Available online at www.ispacs.com/jfsva Volume 2013, Year 2013 Article ID jfsva-00134, 5 Pages
doi:10.5899/2013/jfsva-00134 Research Article
Fuzzy Generalized Variational Like Inequality problems in
Topological Vector Spaces
M. K. Ahmad1∗, Salahuddin1
(1)Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India
Copyright 2013 c⃝M. K. Ahmad and Salahuddin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.
Keywords:Generalized variational like inequalities; topological vector spaces; KKM-Theorems; fuzzy mappings
1 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. LetE be a nonempty subset of a vector spaceX andDbe a nonempty set. A functionF fromDinto the collectionF(E)of all fuzzy sets ofEis called a fuzzy mapping. IfF:D→F(E)is a fuzzy mapping, thenF(x),x∈D(denoted byFxin sequel) is a fuzzy set inF(E)andFx(y),y∈Eis the degree of membership ofyin
Fx. LetA∈F(E)andα∈(0,1]then the set
(A)α={x∈E:A(x)≥α},
is called anα-cut set ofA.
Definition 1.1. [10] Let X,Y be two topological vector spaces and T:X→2Y 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{xλ}in X such that xλ →x and for any net{yλ}in Y such that yλ→y and yλ∈T(xλ)for anyλ we have y∈T(x).
3. T has a closed graph if the graph of T,Gr(T) ={(x,y)∈X×Y :y∈T(x)}is closed in X×Y .
4. T is compact if cl(T(X))is a compact subset of Y .
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 Fx(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 functionχAof 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 andα :X→(0,1]be a lower semicontinuous function. Let F :K→F(E)be a fuzzy mapping with(Fx)α(x)̸=0for any x∈X . LetFe:K→2Eis a multifunction defined byF(x) = (Fe x)α(x). If F is closed
fuzzy set valued mapping, thenF is a closed multifunction.e
Proof. Let{xλ}be a net inKconverging tox,{yλ}be a net inE converging toyandyλ ∈F(x) = (Fx)e α(x). Then
Fxλ(yλ)≥α(xλ). SinceFx(y)is u.s.c. on X×Y as a real ordinary function,Fx(y)≥limFxλ(yλ)≥limFxλ(yλ)≥ limα(xλ)≥α(x)which showsy∈Fe(x).
2 Preliminaries
Let⟨X,X∗⟩be a dual system of Hausdorff topological vector space andKa nonempty convex subset ofX. Given single-valued mappingsf,g,p:X∗→X∗, a bifunctionη:K×K→Xand a mappingh:K×K→R. LetM,S,T:K→
F(X∗)be the fuzzy mappings induced by the multivalued mappings ˜M,S,˜ T˜ :K→2X∗ where 2X∗ is the multivalued mapping of all nonempty subset ofX∗ anda,b,c:X →[0,1]be the functions. LetG:K→K be a mapping. We consider fuzzy generalized variational like inequality (FGVLI) for findingx∈Ksuch that there existsu∈(Mx)a(x),v∈
(Sx)b(x),w∈(T x)c(x)satisfying
⟨pu−(f v−gw),η(y,Gx)⟩ ≥h(Gx,y),∀y∈K. (2.1)
Next we consider the following fuzzy generalized variational inequality (FGVI) for findingx∈Ksuch that there existsu∈(Mx)a(x),v∈(Sx)b(x),w∈(T x)c(x)satisfying
⟨pu−(f v−gw),y−Gx⟩ ≥h(Gx,y) ∀y∈K. (2.2)
Special cases:
If G is an identity mapping then (2.1) is equivalent to a problem of findingx∈Kthere existu∈M(x),v∈S(x),w∈
T(x)such that
⟨pu−(f v−gw),η(y,x)⟩ ≥h(x,y), for ally∈K,
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→2Y is a set valued mapping then
1. T is closed if and only if for any net{xα},xα→x and any net{yα},yα∈T(xα),yα→y one has y∈T(x).
2. 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{xα} ⊆X such that
Definition 2.1. [7] Let X be a nonempty subset of a topological vector space E.A multifunction F:X →2E is said
to be an KKM mapping if for each nonempty finite set{x1,x2,· · ·,xn} ⊆X , we have conv{x1,x2,· · ·,xn} ⊆ n
∪
j=1
F(xj).
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→2E be an KKM mapping with closed values. Assume that there exists a nonempty compact convex subset B of X such that D= ∩
x∈B
F(x)is
compact, then ∩
x∈X
F(x)̸=/0.
3 Main Results
Theorem 3.1. Let K be a nonempty convex subset of a Hausdorff topological vector space X . LetM,˜ S,˜T˜:K→2X∗be the multivalued mappings defined by M,S,T :K→F(X∗), the upper semicontinuous with nonempty compact values, h:K×K→Rand 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) =0for y∈K,
2. G:K→K is continuous and convex,
3. η:K×K→X is continuous in the second argument such that
η(x,Gx) =0, ∀x∈K,
4. The setψK(x) ={y∈K: for all u∈Mx˜ = (Mx)a(x),v∈Sx˜ = (Sx)b(x),w∈T x˜ = (T x)c(x)such that
⟨pu−(f v−gw),η(y,Gx)⟩<h(Gx,y)}, for all x∈K,
is convex.
There exists a nonempty compact convex subset B of K and a nonempty compact subset D of K such that
ψB(x)̸=/0 for all x∈K\D.
Then the solution set of FGVLI is nonempty and compact.
Proof. We defineΩ:K→2Kas:
Ω(y) ={x∈K: there existsu∈(Mx) = (Mx)˜ a(x),v∈Sx˜ = (Sx)b(x),w∈T x˜ = (T x)c(x)such that⟨pu−(f v−gw),η(y,Gx)⟩ ≥
h(Gx,y)}.
We first show thatΩis an KKM-mapping. Suppose on contrary, there existsy1,y2,· · ·,yn∈Kandz∈conv{y1,y2,· · ·,yn} such thatz̸∈ ∪n
i=1
Ω(yi). Hence for allu∈(My˜ i),v∈(Sy˜ i),w∈(T y˜ i)we have,
⟨pu−(f v−gw),η(yi,Gx)⟩<h(Gx,yi).
Hence by (4),η(z,Gz) =0 andh(Gz,z) =0, we thus deduce that 0<0 which is a contradiction. Next we show thatΩ(y)is a closed subset ofXfor eachy∈K. To see this letxαbe a net inΩ(y)which converges tox0∈X. Since
xα∈Ω(y),by the definition ofΩ(y)there exist nets{uα},{vα}and{wα}withuα∈(Mx˜ α),vα∈(Sx˜ α),wα∈(T x˜ α) such that
⟨puα−(f vα−gwα),η(y,Gxα)⟩ ≥h(Gxα,y).
converges tou,{vα}converges tovand{wα}converges tow. Then by upper semicontinuity of ˜M,S,˜ T˜, we have u0∈(MX˜ 0),v0∈(Sx˜ 0),w0∈(T x˜ 0)and
⟨pu0−(f v0−gw0),η(y,Gx0)⟩ ≥h(Gx0,y).
Note that ˜M,S,˜ T˜ are closed mappings since ˜M,S,˜ T˜ are u.s.c. with compact values andX is Hausdorff. The continuity off,g,pand the upper semicontinuity of⟨·,·⟩imply,
⟨pu0−(f v0−gw0),η(y,Gx0)⟩ ≥lim
α sup⟨puα−(f vα−gwα),η(y,Gxα)⟩
≥lim infh(Gxα,y)≥(Gx0,y).
The last inequality follows from (1). Consequently∩ x0∈Ω(y). HenceΩ(y)is closed inKfor ally∈K. By (4) the set
x∈B
Ω(y)is compact. ThereforeΩsatisfies all the assumptions of Lemma 2.2. Hence, there exists ¯y∈Ksuch that
¯ y∈ ∩
x∈K Ω(x),
and so ¯yis a solution of FGVLI (note that the solution set of FGVLI equals to∩x∈KΩ(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 setD. This completes the proof.
Whenη(y,Gx) =y−Gxin 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 . LetM,˜ S,˜ T˜:K→2X∗
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 mappingx→h(Gx,y)is lower semicontinuous withh(Gy,y) =0 for all y∈K;
2. η:K×K→Xis continuous in the second argument such thatη(x,Gx) =0 for all x∈K;
3. G:K→Kis continuous and convex;
4. The setψK(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 Bof K and a nonempty compact subset Dof K such that ψΩ(x)̸≠/0 for allx∈K\D.
Then the solution set of FGVI (2.2) is nonempty and compact.
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