A New Iterative Algorithm for Pseudomonotone Equilibrium Problem and a Finite Family of Demicontractive Mappings

In this paper, we introduce a new iterative method in a real Hilbert space for approximating a point in the solution set of a pseudomonotone equilibrium problem which is a common fixed point of a finite family of demicontractive mappings. Our result does not require that we impose the condition that the sum of the control sequences used in the finite convex combination is equal to 1. Furthermore, we state and prove a strong convergence result and give some numerical experiments to demonstrate the efficiency and applicability of our iterative method.

Obviously, the class of demicontractive mappings is more general than the class of quasi-nonexpansive mappings. Moreover, If T is κ-strictly pseudononspreading (or κ-strictly pseudocontractive) and F(T) ≠ [, then T is κ-demicontractive. Definition 2. A bifunction f: C × C ⟶ R is (i) Strongly monotone on C with modulus β > 0 (β-strongly monotone on C) if and only if f(x, y) + f(y, x) ≤ − β‖y − x‖ 2 , ∀x, y ∈ C; (6) (ii) monotone on C if and only if f(x, y) + f(y, x) ≤ 0, ∀x, y ∈ C; (7) (iii) pseudomonotone on C if and if Let f: C × C ⟶ R be a bifunction such that f(x, x) � 0, for all x ∈ C. We consider the equilibrium problem (EP) in the sense of Blum and Oettli [4], which is to find x * ∈ C such that f x * , y ≥ 0, ∀y ∈ C. (9) Let EP(C, f) denote the set of solutions of EP (9). If f(x, y) � 〈A(x), y − x〉 ∀x, y ∈ C, where A: C ⟶ H, then EP (9) reduces to the variational inequality problem: find x * ∈ C such that 〈A(x), y − x〉 ≥ 0, ∀y ∈ C. (10) EPs form a very important area of research and have recently been considered in many research papers. EP (9) is applied in solving problems from optimization, variational inequality, Kakutani fixed point, Nash equilibria in noncooperative game theory, and minimax problems [4,5].
A popular method that has been applied to solve EP (9) is the subgradient projection method which is developed from the steepest descent projection method in smooth optimization. If bifunction f is convex, subdifferentiable with respect to the second argument, Lipschitz, and strongly monotone on C, then regularization parameters can be chosen such that the subgradient projection method is linearly convergent [6]. However, when f is only monotone, the subgradient projection method may not be convergent. In order to get a method that guarantees convergence for pseudomonotone equilibrium problems (that is, equilibrium problems for pseudomonotone bifunctions) the extragradient (or double projection) method developed by Korpelevich [7] was extended to equilibrium problems. However, the extragradient algorithms involve two projections on the admissible set C, which may be costly to compute if the nature of the admissible set C is complicated. In the light of the need to obtain a more efficient algorithm, the inexact subgradient algorithms using only one projection [8,9] has been proposed for solving equilibrium problems with paramonotone equilibrium bifunctions. Some other methods that have been utilized to solve equilibrium problems include the auxiliary problem principle method [10], gap function method [11], and the Tikhonov and proximal point regularization methods [12][13][14][15].
Recently, the problem of finding a common point in EP(C, f) and the set of fixed points of mappings has become an attractive and interesting subject [16][17][18][19][20][21][22]. is interest is because of the possible application of these problems to mathematical models whose constraints can be present as fixed points of mappings and/or (EP). Such a problem occurs, in particular, in the practical problems as signal processing, network resource allocation, image recovery (see [23,24]).
In 2007, Tada and Takahashi [22] proposed the following iterative algorithm for approximating a common element of the set of solutions of equilibrium problem for monotone bifunctions and the set of fixed points of a nonexpansive mapping T.
where λ k > 0 is the regularization parameter at iteration k, x 0 ∈ C and P C is the metric projection onto C. ey assume that f is a monotone bifunction and obtained a strong convergence result.
Recently, Anh and Muu [25] proposed a new type of algorithm which uses only one projection and does not require any Lipschitz condition for the bifunctions for finding a common point in the solution set of the class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings. More precisely, they gave an iteration scheme generated as follows.
Algorithm 2. Pick x 1 ∈ C. At each iteration k � 1, 2, . . ., do the following: Inspired by Anh and Muu [25], Wangkeeree et al. [26] presented an iterative method for finding hierarchically an element in F(T) ∩ EP(C, f) with respect to a nonexpansive mapping. Precisely, they considered the following problems: where T and S are nonexpansive mappings.
Other authors have also considered different algorithms which involve either projection mapping or projection mapping and the construction of sequences of sets C n and Q n for approximating a common solution of pseudomonotone equilibrium problems and fixed point problems of nonexpansive mappings ( see, for example, [27][28][29]). ose methods are tasking and difficult to compute.
In 2018, ong and Hieu [30] proposed the following iterative algorithm for the approximation of a common fixed point of a finite family of demicontractive operators. Let x n be a sequence in H defined by Among other standard assumptions, they assumed that w i N i�1 is a finite sequence of positive numbers such that Let K be a nonempty closed and convex subset of a real Hilbert space H. Suppose that T i N i�1 , N ≥ 2 is a countable finite family of mappings T i : K ⟶ K. In [31], the authors consider the horizontal iteration process generated from an arbitrary x 1 for the finite family of mappings T i N i�1 , using a finite family of th T i N i�1 control sequences α i n ∞ n�1 N i�1 as follows.
For N � 2, For N � 3, For an arbitrary but finite N ≥ 2, 1.1. Question. Is it possible to give an iterative algorithm and obtain a strong convergence result for finding a common element in the set of fixed points of a finite family of demicontractive mappings which also solves equilibrium problems for pseudomonotone bifunctions without imposing the type of condition in (15) on the control sequences?
In this paper, motivated by the works of Anh and Muu [25] and Wangkeeree et al. [26], we propose an iterative algorithm for finding a common element in the set of fixed points of a finite family of demicontractive mappings, which also solves equilibrium problems for pseudomonotone bifunctions and prove a strong convergence result which does not require such condition as in (15) on the control sequences. We further give a numerical experiment to demonstrate the performance of our iterative algorithm.

Preliminaries
In the sequel, we shall need the following definitions and lemmas. Let H be a real Hilbert space, and C a nonempty, closed, and convex subset of H. By P C , we denote the metric projection operator onto C, that is, Lemma 1. Suppose that C is a nonempty, closed, and convex subset in H. en, P C has the following properties: Lemma 2 (see [32]). Let a k and b k be two nonnegative real sequences satisfying the following conditions: en, lim k⟶∞ a k exists.
Lemma 3 (see [33]). Let H be a real Hilbert space, C a closed convex subset of H, and let T: C ⟶ C be a continuous pseudocontractive mapping, then Lemma 4 (see [1]). Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let T: C ⟶ C be a ρ-strictly pseudononspreading mapping. If F(T) ≠ [, then it is closed and convex.

Abstract and Applied Analysis
Lemma 5 (see [1]). Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let T: C ⟶ C be a ρ-strictly pseudononspreading mapping. en, (I − T) is demiclosed at 0. Definition 3. Let C be a nonempty closed and convex subset of a Hilbert space E. Let f: ⊆ R be a countable subset of the set of real numbers R, where N ≥ 2 is an arbitrary integer. en, the following holds: Proof. is result has been proved in [31], but for the sake of completeness, we present the proof again here. For N � 2, We assume it is true for N and prove for N + 1.
Lemma 7 (see also [31]). Let t and u be arbitrary elements of a real Hilbert space H, and let N ∈ N be such that 2 ≤ N.
be a countable finite subset of H and R, respectively. Define en, Using the wellknown identity which holds for all x, y ∈ H and for all t ∈ [0, 1], we have □ Abstract and Applied Analysis

Main Results
We now give a strong convergence result for the approximation of a solution of a pseudomonotone equilibrium problem which is also a common fixed point of a finite family of demicontractive mappings.
Let f: C × C ⟶ R be a bifunction that satisfies the following conditions: (B1) f is pseudomonotone on C, f(x, x) � 0, and f(x, ·) is lower semicontinuous for all x ∈ C; (B2) f(·, y) is sequentially weakly upper semicontinuous on C for each fixed point y ∈ C, i.e., if x n ⊂ C is a sequence converging weakly to x ∈ C, then lim sup n⟶∞ f(x n , y) ≤ f(x, y); (B3) f(x, ·) is convex and subdifferentiable on E for every fixed x ∈ C; (B4) If x k ⊆ C is bounded ϵ k ↓0 as k ⟶ ∞, then the sequence y k with y k ∈ z ϵ k f(x k , ·)(x k ) is bounded; (B5) f is pseudomonotone on C with respect to every x ∈ EP(C, f) and satisfies the strict paramonotonicity property, i.e., x ∈ EP(C, f), It has been proved that under the conditions (B1)-(B3), the solution set EP(f, C) of EP (9) is closed and convex [34].

Algorithm 3
Step 1. Let λ k , β k and δ k be sequences of nonnegative real numbers and let T i , i � 1, 2, . . . , N, be ρ i -demicontractive mappings on C. Suppose that the following conditions are satisfied: Step 2 Proof. Let x * ∈ Ω. From w k ∈ P C (x k − α k y k ) and x * ∈ C, we have erefore, Since x k ∈ C, we also have From (33) and (34), we have at is, erefore, Moreover, since x k ∈ C and y k ∈ z ϵ k f(x k , ·)(x k ), we have It then follows from the definitions of α k and c k that us, from (33)-(39), we obtain erefore, from (40) and (41), we get 6 Abstract and Applied Analysis Now, from Proposition 2.9, x * ∈ F(T i ), and T i is ρ i -demicontractive, for i � 1, 2, . . . , N, we have where From (42) and (43), we have Furthermore, since x * ∈ Ω, then f(x * , x) ≥ 0 for all x ∈ C. erefore, by the pseudomonotonicity of f with respect to x * , we have f(x, x * ) ≤ 0, for all x ∈ C. It then follows from x k ∈ C that f(x k , x * ) ≤ 0. Hence, we have from (45) that Applying Lemma 2 to (46), we obtain the existence of Since f is pseudomonotone on C and f(x * , x k ) ≥ 0, we have − f(x k , x * ) ≥ 0. us, from (45), we obtain Summing up the inequality (48), for every k, we have Since the sequences y k and λ k are bounded, then there exists a constant M > 0 such that max λ k , ‖y k ‖ ≤ M.

Abstract and Applied Analysis
For any x * ∈ Ω, suppose that x k τ is the subsequence of x k such that lim sup and without loss of generality, we may assume that x k τ ⇀x as τ ⟶ ∞ for some x ∈ C.
Next, we show that x is in EP(C, f). Observe that since f(·, x * ) is weakly upper semicontinuous, we have From (54) and (55), we conclude that f(x, x * ) � 0. us, by condition B6, we obtain that x is in EP(C, f).

Data Availability
e data used to support the findings of this study are included within the article.

Disclosure
Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

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