ON FUZZY φψ-CONTINUOUS MULTIFUNCTION

In the last three decades, the theory of multifunctions has advanced in a variety of ways and applications of this theory can be found, specially in functional analysis and fixed point theory. Recently many authors, for example, Albrycht and Matłoka [1] and Beg [3] have studied fuzzy multifunctions and have characterized some property of fuzzy multifunctions defined on a fuzzy topological space. Several authors have studied some type of fuzzy continuity for fuzzy functions and fuzzy multifunctions [1–5], [8–12]. In [3] fuzzy φψ-continuous functions have been studied. But this brand of fuzzy continuity has not considered for fuzzy multifunctions which we attempt to study and characterize. The fuzzy set in (on) a universe X is a function with domain X and values in I = [0,1]. The class of all fuzzy sets on X will be denoted by IX and symbols A,B, . . . are used for fuzzy sets on X . 01X is called empty fuzzy set, where 1X is the characteristic function on X . For any fuzzy set A in X , the function value A(x) is called the grade of membership of x in A. We write x ∈ A if A(x) > 0. For any fuzzy set A, the fuzzy set 1−A(x) is called the complement of A which is denoted by Ac. Let A and B be fuzzy sets in X , we write A ≤ B if A(x) ≤ B(x) for all x in X . For any family {Aα}α∈ of fuzzy sets in X , ∨ α∈ Aα and ∧ α∈ Aα are defined by supαAα(x) and infαAα(x), respectively. A family τ of fuzzy sets in X is called a fuzzy topology for X if (i) α1X ∈ τ for each α∈ I ; (ii) A∧B ∈ τ where A,B ∈ τ and (iii) ∨α∈ Aα ∈ τ whenever Aα ∈ τ for all α in . The pair (X ,τ) is called a fuzzy topological space [6]. Every member of τ is called fuzzy open set and its complements


Introduction and preliminaries
In the last three decades, the theory of multifunctions has advanced in a variety of ways and applications of this theory can be found, specially in functional analysis and fixed point theory.Recently many authors, for example, Albrycht and Matłoka [1] and Beg [3] have studied fuzzy multifunctions and have characterized some property of fuzzy multifunctions defined on a fuzzy topological space.Several authors have studied some type of fuzzy continuity for fuzzy functions and fuzzy multifunctions [1][2][3][4][5], [8][9][10][11][12].In [3] fuzzy ϕψ-continuous functions have been studied.But this brand of fuzzy continuity has not considered for fuzzy multifunctions which we attempt to study and characterize.
The fuzzy set in (on) a universe X is a function with domain X and values in I = [0,1].The class of all fuzzy sets on X will be denoted by I X and symbols A,B,... are used for fuzzy sets on X. 01 X is called empty fuzzy set, where 1 X is the characteristic function on X.For any fuzzy set A in X, the function value A(x) is called the grade of membership of x in A. We write x ∈ A if A(x) > 0. For any fuzzy set A, the fuzzy set 1 − A(x) is called the complement of A which is denoted by A c .Let A and B be fuzzy sets in X, we write For any family {A α } α∈Ꮽ of fuzzy sets in X, α∈Ꮽ A α and α∈Ꮽ A α are defined by sup α A α (x) and inf α A α (x), respectively.A family τ of fuzzy sets in X is called a fuzzy topology for X if (i) α1 X ∈ τ for each α ∈ I; (ii) A ∧ B ∈ τ where A,B ∈ τ and (iii) α∈Ꮽ A α ∈ τ whenever A α ∈ τ for all α in Ꮽ.The pair (X,τ) is called a fuzzy topological space [6].Every member of τ is called fuzzy open set and its complements are called fuzzy closed sets [6].In a fuzzy topological space X the interior and the closure of a fuzzy set A (simply int(A) and cl(B), resp.) are defined by for all y in Y , where A is an arbitrary fuzzy set in X [12].A fuzzy function f : For more details about fuzzy multifunctions and their properties, the reader is referred to [1,2,10].Throughout this paper, (X,τ) and (Y ,ν) are fuzzy topological spaces.The symbol f : X →→ Y is used for a fuzzy multifunction from X to Y , while f : X → Y for a fuzzy function from X to Y .

Main results
Definition 1.1.
, where ϕ and ψ are fuzzy operation on X and Y , respectively.f : X → Y is said to be a fuzzy ϕψ-continuous function if it is a fuzzy ϕψ-continuous function at each x ∈ X.
, where ϕ and ψ are fuzzy operation on X and Y , respectively.f : X →→ Y is said to be a fuzzy ϕψ-continuous multifunction if it is a fuzzy ϕψ-continuous multifunction at each x ∈ X.
(ii) f : X →→ Y is called a single valued fuzzy multifunction if f at each x is a fuzzy point 1 {yx} , where y x ∈ Y .In this case it would induce a fuzzy function f : Proposition 1.3.Suppose f : X →→ Y be a single valued fuzzy multifunction.Then for any fuzzy set A in X; f (A) = f (A).Therefore, f is a ϕψ-continuous multifunction if and only if f is a fuzzy ϕψ-continuous function.
Proof.The equivalence for any fuzzy set A of X can be derived from the following fact: (1.4) On the other hand, . Now replacing identity function as a fuzzy operation on X instead of ϕ completes the proof.
From the above result this brand of continuity for fuzzy multifunctions is in fact a generalization of ϕψ-continuity introduced in [3].Next, we would like to present a result showing the relation between fuzzy ϕψ-continuity and fuzzy ϕψ-continuity in respect of nets.First we note to the following results.
Proposition 1.4.Suppose f : X →→ Y be a fuzzy multifunction and A,B ∈ I X such that (1.6) Lemma 1.5.Let f : X →→ Y be a fuzzy multifunction and let x be a fuzzy point in X.Then Proof.It is straightforward.
We say that a net (x α α ) α∈Ꮽ of fuzzy points in a fuzzy topological space X is ϕ-convergent to a fuzzy point x (we will denote it by x α α ϕ − → x ) if for any neighborhood set A of x , there is an α 0 ∈ Ꮽ in which x α α ∈ ϕ(A) for all α ≥ α 0 .Lemma 1.

Consider a fuzzy set A and a convergent net (A α ) of fuzzy sets which
Proof.From the assumption given any fuzzy open neighborhood B of A, there is α 0 ∈ Ꮽ such that for all α ≥ α 0 , (1.7) A fuzzy multifunction f : X →→ Y is called net-fuzzy ϕψ-continuous if for each net of fuzzy points x α α and x in X, f Theorem 1.7.Let X be a fuzzy topological space.For any fuzzy multifunction f : X →→ Y the following are equivalent: (i) f is a fuzzy ϕψ-continuous; (ii) f is a net-fuzzy ϕψ-continuous.

Proof. (i)⇒(ii).
For any fuzzy open neighborhood B of f (x ), there is a fuzzy open neighborhood A of x such that f ϕ(A) ≤ ψ(B). (1.8) From the assumption, there is α 0 ∈ A for which On the contrary, there is a fuzzy point x in X, a fuzzy open neighborhood B of f (x ) such that, there is not a fuzzy neighborhood A of x satisfying in f (ϕ(A)) ≤ ψ(B).This means that there is z A Y with the following property: (1.10) Therefore, (1.12) Consider {A α : α ∈ Ꮽ} as a system of fuzzy neighborhoods at x .The following order makes Ꮽ as a directed set and so it makes {A α : α ∈ Ꮽ} as a net: Applying (1.12) for A α instead of A, there is , which completes the proof.
In the following result we show continuity of the composition of two fuzzy multifunction.Suppose f : X →→ Y and g : Y →→ Z.We define composition go f : X →→ Z by (go f )(x) = g( f (x)) = t∈ f (x) g(t).
Definition 1.9.Let X 0 be a subset of X, let i : X 0 → X be the inclusion map, and let f : X →→ Y be a fuzzy multifunction.Say that f oi is the restriction of f to X 0 .
Lemma 1.10.Assuming ϕ is a fuzzy operation on X and X 0 ⊆ X.Then ϕ(A) = ϕ( A) defines a fuzzy operation on X 0 , where A is the extension of A by zero to X.
Proof.It is easy to see that ϕ is a well-defined map and ϕ(01 X ) = 01 X .ϕ is a fuzzy operation, so int( A) ≤ ϕ( A).But, The following result shows the fuzzy continuity of the restriction of fuzzy multifunction.
Theorem 1.11.Suppose f : X →→ Y be a fuzzy ϕψ-continuous multifunction and X 0 ⊆ X.Then f oi is a ϕψ-fuzzy continuous multifunction, where ϕ is a monotonous fuzzy operation.
Proof.For any fuzzy point x in X 0 , j(x ) is a fuzzy point in X.It shows that for any fuzzy open neighborhood B of f (i(x )), there is a fuzzy open neighborhood A of i(x ) for which f (ϕ(A)) ≤ ψ(B).But Aoi is a fuzzy open neighborhood of x in X 0 , only we must show that f oi( ϕ(Aoi)) ≤ ψ(B).To see this, (1.19) Proposition 1.12.Suppose (X,τ) and (Y ,η) be fuzzy topological spaces, ϕ and ψ are fuzzy operations on X and Y , respectively, where ϕ is a monotonous fuzzy operation.Let f : X →→ Y be any fuzzy multifunction and let Ꮾ be a base for η.Then f is fuzzy ϕψ-continuous multifunction if and only if f is fuzzy ϕψ-continuous multifunction with respect to Ꮾ.
For (⇐) consider any fuzzy point x in X and any fuzzy open neighborhood B of f (x ).C ∈ Ꮾ exists such that f (x ) ≤ C and C ≤ B. From the assumption there is a fuzzy open neighborhood A of x such that f (ϕ(A)) ≤ ψ(C).But ψ is monotonous so f (ϕ(A)) ≤ ψ(B).
[11] neighborhood of a fuzzy set A in a fuzzy topological space X is any fuzzy set B for which there is a fuzzy open setV satisfying A ≤ V ≤ B. Any fuzzy open set V that satisfies A ≤ V is called a fuzzy open neighborhood of A[10].A fuzzy set A is called a fuzzy point if it takes the value 0 for all y ∈ X except one, say x ∈ X.If its value at x is (0 ≤ ≤ 1), we denote this fuzzy point by x[11].For any fuzzy point x and any fuzzy set A we write x ∈ A if and only if ≤ A(x).Let f be a function from X to Y .A fuzzy function f : X → Y is defined by