Mann-Type Inertial Accelerated Subgradient Extragradient Algorithm for Minimum-Norm Solution of Split Equilibrium Problems Induced by Fixed Point Problems in Hilbert Spaces

Abstract

This paper presents the Mann-type inertial accelerated subgradient extragradient algorithm with non-monotonic step sizes for solving the split equilibrium and fixed point problems relating to pseudomonotone and Lipschitz-type continuous bifunctions and nonexpansive mappings in the framework of real Hilbert spaces. By sufficient conditions on the control sequences of the parameters of concern, the strong convergence theorem to support the proposed algorithm, which involves neither prior knowledge of the Lipschitz constants of bifunctions nor the operator norm of the bounded linear operator, is demonstrated. Some numerical experiments are performed to show the efficacy of the proposed algorithm.

1. Introduction

The fixed point problem is a powerful instrument for understanding the fields of chemistry, physics, engineering, and economics in various mathematical models (see [1,2,3,4]). The fixed point problem is expressed as follows:

$$\begin{array}{c}\hfill \mathrm{Find}{p}^{*}\in H\mathrm{such}\mathrm{that}T{p}^{*}=p^{*},\end{array}$$

(1)

where H is a real Hilbert space, and $T:H\to H$ is a mapping. The set of fixed points of the mapping T is denoted by $Fix\left(T\right)$. To find fixed points of a nonexpansive mapping T can be performed by using the widest methods, which was proposed by Mann [5]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_0\in C,\hfill \\ x_{k+1}=(1-{\alpha }_k)x_k+{\alpha }_{k}T{x}_k,\hfill \end{array}\right.\end{array}$$

(2)

where C is a nonempty closed convex subset of H, and $\left\{{\alpha }_k\right\}\subset (0,1)$. In [6], the author showed that if T has a fixed point and ${\sum }_{k=0}^{\infty }{\alpha }_k(1-{\alpha }_k)=\infty$, then the sequence $\left\{x_k\right\}$ generated by (2) converges weakly to a fixed point of T.

On the contrary, after a paper by Blum and Oettli [7] was published, the equilibrium problem began to garner attention and has been utilized for studying a range of mathematical problems, such as variational inequality problems, optimization problems, minimax problems, saddle point problems, and Nash equilibrium problems (see [7,8,9,10]). The following is a formulation of the equilibrium problem:

$$\begin{array}{c}\hfill \mathrm{Find}{p}^{*}\in C\mathrm{such}\mathrm{that}f(p^{*},z)\ge 0,\forall z\in C,\end{array}$$

(3)

where $f:H\times H\to \mathbb{R}$ is a bifunction. The solution set of the equilibrium problem (3) is represented by $EP(f,C)$. An acknowledged approach to solve the equilibrium problem (3) is the proximal point method during which f is a monotone bifunction (see [11]). Unfortunately, in this circumstance, if f is a pseudomonotone bifunction, which is a weaker property than a monotone bifunction, the proximal point method cannot be assured. To maneuver past this obstacle, Tran et al. [12] presented the following extragradient method to solve the equilibrium problem when f is a pseudomonotone and Lipschitz-type continuous bifunction:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_k=argmin\left\{\lambda f(x_k,z)+{\displaystyle \frac{1}{2}}{\parallel z-x_k\parallel }^2:z\in C\right\},\hfill \\ x_{k+1}=argmin\left\{\lambda f(y_k,z)+{\displaystyle \frac{1}{2}}{\parallel z-x_k\parallel }^2:z\in C\right\},\hfill \end{array}\right.\end{array}$$

(4)

where $c_1$ and $c_2$ are Lipschitz constants of f, and $0<\lambda <min\left\{{\displaystyle \frac{1}{2c_1}},{\displaystyle \frac{1}{2c_2}}\right\}$. They showed that the sequence $\left\{x_k\right\}$ generated by (4) converges weakly to a solution of the equilibrium problem. It is important to note that when the feasible set C has a complicated structure, the computing performance of the algorithm is impacted by the extragradient method, a two-step iteration technique. Specifically, each iteration requires solving the optimization problems two times on the feasible set C in order to find $y_k$ and $x_{k+1}$. To gain over this drawback, Hieu [13] expanded the following subgradient extragradient method to solve the equilibrium problem when f is a pseudomonotone and Lipschitz-type continuous bifunction:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_0\in H,\hfill \\ y_k=argmin\left\{{\lambda }_{k}f(x_k,z)+{\displaystyle \frac{1}{2}}{\parallel z-x_k\parallel }^2:z\in C\right\},\hfill \\ B_k=\left\{y\in H:⟨x_k-{\lambda }_k{s}_k-y_k,y-y_k⟩\le 0\right\},s_k\in {\partial }_{2}f(x_k,y_k),\hfill \\ z_k=argmin\left\{{\lambda }_{k}f(y_k,z)+{\displaystyle \frac{1}{2}}{\parallel z-x_k\parallel }^2:z\in B_k\right\},\hfill \\ x_{k+1}={\alpha }_k{x}_0+(1-{\alpha }_k)z_k,\hfill \end{array}\right.\end{array}$$

(5)

where $c_1$ and $c_2$ are Lipschitz constants of f, $0<{\lambda }_k<min\left\{{\displaystyle \frac{1}{2c_1}},{\displaystyle \frac{1}{2c_2}}\right\}$, $\left\{{\alpha }_k\right\}\subset (0,1)$ with ${\displaystyle \sum _{k=0}^{\infty }}{\alpha }_k=+\infty$ such that $\underset{k\to \infty }{lim}{\alpha }_k=0$. The author showed that the sequence $\left\{x_k\right\}$ generated by (5) converges strongly to $P_{EP(f,C)}\left(x_0\right)$. It is highlighted that in the second step, the optimization problem on the feasible set C is transformed to the half-space $B_k$ by the subgradient extragradient method in order to achieve $z_k$ for each iteration. As a result, by only having the optimization problem solved on the feasible set C once in order to acquire $y_k$, the computing performance of the extragradient method is improved by the subgradient extragradient method. Additionally, the step sizes of the previously stated algorithms are dependent on the values $c_1$ and $c_2$. This indicates that the prior knowledge of the values $c_1$ and $c_2$ is necessary for these algorithms. In practical applications, it may be difficult to acquire this kind of data.

At this point, the inertial method, which originates from a discrete version of a second-order dissipative dynamic system [14,15], has been extensively studied to accelerate algorithmic convergence properties, for instance, see [16,17,18,19,20] and the references therein. The key characteristic of this method is that the previous two iterates are applied to establish the next iterate point. In 2023, by combining the techniques of subgradient extragradient and inertial methods in conjunction with the Mann-type method, Panyanak et al. [21] presented the following algorithm to solve the equilibrium and fixed point problems, when T is a $\rho$-demicontractive mapping and f is a pseudomonotone and Lipschitz-type continuous bifunction:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_0,x_1\in C,\hfill \\ w_k=x_k+{\theta }_k(x_k-x_{k-1}),\hfill \\ y_k=argmin\left\{{\lambda }_{k}f(w_k,z)+{\displaystyle \frac{1}{2}}{\parallel z-w_k\parallel }^2:z\in C\right\},\hfill \\ B_k=\left\{y\in H:⟨w_k-{\lambda }_k{s}_k-y_k,y-y_k⟩\le 0\right\},s_k\in {\partial }_{2}f(w_k,y_k),\hfill \\ z_k=argmin\left\{{\lambda }_{k}f(y_k,z)+{\displaystyle \frac{1}{2}}{\parallel z-w_k\parallel }^2:z\in B_k\right\},\hfill \\ x_{k+1}=\left(1-{\alpha }_k-{\beta }_k\right)z_k+{\alpha }_{k}T{z}_k,\hfill \end{array}\right.\end{array}$$

(6)

where the parameter ${\theta }_k$ is chosen in $\left[0,{\overline{\theta }}_k\right]$ with

$$\begin{array}{c}{\overline{\theta }}_k=\left\{\begin{array}{cc}min\left\{{\displaystyle \frac{\gamma }{2}},{\displaystyle \frac{{ϵ}_k}{\parallel x_k-x_{k-1}\parallel }}\right\},\hfill & \mathrm{i}\mathrm{f} x_k\ne x_{k-1},\hfill \\ {\displaystyle \frac{\gamma }{2}},\hfill & \mathrm{otherwise},\hfill \end{array}\right.\end{array}$$

and the step size ${\lambda }_k$ is given as

$$\begin{array}{c}\hfill {\lambda }_{k+1}=\left\{\begin{array}{c}min\left\{{\lambda }_k,{\displaystyle \frac{\mu (\parallel w_k-y_k{\parallel }^2+\parallel z_k-y_k{\parallel }^2)}{2\left[f(w_k,z_k)-f(w_k,y_k)-f(y_k,z_k)\right]}}\right\},\hfill \\ \mathrm{i}\mathrm{f} f(w_k,z_k)-f(w_k,y_k)-f(y_k,z_k)>0,\hfill \\ {\lambda }_k,\mathrm{otherwise},\hfill \end{array}\right.\end{array}$$

when ${\lambda }_1>0$, $\gamma \in (0,1)$, $\mu \in (0,1)$, $\left\{{\beta }_k\right\}\subset \left(0,1\right)$ such that ${\displaystyle \sum _{k=1}^{\infty }}{\beta }_k=+\infty$, $\underset{k\to \infty }{lim}{\beta }_k=0$, $\left\{{ϵ}_k\right\}\subset [0,\infty )$, $\left\{{\alpha }_k\right\}\subset \left(a,b\right)$ with $0<a,b<1-\rho$, $0<a,b<1-{\beta }_k$, and $\underset{k\to \infty }{lim}{\displaystyle \frac{{ϵ}_k}{{\alpha }_k}}\parallel x_k-x_{k-1}\parallel =0$. They showed that the sequence $\left\{x_k\right\}$ generated by (6) converges strongly to $P_{Fix\left(T\right)\cap EP(f,C)}\left(0\right)$. It is apparent that (6) effectively addresses the unknown information of the Lipschitz constants of f through the use of the adaptive step size. Furthermore, the adaptive step size criteria are provided, which depend on previously known information for the uncomplicated calculation of updating the step size of each iteration.

In 2016, Dinh et al. [22] generated the following split equilibrium and fixed point problems:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{Find}{p}^{*}\in Fix\left(T\right)\mathrm{such}\mathrm{that}{f}_1(p^{*},z)\ge 0,\forall z\in C_1,\hfill \\ \mathrm{and}A{p}^{*}\in Fix\left(S\right)\mathrm{solves}{f}_2(A{p}^{*},v)\ge 0,\forall v\in C_2,\hfill \end{array}\right.\end{array}$$

(7)

where $H_1$ and $H_2$ are real Hilbert spaces; $C_1$ and $C_2$ are nonempty closed convex subsets of $H_1$ and $H_2$, respectively; $T:C_1\to C_1$ and $S:C_2\to C_2$ are mappings; $f_1:C_1\times C_1\to \mathbb{R}$ and $f_2:C_2\times C_2\to \mathbb{R}$ are bifunctions; and $A:H_1\to H_2$ is a bounded linear operator. Dinh et al. [22] presented the following algorithm by utilizing the extragradient and proximal point methods to solve the split equilibrium and fixed point problems (7), where S and T are nonexpansive mappings, $f_1$ is a pseudomonotone and Lipschitz-type continuous bifunction, and $f_2$ is a monotone bifunction:

$$\left\{\begin{array}{c}{x}_1\in C_1,\hfill \\ y_k=argmin\left\{{\lambda }_k{f}_1(x_k,z)+{\displaystyle \frac{1}{2}}{\parallel x_k-z\parallel }^2:z\in C_1\right\},\hfill \\ z_k=argmin\left\{{\lambda }_k{f}_1(y_k,z)+{\displaystyle \frac{1}{2}}{\parallel x_k-z\parallel }^2:z\in C_1\right\},\hfill \\ s_k=(1-\alpha )z_k+\alpha T{z}_k,\hfill \\ u_k=T_{t_k}^{f_2}A{s}_k,\hfill \\ x_{k+1}=P_{C_1}(s_k+\eta A^{*}(S{u}_k-A{s}_k)),\hfill \end{array}\right.$$

(8)

where $c_1$ and $c_2$ are Lipschitz constants of $f_1$, $\alpha \in (0,1)$, $\eta \in \left(0,{\displaystyle \frac{1}{{\parallel A\parallel }^2}}\right)$, $\left\{{\lambda }_k\right\}\subset [\underline{\lambda },\overline{\lambda }]$ with $0<\underline{\lambda }\le \overline{\lambda }<min\left\{{\displaystyle \frac{1}{2c_1}},{\displaystyle \frac{1}{2c_2}}\right\}$, $\left\{t_k\right\}\subset (0,+\infty )$ such that $\underset{k\to \infty }{lim\; inf}{t}_k>0$, and $T_{t_k}^{f_2}A{s}_k:=\{u\in C_2|$$f_2(u,v)+{\displaystyle \frac{1}{t_k}}⟨v-u,u-A{s}_k⟩\ge 0,\forall v\in C_2\}$, and $A^{*}$ is the adjoint operator of A. They showed that the sequence $\left\{x_k\right\}$ generated by (8) converges weakly to a solution of the split equilibrium and fixed point problems (7). This kind of problem has attracted a lot of attention due to its broad applications in intensity-modulated radiation therapy, image restoration, network resource allocation, signal processing, phase retrievals, and other real-world applications (see, e.g., [23,24,25]).

Inspired by the above results, Petrot et al. [26] presented the following algorithm by applying the concept of the extragradient method to solve the split equilibrium and fixed point problems (7), where S and T are nonexpansive mappings and $f_1$ and $f_2$ are pseudomonotone and Lipschitz-type continuous bifunctions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_1\in H_1,\hfill \\ u_k=argmin\left\{{\mu }_k{f}_2(P_{C_2}\left(A{x}_k\right),v)+{\displaystyle \frac{1}{2}}{\parallel P_{C_2}\left(A{x}_k\right)-v\parallel }^2:v\in C_2\right\},\hfill \\ v_k=argmin\left\{{\mu }_k{f}_2(u_k,v)+{\displaystyle \frac{1}{2}}{\parallel P_{C_2}\left(A{x}_k\right)-v\parallel }^2:v\in C_2\right\},\hfill \\ y_k=P_{C_1}\left(x_k+{\eta }_k{A}^{*}(S{v}_k-A{x}_k)\right),\hfill \\ t_k=argmin\left\{{\lambda }_k{f}_1(y_k,z)+{\displaystyle \frac{1}{2}}{\parallel y_k-z\parallel }^2:z\in C_1\right\},\hfill \\ z_k=argmin\left\{{\lambda }_k{f}_1(t_k,z)+{\displaystyle \frac{1}{2}}{\parallel y_k-z\parallel }^2:z\in C_1\right\},\hfill \\ x_{k+1}={\alpha }_{k}h\left(x_k\right)+(1-{\alpha }_k)\left({\beta }_k{x}_k+(1-{\beta }_k)T{z}_k\right),\hfill \end{array}\right.\end{array}$$

(9)

where $c_1$ and $c_2$ are Lipschitz constants of $f_1$; $d_1$ and $d_2$ are Lipschitz constants of $f_2$, $\left\{{\lambda }_k\right\}\subset [\underline{\lambda },\overline{\lambda }]$ with $0<\underline{\lambda }\le \overline{\lambda }<min\left\{{\displaystyle \frac{1}{2c_1}},{\displaystyle \frac{1}{2c_2}}\right\}$, $\left\{{\mu }_k\right\}\subset [\underline{\mu },\overline{\mu }]$ with $0<\underline{\mu }\le \overline{\mu }<min\left\{{\displaystyle \frac{1}{2d_1}},{\displaystyle \frac{1}{2d_2}}\right\}$, $\left\{{\eta }_k\right\}\subset [\underline{\eta },\overline{\eta }]$ with $0<\underline{\eta }\le \overline{\eta }<{\displaystyle \frac{1}{{\parallel A\parallel }^2}}$, $\left\{{\beta }_k\right\}\subset (0,1)$ with $0<\underset{k\to \infty }{lim\; inf}{\beta }_k\le \underset{k\to \infty }{lim\; sup}{\beta }_k<1$, $\left\{{\alpha }_k\right\}\subset \left(0,{\displaystyle \frac{1}{2-\rho }}\right)$ with ${\displaystyle \sum _{k=1}^{\infty }}{\alpha }_k=\infty$ such that $\underset{k\to \infty }{lim}{\alpha }_k=0$; and h is a $\rho$-contraction mapping. They showed that the sequence $\left\{x_k\right\}$ generated by (9) converges strongly to a solution of the split equilibrium and fixed point problems (7). It is noteworthy that the step size ${\eta }_k$ of this algorithm is determined by the operator norm of the bounded linear operator A. In order to overcome this drawback, Ezeora et al. [27] presented the following algorithm by employing the ideas of extragradient and inertial methods to solve the split equilibrium and fixed point problems (7), where S and T are nonexpansive mappings and $f_1$ and $f_2$ are pseudomonotone and Lipschitz-type continuous bifunctions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_0,x_1\in H_1,\hfill \\ w_k=x_k+{\theta }_k(x_k-x_{k-1}),\hfill \\ u_k=argmin\left\{{\mu }_k{f}_2(P_{C_2}\left(A{w}_k\right),v)+{\displaystyle \frac{1}{2}}{\parallel v-P_{C_2}\left(A{w}_k\right)\parallel }^2:v\in C_2\right\},\hfill \\ v_k=argmin\left\{{\mu }_k{f}_2(u_k,v)+{\displaystyle \frac{1}{2}}{\parallel v-P_{C_2}\left(A{w}_k\right)\parallel }^2:v\in C_2\right\},\hfill \\ y_k=P_{C_1}\left(w_k+{\eta }_k{A}^{*}(S{v}_k-A{w}_k)\right),\hfill \\ t_k=argmin\left\{{\lambda }_k{f}_1(y_k,z)+{\displaystyle \frac{1}{2}}{\parallel z-y_k\parallel }^2:z\in C_1\right\},\hfill \\ z_k=argmin\left\{{\lambda }_k{f}_1(t_k,z)+{\displaystyle \frac{1}{2}}{\parallel z-y_k\parallel }^2:z\in C_1\right\},\hfill \\ x_{k+1}={\alpha }_{k}h\left(w_k\right)+(1-{\alpha }_k)\left({\beta }_k{w}_k+(1-{\beta }_k)T{z}_k\right),\hfill \end{array}\right.\end{array}$$

(10)

where the parameter ${\theta }_k$ is chosen in $\left[0,{\overline{\theta }}_k\right]$ with

$$\begin{array}{c}\hfill {\overline{\theta }}_k=\left\{\begin{array}{cc}min\left\{\gamma ,{\displaystyle \frac{{ϵ}_k}{\parallel x_k-x_{k-1}\parallel }}\right\},\hfill & \mathrm{if} x_k\ne x_{k-1},\hfill \\ \gamma ,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\end{array}$$

and for $\delta >0$ small enough, the step size ${\eta }_k$ is given as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\eta }_k\in \left(\delta ,{\displaystyle \frac{\parallel A{w}_k-S{v}_k{\parallel }^2}{\parallel A^{*}(A{w}_k-S{v}_k){\parallel }^2}}-\delta \right)\hfill & \mathrm{i}\mathrm{f} A{w}_k-S{v}_k\ne 0,\hfill \\ n,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\end{array}$$

when $c_1$ and $c_2$ are Lipschitz constants of $f_1$; $d_1$ and $d_2$ are Lipschitz constants of $f_2$, $n>0$, $\gamma \in [0,1)$, ${\displaystyle \sum _{k=1}^{\infty }}{ϵ}_k<\infty$, $\left\{{\lambda }_k\right\}\subset [\underline{\lambda },\overline{\lambda }]$ with $0<\underline{\lambda }\le \overline{\lambda }<min\left\{{\displaystyle \frac{1}{2c_1}},{\displaystyle \frac{1}{2c_2}}\right\}$, $\left\{{\mu }_k\right\}\subset [\underline{\mu },\overline{\mu }]$ with $0<\underline{\mu }\le \overline{\mu }<min\left\{{\displaystyle \frac{1}{2d_1}},{\displaystyle \frac{1}{2d_2}}\right\}$, $\left\{{\beta }_k\right\}\subset (0,1)$ with $0<\underset{k\to \infty }{lim\; inf}{\beta }_k\le \underset{k\to \infty }{lim\; sup}{\beta }_k<1$, $\left\{{\alpha }_k\right\}\subset \left(0,{\displaystyle \frac{1}{2(1-\rho )}}\right)$ such that ${\displaystyle \sum _{k=1}^{\infty }}{\alpha }_k=\infty$, and $\underset{k\to \infty }{lim}{\alpha }_k=0$; and h is a $\rho$-contraction mapping. They showed that the sequence $\left\{x_k\right\}$ generated by the EIM Algorithm (10) converges strongly to a solution of the split equilibrium and fixed point problems (7). We observe that, in order to guarantee the convergence of the EIM Algorithm, the maximum values of the scalars $\overline{\lambda }$ and $\overline{\mu }$ depend on the Lipschitz constants of $f_1$ and $f_2$, respectively.

Motivated by the advantageous method presented in the EIM Algorithm by Ezeora et al. [27], this paper continues to focus on methods that solve the split equilibrium and fixed point problems (7). We consider an updated iterative algorithm that does not require prior knowledge of either the Lipschitz constants of the bifunctions or the operator norm of the bounded linear operator. This approach aims to find the minimum-norm solution for the split equilibrium and fixed point problems (7). Numerical experiments and comparisons with some recently developed algorithms are conducted to evaluate the performance of the introduced algorithm.

This paper is arranged as follows: Section 2 reviews some preliminary definitions and properties for use later on. Section 3 presents the Mann-type inertial accelerated subgradient extragradient algorithm and the corresponding strong convergence result. In Section 4, we discuss the performance of the proposed algorithm in comparison to some recently developed algorithms through numerical experiments. In Section 5, we close this paper with some conclusions.

2. Preliminaries

We provide some important definitions and qualifications in this section that are applied throughout the work. Let H be a real Hilbert space endowed with inner product $⟨·,·⟩$, and its corresponding norm $\parallel ·\parallel$. For a sequence $\left\{x_k\right\}$ in H, we denote the strong convergence and the weak convergence of a sequence $\left\{x_k\right\}$ to a point $p^{*}\in H$ by $x_k\to p^{*}$ and $x_k\rightharpoonup p^{*}$, respectively. The sets of the real numbers and the natural numbers are represented by $\mathbb{R}$ and $\mathbb{N}$, respectively.

We begin with some definitions and results concerning the nonlinear mappings.

Definition 1.

Let $T:H\to H$ be a mapping. The mapping T is called nonexpansive if

$$\begin{array}{c}\hfill \parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\forall x,y\in H.\end{array}$$

Remark 1.

We notice that if T is a nonexpansive mapping, $Fix\left(T\right)$ is closed and convex; see [28].

Definition 2.

Let $T:H\to H$ be a mapping. The mapping T is called demiclosed at $z\in H$ if for each sequence $\left\{x_k\right\}\subset H$, $x_k\rightharpoonup p^{*}\in H$, and $T{x}_k\to z$, we have $T{p}^{*}=z$.

Lemma 1

([28]). Let $T:H\to H$ be a nonexpansive mapping having a fixed point. Then, $I-T$ is demiclosed at 0.

In what follows, we collect some fundamental concepts which are needed for the sequels. For each $x,y,z\in H$, it is known that

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2⟨y,x+y⟩,$$

(11)

and

$${\parallel \alpha x+\beta y+\eta z\parallel }^2={\alpha \parallel x\parallel }^2+{\beta \parallel y\parallel }^2+{\eta \parallel z\parallel }^2{-\alpha \beta \parallel x-y\parallel }^2{-\beta \eta \parallel y-z\parallel }^2-\alpha \eta {\parallel x-z\parallel }^2,$$

(12)

for each $\alpha ,\beta ,\eta \in [0,1]$ such that $\alpha +\beta +\eta =1$; see [17].

For a point $x\in H$, the metric projection of a point x onto C is represented by $P_C\left(x\right)$ in the sense of

$$\parallel x-P_C\left(x\right)\parallel \le \parallel x-z\parallel ,\forall z\in C,$$

where C is a nonempty closed convex subset of H.

Lemma 2

([28,29]). Let C be a nonempty closed convex subset of H. Then, the following results hold:

(i) 

For each $x\in H$, $P_C\left(x\right)$ is well-defined and singleton;

(ii) 

$y=P_C\left(x\right)$ if and only if $⟨x-y,z-y⟩$$\le 0$, $\forall z\in C$;

(iii) 

$P_C$ is a nonexpansive mapping.

Let $f:H\to \mathbb{R}$ be a function. The subdifferential of f at $y\in H$ is given by

$$\begin{array}{c}\hfill \partial f\left(y\right)=\{v\in H|f\left(z\right)-f\left(y\right)\ge ⟨v,z-y⟩,\forall z\in H\}.\end{array}$$

The function f is called subdifferentiable at y if $\partial f\left(y\right)\ne \varnothing$.

Lemma 3

([29]). For each $y\in H$, the subdifferentiable $\partial f\left(y\right)$ of a continuous convex function f is a weakly closed and bounded convex set.

Lemma 4

([9]). Let C be a convex subset of H. Suppose that $f:C\to \mathbb{R}$ is subdifferentiable on C. Then, $p^{*}$ is a solution to the following convex problem:

$$\begin{array}{c}\hfill min\left\{f\left(x\right):x\in C\right\}\end{array}$$

if and only if $0\in \partial f\left(p^{*}\right)+N_C\left(p^{*}\right)$, where $N_C\left(p^{*}\right):=\{v\in H|⟨v,z-p^{*}⟩\le 0,\forall z\in C\}$ is the normal cone of C at $p^{*}$.

The following technical lemmas are necessary to obtain the convergence results.

Lemma 5

([30]). Let $\left\{a_k\right\}$ and $\left\{c_k\right\}$ be sequences of non-negative real numbers and $\left\{b_k\right\}$ be a sequence of real numbers satisfying the following relation:

$$\begin{array}{c}\hfill a_{k+1}\le (1-{\beta }_k)a_k+{\beta }_k{b}_k+c_k,\forall k\in \mathbb{N}\cup \left\{0\right\},\end{array}$$

where $\left\{{\beta }_k\right\}\subset (0,1)$ with ${\displaystyle \sum _{k=0}^{\infty }}{\beta }_k=\infty$. If ${\displaystyle \sum _{k=0}^{\infty }}{c}_k<\infty$ and $\underset{k\to \infty }{lim\; sup}{b}_k\le 0$, then $\underset{k\to \infty }{lim}{a}_k=0$.

Lemma 6

([31]). Let $\left\{a_k\right\}$ be a sequence of real numbers in which there exists a subsequence $\left\{a_{k_i}\right\}$ of $\left\{a_k\right\}$ with $a_{k_i}<a_{k_i+1}$, for any $i\in \mathbb{N}$. Then, there exists a non-decreasing sequence $\left\{m_n\right\}$ of positive integers with $\underset{n\to \infty }{lim}{m}_n=\infty$, and the following relations hold:

$$\begin{array}{c}\hfill a_{m_n}\le a_{m_n+1}\mathrm{and}{a}_n\le a_{m_n+1},\end{array}$$

for any (sufficiently large) numbers $n\in \mathbb{N}$. Indeed, $m_n$ is the largest number k in the set $\{1,2,\dots ,n\}$ satisfying the following relation:

$$\begin{array}{c}\hfill a_k<a_{k+1}.\end{array}$$

We end this section by providing some definitions and properties relating to the equilibrium problems.

Definition 3.

Let C be a nonempty closed convex subset of H and $f:H\times H\to \mathbb{R}$ be a bifunction. The bifunction f is called

(i) 

Monotone on C if

$$f(x,y)+f(y,x)\le 0,\forall x,y\in C;$$

(ii) 

Pseudomonotone on C if

$$f(x,y)\ge 0\Rightarrow f(y,x)\le 0,\forall x,y\in C;$$

(iii) 

Lipschitz-type continuous on H if there exists constants $c_1>0$ and $c_2>0$ satisfying

$$f(x,y)+f(y,z)\ge f(x,z)-c_1{\parallel x-y\parallel }^2-c_2{\parallel y-z\parallel }^2,\forall x,y,z\in H.$$

Remark 2.

Note that a monotone bifunction is a pseudomonotone bifunction. However, in general, the converse is not true; see, e.g., [32].

Let C be a nonempty closed convex subset of a real Hilbert space H and $f:H\times H\to \mathbb{R}$ be a bifunction. In this paper, we take into account the following assumptions:

(A1) 

For each fixed $z\in C$, $f(·,z)$ is sequentially weakly upper semicontinuous on C, that is, if $\left\{x_k\right\}$ is a sequence in C with $x_k\rightharpoonup p^{*}\in C$, then $\underset{k\to \infty }{lim\; sup}f(x_k,z)\le f(p^{*},z)$;

(A2) 

For each fixed $x\in H$, $f(x,·)$ is convex, subdifferentiable, and lower semicontinuous on H;

(A3) 

f is psuedomonotone on C;

(A4) 

f is Lipschitz-type continuous on H.

Remark 3.

(i) 

It is well-known that the solution set $EP(f,C)$ is closed and convex, where the bifunction f satisfies assumptions $\left(A1\right)-\left(A3\right)$; see [12,33,34].

(ii) 

We observe that $f(x,x)=0$, for each $x\in C$, where the bifunction f satisfies assumptions $\left(A3\right)$ and $\left(A4\right)$; see [17].

(iii) 

For each fixed $x\in H$, the subdifferential of a bifunction $f(x,·)$ at $y\in H$ is provided by

$${\partial }_{2}f(x,y)=\{v\in H|f(x,z)-f(x,y)\ge ⟨v,z-y⟩,\forall z\in H\}.$$

3. Main Results

Let $H_1$ and $H_2$ be real Hilbert spaces and $C_1$ and $C_2$ be nonempty closed convex subsets of $H_1$ and $H_2$, respectively. We start by recalling the split equilibrium and fixed point problems:

$$\left\{\begin{array}{c}\mathrm{Find}{p}^{*}\in C_1\cap Fix\left(T\right)\mathrm{such}\mathrm{that}{f}_1(p^{*},z)\ge 0,\forall z\in C_1,\hfill \\ \mathrm{and}A{p}^{*}\in C_2\cap Fix\left(S\right)\mathrm{solves}{f}_2(A{p}^{*},v)\ge 0,\forall v\in C_2,\hfill \end{array}\right.$$

(13)

where $f_1:H_1\times H_1\to \mathbb{R}$ and $f_2:H_2\times H_2\to \mathbb{R}$ are bifunctions, $T:H_1\to H_1$ and $S:H_2\to H_2$ are mappings, $A:H_1\to H_2$ is a bounded linear operator, and $A^{*}:H_2\to H_1$ is the adjoint operator of A. We observe that the setting of problem (13) differs from that of problem (7). Specifically, in problem (13), the domain of each operator is the entire space, whereas in problem (7), the domain of the considered operator is a closed convex subset of the corresponding space. Consequently, the fixed point sets of the operators T and S in problem (13) are independent of the sets $C_1$ and $C_2$, respectively. This distinction can lead to different applications and interpretations for problems (7) and (13). The solution set of the split equilibrium and fixed point problems (13) is, henceforth, represented by $\Omega$ in the form of

$$\Omega :=\left\{p^{*}\in EP(f_1,C_1)\cap Fix\left(T\right)|A{p}^{*}\in EP(f_2,C_2)\cap Fix\left(S\right)\right\}.$$

It is noteworthy that, by using Remarks 1, 3(i), and the linearity property of the operator A, the solution set $\Omega$ is closed and convex; see [35].

The following algorithm (Algorithm 1) is presented to solve the split equilibrium and fixed point problems (13).

Algorithm 1. Mann-type inertial accelerated subgradient extragradient algorithm

Initialization. Select ${\lambda }_1>0$, ${\mu }_1>0$, ${\eta }_1>0$, $\omega \in (0,1)$, $\tau \in (0,1)$, $\phi \in (0,1)$, ${\gamma }_k\in [0,1)$ with $\underset{k\to \infty }{lim}{\gamma }_k=0$, $\left\{{\xi }_k\right\}\subset \left[1,\infty \right)$ with $\underset{k\to \infty }{lim}{\xi }_k=1$, $\left\{{\sigma }_k\right\}\subset \left[1,\infty \right)$ with $\underset{k\to \infty }{lim}{\sigma }_k=1$, $\left\{{\rho }_k\right\}\subset \left[0,\infty \right)$ with ${\displaystyle \sum _{k=1}^{\infty }}{\rho }_k<+\infty$, $\left\{{\delta }_k\right\}\subset \left[0,\infty \right)$ with ${\displaystyle \sum _{k=1}^{\infty }}{\delta }_k<+\infty$, $\left\{{\zeta }_k\right\}\subset \left[0,\infty \right)$ with ${\displaystyle \sum _{k=1}^{\infty }}{\zeta }_k<+\infty$, $\left\{{ϵ}_k\right\}\subset [0,\infty )$, $\left\{{\beta }_k\right\}\subset \left(0,1\right)$ such that ${\displaystyle \sum _{k=1}^{\infty }}{\beta }_k=\infty$, $\underset{k\to \infty }{lim}{\beta }_k=0$, $\underset{k\to \infty }{lim}{\displaystyle \frac{{ϵ}_k}{{\beta }_k}}=0$, and $\left\{{\alpha }_k\right\}\subset \left(a,b\right)\subset \left(0,1-{\beta }_k\right)$, for some positive constants a and b. Put $k=1$ and $x_0,x_1\in H$ arbitrarily.

Step 1. Select ${\theta }_k$ satisfying $0\le {\theta }_k\le {\overline{\theta }}_k$, where $$\begin{array}{c}\hfill {\overline{\theta }}_k=\left\{\begin{array}{cc}min\left\{{\gamma }_k,{\displaystyle \frac{{ϵ}_k}{\parallel x_k-x_{k-1}\parallel }}\right\},\hfill & \mathrm{i}\mathrm{f} x_k\ne x_{k-1},\hfill \\ {\gamma }_k,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\end{array}$$ and calculate $$\begin{array}{c}\hfill w_k=x_k+{\theta }_k\left(x_k-x_{k-1}\right).\end{array}$$

Step 2. Compute $$\begin{array}{c}\hfill y_k=argmin\{{\xi }_k{\lambda }_k{f}_1(w_k,z)+{\displaystyle \frac{1}{2}}\parallel z-w_k{\parallel }^2:z\in C_1\}.\end{array}$$

Step 3. Select $s_k\in {△}_k^{f_1}$ and create a half-space $$\begin{array}{c}\hfill B_k=\left\{y\in H_1|⟨w_k-{\xi }_k{\lambda }_k{s}_k-y_k,y-y_k⟩\le 0\right\},\end{array}$$ where $$\begin{array}{c}\hfill {△}_k^{f_1}=\left\{s\in {\partial }_2{f}_1(w_k,y_k)|{\xi }_k{\lambda }_{k}s+y_k=w_k-q,\exists q\in N_{C_1}\left(y_k\right)\right\}.\end{array}$$

Step 4. Compute $$\begin{array}{c}\hfill z_k=argmin\{{\lambda }_k{f}_1(y_k,z)+{\displaystyle \frac{1}{2}}\parallel z-w_k{\parallel }^2:z\in B_k\}.\end{array}$$

Step 5. Calculate $$\begin{array}{c}\hfill t_k=\left(1-{\beta }_k-{\alpha }_k\right)z_k+{\alpha }_{k}T{z}_k.\end{array}$$

Step 6. Compute $$\begin{array}{c}\hfill u_k=argmin\{{\sigma }_k{\mu }_k{f}_2(A{t}_k,v)+{\displaystyle \frac{1}{2}}\parallel v-A{t}_k{\parallel }^2:v\in C_2\}.\end{array}$$

Step 7. Select $r_k\in {△}_k^{f_2}$ and create a half-space $$\begin{array}{c}\hfill D_k=\left\{u\in H_2|⟨A{t}_k-{\sigma }_k{\mu }_k{r}_k-u_k,u-u_k⟩\le 0\right\},\end{array}$$ where $$\begin{array}{c}\hfill {△}_k^{f_2}=\left\{r\in {\partial }_2{f}_2(A{t}_k,u_k)|{\sigma }_k{\mu }_{k}r+u_k=A{t}_k-l,\exists l\in N_{C_2}\left(u_k\right)\right\}.\end{array}$$

Step 8. Compute $$\begin{array}{c}\hfill v_k=argmin\{{\mu }_k{f}_2(u_k,v)+{\displaystyle \frac{1}{2}}\parallel v-A{t}_k{\parallel }^2:v\in D_k\}.\end{array}$$

Step 9. Construct $x_{k+1}$ by using the following expression: $$\begin{array}{c}\hfill x_{k+1}=P_{B_k}(t_k+{\eta }_k{A}^{*}(S{v}_k-A{t}_k)).\end{array}$$

Step 10. Calculate $${\lambda }_{k+1}=\left\{\begin{array}{c}min\left\{{\lambda }_k+{\rho }_k,{\displaystyle \frac{\omega (\parallel w_k-y_k{\parallel }^2+\parallel y_k-z_k{\parallel }^2)}{2\left[f_1(w_k,z_k)-f_1(w_k,y_k)-f_1(y_k,z_k)\right]}}\right\},\hfill \\ \mathrm{i}\mathrm{f} f_1(w_k,z_k)-f_1(w_k,y_k)-f_1(y_k,z_k)>0,\hfill \\ {\lambda }_k+{\rho }_k,\mathrm{otherwise},\hfill \end{array}\right.$$ (14) $${\mu }_{k+1}=\left\{\begin{array}{c}min\left\{{\mu }_k+{\delta }_k,{\displaystyle \frac{\tau (\parallel A{t}_k-u_k{\parallel }^2+\parallel u_k-v_k{\parallel }^2)}{2\left[f_2(A{t}_k,v_k)-f_2(A{t}_k,u_k)-f_2(u_k,v_k)\right]}}\right\},\hfill \\ \mathrm{i}\mathrm{f} f_2(A{t}_k,v_k)-f_2(A{t}_k,u_k)-f_2(u_k,v_k)>0,\hfill \\ {\mu }_k+{\delta }_k,\mathrm{otherwise},\hfill \end{array}\right.$$ (15) and $${\eta }_{k+1}=\left\{\begin{array}{cc}min\left\{{\eta }_k+{\zeta }_k,{\displaystyle \frac{\phi \parallel S{v}_k-A{t}_k{\parallel }^2}{\parallel A^{*}(S{v}_k-A{t}_k){\parallel }^2}}\right\},\hfill & \mathrm{i}\mathrm{f} S{v}_k\ne A{t}_k,\hfill \\ {\eta }_k+{\zeta }_k,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$ (16)

Step 11. Set $k:=k+1$ and return to Step 1.

Remark 4.

(i) 

The auxiliary sequences $\left\{{\xi }_k\right\}$ and $\left\{{\sigma }_k\right\}$ in Algorithm 1 are introduced to account for bias in the bifunctions $f_1$ and $f_2$ in steps 2 and 6, respectively. These auxiliary biases contribute to the acceleration of Algorithm 1. We highlight that the superior numerical behavior of Algorithm 1 can be affected by choosing sequences $\left\{{\xi }_k\right\}$ and $\left\{{\sigma }_k\right\}$. Take note that the accelerated subgradient extragradient method contained in Algorithm 1 reduces to the situation as presented in [13,21] if ${\xi }_k=1$ and ${\sigma }_k=1$, for each $k\in \mathbb{N}$.

(ii) 

The self-adaptive step sizes ${\lambda }_k$, ${\mu }_k$, and ${\eta }_k$ are implemented without requiring prior knowledge of the Lipschitz constants of the bifunctions $f_1$ and $f_2$ or the operator norm of the bounded linear operator A, respectively. This demonstrates that Algorithm 1 automatically updates the iteration step sizes ${\lambda }_k$, ${\mu }_k$, and ${\eta }_k$ by utilizing some previously existing data. We emphasize that the step sizes ${\lambda }_k$, ${\mu }_k$, and ${\eta }_k$ in Algorithm 1 reduce to the non-increasing step sizes in the case of ${\rho }_k=0$, ${\delta }_k=0$, and ${\zeta }_k=0$ for each $k\in \mathbb{N}$, as presented in [21,36]. These may have a significant impact on the numerical results; see Section 4 for further discussion.

(iii) 

It is important to note that step 9 in Algorithm 1 is improved as the metric projection onto the half-space $B_k$ instead of the metric projection onto the nonempty closed convex subset $C_1$, as highlighted in [22,26,27]. This is an advantage of Algorithm 1 given that the metric projection onto the half-space $B_k$ has an explicit formula. Meanwhile, the metric projection onto $C_1$ may be difficult if $C_1$ has a complicated structure.

(iv) 

In step 3, the nonemptiness of the set ${△}_k^{f_1}$ is always guaranteed. Indeed, by the definition of $y_k$ and Lemma 4, one sees that

$$\begin{array}{c}\hfill 0\in {\partial }_2\left\{{\xi }_k{\lambda }_k{f}_1(w_k,y_k)+{\displaystyle \frac{1}{2}}{\parallel y_k-w_k\parallel }^2\right\}+N_{C_1}\left(y_k\right).\end{array}$$

Thus, there exist $s\in {\partial }_2{f}_1(w_k,y_k)$ and $q\in N_{C_1}\left(y_k\right)$ such that

$$\begin{array}{c}\hfill {\xi }_k{\lambda }_{k}s+y_k-w_k+q=0.\end{array}$$

This ensures that there exists a solution within ${△}_k^{f_1}$, confirming its nonemptiness. Similarly, in step 7, the nonemptiness of the set ${△}_k^{f_2}$ is guaranteed by utilizing the definition of $u_k$ and Lemma 4.

The following lemmas are very useful for analyzing the convergence of Algorithm 1.

Lemma 7.

Let $f_1:H_1\times H_1\to \mathbb{R}$ and $f_2:H_2\times H_2\to \mathbb{R}$ be bifunctions that satisfy $\left(A1\right)$–$\left(A4\right)$, $T:H_1\to H_1$ and $S:H_2\to H_2$ be nonexpansive mappings, $A:H_1\to H_2$ be a bounded linear operator, and $A^{*}:H_2\to H_1$ be the adjoint operator of A. Let $w_k,y_k,z_k\in H_1$ and $A{t}_k,u_k,v_k\in H_2$. Assume that the sequences $\left\{{\lambda }_k\right\}$, $\left\{{\mu }_k\right\}$, and $\left\{{\eta }_k\right\}$ are generated by (14), (15), and (16), respectively. Then, the following results hold:

$$\underset{k\to \infty }{lim}{\lambda }_k=\lambda \in \left[min\left\{{\lambda }_1,{\displaystyle \frac{\omega }{2max\left\{c_1,c_2\right\}}}\right\},{\lambda }_1+\rho \right],\mathit{where}\rho =\sum _{k=1}^{\infty }{\rho }_k,$$

$$\underset{k\to \infty }{lim}{\mu }_k=\mu \in \left[min\left\{{\mu }_1,{\displaystyle \frac{\tau }{2max\left\{d_1,d_2\right\}}}\right\},{\mu }_1+\delta \right],\mathit{where}\delta =\sum _{k=1}^{\infty }{\delta }_k,$$

and

$$\underset{k\to \infty }{lim}{\eta }_k=\eta \in \left[min\left\{{\eta }_1,{\displaystyle \frac{\phi }{{\parallel A\parallel }^2}}\right\},{\eta }_1+\zeta \right],\mathit{where}\zeta =\sum _{k=1}^{\infty }{\zeta }_k.$$

Proof. 

The proof of this Lemma follows the technique in (Lemma 3.1 of [37]). Firstly, we note that

$$\begin{array}{c}\hfill {\displaystyle \frac{\phi \parallel S{v}_k-A{t}_k{\parallel }^2}{\parallel A^{*}(S{v}_k-A{t}_k){\parallel }^2}}\ge {\displaystyle \frac{\phi \parallel S{v}_k-A{t}_k{\parallel }^2}{{\parallel A\parallel }^2{\parallel S{v}_k-A{t}_k\parallel }^2}}={\displaystyle \frac{\phi }{{\parallel A\parallel }^2}}.\end{array}$$

This, together with the definition of ${\eta }_{k+1}$ and the assumptions on the parameter ${\zeta }_k$, yields

$$\begin{array}{c}\hfill {\eta }_{k+1}\ge min\left\{{\eta }_k+{\zeta }_k,{\displaystyle \frac{\phi }{{\parallel A\parallel }^2}}\right\}\ge min\left\{{\eta }_k,{\displaystyle \frac{\phi }{{\parallel A\parallel }^2}}\right\}.\end{array}$$

Thus, the sequence $\left\{{\eta }_k\right\}$ has a lower bound as $min\left\{{\eta }_1,{\displaystyle \frac{\phi }{{\parallel A\parallel }^2}}\right\}$ by induction.

Additionally, in view of the definition of ${\eta }_{k+1}$, we obtain ${\eta }_{k+1}\le {\eta }_k+{\zeta }_k$ for each $k\in \mathbb{N}$. Then, by induction, the sequence $\left\{{\eta }_k\right\}$ has an upper bound as ${\eta }_1+\zeta$, where $\zeta ={\displaystyle \sum _{k=1}^{\infty }}{\zeta }_k$. Consequently, we conclude that $\left\{{\eta }_k\right\}$ is a bounded sequence and ${\eta }_k\in \left[min\left\{{\eta }_1,{\displaystyle \frac{\phi }{{\parallel A\parallel }^2}}\right\},{\eta }_1+\zeta \right]$.

In what follows, we set

$$\begin{array}{c}\hfill {\left({\eta }_{k+1}-{\eta }_k\right)}^{-}:=max\left\{0,-\left({\eta }_{k+1}-{\eta }_k\right)\right\}\mathrm{and}{\left({\eta }_{k+1}-{\eta }_k\right)}^{+}:=max\left\{0,{\eta }_{k+1}-{\eta }_k\right\}.\end{array}$$

Combining it with the definition of sequence $\left\{{\eta }_k\right\}$, we have

$$\begin{array}{c}\hfill \sum _{k=1}^{\infty }{\left({\eta }_{k+1}-{\eta }_k\right)}^{+}\le \sum _{k=1}^{\infty }{\zeta }_k\le +\infty .\end{array}$$

Thus, the series ${\displaystyle \sum _{k=1}^{\infty }}{\left({\eta }_{k+1}-{\eta }_k\right)}^{+}$ is convergent. Next, we assert the convergence of the series ${\displaystyle \sum _{k=1}^{\infty }}{\left({\eta }_{k+1}-{\eta }_k\right)}^{-}$. Suppose that ${\displaystyle \sum _{k=1}^{\infty }}{\left({\eta }_{k+1}-{\eta }_k\right)}^{-}=+\infty$. We observe that

$$\begin{array}{c}\hfill {\eta }_{k+1}-{\eta }_k={\left({\eta }_{k+1}-{\eta }_k\right)}^{+}-{\left({\eta }_{k+1}-{\eta }_k\right)}^{-}.\end{array}$$

This implies that

$$\begin{array}{c}\hfill {\eta }_{n+1}-{\eta }_1=\sum _{k=1}^n\left({\eta }_{k+1}-{\eta }_k\right)=\sum _{k=1}^n{\left({\eta }_{k+1}-{\eta }_k\right)}^{+}-\sum _{k=1}^n{\left({\eta }_{k+1}-{\eta }_k\right)}^{-}.\end{array}$$

(17)

Then, by taking $n\to +\infty$ in (17), we have ${\eta }_n\to -\infty$, as $n\to \infty$. This is a contradiction with the boundedness property of $\left\{{\eta }_k\right\}$. Owing to the convergence of the series ${\displaystyle \sum _{k=1}^{\infty }}{\left({\eta }_{k+1}-{\eta }_k\right)}^{+}$ and ${\displaystyle \sum _{k=1}^{\infty }}{\left({\eta }_{k+1}-{\eta }_k\right)}^{-}$, by taking $n\to +\infty$ in (17), we deduce that $\underset{k\to \infty }{lim}{\eta }_k=\eta$ and $\eta \in \left[min\left\{{\eta }_1,{\displaystyle \frac{\phi }{{\parallel A\parallel }^2}}\right\},{\eta }_1+\zeta \right]$.

On the other hand, from the Lipschitz-type continuity of $f_1$ on $H_1$ and of $f_2$ on $H_2$, there exists some positive constants $\{c_1,c_2\}$ and $\{d_1,d_2\}$, respectively, such that

$$f_1(w_k,z_k)-f_1(w_k,y_k)-f_1(y_k,z_k)\le max\left\{c_1,c_2\right\}(\parallel w_k-y_k{\parallel }^2+\parallel y_k-z_k{\parallel }^2),$$

and

$$f_2(A{t}_k,v_k)-f_2(A{t}_k,u_k)-f_2(u_k,v_k)\le max\left\{d_1,d_2\right\}(\parallel A{t}_k-u_k{\parallel }^2+\parallel u_k-v_k{\parallel }^2).$$

These, together with the definitions of ${\lambda }_{k+1}$ and ${\mu }_{k+1}$, and the conditions on the sequences $\left\{{\rho }_k\right\}$ and $\left\{{\delta }_k\right\}$, yield

$${\lambda }_{k+1}\ge min\left\{{\lambda }_k+{\rho }_k,{\displaystyle \frac{\omega }{2max\left\{c_1,c_2\right\}}}\right\}\ge min\left\{{\lambda }_k,{\displaystyle \frac{\omega }{2max\left\{c_1,c_2\right\}}}\right\},$$

and

$${\mu }_{k+1}\ge min\left\{{\mu }_k+{\delta }_k,{\displaystyle \frac{\tau }{2max\left\{d_1,d_2\right\}}}\right\}\ge min\left\{{\mu }_k,{\displaystyle \frac{\tau }{2max\left\{d_1,d_2\right\}}}\right\}.$$

Hence, by induction, we find that the sequences $\left\{{\lambda }_k\right\}$ and $\left\{{\mu }_k\right\}$ have a lower bound as $min\left\{{\lambda }_1,{\displaystyle \frac{\omega }{2max\left\{c_1,c_2\right\}}}\right\}$ and $min\left\{{\mu }_1,{\displaystyle \frac{\tau }{2max\left\{d_1,d_2\right\}}}\right\}$, respectively. Similar to the above technique, we can show that

$$\underset{k\to \infty }{lim}{\lambda }_k=\lambda \in \left[min\left\{{\lambda }_1,{\displaystyle \frac{\omega }{2max\left\{c_1,c_2\right\}}}\right\},{\lambda }_1+\rho \right],\mathrm{where}\rho =\sum _{k=1}^{\infty }{\rho }_k,$$

and

$$\underset{k\to \infty }{lim}{\mu }_k=\mu \in \left[min\left\{{\mu }_1,{\displaystyle \frac{\tau }{2max\left\{d_1,d_2\right\}}}\right\},{\mu }_1+\delta \right],\mathrm{where}\delta =\sum _{k=1}^{\infty }{\delta }_k.$$

This completes the proof. □

Lemma 8.

Let $f_1:H_1\times H_1\to \mathbb{R}$ and $f_2:H_2\times H_2\to \mathbb{R}$ be bifunctions that satisfy $\left(A1\right)$–$\left(A4\right)$, $T:H_1\to H_1$ and $S:H_2\to H_2$ be nonexpansive mappings, $A:H_1\to H_2$ be a bounded linear operator, and $A^{*}:H_2\to H_1$ be the adjoint operator of A. Assume that the solution set Ω is nonempty. Let $w_k\in H_1$ and $A{t}_k\in H_2$. If $y_k$, $z_k$, $u_k$, $v_k$, ${\lambda }_{k+1}$, and ${\mu }_{k+1}$ are created by the process of Algorithm 1, then the following results hold:

$$\parallel z_k-p^{*}{\parallel }^2\le \parallel w_k-p^{*}{\parallel }^2-\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right)\parallel w_k-y_k{\parallel }^2-\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right){\parallel y_k-z_k\parallel }^2,$$

and

$$\parallel v_k-A{p}^{*}{\parallel }^2\le \parallel A{t}_k-A{p}^{*}{\parallel }^2-\left({\displaystyle \frac{1}{{\sigma }_k}}-{\displaystyle \frac{\tau {\mu }_k}{{\mu }_{k+1}}}\right)\parallel A{t}_k-u_k{\parallel }^2-\left({\displaystyle \frac{1}{{\sigma }_k}}-{\displaystyle \frac{\tau {\mu }_k}{{\mu }_{k+1}}}\right){\parallel u_k-v_k\parallel }^2,$$

$\forall p^{*}\in \Omega$.

Proof. 

First, we show that $C_1\subset B_k$ for every $k\in \mathbb{N}$. Let $k\in \mathbb{N}$ be fixed and $z\in C_1$. Since $s_k\in {△}_k^{f_1}$, it follows that there exists $q_k\in N_{C_1}\left(y_k\right)$ such that

$$\begin{array}{c}\hfill {\xi }_k{\lambda }_k{s}_k+y_k=w_k-q_k.\end{array}$$

(18)

So, from $q_k\in N_{C_1}\left(y_k\right)$, we obtain

$$\begin{array}{c}\hfill ⟨w_k-{\xi }_k{\lambda }_k{s}_k-y_k,z-y_k⟩=⟨q_k,z-y_k⟩\le 0.\end{array}$$

(19)

This implies that $z\in B_k$. Hence, we conclude that $C_1\subset B_k$ for every $k\in \mathbb{N}$. Similarly, since $r_k\in {△}_k^{f_2}$, we can show that $C_2\subset D_k$ for every $k\in \mathbb{N}$. As a result, we ensure that Algorithm 1 is well-defined.

Afterwards, by utilizing the above facts, we display the results of the Lemma. Let $p^{*}\in \Omega$. That is, $p^{*}\in EP(f_1,C_1)$, $p^{*}\in Fix\left(T\right)$, and $A{p}^{*}\in EP(f_2,C_2)$, $A{p}^{*}\in Fix\left(S\right)$. Due to $s_k\in {\partial }_2{f}_1(w_k,y_k)$ and the subdifferentiability of $f_1$, we obtain

$$\begin{array}{c}\hfill f_1(w_k,z)-f_1(w_k,y_k)\ge ⟨s_k,z-y_k⟩,\forall z\in H_1.\end{array}$$

It follows from $z_k\in B_k\subset H_1$ that

$$\begin{array}{c}\hfill f_1(w_k,z_k)-f_1(w_k,y_k)\ge ⟨s_k,z_k-y_k⟩.\end{array}$$

(20)

Likewise, by using the definition of $B_k$ and $z_k\in B_k$, we obtain

$$\begin{array}{c}\hfill ⟨w_k-{\xi }_k{\lambda }_k{s}_k-y_k,z_k-y_k⟩\le 0.\end{array}$$

Combined with relation (20), one sees that

$$\begin{array}{c}\hfill {\xi }_k{\lambda }_k[f_1(w_k,z_k)-f_1(w_k,y_k)]\ge ⟨y_k-w_k,y_k-z_k⟩.\end{array}$$

(21)

Furthermore, from the definition of $z_k$ and Lemma 4, we obtain

$$\begin{array}{c}\hfill 0\in {\partial }_2\left\{{\lambda }_k{f}_1(y_k,z_k)+{\displaystyle \frac{1}{2}}{\parallel z_k-w_k\parallel }^2\right\}+N_{B_k}\left(z_k\right).\end{array}$$

Then, there exists $s^{*}\in {\partial }_2{f}_1(y_k,z_k)$ and $q^{*}\in N_{B_k}\left(z_k\right)$ such that

$$\begin{array}{c}\hfill {\lambda }_k{s}^{*}+z_k-w_k+q^{*}=0.\end{array}$$

(22)

This, together with the subdifferentiability of $f_1$, yields

$$\begin{array}{c}\hfill f_1(y_k,z)-f_1(y_k,z_k)\ge ⟨s^{*},z-z_k⟩,\forall z\in H_1.\end{array}$$

(23)

Additionally, from $q^{*}\in N_{B_k}\left(z_k\right)$, we obtain

$$\begin{array}{c}\hfill ⟨q^{*},z_k-z⟩\ge 0,\forall z\in B_k.\end{array}$$

It follows from equality (22) that

$$\begin{array}{c}\hfill ⟨w_k-z_k,z_k-z⟩\ge {\lambda }_k⟨s^{*},z_k-z⟩,\forall z\in B_k.\end{array}$$

(24)

Due to relation (23), we obtain

$$\begin{array}{c}\hfill ⟨w_k-z_k,z_k-z⟩\ge {\lambda }_k[f_1(y_k,z_k)-f_1(y_k,z)],\forall z\in B_k.\end{array}$$

(25)

In particular, from $p^{*}\in C_1\subset B_k$, one sees that

$$\begin{array}{c}\hfill ⟨w_k-z_k,z_k-p^{*}⟩\ge {\lambda }_k[f_1(y_k,z_k)-f_1(y_k,p^{*})].\end{array}$$

Combined with the pseudomonotonic of $f_1$, we have

$$\begin{array}{c}\hfill ⟨w_k-z_k,z_k-p^{*}⟩\ge {\lambda }_k{f}_1(y_k,z_k).\end{array}$$

(26)

So, relations (21) and (26) imply that

$$\begin{array}{c}\hfill {\xi }_k{\lambda }_k[f_1(w_k,z_k)-f_1(w_k,y_k)-f_1(y_k,z_k)]\ge {\xi }_k⟨z_k-w_k,z_k-p^{*}⟩+⟨y_k-w_k,y_k-z_k⟩.\end{array}$$

(27)

On the other hand, we observe from the definition of ${\lambda }_{k+1}$ that

$$\begin{array}{c}\hfill f_1(w_k,z_k)-f_1(w_k,y_k)-f_1(y_k,z_k)\le {\displaystyle \frac{\omega (\parallel w_k-y_k{\parallel }^2+\parallel y_k-z_k{\parallel }^2)}{2{\lambda }_{k+1}}}.\end{array}$$

(28)

Using this together with relation (27), we obtain

$$\begin{array}{c}\hfill {\xi }_k⟨w_k-z_k,z_k-p^{*}⟩\ge ⟨y_k-w_k,y_k-z_k⟩-{\displaystyle \frac{\omega {\xi }_k{\lambda }_k(\parallel w_k-y_k{\parallel }^2+\parallel y_k-z_k{\parallel }^2)}{2{\lambda }_{k+1}}}.\end{array}$$

Owing to the above relation, we have the following facts:

$$\begin{array}{ccc}\hfill {\xi }_k\left(\parallel w_k-p^{*}{\parallel }^2-\parallel w_k-z_k{\parallel }^2-{\parallel z_k-p^{*}\parallel }^2\right)& =& 2{\xi }_k⟨w_k-z_k,z_k-p^{*}⟩\hfill \\ & \ge & 2⟨y_k-w_k,y_k-z_k⟩\hfill \\ & & -{\displaystyle \frac{\omega {\xi }_k{\lambda }_k(\parallel w_k-y_k{\parallel }^2+\parallel y_k-z_k{\parallel }^2)}{{\lambda }_{k+1}}}.\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}\hfill \parallel z_k-p^{*}{\parallel }^2& \le & \parallel w_k-p^{*}{\parallel }^2-{\parallel w_k-z_k\parallel }^2-{\displaystyle \frac{2}{{\xi }_k}}⟨y_k-w_k,y_k-z_k⟩\hfill \\ & & +{\displaystyle \frac{\omega {\lambda }_k(\parallel w_k-y_k{\parallel }^2+\parallel y_k-z_k{\parallel }^2)}{{\lambda }_{k+1}}}\hfill \\ & =& \parallel w_k-p^{*}{\parallel }^2-\parallel w_k-z_k{\parallel }^2+{\displaystyle \frac{1}{{\xi }_k}}\parallel w_k-z_k{\parallel }^2-{\displaystyle \frac{1}{{\xi }_k}}\parallel w_k-y_k{\parallel }^2-{\displaystyle \frac{1}{{\xi }_k}}{\parallel y_k-z_k\parallel }^2\hfill \\ & & +{\displaystyle \frac{\omega {\lambda }_k(\parallel w_k-y_k{\parallel }^2+\parallel y_k-z_k{\parallel }^2)}{{\lambda }_{k+1}}}\hfill \\ & =& \parallel w_k-p^{*}{\parallel }^2-\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right)\parallel w_k-y_k{\parallel }^2-\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right){\parallel y_k-z_k\parallel }^2\hfill \\ & & -\left(1-{\displaystyle \frac{1}{{\xi }_k}}\right){\parallel w_k-z_k\parallel }^2.\hfill \end{array}$$

(29)

Hence, by applying the choice of the parameter ${\xi }_k\in \left[1,\infty \right)$, we deduce that

$$\parallel z_k-p^{*}{\parallel }^2\le \parallel w_k-p^{*}{\parallel }^2-\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right)\parallel w_k-y_k{\parallel }^2-\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right){\parallel y_k-z_k\parallel }^2.$$

Similarly, we can show that

$$\parallel v_k-A{p}^{*}{\parallel }^2\le \parallel A{t}_k-A{p}^{*}{\parallel }^2-\left({\displaystyle \frac{1}{{\sigma }_k}}-{\displaystyle \frac{\tau {\mu }_k}{{\mu }_{k+1}}}\right)\parallel A{t}_k-u_k{\parallel }^2-\left({\displaystyle \frac{1}{{\sigma }_k}}-{\displaystyle \frac{\tau {\mu }_k}{{\mu }_{k+1}}}\right){\parallel u_k-v_k\parallel }^2.$$

This completes the proof. □

We are presently equipped to consider the strong convergence of Algorithm 1.

Theorem 1.

Let $f_1:H_1\times H_1\to \mathbb{R}$ and $f_2:H_2\times H_2\to \mathbb{R}$ be bifunctions that satisfy $\left(A1\right)$–$\left(A4\right)$, $T:H_1\to H_1$ and $S:H_2\to H_2$ be nonexpansive mappings, $A:H_1\to H_2$ be a bounded linear operator, and $A^{*}:H_2\to H_1$ be the adjoint operator of A. Assume that the solution set Ω is nonempty. Then, the sequence $\left\{x_k\right\}$ that is generated by Algorithm 1 converges strongly to the minimum-norm element of Ω.

Proof. 

Let $p^{*}\in \Omega$. That is, $p^{*}\in EP(f_1,C_1)$, $p^{*}\in Fix\left(T\right)$, and $A{p}^{*}\in EP(f_2,C_2)$, $A{p}^{*}\in Fix\left(S\right)$. We start by regarding the definition of $x_{k+1}$ and using the nonexpansivity of $P_{B_k}$ as follows:

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p^{*}{\parallel }^2& \le & \parallel (t_k-p^{*})+{\eta }_k{A}^{*}(S{v}_k-A{t}_k){\parallel }^2\hfill \\ & =& \parallel t_k-p^{*}{\parallel }^2+{\eta }_k^2{\parallel A^{*}(S{v}_k-A{t}_k)\parallel }^2\hfill \\ & & +2{\eta }_k⟨A{t}_k-A{p}^{*},S{v}_k-A{t}_k⟩.\hfill \end{array}$$

(30)

Consider,

$$\begin{array}{ccc}\hfill 2⟨A{t}_k-A{p}^{*},S{v}_k-A{t}_k⟩& =& 2⟨S{v}_k-A{p}^{*},S{v}_k-A{t}_k⟩-2{\parallel S{v}_k-A{t}_k\parallel }^2\hfill \\ & =& \parallel S{v}_k-A{p}^{*}{\parallel }^2-\parallel S{v}_k-A{t}_k{\parallel }^2-{\parallel A{t}_k-A{p}^{*}\parallel }^2.\hfill \end{array}$$

Using this one together with relation (30), we have

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p^{*}{\parallel }^2& \le & \parallel t_k-p^{*}{\parallel }^2+{\eta }_k^2\parallel A^{*}(S{v}_k-A{t}_k){\parallel }^2-{\eta }_k{\parallel S{v}_k-A{t}_k\parallel }^2\hfill \\ & & +{\eta }_k(\parallel S{v}_k-A{p}^{*}{\parallel }^2-\parallel A{t}_k-A{p}^{*}{\parallel }^2).\hfill \end{array}$$

It follows from the nonexpansivity of S and the definition of ${\eta }_{k+1}$ that

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p^{*}{\parallel }^2& \le & \parallel t_k-p^{*}{\parallel }^2+{\displaystyle \frac{\phi {\eta }_k^2}{{\eta }_{k+1}}}\parallel S{v}_k-A{t}_k{\parallel }^2-{\eta }_k{\parallel S{v}_k-A{t}_k\parallel }^2\hfill \\ & & +{\eta }_k(\parallel S{v}_k-A{p}^{*}{\parallel }^2-\parallel A{t}_k-A{p}^{*}{\parallel }^2)\hfill \\ & \le & \parallel t_k-p^{*}{\parallel }^2-{\eta }_k\left(1-{\displaystyle \frac{\phi {\eta }_k}{{\eta }_{k+1}}}\right){\parallel S{v}_k-A{t}_k\parallel }^2\hfill \\ & & +{\eta }_k(\parallel v_k-A{p}^{*}{\parallel }^2-\parallel A{t}_k-A{p}^{*}{\parallel }^2).\hfill \end{array}$$

(31)

Combined with the condition on the parameter $\phi \in (0,1)$ and Lemma 7, we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\left(1-{\displaystyle \frac{\phi {\eta }_k}{{\eta }_{k+1}}}\right)=1-\phi >0.\end{array}$$

So, there exists $k_1\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill 1-{\displaystyle \frac{\phi {\eta }_k}{{\eta }_{k+1}}}>0,\forall k\ge k_1.\end{array}$$

(32)

On the other hand, from the assumption of the parameter $\omega \in (0,1)$, the fact that $\underset{k\to \infty }{lim}{\xi }_k=1$, and Lemma 7, we have

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right)=1-\omega >0.\end{array}$$

Thus, there exists $k_2\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}>0,\forall k\ge k_2.\end{array}$$

(33)

Moreover, considering the choice of the parameter $\tau \in (0,1)$, the fact that $\underset{k\to \infty }{lim}{\sigma }_k=1$, and Lemma 7, we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\left({\displaystyle \frac{1}{{\sigma }_k}}-{\displaystyle \frac{\tau {\mu }_k}{{\mu }_{k+1}}}\right)=1-\tau >0.\end{array}$$

Then, there exists $k_3\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\sigma }_k}}-{\displaystyle \frac{\tau {\mu }_k}{{\mu }_{k+1}}}>0,\forall k\ge k_3.\end{array}$$

(34)

Choose $k_0=max\{k_1,k_2,k_3\}$. Let us now consider the case for every $k\in \mathbb{N}$ satisfying $k\ge k_0$. So, by applying relations (33) and (34) to Lemma 8, we have the following relations:

$$\begin{array}{ccc}\hfill \parallel z_k-p^{*}\parallel & \le & \parallel w_k-p^{*}\parallel ,\hfill \end{array}$$

(35)

and

$$\begin{array}{ccc}\hfill \parallel v_k-A{p}^{*}\parallel & \le & \parallel A{t}_k-A{p}^{*}\parallel .\hfill \end{array}$$

(36)

Using this together with relation (31), we have

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p^{*}{\parallel }^2& \le & \parallel t_k-p^{*}{\parallel }^2-{\eta }_k\left(1-{\displaystyle \frac{\phi {\eta }_k}{{\eta }_{k+1}}}\right){\parallel S{v}_k-A{t}_k\parallel }^2.\hfill \end{array}$$

(37)

This, together with the assumption on the parameter ${\eta }_k$ and the fact (32), yields

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p^{*}{\parallel }^2& \le & \parallel t_k-p^{*}{\parallel }^2.\hfill \end{array}$$

(38)

On the other hand, we note from the definition of $t_k$, the nonexpansivity of T, and relation (35) that

$$\begin{array}{ccc}\hfill \parallel t_k-p^{*}\parallel & =& \parallel (1-{\beta }_k-{\alpha }_k)(z_k-p^{*})+{\alpha }_k(T{z}_k-p^{*})-{\beta }_k{p}^{*}\parallel \hfill \\ & \le & (1-{\beta }_k-{\alpha }_k)\parallel z_k-p^{*}\parallel +{\alpha }_k\parallel T{z}_k-p^{*}\parallel +{\beta }_k\parallel p^{*}\parallel \hfill \\ & \le & (1-{\beta }_k)\parallel w_k-p^{*}\parallel +{\beta }_k\parallel p^{*}\parallel .\hfill \end{array}$$

It follows from the definition of $w_k$ that

$$\begin{array}{ccc}\hfill \parallel t_k-p^{*}\parallel & \le & (1-{\beta }_k)\parallel x_k-p^{*}\parallel +(1-{\beta }_k){\theta }_k\parallel x_k-x_{k-1}\parallel +{\beta }_k\parallel p^{*}\parallel \hfill \\ & =& (1-{\beta }_k)\parallel x_k-p^{*}\parallel +{\beta }_k\left[(1-{\beta }_k){\displaystyle \frac{{\theta }_k}{{\beta }_k}}\parallel x_k-x_{k-1}\parallel +\parallel p^{*}\parallel \right].\hfill \end{array}$$

(39)

Owing to the condition on the sequence $\left\{{\theta }_k\right\}$, we obtain

$$\begin{array}{c}\hfill (1-{\beta }_k){\displaystyle \frac{{\theta }_k}{{\beta }_k}}\parallel x_k-x_{k-1}\parallel \le (1-{\beta }_k){\displaystyle \frac{{ϵ}_k}{{\beta }_k}}.\end{array}$$

This, together with the facts $\underset{k\to \infty }{lim}{\displaystyle \frac{{ϵ}_k}{{\beta }_k}}=0$ and $\underset{k\to \infty }{lim}{\beta }_k=0$, yields

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}(1-{\beta }_k){\displaystyle \frac{{\theta }_k}{{\beta }_k}}\parallel x_k-x_{k-1}\parallel =0.\end{array}$$

(40)

Then, there exists a constant $M_1>0$ such that

$$\begin{array}{c}\hfill (1-{\beta }_k){\displaystyle \frac{{\theta }_k}{{\beta }_k}}\parallel x_k-x_{k-1}\parallel \le M_1.\end{array}$$

(41)

Combined with relations (38) and (39), we have

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p^{*}\parallel & \le & (1-{\beta }_k)\parallel x_k-p^{*}\parallel +{\beta }_k\left(M_1+\parallel p^{*}\parallel \right)\hfill \\ & \le & max\left\{\parallel x_k-p^{*}\parallel ,M_1+\parallel p^{*}\parallel \right\}\hfill \\ & \le & \cdots \hfill \\ & \le & max\left\{\parallel x_{k_0}-p^{*}\parallel ,M_1+\parallel p^{*}\parallel \right\}.\hfill \end{array}$$

It follows that $\{\parallel x_k-p^{*}\parallel \}$ is a bounded sequence. As such, the sequence $\left\{x_k\right\}$ is bounded.

On the other hand, in view of the definition of $w_k$, one sees that

$$\begin{array}{ccc}\hfill \parallel w_k-p^{*}{\parallel }^2& =& \parallel (1+{\theta }_k)(x_k-p^{*})-{\theta }_k(x_{k-1}-p^{*}){\parallel }^2\hfill \\ & =& (1+{\theta }_k)\parallel x_k-p^{*}{\parallel }^2-{\theta }_k\parallel x_{k-1}-p^{*}{\parallel }^2+{\theta }_k(1+{\theta }_k){\parallel x_k-x_{k-1}\parallel }^2\hfill \\ & \le & (1+{\theta }_k)\parallel x_k-p^{*}{\parallel }^2-{\theta }_k\parallel x_{k-1}-p^{*}{\parallel }^2+2{\theta }_k{\parallel x_k-x_{k-1}\parallel }^2\hfill \\ & =& \parallel x_k-p^{*}{\parallel }^2+{\theta }_k(\parallel x_k-p^{*}{\parallel }^2-\parallel x_{k-1}-p^{*}{\parallel }^2)+2{\theta }_k{\parallel x_k-x_{k-1}\parallel }^2.\hfill \end{array}$$

(42)

Furthermore, from the definition of $t_k$ and (12), we obtain

$$\begin{array}{ccc}\hfill \parallel t_k-p^{*}{\parallel }^2& =& \parallel (1-{\beta }_k-{\alpha }_k)(z_k-p^{*})+{\alpha }_k(T{z}_k-p^{*})+{\beta }_k(-p^{*}){\parallel }^2\hfill \\ & \le & (1-{\beta }_k-{\alpha }_k)\parallel z_k-p^{*}{\parallel }^2+{\alpha }_k\parallel T{z}_k-p^{*}{\parallel }^2+{\beta }_k{\parallel p^{*}\parallel }^2\hfill \\ & & -{\alpha }_k(1-{\beta }_k-{\alpha }_k){\parallel z_k-T{z}_k\parallel }^2.\hfill \end{array}$$

(43)

Thus, by utilizing relations (35), (38), (42), and (43), the nonexpansivity of T, and Lemma 8, we have

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p^{*}{\parallel }^2& \le & (1-{\beta }_k-{\alpha }_k)\parallel w_k-p^{*}{\parallel }^2+{\alpha }_k\parallel z_k-p^{*}{\parallel }^2+{\beta }_k{\parallel p^{*}\parallel }^2\hfill \\ & & -{\alpha }_k(1-{\beta }_k-{\alpha }_k){\parallel z_k-T{z}_k\parallel }^2\hfill \\ & \le & (1-{\beta }_k)\parallel w_k-p^{*}{\parallel }^2+{\beta }_k\parallel p^{*}{\parallel }^2-{\alpha }_k(1-{\beta }_k-{\alpha }_k){\parallel z_k-T{z}_k\parallel }^2\hfill \\ & & -{\alpha }_k\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right)\parallel w_k-y_k{\parallel }^2-{\alpha }_k\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right){\parallel y_k-z_k\parallel }^2\hfill \\ & \le & (1-{\beta }_k)\parallel x_k-p^{*}{\parallel }^2+{\theta }_k(1-{\beta }_k)(\parallel x_k-p^{*}{\parallel }^2-\parallel x_{k-1}-p^{*}{\parallel }^2)\hfill \\ & & +2{\theta }_k(1-{\beta }_k)\parallel x_k-x_{k-1}{\parallel }^2+{\beta }_k\parallel p^{*}{\parallel }^2-{\alpha }_k(1-{\beta }_k-{\alpha }_k){\parallel z_k-T{z}_k\parallel }^2\hfill \\ & & -{\alpha }_k\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right)\parallel w_k-y_k{\parallel }^2-{\alpha }_k\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right){\parallel y_k-z_k\parallel }^2.\hfill \end{array}$$

(44)

This implies that

$$\begin{array}{ccc} & & {\alpha }_k(1-{\beta }_k-{\alpha }_k)\parallel z_k-T{z}_k{\parallel }^2+{\alpha }_k\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right)\parallel w_k-y_k{\parallel }^2+{\alpha }_k\left({\displaystyle \frac{1}{{\xi }_k}}-{\displaystyle \frac{\omega {\lambda }_k}{{\lambda }_{k+1}}}\right){\parallel y_k-z_k\parallel }^2\hfill \\ & \le & \parallel x_k-p^{*}{\parallel }^2-\parallel x_{k+1}-p^{*}{\parallel }^2+{\theta }_k(1-{\beta }_k)(\parallel x_k-p^{*}{\parallel }^2-\parallel x_{k-1}-p^{*}{\parallel }^2)\hfill \\ & & +2{\theta }_k(1-{\beta }_k)\parallel x_k-x_{k-1}{\parallel }^2+{\beta }_k{\parallel p^{*}\parallel }^2.\hfill \end{array}$$

(45)

Now, we are ready to assert that the sequence $\left\{x_k\right\}$ converges strongly to $\tilde{p}:=P_{\Omega }\left(0\right)$ by investigating the proof in two cases.

Case 1. Suppose that $\parallel x_{k+1}-\tilde{p}\parallel \le \parallel x_k-\tilde{p}\parallel$, for every $k\ge k_0$. This means the sequence $\{\parallel x_k-\tilde{p}{\parallel \}}_{k\ge k_0}$ is non-increasing. Since the sequence $\{\parallel x_k-\tilde{p}\parallel \}$ is bounded, we can confirm that the limit of $\parallel x_k-\tilde{p}\parallel$ exists. Combining with relation (45) the facts that $\underset{k\to \infty }{lim}{\beta }_k=0$ and $\underset{k\to \infty }{lim}{\theta }_k{\parallel x_k-x_{k-1}\parallel }^2=0$, and the properties of the sequences $\left\{{\alpha }_k\right\}$ and $\left\{{\beta }_k\right\}$, we have

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel w_k-y_k\parallel =0,\end{array}$$

(46)

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel y_k-z_k\parallel =0,\end{array}$$

(47)

and

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel T{z}_k-z_k\parallel =0.\end{array}$$

(48)

These imply that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel w_k-z_k\parallel =0.\end{array}$$

(49)

In addition, from $\underset{k\to \infty }{lim}{\theta }_k\parallel x_k-x_{k-1}\parallel =0$, we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel w_k-x_k\parallel =0.\end{array}$$

(50)

This, together with (49), yields

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel x_k-z_k\parallel =0.\end{array}$$

(51)

It follows from (47) that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel x_k-y_k\parallel =0.\end{array}$$

(52)

Consider

$$\begin{array}{ccc}\hfill \parallel t_k-z_k\parallel & =& \parallel {\alpha }_k(T{z}_k-z_k)-{\beta }_k{z}_k\parallel \hfill \\ & \le & {\alpha }_k\parallel T{z}_k-z_k\parallel +{\beta }_k\parallel z_k\parallel .\hfill \end{array}$$

So, by using (48) and the fact that $\underset{k\to \infty }{lim}{\beta }_k=0$, we have

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel t_k-z_k\parallel =0.\end{array}$$

(53)

Combining it with (51), we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel x_k-t_k\parallel =0.\end{array}$$

(54)

On the other hand, in view of relation (37), one sees that

$$\begin{array}{ccc}\hfill {\eta }_k\left(1-{\displaystyle \frac{\phi {\eta }_k}{{\eta }_{k+1}}}\right){\parallel S{v}_k-A{t}_k\parallel }^2& \le & \parallel t_k-\tilde{p}{\parallel }^2-{\parallel x_{k+1}-\tilde{p}\parallel }^2\hfill \\ & \le & \left(\parallel t_k-z_k\parallel +\parallel z_k-x_k\parallel +\parallel x_k-\tilde{p}\parallel -\parallel x_{k+1}-\tilde{p}\parallel \right)(\parallel t_k-\tilde{p}\parallel \hfill \\ & & +\parallel x_{k+1}-\tilde{p}\parallel ).\hfill \end{array}$$

Thus, by utilizing (51) and (53), and the existence of $\underset{k\to \infty }{lim}\parallel x_k-\tilde{p}\parallel$, we have

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel S{v}_k-A{t}_k\parallel =0.\end{array}$$

(55)

Furthermore, from Lemma 8 and the nonexpansivity of S, we obtain

$$\begin{array}{ccc} & & \left({\displaystyle \frac{1}{{\sigma }_k}}-{\displaystyle \frac{\tau {\mu }_k}{{\mu }_{k+1}}}\right)\parallel A{t}_k-u_k{\parallel }^2+\left({\displaystyle \frac{1}{{\sigma }_k}}-{\displaystyle \frac{\tau {\mu }_k}{{\mu }_{k+1}}}\right){\parallel u_k-v_k\parallel }^2\hfill \\ & \le & \parallel A{t}_k-A\tilde{p}{\parallel }^2-{\parallel v_k-A\tilde{p}\parallel }^2\hfill \\ & \le & (\parallel A{t}_k-S{v}_k\parallel +\parallel S{v}_k-A\tilde{p}\parallel -\parallel v_k-A\tilde{p}\parallel \left)\right(\parallel A{t}_k-A\tilde{p}\parallel +\parallel v_k-A\tilde{p}\parallel )\hfill \\ & \le & \parallel A{t}_k-S{v}_k\parallel (\parallel A{t}_k-A\tilde{p}\parallel +\parallel v_k-A\tilde{p}\parallel ).\hfill \end{array}$$

So, by using (55) and the above relation, we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel A{t}_k-u_k\parallel =0,\end{array}$$

(56)

and

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel u_k-v_k\parallel =0.\end{array}$$

(57)

These imply that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel A{t}_k-v_k\parallel =0.\end{array}$$

(58)

Since $\parallel S{v}_k-v_k\parallel \le \parallel S{v}_k-A{t}_k\parallel +\parallel A{t}_k-v_k\parallel$, it follows from (55) and (58) that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel S{v}_k-v_k\parallel =0.\end{array}$$

(59)

Now, let $x^{*}\in {\omega }_w\left(x_k\right)$ and $\left\{x_{k_n}\right\}$ be a subsequence of $\left\{x_k\right\}$ with $x_{k_n}\rightharpoonup x^{*}$, as $n\to \infty$. By using (50)–(52) and (54), we have $w_{k_n}\rightharpoonup x^{*}$, $y_{k_n}\rightharpoonup x^{*}$, $z_{k_n}\rightharpoonup x^{*}$, and $t_{k_n}\rightharpoonup x^{*}$, as $n\to \infty$. It follows that $A{t}_{k_n}\rightharpoonup A{x}^{*}$, as $n\to \infty$. Combining (56) and (58), we have $u_{k_n}\rightharpoonup A{x}^{*}$ and $v_{k_n}\rightharpoonup A{x}^{*}$ as $n\to \infty$. Since $C_1$ and $C_2$ are closed convex subsets, $C_1$ and $C_2$ are weakly closed. Hence, $x^{*}\in C_1$ and $A{x}^{*}\in C_2$.

Next, due to relations (21), (25) and (28), we obtain

$$\begin{array}{ccc}\hfill {\lambda }_{k_n}{f}_1(y_{k_n},z)& \ge & {\lambda }_{k_n}{f}_1(y_{k_n},z_{k_n})+⟨w_{k_n}-z_{k_n},z-z_{k_n}⟩\hfill \\ & \ge & {\lambda }_{k_n}{f}_1(w_{k_n},z_{k_n})-{\lambda }_{k_n}{f}_1(w_{k_n},y_{k_n})-{\displaystyle \frac{\omega {\lambda }_{k_n}}{2{\lambda }_{k_n+1}}}{\parallel w_{k_n}-y_{k_n}\parallel }^2\hfill \\ & & -{\displaystyle \frac{\omega {\lambda }_{k_n}}{2{\lambda }_{k_n+1}}}{\parallel y_{k_n}-z_{k_n}\parallel }^2+⟨w_{k_n}-z_{k_n},z-z_{k_n}⟩\hfill \\ & \ge & {\displaystyle \frac{1}{{\xi }_{k_n}}}⟨y_{k_n}-w_{k_n},y_{k_n}-z_{k_n}⟩-{\displaystyle \frac{\omega {\lambda }_{k_n}}{2{\lambda }_{k_n+1}}}\parallel w_{k_n}-y_{k_n}{\parallel }^2-{\displaystyle \frac{\omega {\lambda }_{k_n}}{2{\lambda }_{k_n+1}}}{\parallel y_{k_n}-z_{k_n}\parallel }^2\hfill \\ & & +⟨w_{k_n}-z_{k_n},z-z_{k_n}⟩,\hfill \end{array}$$

(60)

for each $z\in C_1$. So, by utilizing the facts (46), (47) and (49), and the boundedness of $\left\{z_k\right\}$, we find that the right-hand side of the inequality (60) tends to zero. It follows from the sequentially weakly upper semicontinuity of $f_1$ and ${\lambda }_{k_n}>0$ that

$$\begin{array}{c}\hfill 0\le \underset{n\to \infty }{lim\; sup}{f}_1(y_{k_n},z)\le f_1(x^{*},z),\forall z\in C_1.\end{array}$$

Similarly, we can show that

$$\begin{array}{ccc}\hfill {\mu }_{k_n}{f}_2(u_{k_n},v)& \ge & {\displaystyle \frac{1}{{\sigma }_{k_n}}}⟨u_{k_n}-A{t}_{k_n},u_{k_n}-v_{k_n}⟩-{\displaystyle \frac{\tau {\mu }_{k_n}}{2{\mu }_{k_n+1}}}{\parallel A{t}_{k_n}-u_{k_n}\parallel }^2\hfill \\ & & -{\displaystyle \frac{\tau {\mu }_{k_n}}{2{\mu }_{k_n+1}}}{\parallel u_{k_n}-v_{k_n}\parallel }^2+⟨A{t}_{k_n}-v_{k_n},v-v_{k_n}⟩,\hfill \end{array}$$

(61)

for each $v\in C_2$. Using this one together with (56)–(58), and the boundedness of $\left\{v_k\right\}$, we find that the right-hand side of the inequality (61) tends to zero. Thus, by using the sequentially weakly upper semicontinuity of $f_2$ and ${\mu }_{k_n}>0$, we have

$$\begin{array}{c}\hfill 0\le \underset{n\to \infty }{lim\; sup}{f}_2(u_{k_n},v)\le f_2(A{x}^{*},v),\forall v\in C_2.\end{array}$$

On the other hand, since $z_{k_n}\rightharpoonup x^{*}$, as $n\to \infty$, and considering (48), by the demiclosedness at zero of $I-T$, we have $x^{*}\in Fix\left(T\right)$. Similarly, since $v_{k_n}\rightharpoonup A{x}^{*}$, as $n\to \infty$, and considering (59), it follows from the demiclosedness at zero of $I-S$ that $A{x}^{*}\in Fix\left(S\right)$. Hence, we can conclude that $x^{*}\in \Omega$.

Put $h_k=(1-{\alpha }_k)z_k+{\alpha }_{k}T{z}_k$. Relation (35) and the nonexpansivity of T imply that

$$\begin{array}{ccc}\hfill \parallel h_k-\tilde{p}\parallel & \le & (1-{\alpha }_k)\parallel z_k-\tilde{p}\parallel +{\alpha }_k\parallel T{z}_k-\tilde{p}\parallel \hfill \\ & \le & \parallel w_k-\tilde{p}\parallel .\hfill \end{array}$$

(62)

From the definition of $t_k$, one sees that

$$\begin{array}{ccc}\hfill t_k& =& h_k-{\beta }_k{z}_k\hfill \\ & =& (1-{\beta }_k)h_k-{\beta }_k(z_k-h_k)\hfill \\ & =& (1-{\beta }_k)h_k-{\beta }_k{\alpha }_k(z_k-T{z}_k).\hfill \end{array}$$

It follows from (11) that

$$\begin{array}{ccc}\hfill \parallel t_k-\tilde{p}{\parallel }^2& =& \parallel (1-{\beta }_k)(h_k-\tilde{p})-{\beta }_k{\alpha }_k(z_k-T{z}_k)-{\beta }_k\tilde{p}{\parallel }^2\hfill \\ & \le & (1-{\beta }_k){\parallel h_k-\tilde{p}\parallel }^2-2{\beta }_k{\alpha }_k⟨z_k-T{z}_k,t_k-\tilde{p}⟩\hfill \\ & & -2{\beta }_k⟨\tilde{p},t_k-\tilde{p}⟩.\hfill \end{array}$$

(63)

Using this together with relations (38) and (62), we obtain

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-\tilde{p}{\parallel }^2& \le & (1-{\beta }_k){\parallel w_k-\tilde{p}\parallel }^2+{\beta }_k[2{\alpha }_k⟨T{z}_k-z_k,t_k-\tilde{p}⟩\hfill \\ & & +2⟨t_k-\tilde{p},-\tilde{p}⟩].\hfill \end{array}$$

(64)

Consider

$$\begin{array}{ccc}\hfill \parallel w_k-\tilde{p}{\parallel }^2& \le & (\parallel x_k-\tilde{p}\parallel +{\theta }_k\parallel x_k-x_{k-1}{\parallel )}^2\hfill \\ & \le & \parallel x_k-\tilde{p}{\parallel }^2+2{\theta }_k\parallel x_k-\tilde{p}\parallel \parallel x_k-x_{k-1}\parallel +{\theta }_k{\parallel x_k-x_{k-1}\parallel }^2\hfill \\ & \le & \parallel x_k-\tilde{p}{\parallel }^2+3M_2{\theta }_k\parallel x_k-x_{k-1}\parallel ,\hfill \end{array}$$

(65)

where $M_2=\underset{k\ge k_0}{sup}\{\parallel x_k-\tilde{p}\parallel ,\parallel x_k-x_{k-1}\parallel \}$. It follows from relation (64) that

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-\tilde{p}{\parallel }^2& \le & (1-{\beta }_k)\parallel x_k-\tilde{p}{\parallel }^2+3M_2{\theta }_k(1-{\beta }_k)\parallel x_k-x_{k-1}\parallel \hfill \\ & & +{\beta }_k\left[2{\alpha }_k⟨T{z}_k-z_k,t_k-\tilde{p}⟩+2⟨t_k-\tilde{p},-\tilde{p}⟩\right]\hfill \\ & =& (1-{\beta }_k)\parallel x_k-\tilde{p}{\parallel }^2+{\beta }_k[3M_2(1-{\beta }_k){\displaystyle \frac{{\theta }_k}{{\beta }_k}}\parallel x_k-x_{k-1}\parallel \hfill \\ & & +2{\alpha }_k⟨T{z}_k-z_k,t_k-\tilde{p}⟩+2⟨t_k-\tilde{p},-\tilde{p}⟩].\hfill \end{array}$$

(66)

Indeed, from $x^{*}\in {\omega }_w\left(x_k\right)\subset \Omega$ and the properties of $\tilde{p}:=P_{\Omega }\left(0\right)$, we have

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; sup}⟨t_k-\tilde{p},-\tilde{p}⟩=\underset{n\to \infty }{lim}⟨t_{k_n}-\tilde{p},-\tilde{p}⟩=⟨x^{*}-\tilde{p},-\tilde{p}⟩\le 0.\end{array}$$

(67)

Therefore, by using (40), (48), (66), (67), and Lemma 5, we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel x_k-\tilde{p}\parallel =0.\end{array}$$

(68)

Case 2. Suppose that there exists a subsequence $\{\parallel x_{k_i}-\tilde{p}\parallel \}$ of $\{\parallel x_k-\tilde{p}\parallel \}$ satisfying

$$\begin{array}{c}\hfill \parallel x_{k_i}-\tilde{p}\parallel <\parallel x_{k_i+1}-\tilde{p}\parallel ,\forall i\in \mathbb{N}.\end{array}$$

According to Lemma 6, there exists a non-decreasing sequence $\left\{m_n\right\}\subset \mathbb{N}$ with $\underset{n\to \infty }{lim}{m}_n=\infty$, and

$$\begin{array}{c}\hfill \parallel x_{m_n}-\tilde{p}\parallel \le \parallel x_{m_n+1}-\tilde{p}\parallel \mathrm{and}\parallel x_n-\tilde{p}\parallel \le \parallel x_{m_n+1}-\tilde{p}\parallel ,\forall n\in \mathbb{N}.\end{array}$$

(69)

Using this one together with relation (45), we have

$$\begin{array}{ccc} & & {\alpha }_{m_n}(1-{\beta }_{m_n}-{\alpha }_{m_n})\parallel z_{m_n}-T{z}_{m_n}{\parallel }^2+{\alpha }_{m_n}\left({\displaystyle \frac{1}{{\xi }_{m_n}}}-{\displaystyle \frac{\omega {\lambda }_{m_n}}{{\lambda }_{m_n+1}}}\right){\parallel w_{m_n}-y_{m_n}\parallel }^2\hfill \\ & & +{\alpha }_{m_n}\left({\displaystyle \frac{1}{{\xi }_{m_n}}}-{\displaystyle \frac{\omega {\lambda }_{m_n}}{{\lambda }_{m_n+1}}}\right){\parallel y_{m_n}-z_{m_n}\parallel }^2\hfill \\ & \le & \parallel x_{m_n}-\tilde{p}{\parallel }^2-\parallel x_{m_n+1}-\tilde{p}{\parallel }^2+{\theta }_{m_n}(1-{\beta }_{m_n})(\parallel x_{m_n}-\tilde{p}{\parallel }^2-\parallel x_{m_n-1}-\tilde{p}{\parallel }^2)\hfill \\ & & +2{\theta }_{m_n}(1-{\beta }_{m_n})\parallel x_{m_n}-x_{m_n-1}{\parallel }^2+{\beta }_{m_n}{\parallel \tilde{p}\parallel }^2\hfill \\ & \le & {\theta }_{m_n}(1-{\beta }_{m_n})\parallel x_{m_n}-x_{m_n-1}\parallel (\parallel x_{m_n}-\tilde{p}\parallel +\parallel x_{m_n-1}-\tilde{p}\parallel )\hfill \\ & & +2{\theta }_{m_n}(1-{\beta }_{m_n})\parallel x_{m_n}-x_{m_n-1}{\parallel }^2+{\beta }_{m_n}{\parallel \tilde{p}\parallel }^2.\hfill \end{array}$$

Following the line proof of Case 1, we can check that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel w_{m_n}-y_{m_n}\parallel =0,\underset{n\to \infty }{lim}\parallel y_{m_n}-z_{m_n}\parallel =0,\end{array}$$

(70)

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel x_{m_n}-z_{m_n}\parallel =0,\underset{n\to \infty }{lim}\parallel x_{m_n}-t_{m_n}\parallel =0,\end{array}$$

(71)

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel T{z}_{m_n}-z_{m_n}\parallel =0,\underset{n\to \infty }{lim}\parallel S{v}_{m_n}-v_{m_n}\parallel =0,\end{array}$$

(72)

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel A{t}_{m_n}-v_{m_n}\parallel =0,\underset{n\to \infty }{lim}\parallel A{t}_{m_n}-u_{m_n}\parallel =0,\end{array}$$

(73)

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}⟨t_{m_n}-\tilde{p},-\tilde{p}⟩\le 0,\end{array}$$

(74)

and

$$\begin{array}{ccc}\hfill \parallel x_{m_n+1}-\tilde{p}{\parallel }^2& \le & (1-{\beta }_{m_n})\parallel x_{m_n}-\tilde{p}{\parallel }^2+{\beta }_{m_n}[3M_2(1-{\beta }_{m_n}){\displaystyle \frac{{\theta }_{m_n}}{{\beta }_{m_n}}}\parallel x_{m_n}-x_{m_n-1}\parallel \hfill \\ & & +2{\alpha }_{m_n}⟨T{z}_{m_n}-z_{m_n},t_{m_n}-\tilde{p}⟩+2⟨t_{m_n}-\tilde{p},-\tilde{p}⟩].\hfill \end{array}$$

This, together with relation (69), yields

$$\begin{array}{ccc}\hfill \parallel x_{m_n+1}-\tilde{p}{\parallel }^2& \le & (1-{\beta }_{m_n})\parallel x_{m_n+1}-\tilde{p}{\parallel }^2+{\beta }_{m_n}[3M_2(1-{\beta }_{m_n}){\displaystyle \frac{{\theta }_{m_n}}{{\beta }_{m_n}}}\parallel x_{m_n}-x_{m_n-1}\parallel \hfill \\ & & +2{\alpha }_{m_n}⟨T{z}_{m_n}-z_{m_n},t_{m_n}-\tilde{p}⟩+2⟨t_{m_n}-\tilde{p},-\tilde{p}⟩].\hfill \end{array}$$

It follows from relation (69), again, that

$$\begin{array}{ccc}\hfill \parallel x_n-\tilde{p}{\parallel }^2& \le & 3M_2(1-{\beta }_{m_n}){\displaystyle \frac{{\theta }_{m_n}}{{\beta }_{m_n}}}\parallel x_{m_n}-x_{m_n-1}\parallel +2{\alpha }_{m_n}⟨T{z}_{m_n}-z_{m_n},t_{m_n}-\tilde{p}⟩\hfill \\ & & +2⟨t_{m_n}-\tilde{p},-\tilde{p}⟩\hfill \end{array}$$

Then, by using (40), (72) and (74), we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\parallel x_n-\tilde{p}\parallel }^2\le 0.\end{array}$$

Therefore, the sequence $\left\{x_n\right\}$ converges strongly to $\tilde{p}=P_{\Omega }\left(0\right)$. This completes the proof. □

4. Numerical Experiments

This section presents some numerical experiments occurring in finite- and infinite-dimensional Hilbert spaces to illustrate the effectiveness of the introduced Algorithm 1 and compare it with the EIM Algorithm. We focus on the effectiveness of two groups of auxiliary parameters: ${\xi }_k,{\sigma }_k$, and ${\rho }_k,{\delta }_k,{\zeta }_k$, based on different choices of these parameters. All the numerical computations were implemented in Matlab R2021b and performed on an Apple M1 with 8.00 GB RAM.

Example 1.

Let $H_1={\mathbb{R}}^n$ and $H_2={\mathbb{R}}^m$ be equipped with the Euclidean norm. We consider the functions $g_1:{\mathbb{R}}^n\to \mathbb{R}$ and $g_2:{\mathbb{R}}^m\to \mathbb{R}$, which are formed by $g_{1}x={\displaystyle \frac{1}{2}}{x}^{T}Vx$ and $g_{2}x=\parallel x\parallel$, where $V\in {\mathbb{R}}^{n\times n}$ is an invertible symmetric positive semidefinite matrix. The mappings $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ and $S:{\mathbb{R}}^m\to {\mathbb{R}}^m$ with respect to the functions $g_1$ and $g_2$, respectively, are given as follows:

$$\begin{array}{c}\hfill Tx={(I_n+V)}^{-1}\left(x\right),\end{array}$$

and

$$\begin{array}{c}\hfill Sx=\left\{\begin{array}{c}\left(1-{\displaystyle \frac{1}{\parallel x\parallel }}\right)x,if\parallel x\parallel \ge 1,\hfill \\ 0,otherwise,\hfill \end{array}\right.\end{array}$$

where $I_n$ is the identity matrix of dimension n. Observe that T and S are nonexpansive mappings, so that $Fix\left(T\right)=argmin{g}_1$ and $Fix\left(S\right)=argmin{g}_2$; see [38].

On the other hand, we regard the bifunctions $\widetilde{f_1}$ and $\widetilde{f_2}$, which are manufactured from the Nash–Cournot oligopolistic equilibrium models of electricity markets; see [33,39],

$$\begin{array}{c}\widetilde{f_1}(x,y)=⟨P_{1}x+P_{2}y,y-x⟩,\forall x,y\in {\mathbb{R}}^n,\hfill \\ \widetilde{f_2}(u,v)=⟨Q_{1}u+Q_{2}v,v-u⟩,\forall u,v\in {\mathbb{R}}^m,\hfill \end{array}$$

where $P_1$, $P_2\in {\mathbb{R}}^{n\times n}$ and $Q_1$, $Q_2\in {\mathbb{R}}^{m\times m}$ are matrices such that $P_2$, $Q_2$ are symmetric positive semidefinite and $P_2-P_1$, $Q_2-Q_1$ are negative semidefinite. We notice that $\widetilde{f_1}(x,y)+\widetilde{f_1}(y,x)={(x-y)}^T(P_2-P_1)(x-y),\forall x,y\in {\mathbb{R}}^n$. So, by utilizing the qualification of $P_2-P_1$, we find that $\widetilde{f_1}$ is monotone. Likewise, we find that $\widetilde{f_2}$ is monotone.

Afterwards, the bifunctions $f_1$ and $f_2$ are imposed as follows:

$$\begin{array}{c}\hfill f_1(x,y)=\left\{\begin{array}{c}\widetilde{f_1}(x,y),if(x,y)\in C_1\times C_1,\hfill \\ 0,otherwise,\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill f_2(u,v)=\left\{\begin{array}{c}\widetilde{f_2}(u,v),if(u,v)\in C_2\times C_2,\hfill \\ 0,otherwise,\hfill \end{array}\right.\end{array}$$

where $C_1={\prod }_{i=1}^n[-5,5]$ and $C_2={\prod }_{j=1}^m[-20,20]$ are the constrained boxes. Notice that $f_1$ and $f_2$ are Lipschitz-type continuous; see [12,19].

Through this numerical experiment, the matrices $P_1$, $P_2$, $Q_1$, and $Q_2$ were generated randomly in the interval of $[-5,5]$ such that they satisfy the qualifications mentioned above. The linear operator $A:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is an $m\times n$ matrix, in which each of its entries is generated randomly in the interval of $[-2,2]$. Notice that the solution set Ω is nonempty due to $0\in \Omega$. Algorithm 1 was tested in conjunction with the EIM Algorithm by applying the stopping criteria ${\displaystyle \frac{\parallel x_{k+1}-x_k\parallel }{\parallel x_k\parallel +1}}<{10}^{-6}$. The starting points $x_0=x_1\in {\mathbb{R}}^n$ were generated randomly in the interval of $[-5,5]$. We randomly selected 10 starting points, and the average results are shown, where $n=10$ and $m=20$. Also, we took into account the control parameters of Algorithm 1 and the EIM Algorithm as follows.

• In Algorithm 1, we chose $\omega =\tau =\phi =0.6$, ${\lambda }_1={\mu }_1={\eta }_1=0.9$, ${\gamma }_k={\displaystyle \frac{1}{k+1}}$, ${ϵ}_k={\displaystyle \frac{1}{{(k+1)}^2}}$, ${\beta }_k={\displaystyle \frac{1}{k+1}}$, ${\alpha }_k=0.8(1-{\beta }_k)$, and ${\theta }_k={\overline{\theta }}_k$.
• In the EIM Algorithm, we set $\gamma =0.5$, ${\eta }_k={\displaystyle \frac{1}{{2\parallel A\parallel }^2}}$, ${ϵ}_k={\displaystyle \frac{10}{k^2}}$, ${\beta }_k={10}^{-10}+{\displaystyle \frac{1}{k+1}}$, ${\alpha }_k={\displaystyle \frac{1}{k+3}}$, and ${\theta }_k={\overline{\theta }}_k$, which are the appropriate values as presented in [27]. Indeed, $c_1=c_2={\displaystyle \frac{1}{2}}\parallel P_1-P_2\parallel$ and $d_1=d_2={\displaystyle \frac{1}{2}}\parallel Q_1-Q_2\parallel$ are Lipschitz constants of $f_1$ and $f_2$, respectively. Observe that, if $b_1=max\{c_1,d_1\}$ and $b_2=max\{c_2,d_2\}$, then $b_1$ and $b_2$ are Lipschitz constants of both $f_1$ and $f_2$. Thus, we set ${\mu }_k={\lambda }_k={\displaystyle \frac{1}{max\{b_1,b_2\}}}$.

In order to evaluate the optimal values of the control parameters, for the first experiment, the numerical results were obtained by taking the variation of the parameters ${\rho }_k$, ${\delta }_k$, and ${\zeta }_k$. The results of the numerical comparison are presented in Table 1 for any collections of parameters ${\rho }_k=0$, ${\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, ${\delta }_k=0$, ${\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, and ${\zeta }_k=0$, and ${\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, where the parameters ${\xi }_k={\sigma }_k=1$ are fixed. We observe that when ${\rho }_k=0$, ${\delta }_k=0$, and ${\zeta }_k=0$, the step sizes taken into consideration in Algorithm 1 reduces to forms equivalent to the step sizes provided in [21,36,40].

Table 1. For ${\xi }_k={\sigma }_k=1,$ for all $k\in \mathbb{N}$. Numerical performance for Example 1 based on the different choices of parameters ${\rho }_k$, ${\delta }_k$, and ${\zeta }_k$.

Algorithm 1	${\mathit{\rho }}_{\mathit{k}}=0$				${\mathit{\rho }}_{\mathit{k}}=\frac{1}{{(\mathit{k}+1)}^{1.2}}$				EIM Algorithm
	${\mathit{\delta }}_{\mathit{k}}=\mathbf{0}$		${\mathit{\delta }}_{\mathit{k}}={\displaystyle \frac{\mathbf{1}}{{(\mathit{k}+\mathbf{1})}^{\mathbf{1.2}}}}$		${\mathit{\delta }}_{\mathit{k}}=\mathbf{0}$		${\mathit{\delta }}_{\mathit{k}}={\displaystyle \frac{\mathbf{1}}{{(\mathit{k}+\mathbf{1})}^{\mathbf{1.2}}}}$
	Iter	Time	Iter	Time	Iter	Time	Iter	Time	Iter	Time
${\zeta }_k=0$	16.8	0.07	16.6	0.06	16.4	0.06	16.3	0.05	22.5	0.08
${\zeta }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$	15.8	0.05	15.7	0.05	15.6	0.04	15.4	0.04

In Table 1, the number of iterations (Iter) and the CPU time (Time) in seconds are displayed. Also, we may suggest that the parameters ${\rho }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, ${\delta }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, and ${\zeta }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$ better demonstrate the number of iterations and the CPU time than the parameters ${\rho }_k=0$, ${\delta }_k=0$, and ${\zeta }_k=0$, respectively. This illustrates that Algorithm 1 can be improved efficiently by selecting parameters ${\xi }_k,{\rho }_k$, and ${\delta }_k$. Additionally, the results indicate that Algorithm 1 performs most effectively according to the suggested parameters ${\rho }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, ${\delta }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, and ${\zeta }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$. Eventually, when the parameters ${\xi }_k=1$ and ${\sigma }_k=1$, Algorithm 1 is superior to the EIM Algorithm in terms of the number of iterations and the CPU time.

In the ensuing experiment, we regard the numerical results of the control parameters ${\xi }_k$ and ${\sigma }_k$, which are included to allow for bias in the bifunctions $f_1$ and $f_2$, respectively. The numerical computations are presented in Table 2 for different choices of control parameters ${\xi }_k=1+{\displaystyle \frac{1}{k^{100}}}$, $1+{\displaystyle \frac{1}{k}}$, $1+{\displaystyle \frac{1}{\sqrt{k}}}$, $1+{\displaystyle \frac{1}{log(k+1)}}$, and ${\sigma }_k=1+{\displaystyle \frac{1}{k^{100}}}$, $1+{\displaystyle \frac{1}{k}}$, $1+{\displaystyle \frac{1}{\sqrt{k}}}$, $1+{\displaystyle \frac{1}{log(k+1)}}$, while preserving fixed values of the control parameters ${\rho }_k={\delta }_k={\zeta }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$.

Table 2. For ${\rho }_k={\delta }_k={\zeta }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, for all $k\in \mathbb{N}$. Numerical performance for Example 1 based on the different choices of parameters ${\xi }_k$ and ${\sigma }_k$.

Algorithm 1	Number of Iterations
	${\mathit{\xi }}_{\mathit{k}}=\mathbf{1}+{\displaystyle \frac{\mathbf{1}}{{\mathit{k}}^{\mathbf{100}}}}$	${\mathit{\xi }}_{\mathit{k}}=\mathbf{1}+{\displaystyle \frac{\mathbf{1}}{\mathit{k}}}$	${\mathit{\xi }}_{\mathit{k}}=\mathbf{1}+{\displaystyle \frac{\mathbf{1}}{\sqrt{\mathit{k}}}}$	${\mathit{\xi }}_{\mathit{k}}=\mathbf{1}+{\displaystyle \frac{\mathbf{1}}{log(\mathit{k}+\mathbf{1})}}$
${\sigma }_k=1+{\displaystyle \frac{1}{k^{100}}}$	13.1	13.9	13.9	15.3
${\sigma }_k=1+{\displaystyle \frac{1}{k}}$	13.1	13.8	14.0	15.3
${\sigma }_k=1+{\displaystyle \frac{1}{\sqrt{k}}}$	13.4	13.8	14.0	15.2
${\sigma }_k=1+{\displaystyle \frac{1}{log(k+1)}}$	13.9	13.9	14.3	15.3

Table 2 points out that the choices of pertinent parameters, especially the values of parameters ${\xi }_k$ and ${\sigma }_k$, are important. The parameters ${\xi }_k$ and ${\sigma }_k$, which converge to 1 the fastest, such as ${\xi }_k=1+{\displaystyle \frac{1}{k^{100}}}$ and ${\sigma }_k=1+{\displaystyle \frac{1}{k^{100}}}$, exhibit better performance than other cases when considering the number of iterations. Nevertheless, we point out that the fastest converging sequence to 1 is the constant sequence 1, but the numerical results turn out to be slow (see Table 1). This demonstrates that the proposed parameters ${\xi }_k$ and ${\sigma }_k$ in Algorithm 1 lead to the improved performance of Algorithm 1.

Example 2.

Let $H_1=H_2=L^2\left([0,1]\right)$ be two infinite-dimensional Hilbert spaces endowed with inner product

$$\begin{array}{c}\hfill ⟨x,y⟩=\underset{0}{\overset{1}{\int }}x\left(t\right)y\left(t\right)dt,\forall x,y\in L^2\left([0,1]\right),\end{array}$$

and its corresponding norm

$$\begin{array}{c}\hfill \parallel x\parallel ={\left(\underset{0}{\overset{1}{\int }}\mid x\left(t\right){\mid }^{2}dt\right)}^{{\displaystyle \frac{1}{2}}},\forall x\in L^2\left([0,1]\right).\end{array}$$

Assume that $R_1$, $R_2$, and $r_1$, $r_2$ are positive real numbers such that $R_1/(k_1+1)<r_1/k_1<r_1<R_1$, for some $k_1>1$ and $R_2/(k_2+1)<r_2/k_2<r_2<R_2$, for some $k_2>1$. Take the feasible sets as $C_1=\left\{x\in L^2\left([0,1]\right)|\parallel x\parallel \le r_1\right\}$ and $C_2=\left\{x\in L^2\left([0,1]\right)|\parallel x\parallel \le r_2\right\}$. To be considered here are the operators $F_1:L^2\left([0,1]\right)\to L^2\left([0,1]\right)$ and $F_2:L^2\left([0,1]\right)\to L^2\left([0,1]\right)$, which are defined by

$$\begin{array}{c}\hfill F_{1}x=\left(R_1-\parallel x\parallel \right)x,\mathit{and}{F}_{2}x=\left(R_2-\parallel x\parallel \right)x,\forall x\in L^2\left([0,1]\right).\end{array}$$

Set

$$\begin{array}{c}\hfill f_1(x,y)=⟨F_{1}x,y-x⟩,\mathit{and}{f}_2(x,y)=⟨F_{2}x,y-x⟩,\forall x,y\in L^2\left([0,1]\right).\end{array}$$

Observe that the operators $F_1$ and $F_2$ are psuedomonotone rather than monotone and satisfy Lipschitz continuity; see [41]. These imply that $f_1$ and $f_2$ are psuedomonotone and Lipschitz-type continuous bifunctions. Here, we consider the nonexpansive mappings $T:L^2\left([0,1]\right)\to L^2\left([0,1]\right)$ and $S:L^2\left([0,1]\right)\to L^2\left([0,1]\right)$, which are defined by $Tx={\displaystyle \frac{x}{3}}$ and $Sx={\displaystyle \frac{x}{5}}$, $x\in L^2\left([0,1]\right)$. Also, the linear operator $A:L^2\left([0,1]\right)\to L^2\left([0,1]\right)$ is imposed by $Ax=x$, $x\in L^2\left([0,1]\right)$.

During this numerical experiment, the control parameters of Algorithm 1 are determined as in Example 1. Meanwhile, for the EIM Algorithm, we are required to know the Lipschitz constants of $f_1$ and $f_2$. Indeed, $c_1=c_2={\displaystyle \frac{1}{2}}(2R_1+r_1)$ and $d_1=d_2={\displaystyle \frac{1}{2}}(2R_2+r_2)$ are Lipschitz constants of $f_1$ and $f_2$, respectively. Similar to Example 1, if $b_1=max\{c_1,d_1\}$ and $b_2=max\{c_2,d_2\}$, then $b_1$ and $b_2$ are Lipschitz constants of both $f_1$ and $f_2$. So, we choose ${\mu }_k={\lambda }_k={\displaystyle \frac{1}{max\{b_1,b_2\}}}$ for the EIM Algorithm. The remaining parameters of the EIM Algorithm are given as in Example 1. Here, we take $R_1=1.5$, $r_1=1$, $k_1=1.1$, and $R_2=1.7$, $r_2=1$, $k_2=1.2$. In the above setting, the solution set Ω is $x^{*}\left(t\right)=0$; see [41]. Algorithm 1 was tested along with the EIM Algorithm by utilizing the stopping criteria ${\displaystyle \frac{\parallel x_{k+1}-x_k\parallel }{\parallel x_k\parallel +1}}<{10}^{-6}$ with the starting points $x_0\left(t\right)=x_1\left(t\right)=3exp\left(t\right)$. The numerical results are reported in Table 3 for any collections of parameters ${\rho }_k=0$, ${\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, ${\delta }_k=0$, ${\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, and ${\zeta }_k=0$, ${\displaystyle \frac{1}{{(k+1)}^{1.2}}}$ by fixing the parameters ${\xi }_k={\sigma }_k=1$ as in Example 1.

Table 3. For ${\xi }_k={\sigma }_k=1,$ for all $k\in \mathbb{N}$. Numerical behavior for Example 2 based on the different choices of parameters ${\rho }_k$, ${\delta }_k$, and ${\zeta }_k$.

Algorithm 1	${\mathit{\rho }}_{\mathit{k}}=0$				${\mathit{\rho }}_{\mathit{k}}=\frac{1}{{(\mathit{k}+1)}^{1.2}}$				EIM Algorithm
	${\mathit{\delta }}_{\mathit{k}}=\mathbf{0}$		${\mathit{\delta }}_{\mathit{k}}={\displaystyle \frac{\mathbf{1}}{{(\mathit{k}+\mathbf{1})}^{\mathbf{1.2}}}}$		${\mathit{\delta }}_{\mathit{k}}=\mathbf{0}$		${\mathit{\delta }}_{\mathit{k}}={\displaystyle \frac{\mathbf{1}}{{(\mathit{k}+\mathbf{1})}^{\mathbf{1.2}}}}$
	Iter	Time	Iter	Time	Iter	Time	Iter	Time	Iter	Time
${\zeta }_k=0$	9	0.05	9	0.05	9	0.05	9	0.05	17	0.11
${\zeta }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$	9	0.05	9	0.05	9	0.05	9	0.05

Table 3 indicates that the number of iterations and the CPU time of the concerned parameters ${\rho }_k$, ${\delta }_k$, and ${\zeta }_k$ in all cases are equal. Ultimately, when the parameters ${\xi }_k=1$ and ${\sigma }_k=1$, Algorithm 1 outperforms the EIM Algorithm in terms of both the number of iterations and the CPU time.

To see the optimum values of the control parameters, the following numerical comparison results were obtained while accounting for the different choices of the control parameters ${\xi }_k=1+{\displaystyle \frac{1}{k^{100}}}$, $1+{\displaystyle \frac{1}{k}}$, $1+{\displaystyle \frac{1}{\sqrt{k}}}$, $1+{\displaystyle \frac{1}{log(k+1)}}$, and ${\sigma }_k=1+{\displaystyle \frac{1}{k^{100}}}$, $1+{\displaystyle \frac{1}{k}}$, $1+{\displaystyle \frac{1}{\sqrt{k}}}$, $1+{\displaystyle \frac{1}{log(k+1)}}$ by choosing the appropriate values of parameters ${\rho }_k={\delta }_k={\zeta }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$ as in Example 1.

From Table 4, the number of iterations in the case of the optimal parameters ${\xi }_k=1+{\displaystyle \frac{1}{k^{100}}}$ and ${\sigma }_k=1+{\displaystyle \frac{1}{k^{100}}}$, in which these parameters converge to 1 the fastest, illustrates a better behavior than other cases.

Table 4. For ${\rho }_k={\delta }_k={\zeta }_k={\displaystyle \frac{1}{{(k+1)}^{1.2}}}$, for all $k\in \mathbb{N}$. Numerical behavior for Example 2 based on the different choices of parameters ${\xi }_k$ and ${\sigma }_k$.

Algorithm 1	Number of Iterations
	${\mathit{\xi }}_{\mathit{k}}=\mathbf{1}+{\displaystyle \frac{\mathbf{1}}{{\mathit{k}}^{\mathbf{100}}}}$	${\mathit{\xi }}_{\mathit{k}}=\mathbf{1}+{\displaystyle \frac{\mathbf{1}}{\mathit{k}}}$	${\mathit{\xi }}_{\mathit{k}}=\mathbf{1}+{\displaystyle \frac{\mathbf{1}}{\sqrt{\mathit{k}}}}$	${\mathit{\xi }}_{\mathit{k}}=\mathbf{1}+{\displaystyle \frac{\mathbf{1}}{log(\mathit{k}+\mathbf{1})}}$
${\sigma }_k=1+{\displaystyle \frac{1}{k^{100}}}$	7	9	8	10
${\sigma }_k=1+{\displaystyle \frac{1}{k}}$	7	9	8	10
${\sigma }_k=1+{\displaystyle \frac{1}{\sqrt{k}}}$	9	8	8	10
${\sigma }_k=1+{\displaystyle \frac{1}{log(k+1)}}$	8	9	10	11

Concentrating on these observations, we deduce that making choices for auxiliary parameters ${\xi }_k$, ${\sigma }_k$, and ${\rho }_k$, ${\delta }_k$, ${\zeta }_k$, as allowed in Algorithm 1, can improve the effectiveness of Algorithm 1 in finite- and infinite-dimensional Hilbert spaces.

5. Conclusions

This paper introduces the Mann-type inertial accelerated subgradient extragradient algorithm with non-monotonic step sizes for finding the minimum-norm solution of split equilibrium and fixed point problems, specifically involved with pseudomonotone and Lipschitz-type continuous bifunctions and nonexpansive mappings within the context of real Hilbert spaces. Without requiring prior knowledge of the Lipschitz constants of bifunctions as well as the operator norm of the bounded linear operator, we take into consideration the Mann-type and inertial methods together with the accelerated subgradient extragradient method to propose a sequence that is strongly convergent to the minimum-norm solution of the split equilibrium and fixed point problems by sufficient conditions on the control sequences of the parameters of concern. Numerical experiments were conducted to demonstrate the efficacy of the proposed algorithm in both finite- and infinite-dimensional Hilbert spaces. The results confirm that focusing on the fine-tuning of auxiliary control parameters within algorithms is a promising and important research direction, as it tends to lead to significant improvements in performance.