A New Notion of Fuzzy Local Function and Some Applications

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


Introduction
Zadeh [1] introduced the basic idea of a fuzzy set as an extension of classical set theory. e basic notions of fuzzy sets have been improved and applied in di erent directions. Along this direction, we can refer [2][3][4]. A fuzzy multifunction (multivalued mapping) is a fuzzy set valued function [5][6][7][8]. e di erence between fuzzy multifunctions and fuzzy functions has to do with the de nition 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 de nitions of inverse, we can de ne the continuity of fuzzy multifunction. Fuzzy multifunctions have many applications, for instance, decision theory and arti cial intelligence. Taha [9][10][11] introduced and studied the concepts of r-fuzzy ℓ-open, r-generalized fuzzy ℓ-open, and r-fuzzy δ-ℓ-open sets in a fuzzy ideal topological space (X, τ, ℓ) iň Sostak sense [12]. Moreover, Taha [10,11,13] introduced and studied the concepts of fuzzy upper and lower α-ℓ-continuous (resp., β-ℓ-continuous, δ-ℓ-continuous, and generalized ℓ-continuous) multifunctions via fuzzy ideals [14].
We lay out the remainder of this article as follows. Section 2 contains some basic de nitions and results that help in understanding the obtained results. In Section 3, we introduce and study the notion of r-fuzzy local function by using fuzzy ideal topological space and the de nition of fuzzy di erence between two fuzzy sets. Additionally, from the de nition of r-fuzzy local function, we introduce a stronger form of fuzzy upper (resp., lower) precontinuous multifunctions [15], namely, fuzzy upper (resp., lower) ℓ-continuous multifunctions. In Sections 4, 5 and 6 we introduce the concepts of fuzzy upper (resp., lower) almost ℓ-continuous, weakly ℓ-continuous, and almost weakly ℓ-continuous multifunctions. Several properties of these new multifunctions are established. Finally, Section 7 gives some conclusions and suggests some future works.

Preliminaries
In this section, we present the basic de nitions which we need in the next sections. roughout this paper, X refers to an initial universe. e family of all fuzzy sets in X is denoted by I X and for λ ∈ I X , λ c (x) 1 − λ(x) for all x ∈ X (where I [0, 1] and (I° (0, 1]). For t ∈ I, t(x) t for all x ∈ X. All other notations are standard notations of fuzzy set theory. Let us de ne the fuzzy di erence between two fuzzy sets λ, μ ∈ I X as follows: Recall that a fuzzy idea ℓ on X [14] is a map ℓ: I X ⟶ I that satisfies the following conditions: (i) ∀λ, μ ∈ I X and λ ≤ μ⇒ℓ(μ) ≤ ℓ(λ); (ii) ∀λ, μ ∈ I X ⇒ℓ(λ∨μ) ≥ ℓ(λ)∧ℓ (μ). Also, ℓ is called proper if ℓ(1 ) � 0 and there exists μ ∈ I X such that ℓ(μ)〉0. e simplest fuzzy ideals on X, ℓ 0 , and ℓ 1 are defined as follows: and ℓ 1 (λ) � 1 ∀ λ ∈ I X . If ℓ 1 and ℓ 2 are fuzzy ideals on X, we say that ℓ 1 is finer than ℓ 2 (ℓ 2 is coarser than ℓ 1 ), denoted by Let τ be a fuzzy topology on X inŠostak sense, the triple (X, τ, ℓ) is called fuzzy ideal topological space. A mapping F: X⊸Y is called a fuzzy multifunction [15,16] . All definitions and properties of image, lower, and upper are found in [15].

Fuzzy Ideal and r-Fuzzy Local Function
Definition 1. Let (X, τ, ℓ) be a fuzzy ideal topological space, λ ∈ I X and r ∈ I°. en, the r-fuzzy local function λ * r of λ is defined as follows:

Lemma 1.
Let (X, τ, ℓ) be a fuzzy ideal topological space, and Other case is similarly proved.
en, for each λ ∈ I X and r ∈ I°, we define an operator int * : I X × I°⟶ I X as follows: For each λ, ] ∈ I X , the operator int * satisfies the following properties: (2) Fuzzy upper ℓ-continuous (resp., fuzzy lower ℓ-continuous) iff it is fuzzy upper ℓ-continuous (resp., fuzzy lower ℓ-continuous) at every x t ∈ dom(F).

Remark 2.
If F is normalized, then F is fuzzy upper ℓ-continuous at a fuzzy point
Proof. (⇒) Let F be a fuzzy lower almost ℓ-continuous.
us, F is fuzzy lower weakly ℓ-continuous. e following theorem is similarly proved as in eorem 17.
en, F is fuzzy upper almost ℓ-continuous.
24. e following theorem is similarly proved as in eorem. □ Theorem 25. Let F: (X, τ)⊸(Y, η, ℓ) be a normalized fuzzy multifunction and F be a fuzzy lower almost weakly ℓ-continuous and fuzzy upper almost ℓ-continuous. en, F is fuzzy lower weakly ℓ-continuous.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.