New kinds of fuzzy pairwise multifunctions

Abstract

This paper is devoted to introduce and study new kinds of FP-multifunctions, namely FP-lower (upper) α-continuous, FP-lower (upper) almost α-continuous, and FP-lower (upper) weakly α-continuous multifunctions. Various properties of these multifunctions were investigated. We provided the relationships between these multifunctions and presented contrary examples. Finally, we gave the conditions which make these multifunctions equivalent.

1 Introduction and preliminaries

Kubiak [21] and Šostak [37], introduced the fundamental concept of L-fuzzy topological structure as an extension of both crisp topology and Chang’s fuzzy topology [9], and it has been evolved in several directions ([2–4 , 33]). Kim [18] introduced the concept of L-fuzzy bitopological space as a generalization ofL-fuzzy topological space and Kandil’s fuzzy bitopological space [16]. The working in L-fuzzy bitopological space has attracted the attention of a number of researchers ([15 , 39]).

Papageorgiou [27] introduced and studied the notion of fuzzy multifunction and extended the concepts of fuzzy continuous functions to the fuzzy multivalued case by introducing of fuzzy upper and fuzzy lower semi-continuous multifunctions. Mukherjee and Malakar [26] redefined the concepts of lower inverse and lower semi-continuity of a fuzzy multifunction in terms of the notion of quasi-coincidence due to Pu and Liu [30]. A number of studies examined the different kinds of fuzzy multifunctions from distinct angles ([1 , 36]). Ramadan and Abd El-latif [35] and Abd El-latif [5], introduced the concepts of FP-lower (upper) almost continuous (resp. almost weakly continuous, weakly continuous, semi-continuous, δ-continuous, irresolute and semi-irresolute) multifunctions, where the domain of these functions is a classical bitopological space in view of Kelly [17] with their values as arbitrary fuzzy sets in L-fuzzy bitopological space in view of Šostak’s sense. In this paper, we continue studying fuzzy pairwise multifunctions through introducing and studying new kinds of FP-multifunctions which are useful to study continuous functions, namely FP-lower (upper) α-continuous, FP-lower (upper) almost α-continuous, and FP-lower (upper) weakly α-continuous multifunctions. Various properties of these multifunctions were investigated. We give the relationships between these multifunctions and presented contrary examples. Finally, we gave the conditions which make these multifunctions equivalent.

Throughout this paper, let L = (L, ≤ , ∨ , ∧ , ′) be a completely distributive lattice with an order-reversing involution ^′ with the smallest element 0_L and the largest element 1_L, L₀ = L - {0_L}. The family of all fuzzy sets on Y will be denoted by L^Y ([11, 40]). The smallest element and the largest element of L^Y will be denoted by 0_Y and 1_Y respectively. A fuzzy point x_t for t ∈ L₀ is an element of L^Y such that, for y ∈ Y: $x_{t} (y) = {\begin{matrix} t, & if x = y \\ 0_{L}, & if x \neq y . \end{matrix}$ The set of all fuzzy points in Y is denoted by Pt (Y). For α ∈ L, $\underline{α} (y) = α$ for all y ∈ Y. A fuzzy set λ is said to be quasi-coincident with a fuzzy set ν, denoted by λqν if there exists y ∈ Y such that λ (y) notleν′ (y), otherwise λ q ν [24]. Also, in this paper, the indices i, j ∈ {1, 2} and i ≠ j.

Let A be a subset of an ordinary bitopological space (X, T₁, T₂). The interior (resp. closure) of A with respect to T_i, will be denoted by int_i (A) (resp. cl_i (A)). Then, we have the following

(i) A is said to be (T_i, T_j)-semi-open (briefly, (T_i, T_j)-so) set if A ⊆ cl_j (int_i (A)),

(ii) A is said to be (T_i, T_j)-α-open (briefly, (T_i, T_j)-αo) set if there exists an T_i-open set G such that G ⊆ A ⊆ int_i (cl_j (G))(equivalent A ⊆ int_i (cl_j (int_i (A))).

(iii) The union of any (T_i, T_j)-α-open sets is an (T_i, T_j)-α-open set.

(iv) Every (T_i, T_j)-α-open set is (T_i, T_j)-semi-open.

(v) The complement of (T_i, T_j)-semi-open (resp. (T_i, T_j)-α-open) set is said to be (T_i, T_j)-semi-closed (resp. (T_i, T_j)-α-closed)set.

(vi) (T_i, T_j) - sint (A) = ∪ {B : B ⊆ A, B is (T_i, T_j) - semi - open on X},

(vii) (T_i, T_j) - scl (A) = ∩ {B : B ⊇ A, B is (T_i, T_j) - semi - closed on X},

(viii) (T_i, T_j) - αint (A) = ∪ {B : B ⊆ A, B is (T_i, T_j) - α - open on X},

(ix) (T_i, T_j) - αcl (A) = ∩ {B : B ⊇ A, B is (T_i, T_j) - α - closed on X},

(x) (T_i, T_j) - sint (X - A) = X - ((T_i, T_j) - scl (A)),

(xi) (T_i, T_j) - αint (X - A) = X - ((T_i, T_j) - αcl (A)).

We denote:

(T_i, T_j) - SO (X) = {G : G is (T_i, T_j) - semi - open set on X},

(T_i, T_j) - αO (X) = {G : G is (T_i, T_j) - α - open set on X} and (T_i, T_j) - αC (X) = {F : F is (T_i, T_j) - α - closed set on X}. Moreover, (T_i, T_j) - SO (X, x) = {G : G is (T_i, T_j) - semi - open set onX, x ∈ G}, (T_i, T_j) - αO (X, x) = {G : G is (T_i, T_j) - α - open set onX, x ∈ G}, and T_i (x) = {U : U ∈ T_i, x ∈ U}.

Lemma 1.1. Let (X, T₁, T₂) be a bitopological space and A ⊆ X. Then,

(i) (T_i, T_j) - sint (cl_i (A)) = cl_j (int_i (cl_j (A))).

(ii) (T_i, T_j) - scl (int_i (A)) = int_j (cl_i (int_j (A))).

Definition 1.2. [37] A mapping $T : L^{Y} \to L$ is called an L-fuzzy topology on Y if it satisfies the following conditions:

(LO1) $T (0_{Y}) = T (1_{Y}) = 1_{L}$ ,

(LO2) $T (μ_{1} \land μ_{2}) \geq T (μ_{1}) \land T (μ_{2})$ , for each μ₁, μ₂ ∈ L^Y,

(LO3) $T (⋁_{i \in Γ} μ_{i}) \geq ⋀_{i \in Γ} T (μ_{i})$ , for each {μ_i} _i∈Γ ⊆ L^Y.

The pair $(Y, T)$ is called an L-fuzzy topological space.

The triple $(Y, T_{1}, T_{2})$ is called L-fuzzy bitopological space, where $T_{1}$ and $T_{2}$ are L-fuzzy topologies on Y [18].

Definition 1.3. [10] Let $(Y, T)$ be an L-fuzzy topological space. Then for each λ ∈ L^Y and r ∈ L₀, we define an operator $C_{T} : L^{Y} \times L_{0} ⟶ L^{Y}$ as follows: $C_{T} (λ, r) = ⋀ {μ : λ \leq μ, T (μ^{'}) \geq r} .$

For each λ, μ ∈ L^Y and r, s ∈ L₀, the operator $C_{T}$ satisfies the following conditions:

(C1) $C_{T} (0_{Y}, r) = 0_{Y}$ .

(C2) $λ \leq C_{T} (λ, r)$ .

(C3) $C_{T} (λ, r) \lor C_{T} (μ, r) = C_{T} (λ \lor μ, r)$ .

(C4) $C_{T} (λ, r) \leq C_{T} (λ, s)$ if r ≤ s.

(C5) $C_{T} (C_{T} (λ, r), r) = C_{T} (λ, r)$ .

Theorem 1.4. [22] Let $(Y, T)$ be an L-fuzzy topological space. Then for each λ ∈ L^Y and r ∈ L₀, we define an operator $I_{T} : L^{Y} \times L_{0} ⟶ L^{Y}$ as follows: $I_{T} (λ, r) = ⋁ {μ : μ \leq λ, T (μ) \geq r} .$ Then, $I_{T} (λ^{'}, r) = (C_{T} (λ, r))^{'} .$

For each λ, μ ∈ L^Yand r, s ∈ L₀, the operator $I_{T}$ satisfies the following conditions:

(I1) $I_{T} (1_{Y}, r) = 1_{Y}$ .

(I2) $I_{T} (λ, r) \leq λ$ .

(I3) $I_{T} (λ, r) \land I_{T} (μ, r) = I_{T} (λ \land μ, r)$ .

(I4) $I_{T} (λ, r) \geq I_{T} (λ, s)$ if r ≤ s.

(I5) $I_{T} (I_{T} (λ, r), r) = I_{T} (λ, r)$ .

Definition 1.5. [18, 19] Let $(Y, T_{1}, T_{2})$ be an L-fuzzy bitopological space. For λ ∈ L^Y and r ∈ L₀

(i) λ is called $r - (T_{i}, T_{j})$ -fuzzy semi-open (briefly, $r - (T_{i}, T_{j})$ -fso) set if there exists μ ∈ L^Y with $T_{i} (μ) \geq r$ such that $μ \leq λ \leq C_{T_{j}} (μ, r)$ . A fuzzy set λ is $r - (T_{i}, T_{j})$ -fso iff $λ \leq C_{T_{j}} (I_{T_{i}} (λ, r), r)$ . By $r - (T_{i}, T_{j}) - FSO (Y)$ , we denote the set of all $r - (T_{i}, T_{j})$ -fso sets on Y.

(ii) λ is called $r - (T_{i}, T_{j})$ -fuzzy semi-closed (briefly, $r - (T_{i}, T_{j})$ -fsc) set if λ′ is $r - (T_{i}, T_{j})$ -fso set. A fuzzy set λ is $r - (T_{i}, T_{j})$ -fsc iff $I_{T_{j}} (C_{T_{i}} (λ, r), r) \leq λ$ .

(iii) λ is called $r - (T_{i}, T_{j})$ -fuzzy preopen (briefly, $r - (T_{i}, T_{j})$ -fpo) if $λ \leq I_{T_{i}} (C_{T_{j}} (λ, r), r)$ .

(iv) λ is called $r - (T_{i}, T_{j})$ -fuzzy regular open (briefly, $r - (T_{i}, T_{j})$ -fro) set if $λ = I_{T_{i}} (C_{T_{j}} (λ, r), r)$ .

(v) λ is called $r - (T_{i}, T_{j})$ -fuzzy regular closed (briefly, $r - (T_{i}, T_{j})$ -frc) set if $λ = C_{T_{i}} (I_{T_{j}} (λ, r), r)$ .

Definition 1.6. [18] Let $(Y, T_{1}, T_{2})$ be an L-fuzzy bitopological space. For λ ∈ L^Y and r ∈ L₀ the $(T_{i}, T_{j}) - sint (λ, r)$ is defined by: $(T_{i}, T_{j}) - sint (λ, r) = ⋁ {μ : μ \leq λ, μ is r - (T_{i}, T_{j}) - fso set},$ and the $(T_{i}, T_{j}) - scl (λ, r)$ is defined by: $(T_{i}, T_{j}) - scl (λ, r) = ⋀ {μ : μ \geq λ, μ is r - (T_{i},$ $T_{j}) - fsc set} .$ Obviously, $(T_{i}, T_{j}) - sint (λ, r) = λ$ for any $r - (T_{i}, T_{j})$ -fso set λ and $(T_{i}, T_{j}) - scl (λ, r) = λ$ for any $r - (T_{i}, T_{j})$ -fsc set λ. Moreover, we have $(T_{i}, T_{j})$ $- sint (λ^{'}, r) = ((T_{i}, T_{j}) - scl (λ, r))^{'} .$

Definition 1.7. [19] Let $(Y, T_{1}, T_{2})$ be an L-fuzzy bitopological space. For λ ∈ L^Y, x_t ∈ Pt (Y) and r ∈ L₀,

(i) μ is called r-open $R_{T_{j}}^{T_{i}}$ -neighborhood of x_t if x_tqμ with $μ = I_{T_{i}} (C_{T_{j}} (μ, r), r)$ .

We denote $R_{T_{j}}^{T_{i}} (x_{t}, r) = {μ \in L^{Y} : x_{t} q μ = I_{T_{i}} (C_{T_{j}} (μ, r), r)}$ .

(ii) x_t is called $r - (T_{i}, T_{j})$ δ-cluster point of λ if for every $μ \in R_{T_{j}}^{T_{i}} (x_{t}, r)$ , we have μqλ.

(iii) A $(T_{i}, T_{j})$ δ-closure operator is a mapping $D_{T_{j}}^{T_{i}} : L^{Y} \times L_{0} ⟶ L^{Y}$ defined as: $D_{T_{j}}^{T_{i}} (λ, r) = ⋁ {x_{t} : x_{t} is r - (T_{i}, T_{j}) δ - cluster point of λ} .$

(iv) λ is called $r - (T_{i}, T_{j})$ fuzzy δ-closed iff $λ = D_{T_{j}}^{T_{i}} (λ, r)$ and its complement is called $r - (T_{i}, T_{j})$ fuzzy δ-open.

Theorem 1.8. [19] Let $(Y, T_{1}, T_{2})$ be an L-fuzzy bitopological space. For λ ∈ L^Y, x_t ∈ Pt (Y) and r ∈ L₀ we have the following properties:

(i) x_t is $r - (T_{i}, T_{j})$ -δ cluster point of λ iff $x_{t} \in D_{T_{j}}^{T_{i}} (λ, r)$ .

(ii) If $λ = C_{T_{i}} (I_{T_{j}} (λ, r), r)$ , then $D_{T_{j}}^{T_{i}} (λ, r) = λ$ .

(iii) $λ \leq C_{T_{i}} (λ, r) \leq D_{T_{j}}^{T_{i}} (λ, r)$ .

(iv) If $T_{j} (λ) \geq r$ , then $C_{T_{i}} (λ, r) = D_{T_{j}}^{T_{i}} (λ, r)$ .

Definition 1.9. [26] A mapping F : X → Y is called a fuzzy multifunction if for each x ∈ X, F (x) is a fuzzy set in Y. The upper inverse F⁺ (λ) and lower inverse F^- (λ) of λ ∈ L^Y are defined as follows: F⁺ (λ) = {x ∈ X : F (x) ≤ λ} and F^- (λ) = {x ∈ X : F (x) qλ}. For A ⊆ X, F (A) = ⋁ {F (x) : x ∈ A}.

Theorem 1.10. [26] For a fuzzy multifunction F : X→Y, $F^{-} (λ^{'}) = X - F^{+} (λ)$ for each λ ∈ L^Y.

Definition 1.11. [35] Let (X, T₁, T₂) be an ordinary bitopological space and $(Y, T_{1}, T_{2})$ be an L-fuzzy bitoplogical space. By $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ we mean that F is a fuzzy multifunction between X and Y, and we call it FP-multifunction.

Definition 1.12. [35] A FP-multifunction F : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ is called:

(i) FP-lower (resp. upper) semi-continuous at some point x₀ ∈ X if for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x₀ ∈ F^- (μ) (resp. x₀ ∈ F⁺ (μ)), there exists U ∈ T_i (x₀) such that U ⊆ F^- (μ) (resp. U ⊆ F⁺ (μ)).

(ii) FP-lower (resp. upper) semi-continuous if F is FP-lower (resp. upper) semi-continuous at each x ∈ X.

Theorem 1.13. [35] A FP-multifunction $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ is FP-lower semi-continuous (resp. FP-upper semi-continuous) iff for any μ ∈ L^Y with $T_{i} (μ) \geq r$ , F^- (μ) (resp. F⁺ (μ)) is an T_i-open set in X.

Definition 1.14. [5] A FP-multifunction F : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ is called

(i) FP-lower semi-irresolute at some point x₀ ∈ X if for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x₀ ∈ F^- (μ), there exists U ∈ (T_i, T_j) - SO (X, x₀) such that U ⊆ F^- (μ).

(ii) FP-upper semi-irresolute at some point x₀ ∈ X if for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x₀ ∈ F⁺ (μ), there exists U ∈ (T_i, T_j) - SO (X, x₀) such that U ⊆ F⁺ (μ).

(iii) FP-lower (resp. upper) semi-irresolute if F is FP-lower (resp. upper) semi-irresolute at eachx ∈ X.

Theorem 1.15. [5] A FP-multifunction F : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ is FP-lower (resp. FP-upper) semi-irresolute iff F^- (μ) ∈ (T_i, T_j) - SO (X) (resp. F⁺ (μ) ∈ (T_i, T_j) - SO (X)) for any μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀.

2 Upper and lower α-continuous FP-multifunctions

Definition 2.1. A FP-multifunction F : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ is called:

(i) FP-lower α-continuous at some point x₀ ∈ X if for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x₀ ∈ F^- (μ), there exists U ∈ (T_i, T_j) - αO (X, x₀) such that U ⊆ F^- (μ).

(ii)FP-upper α-continuous at some point x₀ ∈ X if for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x₀ ∈ F⁺ (μ), there exists U ∈ (T_i, T_j) - αO (X, x₀) such that U ⊆ F⁺ (μ).

(iii) FP-lower (resp. upper) α-continuous if F is FP-lower (resp. upper) α-continuous at eachx ∈ X.

Theorem 2.2. A FP-multifunction $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ is FP-lower (resp. FP-upper) α-continuous iff for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀, we have F^- (μ) (resp. F⁺ (μ)) is an (T_i, T_j) - αo set in X.

Proof. Suppose that F is a FP-lower α-continuous multifunction and μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀. Let x ∈ F^- (μ). Then, there exists U_x ∈ (T_i, T_j) - αO (X, x) such that U_x ⊆ F^- (μ). Therefore, x ∈ U_x ⊆ F^- (μ) . Then, F^- (μ) = ∪ {U_x : x ∈ F^- (μ)} . Hence F^- (μ) is an (T_i, T_j) - αo set in X.

Conversely, let x ∈ X be arbitrary and μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x ∈ F^- (μ). Then, U = F^- (μ) ∈ (T_i, T_j) - αO (X, x) and U ⊆ F^- (μ). Hence, F is a FP-lower α-continuous multifunction. The proof of FP-upper α-continuous is similar.□

Remark 2.3. From Definitions 1.12, 2.1 and 1.14, we have the following implications,

FP-lower (resp. upper) semi-continuous multifunction⇒ FP-lower (resp. upper) α-continuous multifunction⇒ FP-lower (resp. upper) semi-irresolute multifunction

The converse implications in Remark 2.3 are not true in general as the following examples show.

Example 2.4. Let X = {a, b, c} be a set. Let Y = L = I. Let T₁ = {X, φ, {b}} and T₂ = {X, φ, {a} , {b, c}} be two ordinary topologies on X. Define L-fuzzy topologies $T_{1}, T_{2} : L^{Y} \to L$ as follows:

$T_{1} (λ) = {\begin{matrix} 1, & if λ = \underline{0}, \underline{1} \\ 0.5, & if λ = \underline{0.5} \\ 0, & otherwise . \end{matrix}$

$T_{2} (λ) = {\begin{matrix} 1, & if λ = \underline{0}, \underline{1} \\ 0.6, & if λ = \underline{0.4} \\ 0, & otherwise . \end{matrix}$ Define the FP-multifunctions F₁, F₂ : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ by: $F_{1} (a) = \underline{0.9} F_{1} (b) = \underline{0.4} F_{1} (c) = \underline{0.4}$ $F_{2} (a) = \underline{0.4} F_{2} (b) = \underline{0.8} F_{2} (c) = \underline{0.7}$ For 0 < r ≤ 0.5: F₁ is FP- upper semi-irresolute but not FP-upper α-continuous, since $T_{1} (\underline{0.5}) \geq r$ but $F_{1}^{+} (\underline{0.5}) = {b, c} \notin (T_{1}, T_{2}) - α O (X)$ . Also, F₂ is FP- lower semi-irresolute but not FP-lower α-continuous, since $T_{1} (\underline{0.5}) \geq r$ but $F_{2}^{-} (\underline{0.5}) = {b, c} \notin (T_{1}, T_{2}) - α O (X)$ .

Example 2.5. In Example 2.4, if we change T₂ by T₂ = {X, φ, {b} , {b, c}}, we have: F₁ is FP- upper α-continuous but not FP-upper semi-continuous, since $T_{1} (\underline{0.5}) \geq r$ but $F_{1}^{+} (\underline{0.5}) = {b, c} \notin T_{1}$ . Also, F₂ is FP- lower α-continuous but not FP-lower semi-continuous, since $T_{1} (\underline{0.5}) \geq r$ but $F_{2}^{-} (\underline{0.5}) = {b, c} \notin T_{1}$ .

Theorem 2.6. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction. Then the following statements are equivalent:

(i) F is FP-lower α-continuous at a point x ∈ X.

(ii) For each λ ∈ L^Y with $T_{i} (λ) \geq r$ , r ∈ L₀ such that x ∈ F^- (λ), we have x ∈ int_i (cl_j (int_i (F^- (λ)))).

(iii) For each λ ∈ L^Y with $T_{i} (λ) \geq r$ , r ∈ L₀ such that x ∈ F^- (λ), there exists U ∈ T_i with U ⊆ F^- (λ) such that x ∈ F^- (λ) ⊆ int_i (cl_j (U)).

Proof. (i)⇒ (ii) Let x ∈ X and λ ∈ L^Y with $T_{i} (λ) \geq r$ , r ∈ L₀ such that x ∈ F^- (λ). By (i), there exists V ∈ (T_i, T_j) - αO (X, x) such that V ⊆ F^- (λ). Then,

$\begin{matrix} x \in V & \subseteq {int}_{i} ({cl}_{j} ({int}_{i} (V))) \\ \subseteq {int}_{i} ({cl}_{j} ({int}_{i} (F^{-} (λ)))) . \end{matrix}$ (ii)⇒ (iii) Let λ ∈ L^Y with $T_{i} (λ) \geq r$ , r ∈ L₀ such that x ∈ F^- (λ). By (ii) we have, x ∈ int_i (cl_j (int_i (F^- (λ)))). Thus, F^- (λ) ⊆ int_i (cl_j (int_i (F^- (λ)))). Put U = int_i (F^- (λ)). Then U ∈ T_i, U ⊆ F^- (λ) and x ∈ F^- (λ) ⊆ int_i (cl_j (U)).

(iii)⇒ (i) Let λ ∈ L^Y with $T_{i} (λ) \geq r$ , r ∈ L₀ such that x ∈ F^- (λ). By (iii), there exists U ∈ T_i with U ⊆ F^- (λ) such that x ∈ F^- (λ) ⊆ int_i (cl_j (U)). Then,

$\begin{matrix} x \in F^{-} (λ) & \subseteq {int}_{i} ({cl}_{j} ({int}_{i} (U))) \\ \subseteq {int}_{i} ({cl}_{j} ({int}_{i} (F^{-} (λ)))) . \end{matrix}$ Then, F^- (λ) ∈ (T_i, T_j) - αO (X, x). Thus, F is FP-lower α-continuous at a point x ∈ X.□

Theorem 2.7. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction. Then the following statements are equivalent:

(i) F is FP-upper α-continuous at a point x ∈ X.

(ii) For each λ ∈ L^Y with $T_{i} (λ) \geq r$ , r ∈ L₀ such that x ∈ F⁺ (λ), we have x ∈ int_i (cl_j (int_i (F⁺ (λ)))).

(iii) For each λ ∈ L^Y with $T_{i} (λ) \geq r$ , r ∈ L₀ such that x ∈ F⁺ (λ), there exists U ∈ T_i with U ⊆ F⁺ (λ) such that x ∈ F⁺ (λ) ⊆ int_i (cl_j (U)).

Proof. It is similar to the proof of Theorem 2.6.□

Theorem 2.8. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction. Then the following statements are equivalent:

(i) F is FP-lower α-continuous.

(ii) F^- (λ) ∈ (T_i, T_j) - αO (X), for every λ ∈ L^Y with $T_{i} (λ) \geq r$ , r ∈ L₀.

(iii) F⁺ (μ) ∈ (T_i, T_j) - αC (X), for every μ ∈ L^Y with $T_{i} (μ^{'}) \geq r$ , r ∈ L₀.

(iv) $(T_{j}, T_{i}) - sint ({cl}_{i} (F^{+} (λ))) \subseteq F^{+} (C_{T_{i}} (λ, r))$ , for every λ ∈ L^Y, r ∈ L₀.

(v) $F ((T_{j}, T_{i}) - sint ({cl}_{i} (A))) \leq C_{T_{i}} (F (A), r)$ , for every A ⊆ X, r ∈ L₀.

(vi) $F ((T_{i}, T_{j}) - α cl (A)) \leq C_{T_{i}} (F (A), r)$ , for every A ⊆ X, r ∈ L₀.

(vii) $(T_{i}, T_{j}) - α cl (F^{+} (λ)) \subseteq F^{+} (C_{T_{i}} (λ, r))$ , for every λ ∈ L^Y, r ∈ L₀.

(viii) $F ({cl}_{i} ({int}_{j} ({cl}_{i} (A)))) \leq C_{T_{i}} (F (A), r)$ , for every A ⊆ X, r ∈ L₀.

Proof. (i)⇔ (ii) Follows from Theorem 2.2.

(ii)⇔ (iii) It is obvious.

(iii)⇒ (iv) Let λ ∈ L^Y, r ∈ L₀. Then, $T_{i} ((C_{T_{i}} (λ, r))^{'}) \geq r$ . By (iii), we have $F^{+} (C_{T_{i}} (λ, r)) \in (T_{i}, T_{j}) - α C (X)$ . Then, ${cl}_{i} ({int}_{j} ({cl}_{i} (F^{+} (C_{T_{i}} (λ, r))))) \subseteq F^{+} (C_{T_{i}} (λ, r)) .$ This implies that, ${cl}_{i} ({int}_{j} ({cl}_{i} (F^{+} (λ)))) \subseteq F^{+} (C_{T_{i}} (λ, r)) .$ From Lemma 1.1, we have $(T_{j}, T_{i}) - sint ({cl}_{i} (F^{+} (λ))) \subseteq F^{+} (C_{T_{i}} (λ, r)) .$

(iv)⇒ (v) Let A ⊆ X, r ∈ L₀. Then F (A) ∈ L^Y, and by (iv), we have $\begin{matrix} (T_{j}, T_{i}) - & sint ({cl}_{i} (A)) \\ \subseteq (T_{j}, T_{i}) - sint ({cl}_{i} (F^{+} (F (A)))) \\ \subseteq F^{+} (C_{T_{i}} (F (A), r)) . \end{matrix}$ Thus, $\begin{matrix} F ((T_{j}, T_{i}) - & sint ({cl}_{i} (A))) \\ \leq F (F^{+} (C_{T_{i}} (F (A), r))) \\ \leq C_{T_{i}} (F (A), r) . \end{matrix}$ (v)⇒ (viii) Follows directly from Lemma 1.1.

(viii)⇒ (i) Let x ∈ X and λ ∈ L^Y with $T_{i} (λ) \geq r$ , r ∈ L₀ such that x ∈ F^- (λ). By (viii), we have $\begin{matrix} F ({cl}_{i} ({int}_{j} ({cl}_{i} (F^{+} (λ^{'}))))) & \leq C_{T_{i}} (F (F^{+} (λ^{'})), r) \\ \leq C_{T_{i}} (λ^{'}, r) \\ = λ^{'} . \end{matrix}$ This implies that $\begin{matrix} {cl}_{i} ({int}_{j} ({cl}_{i} ( & F^{+} (λ^{'})))) \\ \subseteq F^{+} (F ({cl}_{i} ({int}_{j} ({cl}_{i} (F^{+} (λ^{'})))))) \\ \subseteq F^{+} (λ^{'}) = X - F^{-} (λ) . \end{matrix}$ Then, $\begin{matrix} F^{-} (λ) & \subseteq X - ({cl}_{i} ({int}_{j} ({cl}_{i} (F^{+} (λ^{'}))))) \\ = {int}_{i} ({cl}_{j} ({int}_{i} (X - F^{+} (λ^{'})))) \\ = {int}_{i} ({cl}_{j} ({int}_{i} (F^{-} (λ)))) . \end{matrix}$ Thus, F^- (λ) ∈ (T_i, T_j) - αO (X). Therefore by Theorem 2.2, F is FP-lower α-continuous.

(iii)⇒ (vi) Let A ⊆ X, r ∈ L₀. Since A ⊆ F⁺ (F (A)), then $A \subseteq F^{+} (C_{T_{i}} (F (A), r))$ . Since $T_{i} ((C_{T_{i}} (F (A), r))^{'}) \geq r$ , by (iii) we have $F^{+} (C_{T_{i}} (F (A), r)) \in (T_{i}, T_{j}) - α C (X)$ . Thus $\begin{matrix} (T_{i}, T_{j}) - α cl (A) & \subseteq (T_{i}, T_{j}) - α cl (F^{+} (C_{T_{i}} (F (A), r))) \\ = F^{+} (C_{T_{i}} (F (A), r)) . \end{matrix}$ This implies that $F ((T_{i}, T_{j}) - α cl (A)) \leq F (F^{+} (C_{T_{i}} (F (A), r))) \leq C_{T_{i}} (F (A), r) .$ (vi)⇒ (iii) Let μ ∈ L^Y, r ∈ L₀ with $T_{i} (μ^{'}) \geq r$ . By (vi), we have $\begin{matrix} F ((T_{i}, T_{j}) - α cl (F^{+} (μ))) & \leq C_{T_{i}} (F (F^{+} (μ)), r) \\ \leq C_{T_{i}} (μ, r) = μ . \end{matrix}$ Then, (T_i, T_j) - αcl (F⁺ (μ)) ⊆ F⁺ (F ((T_i, T_j) - αcl (F⁺ (μ)))) ⊆ F⁺ (μ) . Thus, F⁺ (μ) ∈ (T_i, T_j) - αC (X).

(vi)⇒ (vii) Let λ ∈ L^Y, r ∈ L₀. By (vi), we have $F ((T_{i}, T_{j}) - α cl (F^{+} (λ))) \leq C_{T_{i}} (F (F^{+} (λ)), r) \leq C_{T_{i}} (λ, r) .$ Thus, $(T_{i}, T_{j}) - α cl (F^{+} (λ)) \leq F^{+} (F ((T_{i}, T_{j}) - α cl (F^{+} (λ)))) \leq F^{+} (C_{T_{i}} (λ, r)) .$ (vii)⇒ (vi) Let A ⊆ X, r ∈ L₀. By (vii), we have $(T_{i}, T_{j}) - α cl (A) \subseteq (T_{i}, T_{j}) - α cl (F^{+} (F (A))) \subseteq F^{+} (C_{T_{i}} (F (A), r)) .$ Therefore, $F ((T_{i}, T_{j}) - α cl (A)) \leq F (F^{+} (C_{T_{i}} (F (A), r))) \leq C_{T_{i}} (F (A), r) .$ □

Theorem 2.9. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction. Then the following statements are equivalent:

(i) F is FP-upper α-continuous.

(ii) F⁺ (λ) ∈ (T_i, T_j) - αO (X), for every λ ∈ L^Y with $T_{i} (λ) \geq r$ , r ∈ L₀.

(iii) F^- (μ) ∈ (T_i, T_j) - αC (X), for every μ ∈ L^Y with $T_{i} (μ^{'}) \geq r$ , r ∈ L₀.

(iv) $(T_{j}, T_{i}) - sint ({cl}_{i} (F^{-} (λ))) \subseteq F^{-} (C_{T_{i}} (λ, r))$ , for every λ ∈ L^Y, r ∈ L₀.

(v) $F ((T_{j}, T_{i}) - sint ({cl}_{i} (A))) \leq C_{T_{i}} (F (A), r)$ , for every A ⊆ X, r ∈ L₀.

(vi) $F ((T_{i}, T_{j}) - α cl (A)) \leq C_{T_{i}} (F (A), r)$ , for every A ⊆ X, r ∈ L₀.

(vii) $(T_{i}, T_{j}) - α cl (F^{-} (λ)) \subseteq F^{-} (C_{T_{i}} (λ, r))$ , for every λ ∈ L^Y, r ∈ L₀.

(viii) $F ({cl}_{i} ({int}_{j} ({cl}_{i} (A)))) \leq C_{T_{i}} (F (A), r)$ , for every A ⊆ X, r ∈ L₀.

Proof. It is similar to the proof of Theorem 2.8.□

3 Upper and lower almost α-continuous FP-multifunctions

Definition 3.1. A FP-multifunction F : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ is called:

(i) FP-lower almost α-continuous at some point x₀ ∈ X if for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x₀ ∈ F^- (μ), there exists U ∈ (T_i, T_j) - SO (X, x₀) such that $U \subseteq F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r))$ .

(ii) FP-upper almost α-continuous at some point x₀ ∈ X if for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x₀ ∈ F⁺ (μ), there exists U ∈ (T_i, T_j) - SO (X, x₀) such that $U \subseteq F^{+} (I_{T_{i}} (C_{T_{j}} (μ, r), r))$ .

(iii) FP-lower (resp. upper) almost α-continuous if F is FP-lower (resp. upper) almost α-continuous at each x ∈ X.

Theorem 3.2. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction. Then the following statements are equivalent:

(i) F is FP-lower almost α-continuous.

(ii) $F^{-} (μ) \subseteq {cl}_{j} ({int}_{i} (F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r))))$ , for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀.

(iii) F^- (μ) is (T_i, T_j) - so set in X for every $r - (T_{i}, T_{j})$ -fro set μ in Y, r ∈ L₀.

(iv) $F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r))$ is (T_i, T_j) - so set in X for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀.

Proof.(i)⇒ (ii) Let μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀. Let x ∈ F^- (μ) be arbitrary. Then there exists U ∈ (T_i, T_j) - SO (X, x) such that $U \subseteq F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r))$ . Consequently $\begin{matrix} x \in U & \subseteq {cl}_{j} ({int}_{i} (U)) \\ \subseteq {cl}_{j} ({int}_{i} (F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r)))) . \end{matrix}$ Thus $F^{-} (μ) \subseteq {cl}_{j} ({int}_{i} (F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r)))) .$ (ii)⇒ (iii) Let μ be an $r - (T_{i}, T_{j})$ -fro set in Y, r ∈ L₀. Then, $μ = I_{T_{i}} (C_{T_{j}} (μ, r), r))$ and therefore $T_{i} (μ) \geq r$ . By (ii) we have $\begin{matrix} F^{-} (μ) & \subseteq {cl}_{j} ({int}_{i} (F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r)))) \\ = {cl}_{j} ({int}_{i} (F^{-} (μ))) . \end{matrix}$ Hence, F^- (μ) is an (T_i, T_j) - so set in X.

(iii)⇒ (iv) Let μ ∈ L^Y with $T_{i} (μ) \geq r$ . Then, $I_{T_{i}} (C_{T_{j}} (μ, r), r))$ is an $r - (T_{i}, T_{j})$ -fro. By (iii), $F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r))$ is an (T_i, T_j) - so set in X.

(iv)⇒ (i) Let x ∈ X and μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ such that x ∈ F^- (μ). Then by (iv) we have $F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r)) = U$ (say) is (T_i, T_j) - so in X. Also since $μ \leq I_{T_{i}} (C_{T_{j}} (μ, r), r)$ and F (x) qμ we have $F (x) q (I_{T_{i}} (C_{T_{j}} (μ, r), r))$ . Thus, $x \in F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r)) = U .$ Hence F is FP-lower almost α-continuous at a point x. Since x is an arbitrary point, F is FP-lower almost α-continuous.

Theorem 3.3. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction. Then the following statements are equivalent:

(i) F is FP-upper almost α-continuous.

(ii) $F^{+} (μ) \subseteq {cl}_{j} ({int}_{i} (F^{+} (I_{T_{i}} (C_{T_{j}} (μ, r), r))))$ , for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀.

(iii) F⁺ (μ) is (T_i, T_j) - so set in X for every $r - (T_{i}, T_{j})$ -fro set μ in Y, r ∈ L₀.

(iv) $F^{+} (I_{T_{i}} (C_{T_{j}} (μ, r), r))$ is (T_i, T_j) - so set in X for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀.

Proof. It is similar to the proof of Theorem 3.2.

Theorem 3.4. A FP-multifunction $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ is FP-lower almost α-continuous iff for any $r - (T_{j}, T_{i})$ -fso set λ in Y,r ∈ L₀, ${int}_{j} ({cl}_{i} ((F^{+} (λ)) \subseteq F^{+} (C_{T_{i}} (λ, r)))) .$

Proof. Let F be FP-lower almost α-continuous and λ be an $r - (T_{j}, T_{i})$ -fso set in Y, r ∈ L₀. Then, $λ \leq C_{T_{i}} (I_{T_{j}} (λ, r), r) = η (say) .$ So, η is an $r - (T_{i}, T_{j})$ -frc in Y. Then, η′ is an $r - (T_{i}, T_{j})$ -fro. By Theorem 3.2, F^- (η′) = X - F⁺ (η) is an (T_i, T_j) - so set and therefore F⁺ (η) is (T_i, T_j) - sc set. Thus $\begin{matrix} {int}_{j} ({cl}_{i} ((F^{+} (λ)))) & \subseteq {int}_{j} ({cl}_{i} ((F^{+} (η)))) \\ \subseteq F^{+} (η) \\ = F^{+} (C_{T_{i}} (I_{T_{j}} (λ, r), r)) \\ \subseteq F^{+} (C_{T_{i}} (λ, r)) . \end{matrix}$

Conversely, let λ be any $r - (T_{i}, T_{j})$ -fro set in Y, r ∈ L₀. Then $T_{i} (λ) \geq r$ . Also, since λ is an $r - (T_{i}, T_{j})$ -fro set, we have λ is an $r - (T_{j}, T_{i})$ -fsc set in Y. Since λ′ is an $r - (T_{j}, T_{i})$ -fso set in Y, we have $\begin{matrix} {int}_{j} ({cl}_{i} ((F^{+} (λ^{'})))) & \subseteq F^{+} (C_{T_{i}} (λ^{'}, r)) \\ = F^{+} (λ^{'}) \\ = X - F^{-} (λ) . \end{matrix}$ Then, $F^{-} (λ) \subseteq {cl}_{j} ({int}_{i} ((F^{-} (λ)))) .$ Thus, F^- (λ) is an (T_i, T_j) - so set in X. Hence by Theorem 3.2, F is FP-lower almost α-continuous.

Theorem 3.5. A FP-multifunction $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ is FP-upper almost α-continuous iff for any $r - (T_{j}, T_{i})$ -fso set λ in Y,r ∈ L₀, ${int}_{j} ({cl}_{i} ((F^{-} (λ)) \subseteq F^{-} (C_{T_{i}} (λ, r)))) .$ Proof. It is similar to the proof of Theorem 3.4.

Theorem 3.6. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be FP-multifunction. Then the following statements are equivalent:

(i) F is FP-lower almost α-continuous.

(ii) $(T_{i}, T_{j}) - scl (F^{+} (λ)) \subseteq F^{+} (D_{T_{j}}^{T_{i}} (λ, r))$ for every λ ∈ L^Y, r ∈ L₀.

(iii) F⁺ (λ) is (T_i, T_j)-sc set in X, for every λ ∈ L^Y such that $λ = D_{T_{j}}^{T_{i}} (λ, r)$ , r ∈ L₀.

(iv) F^- (λ) is (T_i, T_j)-so set in X, for every λ ∈ L^Y such that $λ^{'} = D_{T_{j}}^{T_{i}} (λ^{'}, r)$ , r ∈ L₀.

Proof. (i)⇒ (ii) Let x ∈ (T_i, T_j) - scl (F⁺ (λ)), y_α ∈ F (x) and $μ \in R_{T_{j}}^{T_{i}} (y_{α}, r)$ . Then, F (x) qμ i.e., x ∈ F^- (μ). Since $T_{i} (μ) \geq r$ and F is FP-lower almost α-continuous, then there exits U ∈ (T_i, T_j) - SO (X, x) such that $U \subseteq F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r)) .$ Then, $F (z) q I_{T_{i}} (C_{T_{i}} (μ, r), r) = μ for all z \in U .$ Since x ∈ (T_i, T_j) - scl (F⁺ (λ)), U ∩ F⁺ (λ) ≠ φ. So, there exists z₀ ∈ U ∩ F⁺ (λ). Therefore, z₀ ∈ U and z₀ ∈ F⁺ (λ). Since z₀ ∈ U and z₀ ∈ F⁺ (λ), we have F (z₀) qμ and F (z₀) ≤ λ. So, μqλ. Thus, y_α is $r - (T_{i}, T_{j})$ -δ-cluster point of λ. Then we have, $y_{α} \in D_{T_{j}}^{T_{i}} (λ, r)$ . Thus, $F (x) \leq D_{T_{j}}^{T_{i}} (λ, r)$ , consequently $x \in F^{+} (D_{T_{j}}^{T_{i}} (λ, r))$ . Hence, $(T_{i}, T_{j}) - scl (F^{+} (λ)) \subseteq F^{+} (D_{T_{j}}^{T_{i}} (λ, r))$ .

(ii)⇒ (iii) Let λ ∈ L^Y such that $λ = D_{T_{j}}^{T_{i}} (λ, r)$ . By (ii) we have $(T_{i}, T_{j}) - scl (F^{+} (λ)) \subseteq F^{+} (D_{T_{j}}^{T_{i}} (λ, r)) = F^{+} (λ) .$ Thus, F⁺ (λ) = (T_i, T_j) - scl (F⁺ (λ)). Hence, F⁺ (λ) is (T_i, T_j)-sc set in X.

(iii)⇒ (iv) It is clear.

(iv)⇒ (i) Let λ be $r - (T_{i}, T_{j})$ -fro set in Y. Then, λ′ is $r - (T_{i}, T_{j})$ -frc. Then, $λ^{'} = C_{T_{i}} (I_{T_{j}} (λ^{'}, r), r)$ . By Theorem 1.8(ii), we have $λ^{'} = D_{T_{j}}^{T_{i}} (λ^{'}, r) .$ Thus, by (iv) we have F^- (λ) is (T_i, T_j)-so set in X. Hence, by Theorem 3.2, we have F is FP-lower almost continuous.

4 Upper and lower weakly α-continuous FP-multifunctions

Definition 4.1. A FP-multifunction F : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ is called:

(i) FP-lower weakly α-continuous at some point x₀ ∈ X if for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x₀ ∈ F^- (μ), there exists U ∈ (T_i, T_j) - SO (X, x₀) such that $U \subseteq {cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ .

(ii) FP-upper weakly α-continuous at some point x₀ ∈ X if for every μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x₀ ∈ F⁺ (μ), there exists U ∈ (T_i, T_j) - SO (X, x₀) such that $U \subseteq T_{j} - cl (F^{+} (C_{T_{j}} (μ, r)))$ .

(iii) FP-lower (resp. upper) weakly α-continuous if F is FP-lower (resp. upper) weakly α-continuous at each x ∈ X.

Theorem 4.2. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction. Then the following statements are equivalent:

(i) F is FP-lower weakly α-continuous at a point x ∈ X.

(ii) $x \in (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ , for each μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ such that x ∈ F^- (μ).

(iii) $x \in {cl}_{j} ({int}_{i} ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))))$ , for each μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ such that x ∈ F^- (μ).

Proof. (i)⇒ (ii) Let μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ such that x ∈ F^- (μ). By (i), there exists U ∈ (T_i, T_j) - SO (X, x) such that $U \subseteq {cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ . Then, $\begin{matrix} x \in U & = (T_{i}, T_{j}) - sint (U) \\ \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r))) . \end{matrix}$ (ii)⇒ (iii) Let μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ such that x ∈ F^- (μ). By (ii), $x \in (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ . Put $U = (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ . Then U ∈ (T_i, T_j) - SO (X, x) and $U \subseteq {cl}_{j} (F^{-} (C_{T_{j}} (μ, r))$ . Thus we have $\begin{matrix} x \in U & \subseteq {cl}_{j} ({int}_{i} (U)) \\ \subseteq {cl}_{j} ({int}_{i} ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r))))) . \end{matrix}$

(iii)⇒ (i) Let μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ such that x ∈ F^- (μ). By (iii), we have $x \in {cl}_{j} ({int}_{i} ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))))$ . From Lemma 1.1, we have $x \in (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ . Put $U = (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ . Then U ∈ (T_i, T_j) - SO (X, x) and $U \subseteq {cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ . Therefore, F is FP-lower weakly α-continuous at a point x ∈ X.

Theorem 4.3. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction. Then the following statements are equivalent:

(i) F is FP-upper weakly α-continuous at a point x ∈ X.

(ii) $x \in (T_{i}, T_{j}) - sint ({cl}_{j} (F^{+} (C_{T_{j}} (μ, r)))$ , for each μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ such that x ∈ F⁺ (μ).

(iii) $x \in {cl}_{j} ({int}_{i} ({cl}_{j} (F^{+} (C_{T_{j}} (μ, r)))))$ , for each μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ such that x ∈ F⁺ (μ).

Proof. The proof is similar to that of Theorem 4.2.□

Theorem 4.4. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction. Then the following statements are equivalent:

(i) F is FP-lower weakly α-continuous.

(ii) $F^{-} (μ) \subseteq {cl}_{j} ({int}_{i} ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))))$ , for each μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀.

(iii) ${int}_{j} ({cl}_{i} ({int}_{j} (F^{+} (I_{T_{j}} (λ, r))))) \subseteq F^{+} (λ)$ , for each λ ∈ L^Y with $T_{i} (λ^{'}) \geq r$ , r ∈ L₀.

(iv) $(T_{i}, T_{j}) - scl ({int}_{j} (F^{+} (I_{T_{j}} (λ, r))) \subseteq F^{+} (λ)$ , for each λ ∈ L^Y with $T_{i} (λ^{'}) \geq r$ , r ∈ L₀.

(v) $(T_{i}, T_{j}) - scl ({int}_{j} (F^{+} (I_{T_{j}} (C_{T_{i}} (ν, r), r))) \subseteq F^{+} (C_{T_{i}} (ν, r))$ , for each ν ∈ L^Y, r ∈ L₀.

(vi) $F^{-} (I_{T_{i}} (ν, r)) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (I_{T_{i}} (ν, r), r)))$ , for each ν ∈ L^Y, r ∈ L₀.

(vii) $F^{-} (μ) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ , for each μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀.

Proof. (i)⇒ (ii) Let μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ and x ∈ F^- (μ). Since F is FP-lower weakly α-continuous and by Theorem 4.2, we have $x \in {cl}_{j} ({int}_{i} ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))))$ . Thus,

$F^{-} (μ) \subseteq {cl}_{j} ({int}_{i} ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))))$ .

(ii)⇒ (iii) Let λ ∈ L^Y with $T_{i} (λ^{'}) \geq r$ , r ∈ L₀. By (ii), we have $F^{-} (λ^{'}) \subseteq {cl}_{j} ({int}_{i} ({cl}_{j} (F^{-} (C_{T_{j}} (λ^{'}, r)))))$ . This implies that $\begin{matrix} X - F^{+} (λ) & = F^{-} (λ^{'}) \\ \subseteq {cl}_{j} ({int}_{i} ({cl}_{j} (F^{-} (C_{T_{j}} (λ^{'}, r))))) \\ = {cl}_{j} ({int}_{i} (X - ({int}_{j} (F^{+} (I_{T_{j}} (λ, r))))) \\ = X - ({int}_{j} ({cl}_{i} ({int}_{j} (F^{+} (I_{T_{j}} (λ, r)))))) . \end{matrix}$ Therefore, ${int}_{j} ({cl}_{i} ({int}_{j} (F^{+} (I_{T_{j}} (λ, r))))) \subseteq F^{+} (λ)$ .

(iii)⇒ (iv) Let λ ∈ L^Y with $T_{i} (λ^{'}) \geq r$ , r ∈ L₀. By (iii), we have ${int}_{j} ({cl}_{i} ({int}_{j} (F^{+} (I_{T_{j}} (λ, r))))) \subseteq F^{+} (λ)$ . From Lemma 1.1, we find $(T_{i}, T_{j}) - scl ({int}_{j} (F^{+} (I_{T_{j}} (λ, r)) \subseteq F^{+} (λ)$ .

(iv)⇒ (v) Let ν ∈ L^Y, r ∈ L₀. Then, $T_{i} ((C_{T_{i}} (ν, r))^{'}) \geq r$ . By (iv), $(T_{i}, T_{j}) - scl ({int}_{j} (F^{+} (I_{T_{j}} (C_{T_{i}} (ν, r), r)))) \subseteq F^{+} (C_{T_{i}} (ν, r))$ .

(v)⇒ (vi) Let ν ∈ L^Y, r ∈ L₀. Then, $\begin{matrix} X - F^{-} (I_{T_{i}} (ν, r)) = F^{+} (C_{T_{i}} (ν^{'}, r)) \\ \supseteq (T_{i}, T_{j}) - scl ({int}_{j} (F^{+} (I_{T_{j}} (C_{T_{i}} (ν^{'}, r), r)))) \\ = (T_{i}, T_{j}) - scl ({int}_{j} (F^{+} ((C_{T_{j}} \\ (I_{T_{i}} (ν, r), r))^{'}))) \\ = (T_{i}, T_{j}) - scl (X - ({cl}_{j} (F^{-} \\ (C_{T_{j}} (I_{T_{i}} (ν, r), r))))) \\ = X - ((T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} \\ (C_{T_{j}} (I_{T_{i}} (ν, r), r))))) . \end{matrix}$ Hence, $F^{-} (I_{T_{i}} (ν, r)) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (I_{T_{i}} (ν, r), r))))$ .

(vi)⇒ (vii) Let μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀. Then, $\begin{matrix} F^{-} (μ) = F^{-} (I_{T_{i}} (μ, r)) \\ \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (I_{T_{i}} (μ, r), r)))) \\ = (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))) . \end{matrix}$

(vii)⇒ (i) Let x ∈ X and μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ such that x ∈ F^- (μ). Then from (vii) we have, $x \in F^{-} (μ) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r))))$ . Put $U = (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r))))$ . Then, U ∈ (T_i, T_j) - SO (X, x) and $U \subseteq {cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ . Then F is FP-lower weakly α-continuous at a point x. Since x is arbitrary, F is FP-lower weakly α-continuous.

Theorem 4.5. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction. Then the following statements are equivalent:

(i) F is FP-upper weakly α-continuous.

(ii) $F^{+} (μ) \subseteq {cl}_{j} ({int}_{i} ({cl}_{j} (F^{+} (C_{T_{j}} (μ, r)))))$ , for each μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀.

(iii) ${int}_{j} ({cl}_{i} ({int}_{j} (F^{-} (I_{T_{j}} (λ, r))))) \subseteq F^{-} (λ)$ , for each λ ∈ L^Y with $T_{i} (λ^{'}) \geq r$ , r ∈ L₀.

(iv) $(T_{i}, T_{j}) - scl ({int}_{j} (F^{-} (I_{T_{j}} (λ, r))) \subseteq F^{-} (λ)$ , for each λ ∈ L^Y with $T_{i} (λ^{'}) \geq r$ , r ∈ L₀.

(v) $(T_{i}, T_{j}) - scl ({int}_{j} (F^{-} (I_{T_{j}} (C_{T_{i}} (ν, r), r))) \subseteq F^{-} (C_{T_{i}} (ν, r))$ , for each ν ∈ L^Y, r ∈ L₀.

(vi) $F^{+} (I_{T_{i}} (ν, r)) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{+} (C_{T_{j}} (I_{T_{i}} (ν, r), r)))$ , for each ν ∈ L^Y, r ∈ L₀.

(vii) $F^{+} (μ) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{+} (C_{T_{j}} (μ, r)))$ , for each μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀.

Proof. The proof is similar to that of Theorem 4.4.□

Theorem 4.6. If the FP-multifunction F : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ is FP-lower weakly α-continuous, then for every $r - (T_{i}, T_{j})$ -fpo set μ in Y we have $F^{-} (μ) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))) .$

Proof. Suppose that F is FP-lower weakly α-continuous and μ is any $r - (T_{i}, T_{j})$ -fpo set in Y. Let x ∈ F^- (μ). Then, F (x) qμ. Since $μ \leq I_{T_{i}} (C_{T_{j}} (μ, r), r)$ , then $F (x) q (I_{T_{i}} (C_{T_{j}} (μ, r), r)) .$ This implies that $x \in F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r))$ . Since $T_{i} (I_{T_{i}} (C_{T_{j}} (μ, r), r)) \geq r$ , by Theorem 4.4, we have $F^{-} (I_{T_{i}} (I_{T_{i}} (C_{T_{j}} (μ, r), r), r)) \subseteq (T_{i}, T_{j}) - sint (T_{j} - cl (F^{-} (C_{T_{j}} (I_{T_{i}} (I_{T_{i}} (C_{T_{j}} (μ, r), r), r), r))))$ which implies that $F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r)) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (I_{T_{i}} ((C_{T_{j}} (μ, r), r)))))$ . Since μ is $r - (T_{i}, T_{j})$ -fpo, $F^{-} (μ) \subseteq F^{-} (I_{T_{i}} (C_{T_{j}} (μ, r), r)) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (I_{T_{i}} (C_{T_{j}} (μ, r), r), r))))$ . Since $I_{T_{i}} (C_{T_{j}} (μ, r), r) \leq C_{T_{j}} (μ, r)$ , $C_{T_{j}} (I_{T_{i}} (C_{T_{j}} (μ, r), r), r) \leq C_{T_{j}} (μ, r)$ . Therefore, $F^{-} (μ) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))) .$

Theorem 4.7. If the FP-multifunction F : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ is FP-upper weakly α-continuous, then for every $r - (T_{i}, T_{j})$ -fpo set μ in Y we have $F^{+} (μ) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{+} (C_{T_{j}} (μ, r)))) .$

Proof. The proof is similar to that of Theorem 4.6.□

5 Relationships between FP-lower (upper) (α-continuous, almost α-continuous, weakly α-continuous) multifunctions

Remark 5.1. From the definition of FP-lower (resp. upper) α-continuous, FP-lower (resp. upper) almost α-continuous and FP-lower (resp. upper) weakly α-continuous multifunctions, we have the following implications. $\begin{matrix} FP - lower (resp . upper) α - continuous & \Rightarrow \\ FP - lower (resp . upper) almost α - continuous & \Rightarrow \\ FP - lower (resp . upper) weakly α - continuous \end{matrix}$

The converses of the above implications are not true in general as the following examples show.

Example 5.2. Let X = {a, b, c} be a set. Let Y = L = I. Let T₁ = {X, φ, {b}} and T₂ = {X, φ, {a} , {b, c}} be two ordinary topologies on X. Define L-fuzzy topologies $T_{1}, T_{2} : L^{Y} \to L$ as follows:

$T_{1} (λ) = {\begin{matrix} 1, & if λ = \underline{0}, \underline{1} \\ 0.7, & if λ = \underline{0.5} \\ 0, & otherwise . \end{matrix}$

$T_{2} (λ) = {\begin{matrix} 1, & if λ = \underline{0}, \underline{1} \\ 0.6, & if λ = \underline{0.3} \\ 0, & otherwise . \end{matrix}$ Define the FP-multifunctions F₁, F₂ : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ by: $F_{1} (a) = \underline{0.8} F_{1} (b) = \underline{0.4} F_{1} (c) = \underline{0.3}$ $F_{2} (a) = \underline{0.2} F_{2} (b) = \underline{0.6} F_{2} (c) = \underline{0.9}$ For 0 < r ≤ 0.6: F₁ is FP- upper weakly α-continuous but not FP-upper almost α-continuous, since $T_{2} (\underline{0.3}) \geq r$ but $F_{1}^{+} (\underline{0.3}) = {c} T_{1} - cl (T_{2} - int (F_{1}^{+} (I_{T_{2}} (C_{T_{1}} (\underline{0.3}, r), r)))) = φ$ (see Theorem 3.3). Also, F₂ is FP-lower weakly α-continuous but not FP-lower almost α-continuous, since $T_{2} (\underline{0.3}) \geq r$ but $F_{2}^{-} (\underline{0.3}) = {c} T_{1} - cl (T_{2} - int (F_{2}^{-} (I_{T_{2}} (C_{T_{1}} (\underline{0.3}, r), r)))) = φ$ (see Theorem 3.2).

Example 5.3. Let X = {a, b, c} be a set. Let Y = L = I. Let T₁ = {X, φ, {a} , {b, c}} and T₂ = {X, φ, {c}} be two ordinary topologies on X. Define L-fuzzy topologies $T_{1}, T_{2} : L^{Y} \to L$ as follows:

$T_{1} (λ) = {\begin{matrix} 1, & if λ = 0_, 1_ \end{matrix} 0.5, & if λ = \underline{0.6} 0, & otherwise,$

$T_{2} (λ) = {\begin{matrix} 1, & if λ = \underline{0}, \underline{1} \\ 0.4, & if λ = \underline{0.7} \\ 0, & otherwise . \end{matrix}$ Define the FP-multifunctions F₁, F₂ : (X, T₁, $T_{2}) \to (Y, T_{1}, T_{2})$ by: $F_{1} (a) = \underline{0.2} F_{1} (b) = \underline{0.3} F_{1} (c) = \underline{0.8}$ $F_{2} (a) = \underline{0.9} F_{2} (b) = \underline{0.8} F_{2} (c) = \underline{0.2}$ For 0 < r ≤ 0.4: F₁ is FP- upper almost α-continuous but not FP-upper α-continuous, since $T_{1} (\underline{0.6}) \geq r$ but $F_{1}^{+} (\underline{0.6}) = {a, b} \notin (T_{1}, T_{2}) - α O (X)$ (see Theorem 2.2). Also, F₂ is FP-lower almost α-continuous but not FP-lowerα-continuous, since $T_{1} (\underline{0.6})$ ≥r but $F_{2}^{-} (\underline{0.6}) = {a, b} \notin (T_{1}, T_{2}) - α O (X)$ (see Theorem 2.2).

In the following two theorems we give the conditions which make the concepts in Remark 5.1 equivalent.

Theorem 5.4. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction such that x ∈ int_i (cl_j (int_i (U))), for each U ∈ (T_i, T_j) - SO (X, x) and $F^{-} (μ) \supseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ , for each μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀. Then the following are equivalent:

(i) F is FP-lower α-continuous,

(ii) F is FP-lower almost α-continuous,

(iii) F is FP-lower weakly α-continuous.

Proof. From Remark 5.1, (i)⇒(ii) ⇒(iii), so we will prove only (iii)⇒(i). Let x ∈ X and μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀ such that x ∈ F^- (μ). Since F is FP-lower weakly α-continuous, there exists U ∈ (T_i, T_j) - SO (X, x) such that $U \subseteq {cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ . From the hypotheses x ∈ int_i (cl_j (int_i (U))). Put G = int_i (U). Then x ∈ int_i (cl_j (G)). Since G ∈ T_i, G ⊆ int_i (cl_j (G)). Put S = G ∪ {x}. Then G ⊆ S ⊆ int_i (cl_j (G)). Therefore, S ∈ (T_i, T_j) - αO (X, x). It remains to prove that, S ⊆ F^- (μ). Since $U \subseteq {cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))$ , $U = (T_{i}, T_{j}) - sint (U) \subseteq (T_{i}, T_{j}) - sint ({cl}_{j} (F^{-} (C_{T_{j}} (μ, r)))) \subseteq F^{-} (μ)$ . Since G = int_i (U) ⊆ U ⊆ F^- (μ) and x ∈ F^- (μ), S ⊆ F^- (μ). Hence, F is FP-lower α-continuous.

Theorem 5.5. Let $F : (X, T_{1}, T_{2}) \to (Y, T_{1}, T_{2})$ be a FP-multifunction such that x ∈ int_i (cl_j (int_i (U))), for each U ∈ (T_i, T_j) - SO (X, x) and F⁺ (μ)⊇ $(T_{i}, T_{j}) - sint ({cl}_{j} (F^{+} (C_{T_{j}} (μ, r)))$ , for each μ ∈ L^Y with $T_{i} (μ) \geq r$ , r ∈ L₀. Then the following are equivalent:

(i) F is FP-upper α-continuous,

(ii) F is FP-upper almost α-continuous,

(iii) F is FP-upper weakly α-continuous.

Proof. The proof is similar to that of Theorem 5.4.□

6 Conclusion

In this paper, we studied fuzzy pairwise multifunctions through introducing and studying new kinds of FP-multifunctions, namely FP-lower (upper) α-continuous, FP-lower (upper) almost α-continuous, and FP-lower (upper) weakly α-continuous multifunctions. Various properties of these multifunctions were investigated. We gave the relationships between these multifunctions and presented contrary examples. Finally, we gave the conditions which made these multifunctions equivalent. Fuzzy multifunctions can be extended to fuzzy rough set theory and fuzzy soft set theory, and therefore they can be useful for decision making [41 –46].

Footnotes

Acknowledgments

The author wish to thank the associate editor Prof. Jianming Zhan and the reviewers for their valuable suggestions.

References

S.E.

Abbas ,

M.A.

Hebeshi and

I.M.

Taha , On upper and lower almost weakly continuous fuzzy multifunctions, Iranian Journal of Fuzzy Systems 12(1) (2015), 101–114.

A.A.

Abd El-latif , Fuzzy R-convergence in fuzzy topological molecular lattice, Sylwan 159(2) (2015), 177–199.

A.A.

Abd El-latif and

Y.C.

Kim , On (L,M)-double fuzzy remote neighborhood systems in (L,M)-DFTML, Journal of Intelligent and Fuzzy Systems 30(6) (2016), 13321–13333.

A.A.

Abd El-latif , Convergence structures in (L,M)-DFTML, Journal of Intelligent and Fuzzy Systems 33(3) (2017), 1513–1526.

A.A.

Abd El-latif , Irresolute fuzzy pairwise multifunctions, Journal of Intelligent and Fuzzy Systems 33(6) (2017), 3173–3180.

K.M.A.

Al-Hamadi and

S.B.

Nimse , On fuzzy α-continuous multifunctions, Miskolc Math Notes 11(2) (2010), 105–112.

Alimohammady ,

Ekici ,

Jafari and

Roohi , On fuzzy upper and lower contra-continuous multifunctions, Iranian Journal of Fuzzy Systems 8(3) (2011), 149–158.

Bhattacharyya , Weakly α-continuous fuzzy multifunctions, International Journal of Pure and Applied Mathematics 115(4) (2017), 693–712.

C.L.

Chang , Fuzzy topological spaces, J Math Anal Appl 24 (1968), 182–190.

10.

K.C.

Chattopadhyay and

S.K.

Samanta , Fuzzy topology: Fuzzy closure operator, fuzzy compactness and fuzzy connectedness, Fuzzy Sets and Systems 54 (1993), 207–212.

11.

J.A.

Goguen , L-fuzzy sets, J Math Anal Appl 18 (1967), 145–175.

12.

M.A.

Hebeshi and

I.M.

Taha , On upper and lower α-continuous fuzzy multifunctions, Journal of Intelligent and Fuzzy Systems 28(6) (2015), 2537–2546.

13.

Höhle , Upper semicontinuous fuzzy set and applications, J Math Anal Appl 78 (1980), 659–673.

14.

Höhle and

S.E.

Rodabaugh , Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Boston, 1999.

15.

Ibedou , Separation axioms in fuzzy bitopological spaces, Journal of Intelligent and Fuzzy Systems 27(2) (2014), 943–951.

16.

Kandil , Biproximities and fuzzy bitopological spaces, Simon Stevin 63 (1989), 45–66.

17.

J.C.

Kelly , Bitopological spaces, Proc Lodon Math Soc 3 (1963), 71–89.

18.

Y.C.

Kim , r-fuzzy semi-open sets in fuzzy bitopological spaces, Far East J Math Sci Special(FJMS) 2 (2000), 221–236.

19.

Y.C.

Kim , δ-closure operators in fuzzy bitopological spaces, Far East J Math Sci(FJMS) 2(5) (2000), 791–808.

20.

Y.C.

Kim , Join-meet preserving maps and Alexandrov fuzzy topologies, Journal of Intelligent and Fuzzy Systems 28(1) (2015), 457–467.

21.

Kubiak , On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985.

22.

S.J.

Lee and

E.P.

Lee , Fuzzy r-continuous and fuzzy rsemicontinuous maps, International Journal of Mathematics and Mathematical Sciences 27(1) (2001), 53–63.

23.

E.P.

Lee , Preopen sets in smooth bitopological spaces, Commun Korean Math Soc 18(3) (2003), 521–532.

24.

Y.-M.

Liu and

M.-K.

Luo , Fuzzy topology, World Scientific Publishing Co., Singapore, 1997.

25.

R.A.

Mahmoud , An application of continuous fuzzy multifunctions, Chaos, Solitons and Fractals 17 (2003), 833–841.

26.

M.N.

Mukherjee and

Malakar , On almost continuous and weakly continuous fuzzy multifunctions, Fuzzy Sets and Systems 41 (1991), 113–125.

27.

N.S.

Papageorgiou , Fuzzy topology and fuzzy multifunctions, J Math Anal Appl 109 (1985), 397–425.

28.

Paul ,

Bhattacharya and

Chakraborty , γ Hyperconnectedness and fuzzy mappings in fuzzy bitopological spaces, Journal of Intelligent and Fuzzy Systems 32(3) (2017), 1815–1820.

29.

Popa and

Noiri , On Upper and lower weakly α-continuous multifunctions, Novi Asd J Math 32(1) (2002), 7–24.

30.

P.M.

Pu and

Y.M.

Liu , Fuzzy topology I. Neighborhood structure of a fuzzy point and Moor-Smith convergence, J Math Anal Appl 76 (1980), 571–599.

31.

A.A.

Ramadan ,

S.E.

Abbas and

A.A.

Abd El-latif , On fuzzy bitopological spaces in Šostak’s sense, Commun Korean Math Soc 21(3) (2006), 497–514.

32.

A.A.

Ramadan and

A.A.

Abd El-latif , Supra fuzzy convergence of fuzzy filters, Bull Korean Math Soc 45(2) (2008), 207–220.

33.

A.A.

Ramadan and

A.A.

Abd El-latif , Images and preimages of L-fuzzy ideal bases, J Fuzzy Math 17(3) (2009), 611–632.

34.

A.A.

Ramadan ,

S.E.

Abbas and

A.A.

Abd El-latif , On fuzzy bitopological spaces in Šostak’s sense (II), Commun Korean Math Soc 25(3) (2010), 457–475.

35.

A.A.

Ramadan and

A.A.

Abd El-latif , Fuzzy pairwise multifunctions, Asian Journal of Mathematics and Computer Research 2(4) (2015), 219–234.

36.

Seenivasan and

Balasubramanian , On upper and lower semiirresolute fuzzy multifunctions, East Asian Math J 23(1) (2007), 9–22.

37.

A.P.

Šostak , On a fuzzy toplogical structure, Supp Rend Circ Math Palermo(SerII) 11 (1985), 89–103.

38.

B.C.

Tripathy and

Debnath , γ-Open sets and –continuous mappings in fuzzy bitopological spaces, Journal of Intelligent and Fuzzy Systems 24(3) (2013), 631–635.

39.

Vadivel and

Elavarasan , r-(i, j)-fuzzy regular semi open sets in smooth bitipological spaces, Ann Fuzzy Math Inform 13(2) (2017), 199–211.

40.

L.A.

Zadeh , Fuzzy sets, Information and Control 8 (1965), 338–353.

41.

Zhan and

Xu , Two types of coverings based multigranulation rough fuzzy sets and applications to decision making, Artifcial Intelligence Review (2018). https://doi.org/10.1007/s10462-018-9649-8.

42.

Zhan and

Wang , Certain types of soft coverings based rough sets with applications, Int J Mach Learn Cybern (2018). https://doi.org/10.1007/s13042-018-0785-x.

43.

Zhan ,

Sun and

J.C.R.

Alcantud , Covering based multigranulation (I, T)-fuzzy rough set models and applications in multi-attribute group decision-making, Information Sciences 476 (2019), 290–318.

44.

Zhang and

Zhan , Novel classes of fuzzy soft β- coverings-based fuzzy rough sets with applications to multicriteria fuzzy group decision making, Soft Computing (2018). https://doi.org/10.1007/s00500-018-3470-9.

45.

Zhang and

Zhan , Fuzzy soft β-covering based fuzzy rough sets and corresponding decision-making applications, Int J Mach Learn Cybern (2018). https://doi.org/10.1007/s13042-018-0828-3.

46.

Zhang ,

Zhan and

Xu , Covering-based generalized IF rough sets with applications to multi-attribute decision-making, Information Sciences 478 (2019), 275–302.