#### Abstract

In this paper, a new notion of fuzzy local function called -fuzzy local function is introduced, and some properties are given in a fuzzy ideal topological space in Šostak sense. After that, the concepts of fuzzy upper (resp., lower) almost -continuous, weakly -continuous, and almost weakly -continuous multifunctions are introduced. Some properties and characterizations of these multifunctions along with their mutual relationships are discussed with the help of examples.

#### 1. Introduction

Zadeh [1] introduced the basic idea of a fuzzy set as an extension of classical set theory. The basic notions of fuzzy sets have been improved and applied in different directions. Along this direction, we can refer [24]. A fuzzy multifunction (multivalued mapping) is a fuzzy set valued function [58]. The difference between fuzzy multifunctions and fuzzy functions has to do with the definition of an inverse image. For a fuzzy function, there is one inverse but for a fuzzy multifunction there are two types of inverses. By these two definitions of inverse, we can define the continuity of fuzzy multifunction. Fuzzy multifunctions have many applications, for instance, decision theory and artificial intelligence. Taha [911] introduced and studied the concepts of -fuzzy -open, -generalized fuzzy -open, and -fuzzy --open sets in a fuzzy ideal topological space in Šostak sense [12]. Moreover, Taha [10, 11, 13] introduced and studied the concepts of fuzzy upper and lower --continuous (resp., --continuous, --continuous, and generalized -continuous) multifunctions via fuzzy ideals [14].

We lay out the remainder of this article as follows. Section 2 contains some basic definitions and results that help in understanding the obtained results. In Section 3, we introduce and study the notion of -fuzzy local function by using fuzzy ideal topological space and the definition of fuzzy difference between two fuzzy sets. Additionally, from the definition of -fuzzy local function, we introduce a stronger form of fuzzy upper (resp., lower) precontinuous multifunctions [15], namely, fuzzy upper (resp., lower) -continuous multifunctions. In Sections 4, 5 and 6 we introduce the concepts of fuzzy upper (resp., lower) almost -continuous, weakly -continuous, and almost weakly -continuous multifunctions. Several properties of these new multifunctions are established. Finally, Section 7 gives some conclusions and suggests some future works.

#### 2. Preliminaries

In this section, we present the basic definitions which we need in the next sections. Throughout this paper, refers to an initial universe. The family of all fuzzy sets in is denoted by and for , for all (where and . For , for all . All other notations are standard notations of fuzzy set theory. Let us define the fuzzy difference between two fuzzy sets as follows:

Recall that a fuzzy idea on [14] is a map that satisfies the following conditions: (i) and ; (ii) . Also, is called proper if and there exists such that . The simplest fuzzy ideals on , , and are defined as follows:and . If and are fuzzy ideals on , we say that is finer than ( is coarser than ), denoted by , iff . Let be a fuzzy topology on in Šostak sense, the triple is called fuzzy ideal topological space.

A mapping is called a fuzzy multifunction [15, 16] iff for each . The degree of membership of in is denoted by for any . Also, is normalized iff for each , there exists such that and is Crisp iff for each and . The upper inverse , the lower inverse of , and the image of are defined as follows: , , and . All definitions and properties of image, lower, and upper are found in [15].

#### 3. Fuzzy Ideal and -Fuzzy Local Function

Definition 1. Let be a fuzzy ideal topological space, and . Then, the -fuzzy local function of is defined as follows:

Remark 1. .(1)If we take for each , we have(2)If we take (resp., ) for each , we have .We will occasionally write or for .

Theorem 1. Let be a fuzzy ideal topological space and and be two fuzzy ideals on X. Then, for any fuzzy sets ,(1)(2)If (3)If (4)  =  (5) and (6) and (7)If (8)

Proof. (1)From Definition 1, we have .(2)Suppose that if . By the definition of , there exists with , , and such that .Hence, implies and . Hence, ; it is a contradiction. Then, .(3)Suppose that if . By the definition of , there exists with , , and such that . Hence, implies . Hence, ; it is a contradiction. Then, .(4)From Definition 2, we have  =  . Since for any fuzzy ideal , . Thus,  =  .(5)By (4), we have . In general, the converse is not true, as shown by Example 1.(6)Hence, and implies and . Thus, . Now, we show that .Suppose implies and .Case 1. Since , by the definition of , there exists with , , and such that . Hence, implies . Hence, , which is a contradiction. Thus, .Case 2. Since , by the definition of , there exists with , , and such that . Hence, implies . Hence, , which is a contradiction. Thus, . Then, .Also, and imply and . Thus, .(7)Hence, implies . Thus, .(8)By (3), , i.e., .

Example 1. Define as follows:In , and .

Lemma 1. Let be a fuzzy ideal topological space, and . Then, (resp., ).

Proof. Since for each , by Theorem 1 (2), we have for each . This implies . Other case is similarly proved.

Definition 2. Let be a fuzzy ideal topological space. Then, for each and , we define an operator as follows:The fuzzy topology generated by is denoted by , i.e., and . Now, if , then for each . So, . Again if (resp., ), then for each .

Theorem 2. Let be a fuzzy ideal topological space. Then, for any fuzzy sets and , the operator satisfies the following properties:(1)(2)(3)If , then (4)(5)(6)

Proof. (1)Hence, and implies .(2)Hence, implies . Since and from Theorem 1(4), we have implying . Thus, .(3)From and Theorem 1 (2), we have , i.e., .(4)By the definition of and from Theorem 1 (6), we have .(5)Hence, and imply and . Thus, .(6)From (2) and (3), we have . Now, we show that . By (4) and the definition of , we have .The following theorem is similarly proved, as in Theorem 2.

Theorem 3. Let be a fuzzy ideal topological space. Then, for each and , we define an operator as follows:

For each , the operator satisfies the following properties:(1)(2)(3)If , then (4)(5)(6), if (7)

Definition 3. A fuzzy multifunction is called(1)Fuzzy upper (resp., lower) -continuous at a fuzzy point iff (resp., ) for each , , there exists , , and such that (resp., ).(2)Fuzzy upper -continuous (resp., fuzzy lower -continuous) iff it is fuzzy upper -continuous (resp., fuzzy lower -continuous) at every .

Remark 2. If is normalized, then is fuzzy upper -continuous at a fuzzy point iff for each , there exists , and such that .

Theorem 4. Let be a fuzzy multifunction (resp., normalized fuzzy multifunction). Then, is fuzzy lower (resp., upper) -continuous iff (resp., ) for each with and .

Proof.. Let , with and . Then, there exists with and such that . Thus, and hence .
Let , with and . Then, and hence . Thus, is fuzzy lower -continuous. Other case is similarly proved.

Remark 3. .(1)Every fuzzy lower (resp., upper) -continuous multifunction is fuzzy lower (resp., upper) precontinuous [15](2)If we take , we have as fuzzy lower (resp., upper) -continuous iff it is fuzzy lower (resp., upper) precontinuous(3)Fuzzy upper (resp., lower) -continuity and fuzzy upper (resp., lower) semicontinuity [15] are independent notions, as shown by example 2.

Example 2. Let , , and be a fuzzy multifunction defined by , and . Define and as follows:Then, (1) is fuzzy upper (resp., lower) semicontinuous, but it is not fuzzy upper (resp., lower) -continuous because (resp., ) and , but (2) is fuzzy upper (resp., lower) -continuous, but it is not fuzzy upper (resp., lower) semicontinuous because (resp., ) and , but

Corollary 1. Let and be two fuzzy multifunctions (resp., normalized fuzzy multifunctions). Then, is fuzzy lower (resp., upper) -continuous if is fuzzy lower (resp., upper) -continuous and is fuzzy lower (resp., upper) semicontinuous.

#### 4. Fuzzy Upper and Lower Almost -Continuity

Definition 4. A fuzzy multifunction is called(1)Fuzzy upper almost -continuous at a fuzzy point iff for each and , there exists with and such that .(2)Fuzzy lower almost -continuous at a fuzzy point iff for each and , there exists with and such that .(3)Fuzzy upper almost -continuous (resp., fuzzy lower almost -continuous) iff it is fuzzy upper almost -continuous (resp., fuzzy lower almost -continuous) at every .

Remark 4. .(1)If is normalized, is fuzzy upper almost -continuous at a fuzzy point iff for each and , there exists , and such that .(2)Fuzzy upper (lower) semicontinuity fuzzy upper (lower) almost -continuity fuzzy upper (lower) almost continuity [17].(3)Fuzzy upper (lower) almost -continuity fuzzy upper (lower) almost continuity.

Theorem 5. For a fuzzy multifunction , , and , the following statements are equivalent:(1) is fuzzy lower almost -continuous(2), if (3), if

Proof. (1) (2) Let , with and . Then, there exists with and such that . Thus, , i.e., .(2) (3) Let with . Then, by (2), . Thus,(3) (1) Let , with and . Then, by (3), , i.e., . Thus, . Hence, is fuzzy lower almost -continuous.The following theorem is similarly proved as in Theorem 5.

Theorem 6. For a normalized fuzzy multifunction , and , the following statements are equivalent:(1) is fuzzy upper almost -continuous(2), if (3), if

Example 3. Define a fuzzy multifunction as in example 2, and as follows:Then,(1) is fuzzy upper (resp., lower) almost -continuous, but it is not fuzzy upper (resp., lower) semicontinuous because and , but ,(2) is fuzzy upper (resp., lower) almost continuous, but it is not fuzzy upper (resp., lower) almost -continuous because and , but .

Theorem 7. For a fuzzy multifunction , , and , the following statements are equivalent:(1) is fuzzy lower almost -continuous(2), if (3), if

Proof. (1) (2) If , then . By Theorem 5 (2), . Thus, .(2) (3) is obvious.(3) (1) Let , with and . Then, by (3) and , , and . Thus, is fuzzy lower almost -continuous.The following theorems are similarly proved as in Theorem 7.

Theorem 8. For a fuzzy multifunction , , and , the following statements are equivalent:(1) is fuzzy lower almost -continuous(2) if (3) if

Theorem 9. For a normalized fuzzy multifunction and , the following statements are equivalent:(1) is fuzzy upper almost -continuous(2) if (3) if

Theorem 10. For a normalized fuzzy multifunction , , and , the following statements are equivalent:(1) is fuzzy upper almost -continuous(2) if (3) if

Theorem 11. Let be a fuzzy multifunction. Then, is fuzzy lower almost -continuous iff for any with and .

Proof. Let be a fuzzy lower almost -continuous. Then, for any with (say), where . By Theorem 8, and thus .
Let with . Then, and . Therefore, we obtain . Thus, by Theorem 8, is fuzzy lower almost -continuous.
11. The following theorem is similarly proved as in Theorem.

Theorem 12. Let be a normalized fuzzy multifunction. Then, is fuzzy upper almost -continuous iff for any with and .

Theorem 13. Let be a family of fuzzy lower almost -continuous. Then, is fuzzy lower almost -continuous.

Proof. Let . Then, . Since is a family of fuzzy lower almost -continuous, for any and . Then, . Hence, is fuzzy lower almost -continuous.

Theorem 14. Let and be two normalized fuzzy upper almost -continuous. Then, is fuzzy upper almost -continuous.

Proof. Let . Then, . Since and are two normalized fuzzy upper almost -continuous, for any and . Then, . Hence, is fuzzy upper almost -continuous.

Corollary 2. Let