The fuzzy concept has penetrated nearly all branches of mathematics since the concept has been defined by Zadeh [1]. Fuzzy sets have applications in several fields, such as information [2] and control [3]. The theory of fuzzy topological spaces was first defined and developed by Chang [2], and since then various notions of general topology have been generalized to the fuzzy topological spaces defined by Chang. Šostak [4] and Kubiak [5] developed fuzzy topology as an extension of the fuzzy topology introduced by Chang [2]. It has since been developed in several ways. Šostak [6] published a survey article on the areas of development of fuzzy topological spaces. Topologists often refer to the concept of fuzzy topology developed by Chang as “”and the concept of fuzzy topology proposed by Kubiak-Šostak as “”, where L is an appropriate lattice. In a previous study [3], The author introduced the idea of intuitionistic fuzzy sets; subsequently, Coker [7,8] introduced the concept of intuitionistic fuzzy topological spaces. Furthermore, as a generalization of fuzzy topological spaces, Montal and Samanta [9] introduced the concept of the intuitionistic gradation of openness. In 2005, Garcia and Rodabaugh [10] proposed the termination of the term intuitionistic. They proved that the term intuitionistic is unsuitable in mathematics and related applications and replaced it with double. Several topologists have studied various concepts in a double fuzzy topological space [11–22]. In this study, we introduce and examine the concepts of double fuzzy upper and fuzzy lower α-irresolute multifunctions. The characterizations and properties of these multifunctions are established in double fuzzy topological spaces.

Throughout this paper, let X be a non-empty set and I be the unit interval [0, 1]; I₀ = (0, 1] and I₁ = [0, 1). The family of all fuzzy sets in X is denoted by I^X. We denote the smallest and greatest fuzzy sets in X by 0̄ and 1̄, respectively. For a fuzzy set λ ∈ I^X, 1̄ – λ denotes the complement. Considering a function f : I^X –→ I^Y and its inverse f⁻¹ : I^Y –→ I^X, they can be defined by f(λ)(y) = ∨_f(x)=yλ(x) and f⁻¹(μ)(x) = μ(f(x)) for each λ ∈ I^X, μ ∈ I^Y and x ∈ X. All other notations are standard notations of the fuzzy set theory.

Definition 1 [23]

Let F : (X, τ) → (Y, σ), then F is called a fuzzy multifunction if and only if F(x) ∈ I^Y for each x ∈ X. The degree of membership ofy in F(x) is denoted by F(x)(y) = G_F (x, y) for any (x, y) ∈ X × Y. The domain of F, denoted by dom(F), and the range of F, denoted by rng(F), for any x ∈ X and y ∈ Y are defined by dom(F)(x)=⋁y∈YGF(x,y) and rng(F)(x)=⋁x∈XGF(x,y), respectively.

Definition 2 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then F is termed:

• (1) normalized if and only if for each x ∈ X, there exists a y₀ ∈ Y such that G_F (x, y₀) = 1̄.

• (2) crisp if and only if G_F (x, y₀) = 1̄ for each x ∈ X and y ∈ Y.

Definition 3 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then:

• (1) the lower inverse of μ ∈ I^Y is a fuzzy set F^l(μ) ∈ I^X defined by Fl(μ)(x)=⋁y∈Y(GF(x,y)⋀μ(y)).

• (2) the upper inverse of μ ∈ I^Y is a fuzzy set F^u(μ) ∈ I^X defined by Fu(μ)(x)=⋀y∈Y(GFc(x,y)∨μ(y)).

Definition 4 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then

• (1) F(λ₁) ≤ F(λ₂) if λ₁ ≤ λ₂.

• (2) F^l(μ₁) ≤ F^l(μ₂) and F^u(μ₁) ≤ F^u(μ₂) if μ₁ ≤ μ₂.

• (3) F^l(μ^c) ≤ (F^u(μ))^c.

• (4) F^u(μ^c) ≤ (F^l(μ))^c.

• (5) F(F^u(μ)) ≤ μ if F is a crisp.

• (6) F^u(F(λ)) ≥ λ if F is a crisp.

Definition 5 [23]

Let F : (X, τ) → (Y, σ) and H : (Y, σ) → (Z, η) be two fuzzy multifunctions. Then, the composition H ∘ F is defined as ((H∘F)(x))(z)=⋁y∈Y(GF(x,y)⋀GH(y,z)).

Theorem 1 [23]

Let F : (X, τ) → (Y, σ) and H : (Y, σ) → (Z, η) be two fuzzy multifunctions. Thus, the following can be obtained:

• (1) H ∘ F = F(H).

• (2) (H ∘ F)^u = F^u(H^u).

• (3) (H ∘ F)^l = F^l(H^l).

Theorem 2 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then, the following can be obtained:

• (1) (∪i∈ΓFi)(λ)=⋁i∈ΓFi(λ).

• (2) (∪i∈ΓFi)l(μ)=⋁i∈ΓFil(μ).

• (3) (∪i∈ΓFi)u(μ)=⋁i∈ΓFiu(μ).

Definition 6

A fuzzy point x_t in X is a fuzzy set that takes the value t ∈ I₀ at x and zero elsewhere such that x_t ∈ λ if and only if t ≤ λ(x).

Definition 7 [8,9]

A double fuzzy topology on X is a pair of maps τ, τ^★ : I^X → I, which satisfies the following properties:

• (1) τ (λ) ≤ 1̄ – τ^★ (λ) for each λ ∈ I^X.

• (2) τ (λ₁ ∧ λ₂) ≥ τ (λ₁) ∧ τ (λ₂) and τ^★ (λ₁ ∧ λ₂) ≤ τ^★ (λ₁) ∨ τ^★ (λ₂) for each λ₁, λ₂ ∈ I^X.

• (3) τ(⋁i∈Γλi)≥⋀i∈Γτ(λi) and τ★(⋁i∈Γλi)≤∨i∈Γτ★(λi) for each λ_i ∈ I^Xandi ∈ Γ.

The triplet (X, τ, τ^★) is called a double fuzzy topological space.

Definition 8 [8,9]

A fuzzy set λ is termed (r, s)-fuzzy open if τ (λ) ≥ r and τ^★ (λ) ≤ s; λ is termed (r, s)-fuzzy closed if and only if 1̄ – λ is an (r, s)-fuzzy open set.

Theorem 3 [24,25]

Let (X, τ, τ^★) be a double fuzzy topological space. Then, the double fuzzy closure and double fuzzy interior operators of λ ∈ I^X are defined by: C_τ,τ^★ (λ, r, s) = ∧{μ ∈ I^X | λ ≤ μ, τ (1̄ – μ) ≥ randτ^★(1̄ – μ) ≤ s}, I_τ,τ^★ (λ, r, s) = ∨{μ ∈ I^X | μ ≤ λ, τ (μ) ≥ r, τ^★(μ) ≤ s},, respectively, where r ∈ I₀ and s ∈ I₁ such that r + s ≤ 1.

Definition 9

Let (X, τ, τ^★) be a double fuzzy topological space. For each λ ∈ I^X, r ∈ I₀, and s ∈ I₁,

• (1) λ is called (r, s)-fuzzy α-open if λ ≤ I_τ,τ^★ (C_τ,τ^★ (I_τ,τ^★ (λ, r, s), r, s), r, s).

• (2) λ is called (r, s)-fuzzy α-closed if 1̄ – λ is (r, s)-fuzzy α-open.

• (3) αC_τ,τ^★ (λ, r, s) = ∧{μ ∈ I^X : μ ≥ λ and μ is (r, s)-fuzzy α-closed} is called (r, s)-fuzzy α-closure of λ.

• (4) αI_τ,τ^★ (λ, r, s) = ∨{μ ∈ I^X : μ ≤ λ and μ is (r, s)-fuzzy α-open} is called (r, s)-fuzzy α-interior of λ.

Definition 10 [?]

Let F : (X, τ, τ^★) → (Y, σ, σ^★) be a fuzzy multifunction. Then, F is called:

• (1) double fuzzy upper α-continuous at a fuzzy point x_t ∈ dom(F) if x_t ∈ F^u(μ) for each μ ∈ I^Y, σ(μ) ≥ r, and σ^★(μ) ≤ s, there exists an (r, s)-fuzzy α-open set λ ∈ I^X and x_t ∈ λ such that λ ∧ dom(F) ≤ F^u(μ).

• (2) double fuzzy lower α-continuous at a fuzzy point x_t ∈ dom(F) if x_t ∈ F^u(μ) for each μ ∈ I^Y, σ(μ) ≥ r and σ^★(μ) ≤ s, there exists an (r, s)-fuzzy α-open set λ ∈ I^X and x_t ∈ λ such that λ ≤ F^l(μ).

• (3) double fuzzy upper (lower) α-continuous if it is fuzzy upper (lower) α-continuous at every x_t ∈ dom(F).

Definition 11

Let F : (X, τ, τ^★) → (Y, σ, σ^★) be a fuzzy multifunction. Then, F is called:

• (1) double fuzzy upper α-irresolute at a fuzzy point x_t ∈ dom(F) if x_t ∈ F^u(μ) for each (r, s)-fuzzy α-open set μ ∈ I^Y, there exists an (r, s)-fuzzy α-open set λ ∈ I^X and x_t ∈ λ such that λ ∧ dom(F) ≤ F^u(μ).

• (2) double fuzzy lower α-irresolute at a fuzzy point x_t ∈ dom(F) if x_t ∈ F^u(μ) for each (r, s)-fuzzy α-open set μ ∈ I^Y, there exists an (r, s)-fuzzy α-open set λ ∈ I^X and x_t ∈ λ such that λ ≤ F^l(μ).

• (3) double fuzzy upper (lower) α-irresolute if it is fuzzy upper (lower) α-irresolute at every x_t ∈ dom(F).

Proposition 1

If F is normalized, then F is a double fuzzy upper α-irresolute at a fuzzy point x_t ∈ dom(F) if and only if x_t ∈ F^u(μ) for each (r, s)-fuzzy α-open set μ ∈ I^Y, there exists an (r, s)-fuzzy α-open set λ ∈ I^X and x_t ∈ λ such that λ ≤ F^u(μ).

Theorem 4

For a fuzzy multifunction F : (X, τ, τ^★) → (Y, σ, σ^★), the following statements are equivalent:

• (1) F is double fuzzy lower α-irresolute,

• (2) F^l(μ) is (r, s)-fuzzy α-open for any (r, s)-fuzzy α-open set μ ∈ I^Y,

• (3) F^u(μ) is (r, s)-fuzzy α-closed for any (r, s)-fuzzy α-closed set μ ∈ I^Y,

• (4) C_τ,τ^★ (I_τ,τ^★ (C_τ,τ^★ (F^u(μ), r, s), r, s), r, s) ≤ F^u(αC_σ,σ^★ (μ, r, s)) for any μ ∈ I^Y,

• (5) αC_τ,τ^★ (F^u(μ), r, s) ≤ F^u(αC_σ,σ^★ (μ, r, s)) for any μ ∈ I^Y.

Theorem 5

For a fuzzy multifunction F : (X, τ, τ^★) → (Y, σ, σ^★), the following statements are equivalent:

• (1) F is double fuzzy upper α-irresolute.

• (2) F^u(μ) is r-fuzzy α-open for any (r, s)-fuzzy α-open set μ ∈ I^Y.

• (3) F^l(μ) is r-fuzzy α-closed for any (r, s)-fuzzy α-closed set μ ∈ I^Y.

• (4) C_τ,τ^★ (I_τ,τ^★ (C_τ,τ^★ (F^l(μ), r, s), r, s), r, s) ≤ F^l(αC_σ,σ^★ (μ, r, s)) for any μ ∈ I^Y.

• (5) αC_τ,τ^★ (F^l(μ), r, s) ≤ F^l(αC_σ,σ^★ (μ, r, s)) for any μ ∈ I^Y.

Corollary 1

Let F : (X, τ, τ^★) → (Y, σ, σ^★) be a fuzzy multifunction. Then, the following can be postulated:

• (1) F as normalized implies that it is double fuzzy upper α-irresolute at a fuzzy point x_t if and only if x_t ∈ I_τ,τ^★ (C_τ,τ^★ (I_τ,τ^★ (F^u(μ), r, s), r, s), r, s) for (r, s)-fuzzy α-open sets μ ∈ I^Y and x_t ∈ F^u(μ).

• (2) F is double fuzzy lower α-irresolute at a fuzzy point x_t if and only if x_t ∈ I_τ,τ^★ (C_τ,τ^★ (I_τ,τ^★ (F^l(μ), r, s), r, s), r, s) for (r, s)-fuzzy α-open sets μ ∈ I^Y and x_t ∈ F^l(μ).

Proposition 2

For a fuzzy multifunction F : (X, τ, τ^★) → (Y, σ, σ^★), we obtain the following:

• (1) Every double fuzzy upper α-continuous multifunction is double fuzzy upper α-irresolute.

• (2) Every double fuzzy lower α-continuous multifunction is double fuzzy lower α-irresolute.

The following example demonstrates that the converses of the above propositions are not true.

Example 1

Let X = {x₁, x₂}, Y = {y₁, y₂, y₃}, and F : (X, τ, τ^★) → (Y, σ, σ^★) be fuzzy multifunctions defined by G_F (x₁, y₁) = 0.2, G_F (x₁, y₂) = 1̄, G_F (x₁, y₃) = 0.3, G_F (x₂, y₁) = 0.5, G_F (x₂, y₂) = 0.3, and G_F (x₂, y₃) = 1̄. We define fuzzy topologies τ, τ^★ : I^X → I, and σ, σ^★ : I^Y → I as follows:

Then, F is (13,23)-fuzzy upper (lower) α-irresolute, but not (13,23)-fuzzy upper (lower) α-continuous.

Theorem 6

Let F: (X, τ, τ^★)→(Y, σ, σ^★) and H: (Y, σ, σ^★) → (Z, η, η^★) be fuzzy multifunctions. Thus, the following can be concluded:

• (1) If F and H are double fuzzy lower α-irresolute, then, H ∘ F is also double fuzzy lower α-irresolute.

• (2) If F is double fuzzy lower α-irresolute and H is double fuzzy lower α-continuous, then, H ∘ F is also double fuzzy lower α-irresolute.

Theorem 7

Let {F_i : i ∈ Γ} be a family of double fuzzy lower α-irresolute functions. Then, (∪i∈ΓFi)l(μ)=⋁i∈ΓFil(μ) is double fuzzy lower α-irresolute.

Definition 12

A fuzzy multifunction F : (X, τ, τ^★) → (Y, σ, σ^★) is called (r, s)-fuzzy α-compact-valued if and only if F(x_t) is (r, s)-fuzzy α-compact for each x_t ∈ dom (F).

Theorem 8

Let F : (X, τ, τ^★) → (Y, σ, σ^★) be a crisp double fuzzy upper α-irresolute and α-compact-valued multifunction. If λ is (r, s)-fuzzy α-compact, then, F(λ) is (r, s)-fuzzy α-compact.