

Moreover, $m_{j}$ is the largest number $n$ in the set $\{1, 2, 3, ..., j\}$ such that $b_{n} < b_{n+1}$.

Lemma 2.2. ^[19] Assume that $D$ is a strongly positive, self-adjoint, and bounded linear operator on a Hilbert space $Y_1$ with a positive coefficient $\overline{\gamma} > 0$ and $0 < \rho\leq \|D\|^{-1}$. Then, it follows that $\|I-\rho D\|\leq 1-\rho \overline{\gamma}$.

Assumption 2.1. Let $f_{1}, \phi_{1} :Q_{1}\times Q_{1}\to \mathbb {R}$ be bimappings satisfying the following conditions:

(1) $f_{1}(v_{1}, v_{1}) = 0, \; \; \; \; \forall v_{1} \in Q_{1};$

(2) $f_{1}$ is monotone, i.e.,

(3) For each $v_{2}\in Q_{1}$, $v_{1}\rightarrow f_{1}(v_{1}, v_{2})$ is weakly upper semicontinuous;

(4) For each $v_{1}\in Q_{1}$, $v_{2}\rightarrow f_{1}(v_{1}, v_{2})$ is convex and lower semicontinuous;

(5) $\phi_{1}(., .)$ is weakly continuous and $\phi_{1}(., v_{2})$ is convex;

(6) $\phi_{1}$ is skew-symmetric, i.e.,

Now, we define $\digamma_{r}^{(f_{1}, \phi_{1})}:Y_{1}\to Q_{1}$ by

where $r$ is a positive real number.

Lemma 2.3. ^[28] Let $f_{1}, \phi_{1}$ satisfy Assumption 2.1. Suppose that for each $w \in Y_{1}$ and for each $v_{1}\in Q_{1}$, there exist a bounded subset $D_{v_{1}}\subseteq Q_{1}$ and $w_{v_{1}}\in Q_{1}$ such that for any $v_{2}\in Q_{1}\setminus D_{v_{1}}$,

Let the mapping $\digamma_{r}^{(f_{1}, \phi_{1})}$ be defined by (2.3). Then, the following properties hold:

${\rm{(i)}}$ $\digamma_{r}^{(f_{1}, \phi_{1})}(w)$ is nonempty for each $w\in Y_{1}$;

${\rm{(ii)}}$ $\digamma_{r}^{(f_{1}, \phi_{1})}$ is single-valued;

${\rm{(iii)}}$ $\digamma_{r}^{(f_{1}, \phi_{1})}$ is a firmly nonexpansive mapping, i.e., for all $w_{1}, w_{2}\in Y_{1}$,

${\rm{(iv)}}$ ${\rm Fix}(\digamma_{r}^{(f_{1}, \phi_{1})})$ = Sol(GEP(1.3));

${\rm{(v)}}$ Sol(GEP(1.3)) is closed and convex.

Further, suppose $f_{2}, \phi_{2}:Q_{2}\times Q_{2}\to \mathbb{R}$ satisfies Assumption 2.1. For $s > 0$ and $v_{1}\in Y_{2}$, define the mapping $\digamma_{s}^{(f_{2}, \phi_{2})}:Y_{2}\to Q_{2}$ as follows:

It follows that $\digamma_{s}^{(f_{2}, \phi_{2})}$ is nonempty, single-valued, and firmly nonexpansive, ${\rm Fix}(\digamma_{s}^{(f_{2}, \phi_{2})})$ = Sol(GEP(1.4)), and Sol(GEP(1.4)) is closed and convex.

Lemma 2.4. ^[28] Let $f_{1}$ and $\phi_{1}$ satisfy Assumption 2.1 and let $\digamma_{r}^{(f_{1}, \phi_{1})}$ be defined by (2.3). Then, for $v_{1}, v_{2}\in Y_{1}$ and $r_{1}, r_{2} > 0$, we have

Lemma 2.5. ^[35] Let $S: Q_{1} \to Y_{1}$ be a $\epsilon_{i}$-strictly pseudo-contractive mapping. Then, ${\rm Fix}(S)$ is closed convex and it yields that $P_{{\rm Fix}(S)}$ is well defined.

Lemma 2.6. ^[26] For any $u, v, w\in Y_{1}$, we have

where $\sigma, \; \gamma, \; \mu \in [0, 1]$ with $\sigma +\gamma +\mu = 1$.

Lemma 2.7. ^[33] For each $i = 1, 2, 3, ..., \mathbb{N}$, where $\mathbb{N}$ is a natural number, consider $S_{i}: Q_{1} \to Y_{1}$ to be a $\epsilon_{i}$-strictly pseudo-contractive mapping for some $0\leq\epsilon_{i} < 1$ with $\cap_{i = 1}^{\mathbb{N}}{\rm Fix}(S_{i})\neq\emptyset$. Let $\{\xi_{i}\}_{i = 1}^{\mathbb{N}}$ be a positive sequence with $\sum\limits_{i = 1}^{\mathbb{N}}\xi_{i = 1}^{n} = 1$. Then, $\sum\limits_{i = 1}^{\mathbb{N}}\xi_{i}S_{i}:Q_{1} \to Y_{1}$ is $\epsilon$-strictly pseudo-contractive with coefficient $\epsilon = \max\limits_{1\leq i < \mathbb{N}}\epsilon_{i}$ and ${\rm Fix}(\sum\limits_{i = 1}^{\mathbb{N}}\xi_{i}S_{i}) = \cap_{i = 1}^{\mathbb{N}}{\rm Fix}(S_{i})$.

Lemma 2.8. ^[32] Assume that $\{a_{n}\}$ is a sequence of nonnegative real numbers such that

where $\{\gamma_{n}\}$ is a sequence in $(0, 1)$ and $\{\delta_{n}\}$ is a sequence in $\mathbb{R}$ such that

${\rm{(i)}}$ $\sum\limits_{n = 1}^{\infty}\gamma_{n} = \infty$;

${\rm{(ii)}}$ $\limsup\limits_{n\to \infty}\frac{\delta_{n}}{\gamma_{n}}\leq 0 \; {\rm or} \; \sum\limits_{n = 1}^{\infty}|\delta_{n}| < +\infty$.

Then, $\lim\limits_{n\to \infty}a_{n} = 0$.

3.   Results

Suppose $f_{j}, \phi_{j}: Q_{j}\times Q_{j}\to \mathbb{R}$ for $j = 1, 2$ are nonlinear bifunctions, and $B:Y_{1}\to Y_{2}$ is a bounded linear operator. Let $A:Q_{1} \to Y_{1}$ be a $\sigma$-inverse strongly monotone mapping and $h: Q_{1} \to Q_{1}$ a $\delta$-contraction mapping. Additionally, assume $T:Q_{1} \to Y_{1}$ is a nonexpansive mapping, and for each $i = 1, 2, 3, ..., \mathbb{N}$, let $S_{i}: Q_{1} \to Y_{1}$ be an $\epsilon_{i}$-strictly pseudo-contractive mapping. Let $\{\xi_{i}^{n}\}_{i = 1}^{\mathbb{N}}$ be a finite sequence of positive numbers satisfying $\sum\limits_{i = 1}^{\mathbb{N}}\xi_{i}^{n} = 1$. The algorithm we propose is as follows:

Theorem 3.1. Let $Q_{1}$ and $Q_{2}$ be non-empty closed convex subsets of Hilbert spaces $Y_{1}$ and $Y_{2}$, respectively. Let $f_{j}, \phi_{j}: Q_{j}\times Q_{j}\to \mathbb{R}$, where $j = 1, 2,$ be non-linear bifunctions that satisfy Assumption 2.1, and $B:Y_{1}\to Y_{2}$ be a bounded linear operator. Let $A:Q_{1} \to Y_{1}$ and $h: Q_{1} \to Q_{1}$ be a $\sigma$-inverse strongly monotone mapping and $\delta$-contraction mapping, respectively. Further, assume that $T:Q_{1} \to Y_{1}$ is a nonexpansive mapping and for each $i = 1, 2, 3, ..., \mathbb{N}$, $S_{i}: Q_{1} \to Y_{1}$ is a $\epsilon_{i}$-strict pseudo-contractive mapping. Let $\{\xi_{i}^{n}\}_{i = 1}^{\mathbb{N}}$ be a finite sequence of positive numbers with $\sum\limits_{i = 1}^{\mathbb{N}}\xi_{i}^{n} = 1$. Assume $\Omega: = \cap_{i = 1}^{\mathbb{N}}{\rm Fix}(S_{i})\cap{\rm Fix}(T) \cap{\rm{Sol(SGEP(1.3-1.4))}}\cap {\rm{Sol(VIP(1.1))}} \neq\emptyset$. Then, the sequence $\{v_{n}\}$ generated by Algorithm 3.1 converges strongly to $\bar{v}\in \Omega$, where $\bar{v} = P_{\Omega}h(\bar{v})$.

For convenience, we split the proof of our main Theorem 3.1 into some lemmas as follows:

Lemma 3.1. The sequences $\{v_{n}\}$, $\{\mathfrak{z}_{n}\}$, and $\{\varrho_{n}\}$ generated by iterative Algorithm 3.1 are bounded.

Proof. We claim that $\{v_{n}\}$ is bounded. We set $s_{n} = \sigma_{n}v_{n}+\gamma_{n}T\varrho_{n}+\mu_{n}\sum\limits_{i = 1}^{\mathbb{N}}\xi_{i}^{n}S_{i}v_{n}$. Let $s^*\in \Omega$. Using the concept of non-expansivity of $I-\alpha_{n}A$ and $T$, we compute

and thus

We calculate

Thus, we have

Thus,

Assume that $\Pi : = 2\eta\langle v_{n}-s^*, B^{*}(\digamma_{r_{n}}^{(f_{2}, \phi_{2})}-I)Bv_{n}\rangle$, and we have

By (3.2)–(3.4), we get

As $\eta\in (0, \frac{1}{L})$, we have

We compute

Now,

Setting $G_{n} = \sum\limits_{i = 1}^{\mathbb{N}}\xi_{i}^{n}S_{i}$ and using Lemma 2.7, we observe that the mapping $G_{n}:Q_{1}\to Y_{1}$ is $\epsilon$-strictly pseudo-contractive with $\epsilon = \max\limits_{1\leq i < \mathbb{N}}\epsilon_{i}$ and ${\rm Fix}(G_{n}) = \cap_{i = 1}^{\mathbb{N}}{\rm Fix}(S_{i})$. Thus, by Lemma 2.6, we estimate

that implies

Thus, by (3.6), (3.7), and (3.10), we have

By induction, we get

which shows that $\{v_{n}\}$ is bounded and hence, $\{\mathfrak{z}_{n}\}$ and $\{\varrho_{n}\}$ are also bounded. □

Lemma 3.2. For each $n\geq 1$, prove that $\lim\limits_{n\to \infty}\|v_{n+1}-v_{n}\| = 0$, $\lim\limits_{n\to \infty}\|(\digamma_{r_{n}}^{(f_{2}, \phi_{2})}-I)Bv_{n}\| = 0$, $\lim\limits_{n\to \infty}\|\mathfrak{z}_{n}-v_{n}\| = 0$, and $\lim\limits_{n\to \infty}\|\varrho_{n}-\mathfrak{z}_{n}\| = 0$. Also, show that the sequence $\{v_{n}\}$ strongly converges to $s^*$, where $s^* = P_{\Omega}h(s^*)$.

Proof. As $s^*\in \Omega$, therefore we compute

Also, we estimate

where $M = \sup\limits_{n}\|h(v_{n})-s^*\|$. Using (3.9), (3.11), and (3.12), we have

Set $q_{n} = \|v_{n}-s^*\|^{2}$. Consider the two cases on $\{q_{n}\}$ as:

Case 1. For every $n\geq m_{0}$ where $m_{0}\in \mathbb{N}$, consider the sequence $\{q_{n}\}$ as decreasing, therefore it must be convergent. Applying the conditions in (3.13), we get

Notice that $\{v_{n}\}$ is bounded, therefore $\exists$ a subsequence $\{v_{n_{j}}\}$ of $\{v_{n}\}$ with $v_{n_{j}}\rightharpoonup p \in Q_{1}$ and satisfies

Define $H_{n} = \kappa_{n}v+(1-\kappa_{n})G_{n}v, \; \; \forall v\in Q_{1}$ and $\kappa_{n} \in [\delta, 1)$. Applying Lemma 2.5, $H_{n}:Q_{1}\to Y_{1}$ is nonexpansive and we have

Thus,

Applying the given conditions, we may consider that $\xi_{i}^{n}\to \xi_{i}\; {\rm as}\; n\to \infty, \; \; \forall i$. By Lemma 2.7, the map $G:Q_{1}\to Y_{1}$ with $Gv = (\sum\limits_{i = 1}^{\mathbb{N}}\xi_{i}S_{i})v, \; \forall v\in Q_{1}$, is $\epsilon$-strict pseudo-contractive and ${\rm Fix}(G) = \cap_{i = 1}^{\mathbb{N}}{\rm Fix}(S_{i})$. Applying Lemma 2.7, given the conditions and boundedness of $v_{n}$, we get

and thus

As

this yields by (3.14) and (3.18) that

Again, we notice that the map $H = tv+(1-t)Gv, \; \; \forall v\in Q_{1}$ and $t \in [\delta, 1)$, and ${\rm Fix} H = {\rm Fix}G$. Thus, we obtain

Applying (3.17)–(3.19), we have

As $v_{n}\in Q_{1}$, therefore

Using the given conditions and (3.14), we get

Applying (3.11) and (3.12), we estimate

Using (3.1), (3.5), and (3.8), we compute

By (3.21) and (3.23), we get

Applying the given condition and (3.20), we get

Next, we compute

Thus,

Using (3.22) and (3.25) in (3.21), we get

Applying the given conditions, (3.20) and (3.24) in (3.26), we get

Further, we estimate

From (3.8), we obtain

From (3.21) and (3.29), we estimate

which implies

Applying the given conditions, (3.15) and (3.20) in (3.30), we obtain

Using (3.11), (3.12), (3.28), and (3.29), we compute

Applying the given conditions, (3.14), (3.20), and (3.31), we get

Now, we show that $s^*\in {\rm Fix}(H) = {\rm Fix}(G) = {\rm Fix}(G_{n}) = \cap_{i = 1}^{\mathbb{N}}{\rm Fix}(S_{i})$. Let $s^*\notin {\rm Fix}(H)$. As $v_{n_{j}}\rightharpoonup s^*$ and $s^*\neq Hs^*$, then by the Opial condition, we obtain

which contradicts to our supposition. Hence, $s^*\in {\rm Fix}(H) = {\rm Fix}(G) = {\rm Fix}(G_{n}) = \cap_{i = 1}^{\mathbb{N}}{\rm Fix}(S_{i})$. By (3.14), we observe that $\{v_{n}\}$ and $\{\varrho_{n}\}$ have the same asymptotic behavior, and therefore $\exists$ a subsequence $\{\varrho_{n_{j}}\}$ of $\{\varrho_{n}\}$ with $\varrho_{n_{j}}\rightharpoonup s^*$. Again from (3.14) and the opial condition we have that $s^*\in {\rm Fix}(T)$. Next, we prove that $s^*\in {\rm{Sol(SGEP(1.3-1.4))}}$. Set $\tau_{n}: = v_{n}+\eta B^{*}((\digamma_{r_{n}}^{(f_{2}, \phi_{2})}-I)Bv_{n}$. Then, $\mathfrak{z}_{n} = \digamma_{r_{n}}^{(f_{1}, \phi_{1})}\tau_{n}$. For any $v\in Q_{1}$, we get

Assume $\omega_{\varsigma}: = (1-\varsigma)s^*+\varsigma v, \; \; \forall \varsigma\in (0, 1]$. As $v, s^*\in Q_{1}$, therefore $\omega_{\varsigma}\in Q_{1}$. Hence, by (3.33)

Using the given conditions (3.24) and (3.27), we get

Thus,

Assuming $\varsigma\to 0$, we obtain

This implies that $s^*\in {\rm{Sol(GEP(1.3))}}$. Further, we prove that $Bs^*\in {\rm Sol(GEP(1.4))}$. As $\|z_n-v_n\| \to 0, \; z_n \rightharpoonup s^*$ as $n \to \infty$ and $\{v_n\}$ is bounded and, $\exists$ a subsequence $\{v_{n_{j}}\}$ of $\{v_n\}$ with $v_{n_{j}}\rightharpoonup s^*$ and $Bv_{n_{j}} \rightharpoonup Bs^*$ because $B$ is a bounded linear operator.

Set $q_{n_{j}} = Bv_{n_{j}}-\digamma_{r_{n}}^{(f_{2}, \phi_{2})}Bv_{n_{j}}$. Using (3.24), we get $\lim\limits_{j\to\infty}q_{n_{j}} = 0$ and $Bv_{n_{j}}-q_{n_{j}} = \digamma_{r_{n}}^{(f_{2}, \phi_{2})}Bv_{n_{j}}$. Applying Lemma 2.3, we get

Taking the limit superior in (3.34) as $j \to \infty$, using the concept of upper semicontinuity in the first argument of $f_{2}$, and applying the given conditions, we get

which implies $Bs^*\in {\rm{Sol(GEP(1.3))}}$. Thus, $s^*\in {\rm{Sol(SGEP(1.3-1.4))}}$.

Next, we show that $s^*\in {\rm{Sol(VIP(1.1))}}$. As $\lim_{n\to \infty}\|\mathfrak{z}_{n}-\varrho_{n}\| = 0$, $\exists$ $\{\mathfrak{z}_{n_{j}}\}$ and $\{\varrho_{n_{j}}\}$ subsequences of $\{\mathfrak{z}_{n}\}$ and $\{\varrho_{n}\}$ with $\mathfrak{z}_{n_{j}}\rightharpoonup s^*$ and $\varrho_{n_{j}}\rightharpoonup s^*$.

Let

where $N_{Q_{1}}(s^*): = \{t\in Y_{1}:\langle s^*-v, t\rangle\geq 0, \; \forall t\in Q_{1}\}$ is the normal cone to $Q_{1}$ at $s^* \in Y_{1}$. Hence, $\Delta$ is maximal monotone and $0\in \Delta s^*$ $\Leftrightarrow$ $s^*\in {\rm{Sol(VIP}}(1.1))$. Let $(s^*, w)\in {\rm graph}(\Delta)$. Then, $w\in \Delta s^* = As^*+N_{Q_{1}}(s^*)$ and hence $w-As^*\in N_{Q_{1}}(s^*)$. Thus, $\langle s^*-t, w-As^*\rangle\geq 0, \; \; \forall t\in Q_{1}$. Since, $\varrho_{n} = P_{Q_{1}}(\mathfrak{z}_{n}-\alpha_{n}A\mathfrak{z}_{n})$ and $s^*\in Q_{1}$, therefore

As $\langle p-t, w-Ap\rangle\geq 0$, for all $p\in Q_{1}$ and $\varrho_{n_{j}}\in Q_{1}$, monotonicity of $A$, we obtain

Taking $j\to \infty$ and by the continuity of $A$, we get $\langle p-s^*, w \rangle \geq 0$. As $\Delta$ is maximal monotone, $s^*\in \Delta^{-1}(0)$ and hence $s^*\in {\rm{Sol(VIP}}(1.1))$. Hence, $s^*\in \Omega$.

As $s^* = P_{\Omega}h(s^*)$, therefore by (3.16)

Applying the given conditions, (3.13), (3.20), (3.35), and Lemma 2.8, we obtain $q_{n}\to 0\; \; {\rm as}\; n\to \infty$. Hence, $\{v_{n}\}$ strongly converges to $s^* = P_{\Omega}h(s^*)$.

Case 2. Consider $\{q_{t_{j}}\}$ to be a subsequence of $\{q_{t}\}$ with $q_{t_{j}} < q_{t_{j+1}}, \; \; \forall j\geq 0$. Then followed by Lemma 2.1, construct a nondecreasing sequence $\{m_{t}\}\subset \mathbb{N}$ with $m_{t}\to \infty, \; \; {\rm as}\; t\to \infty$ and $\max \{ q_{m_{t}}, q_{t}\}\leq q_{m_{t+1}}, \; \forall t$. As $r_{t}\in [c, d]\subset (0, \sigma^{-1}), \; t\geq 0$, $\sigma_{t}, \gamma_{t}, \mu_{t}\in (0, 1)$ with the given condition and (3.13), and we get

By applying the same steps as in Case 1, we get

As $\{v_{t}\}$ is bounded and $\lim\limits_{t\to \infty}\beta_{t} = 0$, we obtain from (3.15), (3.17), and (3.20) that

As $q_{m_{t}}\leq q_{m_{t+1}}, \; \forall t$, we obtain from (3.14) that

Taking $t\to \infty$, we get $q_{m_{t+1}} \to 0$. As $q_{m_{t}}\leq q_{m_{t+1}}, \; \forall t$, therefore $q_{t} \to 0\; \; {\rm as}\; t\to \infty$. Thus, $v_{t} \to 0\; \; {\rm as}\; t\to \infty$. Hence, we have proved that the sequence $\{v_{n}\}$ strongly converges to $s^* = P_{\Omega}h(s^*)$. □

Following this approach, we present several remarks that stem from the conclusions of Theorem 3.1. These remarks provide a concise overview of the theoretical results and pave the way for broader exploration and application of the proposed iterative scheme across various mathematical and computational settings.

Remark 3.1. Let $T = I$, where $I$ is the identity mapping and $\epsilon_{i} = 0$, that is, $S_{i}$ is a finite family of nonexpansive mappings in Theorem 3.1. Then, $\Omega: = {\rm Fix}(S_{i})\cap{\rm{Sol(SGEP(1.3-1.4))}}\cap {\rm{Sol(VIP(1.1))}} \neq\emptyset$.

Remark 3.2. Let $B = I$, where $I$ is the identity mapping, $Y_{1} = Y_{2}, \; Q_{1} = Q_{2}, \; f_{1} = f_{2}$, and $\phi_{1} = \phi_{2}$ in Theorem 3.1. Then, $\Omega: = \cap_{i = 1}^{\mathbb{N}}{\rm Fix}(S_{i})\cap{\rm Fix}(T) \cap{\rm{Sol(GEP(1.3))}}\cap {\rm{Sol(VIP(1.1))}} \neq\emptyset$.

4.   Numerical example

We now provide examples to illustrate the main theorem.

Example 4.1. Let $Y_{1} = Y_{2} = \mathbb R$ and $Q_{1} = Q_{2} = [0, +\infty)$. Define the mappings: $f_{1}(v_{1}, v_{2}) = v_{1}(v_{2}-v_{1}), \; \forall v_{1}, v_{2}\in Q_{1}$; $f_{2}(t_{1}, t_{2}) = t_{1}(t_{2}-t_{1}), \; \forall t_{1}, t_{2}\in Q_{2}$, and $\phi_{1}(v_{1}, v_{2}) = \phi_{2}(v_{1}, v_{2}) = v_{1}v_{2}, \; \forall v_{1}, v_{2}\in Q_{1}$. It is straightforward to verify that the functions $f_{1}, f_{2}, \phi_{1}$, and $\phi_{2}$ satisfy the conditions of Assumption 2.1. Now, consider the additional mappings: $h(v) = \frac{v}{5}, \; Av = 3v, \; v\in Q_{1}$; $B(s) = \frac{1}{2}s, \; s\in Y_{1}$; $T(v) = \frac{v}{4}, \; v\in Q_{1}$, and $S_{i}(v) = -(1+i)v, \; v\in Q_{1}, \; i = 1, 2, 3$. These mappings can also be easily checked to satisfy the requirements of Theorem 3.1. The execution of the algorithms involves specific parameter settings. Let $r_{n} = 1$, $\alpha_{n} = \{\frac{1}{5}\}, \; \eta = \frac{1}{6}$, $\beta_{n} = \{\frac{1}{10n}\}$, $\sigma_{n} = 0.7+\frac{0.1}{n^{2}}, \; \gamma_{n} = 0.2-\frac{0.2}{n^{2}}, \; \mu_{n} = 0.1+\frac{0.1}{n^{2}}$, and $\{\xi_{i}^{n}\} = \{\frac{1}{3}\}$. Under these configurations, the sequence produced by Algorithm 3.1 converges to $q = \{0\}\in \Omega$.

The computations and graphical visualizations for this algorithm were carried out using MATLAB R2015a on a standard HP laptop featuring an Intel Core i7 processor and 8 GB of RAM. The stopping criterion is set as $\|v_{n+1}-v_{n}\| < 10^{-10}$. Various initial points $v_{1}$ are tested, and the results are summarized in Tables 1 and 2, where we also compare our findings with those in ^[16,21]. Additionally, the convergence behavior is illustrated in Figures 1 and 2. Upon analyzing the figures and the table, on taking distinct initial points, we observe that our proposed algorithm tends to complete tasks more quickly, typically measured in seconds, compared to other methods. However, it is challenging to identify a clear trend from these results.

Example 4.2. Let $Y_{1} = Y_{2} = l_{2}$ be real Hilbert spaces, where $l_{2}$ consists of square-summable infinite sequences of real numbers. Define $Q_{1} = Q_{2} = \{w\in l_{2}: \|w\|\leq 3 \}$. The mappings are defined as follows: $f_{1}(u, v) = (4v+5u)(v-u), \; f_{2}(u, v) = (2v+3u)(v-u)$, where $\forall u = \{ u_{1}, u_{2}, ..., u_{n}, ...\}, v = \{ v_{1}, v_{2}, ..., v_{n}, ...\}$. The norm and inner product on $l_{2}$ are defined by: $\|u \| = (\sum\limits_{j = 1}^{\infty}|u_{j}|^{2})^{\frac{1}{2}}$, $\langle u, v \rangle = \sum\limits_{j = 1}^{\infty}u_{j}v_{j}$. Additional mappings are given as: $\phi_{1}(u, v) = (5v-4u)u, \; \phi_{2}(u, v) = (3v-2u)u$. It is straightforward to verify that the functions $f_{1}, f_{2}, \phi_{1}$, and $\phi_{2}$ satisfy the conditions of Assumption 2.1. Now, consider the additional mappings: $h(u) = \frac{1}{2}u, \; Au = 10u, \; u\in Q_{1}$; $B(s) = \frac{1}{5}s, \; s\in Y_{1}$; $T(u) = \frac{1}{100}u, \; u\in Q_{1}$, and $S_{i}(v) = \frac{1}{2(i+1)}u, \; u\in Q_{1}, \; i = 1, 2, 3$. These mappings can also be verified to satisfy the requirements of Theorem 3.1. The execution of the algorithms involves specific parameter settings. Let $r_{n} = 1$, $\alpha_{n} = \{\frac{1}{13}\}, \; \eta = \frac{1}{7}$, $\beta_{n} = \{\frac{1}{10n}\}$, $\sigma_{n} = 0.7+\frac{0.1}{n^{2}}, \; \gamma_{n} = 0.2-\frac{0.2}{n^{2}}, \; \mu_{n} = 0.1+\frac{0.1}{n^{2}}$, and $\{\xi_{i}^{n}\} = \{\frac{1}{3}\}$. Under these configurations, the sequence produced by Algorithm 3.1 converges to $q = \{0\}\in \Omega$.

The computations and graphical visualizations for this algorithm were carried out using MATLAB R2015a on a standard HP laptop featuring an Intel Core i7 processor and 8 GB of RAM. The stopping criterion is set to $\|v_{n+1}-v_{n}\| < 10^{-10}$. Several initial points $v_{1}$ are tested, and the convergence behavior is illustrated in Figures 3 and 4.

Application in optimization problems: We explore the application of our algorithms to optimization problems. Let $M_{1}:Q_{1}\to \mathbb{R}$ and $M_{2}:Q_{2}\to \mathbb{R}$ be two functions. Define $f_{1}(u_{1}, v_{1}) = M_{1}(v_{1})-M_{1}(u_{1}), \; \forall u_{1}, v_{1}\in Q1$, and $f_{2}(u_{2}, v_{2}) = M_{2}(v_{2})-M_{2}(u_{2}), \; \forall u_{2}, v_{2}\in Q2$. The objective is to determine $u\in Q1$ such that

and ensure that

Denote the solution set of these optimization problems (4.1) and (4.2) by $\Gamma$ and assume that $\Gamma \neq \emptyset$. It is straightforward to verify that Assumption 2.1, $1-4$, hold. Consequently, we have $\Gamma = \Omega$.

5.   Conclusions

In this paper, we proposed a viscosity-based extragradient iterative algorithm for solving the split generalized equilibrium problem, the variational inequality problem, and the fixed point problem for a finite family of $\epsilon$-strict pseudo-contractive and a nonexpansive mapping in Hilbert space. The strong convergence of the algorithm was established under appropriate assumptions. To demonstrate the practical applicability of the proposed algorithm, we presented results in the form of two comprehensive tables and four illustrative figures. These include comparisons with existing methods and a detailed analysis of convergence behavior, highlighting the effectiveness and efficiency of our approach.

This study extends and unifies various well-known results in the literature, offering a versatile tool for tackling a range of problems in optimization and computational mathematics.

However, the algorithm has certain limitations. Its convergence heavily depends on precise parameter tuning, which may pose challenges in practical applications. Additionally, the framework is currently restricted to Hilbert spaces, limiting its generalization to Banach spaces or other settings. Despite these limitations, the results presented in this paper extend and unify numerous previously established outcomes in this particular research domain.

Author contributions

Mohammad Farid: Writing $-$ Original Draft, Software; Saud Fahad Aldosary: Review and Editing. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025).

Conflict of interest

The authors declare that they have no competing interests.