Generalized viscosity approximation method for solving split generalized mixed equilibrium problem with application to compressed sensing

1.   Introduction

Consider $H$ a Hilbert space and $Q$ a nonempty, closed and convex subset of $H$. Let $F:Q \times Q \to \mathbb{R}$ be a bifunction, $g:Q \to H$ a nonlinear mapping and $\psi:Q \to \mathbb{R}$ a function. Then, the generalized mixed equilibrium problem (GMEP) identifies $\xi \in Q$ such that

If $g = 0$, Problem (1.1) becomes a mixed equilibrium problem to identify $\xi \in Q$ such that

If $\psi = 0$, Problem (1.2) becomes a mixed equilibrium problem (MEP), which is to identify $\xi \in Q$ such that

If $F(t, z) = 0$ for all $t, z\in Q$, Problem (1.1) becomes a generalized vector variational inequality problem which identifying $\xi \in Q$ such that

Censor et al. ^[1] proposed the split feasibility problem (SFP) for modeling inverse problems for the first time in 1994. SFPs are used in various applications, including signal processing, image restoration, computer tomography, intensity-modulated radiation therapy (IMRT) and so on; see ^[2]. SFP involves the use of a bounded linear operator for identifying a point in a nonempty closed and convex set in the space whose image corresponds to another nonempty closed and convex set in the image space.

Suppose that $H_1$ and $H_2$ are Hilbert spaces and $Q_1$ and $Q_2$ are nonempty, closed and convex subsets of $H_1$ and $H_2$, respectively. Suppose that $D : H_1 \to H_2$ is a bounded linear operator. Let $F_1:Q_1 \times Q_1 \to \mathbb{R}$, $F_2:Q_2 \times Q_2 \to \mathbb{R}$ be bifunctions, $g_1:Q_1 \to H_1$, $g_2:Q_2 \to H_2$ be nonlinear mappings and $\psi_1:Q_1 \to \mathbb{R}$, $\psi_2:Q_2 \to \mathbb{R}$ be functions. Then, the split generalized mixed equilibrium problem (SGMEP), which involves finding $\xi \in Q_1$ such that

and $w^* = D\xi\in Q_2$ solve

Let the solution set of (1.5), (1.6) and SGMEP be denoted by $GMEP(F_1, g_1, \psi_1, Q_1)$, $GMEP(F_2, g_2, \psi_2, Q_2)$ and $\Theta$, respectively.

Fan ^[3] was the first to introduce the equilibrium problem in 1972, but Blum and Oettli ^[4] made the most significant contributions to the issue in 1994. They studied variational principles and existence theorems for equilibrium problems, which have a significant role on the establishment of numerous domains in both pure and applied sciences; see ^[5,6]. These equilibrium problems serve as generalizations of various mathematical problems, including Nash equilibrium, optimization, variational inequality, minimization, saddle point problems and so on. Equilibrium problems have several applications in image reconstruction, networks, engineering, physics, game theory, economics, transportation and elasticity. As a result, the equilibrium problem has been expanded to broader issues in various ways.

GMEP was introduced by Peng and Yao ^[7] in 2008 and it includes the variational inequality problem (VIP), minimization problem (MP), fixed point problem (FPP) and many more as its special cases, see ^[8,9]. SGMEP includes split monotone variational inclusion problem (SMVIP), generalized mixed equilibrium problem (GMEP), mixed equilibrium problem (MEP), equilibrium problem (EP), variational inequality (Ⅵ), minimization problem, mixed variational inequality (MVI), split mixed equilibrium problem (SMEP), split generalized equilibrium problem (SGEP), split variational inequality (SVI), split minimization problem, split feasibility problem (SFP), split equilibrium problem (SEP) and many more as its special cases, see ^[10,11].

Moudafi and Mainge ^[12] initiated the hierarchical fixed point problem (HFPP) for a nonexpansive mapping $S$ related to another nonexpansive mapping $U$ on $Q_1$, which can be defined as finding $\xi \in$ Fix($U$), such that

Let $\Omega$ represent the solution set of HFPP. Using the normal cone's definition

one can easily see that $\xi \in$ Fix($U$) satisfies a VIP by using a criterion $S$, namely: Identify $\xi \in$ Fix($U$) and

The HFPP (1.7) is clearly identical to the problem of identifying the fixed point of a map $G = P_{F ix(U)}\circ S$, see ^[12], which includes monotone problems over equilibrium constraints, monotone variational inequality problems, and many more; see ^[13] and references therein.

The following mapping was described by Kangtunyakarn and Suantai ^[14] in 2009 as

where $S_j:Q_1 \to Q_1$ represents a finite collection of nonexpansive mappings, $\{\eta_{n, j}\}_{j = 1}^{M} \subset (0, 1]$ with $\eta_{n, j}\to \eta_j$ and $\sum_{n = 0}^{+\infty}|\eta_{n, j}-\eta_{n-1, j}| < +\infty$ for $1\le j\le M$. The mapping $K_n$ is the $K$-mapping generated by $S_1, S_2, ..., S_M$ and ${\eta_{n, 1}}, {\eta_{n, 2}}, ..., {\eta_{n, M}}$.

Recently, various common problems, namely the common solution of fixed point ^[15,16], variational inequality ^[17], variational inclusion ^[18], equilibrium ^[19,20], hierarchical fixed point ^[14] and split feasibility ^[21,22] problems with fixed point problems have been investigated by numerous authors. In 2009, Kangtunyakarn and Suantai ^[14] introduced an iterative technique and established a strong convergence theorem. In 2017, Kazmi et al. ^[23] proposed the following Krasnosel'skii-Mann iteration method to find common solutions of HFPP and SMEP.

where $K_{Q_1} = T_{r_n}^{F_1}(I-r_ng_1)$, $K_{Q_2} = T_{r_n}^{F_2}(I-r_ng_2)$ and $\delta \in (0, \frac{1}{\|D\|^2})$. In 2017, Majee and Nahak ^[24] initiated the following hybrid viscosity algorithm to find a common solution of SEP and FPP with the finite family of nonexpansive mappings.

where $K_{Q_1} = T_{r_n}^{F_1}$, $K_{Q_2} = T_{r_n}^{F_2}$, $U_{i}^n = (1- \kappa_n^i)I+ \kappa_n^iU_i$ and $\delta \in (0, \frac{1}{\|D\|^2})$. In 2018, Majee and Nahak ^[25] proposed the following viscosity approximation hybrid steepest-descent method to find a common solution of a SGEP and FPP for a finite collection of nonexpansive mappings.

where $K_{Q_1} = T_{r_n}^{F_1}$, $K_{Q_2} = T_{r_n}^{F_2}$, $U_{i}^n = (1- \kappa_n^i)I+ \kappa_n^iU_i$ and $\delta \in (0, \frac{1}{\|D\|^2})$. In 2020, Kim and Majee ^[26] proposed the following modified Krasnosel'skii-Mann type iterative method in order to identify a common solution of SMEP and HFPP of a finite collection of $k$-strictly pseudocontractive operators.

where $K_{Q_1} = T_{r_n}^{F_1}$, $K_{Q_2} = T_{r_n}^{F_2}$ and $U_{i}^n = (1- \kappa_n^i)I+ \kappa_n^iP_{Q_1}(\zeta_{n}^iI+(1-\zeta_{n}^i)U_i)$. They proved its strong and weak convergence. In 2022, Yazdi and Sababe ^[27] proposed the following method in order to identify a common solution of a GMEP, common fixed points of a finite collection of nonexpansive mappings and a general system of variational inequalities.

where $U_{i}^n = (1- \kappa_n^i)I+ \kappa_n^iU_i$. They proved its strong convergence by taking some conditions on parameters.

The fixed point problem and its applications are very important in nonlinear analysis. In recent years, significant progress has also been made in research results; see ^{[28,29,30,31,32,33]}. We have applied our result for solving compressed sensing, and one can solve various nonlinear analysis problems using our algorithm. But, the applicability of our algorithm is not limited to the problems discussed above. It can be further used to solve many important problems, for instance, uncertain fractional-order differential equation with Caputo type ^[34,35].

In recent times, numerous researchers have explored inertial-type methods, drawing inspiration from the concept of heavy ball techniques. Polyak, in their work from 1964 ^[27] introduced an iterative approach aimed at enhancing the convergence rate of iterative sequences through the incorporation of an inertial extrapolation factor. Inertial approaches typically involve a two-step iterative process, where the next iteration is determined based on the previous two iterations. In 2021, Rehman et al. ^[19] introduced an innovative approach by combining an inertial term with a subgradient extragradient algorithm. They provided a proof of weak convergence for their proposed method. In the same year, Chuasuk and Kaewcharoen ^[37] introduced a Krasnosel'skii-Mann-type inertial technique designed for solving SGMEP and HFPP involving $k$-strictly pseudocontractive operators. They demonstrated its weak convergence properties. Recently, a variety of inertial techniques have emerged to address a wide range of equilibrium problems, as documented in the literature ^[10,37]. In 2023, Ugwunnadi et al. ^[38] introduced a Krasnosel'skii-Mann-type inertial technique for solving SMVIP and HFPP. These techniques offer valuable tools for solving mathematical problems efficiently and effectively.

In this study, influenced and inspired by aforementioned work, we give a new generalized viscosity approximation method for solving an SGMEP, fixed point problem for a finite collection of nonexpansive mappings $S_j$ with $1 \le j\le M$ and an HFPP for a finite collection of $\mu_i$-strictly pseudocontractive mappings which involve finding a point $\xi\in Q_1$ and

Let the solution set of problem (1.17) be represented by $\Gamma$. We will prove strong convergence for Problem (1.17).

Remark 1.1. In this paper, our contribution can be highlighted as

$1)$ For proving the convergence result, we have embedded an inertial term which accelerates the convergence speed of the algorithm. Majee and Nahak ^[24,25], Kim and Majee ^[26], and Yazdi and Sababe ^[27] do not consider the inertial approach in their method.

$2)$ We consider $K_{Q_1} = T_{r_n}^{F_1}(I-r_ng_1)$, $K_{Q_2} = T_{r_n}^{F_2}(I-r_ng_2)$ in our algorithm, and if we take $g_1 = g_2 = 0$, then various results are special cases of our result.

$3)$ We consider $\tau$-strictly pseudocontractive mappings for solving HFPP which include various mappings like pseudocontractive and nonexpansive mappings. Additionally, $\tau$-strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems.

$4)$ Yazdi and Sababe ^[27] take the condition $\lim_{n\rightarrow +\infty}|t_{n+1}-t_{n}| = 0$, whereas our main proof does not require such a condition.

$5)$ Our result improved the results of Kazmi et al. ^[23] from the common solution of HFPP and SMEP, Majee and Nahak ^[24] from common solution of SEP and finite family of FPP, Majee and Nahak ^[25] from the common solution of a SGEP and finite family of FPP, Kim and Majee ^[26] from common solution of SMEP and HFPP to common solution of SGMEP, HFPP and finite family of FPP.

$6)$ We provide a real-life application to compressed sensing for our problem and show that our method requires less computation time to recover the signal in comparison with other methods.

$7)$ We compare our iterative technique to other approaches and present numerical examples to show the effectiveness of our algorithm.

$8)$ Our result generalizes the result of Kazmi et al. ^[23] from weak convergence to strong convergence.

In Section 1, we introduce the background and motivation for our research, highlighting the significance of GMEP and HFPP in real-world applications. Section 2 provides a comprehensive literature review, discussing previous methods and techniques proposed for solving GMEP, HFPP, and related problems. Section 3 outlines our proposed method and Algorithm 1, and we prove our main result. Section 4 discusses the practical applicability of our approach in compressed sensing. Section 5 presents numerical experiments to validate the effectiveness of the algorithm and compare it with other existing approaches.

2.   Preliminaries

In this section, we consider a real Hilbert space denoted as $H$, equipped with an inner product denoted as $\langle., .\rangle$ and the corresponding norm denoted as $\|.\|$. We assume that $Q$ is a nonempty, closed and convex subset of this real Hilbert space $H$. We will use the notations $\chi_n \rightharpoonup \chi$ and $\chi_n \rightarrow \chi$ to signify weak and strong convergence, respectively, of the sequence $\{\chi_n\}$ to the limit $\chi$. Furthermore, we denote the set of all fixed points of the mapping $U$ as ${\rm{Fix}}(U)$.

Definition 2.1. ^[39] A $\{graph (D_n) \}$ converges to $\{graph (D) \}$ in the Kuratowski-Painleve sense, if

where $D_n$ is a sequence of maximal monotone mappings and $D$ is a multivalued mapping.

Definition 2.2. ^[40] The metric projection $P_{Q}: H\rightarrow Q$ is defined as

Definition 2.3. ^[41] Suppose that $U:H\rightarrow H$ is an operator. Then $U$ is called

$1)$ contraction on $H$ if there is a constant $\mu \in [0, 1)$ and

$2)$ $L$-Lipschitz continuous on $H$ if

$3)$ monotone on $Q$ if

$4)$ $\gamma$-inverse strongly monotone on $Q$ if

$5)$ $\tau$-strictly pseudocontractive mapping if there exists $\tau \in [0, 1)$, such that

$6)$ nonexpansive if

Definition 2.4. ^[42] The monotone bifunction $g :Q\times Q\rightarrow \mathbb{R}$ on $Q$ is defined as

Definition 2.5. ^[43] The normal cone of $Q$ at $z' \in Q$ is defined as

Definition 2.6. ^[44] A bounded linear operator $D$ defined on $H$ is called strongly positive if there is a constant $\gamma > 0$ such that

Lemma 2.7. ^[45] Consider a strongly positive bounded linear, self-adjoint operator denoted as $D$. This operator has a positive coefficient $\gamma > 0$, and $0 < \rho \le \|D\|^{-1}$. Then, $\|I - \rho D\| \le 1 - \rho \gamma$.

Lemma 2.8. ^[46] For $u' \in H$ and $y' \in Q$, $y' = P_Qu'$ iff $\langle u'-y', y'-z' \rangle\leq 0$for all$z'\in Q$, where $P_Q$ is a metric projection.

Lemma 2.9. ^[47] Assume that $\{U_i\}_{i = 1}^N$ are averaged mappings with a common fixed point. Then,

Lemma 2.10. ^[48] Let $u', v', z'\in H$. Then, the following conditions hold:

$1)$ $\|\xi u'+(1-\xi)z'\|^2 = \xi \|u'\|^2+ (1-\xi) \|z'\|^2-\xi (1-\xi) \|u'-z'\|^2$for all$u', z' \in H$ and $\xi \in [0, 1].$

$2)$ $\|u'+z'\|^2 \leq \|u'\|^2+2\langle z', u'+z'\rangle$ for all$u', z'\in H.$

$3)$ (Opial's condition) Consider a sequence $y_n$ with $y_n\rightharpoonup z'$, then the following conclusions hold:

Lemma 2.11. ^[49] Suppose that $U:Q \rightarrow H$ is a $\eta$-strictly pseudocontractive mapping with $Fix(U)\ne \phi$. Consider a mapping $S$ as $Sv = \tau v+(1-\tau)Uv$ for all $v\in H$, where $\tau\in [\eta, 1)$. Then, the following conclusions hold:

$1)$ $Fix(P_{Q}U) = Fix(U).$

$2)$ $S$ is nonexpansive and $Fix(U) = Fix(S).$

Lemma 2.12. ^[50] If $\{v_n\}\subset[0, +\infty)$, $\{w_n\} \subset(0, 1)$, $\{\tau_n\}\subset(0, 1)$ and $\{\eta_n\}$ are real sequences satisfying the inequality

Suppose $\sum_{n = 0}^{+\infty}\tau_n < +\infty,$ then the conclusions stated below hold:

$1)$ If $\eta_n \le w_n M$ for some $M \ge 0$, then $\{v_n\}$ is bounded sequence.

$2)$ If $\sum_{n = 0}^{+\infty}w_n = +\infty$ and $\lim\limits_{n\rightarrow +\infty} \frac{\eta_n}{w_n} \le0$, then $\lim\limits_{n\rightarrow +\infty}v_n = 0.$

We need the following assumptions on bifunction $g: Q\to Q$ to solve the split generalized mixed equilibrium problem:

Assumption 1.

1) $g$ is monotone.

2) $g(u', u')\ge 0$ for all $u' \in Q$.

3) For each $u', w', y'\in Q$, $\limsup\limits_{t\rightarrow 0^+} g(t'u'+(1-t')w', y') \leq h(w', y')$.

4) For each $u' \in Q, y' \rightarrow h(u', y')$ is lower semi-continuous and convex.

Now, we mention the following lemma which will be utilized for solving the monotone split generalized mixed equilibrium problem.

Lemma 2.13. ^[51] Assume that $g:Q \times Q \rightarrow \mathbb{R}$ is a bifunction satisfying Assumption 1. Consider $g_1:Q \to H$ a nonlinear mapping, $\psi: Q \to \mathbb{R} \cup \{+\infty\}$ a convex and proper lower semicontinuous function. Define $S_r^{g}(w)$ as follows:

where $w\in H$ and $r > 0$. Then, the following statements hold:

$1)$ For every $u' \in H$, $S_r^g(u') \ne \phi$.

$2)$ $S_r^g$ is single-valued.

$3)$ Fix($S_r^g$) = GMEP($g, g_1, \psi$).

$4)$ Solution set GMEP($g, g_1, \psi$) is closed and convex.

$5)$ $S_r^g$ is firmly nonexpansive, i.e., for any $u', y' \in H$

Lemma 2.14. ^[52] Consider $C$ a Lipschitz maximal monotone mapping and $\{D_n\}$ a sequence of maximal monotone mappings defined on $H$. The statements are as follows:

$1)$ If $D_n$ is graph convergent to a mapping $D$ on $H$, then $C+D$ is maximal monotone and $\{C+D_n\}$ is also graph convergent to $C+D$.

$2)$ In addition, if $D$ is a maximal monotone mapping defined on $H$ and $D^{-1}0 \ne \phi$, then $\{s_n^{-1}D\}$ is graph convergent to $N_{D^{-1}0}$ as $s_n \to +\infty$.

Lemma 2.15. ^[53] Suppose $\{v_n\}$, $\{w_n\}$ and $\{\tau_n\}$ are bounded sequences in a Hilbert space $H$ such that $\{\tau_n\}\subset (0, 1)$ with $0 < \liminf_{n \to +\infty}\tau_n\le \limsup_{n \to +\infty}\tau_n < 1$. If $v_{n+1} = (1-\tau_n)w_n+\tau_nv_n$ for all integers $n \ge 0$ and $\limsup_{n \to +\infty}\|w_{n+1}-w_n\|-\|v_{n+1}-v_{n}\|\le 0$, then $\lim_{n\rightarrow +\infty}\|w_n-v_n\| = 0$.

Lemma 2.16. ^[54] Suppose that $\{U_j\}:Q \to Q$ is a finite family of nonexpansive mappings with $1\le j\le M$ and ${\bigcap^M_{j = 1} {{Fix}}(U_j)}\ne \phi$. Assume that the sequence $\{\eta_{n, j}\}$ converges to $\{\eta_{j}\}$, where $\eta_{n, j} \subset [0, 1]$ for $1\le j\le M$, $\eta_j \subset (0, 1)$ for $1\le j\le M-1$ and $\eta_M \subset (0, 1]$. Consider a $K$-mapping generated by $U_1, U_2, ..., U_M$ and $\eta_1, \eta_2, ..., \eta_n$. Let $K_n$ be the $K$-mapping generated by $U_1, U_2, ..., U_M$ and $\eta_{n, 1}, \eta_{n, 2}, ..., \eta_{n, M}$. Then, the conclusions stated below hold:

$1)$ Fix($K$) = ${\bigcap^M_{j = 1} {\rm{Fix}}(U_j)}$.

$2)$ $\lim_{n\rightarrow +\infty}\|K_nv-Kv\| = 0$ for each $v \in Q_1.$

Lemma 2.17. ^[55] Suppose $U: Q \to Q$ is a nonexpansive mapping. Let $\{v_n\}$ be a sequence in $Q$ converging weakly to $v \in Q$ and $\{(I-U)v_n\}$ converging strongly to $w \in Q$, then $(I -U)v = w$ and if $w = 0$, then $v \in Fix(U)$.

3.   Main result

In this section, we propose a new inertial generalized viscosity approximation method and prove a strong convergence theorem for solving split generalized mixed equilibrium problem, common fixed point problem of a finite family of nonexpansive mappings and hierarchical fixed point problem. Let $F_1:Q_1\times Q_1 \to \mathbb{R}, F_2 :Q_2\times Q_2 \to \mathbb{R}$ be bifunctions satisfying Assumption 1 and $F_2$ be upper semicontinuous. Suppose $D:H_1\rightarrow H_2$ is a bounded linear operator with adjoint $D^*$ such that $\delta \in (0, \frac{1}{L})$, where $L$ is the spectral radius of $D$ Let $g_1:Q_1 \to H_1$, $g_2:Q_2 \to H_2$ be $\alpha_1$, $\alpha_2$-ism mappings respectively, $h:Q_1 \to Q_1$ be a $\nu$-contraction mapping, $U_i:Q_1 \to Q_1$ be $\mu_i$-strictly pseudocontractive mappings for $1\le i \le N$, $\psi_1: Q_1 \to \mathbb{R} \cup \{+\infty\}$, $\psi_2: Q_2 \to \mathbb{R} \cup \{+\infty\}$ be convex and proper lower semicontinuous functions, $U:Q_1 \to Q_1$ be nonexpansive mapping and $S_j:Q_1 \to Q_1$ be nonexpansive mappings for $1 \le j\le M$ and $A$ is a strongly positive bounded linear self-adjoint operator on $H_1$ with constant $\bar{\gamma} > 0$ such that $0 < \gamma < \frac{\bar{\gamma}}{\nu} < \gamma+\frac{1}{\nu}$.

Algorithm 3.1. Consider $\lambda_n \subset [0, +\infty)$ with $\sum_{n = 0}^{+\infty}\lambda_n < +\infty$, $\tau \in [0, 1)$, $\varphi_n, \sigma_n, \kappa_n^i, \omega_n, \rho_n \in (0, 1)$, $\rho = \sup\{\rho_n; n\in \mathbb{{N}}\}$ with $\lim\limits_{n\rightarrow +\infty}|\varphi_{n+1}-\varphi_n| = 0$, $\lim\limits_{n\rightarrow +\infty}|\sigma_{n+1}-\sigma_n| = 0$ and $\sum_{n = 0}^{+\infty}\sigma_n < +\infty$. Set $n = 1$. Choose $x_0, x_1 \in Q_1$ and $\tau_n$ such that $0 \le \tau_n\le \bar{\tau_n},$ where

Step 1: Compute

where $K_{Q_1} = T_{r_n}^{F_1}(I-r_ng_1)$, $K_{Q_2} = T_{r_n}^{F_2}(I-r_ng_2)$ with $r_n\subset\min\{\alpha_1, \alpha_2\} = 2\alpha$, $\liminf_{n \to +\infty}r_n > 0$ and $\lim\limits_{n\rightarrow +\infty}|r_{n+1}-r_n| = 0$.

Step 2: Compute

where $U_{i}^n = (1- \kappa_n^i)I+ \kappa_n^iP_{Q_1}(\zeta_{n}^iI+(1-\zeta_{n}^i)U_i)$ with $0 \le \mu_i \le\zeta_{n}^i < 1$ and $\lim\limits_{n\rightarrow +\infty}|\kappa^i_{n+1}-\kappa_n^i| = 0$ for $i\le i\le M$.

Step 3: Evaluate

Step 4: If $\chi_{n+1} = \chi_n$, terminate the process. Otherwise, set $n: = n+1$ and return to Step 1.

Remark 3.2. From Eq (3.1) and $\sum_{n = 0}^{+\infty}\lambda_n < +\infty$, we get $\sum_{n = 0}^{+\infty}\tau_n(\chi_n-\chi_{n-1}) < +\infty$.

Theorem 3.3. Let $\Gamma$, the solution set defined in Eq (1.17) be nonempty. Suppose that Assumption 1 holds and the following conditions are satisfied:

$1)$ $\lim\limits_{n\rightarrow +\infty}\omega_n = 0$ and $\sum_{n = 0}^{+\infty}\omega_n = +\infty$,

$2)$ $0 < \liminf_{n \to +\infty}\rho_n\le \limsup_{n \to +\infty}\rho_n < 1,$

$3)$ $\lim\limits_{n\rightarrow +\infty}\frac{\|t_n-u_n\|}{\sigma_n\varphi_n} = 0.$

Then, $\{\chi_n\}$ generated by Algorithm 3.1 converges strongly to $\varrho$, where $\varrho\in \Gamma$ and $\varrho$ is the unique fixed point of contraction mapping $P_{\Gamma}(I+\gamma h-A).$

Proof. We have divided the proof in various steps. We will establish the theorem for the case when $N = 2$ and, subsequently, we will illustrate how the procedure can be readily applied to the general case.

Claim 1: The sequence $\{\chi_n\}$ is bounded.

Let $\varrho\in \bigcap^M_{j = 1} {\rm{Fix}}(S_j) \cap \Omega\cap \Theta = \Gamma$. Also, with $\omega_n\to 0$ as $n\to +\infty$, we can assume that

and then

Using Lemma 2.7, we get

As we know that $A$ is strongly positive bounded linear operator, then $\langle Av, v\rangle \ge \bar{\gamma}\|v\|^2$ and $\|A\| = \sup\{|\langle Av, v\rangle|; \|v\| = 1, v \in H_1 \}.$ Now consider

Thus, using Eq (3.5), we get $(1-\rho_n)I-\omega_n A$ is positive operator. Also,

Consider

Using Lemma 2.9, $I - r_n g_1$ is a nonexpansive mapping and hence $K_{Q_1}$ is a nonexpansive mapping. From Eq (3.2), we have

Similarly,

Using Eq (3.12), we have

From Eqs (3.2), (3.11), (3.13) and $\delta \in(0, \frac{1}{L})$, we have

From Eqs (3.9) and (3.15), we have

Using Lemmas 2.9 and 2.11, we have $U_{2}^nU_{1}^n\varrho = \varrho$. From Eqs (3.3) and (3.16), we have

Using Eqs (3.4) and (3.18), we have

Let $v_n = \|\chi_n- \varrho\|$, $w_n = \omega_n(\bar{\gamma}-\gamma \nu)$, $\eta_n\le w_n M = \frac{\omega_n(\bar{\gamma}-\gamma \nu)\|\gamma h(\varrho)- A\varrho\|}{(\bar{\gamma}-\gamma \nu)}$ and $\tau_n^1 = \tau_n\|\chi_n-\chi_{n-1}\|+\sigma_n\|U\varrho-\varrho\|$. Thus, we have

Using Lemma 2.12, Remark 3.2, condition $(i)$ and $\sum_{n = 0}^{+\infty}\sigma_n < +\infty$, we get that $\{v_n\}$ is bounded, which implies $\{\|\chi_n- \varrho\|\}$ is bounded. Hence, $\{\chi_n\}$ is bounded. Consequently, $\{t_n\}$, $\{u_n\}$, $\{w_n\}$, $\{y_n\}$, $\{h(K_n \chi_n)\}$ are also bounded.

Claim 2: $\limsup_{n \to +\infty} (\|f_{n+1}-f_{n}\|-\|\chi_{n+1}-\chi_n\|)\le0.$

Consider

where $M = \sup\{\tau_n\|\chi_n-\chi_{n-1}\|+2\|\chi_n-\varrho\|; n\in \mathbb{{N}} \}$.

Also,

As $y_n = T_{r_n}^{F_1}(I-r_ng_1)(w_n)$ and $y_{n+1} = T_{r_{n+1}}^{F_1}(I-r_{n+1}g_1)(w_{n+1})$, we get

and

Putting $z = y_{n+1}$ and $z = y_n$ in Eqs (3.22) and (3.23), respectively, we get

and

Adding Eqs (3.24) and (3.25) and using the monotonicity of $F_1$, we get

Upon rearranging the terms in Eq (3.26), we get

Hence, we get

Subsequently, we have

Assume that for any $n > 0$, there is a real number $c_1$ such that $r_n > c_1 > 0$. From Eq (3.29), we get

where $M_1 = \sup \{\|w_{n+1}-y_{n+1}\|; n\in \mathbb{{N}}\}$. In a similar way, we can deduce that

where $M_2 = \sup \{\|l_{n+1}-Dy_{n+1}\|; n\in \mathbb{{N}}\}$, $l_{n} = T_{r_{n}}^{F_2}(I-r_{n}g_2)Dy_n$ and $l_{n+1} = T_{r_{n+1}}^{F_2}(I-r_{n+1}g_2)Dy_{n+1}.$ Also,

Using the inequality $\sqrt{|c'|+|d'|}\le \sqrt{|c'|}+\sqrt{|d'|}$, we get

Using Eq (3.30), we get

Choose constant $M_3$ such that

From Eqs (3.34) and (3.35), we have

Let $s_n = \sigma_nUu_n+(1-\sigma_n)U_{2}^nU_{1}^nu_n$, then $t_n = (1-\varphi_n)u_n+\varphi_ns_n$, and we estimate

In a similar way,

Using Eqs (3.37) and (3.38), we get

Now, consider

Using the definition of $U_i^n$, we estimate that

where $J_1 = \|P_{Q_1}(\zeta_{n}^1I+(1-\zeta_{n}^1)U_1)u_{n+1}\|+\|u_{n+1}\|$. As $\lim\limits_{n\rightarrow +\infty}|\kappa_{n+1}^1-\kappa_{n}^1| = 0$, $\{u_n\}$ and $\{P_{Q_1}(\zeta_{n}^1I+(1-\zeta_{n}^1)U_1)u_{n+1}\}$ are bounded, we have

Similarly,

where $J_2 = \|U_1^nu_{n+1}\|+\|P_{Q_1}(\zeta_{n}^2I+(1-\zeta_{n}^2)U_1)u_{n+1}\|$. Using Eqs (3.36), (3.39), (3.40), (3.41) and (3.43), we get

Let $\chi_{n+1} = (1-\rho_n)f_n+\rho_n\chi_n$. Then, $f_n = \frac{\chi_{n+1}-\rho_n\chi_n}{1-\rho_n}$ and

Now, calculating $\|K_{n+1}t_{n}-K_{n}t_{n}\|$ for each $j \in\{2, 3, ..., M-2\}$, we get

where $M_4 = \sup\{\sum_{j = 2}^{M}\|S_jT_{n, j-1}\|+\|T_{n, j-1}t_n\|+\|S_1t_n\|+\|t_n\|\}$.

Consider

Also,

From Eqs (3.44), (3.45) and (3.48), we get

Using Remark 3.2, $\lim\limits_{n\rightarrow +\infty}|\kappa_{n+1}^i-\kappa_{n}^i| = 0$, for $i = 1, 2$, $\lim\limits_{n\rightarrow +\infty}\omega_n = 0$, $\lim\limits_{n\rightarrow +\infty}|r_{n+1}-r_n| = 0$, $\lim\limits_{n\rightarrow +\infty}|\sigma_{n+1}-\sigma_{n}| = 0$, $\lim\limits_{n\rightarrow +\infty}|\varphi_{n+1}-\varphi_n| = 0$ and taking lim sup in Eq (3.49), we have

Claim 3: $\lim_{n\rightarrow +\infty}\|\chi_n-\chi_{n-1}\| = \lim\limits_{n\rightarrow +\infty}\|f_n-\chi_n\| = 0$ and $\lim_{n\rightarrow +\infty}\|t_n-u_{n}\| = \lim\limits_{n\rightarrow +\infty} \|w_n-\chi_n\| = \lim\limits_{n\rightarrow +\infty}\|\chi_n-K_nt_n\| = 0.$

Using Lemma 2.15, we have

Also, $\chi_{n+1} = (1-\rho_n)f_n+\rho_n\chi_n$, which implies $\|\chi_{n+1}-\chi_n\| = \|(1-\rho_n)(f_n-\chi_n)\|$. Now using Eq (3.50), we have

From Eq (3.2), we have $\|w_n-\chi_n\| = \|\tau_n(\chi_n-\chi_{n-1})\|$. Taking the limit $n \to +\infty$, we get

Also,

which implies

Taking the limit $n\to +\infty$ and using $\lim\limits_{n\rightarrow +\infty}\omega_n = 0$, we have

Consider

Also,

which implies

Using Eq (3.14), we have

which implies

Using Eqs (3.11), (3.14), (3.17) and (3.55), we get

Using Eqs (3.20) and (3.60), we get

and using Eqs (3.20), (3.59) and (3.60), we estimate

Using Eqs (3.14), (3.20) and (3.57), we have

Using Eqs (3.20), (3.60) and (3.57), we have

From Eqs (3.54), (3.62) and (3.64) and using $\lim_{n\rightarrow +\infty}\sigma_n < +\infty$, $\lim_{n\rightarrow +\infty}\|\chi_n-\chi_{n-1}\| = 0$, $\lim_{n\rightarrow +\infty}\omega_n = 0$ and Remark 3.2, we get

Using Eqs (3.54), (3.62) and (3.66) and using $\lim_{n\rightarrow +\infty}\sigma_n < +\infty$, $\lim_{n\rightarrow +\infty}\|\chi_n-\chi_{n-1}\| = 0$, $\lim_{n\rightarrow +\infty}\omega_n = 0$ and Remark 3.2, we get

Using the triangle inequality and Eqs (3.52) and (3.67), we get

As we know that $U$ and $U_{2}^nU_{1}^n$ are nonexpansive mappings and $\{u_n\}$ is bounded, one may suppose that there is a nonnegative real number $k$ such that $\|Uu_n-U_{2}^nU_{1}^nu_n\|\le k$ for all $n\ge0$. Now, consider

Subsequently, we have

Using condition $(iii)$ and $\lim_{n\rightarrow +\infty}\sigma_n < +\infty$, we get

Also,

Hence, we have

From Eq (3.71) and $\lim_{n\rightarrow +\infty}\sigma_n = 0$, we have

Claim 4: $u' \in GMEP(F_1, g_1, \psi_1, Q_1)$.

As we know that $\{u_n\}$ is bounded, there exists a subsequence $\{u_{n_i}\}$ of $\{u_n\}$ that converges weakly to some $u'\in Q_1$. Also, from Eq (3.54), we have $K_nt_{n_i}\rightharpoonup u'$. Now, we show $u' \in GMEP(F_1, g_1, \psi_1, Q_1)$. Using Lemma 2.13, we have

Using the monotonicity of $F_1$, we have

Replacing $n$ by $n_k$, we have

Let $m$ with $0 < m \le 1$ and $u \in Q_1$ satisfying $u_m = mu+(1-m)u'$, then $u_m \in Q_1$ and, from the above inequality, we have

Using the Lipschitz continuity of $g_1$ and Eq (3.67), we have $\|g_1u_{n_k}-g_1z_{n_k}\| = 0$ as $k\to +\infty$. Further, as $F_1$ is monotone and $\phi_1$ is convex and lower semicontinuous, the above equation implies

Consider for $m > 0$

Taking $m \to 0_+$, we get

Hence, $u' \in GMEP(F_1, g_1, \psi_1, Q_1)$.

Claim 5: Now we will prove $Du' \in GMEP(F_2, g_2, \psi_2, Q_2)$.

As $D$ is a bounded linear operator, and using Eqs (3.66) and (3.67), this implies $Dz_{n_k}\rightharpoonup Du'$. Taking $l'_{n_k} = Dz_k-T_{r_{n_k}}^{F_2}(I-r_{n_k}g_2)Dz_k$ and using Eq (3.66) we have $\lim_{n\rightarrow +\infty}l'_{n_k} = 0$ and $T_{r_{n_k}}^{F_2}(I-r_{n_k}g_2)Dz_k = Dz_k-l'_{n_k}$. Now, using Lemma 2.13, we get

As $F_2$ is upper semicontinuous, we use $\limsup$ in the above equation as $k\to +\infty$. Also, with $\liminf_{n \to +\infty} r_n > 0$, we have

Hence, $Du' \in GMEP(F_2, g_2, \psi_2, Q_2)$.

Claim 6: Now we will prove $u' \in Fix (K)$.

Assume that $K$ is the $K$-mappings generated by $S_1, S_2, ..., S_M$ and $\eta_1, \eta_2, ..., \eta_M$. Now, using Lemma 2.16, we have

We have to show $u'\in$Fix($K$). We will do it by contradiction. Assume that $u'\notin$Fix($K$), which implies $Ku'\neq u'$. Now using opial conditions, we get

which is a contradiction. Thus, $u'\in$Fix($K$)$= {\bigcap^M_{j = 1} {\rm{Fix}}(S_j)}$.

Claim 7: We claim that $u' \in {\rm{Fix}}(U_1)\cap {\rm{Fix}}(U_2)$. As the sequence $\{\chi_n\}$ is bounded, then there is a subsequence $\{x_{n_k}\}$ of $\{\chi_n\}$ such that $\{x_{n_k}\}\rightharpoonup u'$ as $k\to +\infty.$ Also, $\kappa_{n}^i$ is bounded, which implies $\kappa_{n_k}^i\to \kappa_{+\infty}^i$ for $i = 1, 2$ and $k\to +\infty$, where $0 < \kappa_{+\infty}^i < 1$. Consider $U_{i}^{+\infty} = (1- \kappa_{+\infty}^i)I+ \kappa_{+\infty}^iP_{Q_1}(\zeta_{+\infty}^iI+(1-\zeta_{+\infty}^i)U_i)$ for $i = 1, 2$. Using Lemma 2.11, we conclude that ${\rm{Fix}}(P_{Q_1}(\zeta_{+\infty}^iI+(1-\zeta_{+\infty}^i)U_i)) = {\rm{Fix}}(U_i).$ As $P_{Q_1}(\zeta_{+\infty}^iI+(1-\zeta_{+\infty}^i)U_i)$ is a nonexpansive mapping, ${\rm{Fix}}(U_i^{+\infty}) = {\rm{Fix}}(U_i)$ and $U_i^{+\infty}$ is averaged. Further,

Using Lemma 2.9, we get

Additionally,

Subsequently, we get

where $K$ is any bounded subset of $H_1$. Note that

where $K_1$ and $K_2$ are bounded subsets including $\{U_1^{n_k}x_{n_k}\}$ and $\{x_{n_k}\}$ respectively. From Eqs (3.71), (3.86) and (3.87), we conclude that

Subsequently, using Lemma 2.17, we get $u'\in {\rm{Fix}}(U_1^{+\infty}U_2^{+\infty}) = {\rm{Fix}}(U_1)\cap {\rm{Fix}}(U_2)$.

Claim 8: Next, we will show $u' \in\Omega$. From Eq (3.3), we get

and hence

Using Lemma 2.14 ($i$), the sequence $\bigg\{\frac{(1-\sigma_n)}{\sigma_n}(I-U_{2}^nU_{1}^n)\bigg\}$ is graph convergent to $N_{ {\rm{Fix}}(U_1)\cap {\rm{Fix}}(U_2)}$, and using Lemma 2.14 ($ii$), one can conclude that the sequence $(I-U)+\bigg\{\frac{(1-\sigma_n)}{\sigma_n}(I-U_{2}^nU_{1}^n)\bigg\}$ is graph convergent to $(I-U)+N_{ {\rm{Fix}}(U_1)\cap {\rm{Fix}}(U_2)}$. Replacing $n$ by $n_j$ and taking the limit $j \to +\infty$ in Eq (3.90) and using condition ($iii$), we have

By substituting $n_j$ for $n$ and taking the limit as $j$ tends to infinity in Eq (3.90) while utilizing condition ($iii$), we obtain:

which implies $u' \in\Omega$. From Claims 5–8, we have $u' \in\Gamma$.

Claim 9: Now we show $\limsup_{n \to +\infty}\langle (\gamma h-A)u', \chi_n-v \rangle \le 0$, where $u' = P_{\Gamma}(I+\gamma h-A)u'$. As the sequence $\{t_n\}$ weakly converges to $u'$ and using Lemma 2.8, we have

As $h$ is a contraction mapping, one can easily prove $P_{\Gamma}(I+\gamma h-A)$ is also a contraction mapping from $H_1$ to itself. Using the Banach contraction principle, there exists a $u'\in H_1$ such that $u' = P_{\Gamma}(I+\gamma h-A)u'$.

Claim 10: Next we show $\chi_n \to \varrho$.

Consider

which implies

where $M_5 = \sup\{\|\chi_n-\varrho\|: n\in \mathbb{N}\}$. Hence, we get

where $a_n = \|\chi_n-\varrho\|^2$, $b_n = \omega_n(\bar{\gamma}-\gamma\nu)$, $d_n = \frac{1}{(\bar{\gamma}-\gamma\nu)}(\bar{\gamma}-\gamma\nu)\omega_n\langle\gamma h\varrho-A\varrho, \chi_{n+1}-\varrho \rangle$ and $c_n = M_5(\tau_n\|\chi_n-\chi_{n-1}\|+\sigma_n\|U\varrho-\varrho\|)$. From Remark (3.2) and $\sum_{n = 0}^{+\infty}\sigma_n < +\infty$, we have $\sum_{n = 0}^{+\infty}c_n < +\infty$. From Eq (3.92), we get $\limsup_{n \to +\infty}\frac{d_n}{b_n} \le 0$. Also, $\sum_{n = 0}^+\infty b_n = +\infty$ and from Lemma 2.12 ($ii$), we obtain

Therefore, $\chi_n\to \varrho.$ □

Corollary 3.4. Let $x_0, x_1 \in Q_1$ and $\tau_n$ such that $0 \le \tau_n\le \bar{\tau_n}.$ Define a sequence $\left\lbrace \chi_n \right\rbrace$ as:

where $K_{Q_1} = T_{r_n}^{F_1}(I-r_ng_1)$, $\liminf_{n \to +\infty}r_n > 0$ and $\lim\limits_{n\rightarrow +\infty}|r_{n+1}-r_n| = 0$, $U_{i}^n = (1- \kappa_n^i)I+ \kappa_n^iP_{Q_1}(\zeta_{n}^iI+(1-\zeta_{n}^i)U_i)$ with $0 \le \mu_i \le\zeta_{n}^i < 1$ and $\lim\limits_{n\rightarrow +\infty}|\kappa^i_{n+1}-\kappa_n^i| = 0$ for $i\le i\le M$. Also $\lambda_n \subset [0, +\infty)$ with $\sum_{n = 0}^{+\infty}\lambda_n < +\infty$, $\tau \in [0, 1)$, $\varphi_n, \sigma_n, \kappa_n^i, \omega_n, \rho_n \in (0, 1)$, $\rho = \sup\{\rho_n; n\in \mathbb{{N}}\}$ with $\lim\limits_{n\rightarrow +\infty}|\varphi_{n+1}-\varphi_n| = 0$, $\eta_{n, j}\to \eta_j$, $\sum_{n = 0}^{+\infty}|\eta_{n, j}-\eta_{n-1, j}| < +\infty$, $\lim\limits_{n\rightarrow +\infty}|\sigma_{n+1}-\sigma_n| = 0$ and $\sum_{n = 0}^{+\infty}\sigma_n < +\infty$. Under the assumptions that conditions $(i)$–$(iii)$ of Theorem 3.3 are satisfied, we can conclude that the sequence ${\chi_n}$ generated by Eq (3.97) strongly converges to the element $\xi\in \Delta$. This element $\xi$ represents the unique solution to the fixed-point problem associated with the contraction mapping $P_{\Delta}(I+\gamma h-A)$. In other words, $\xi$ is the solution to the variational inequality stated below:

Proof. By taking $D = O$, $H_1 = H_2$, $Q_1 = Q_2$, $F_1 = F_2$, $g_1 = g_2$ and $\psi_1 = \psi_2$ in Theorem 3.3, we get the required conclusion. □

Corollary 3.5. Let $x_0, x_1 \in Q_1$ and $\tau_n$ such that $0 \le \tau_n\le \bar{\tau_n}.$ Define a sequence $\left\lbrace \chi_n \right\rbrace$ as:

where $K_{Q_1} = T_{r_n}^{F_1}(I-r_ng_1)$, $\liminf_{n \to +\infty}r_n > 0$ and $\lim\limits_{n\rightarrow +\infty}|r_{n+1}-r_n| = 0$, $U_{i}^n = (1- \kappa_n^i)I+ \kappa_n^iP_{Q_1}(\zeta_{n}^iI+(1-\zeta_{n}^i)U_i)$ with $0 \le \mu_i \le\zeta_{n}^i < 1$ and $\lim\limits_{n\rightarrow +\infty}|\kappa^i_{n+1}-\kappa_n^i| = 0$ for $i\le i\le M$. Also $\lambda_n \subset [0, +\infty)$ with $\sum_{n = 0}^{+\infty}\lambda_n < +\infty$, $\tau \in [0, 1)$, $\varphi_n, \sigma_n, \kappa_n^i, \omega_n, \rho_n \in (0, 1)$, $\rho = \sup\{\rho_n; n\in \mathbb{{N}}\}$ with $\lim\limits_{n\rightarrow +\infty}|\varphi_{n+1}-\varphi_n| = 0$, $\eta_{n, j}\to \eta_j$, $\sum_{n = 0}^{+\infty}|\eta_{n, j}-\eta_{n-1, j}| < +\infty$, $\lim\limits_{n\rightarrow +\infty}|\sigma_{n+1}-\sigma_n| = 0$ and $\sum_{n = 0}^{+\infty}\sigma_n < +\infty$. Under the assumptions that conditions $(i)$–$(iii)$ of Theorem 3.3 are satisfied, we can conclude that the sequence ${\chi_n}$ generated by Eq (3.97) strongly converges to the element $\xi\in \Delta$. This element $\xi$ represents the unique solution to the fixed-point problem associated with the contraction mapping $P_{\Delta}(I+\gamma h-A)$. In other words, $\xi$ is the solution to the variational inequality stated below:

Proof. By taking $D = I$, $H_1 = H_2$, $Q_1 = Q_2$, $F_1 = F_2$, $g_1 = g_2$ and $\psi_1 = \psi_2$ in Theorem 3.3, we get the required conclusion. □

Remark 3.6.

1) Theorem 3.3 generalizes and enhances the findings of Rizvi ^[56] from a nonexpansive mapping to a finite family of nonexpansive mappings. Furthermore, our findings extend the outcomes of Rizvi ^[56] from a common solution of SMEP and HFPP to a common solution of HFPP, SGMEP and FPP for a finite collection of nonexpansive mappings.

2) Theorem 3.3 generalizes the Husain and Singh ^[57] result from a common solution of SMEP and HFPP to a common solution of HFPP, SGMEP and FPP for a finite family of nonexpansive operators. In addition, we consider HFPP for a finite collection of strictly pseudocontractive operators, which is more general than the nonexpansive mappings taken in Husain and Singh ^[57] result.

3) Theorem 3.3 generalizes and enhances the findings of Kim and Majee ^[26] (Theorem 3.6) from a common solution of SEP and HFPP to a common solution of HFPP, SGMEP and FPP for a finite collection of nonexpansive operators.

4) Theorem 3.3 generalizes and enhances the result of Majee and Nahak ^[24] from a common solution of SEP and HFPP to a common solution of HFPP, SGMEP and FPP for a finite collection of nonexpansive operators.

4.   Application in compressed sensing in signal processing

Compressed sensing in signal processing ^[58] can be represented by the following linear equation:

Here, $\epsilon$ is the noise, $D$ is an $M \times N$ matrix with $M < N$, $x \in \mathbb{R}^N$ is a recovered vector with $m$ non-zero components and $y \in \mathbb{R}^M$ is the observed data. The problem described in Eq (4.1) can be considered as a LASSO problem:

Here, $u > 0$ is constant.

In this case, a uniform distribution in the interval [-1, 1] is used to construct the sparse vector $x \in \mathbb{R}^N$, which has $m$ non-zero members. A normal distribution with a zero mean and a unit variance is used to produce the matrix $D$. As $\delta\in(0, 1/L)$, it is randomly generated in MATLAB. By applying white Gaussian noise with a signal-to-noise ratio (SNR) of 40, the observation $y$ is produced. The process starts with an initial point $x_1 = ones_{N \times 1}$ and $u = m$. Specifically, the LASSO problem can be seen as an SFP (Split Feasibility Problem) if $Q_1 = \{x \in \mathbb{R}^N : \|x\|_1 \leq u\}$ and $Q_2 = \{y\}$. In this connection, we can solve Eq (4.2) using the CQ technique. The stopping criterion is given by the mean squared error (MSE):

where $\Lambda$ is a tolerance and $\chi_n$ is the estimated signal of $x$. Note that if in Problem (1.5)–(1.6) we set $g_1 = g_2 = \psi_1 = \psi_2 = 0$, we obtain the split equilibrium problems (SEQ) and if, in addition, $F_1(v, w) = I_{Q_1}(v)-I_{Q_1}(w)$ and $F_2(v', w') = I_{Q_2}(v')-I_{Q_2}(w')$, where $I_{Q_1}$ and $I_{Q_2}$ are identity operators on $Q_1$ and $Q_2$ respectively, then SEQ becomes SFP. Hence, we can apply our algorithm to the SFP with the resolvent operator $T_r^{F_1}$ and $T_r^{F_2}$ being the projection onto $Q_1$ and $Q_2$, respectively. In order to implement our algorithm, we choose the following parameters: $\lambda_n = \frac{1}{n^2}$, $\tau = 0.5$, $U = U_i = S_j = I$ for all $i, j$ so that $K_n = I$ (Identity mapping), $A = I$ (Identity operator), $\rho_n = \omega_n = \frac{1}{n+1}$ and $h(x) = x/2$ so that $\gamma = 1$ and $\Lambda = 10^{-10}.$

Figure 1 represents the original signal, observed value and recovered signals by Algorithm 3.1, the Chuasuk Algorithm ^[37], the Kim Algorithm ^[26] and the Majee Algorithm ^[24]. Table 1 and Figure 2 give the mean square error of Algorithm 3.1, the Chuasuk Algorithm ^[37], the Kim Algorithm ^[26] and the Majee Algorithm ^[24]. The experiment shows that all three methods are effective in recovering the signal, however, the time taken by the Chuasuk Algorithm ^[37] (Average time = 7.8654s), the Majee Algorithm ^[24] (Average time = 10.9854s) and the Kim Algorithm ^[26] (Average time = 13.5864s) is more than the time taken by the proposed algorithm (Average time = 5.8754s).

5.   Numercial examples

In this section, we first conduct a comparison of the convergence rates between our algorithm and those presented in the Chuasuk Algorithm ^[37], the Kim Algorithm ^[26] (Theorem 3.6), and the Majee Algorithm ^[24]. We implemented the proposed algorithm using MATLAB $9.10.0$ ($R2021a$) on a laptop equipped with an Intel Core i5 CPU running at $1.60 GHz$, $256$ GB SSD and $1$ TB hard-disk capacity. The operating system used is Microsoft Windows $11$, version $21 H2$. Secondly, we present numerical experiments related to compressed sensing.

Example 5.1. Assume that $H_1 = H_2 = \mathbb{R}^5$, and

Let $g_1: Q_1 \to \mathbb{R}$, $g_2: Q_2 \to \mathbb{R}$ be inverse strongly monotone mappings defined by $g_1(x) = 3x$ and $g_2(x) = 3x$. Suppose $F_1:Q_1\times Q_1 \to \mathbb{R}, F_2:Q_2\times Q_2 \to \mathbb{R}$ are the bifunctions defined by $F_1(x, y) = F_2(x, y) = \langle Px+Qy+q, y-x\rangle$, arising from Nash Cournot Oligopolistic market equilibrium model ^[17] where $q\in \mathbb{R}^5$ and $P, Q \in \mathbb{R}^{5\times 5}$ are two matrices of order 5 with $Q$ being symmetric, positive semidefinite and $Q-P$ being negative semidefinite. Obviously, bifunction $g$ satisfies Assumption 1 and $A: \mathbb{R}\to \mathbb{R}$ is defined by $A(x) = x$ with constant $\bar{\gamma} = 1$. Let $\psi_1: Q_1 \to \mathbb{R} \cup \{+\infty\}$, $\psi_2: Q_2 \to \mathbb{R} \cup \{+\infty\}$ be defined by $\psi_1(x) = \psi_2(x) = 0$, $D: \mathbb{R}\to \mathbb{R}$ be defined by $D(x) = x$, $D^*(x) = x$, then $T_r^{F_1}(x) = T_r^{F_2}(x) = ((P+Q+3I)r+I)^{-1}x$. Let $h:Q_1 \to Q_1$ be $\frac{1}{2}$-contraction defined by $h(x) = \frac{x}{2}$ and $S_j:Q_1 \to Q_1$ be pseudocontractive mappings defined by $S_j(x) = \frac{x}{6(j+1)}$, for $j = 1, 2$. Assume that $U:Q_1 \to Q_1$ and $U_i:Q_1 \to Q_1$ are nonexpansive mappings defined by $U(x) = \frac{x}{4}$ and $U_i(x) = \frac{x}{10i}$, for $i, = 1, 2$, $x = (x_1, x_2, x_3, x_4, x_5)^{T}$. Choose $\delta = \frac{1}{16}$, $r_n = 1$, $\lambda_n = \frac{1}{n^2}$, $\tau = 0.5$, $\kappa_n^i = \frac{n+i}{n+5+i}$, $\zeta_{n}^i = \frac{1}{20}$, $\eta_n^j = \frac{1}{20n+5j}$ for $i, j = 1, 2$, $\sigma_n = \frac{1}{(n+1)^2}$, $\varphi_n = \frac{5}{6}$, $\rho_n = \frac{n+1}{2(n+50)}$ and $\omega_n = \frac{1}{n+200}$. One can easily see that Fix$(\Gamma) = \{0\}\ne \phi.$ We can obtain $K_{Q_1}(x) = K_{Q_2}(x) = -2(P+Q+4I)^{-1}x$. Take $P = I_5$, $Q = 0_{5\times 5}$, $x_0 = (0.5, 0.5, 0.5, 0.5, 0.5)^{T}, x_1 = (0.8, 0.8, 0.8, 0.8, 0.8)^{T}$ and $q = [0, 0, 0, 0, 0]^\mathit{T}$. We take a stopping criterion of $E_n = \|\chi_n-\chi_{n+1}\| < 10^{-4}$ and plot the graphs between number of iterations $n$ and errors $E_n$. We do comparative analysis of the numerical result of Algorithm 3.1 with the Chuasuk Algorithm ^[37], the Kim Algorithm ^[26] (Theorem (3.6)) and the Majee Algorithm ^[24]. Table 2 and Figure 3 represent the comparative analysis.

Example 5.2. Assume that $H_1 = H_2 = l_2$ are real Hilbert spaces with square-summable infinite sequences of real numbers as its elements and $Q_1 = Q_2 = \{v\in l_2:\|v\| \le 3 \}$. Let $g_1: [-5, 5] \to \mathbb{R}$, $g_2: [-5, 5] \to \mathbb{R}$ be ism mappings defined by $g_1(x) = 10x$ and $g_2(x) = 2x$. Suppose $F_1:Q_1\times Q_1 \to \mathbb{R}, F_2:Q_2\times Q_2 \to \mathbb{R}$ are the bifunctions defined by $F_1(x, y) = -5x^2+xy+4y^2$, $F_2(x, y) = -3x^2+xy+2y^2$ for all $x = (x_1, x_2, x_3, ..., x_n, ...)$ and $y = (y_1, y_2, y_3, ..., y_n, ...)$ with $\|.\|:l_2 \rightarrow \mathbb{R}$ and $\langle., .\rangle:l_2\times l_2 \rightarrow \mathbb{R}$ given by $\|x\| = (\sum_{k = 1}^{+\infty}|x_k|^2)^\frac{1}{2}$ and $\langle x, y\rangle = \sum_{k = 1}^{+\infty}x_ky_k$, where $x = \{x_k\}_{k = 1}^{+\infty}$, $y = \{y_k\}_{k = 1}^{+\infty}$. Suppose that $A: \mathbb{R}\to \mathbb{R}$ is defined by $A(x) = x$ for all $x = (x_1, x_2, x_3, ..., x_n, ...)$ with constant $\bar{\gamma} = 1$. Let $\psi_1: Q_1 \to \mathbb{R} \cup \{+\infty\}$, $\psi_2: Q_2 \to \mathbb{R} \cup \{+\infty\}$ be given by $\psi_1(x) = x^2$, $\psi_2(x) = 2x^2$, $D: \mathbb{R}\to \mathbb{R}$ be defined by $D(x) = -5x$, $D^* = -5x$, then $T_r^{F_1}(x) = \frac{x}{21r+1}$ and $T_r^{F_2}(x) = \frac{x}{11r+1}$. Let $h:Q_1 \to Q_1$ be $\frac{1}{2}$-contraction defined by $h(x) = \frac{x}{2}$ and $S_j:Q_1 \to Q_1$ be pseudocontractive mappings defined by $S_j(x) = \frac{x}{2(j+1)}$, for $j = 1, 2$. Assume that $U:Q_1 \to Q_1$ and $U_i:Q_1 \to Q_1$ are nonexpansive mappings defined by $U(x) = x$ and $U_i(x) = \frac{x}{100i}$, for $i, = 1, 2$. Choose $\delta = \frac{1}{16}$, $r_n = 1$, $\lambda_n = \frac{1}{n^2}$, $\tau = 0.9$, $\kappa_n^i = \frac{n+i}{n+5+i}$, $\zeta_{n}^i = \frac{7}{8i}$, $\eta_n^j = \frac{1}{20n+5j}$ for $i, j = 1, 2$, $\sigma_n = \frac{1}{n^2}$, $\varphi_n = \frac{5}{6}$, $\rho_n = \frac{n+1}{2000(n+5)}$ and $\omega_n = \frac{1}{700n+4}$. One can easily see that Fix$(\Gamma) = \{0\}\ne \phi.$ We can obtain $K_{Q_1}(x) = \frac{-9x}{21}$ and $K_{Q_2}(x) = \frac{-x}{12}$. We take a stopping criterion of $E_n = \|\chi_n-\chi_{n+1}\| < 10^{-4}$ and plot the graphs between errors $E_n$ and the number of iterations $n$. Take initial values $x_0 = (0.5, 0.5, 0.5, 0.5, ..., 0.5, ...)$ and $x_1 = (0.8, 0.8, 0.8, 0.8, ..., 0.8, ...)$. We do comparative analysis of the numerical result obtain from Algorithm 3.1 with the Chuasuk ^[37], the Kim ^[26] (Theorem (3.6)) and the Majee ^[24] algorithms. Table 3 and Figure 4 show the numerical results

6.   Conclusions

This paper discussed a new inertial generalized viscosity approximation method for solving split generalized mixed equilibrium problem, fixed point problem for a finite family of nonexpansive mappings and hierarchical fixed point problem in real Hilbert spaces. Under certain appropriate conditions, we have established the result of strong convergence. We have demonstrated the use of our main finding with compressed sensing in signal processing. We have explained the numerical effectiveness of our approach in comparison to another method. The results discussed in this paper enhance and summarize previously published findings in the literature.

Use of AI tools declaration

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

Acknowledgement

Researchers would like to thank the Deanship of Scientific Research, Qassim University for funding publication of this project.

Conflict of interest

The authors declare that they have no competing interests.