A new weak convergence non-monotonic self-adaptive iterative scheme for solving equilibrium problems

1 Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand 2 Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, Thanyaburi, Pathumthani 12110, Thailand 3 Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand 4 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan 5 Department of Mathematics College of Science & Arts, King Abdulaziz University, P. O. Box 344, Rabigh 21911, Saudi Arabia


Introduction
Assume that C is a non-empty, convex and closed subset of real Hilbert space H and f : H ×H → R be a bi-function with f (v, v) = 0, for every v ∈ C. A equilibrium problem (EP) for f on C is stated in the following manner: Find ℘ * ∈ C such that f (℘ * , v) ≥ 0, ∀ v ∈ C. (EP) In this paper, we study a novel numerical method to solve equilibrium problem based on the following hypothesis. A bi-function f : H × H → R is called to be (see [4,6]): (c2) Lipschitz-type continuous [23] on C if there exist two constants c 1 , c 2 > 0 such that for all v ∈ C and {v n } ⊂ C satisfies v n v * ; (c4) f (v 1 , ·) is sub-differentiable and convex on H for each fixed v 1 ∈ H.
Furthermore, E p( f, C) denotes the solution set of the problem (EP). To the best of our understanding, the term "equilibrium problem" is introduced in 1992 by Muu and Oettli [27] and has been further studied by Blum and Oettli [6]. The problem (EP) is also known as the Ky Fan inequality due to his contribution [10]. In particular, the equilibrium problem is a general mathematical framework in the sense that it brings together different mathematical problems, i.e., the fixed point problems, the vector and scalar minimization problems, the variational inequality problems, the complementarity problems, the saddle point problems, the Nash equilibrium problems in non-cooperative games and the inverse optimization problems [5,6,20,27]. A comprehensive study on equilibria and the detailed description of numerical methods for equilibrium problems can be found in [5,6,9,17]. Iterative schemes are useful mechanisms for finding an estimated solution to the equilibrium problems. Many methods have been used to solve the problem (EP) in real Hilbert spaces, i.e., proximal point-like methods [12,18,19,26], the extragradient-like methods [2, 15, 16, 22, 30, 31, 33, 34, 37-39, 41, 42, 44] and others in [1,3,11,14,24,25,28,32,43].
Meanwhile, by using the Korpelevich extragradient method [21] Flaam et al. [12] and Quoc et al. [31] was set up the following method for the solution of equilibrium problems involving pseudomonotone and Lipschitz-type bi-function: where 0 < χ < min 1 2c 1 , 1 2c 2 and c 1 , c 1 are Lipschitz-type constants. The iterative schemes in [12,31] are usually recognised as the extragradient method initially due to the result of Korpelevich in [21] to solve the saddle point problems.
It is important to note that most previously established methods used a constant step size rule that is dependent on the Lipschitz-type constants of the bi-functions [14,22,31]. This can lead to some limitations in applications because the Lipschitz-type constants are usually unknown or complicated to estimate. Very recently, Hieu et al. [16] introduced the modifications of the gradient method for solving pseudo-monotone equilibrium problems with the new step size rule. But the step size rule is non-increasing, monotone, and the methods in [16] may depend on the choice of initial step size.
A natural question arises, i.e., "Is it possible to introduce a new weakly convergent extragradient algorithm with non-monotone step size rule for approximate the solution of problem (EP) involving pseudo-monotone bi-function that does not depend on the Lipschitz-type constants of the bi-function"?
In this work, we provide a positive answer to the above question, i.e., the gradient methods still hold in case of non-monotonic step size rule for solving equilibrium problems associated with pseudomonotone bi-functions. Inspired by the works of Censor et al. [8] and Hieu et al. [16], we introduce a new extragradient-type method for evaluating a numerical solution of the problem (EP) in the context of infinite-dimensional real Hilbert spaces. Our main contributions in this study are as follows: • We introduce a subgradient extragradient method with a non-monotone step size rule to solve the (EP) equilibrium problem in a real Hilbert space. In our suggested method, we do assume that the bi-function in the (EP) equilibrium problem is pseudo-monotone, that involves monotone bi-function.
• A weak convergence result is provided, i.e., that the sequence of iterates generated by our proposed method is weakly convergent to a solution of the problem (EP) in the setting of mild condition on the bi-functions and the iterative control parameters.
• We give numerical descriptions of our method for confirming the theoretical findings and comparing the results in [Algorithm 1 in [16]] and [Algorithm 2 in [16]]. Our numerical findings show that our method is efficient and effective compared to the current ones.
The remaining part of this paper is organized as follows. Section 2 recalls some basic definitions and lemmas to be used later. Section 3 introduces a new subgradient extragradient algorithm and provides its weak convergence result, which combines a non-monotonic step size rule. Section 4, provide the applications of our results to the particular classes of the problem (EP). Finally, many numerical results are reported to explain the behaviour of the new algorithm and also to compare it with existing algorithms.

Preliminaries
A metric projection P C (v 1 ) of v 1 ∈ H onto a closed and convex subset C of H is defined by The following are the key properties of projection mapping.
Lemma 2.1. [13] Let P C : H → C be a metric projection such that A normal cone of C at v 1 ∈ C is define by Let : C → R be a convex function and sub-differential of at v 1 ∈ C is defined by Lemma 2.2. [36] Let : C → R be a sub-differentiable, convex and lower semi-continuous function on C. An element v 1 ∈ C is a minimizer of a function iff where ∂ (v 1 ) denotes the sub-differential of at v 1 ∈ C and N C (v 1 ) is the normal cone of C at v 1 .

Lemma 2.3. [29]
Let C be a non-empty subset of H and {v n } is a sequence in H such that the following two conditions are held: (i) for every v 1 ∈ C, lim n→∞ v n − v 1 exists; (ii) every sequentially weak cluster point of {v n } is in C.
Then, {v n } converges weakly to a point in C.

Lemma 2.4. [35]
Let {l n } and {m n } are sequences of non-negative real numbers meet the following inequality l n+1 ≤ l n + m n , ∀ n ∈ N.
and construct a half-space Step 2: Compute Step 3: Keep updating the step size rule in the following way: else. (3.1) Step 4: If u n+1 = u n = v n , then STOP. Otherwise, set n := n + 1 and go back to Step 1. ϕ n such that Proof. Due to the Lipschitz-type condition on a bi-function f with two positive consonants c 1 and c 2 .
By using mathematical induction on the definition of χ n+1 , we have By the definition of {χ n } we get That is, the series +∞ n=1 (χ n+1 − χ n ) + is convergent. Next, we need to prove the convergence of the  Proof. From Lemma 2.2, we can write The above implies that By combining (3.7) and (3.8) we get Due to the definition of half-space H n implies that Thus, implies that By combining (3.10) with (3.11), we have Due to the definition of χ n+1 we get with χ n > 0, provides that By combining (3.14) and (3.15) we accomplish the following Due to (3.12) and (3.16), we obtain In addition, we have the following formulas: Using the facts and the expression (3.17), we have (3.18) To prove the boundedness, let fixed a number m ≥ n 0 and using (3.18), for every numbers n 0 , n 0 + 1, · · · , m, such that Thus, expression (3.19) is obtained due to the following facts. Due to definition χ n+1 we have χ n χ n+1 → 1 as n → ∞, and due to µ ∈ (0, 1 3 ) there exist a fixed number ∈ (0, 1 − 3µ) such that Due to the above facts there exists a fixed natural number n 0 ∈ N such that The relation (3.19) imply that the sequence {u n } is bounded. The following results are also deduced: The remaining part of the proof is to show that each cluster point of sequence {u n } belongs to the solution set EP( f, C). Let us take a pointû to be a weak cluster point of {u n }. It implies that there exists a subsequence {u n k } of {u n } such that {u n k } û. Due to u n − v n −→ 0, we have {v n k } û. Due to expression (3.9), we have (3.24) By expression (3.15), we obtain Combining the relations (3.24), (3.25) and (3.12) we write where v is be an arbitrary element in H n . By using the boundedness of the sequence and expressions (3.20), (3.21) and (3.23) that right-hand of the last inequality goes to zero. By the use of χ n k ≥ χ > 0, It gives thatû ∈ EP( f, C). Then, Lemma 2.3, guarantees that {u n } and {v n } weakly converge to ℘ * as n → ∞.
The second part of the proof is to show that lim n→∞ P EP( f,C) (u n ) = ℘ * . Let consider p n := P EP( f,C) (u n ) for every n ∈ N. Given that ℘ * ∈ EP( f, C), we have p n ≤ p n − u n + u n ≤ ℘ * − u n + u n . (3.27) Due to the above expression we obtain the boundedness of the sequence {p n }. Due to the expression (3.18) for every n ≥ n 0 , we can infer that By using expression (3.28) and Lemma 2.4 gives the existence of lim n→∞ u n − p n . Consider that Next, to show that {p n } is a Cauchy sequence. For this, let p m , p n ∈ EP( f, C), for m > n ≥ n 0 , and Lemma 2.1(i) and (3.29) such that By the use of lim n→∞ p n −u n and the summability of n u n −v n−1 implies that lim n→∞ p n − p m = 0, for every m ≥ n. As a results {p n } is a Cauchy sequence. Since EP( f, C) is closed set and thus implies that {p n } converges strongly to p * ∈ EP( f, C). Now, we need to prove that p * = ℘ * . By Lemma 2.1(ii) and ℘ * , p * ∈ EP( f, C), we have u n − p n , ℘ * − p n ≤ 0. (3.31) Since p n → p * and u n ℘ * , we have that implies that ℘ * = p * = lim n→∞ P EP( f,C) (u n ). Furthermore, u n − v n → 0, as n → ∞ implies that lim n→∞ P EP( f,C) (v n ) = ℘ * .
It is worth noting that under the presumptions of Lipschitz-type continuity and pseudo-monotonicity, there is still a need to solve two minimization problems on C. If the set C is simple enough so that minimization problem onto it can be easily solved, then this method is particularly effective; but in case of C is a more general closed and convex set, then a minimal distance problem has to be figure out twice in order to obtain the next iterate. This could have a serious impact on the efficiency of the extra-gradient method. On other hand, the subgradient extragradient method involves the replacement of the second minimization problem onto C by a specific subgradient projection.
By the use of Algorithm 1 and Theorem 3.3, we obtain the modification of the Algorithm 1 in [31] with non-monotonic step size rule. 1 3 ) and select a non-negative real sequence {ϕ n } such that ∞ n ϕ n < +∞. Let {u n } be the sequence generated in the following way:

32)
where Then, the sequence {u n } converges weakly to some ℘ * ∈ E p( f, C).

Applications
In this section, we derive some results to solve variational inequalities and the fixed point problems. In the last few years, variational inequalities have attracted a great deal of attention from both researchers and readers. It is well known that variational inequalities cover a number of subjects such as partial differential equations, optimal control, optimization techniques, applied mathematics, engineering, finance and operational science. On the other hand, the solution of a several problems in pure and applied mathematics is the fixed point of some mapping S. As a result, many iterative schemes in the field of numerical analysis and approximation theory use the accomplishment of approximating to the fixed point of mapping. The significance of fixed point theory primarily lies in the fact that many of the equations that emerge in the various physical phenomena can be converted into fixed point formulae or inclusions.
The problem of classical variational inequalities for an operator F : C → H is defined in the following manner: Find ℘ * ∈ C such that (VIP) Suppose that the following conditions are met in order to obtain the convergence results of variational inequalities. A mapping F : C → H is said to be (F 1) pseudo-monotone on C such that  is non-empty. Choose u 0 , v 0 ∈ C, χ 0 > 0, µ ∈ (0, 1 3 ) and choose a non-negative real sequence {ϕ n } such that ∞ n ϕ n < +∞. Then, the sequence {u n } is generated in the following way: where H n = {z ∈ H : u n − χ n F (v n−1 ) − v n , z − v n ≤ 0} and Then, {u n } weakly converges to ℘ * ∈ V I(F , C).
and choose a non-negative real sequence {ϕ n } such that ∞ n ϕ n < +∞. Then, the sequence {u n } is generated in the following way: Then, the sequence {u n } converges weakly to ℘ * ∈ V I(F , C).
The problem of fixed point for a mapping S : C → H is defined in the following manner: Find ℘ * ∈ C such that S(℘ * ) = ℘ * .
Consider the following conditions are satisfied in order to achieve the convergence analysis of fixed point algorithms: A mapping S : C → H is called to be (S1) κ-strict pseudo-contraction [7] on C if (S2) sequentially weakly continuous on C if Then, the problem (EP) turns into the problem of fixed point where 2c 1 = 2c 2 = 3−2κ 1−κ . Corollary 4.3. Let a mapping S : C → H meet the conditions (S1) and (S2) and the solution set Fix(S, C) is non-empty. Choose u 0 , v 0 ∈ C, χ 0 > 0, µ ∈ (0, 1 3 ) and choose a non-negative real sequence {ϕ n } such that ∞ n ϕ n < +∞. Then, the sequence {u n } is generated in the following way: where H n = {z ∈ H : (1 − χ n )u n + χ n S(v n−1 ) − v n , z − v n ≤ 0} and Then, the sequence {u n } converges weakly to ℘ * ∈ Fix(S, C).
Corollary 4.4. Let a mapping S : C → H meet the conditions (S1) and (S2) and the solution set and choose a non-negative real sequence {ϕ n } such that ∞ n ϕ n < +∞. Then, the sequence {u n } is generated in the following way: Then, the sequence {u n } converges weakly to ℘ * ∈ Fix(S, C).

Numerical illustrations
In this section, we provide a numerical example to show the implementations of the proposed method. All computations are done in MATLAB R2018b and run on HP Core(TM)i5-6200 (7.78 GB usable) RAM 8.00 GB laptop.
Example 5.1. Consider a test problem where a bi-function f is defined as follows where P = (p i j ) N×N and Q = (q i j ) N×N are N × N symmetric positive semi-definite matrices such that P − Q is also positive semi-definite and r ∈ R N . The bi-function f has the form of the one arising from a Nash-Cournot oligopolistic electricity market equilibrium model [31] and that f is Lipschitz-type continuous with constants c 1 = c 2 = 1 2 P − Q and the positive semi-definition of P − Q gives that f is pseudo-monotone. P and Q are matrices of the form: Choose two diagonal matrices Experiment 1: In this experiment, the numerical performance of Algorithm 1 with Algorithm 1 in [16] and Algorithm 2 in [16] is provided by letting the starting points u 0 , v −1 , v 0 are randomly generated in [−N, N]. We assume the feasible set in the following manner:      [16] and Algorithm 2 in [16] in R 100 for first 50 iterations.

Experiment 2:
In this experiment, the numerical performance of Algorithm 1 with Algorithm 1 in [16] and Algorithm 2 in [16] is considered by taking the starting points u 0 , v −1 , v 0 are randomly generated in [−N, N]. We consider the feasible set as follows: where C ⊂ R 5 is taken as follows: Thus, f is Lipschitz-type continuous with c 1 = c 2 = 2 and satisfies the conditions (c1)-(c4).

Conclusions
The paper has introduced a new modified subgradient extragradient method to approximate the solution of the equilibrium problem in Hilbert spaces. A non-monotonic step size rule has been added that is not dependent on the Lipschitz-type constant information. A weak convergence theorem is well established under mild bi-functional conditions. Applications of the proposed algorithm are presented to solve variational inequalities and fixed-point problems. Many experiments have been reported to demonstrate the numerical behaviour of the proposed algorithm and to compare it to other algorithms.