Viscosity-type method for solving pseudomonotone equilibrium problems in a real Hilbert space with applications

1.   Introduction

Assume that $\mathcal{C}$ is a closed and convex subset of a real Hilbert space $\mathcal{H}$ with the inner product and the induced norm are denoted by $\langle \cdot, \cdot\rangle$ and $\|\cdot\|$, respectively. Moreover, $\mathcal{R}$ be a set of real numbers during whole article. Let $f : \mathcal{H} \times \mathcal{H} \rightarrow \mathcal{R}$ be a bifunction and satisfy $f (v, v) = 0$ for all $v \in \mathcal{C},$ the equilibrium problem (EP) ^[6,11] for a bifunction $f$ on $\mathcal{C}$ is defined in the following way:

Moreover, $S_{EP(f, \mathcal{C})}$ denotes the solution set of an equilibrium problem over the set $\mathcal{C}$ and $u^{*}$ is an arbitrary element of $S_{EP(f, \mathcal{C})}.$ A metric projection $P_{\mathcal{C}}(u)$ of $u \in \mathcal{H}$ onto a closed and convex subset $\mathcal{C}$ of $\mathcal{H}$ is defined by $P_{\mathcal{C}}(u) = \underset{v \in \mathcal{C}}{\arg\min} \|v - u\|.$

In this article, the equilibrium problem is studied based on the following hypothesis:

${\rm(a1)}$ A bifunction $f : \mathcal{H} \times \mathcal{H} \rightarrow \mathcal{R}$ is said to be (see ^[3,6]) pseudomonotone on $\mathcal{C}$ if

${\rm(a2)}$ A bifunction $f : \mathcal{H} \times \mathcal{H} \rightarrow \mathcal{R}$ is said to be Lipschitz-type continuous ^[21] on $\mathcal{C}$ if there exist two constants $c_{1}, c_{2} > 0$ such that

${\rm(a3)} \limsup\limits_{n\rightarrow \infty} f(u_{n}, v) \leq f(p^{*}, v)$ for all $v \in \mathcal{C}$ and $\{u_{n}\} \subset \mathcal{C}$ satisfies $u_{n} \rightharpoonup p^{*};$

${\rm(a4)} f (u, \cdot)$ is subdifferentiable and convex on $\mathcal{H}$ for every each $u \in \mathcal{H}.$

The above-defined problem (EP) is a general mathematical problem in the sense that it unifies a number of mathematical problems, i.e., the fixed point problems, the vector and scalar minimization problems, the problems of variational inequalities (VIP), the complementarity problems, the saddle point problems, the Nash equilibrium problems in non-cooperative games and the inverse optimization problems ^[4,6,24,41]. The problem (EP) is also known as the well-known Ky Fan inequality due to his initial contribution ^[11]. Many authors have developed and generalized many results on the existence and nature of the solution of an equilibrium problem (see for more detail ^[2,4,11]). Due to the significance of the problem (EP) and its applications in both pure and applied sciences, many researchers have studied it extensively in recent years ^[5,10,14].

A proximal point method is used to solve the problem (EP) based on mathematical programming ^[13]. This method was also known as the two-step extragradient method in ^[34] due to initial contribution of Korpelevich ^[18] to solve saddle point problems. Tran et al. in ^[34] established an iterative sequence $\{u_{n}\}$ in the following way:

where $0 < \rho < \min \big\{\frac{1}{2 c_{1}}, \frac{1}{2 c_{2}} \big\}.$ Recently, many existing methods have been extended in the case of problem (EP) in finite and infinite-dimensional spaces, such as the proximal point-like methods ^[13,22], the extragradient-like methods ^{[17,19,26,27,32]}, the subetaadient extragradient methods ^[1,29,36,37], the inertial methods ^[35,39] and others in ^{[9,12,16,25,30,33,38]}.

Inspired by the results in ^[8,16,23], in this paper, we introduce a viscosity-type subetaadient extragradient algorithm to solve the equilibrium problems involving pseudomonotone bifunction. A strong convergence theorem for the proposed algorithm is well-established by considering certain mild conditions on bifunction and control parameters. Some applications for our main results are studied to solve two particular classes of an equilibrium problem. In the end, the computational studies show that the new method is more efficient than the existing ones ^[16,34].

The remainder of this article has been organized as follows: Section 2 includes some preliminary and basic results. Section 3 contains proposed algorithm and corresponding strong convergence result. Section 4 contains applications of our main results. Section 5 involves the numerical discussion of the proposed method compared to existing ones.

2.   Background

A normal cone of $\mathcal{C}$ at $u \in \mathcal{C}$ is defined by

Let $\varphi : \mathcal{C} \rightarrow \mathcal{R}$ is convex function. The subdifferential of $\varphi$ at $u \in \mathcal{C}$ is defined by

Lemma 2.1. ^[15] Assume that $P_{\mathcal{C}} : \mathcal{H} \rightarrow \mathcal{C}$ is a metric projection such that

(ⅰ)

(ⅱ) $u_{3} = P_{\mathcal{C}}(u_{1})$ if and only if

(ⅲ)

Lemma 2.2. ^[40] Assume that $\{a_{n}\} \subset (0, +\infty)$ is a sequence satisfying

where $\{\gamma_{n}\} \subset (0, 1)$ and $\{\delta_{n}\} \subset \mathcal{R}$ satisfies the following conditions:

Then, $\lim_{n \rightarrow \infty} a_{n} = 0.$

Lemma 2.3. ^[20] Assume that a sequence $\{a_{n}\} \subset \mathcal{R}$ and there exists a subsequence $\{n_{i}\}$ of $\{n\}$ such that $a_{n_{i}} < a_{n_{i+1}}$, for all $i \in \mathbb{N}.$ Then, there is a non decreasing sequence $\{m_{k}\} \subset \mathbb{N}$ such that $m_{k} \rightarrow +\infty$ as $k \rightarrow \infty,$ and the following conditions are fulfilled by all (sufficiently large) numbers $k \in \mathbb{N}$,

In fact $m_{k}$ is the largest number $n$ in the set $\{1, 2, \cdots, k \}$ such that $a_{n} \leq a_{n+1}.$

Lemma 2.4. ^[15] For each $u_{1}, u_{2} \in \mathcal{H}$ and $\delta \in \mathcal{R},$ then the following relationships are true.

(ⅰ)

(ⅱ)

Lemma 2.5. (Theorem 27.4 ^[28]) Let $\varphi: \mathcal{C} \rightarrow \mathcal{R}$ be a proper convex, subdifferentiable and lower semi-continuous function on $\mathcal{C}.$ An element $u \in \mathcal{C}$ is a minimizer of a function $\varphi$ iff

where $\partial \varphi(u)$ stands for the sub-differential of $\varphi$ at $u \in \mathcal{C}$ and $N_{\mathcal{C}}(u)$ the normal cone of $\mathcal{C}$ at $u.$

3.   Main results

In this section, we present an iterative scheme for solving pseudomonotone equilibrium problems that is based on Tran et al. in ^[34] and viscosity scheme ^[23]. It is important to note that the proposed method has a straightforward structure for achieving strong convergence. Suppose that $\mathrm{g} : \mathcal{H} \rightarrow \mathcal{H}$ be a strict contraction function with constant $\xi \in [0, 1).$ The main algorithm has been presented as follows:

Remark 3.1. It can be easily prove that $\mathcal{C} \subset \mathcal{H}_{n}$. By $v_{n}$ and Lemma 2.5, we have

Indeed, for $\omega_{n} \in \partial f(u_{n}, v_{n})$ there exists $\overline{\omega}_{n} \in N_{\mathcal{C}}(v_{n})$ such that

Thus, we have

Due to $\overline{\omega}_{n} \in N_{\mathcal{C}}(v_{n})$ means that $\langle \overline{\omega}_{n}, v - v_{n} \rangle \leq 0,$ for all $v \in \mathcal{C}.$ It implies that

which imply that $\langle u_{n} - \rho \omega_{n} - v_{n}, v - v_{n} \rangle \leq 0, \forall v \in \mathcal{C}.$ It proves that $\mathcal{C} \subset \mathcal{H}_{n}$ for each $n \in \mathbb{N}.$

Theorem 3.1. Assume that $\{u_{n}\}$ is a sequence generated by Algorithm 1 and for some $u^{*} \in S_{EP(f, \mathcal{C})} \neq \emptyset.$ Then, $\{u_{n}\}$ converges strongly to $u^{*} = P_{S_{EP(f, \mathcal{C})}} \circ \mathrm{g} (u^{*}).$

Proof. Claim 1: The $\{u_{n}\}$ sequence is bounded.

By Lemma 2.5, we have

Thus, there exists $\omega_{n} \in \partial f(v_{n}, t_{n})$ and $\overline{\omega}_{n} \in N_{\mathcal{H}_{n}}(t_{n})$ such that $\rho \omega_{n} + t_{n} - u_{n} + \overline{\omega}_{n} = 0.$ Thus,

Since $\overline{\omega}_{n} \in N_{\mathcal{H}_{n}}(t_{n})$ follows that $\langle \overline{\omega}_{n}, v - t_{n} \rangle \leq 0,$ for all $v \in \mathcal{H}_{n}.$ Thus, we have

Since $\omega_{n} \in \partial f(v_{n}, t_{n})$ and using the subdifferential definition, we get

Combining expressions (3.1) and (3.2), we obtain

Substituting $v = u^{*}$ in expression (3.3), we obtain

Since $u^{*} \in S_{EP(f, \mathcal{C})}$, it follows that $f(u^{*}, v_{n}) \geq 0,$ implies that $f(v_{n}, u^{*}) \leq 0$ due to the pseudomonotonicity of the bifunction $f.$ From expression (3.4), we have

Due to the Lipschitz-type condition on a bifunction $f$, we obtain

Combining expressions (3.5) and (3.6), we have

Since $t_{n} \in \mathcal{H}_{n}$ and it gives that $\langle u_{n} - \rho \omega_{n} - v_{n}, t_{n} - v_{n} \rangle \leq 0,$ which implies that

Since $\omega_{n} \in \partial_{2} f(u_{n}, v_{n})$ and using the subdifferential definition, we obtain

Substituting $v = t_{n}$ in the above expression, we get

It follows from inequalities (3.8) and (3.9) that

From (3.7) and (3.10), we have

We have the following equalities:

and

The above facts and (3.11) implies that

It is given that $u^{*} \in S_{EP(f, \mathcal{C})}.$ From the definition of sequence $\{u_{n+1}\}$ and due to the fact that $\mathrm{g}$ is a contraction with $\xi \in [0, 1)$, we have

Combining expressions (3.12) and (3.13) and $\chi_{n} \subset (0, 1)$, we deduce that

Thus, we conclude that the $\{u_{n}\}$ is bounded sequence. Due to the reflexivity of $\mathcal{H}$ and the boundedness of $\{u_{n}\}$ guarantees that there exists a subsequence $\{u_{n_{k}}\}$ of $\{u_{n}\}$ such that $\{u_{n_{k}}\} \rightharpoonup u^{*} \in \mathcal{H}$ as $k \rightarrow \infty.$

Claim 2: The sequence $\{u_{n}\}$ is strongly convergent.

Next, we prove the strong convergence of the iterative sequence $\{u_{n}\}$ generated by Algorithm 1. The Lipschitz-continuity and pseudomonotone property of the bifunction $f$ implies that the solution set $S_{EP(f, \mathcal{C})}$ is a closed and convex set (for more details see ^[34]). Since the mapping is a contraction and so does $P_{S_{EP(f, \mathcal{C})}} \circ \mathrm{g}$. Now, we are in position to use the Banach contraction theorem for the existence of a unique fixed point $u^{*} \in S_{EP(f, \mathcal{C})}$ such that

By using Lemma 2.1 (ⅱ), we have

By Lemma 2.4 (ⅰ) and (3.12), we have

The remainder of the proof shall be split into the following two parts:

Case 1: Assume that there is a fixed number $N_{1} \in \mathbb{N}$ such that

Thus, above implies that $\lim_{n \rightarrow \infty} \| u_{n} - u^{*}\|$ exists and let $\lim_{n \rightarrow \infty} \| u_{n} - u^{*}\| = l.$ From (3.16), we get

Due to the existence of $\lim_{n \rightarrow \infty} \| u_{n} - u^{*}\| = l$ and $\chi_{n} \rightarrow 0$, we infer that

It follows that

We can also obtain

The above expression implies that

Thus, implies that the sequences $\{v_{n}\}$ and $\{t_{n}\}$ are bounded. Let $\{u_{n_k}\}$ be subsequence of $\{u_{n}\}$ such that $\{u_{n_k}\}$ converges weakly to $\hat{u} \in \mathcal{H}.$ Next, we need to prove that $\hat{u} \in S_{EP(f, \mathcal{C})}.$ Due to the inequality (3.3), the Lipschitz-like condition of $f$ and (3.10), we obtain

where $v$ is an arbitrary member in $\mathcal{H}_{n}.$ From (3.19), (3.20) and the boundedness of $\{u_{n}\}$ imply that right-hand side goes to zero. From $\rho > 0,$ the condition (a3) and $v_{n_{k}} \rightharpoonup \hat{u}$, we obtain

It follows that $f(\hat{u}, v) \geq 0,$ for all $v \in \mathcal{C}$, and hence $\hat{u} \in S_{EP(f, \mathcal{C})}.$ Next, we consider

We have $\lim_{n \rightarrow \infty} \big\lVert u_{n+1} - u_{n} \big\rVert = 0.$ We can deduce that

From Lemma 2.4(ⅱ) and (3.12), we have

It follows from expressions (3.26) and (3.27), we obtain

Choose $n \geq N_{2} \in \mathbb{N}$ ($N_{2} \geq N_{1}$) large enough such that $2\chi_{n} (1 - \xi) < 1.$ By using (3.27), (3.28) and applying Lemma 2.2, we conclude that $\big\lVert u_{n} - u^{*} \big\rVert \rightarrow 0$ as $n \rightarrow \infty.$

Case 2: Assume there is a subsequence $\{n_{i}\}$ of $\{n\}$ such that

Thus, by Lemma 2.3 there is a sequence $\{m_{k}\} \subset \mathbb{N}$ as $\{m_{k}\} \rightarrow \infty,$ such that

Similar to Case 1, the expression (3.16) provides that

Due to $\chi_{m_{k}} \rightarrow 0$, we deduce the following:

Also, we can obtain

We have to use the same justification as in the Case 1, such that

By using expressions (3.27) and (3.29), we have

It follows that

Since $\chi_{m_{k}} \rightarrow 0,$ and $\big\lVert u_{m_{k}} - u^{*} \big\rVert$ is a bounded. Thus, expressions (3.33) and (3.35) implies that

The above implies that

Consequently, $u_{n} \rightarrow u^{*}.$ This completes the proof of the theorem.

Remark 3.2. If we define $\mathrm{g}(u) = u_{0}$ in Algorithm 1, we obtain the Halpern subetaadient extragradient method in ^[16].

4.   Applications

Now, we consider the application of our main results to solve the problem of classic variational inequalities ^[31] for an operator $\mathcal{G} : \mathcal{H} \rightarrow \mathcal{H}$ is defined in the following way:

We consider the following conditions to study variational inequalities.

${\rm(b1)}$ The solution set of the problem (VIP) denoted by $VI(\mathcal{G}, \mathcal{C})$ is nonempty.

${\rm(b2)} \mathcal{G} : \mathcal{H} \rightarrow \mathcal{H}$ is said to be a pseudomonotone, i.e.,

${\rm(b3)}$ $\mathcal{G} : \mathcal{H} \rightarrow \mathcal{H}$ is said to be a Lipschitz continuous if there exits a constants $L > 0$ such that

${\rm(b4)} \mathcal{G} : \mathcal{H} \rightarrow \mathcal{H}$ is called to be sequentially weakly continuous, i.e., $\{\mathcal{G}(u_{n})\}$ weakly converges to $\mathcal{G}(u)$ for every sequence $\{u_{n}\}$ converges weakly to $u$.

If we define $f(u, v) : = \big\langle \mathcal{G}(u), v - u \big\rangle,$ for all $u, v \in \mathcal{C}.$ Then, the equilibrium problem becomes the problem of variational inequalities described above where $L = 2c_{1} = 2 c_{2}.$ From the above value of the bifunction $f$, we have

As a consequence of the results in Section 3, we have the following results:

Corollary 4.1. Let $\mathcal{G} : \mathcal{C} \rightarrow \mathcal{H}$ be a mapping satisfying the conditions ${\rm(b1)–(b4)}$. Choose $u_{0} \in \mathcal{C},$ $0 < \rho < \frac{1}{L}$ and a sequence $\chi_{n} \subset (0, 1)$ satisfying the conditions, i.e.,

Assume that $\{u_{n}\}$ is the sequence generated in the following way:

where

Then, the sequence $\{u_{n}\}$ converges strongly to $u^{*} \in VI(\mathcal{G}, \mathcal{C}).$

Note that condition ${\rm(b4)}$ can be deleted in the case when $\mathcal{G}$ is monotone. Indeed, this condition is a particular case of condition ${\rm(a3)}$ is used to prove (3.24). Without condition ${\rm(b4)}$, the inequality (3.23) is obtained by imposing the monotonocity on $\mathcal{G}$. In that case, we obtain

By the use $f(u, v) = \langle \mathcal{G}(u), v - u \rangle$ in (3.23), we get

Combining (4.2) with (4.3), we obtain

Let $v_{s} = (1 - s)\hat{u} + sy$, for all $s \in [0, 1].$ Due to convexity of $\mathcal{C}$ implies that $v_{s} \in \mathcal{C}$ for any $s \in (0, 1).$ Since $v_{n_{k}} \rightharpoonup \hat{u} \in \mathcal{C}$ and $\langle \mathcal{G}(v), v - \hat{u} \rangle \geq 0$ for every $v \in \mathcal{C}$. Thus, we have

Therefore, $\langle \mathcal{G}(v_{s}), v - \hat{u} \rangle \geq 0,$ for all $s\in (0, 1).$ Since $v_{s} \rightarrow \hat{u}$ as $s \rightarrow 0$ and due to continuity of $\mathcal{G}$, we have $\langle \mathcal{G}(\hat{u}), v - \hat{u} \rangle \geq 0,$ for all $v \in \mathcal{C},$ which implies that $\hat{u} \in VI(\mathcal{G}, \mathcal{C}).$

Remark 4.1. From the above discussion, it can be concluded that the Corollary 4.1 still holds, even if we remove the condition ${\rm(b4)}$ in case of monotone variational inequaltiy.

Next, we consider the application of our results to solve fixed-point problems involving $\kappa$-strict pseudocontraction mapping and the fixed point problem for an operator $\mathcal{T} : \mathcal{H} \rightarrow \mathcal{H}$ is defined in the following way:

We assume that the following requirements have been met to study fixed point problems.

${\rm(c1)} \mathcal{T} : \mathcal{C} \rightarrow \mathcal{C}$ is said to be a $\kappa$-strict pseudo-contraction ^[7] on $\mathcal{C}$ if

${\rm(c2)}$ weakly sequentially continuous on $\mathcal{C}$ if

If we consider that the mapping $\mathcal{T}$ is a $\kappa$-strict pseudocontraction and weakly continuous then $f(u, v) = \langle u - \mathcal{T} u, v - u \rangle$ satisfies the conditions ${\rm(a1)–(a4)}$ and $2 c_{1} = 2 c_{2} = \frac{3 - 2 \kappa}{1 - \kappa}.$ The values of $v_{n}$ and $t_{n}$ in Algorithm 1, we have

Corollary 4.2. Let $\mathcal{C}$ be a nonempty, convex and closed subset of a Hilbert space $\mathcal{H}$ and $\mathcal{T} : \mathcal{C} \rightarrow \mathcal{C}$ is a $\kappa$-strict pseudocontraction and weakly continuous with solution set $Fix(\mathcal{T}) \neq \emptyset.$ Choose $u_{0} \in \mathcal{C},$ $0 < \rho < \frac{1 - \kappa}{3 - 2 \kappa}$ and sequence $\chi_{n} \subset (0, 1)$ satisfies the conditions, i.e.,

Let $\{u_{n}\}$ be the sequence generated in the following way:

where

Then, sequence $\{u_{n}\}$ strongly converges to $u^{*} \in Fix(\mathcal{T}).$

5.   Numerical illustrations

Numerical results are presented in this section to show the efficiency of the proposed method. The MATLAB codes was run in MATLAB version 9.5 (R2018b) on the Intel(R) Core(TM)i5-6200 CPU PC @ 2.30GHz 2.40GHz, RAM 8.00 GB.

Example 5.1. Assume that $f: \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{R}$ is defined by

where $c \in \mathcal{R}^{n}$ and $P$, $Q$ are matrices of order $n.$ The matrix $P$ is symmetric positive semi-definite and the matrix $Q - P$ is symmetric negative semi-definite with Lipschitz-type constants $c_{1} = c_{2} = \frac{1}{2}\| P - Q\|$ (see ^[34] for details). The matrices $P, Q$ and vector $c$ are defined by

The constraint set $C \subset \mathcal{R}^{n}$ is considered as $C : = \{ u \in \mathcal{R}^{n} : -5 \leq u_{i} \leq 5 \}.$ Furthermore, control parameters conditions are taken as follows: $\rho = \frac{1}{4c_{1}}$ and $D_{n} = \|u_{n} - v_{n} \|$ for Algorithm 1 (EgM) in ^[34]; $\rho = \frac{1}{4c_{1}}$, $\chi_{n} = \frac{1}{100(n + 2)}$ and $D_{n} = \|u_{n} - v_{n} \|$ for Algorithm 2 (H-EgM) in ^[16]; $\rho = \frac{1}{4c_{1}}, \chi_{n} = \frac{1}{100(n + 2)}$, $\mathrm{g}(u) = \frac{u}{2}$ and $D_{n} = \|u_{n} - v_{n} \|$ for Algorithm 1 (V-EgM). The numerical and graphical results of three methods are shown in Figures 1–4 and Table 1.

Example 5.2. Let $f : \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{R}$ defined in the following way:

where $\mathcal{C} = \big\{ (u_{1}, \cdots, u_{5}) : u_{1} \geq -1, u_{i} \geq 1, i = 2, \cdots, 5 \big\}.$ Thus, the bifunction $f$ is Lipschitz-type continuous with $c_{1} = c_{2} = 2,$ and satisfies the conditions ${\rm(a1)–(a4)}$. The solution set of an equilibrium problem is $EP(f, \mathcal{C}) = \{(u_{1}, 1, 1, 1, 1) : u_{1} > 1 \}$. Furthermore, the control conditions $\rho = \frac{1}{6c_{1}}$ for Algorithm 1 (EgM) in ^[34]; $\rho = \frac{1}{6c_{1}}$ and $\chi_{n} = \frac{1}{200(n + 2)}$ for Algorithm 2 (H-EgM) in ^[16]; $\rho = \frac{1}{6c_{1}}, \chi_{n} = \frac{1}{100(n + 2)}$ and $\mathrm{g}(u) = \frac{u}{3}$ for Algorithm 1 (V-EgM). The numerical results of three methods are shown in Tables 2–4.

Example 5.3. Suppose that $\mathcal{H} = L^{2}([0, 1])$ is a Hilbert space with the inner product $\langle u, v \rangle = \int_{0}^{1} u(t) v(t) dt,$ for all $u, v \in \mathcal{H}$ and the induced norm is

Moreover, assume that $\mathcal{C} : = \{ u \in L^{2}([0, 1]): \|u\| \leq 1 \}$. Assume that $\mathcal{G} : \mathcal{C} \rightarrow \mathcal{H}$ is defined by

where $H(t, s) = \frac{2ts e^{(t+s)}}{e \sqrt{e^{2}-1}}, f(u) = \cos (u)$ and $g(t) = \frac{2t e^{t}}{e \sqrt{e^{2}-1}}.$ We consider the bifunction $f(u, v) = \langle \mathcal{G}(u), v - u \rangle$ with the Lipschitz-type continuous with $c_{1} = c_{2} = 1.$ Furthermore, control conditions $\rho = \frac{1}{5c_{1}}$ for Algorithm 1 (EgM) in ^[34]; $\rho = \frac{1}{5c_{1}}$ and $\chi_{n} = \frac{1}{300(n + 2)}$ for Algorithm 2 (H-EgM) in ^[16]; $\rho = \frac{1}{5c_{1}}, \chi_{n} = \frac{1}{100(n + 2)}$ and $\mathrm{g}(u) = \frac{u}{2}$ for Algorithm 1 (V-EgM). The numerical results of three methods are shown in Figures 5 and 6 and Table 5.

Example 5.4. Assume that $f : \mathcal{H} \times \mathcal{H} \rightarrow \mathcal{R}$ is defined by

where $\mathcal{H} = l_{2}$ is a real Hilbert space having the elements are square-summable sequences and $\mathcal{C} = \{ u \in \mathcal{H} : \|u\| \leq 3 \}.$ The bifunction $f$ is Lipschitz-type continuous and value of Lipschitz-constants are $c_{1} = c_{2} = \frac{11}{2}.$ Furthermore, control conditions $\rho = \frac{1}{4c_{1}}$ for Algorithm 1 (EgM) in ^[34]; $\rho = \frac{1}{4c_{1}}$ and $\chi_{n} = \frac{1}{200(n + 2)}$ for Algorithm 2 (H-EgM) in ^[16]; $\rho = \frac{1}{4c_{1}}, \chi_{n} = \frac{1}{100(n + 2)}$ and $\mathrm{g}(u) = \frac{u}{2}$ for Algorithm 1 (V-EgM). The numerical results of three methods are shown in Figures 7 and 8 and Table 6. The projection onto $\mathcal{C}$ is evaluated in the following way:

6.   Conclusion

We have studied a viscosity type extragradient-like method for determining the solution of pseudomonotone equilibrium problem in real Hilbert spaces and also prove that the generated sequence converges strongly to the solution. Numerical conclusions were drawn to explain the numerical efficiency of our algorithms in comparison to other methods. These numerical studies showed that viscosity influences improve the efficiency of the iterative sequence in this context.

Acknowledgements

This research work was financially supported by King Mongkut's University of Technology Thonburi through the "KMUTT 55th Anniversary Commemorative Fund". In particular, Habib ur Rehman was financed by the Petchra Pra Jom Doctoral Scholarship Academic for Ph.D. Program at KMUTT [grant number 39/2560]. Moreover, this project was supported by Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Also, this project was financially supported by King Mongkut’s University of Technology North Bangkok, Contract no. KMUTNB-BasicR-64-22.

The first author would like to thank the "Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut’s University of Technology Thonburi". We are very grateful to the editor and the anonymous referees for their valuable and useful comments, which helps in improving the quality of this work.

Conflict of interest

No potential conflict of interest was reported by the author(s).