Fuzzy Generalized Variational like Inequality Problems in Topological Vector Spaces

This paper is devoted to the existence of solutions for generalized variational like inequalities with fuzzy mappings in topological vector spaces by using a particular form of the generalized KKM-Theorem.


Introduction
In 1980, Giannessi [9] conceived the idea of vector variational inequality in finite dimensional Euclidean spaces.Since then, Chen et al. [4,5] have extensively studied vector variational inequalities in abstract spaces and have obtained existence theorems for their inequalities.In 1989, Chang et al. [2] introduced the concept of variational inequalities for fuzzy mappings in abstract spaces and investigated existence theorems for some kind of variational inequalities for fuzzy mappings.Recently several types of variational inequalities are initiated by Chang [1], Chang et al. [3], Park et al [15], Lee et al [12,13,14], Khan et al. [10], and Ding et al. [6], wherein they have obtained existence theorems for certain variational inequalities for fuzzy mappings by following the approach of Chang and Zhu [2] and using results of Kim and Tan [11].Our main motivation of this paper is to obtain the fuzzy extension of results of Farajzadeh et al. [8].Further an appropriate fixed point theorem is used to obtain existence result of solution for the considered problem.Let E be a nonempty subset of a vector space X and D be a nonempty set.A function F from D into the collection F(E) of all fuzzy sets of E is called a fuzzy mapping.If F : D → F(E) is a fuzzy mapping, then F(x), x ∈ D (denoted by F x in sequel) is a fuzzy set in F(E) and F x (y), y ∈ E is the degree of membership of y in F x .Let A ∈ F(E) and α ∈ (0, 1] then the set Definition 1.1.[10] Let X,Y be two topological vector spaces and T : X → 2 Y be a multifunction.Then we say that 1. T is upper semicontinuous (u.s.c.) at x ∈ X if for any open set N containing T (x), there exists a neighbourhood M of x such that T (M) ⊂ N, T is u.s.c.if T is u.s.c. at every x ∈ X.
2. T is closed at x ∈ X if for any net {x λ } in X such that x λ → x and for any net {y λ } in Y such that y λ → y and y λ ∈ T (x λ ) for any λ we have y ∈ T (x).

T has a closed graph if the graph of T, Gr
Definition 1.2.[12] Let X,Y be two topological vector spaces and F : X → F(Y ) be a fuzzy mapping, we say that F is a mapping with closed fuzzy set valued if F x (y) is u.s.c. on X ×Y as a real ordinary function.
Lemma 1.1.[12] If A is a closed subset of a topological vector space X, then the characteristic function χ A of A is an u.s.c.real valued function.
Lemma 1.2.[12] Let K and E be two nonempty closed convex subsets of two real Hausdorff topological vector spaces X and Y , respectively and α : X → (0, 1] be a lower semicontinuous function.Let F : K → F(E) be a fuzzy mapping with Proof.Let {x λ } be a net in K converging to x, {y λ } be a net in E converging to y and

Preliminaries
Let ⟨X, X * ⟩ be a dual system of Hausdorff topological vector space and K a nonempty convex subset of X.Given single-valued mappings f , g, p : X * → X * , a bifunction η : K ×K → X and a mapping h : K ×K → R. Let M, S, T : K → F(X * ) be the fuzzy mappings induced by the multivalued mappings M, S, T : K → 2 X * where 2 X * is the multivalued mapping of all nonempty subset of X * and a, b, c : X → [0, 1] be the functions.Let G : K → K be a mapping.We consider fuzzy generalized variational like inequality (FGVLI) for finding x ∈ K such that there exists u (2.1) Next we consider the following fuzzy generalized variational inequality (FGVI) for finding x ∈ K such that there (2.2)

Special cases:
If G is an identity mapping then (2.1) is equivalent to a problem of finding considered and studied by Farajzadeh et al [8] without fuzzy mappings.
Lemma 2.1.[16] Let X and Y be two topological spaces.If T : X → 2 Y is a set valued mapping then 1. T is closed if and only if for any net {x α }, x α → x and any net {y α }, y α ∈ T (x α ), y α → y one has y ∈ T (x).

Y is compact and T (x) is closed for each x ∈ X then T is upper semicontinuous if and only if T is closed.
3. For any x ∈ X, T (x) is compact and T is upper semicontinuous on X then for any net {x α } ⊆ X such that x α → x ∈ X and for every y α ∈ T (x α ) there exists y ∈ T (x) and subnet {y β } of y α such that {y β } → y.
International Scientific Publications and Consulting Services Definition 2.1.[7] Let X be a nonempty subset of a topological vector space E. A multifunction F : X → 2 E is said to be an KKM mapping if for each nonempty finite set The following version of the KKM principle is a special case of the Fan-KKM principle [7].
Lemma 2.2.[7] Let X be a nonempty subset of a topological vector space E and F : X → 2 E be an KKM mapping with closed values.Assume that there exists a nonempty compact convex subset B of X such that D =

Main Results
Theorem 3.1.Let K be a nonempty convex subset of a Hausdorff topological vector space X.Let M, S, T : K → 2 X * be the multivalued mappings defined by M, S, T : K → F(X * ), the upper semicontinuous with nonempty compact values, h : K × K → R and f , g, p : X * → X * be the continuous mappings.Suppose that the following conditions hold: 1.The mapping x → h(Gx, y) is lower semicontinuous with h(Gy, y) = 0 for y ∈ K, There exists a nonempty compact convex subset B of K and a nonempty compact subset D of K such that ψ B (x) ̸ = / 0 for all x ∈ K\D.
Then the solution set of FGVLI is nonempty and compact.
Proof.We define Ω : K → 2 K as: We first show that Ω is an KKM-mapping.Suppose on contrary, there exists Hence by (4), η(z, Gz) = 0 and h(Gz, z) = 0, we thus deduce that 0 < 0 which is a contradiction.Next we show that Ω(y) is a closed subset of X for each y ∈ K. To see this let x α be a net in Ω(y) which converges to x 0 ∈ X.Since x α ∈ Ω(y), by the definition of Ω(y) there exist nets {u α }, {v α } and {w α } with Since the fuzzy mappings M, S, T are upper semicontinuous with compact values.Then by Lemma 2.1, {u α }, {v α } and {w α } have convergent subnets with limits say u, v and w.Without loss of generality we may assume that {u α } International Scientific Publications and Consulting Services converges to u, {v α } converges to v and {w α } converges to w.Then by upper semicontinuity of M, S, T , we have u 0 ∈ ( MX 0 ), v 0 ∈ ( Sx 0 ), w 0 ∈ ( T x 0 ) and ⟨pu 0 − ( f v 0 − gw 0 ), η(y, Gx 0 )⟩ ≥ h(Gx 0 , y).
Note that M, S, T are closed mappings since M, S, T are u.s.c. with compact values and X is Hausdorff.The continuity of f , g, p and the upper semicontinuity of ⟨•, •⟩ imply, The last inequality follows from (1).Consequently x 0 ∈ Ω(y).Hence Ω(y) is closed in K for all y ∈ K.By (4) the set ∩ x∈B Ω(y) is compact.Therefore Ω satisfies all the assumptions of Lemma 2.2.Hence, there exists ȳ and so ȳ is a solution of FGVLI (note that the solution set of FGVLI equals to ∩ x∈K Ω(y)).Hence the solution set of FGVLI is nonempty.By (4) it is clear that the solution set of FGVLI is a closed subset of the compact set D. This completes the proof.
When η(y, Gx) = y − Gx in Theorem 3.1 we can obtain the existence theorem of solutions to the Fuzzy generalized variational inequality in topological vector spaces.Theorem 3.2.Let K be a nonempty convex subset of a Hausdorff topological vector space X.Let M, S, T : K → 2 X * be equipped with fuzzy mappings, M, S, T : K → F(X * ) be the upper semicontinuous with nonempty compact values, h : K × K → R and f , g, p : X * → X * be continuous.Suppose that the following conditions hold: 1.The mapping x → h(Gx, y) is lower semicontinuous with h(Gy, y) = 0 for all y ∈ K; 2. η : K × K → X is continuous in the second argument such that η(x, Gx) = 0 for all x ∈ K; 3. G : K → K is continuous and convex; 4. The set ψ K (x) = {y ∈ K : there exists u ∈ Mx, v ∈ Sx, w ∈ T x such that ⟨pu − ( f v − gw), η(y, Gx)⟩ < h(Gx, y)} for all x ∈ K, is convex.
5. There exists a nonempty compact convex subset B of K and a nonempty compact subset D of K such that ψ Ω (x) ̸ ≠ / 0 for all x ∈ K\D.
Then the solution set of FGVI (2.2) is nonempty and compact.