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

Abstract

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.

1. Introduction

Let 𝐾 be a nonempty subset of a real reflexive Banach space 𝑋. Let 𝜑∶𝐾→ℝ be a real valued function and let 𝑓∶𝐾×𝐾→ℝ be an equilibrium bi-function; that is, 𝑓(𝑥,𝑥)=0, for all 𝑥∈𝐾. Then the mixed equilibrium problem (for short, (MEP)) is to find 𝑥∈𝐾 such that 𝑓𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐾. (1) In particular, if 𝜑≡0, this problem reduces to the classical equilibrium problem (for short, (EP)), which is to find 𝑥∈𝐾 such that 𝑓𝑥,𝑦≥0,∀𝑦∈𝐾. (2)

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.

The monotonicity and generalized monotonicity play an important role in the study of equilibrium problems and mixed equilibrium problems. In recent years, a substantial number of papers on existence results for solving equilibrium problems and mixed equilibrium problems based on different generalization of monotonicity such as pseudomonotonicity, quasimonotonicity, relaxed monotonicity, semimonotonicity, 𝑝-monotonicity, and 𝐶_𝑥-pseudomonotonicity, see [1–17], appeared.

In 1990, Karamardian and Schaible [14] introduced various kinds of generalized monotone mappings. Afterward, several researcher in [2–4] extended the idea of Karamardian and Schaible [14] for bi-functions to study equilibrium problems.

In 2003, Fang and Huang [10] considered two classes of the variational-like inequalities with the relaxed 𝜂−𝛼 monotone and relaxed 𝜂−𝛼 semimonotone mappings. They obtained the existence solutions of variational-like inequalities with relaxed 𝜂−𝛼 monotone and relaxed 𝜂−𝛼 semimonotone mappings in Banach spaces using the KKM technique. Later, Bai et al. [1] introduced a new concept of relaxed 𝜂−𝛼 pseudomonotone mappings and obtained the solutions for the variational-like inequalities. Afterward, Mahato and Nahak [18] defined the weakly relaxed 𝜂−𝛼 pseudomonotone bi-function to study the equilibrium problems.

Recently, Mahato and Nahak [19] introduce the concept of the relaxed 𝛼-monotonicity for bi-functions. They also obtained the existence of solutions for mixed equilibrium problems with the relaxed 𝛼-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 𝛼-monotone bi-functions which contain the class of relaxed 𝛼-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.

2. Preliminaries

In this paper, unless otherwise specified, let 𝐾 be a nonempty closed convex subset of a real reflexive Banach space 𝑋. The following definitions and lemma will be useful in our paper.

Definition 1. A real valued function 𝑓 defined on a convex subset 𝐾 of 𝑋 is said to be hemicontinuous if lim_𝑡→0⁺𝑓(𝑡𝑥+(1−𝑡)𝑦)=𝑓(𝑦), for each 𝑥,𝑦∈𝐾.

Definition 2. Let 𝐹∶𝐾→2^𝑋 be a set-valued mapping. Then 𝐹 is said to be KKM mapping if for any finite subset {𝑦₁,𝑦₂,…,𝑦_𝑛} of 𝐾, we have co{𝑦₁,𝑦₂,…,𝑦_𝑛}⊂⋃^𝑛_𝑖=1𝐹(𝑦_𝑖), where co{𝑦₁,𝑦₂,…,𝑦_𝑛} denotes the convex hull of {𝑦₁,𝑦₂,…,𝑦_𝑛}.

Remark 3. Let 𝐹,𝐺∶𝐾→2^𝑋. If 𝐹 is KKM mapping and 𝐹(𝑦)⊂𝐺(𝑦) for all 𝑦∈𝐾, then 𝐺 also is KKM mapping.

Lemma 4 (see [20]). Let 𝑀 be a nonempty subset of a Hausdorff topological vector space 𝑋 and let 𝐹∶𝑀→2^𝑋 be a KKM mapping. If 𝐹(𝑦) is closed in 𝑋 for all 𝑦∈𝑀 and compact for some 𝑦∈𝑀, then ⋂_𝑦∈𝑀𝐹(𝑦)≠∅.

Definition 5. Let 𝑋 be a Banach space. A function 𝑓∶𝑋→ℝ is lower semicontinuous at 𝑥₀∈𝑋 if 𝑓𝑥₀≤liminf_𝑛→∞𝑓𝑥_𝑛 (3) for any sequence {𝑥_𝑛} in 𝑋 such that {𝑥_𝑛} converges to 𝑥₀.

Definition 6. Let 𝑋 be a Banach space. A function 𝑓∶𝑋→ℝ is weakly upper semicontinuous at 𝑥₀∈𝑋 if 𝑓𝑥₀≥limsup_𝑛→∞𝑓𝑥_𝑛 (4) for any sequence {𝑥_𝑛} in 𝑋 such that {𝑥_𝑛} converges to 𝑥₀ weakly.

Definition 7 (see [19]). A bi-function 𝑓∶𝐾×𝐾→ℝ is said to be a relaxed 𝛼-monotone if there exists a function 𝛼∶𝑋→ℝ with 𝛼(𝑡𝑥)=𝑡^𝑝𝛼(𝑥) for all 𝑡>0 and 𝑥∈𝐾 such that 𝑓𝑥,𝑦+𝑓𝑦,𝑥≤𝛼𝑦−𝑥,∀𝑥,𝑦∈𝐾, (5) where 𝑝>1 is a real constant.

Remark 8. If 𝛼≡0 then from (5), it follows that 𝑓 is monotone; that is, 𝑓𝑥,𝑦+𝑓𝑦,𝑥≤0,∀𝑥,𝑦∈𝐾. (6) Therefore, the monotonicity implies relaxed 𝛼-monotonicity. However, the converse of previous statement is not true in general, which is shown by the following examples.

Example 9. Let 𝑋=ℝ, 𝐾=ℝ and let bi-function 𝑓∶𝐾×𝐾→ℝ be defined by 𝑓𝑥,𝑦=sin(𝑥−𝑦)²+(𝑥−𝑦)², (7) for all 𝑥,𝑦∈𝐾. Then 𝑓𝑥,𝑦+𝑓𝑦,𝑥=2sin(𝑥−𝑦)²+2(𝑥−𝑦)²≰0, (8) when 𝑥≠𝑦. So 𝑓 is not monotone.
However, it easy to see that 𝑓 is a relaxed 𝛼-monotone with 𝛼(𝑥)=5𝑥². In fact, 𝑓𝑥,𝑦+𝑓𝑦,𝑥=2sin(𝑥−𝑦)²+2(𝑥−𝑦)²≤5𝑥−𝑦²=𝛼𝑦−𝑥. (9)

Example 10. Let 𝑋=ℓ^𝑝={𝑥={𝑥_𝑛}⊆ℝ∶‖𝑥‖=(∑_𝑛≥1|𝑥_𝑛|^𝑝)^1/𝑝<∞}. For 1<𝑝<∞, ℓ^𝑝 is a reflexive Banach space. Let 𝐾={𝑥∈ℓ^𝑝∶‖𝑥‖≤1}, which is nonempty, closed, and convex subset of ℓ^𝑝. Define 𝑓∶𝐾×𝐾→ℝ by 𝑓𝑥,𝑦=𝑥−𝑦². (10) Then, 𝑓𝑥,𝑦+𝑓𝑦,𝑥=2𝑥−𝑦²>0, (11) for 𝑥≠𝑦, that is, 𝑓 is not monotone mapping.
But, if we choose 𝛼∶𝑋→ℝ by 𝛼(𝑥)=4‖𝑥‖², then 𝑓 is a relaxed 𝛼-monotone.

3. Mixed Equilibrium Problems with Weakly Relaxed 𝛼-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.

Definition 11. A bi-function 𝑓∶𝐾×𝐾→ℝ is said to be a weakly relaxed 𝛼-monotone if there exists a function 𝛼∶𝑋→ℝ with lim_𝑡→0+𝛼(𝑡𝑥)=0,lim_𝑡→0+𝑑𝛼()𝑑𝑡𝑡𝑥=0, (12)(13) for all 𝑡>0 and 𝑥∈𝐾 such that 𝑓𝑥,𝑦+𝑓𝑦,𝑥≤𝛼𝑦−𝑥,∀𝑥,𝑦∈𝐾. (14)

Remark 12. We obtain that the relaxed 𝛼-monotonicity implies weakly relaxed 𝛼-monotonicity. So the class of relaxed 𝛼-monotone bi-functions is a subclass of weakly relaxed 𝛼-monotone bi-function class.

Next, we discuss the existence solution of the (MEP) (1), using the concept of the weakly relaxed 𝛼-monotonicity.

Theorem 13. Suppose 𝑓∶𝐾×𝐾→ℝ is a weakly relaxed 𝛼-monotone which is hemicontinuous in the first argument, and convex in the second argument let 𝜑∶𝐾→ℝ be a convex function. Then, the (MEP) and the following problem are equivalent: 𝑥(𝑥)𝑦𝑥𝑓𝑖𝑛𝑑𝑥∈𝐾𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑓𝑦,+𝜑−𝜑≤𝛼𝑦−,∀𝑦∈𝐾. (15)

Proof. Suppose that the (MEP) (1) has a solution. So there exists 𝑥∈𝐾 such that 𝑓𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐾. (16) Since 𝑓 is, weakly relaxed 𝛼-monotone, we have 𝑓𝑥,𝑦+𝑓𝑦,𝑥≤𝛼𝑦−𝑥,∀𝑦∈𝐾, (17) and then 𝑓𝑥𝑦(𝑥)𝑥𝑦(𝑥)𝑥−𝑓𝑦(𝑥𝑥𝑦,+𝜑−𝜑≤𝛼𝑦−−𝑓𝑥,𝑦+𝜑−𝜑=𝛼𝑦−𝑥,𝑦−𝜑+𝜑)≤𝛼𝑦−,∀𝑦∈𝐾. (18) Therefore, 𝑥∈𝐾 is a solution of problem (15).
Conversely, suppose 𝑥∈𝐾 is a solution of problem (15) and 𝑦 is any point in 𝐾. For 𝑡∈(0,1], we let 𝑥_𝑡∶=𝑡𝑦+(1−𝑡)𝑥. Since 𝐾 is convex, we obtain that 𝑥_𝑡∈𝐾. From (15), we have 𝑓𝑥_𝑡,𝑥+𝜑(𝑥)−𝜑𝑥_𝑡≤𝛼𝑥_𝑡−𝑥. (19) By the convexity of 𝑓 in the second argument, we have 0=𝑓𝑥_𝑡,𝑥_𝑡≤𝑡𝑓𝑥_𝑡,𝑦+(1−𝑡)𝑓𝑥_𝑡,𝑥; (20) that is, 𝑡𝑓𝑥_𝑡,𝑥−𝑓𝑥_𝑡,𝑦≤𝑓𝑥_𝑡,𝑥. (21) The convexity of 𝜑 implies that 0=𝜑𝑥_𝑡−𝜑𝑥_𝑡≤𝑡𝜑𝑦+(1−𝑡)𝜑(𝑥)−𝜑𝑥_𝑡, (22) and thus 𝑡𝜑(𝑥)−𝜑𝑦≤𝜑(𝑥)−𝜑𝑥_𝑡. (23) From (19), (21), and (23), we have 𝑡𝑓𝑥_𝑡,𝑥−𝑓𝑥_𝑡,𝑦+𝜑(𝑥)−𝜑𝑦≤𝛼𝑥_𝑡−𝑥, (24) and so 𝑓𝑥_𝑡,𝑥−𝑓𝑥_𝑡(𝑥)𝑦≤𝛼𝑡𝑥,𝑦+𝜑−𝜑𝑦−𝑡. (25) Since 𝑓 is hemicontinuous in the first argument and taking 𝑡→0⁺ we get 𝑓(𝑥,𝑥)−𝑓𝑥,𝑦+𝜑(𝑥)−𝜑𝑦≤lim_𝑡→0+𝛼𝑡𝑦−𝑥𝑡. (26) From (12), we get lim_𝑡→0⁺(𝛼(𝑡(𝑦−𝑥)/𝑡)) is indeterminate form. Using L’ Hopital’s rule, we obtain that 𝑓(𝑥,𝑥)−𝑓𝑥,𝑦+𝜑(𝑥)−𝜑𝑦≤lim_𝑡→0+(𝑑/𝑑𝑡)𝛼𝑡𝑦−𝑥1. (27) By property (13) of weakly relaxed 𝛼-monotone 𝑓, we have 𝑓(𝑥,𝑥)−𝑓𝑥,𝑦+𝜑(𝑥)−𝜑𝑦≤0, (28) and then 𝑓𝑥,𝑦−𝜑(𝑥)+𝜑𝑦≥0,∀𝑦∈𝐾. (29) Therefore, 𝑥 is a solution of (MEP).

Theorem 14. Let 𝐾 be a nonempty bounded closed convex subset of a real reflexive Banach space 𝑋. Suppose that 𝑓∶𝐾×𝐾→ℝ is a weakly relaxed 𝛼-monotone and hemicontinuous in the first argument; let 𝜑∶𝐾→ℝ be a convex and lower semicontinuous function. Assume that (a)for fixed 𝑧∈𝐾, the mapping 𝑥↦𝑓(𝑧,𝑥) is convex and lower semicontinuous; (b)𝛼∶𝑋→ℝ is weakly upper semicontinuous. Then the problem (MEP) has a solution.

Proof. Consider the set valued mapping 𝐹∶𝐾→2^𝑋 defined by 𝐹𝑦=𝑥∈𝐾∶𝑓𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0, (30) for all 𝑦∈𝐾.
It is easy to see that 𝑥∈𝐾 solves the problem (MEP) that is 𝑓𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐾, (31) if and only if 𝑥∈⋂_𝑦∈𝐾𝐹(𝑦). Thus it is sufficient to prove that ⋂_𝑦∈𝐾𝐹(𝑦)≠∅.
Next, we show that 𝐹 is a KKM mapping. Assuming this contrary, then there exists {𝑥₁,𝑥₂,…,𝑥_𝑚}⊂𝐾 such that co{𝑥₁,𝑥₂,…,𝑥_𝑚}⊈⋃^𝑚_𝑖=1𝐹(𝑥_𝑖). This implies that there exists 𝑥₀∈co{𝑥₁,𝑥₂,…,𝑥_𝑚} such that 𝑥₀=∑^𝑚_𝑖=1𝑡_𝑖𝑥_𝑖, where 𝑡_𝑖≥0, 𝑖=1,2,…,𝑚, and ∑^𝑚_𝑖=1𝑡_𝑖=1, but 𝑥₀∉⋃^𝑚_𝑖=1𝐹(𝑥_𝑖).
From the construction of 𝐹, we have 𝑓𝑥₀,𝑥_𝑖+𝜑𝑥_𝑖−𝜑𝑥₀<0,for𝑖=1,2,…,𝑚. (32) By the convexity of 𝑓 in the second variable and the convexity of 𝜑 and (32), we obtain that 0=𝑓𝑥₀,𝑥₀=𝑓𝑥₀,^𝑚_𝑖=1𝑡_𝑖𝑥_𝑖≤^𝑚_𝑖=1𝑡_𝑖𝑓𝑥₀,𝑥_𝑖<^𝑚_𝑖=1𝑡_𝑖𝜑𝑥₀−𝜑𝑥_𝑖=𝜑𝑥₀−^𝑚_𝑖=1𝑡_𝑖𝜑𝑥_𝑖≤𝜑𝑥₀−𝜑𝑥₀=0, (33) which is a contradiction. Therefore, 𝐹 is a KKM mapping.
Now we define another set valued mapping 𝐺∶𝐾→2^𝑋 such that 𝐺𝑦=𝑥∈𝐾∶𝑓𝑦,𝑥+𝜑(𝑥)−𝜑𝑦≤𝛼𝑦−𝑥 (34) for all 𝑦∈𝐾.
Next, we will prove that 𝐹(𝑦)⊂𝐺(𝑦) for all 𝑦∈𝐾. For each 𝑦∈𝐾, let 𝑥∈𝐹(𝑦); then 𝑓𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0. (35) From the weakly relaxed 𝛼-monotonicity of 𝑓, we get 𝑓(𝑥)𝑦−𝑓𝑦(𝑥.𝑦,𝑥+𝜑−𝜑≤𝛼𝑦−𝑥𝑥,𝑦+𝜑−𝜑)≤𝛼𝑦−𝑥 (36)
This implies that 𝑥∈𝐺(𝑦) and hence 𝐹(𝑦)⊂𝐺(𝑦) for all 𝑦∈𝐾. So 𝐺 is also a KKM mapping.
By assumption, 𝜑 and 𝑥↦𝑓(𝑧,𝑥) are convex lower semicontinuous functions, where fixed 𝑧∈𝐾. Then it is easy to see that they are both weakly lower semicontinuous. From the definition of 𝐺 and the weakly upper semicontinuity of 𝛼, we get 𝐺(𝑦) is weakly closed for all 𝑦∈𝐾.
Since 𝐾 is closed bounded and convex, it is also weakly compact, and then 𝐺(𝑦) is weakly compact in 𝐾 for each 𝑦∈𝐾. From Lemma 4 and Theorem 13, we obtain that _𝑦∈𝐾𝐹𝑦=_𝑦∈𝐾𝐺𝑦≠∅. (37) So there exists 𝑥∈𝐾, such that 𝑓𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐾, (38) then the problem (MEP) has a solution. This completes the proof.

It easy to see that the relaxed 𝛼-monotonicity implies the weakly relaxed 𝛼-monotonicity. So Theorem 14 can be deduced to the following corollary.

Corollary 15. Let 𝐾 be a nonempty bounded closed convex subset of a real reflexive Banach space 𝑋. Suppose that 𝑓∶𝐾×𝐾→ℝ is relaxed 𝛼-monotone and hemicontinuous in the first argument; let 𝜑∶𝐾→ℝ be a convex and lower semicontinuous function. Assume that (a)for fixed 𝑧∈𝐾, the mapping 𝑥↦𝑓(𝑧,𝑥) is convex and lower semicontinuous; (b)𝛼∶𝑋→ℝ is weakly upper semicontinuous. Then the problem (MEP) has a solution.

Next, we study and prove result for the case of 𝐾 is unbounded set.

Theorem 16. Let 𝐾 be a nonempty unbounded closed convex subset of a real reflexive Banach space 𝑋. Suppose that 𝑓∶𝐾×𝐾→ℝ is a weakly relaxed 𝛼-monotone and hemicontinuous in the first argument; let 𝜑∶𝐾→ℝ be a convex and lower semicontinuous function. Assume that (a)for fixed 𝑧∈𝐾, the mapping 𝑥↦𝑓(𝑧,𝑥) is convex and lower semicontinuous; (b)𝛼∶𝑋→ℝ is weakly upper semicontinuous. (c)𝑓 satisfied the weakly coercivity condition, that is, there exists 𝑥₀∈𝐾 such that 𝑓𝑥,𝑥₀+𝜑𝑥₀−𝜑(𝑥)<0, (39)whenever 𝑥∈𝐾 and ‖𝑥‖is large enough.
Then the problem (MEP) has a solution.

Proof. For 𝜖>0, define 𝐾_𝜖∶={𝑦∈𝐾∶‖𝑦‖≤𝜖}.
Consider the problem: find 𝑥_𝜖∈𝐾_𝜖 such that 𝑓𝑥_𝜖,𝑦+𝜑𝑦−𝜑𝑥_𝜖≥0,∀𝑦∈𝐾_𝜖. (40) Since 𝐾_𝜖 is bounded, by Theorem 14, we get that the problem (40) has at least one solution 𝑥_𝜖∈𝐾_𝜖.
For 𝑥₀ in the weakly coercivity condition (c), we choose 𝜖^′>‖𝑥₀‖. From (40), we have 𝑓𝑥_𝜖′,𝑥₀+𝜑𝑥₀−𝜑𝑥_𝜖′≥0. (41) Since 𝑥_𝜖′∈𝐾_𝜖′, we have ‖𝑥_𝜖′‖≤𝜖^′. If ‖𝑥_𝜖′‖=𝜖^′, we may choose 𝜖^′ large enough so that by the weakly coercivity condition (c), we get 𝑓𝑥_𝜖′,𝑥₀+𝜑𝑥₀−𝜑𝑥_𝜖′<0, (42) which contradicts (41). Therefore, we must have 𝜖^′ such that ‖𝑥_𝜖′‖<𝜖^′.
For each 𝑦∈𝐾, we can choose 0<𝑡<1 small enough such that 𝑡𝑦+(1−𝑡)𝑥_𝜖′∈𝐾_𝜖′. From (40), for each 𝑦∈𝐾, we have 0≤𝑓𝑥_𝜖′,𝑡𝑦+(1−𝑡)𝑥_𝜖′+𝜑𝑡𝑦+(1−𝑡)𝑥_𝜖′−𝜑𝑥_𝜖′≤𝑡𝑓𝑥_𝜖′,𝑦+(1−𝑡)𝑓𝑥_𝜖′,𝑥_𝜖′+𝑡𝜑𝑦+(1−𝑡)𝜑𝑥_𝜖′−𝜑𝑥_𝜖′=𝑡𝑓𝑥_𝜖′,𝑦+𝜑𝑦−𝜑𝑥_𝜖′. (43) This implies that 𝑓𝑥_𝜖′,𝑦+𝜑𝑦−𝜑𝑥_𝜖′≥0∀𝑦∈𝐾. (44) Therefore, the problem (MEP) has a solution. This completes the proof.

Corollary 17. Let 𝐾 be a nonempty unbounded closed convex subset of a real reflexive Banach space 𝑋. Suppose that 𝑓∶𝐾×𝐾→ℝ is relaxed 𝛼-monotone and hemicontinuous in the first argument; let 𝜑∶𝐾→ℝ be a convex and lower semicontinuous function. Assume that (a)for fixed 𝑧∈𝐾, the mapping 𝑥↦𝑓(𝑧,𝑥) is convex and lower semicontinuous; (b)𝛼∶𝑋→ℝ is weakly upper semicontinuous;(c)𝑓 satisfied the weakly coercivity condition; that is, there exists 𝑥₀∈𝐾 such that 𝑓𝑥,𝑥₀+𝜑𝑥₀−𝜑(𝑥)<0, (45)whenever 𝑥∈𝐾 and ‖𝑥‖ large enough.
Then the problem (MEP) has a solution.

Remark 18. Theorems 13, 14, and 16 are improving the results of Fang [10] from the corresponding results of variational-like inequality problems to equilibrium problems. Also, these results are extensions of the main results of Mahato and Nahak [19].