Mixed Equilibrium Problems with Weakly Relaxed α-Monotone Bifunction in Banach Spaces

We introduce the class of mixed equilibrium problems with the weakly relaxed α-monotone bi-function in Banach spaces. Using the KKM technique, we obtain the existence of solutions for mixed equilibrium problem with weakly relaxed α-monotone bi-function in Banach spaces. The results presented in this paper extend and improve the corresponding results in the existing literature.


Introduction
Let  be a nonempty subset of a real reflexive Banach space .Let  :  → R be a real valued function and let  :  ×  → R be an equilibrium bi-function; that is, (, ) = 0, for all  ∈ .Then the mixed equilibrium problem (for short, (MEP)) is to find x ∈  such that  (x, ) +  () −  ( x) ≥ 0, ∀ ∈ . ( In particular, if  ≡ 0, this problem reduces to the classical equilibrium problem (for short, (EP)), which is to find x ∈  such that  (x, ) ≥ 0, ∀ ∈ . ( Equilibrium problems and mixed equilibrium problems play an important role in many fields, such as economics, physics, mechanics, and engineering sciences.Also, the equilibrium problems and mixed equilibrium problems include many mathematical problems as particular cases for example, mathematical programming problems, complementary problems, variational inequality problems, Nash equilibrium problems in noncooperative games, minimax inequality problems, and fixed point problems.Because of their wide applicability, equilibrium problems and mixed equilibrium problems have been generalized in various directions for the past several years.
In 2003, Fang and Huang [10] considered two classes of the variational-like inequalities with the relaxed  −  monotone and relaxed  −  semimonotone mappings.They obtained the existence solutions of variational-like inequalities with relaxed  −  monotone and relaxed  −  semimonotone mappings in Banach spaces using the KKM technique.Later, Bai et al. [1] introduced a new concept of relaxed  −  pseudomonotone mappings and obtained the solutions for the variational-like inequalities.Afterward, Mahato and Nahak [18] defined the weakly relaxed  −  pseudomonotone bi-function to study the equilibrium problems.
Recently, Mahato and Nahak [19] introduce the concept of the relaxed -monotonicity for bi-functions.They also obtained the existence of solutions for mixed equilibrium problems with the relaxed -monotone bi-function in reflexive Banach spaces, by using the KKM technique.
The purpose of this paper is to introduce the class of weakly relaxed -monotone bi-functions which contain the class of relaxed -monotone bi-functions.The existence of solutions for mixed equilibrium problems with bi-function in such class is given.Our results in this paper extend and improve the results of Mahato and Nahak [19] and many results in the literature.

Preliminaries
In this paper, unless otherwise specified, let  be a nonempty closed convex subset of a real reflexive Banach space .The following definitions and lemma will be useful in our paper.Lemma 4 (see [20]).Let  be a nonempty subset of a Hausdorff topological vector space  and let  :  → 2  be a KKM mapping.If () is closed in  for all  ∈  and compact for some  ∈ , then ⋂ ∈ () ̸ = 0.
Definition 5. Let  be a Banach space.A function  :  → R is lower semicontinuous at  0 ∈  if for any sequence {  } in  such that {  } converges to  0 .Definition 6.Let  be a Banach space.A function  :  → R is weakly upper semicontinuous at  0 ∈  if for any sequence {  } in  such that {  } converges to  0 weakly.
Definition 7 (see [19]).A bi-function  : × → R is said to be a relaxed -monotone if there exists a function  :  → R with () =   () for all  > 0 and  ∈  such that where  > 1 is a real constant.
Remark 8.If  ≡ 0 then from (5), it follows that  is monotone; that is, Therefore, the monotonicity implies relaxed -monotonicity.However, the converse of previous statement is not true in general, which is shown by the following examples.
Example 9. Let  = R,  = R and let bi-function  :  ×  → R be defined by for all ,  ∈ .Then when  ̸ = .So  is not monotone.However, it easy to see that  is a relaxed -monotone with () = 5 2 .In fact, Then, for  ̸ = , that is,  is not monotone mapping.But, if we choose  :  → R by () = 4‖‖ 2 , then  is a relaxed -monotone.

Mixed Equilibrium Problems with Weakly
Relaxed -Monotone Bi-Function In this section, we introduce the new class of bi-functions.
Using KKM technique, we study and prove the existence of solutions for mixed equilibrium problem with bi-function in such class in Banach spaces.
Remark 12.We obtain that the relaxed -monotonicity implies weakly relaxed -monotonicity.So the class of relaxed -monotone bi-functions is a subclass of weakly relaxed monotone bi-function class.
Next, we discuss the existence solution of the (MEP) (1), using the concept of the weakly relaxed -monotonicity.
Theorem 13.Suppose  :  ×  → R is a weakly relaxed -monotone which is hemicontinuous in the first argument, and convex in the second argument let  :  → R be a convex function.Then, the (MEP) and the following problem are equivalent: Proof.Suppose that the (MEP) (1) has a solution.So there exists x ∈  such that Since  is, weakly relaxed -monotone, we have and then Therefore, x ∈  is a solution of problem (15).Conversely, suppose x ∈  is a solution of problem (15) and  is any point in .For  ∈ (0, 1], we let   := +(1−) x.Since  is convex, we obtain that   ∈ .From (15), we have By the convexity of  in the second argument, we have that is, The convexity of  implies that for all  ∈ .
It is easy to see that x ∈  solves the problem (MEP) that is if and only if x ∈ ⋂ ∈ ().Thus it is sufficient to prove that ⋂ ∈ () ̸ = 0.
By assumption,  and   → (, ) are convex lower semicontinuous functions, where fixed  ∈ .Then it is easy to see that they are both weakly lower semicontinuous.From the definition of  and the weakly upper semicontinuity of , we get () is weakly closed for all  ∈ .
Since  is closed bounded and convex, it is also weakly compact, and then () is weakly compact in  for each  ∈ .From Lemma 4 and Theorem 13, we obtain that So there exists x ∈ , such that  (, ) +  () −  () ≥ 0, ∀ ∈ , then the problem (MEP) has a solution.This completes the proof.
It easy to see that the relaxed -monotonicity implies the weakly relaxed -monotonicity.So Theorem 14 can be deduced to the following corollary.

Corollary 15.
Let  be a nonempty bounded closed convex subset of a real reflexive Banach space .Suppose that  :  ×  → R is relaxed -monotone and hemicontinuous in the first argument; let  :  → R be a convex and lower semicontinuous function.Assume that (a) for fixed  ∈ , the mapping   → (, ) is convex and lower semicontinuous; (b)  :  → R is weakly upper semicontinuous.
Then the problem (MEP) has a solution.
Next, we study and prove result for the case of  is unbounded set.
For each  ∈ , we can choose 0 <  < 1 small enough such that  + (1 − )   ∈    .From (40), for each  ∈ , we have Remark 18. Theorems 13, 14, and 16 are improving the results of Fang [10] from the corresponding results of variationallike inequality problems to equilibrium problems.Also, these results are extensions of the main results of Mahato and Nahak [19].