On F -implicit minimal vector variational inequalities

In this paper, by introducing some new concepts in minimal spaces, we prove a generalized form of the Fan-KKM theorem in minimal vector spaces. A new class of minimal generalized vector F -implicit variational inequality problems and, as an application of Fan-KKM theorem is investigated. Moreover, an existence theorem for this kind of problems under some suitable assumptions in minimal vector spaces is given.


Introduction
Nonlinear analysis presents many problems that can be resolved by the nonemptyness of the intersection of a certain family of subsets of a underlying set. Each point of the intersection can be a fixed point, a coincidence point, an equilibrium point, a saddle point, an optimal point, a solution point for complementarity problem, a solution point for variational problem, or others of the corresponding problem under consideration. The first result on the nonempty intersection was the celebrated Knaster-Kuratowski-Mazurkiewicz theorem (simply, the KKM principle) in [10], which is concerned with certain types of multimaps called the KKM maps. The KKM theory is the study of KKM maps and their applications. Generalized form of the KKM theorem namely Fan-KKM principle provides a foundation for many of the modern essential results in diverse areas of mathematical sciences (for more details see [15]). However the Fan-KKM theorem has essential role in solving all kinds of implicit complementarity problems or variational inequality problems, particularly in generalized vector F -implicit variational inequality problems [17]. The vector variational inequalities and vector complementarity problems have found many of their applications in vector optimization, set-valued optimization, approximate analysis of vector optimization problems and vector network equilibrium problems.
Huang and Li [7] introduced and studied a class of scalar F -implicit variational inequality problems in Banach spaces. In 2006, Li and Huang [12] generalized the results from the scalar case in [7] to the vector case. The vector variational inequalities and vector complementarity problems have found many of its applications in vector optimization, set-valued optimization, approximate analysis of vector optimization problems and vector network equilibrium problems. Recently, Lee, Khan, and Salahuddin [11] introduced the class of generalized vector Fimplicit variational inequality problems in Banach spaces, which generalized some results of [7] and [12] to a 402 F -IMPLICIT MINIMAL VECTOR VARIATIONAL INEQUALITIES more generalized vector case. Some new existence theorems of solutions for generalized F -implicit variational inequality problems were also proved in [11].
In this paper, motivated by the above mentioned works, we obtain some new results to prove a generalized form of the Fan-KKM theorem in minimal vector spaces. Also we introduce a new class of vector F -implicit variational inequality problems and as an application of our obtained Fan-KKM theorem we derive an existence theorem for this kind of problems.

Preliminaries
The concepts of minimal structures and minimal spaces, as generalization of topology and topological spaces were introduced in [14]. For easy understanding of the material incorporated in this paper, we recall some basic definitions and results. Also some new concepts are introduced in minimal spaces. Further results about minimal spaces can be found in [1,2,3,4,13], [16] and some references cited therein.
A family M ⊆ P(X) is said to be a minimal structure on X if ∅, X ∈ M. In this case (X, M) is called a minimal space. For example, let (X, τ ) be a topological space, then τ , SO(X), P O(X), αO(X) and βO(X) are minimal structures on X [13]. In a minimal space ( is not necessarily m-closed (resp. m-open). The following lemma may be useful to apply in a minimal space.  Definition 2 [1] For two minimal spaces (X, M) and (Y, N ), we define minimal product structure for X × Y as follows:

Lemma 1 [13] For any two sets
Definition 3 [1] A linear minimal structure on a vector space X over the complex field F is a minimal structure M on X such that the two mappings are m-continuous, where F has the usual topology and both F × X and X × X have the corresponding product minimal structures. A linear minimal space (or minimal vector space) is a vector space together with a linear minimal structure.
Obviously, any topological vector space is a minimal vector space but the converse is not true generally. In the following, it is shown that there is a linear minimal space which is not a topological vector space.

Example 1
Consider the real field R and let M = {(a, b) : a, b ∈ R ∪ {±∞}}. Clearly, M is a minimal structure on R. We claim that M is a linear minimal structure on R. For this, we must prove that, two operations + and · are mcontinuous.
Theorem 1 [16] Suppose that X and Y are two minimal spaces and f : X → Y is an m-continuous function. For any m- In the following result, we prove the minimal version of Tychonoff theorem

Theorem 2
The minimal product space

Proof
One direction is an immediate consequence of Theorem 1. For the converse, on the contrary, assume that A ⊆ ∏ α∈I M α is an m-open cover of ∏ α∈I X α without any finite subcover for ∏ α∈I X α . For any α ∈ I, set U α = {V ∈ M α : Since A has no finite subcover for ∏ α∈I X α , no finite subcover of U α can cover X α , for any α ∈ I. Now, m-compactness of X α implies that U α can not cover X α . Therefore there exists

Generalized Fan-KKM Theorem
). It is obvious that an m-closed valued multimap is a minimal transfer closed multimap.
A subset A of a vector space X is convex if we have ty + (1 − t)z ∈ A, whenever y, z ∈ A and t ∈ [0, 1]. Also the convex hull of A, denoted by co(A), is the smallest convex set that contains A, that is, the intersection of all convex sets containing A.

Definition 6
Suppose that D is a convex subset of a minimal vector space X. A multimap F : D X is called a KKM map if co(A) ⊆ F (A) for any A ∈ ⟨D⟩, where the notation ⟨D⟩ means the set of all finite subsets of D.
The following theorem is a generalized form of the Fan-KKM theorem, as a special case of Theorem 4.7 in [3], related to the minimal vector spaces.

Theorem 3
Suppose that X is a minimal vector space. Consider two nonempty valued multimaps F, G : Now, we introduce a new concept in minimal vector spaces with some examples which is useful for the proof of our results.

Definition 7
Suppose that X is a minimal (topological) vector space. A nonempty set A ⊆ X has the minimal (topological) finitely adjoint co-compact property if co (A ∪ B) is an m-compact (compact) subset of X, for any B ∈ ⟨X⟩. Example 2 (a) Consider R with it's usual topology. Then every finite subset of R has the topological finitely adjoint co-compact property. (b) Since every topological space is a minimal space, any set with the topological finitely adjoint co-compact property in a topological vector space has the minimal finitely adjoint co-compact property if we consider the topological structure as minimal structure, but the converse is not true. Consider the minimal structure which is not compact in R when we consider the usual topology. But it is an m-compact subset of R via minimal structure M (just R is it's cover). So there exists a set with the minimal finitely adjoint co-compact property which does not have the topological finitely adjoint co-compact property. (c) Suppose that X is a minimal vector space. Any m-compact convex subset of X has the minimal finitely adjoint co-compact property. To see this, suppose that A is an m-compact convex subset of X and B = is the standard base of R n . It is not hard to see that φ is an onto m-continuous function by Definition 3. Since we can consider ∆ n+1 as an m-compact subset of R n , according to the Theorem 2, From Theorem 3 along with Definition 7 we can achieve another version of the Fan-KKM theorem which is our main task in this section.

Theorem 4
Suppose that in Theorem 3 condition (c) is replaced with the following condition: (c ′ ) there exists a nonempty subset B ⊆ X with the minimal finitely adjoint co-compact property such that m- For Since B has the minimal finitely adjoint co-compact property then L A is m-compact and Lemma 2 implies that m-Cl LA (G A (x)) is m-compact for any x ∈ L A . All conditions of Theorem 3 are satisfied for A ∈ ⟨X⟩} has the finite intersection property. To see this, assume that A 1 , A 2 , . . . , A n ∈ ⟨X⟩. Since which gives the result.
By assertion (e),x

Remark 1
It should be noticed that (a) according to Example 2(c), when B is an m-compact convex subset of X, the first assertion in condition (c ′ ) of Theorem 4 is fullfilled. (b) Theorem 4 generally goes back to the Fan-KKM theorem discussed in [5].

Minimal Generalized Vector F -implicit Variational Inequality Problem
In this section, as an application of the generalized Fan-KKM theorem, we give sufficient conditions to solve a minimal generalized vector F -implicit variational inequality problem in the minimal vector spaces.
A nonempty subset P of a vector space X is called a convex cone if P + P = P and αP ⊆ P for all α ≥ 0, where P + P = {x + y : x, y ∈ P } and αP = {αx : x ∈ P }. A cone is said to be pointed if P ∩ −P = {0}.
The following "generalized vector F-implicit variational inequality problem (GVF-IVIP)" problem has been considered and solved in [11]: Let X be a real Banach space, K ⊆ X be a nonempty closed convex cone and (Y, P ) be an ordered Banach space induced by the pointed closed convex cone P . Denote the space of all continuous linear mappings from X into Y by L(X, Y ) and the value of a linear continuous mapping t ∈ L(X, Y ) at x by ⟨t, x⟩. Let A, T : K → L(X, Y ),

Theorem 5
Suppose that (a) five mappings N, g, A, T and F are continuous, (b) there exists a mapping h : (c) there exists a nonempty compact convex subset C of K such that for all x ∈ K \ C there exists y ∈ C such that ⟨N (Ax, T x), g(y) − g(x)⟩ + F (g(y)) − F (g(x)) ≥ 0.
Then GVF-IVIP has a solution. Furthermore, the solution set of (GVF-IVIP) is closed. Now, suppose that X and Y are two minimal vector spaces and M is a nonempty convex subset of X. Set L m (X, Y ) as the set of all m-continuous linear mappings from X into Y . Let ⟨T, x⟩ be the value of T ∈ L m (X, Y ) at point x. In addition, let A, B : Y has nonempty convex pointed cone values; that is for all y ∈ M , C(y) is a nonempty and convex pointed cone in Y .

Conclusion
Vector variational inequality (VVI) is a base for vector equilibrium problems, which can be applied to traffic networks and migration equilibrium problems (see [6]). Meanwhile, Fan-KKM principle can be utilized to solve some special problems related to vector variational inequalities. Additionally, many new problems in nonlinear analysis can be solved by introducing generalized concepts and extending spaces to provide a proper solution.
The above description motivated us to introduce the concepts of generalized Fan-KKM multimaps and generalized vector F -implicit variational inequality problems (See, also [9]). In this work, we introduced a generalized version of the Fan-KKM theorem for vector F -implicit variational inequality problems in minimal vector spaces that covers many previous results.