EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6009 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Contra-(τ1, τ2)p-continuity and Almost Contra-(τ1, τ2)p-continuity for Multifunctions Chokchai Viriyapong1, Areeyuth Sama-Ae2, Chawalit Boonpok1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand 2 Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand Abstract. This paper presents four new classes of multifunctions called upper contra-(τ1, τ2)p- continuous multifunctions, lower contra-(τ1, τ2)p-continuous multifunctions, upper almost contra- (τ1, τ2)p-continuous multifunctions and lower almost contra-(τ1, τ2)p-continuous multifunctions. Furthermore, several characterizations and some properties concerning upper contra-(τ1, τ2)p- continuous multifunctions, lower contra-(τ1, τ2)p-continuous multifunctions, upper almost contra- (τ1, τ2)p-continuous multifunctions and lower almost contra-(τ1, τ2)p-continuous multifunctions are established. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: upper contra-(τ1, τ2)p-continuous multifunction, lower contra-(τ1, τ2)p- continuous multifunction, upper almost contra-(τ1, τ2)p-continuous multifunction, lower almost contra-(τ1, τ2)p-continuous multifunction 1. Introduction The field of the mathematical science which goes under the name of topology is con- cerned with all questions directly or indirectly related to continuity. In topology, there has been recently significant interest in characterizing and investigating the character- izations of some weak forms of continuity for functions and multifunctions. Weaker and stronger forms of open sets play an important role in the generalization of differ- ent forms of continuity. Using different forms of open sets, several authors have in- troduced and studied various types of continuity. The concepts of (Λ, sp)-open sets, s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets and β(Λ, sp)-open sets were stud- ied in [1]. Viriyapong and Boonpok [2] investigated several characterizations of (Λ, sp)- continuous functions by utilizing the notions of (Λ, sp)-open sets and (Λ, sp)-closed sets. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6009 Email addresses: chokchai.v@msu.ac.th (C. Viriyapong), areeyuth.s@psu.ac.th (A. Sama-Ae), chawalit.b@msu.ac.th (C. Boonpok) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 2 of 18 Dungthaisong et al. [3] introduced and studied the concept of g(m,n)-continuous func- tions. Duangphui et al. [4] introduced and investigated the notion of almost (µ, µ′)(m,n)- continuous functions. Furthermore, several characterizations of almost (Λ, p)-continuous functions, strongly θ(Λ, p)-continuous functions, almost strongly θ(Λ, p)-continuous func- tions, θ(Λ, p)-continuous functions, weakly (Λ, b)-continuous functions, θ(⋆)-precontinuous functions, (Λ, p(⋆))-continuous functions, ⋆-continuous functions, θ-I -continuous func- tions, almost (g,m)-continuous functions, pairwise almostM -continuous functions, (τ1, τ2)- continuous functions, almost (τ1, τ2)-continuous functions, weakly (τ1, τ2)-continuous func- tions and slightly (τ1, τ2)s-continuous functions were presented in [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] and [19], respectively. Kong-ied at al. [20] intro- duced and studied the concept of almost quasi (τ1, τ2)-continuous functions. Chiangpradit et al. [21] introduced and investigated the notion of weakly quasi (τ1, τ2)-continuous functions. Thongmoon et al. [22] introduced and studied the notion of rarely (τ1, τ2)- continuous functions. Srisarakham et al. [23] introduced and investigated the concept of faintly (τ1, τ2)-continuous functions. On the other hand, the present authors introduced and studied the notions of δ(τ1, τ2)-continuous functions [24], quasi θ(τ1, τ2)-continuous functions [25], almost weakly (τ1, τ2)-continuous functions [26] and almost nearly (τ1, τ2)- continuous functions [27]. In 1966, Dontchev [28] introduced the concepts of contra- continuity and strong S-closedness in topological spaces. Moreover, Dontchev [28] ob- tained very interesting and important results concerning contra-continuity, compactness, S-closedness and strong S-closedness. Dontchev et al. [29] defined a new class of func- tions called regular set-connected functions. Dontchev and Noiri [30] introduced and studied the concept of RC-continuity between topological spaces which is weaker than contra-continuity. Jafari and Noiri [31] introduced a new class of functions called contra- precontinuous functions which is weaker than contra-continuous functions and studied several basic properties of contra-precontinuous functions. In 2004, Ekici [32] introduced and studied a new class of functions called almost contra-precontinuous functions which generalize classes of regular set-connected functions [29], contra-precontinuous functions [31], contra-continuous functions [28], almost s-continuous functions [33] and perfectly continuous functions [34]. In 2008, Ekici et al. [35] extended the notion of contra-continuous functions to the set- ting of multifunctions. Noiri and Popa [36] introduced the notion of weakly precontinuous multifunctions. Ekici et al. [37] introduced and studied two new concepts namely contra- precontinuous multifunctions and almost contra-precontinuous multifunctions which are containing the class of contra-continuous multifunctions [35] and contained in the class of weakly precontinuous multifunctions. Laprom et al. [38] introduced and investigated the notion of almost β(τ1, τ2)-continuous multifunctions. Moreover, some characterizations of (τ1, τ2)δ-semicontinuous multifunctions, almost weakly (τ1, τ2)-continuous multifunc- tions, weakly quasi (Λ, sp)-continuous multifunctions, ⋆-continuous multifunctions, β(⋆)- continuous multifunctions, α-⋆-continuous multifunctions, almost α-⋆-continuous multi- functions, almost quasi ⋆-continuous multifunctions, weakly α-⋆-continuous multifunc- tions, sβ(⋆)-continuous multifunctions, weakly sβ(⋆)-continuous multifunctions, θ(⋆)-quasi continuous multifunctions, almost ı⋆-continuous multifunctions, weakly (Λ, sp)-continuous C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 3 of 18 multifunctions, α(Λ, sp)-continuous multifunctions, almost α(Λ, sp)-continuous multifunc- tions, weakly α(Λ, sp)-continuous multifunctions, almost β(Λ, sp)-continuous multifunc- tions, slightly (Λ, sp)-continuous multifunctions, weakly quasi (τ1, τ2)-continuous mul- tifunctions, almost quasi (τ1, τ2)-continuous multifunctions, c-(τ1, τ2)-continuous multi- functions, c-quasi (τ1, τ2)-continuous multifunctions, s-(τ1, τ2)p-continuous multifunctions, slightly α(τ1, τ2)-continuous multifunctions and slightly (τ1, τ2)p-continuous multifunctions were established in [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63] and [64], respectively. On the other hand, the present authors introduced and investigated the notions of rarely s-(τ1, τ2)p- continuous multifunctions [65], almost nearly (τ1, τ2)-continuous multifunctions [66], s- (τ1, τ2)-continuous multifunctions [67], quasi θ(τ1, τ2)-continuous multifunctions [68], al- most nearly quasi (τ1, τ2)-continuous multifunctions [69], weakly s-(τ1, τ2)-continuous mul- tifunctions [70], nearly (τ1, τ2)-continuous multifunctions [71] and almost quasi (τ1, τ2)- continuous multifunctions [72]. In 2023, the present authors [73] introduced and in- vestigated the notion of almost contra-(Λ, sp)-continuous multifunctions. Pue-on et al. [74] introduced and studied the notions of upper (τ1, τ2)-continuous multifunctions and lower (τ1, τ2)-continuous multifunctions. Klanarong et al. [75] introduced and inves- tigated the concepts of upper almost (τ1, τ2)-continuous multifunctions and lower al- most (τ1, τ2)-continuous multifunctions. Thongmoon et al. [76] introduced and studied the notions of upper weakly (τ1, τ2)-continuous multifunctions and lower weakly (τ1, τ2)- continuous multifunctions. In this paper, we introduce the concepts of upper contra- (τ1, τ2)p-continuous multifunctions, lower contra-(τ1, τ2)p-continuous multifunctions, up- per almost contra-(τ1, τ2)p-continuous multifunctions and lower almost contra-(τ1, τ2)p- continuous multifunctions. We also investigate several characterizations of upper contra- (τ1, τ2)p-continuous multifunctions, lower contra-(τ1, τ2)p-continuous multifunctions, up- per almost contra-(τ1, τ2)p-continuous multifunctions and lower almost contra-(τ1, τ2)p- continuous multifunctions. 2. Preliminaries Throughout the present paper, spaces (X, τ1, τ2) and (Y, σ1, σ2) (or simply X and Y ) always mean bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a bitopological space (X, τ1, τ2). The closure of A and the interior of A with respect to τi are denoted by τi-Cl(A) and τi-Int(A), respectively, for i = 1, 2. A subset A of a bitopological space (X, τ1, τ2) is called τ1τ2-closed [77] if A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. The intersection of all τ1τ2-closed sets of X containing A is called the τ1τ2-closure [77] of A and is denoted by τ1τ2-Cl(A). The union of all τ1τ2-open sets of X contained in A is called the τ1τ2-interior [77] of A and is denoted by τ1τ2-Int(A). Let A be a subset of a bitopological space (X, τ1, τ2). The set ∩{G | A ⊆ G and G is τ1τ2-open} is called the τ1τ2-kernel [77] of A and is denoted by τ1τ2-ker(A). Lemma 1. [77] For subsets A,B of a bitopological space (X, τ1, τ2), the following properties hold: C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 4 of 18 (1) A ⊆ τ1τ2-ker(A). (2) If A ⊆ B, then τ1τ2-ker(A) ⊆ τ1τ2-ker(B). (3) If A is τ1τ2-open, then τ1τ2-ker(A) = A. (4) x ∈ τ1τ2-ker(A) if and only if A ∩H ̸= ∅ for every τ1τ2-closed set H containing x. Lemma 2. [77] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2- closure, the following properties hold: (1) A ⊆ τ1τ2-Cl(A) and τ1τ2-Cl(τ1τ2-Cl(A)) = τ1τ2-Cl(A). (2) If A ⊆ B, then τ1τ2-Cl(A) ⊆ τ1τ2-Cl(B). (3) τ1τ2-Cl(A) is τ1τ2-closed. (4) A is τ1τ2-closed if and only if A = τ1τ2-Cl(A). (5) τ1τ2-Cl(X −A) = X − τ1τ2-Int(A). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [78] (resp. (τ1, τ2)s-open [39], (τ1, τ2)p-open [39], (τ1, τ2)β-open [39]) if A = τ1τ2-Int(τ1τ2-Cl(A)) (resp. A ⊆ τ1τ2-Cl(τ1τ2-Int(A)), A ⊆ τ1τ2-Int(τ1τ2-Cl(A)), A ⊆ τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A)))). The complement of a (τ1, τ2)r-open (resp. (τ1, τ2)s-open, (τ1, τ2)p-open, (τ1, τ2)β-open) set is said to be (τ1, τ2)r-closed (resp. (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed). A subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [79] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-open set is called α(τ1, τ2)- closed. A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-δ-open [17] if A is the union of (τ1, τ2)r-open sets of X. The complement of a τ1τ2-δ-open set is called τ1τ2-δ-closed [17]. Let A be a subset of a bitopological space (X, τ1, τ2). The union of all τ1τ2-δ-open sets of X contained in A is called the τ1τ2-δ-interior [17] of A and is denoted by τ1τ2-δ-Int(A). The intersection of all τ1τ2-δ-closed sets of X containing A is called the τ1τ2-δ-closure [17] of A and is denoted by τ1τ2-δ-Cl(A). The intersection of all (τ1, τ2)p-closed (resp. (τ1, τ2)s-closed, α(τ1, τ2)-closed) sets of X containing A is called the (τ1, τ2)p-closure [62] (resp. (τ1, τ2)s-closure [39], α(τ1, τ2)-closure [63]) of A and is denoted by (τ1, τ2)-pCl(A) (resp. (τ1, τ2)-sCl(A), α(τ1, τ2)-Cl(A)). The union of all (τ1, τ2)p-open (resp. (τ1, τ2)s-open, α(τ1, τ2)-open) sets of X contained in A is called the (τ1, τ2)p-interior [62] (resp. (τ1, τ2)s-interior [39], α(τ1, τ2)-interior [63]) of A and is denoted by (τ1, τ2)-pInt(A) (resp. (τ1, τ2)-sInt(A), α(τ1, τ2)-Int(A)). Lemma 3. For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) (τ1, τ2)-pCl(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∪A [62]; (2) (τ1, τ2)-pInt(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∩A [26]; C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 5 of 18 (3) (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∪A [39]; (4) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A [59]. Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called a s(τ1, τ2)θ-cluster point [80] of A if τ1τ2-Cl(U) ∩ A ̸= ∅ for every (τ1, τ2)s-open set U containing x. The set of all s(τ1, τ2)θ-cluster points of A is called the s(τ1, τ2)θ-closure [80] of A and is denoted by s(τ1, τ2)θ-Cl(A). A subset A of a bitopological space (X, τ1, τ2) is called s(τ1, τ2)θ-closed [80] if s(τ1, τ2)θ-Cl(A) = A. The complement of a s(τ1, τ2)θ-closed set is said to be s(τ1, τ2)θ-open [80]. The union of all s(τ1, τ2)θ-open sets of X contained in A is called the s(τ1, τ2)θ-interior [80] of A and is denoted by s(τ1, τ2)θ-Int(A). By a multifunction F : X → Y , we mean a point-to-set correspondence from X into Y , and we always assume that F (x) ̸= ∅ for all x ∈ X. For a multifunction F : X → Y , we shall denote the upper and lower inverse of a set B of Y by F+(B) and F−(B), respectively, that is, F+(B) = {x ∈ X | F (x) ⊆ B} and F−(B) = {x ∈ X | F (x) ∩ B ̸= ∅}. In particular, F−(y) = {x ∈ X | y ∈ F (x)} for each point y ∈ Y . For each A ⊆ X, F (A) = ∪x∈AF (x). 3. Upper and lower contra-(τ1, τ2)p-continuous multifunctions In this section, we introduce the concepts of upper contra-(τ1, τ2)p-continuous mul- tifunctions and lower contra-(τ1, τ2)p-continuous multifunctions. Furthermore, several characterizations of upper contra-(τ1, τ2)p-continuous multifunctions and lower contra- (τ1, τ2)p-continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called upper contra-(τ1, τ2)p- continuous at a point x ∈ X if for each σ1σ2-closed set K of Y with x ∈ F+(K), there exists a (τ1, τ2)p-open set U of X containing x such that U ⊆ F+(K). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called upper contra-(τ1, τ2)p-continuous if F is upper contra- (τ1, τ2)p-continuous at each point x of X. Theorem 1. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper contra-(τ1, τ2)p-continuous; (2) F+(K) is (τ1, τ2)p-open in X for every σ1σ2-closed set K of Y ; (3) F−(V ) is (τ1, τ2)p-closed in X for every σ1σ2-open set V of Y ; (4) for each x ∈ X and each σ1σ2-closed set K of Y containing F (x), there exists a (τ1, τ2)p-open set U of X containing x such that if y ∈ U , then F (y) ⊆ K. Proof. (1) ⇔ (2): Let K be any σ1σ2-closed set of Y and x ∈ F+(K). Since F is upper contra-(τ1, τ2)-continuous, there exists a (τ1, τ2)p-open set U of X containing x such that U ⊆ F+(K). Thus, F+(K) is (τ1, τ2)p-open in X. The converse of the proof is similar. C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 6 of 18 (2) ⇔ (3): This follows from the fact that F+(Y −B) = X − F−(B) for every subset B of Y . (1) ⇔ (4): Obvious. Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower contra-(τ1, τ2)p- continuous at a point x ∈ X if for each σ1σ2-closed set K of Y with x ∈ F−(K), there exists a (τ1, τ2)p-open set U of X containing x such that U ⊆ F−(K). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower contra-(τ1, τ2)p-continuous if F is lower contra- (τ1, τ2)p-continuous at each point x of X. Theorem 2. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower contra-(τ1, τ2)p-continuous; (2) F−(K) is (τ1, τ2)p-open in X for every σ1σ2-closed set K of Y ; (3) F+(V ) is (τ1, τ2)p-closed in X for every σ1σ2-open set V of Y ; (4) for each x ∈ X and each σ1σ2-closed set K of Y such that F (x)∩K ̸= ∅, there exists a (τ1, τ2)p-open set U of X containing x such that if y ∈ U , then F (y) ∩K ̸= ∅. Proof. The proof is similar to that of Theorem 1. Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)s-regular [6] if for each (τ1, τ2)s-closed set F and each x ̸∈ F , there exist disjoint (τ1, τ2)s-open sets U and V such that x ∈ U and F ⊆ V . Lemma 4. [81] Let (X, τ1, τ2) be a (τ1, τ2)s-regular space. Then, the following properties hold: (1) τ1τ2-Cl(A) = τ1τ2-δ-Cl(A) for every subset A of X; (2) every τ1τ2-open set is τ1τ2-δ-open. Theorem 3. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)s- regular, the following properties are equivalent: (1) F is upper contra-(τ1, τ2)p-continuous; (2) F+(σ1σ2-δ-Cl(B)) is (τ1, τ2)p-open in X for every subset B of Y ; (3) F+(K) is (τ1, τ2)p-open in X for every σ1σ2-δ-closed set K of Y ; (4) F−(V ) is (τ1, τ2)p-closed in X for every σ1σ2-δ-open set V of Y . C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 7 of 18 Proof. (1) ⇒ (2): Let B be any subset of Y . Then, σ1σ2-δ-Cl(B) is a σ1σ2-closed set of Y and by Theorem 1, F+(σ1σ2-δ-Cl(B)) is (τ1, τ2)p-open in X. (2) ⇒ (3): Let K be any σ1σ2-δ-closed set of Y . Then, σ1σ2-δ-Cl(K) = K. By (2), F+(K) is (τ1, τ2)p-open in X. (3) ⇒ (4): Let V be any σ1σ2-δ-open set of Y . Then, Y −V is σ1σ2-δ-closed in Y . By (3), F+(Y − V ) = X − F−(V ) is (τ1, τ2)p-open in X. Thus, F−(V ) is (τ1, τ2)p-closed in X. (4) ⇒ (1): Let V be any σ1σ2-open set of Y . Since (Y, σ1, σ2) is (σ1, σ2)s-regular, by Lemma 4 we have V is σ1σ2-δ-open in Y . By (4), F−(V ) is (τ1, τ2)p-closed in X and by Theorem 1, F is upper contra-(τ1, τ2)p-continuous. Theorem 4. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)s- regular, the following properties are equivalent: (1) F is lower contra-(τ1, τ2)p-continuous; (2) F−(σ1σ2-δ-Cl(B)) is (τ1, τ2)p-open in X for every subset B of Y ; (3) F−(K) is (τ1, τ2)p-open in X for every σ1σ2-δ-closed set K of Y ; (4) F+(V ) is (τ1, τ2)p-closed in X for every σ1σ2-δ-open set V of Y . Proof. The proof is similar to that of Theorem 3. 4. Upper and lower almost contra-(τ1, τ2)p-continuous multifunctions In this section, we introduce the concepts of upper almost contra-(τ1, τ2)p-continuous multifunctions and lower almost contra-(τ1, τ2)p-continuous multifunctions. Moreover, some characterizations of upper almost contra-(τ1, τ2)p-continuous multifunctions and lower almost contra-(τ1, τ2)p-continuous multifunctions are considered. Definition 3. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper almost contra-(τ1, τ2)p-continuous at a point x ∈ X if for each (σ1, σ2)r-closed set K of Y with x ∈ F+(K), there exists a (τ1, τ2)p-open set U of X containing x such that U ⊆ F+(K). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper almost contra-(τ1, τ2)p- continuous if F is upper almost contra-(τ1, τ2)p-continuous at each point x of X. Remark 1. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following implication holds: upper contra-(τ1, τ2)p-continuity ⇒ upper almost contra-(τ1, τ2)p-continuity. The converse of the implication is not true in general. We give an example for the implication as follows. C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 8 of 18 Example 1. Let X = {a, b, c, d} with topologies τ1 = {∅, {a}, {c}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}, X} and τ2 = {∅, {a}, {c}, {a, b}, {a, c}, {a, b, c}, {a, b, d}, X}. Let Y = {1, 2, 3, 4} with topolo- gies σ1 = {∅, {1}, {3}, {1, 2}, {1, 3}, {1, 2, 3}, {1, 2, 4}, Y } and σ2 = {∅, {1}, {3}, {1, 2}, {1, 3}, {1, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, Y }. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is defined as follows: F (a) = {4}, F (b) = {3}, F (c) = {1} and F (d) = {2}. Then, F is upper almost contra-(τ1, τ2)p-continuous but F is not upper contra-(τ1, τ2)p-continuous. Theorem 5. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost contra-(τ1, τ2)p-continuous; (2) F+(K) is (τ1, τ2)p-open in X for every (σ1, σ2)r-closed set K of Y ; (3) F−(V ) is (τ1, τ2)p-closed in X for every (σ1, σ2)r-open set V of Y ; (4) F−(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X for every σ1σ2-open set V of Y ; (5) F+(σ1σ2-Cl(σ1σ2-Int(K))) is (τ1, τ2)p-open in X for every σ1σ2-closed set K of Y ; (6) for each x ∈ X and each (σ1, σ2)s-open set V of Y containing F (x), there exists a (τ1, τ2)p-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ); (7) F+(V ) ⊆ (τ1, τ2)-pInt(F +(σ1σ2-Cl(V ))) for every (σ1, σ2)s-open set V of Y . Proof. (1) ⇒ (2): Let K be any (σ1, σ2)r-closed set of Y and x ∈ F+(K). Since F is upper almost contra-(τ1, τ2)p-continuous, there exists a (τ1, τ2)p-open set U of X containing x such that U ⊆ F+(K). Thus, F+(K) is (τ1, τ2)p-open in X. (2) ⇒ (1): The proof is obvious. (2) ⇔ (3) and (4) ⇔ (5): It follows from the fact that F+(Y − B) = X − F−(B) for every subset B of Y . (3) ⇔ (4): Let V be any σ1σ2-open set of Y . Since σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r- open in Y , by (3) we have F−(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X. The converse is obvious. (5) ⇔ (2): It is similar to that of (3) ⇔ (4). (6) ⇒ (7): Let V be any (σ1, σ2)s-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V . By (6), there exists a (τ1, τ2)p-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). This implies that x ∈ U ⊆ F+(σ1σ2-Cl(V )). Thus, x ∈ (τ1, τ2)-pInt(F +(σ1σ2-Cl(V ))) and hence F+(V ) ⊆ (τ1, τ2)-pInt(F +(σ1σ2-Cl(V ))). (7) ⇒ (2): Let K be any (σ1, σ2)r-closed set of Y . Then, K is (σ1, σ2)s-open in Y . By (7), we have F+(K) ⊆ (τ1, τ2)-pInt(F +(σ1σ2-Cl(K))) and hence F+(K) is (τ1, τ2)p-open in X. C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 9 of 18 (2) ⇒ (6): Let x ∈ X and V be any (σ1, σ2)s-open set of Y with F (x) ⊆ V . Since σ1σ2-Cl(V ) is (σ1, σ2)r-closed, by (2) we have F+(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X. Then, there exists a (τ1, τ2)p-open set U of X containing x such that U ⊆ F+(σ1σ2-Cl(V )). Thus, F (U) ⊆ σ1σ2-Cl(V ). Theorem 6. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost contra-(τ1, τ2)p-continuous; (2) F−(K) is (τ1, τ2)p-open in X for every (σ1, σ2)r-closed set K of Y ; (3) F+(V ) is (τ1, τ2)p-closed in X for every (σ1, σ2)r-open set V of Y ; (4) F+(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X for every σ1σ2-open set V of Y ; (5) F−(σ1σ2-Cl(σ1σ2-Int(K))) is (τ1, τ2)p-open in X for every σ1σ2-closed set K of Y ; (6) for each x ∈ X and each (σ1, σ2)s-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a (τ1, τ2)p-open set U of X containing x such that F (z) ∩ σ1σ2-Cl(V ) ̸= ∅ for each z ∈ U ; (7) F−(V ) ⊆ (τ1, τ2)-pInt(F −(σ1σ2-Cl(V ))) for every (σ1, σ2)s-open set V of Y . Proof. The proof is similar to that of Theorem 5. Theorem 7. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost contra-(τ1, τ2)p-continuous; (2) (τ1, τ2)-pCl(F −(σ1σ2-Int(K))) ⊆ F−(K) for every (σ1, σ2)s-closed set K of Y ; (3) (τ1, τ2)-pCl(F −(σ1σ2-Int((σ1, σ2)-sCl(B)))) ⊆ F−((σ1, σ2)-sCl(B)) for every subset B of Y ; (4) F+((σ1, σ2)-sInt(B)) ⊆ (τ1, τ2)-pInt(F +(σ1σ2-Cl((σ1, σ2)-sInt(B)))) for every subset B of Y . Proof. (1) ⇒ (2): LetK be any (σ1, σ2)s-closed set of Y . Then, Y −K is (σ1, σ2)s-open in Y . By Theorem 5, F+(Y −K) ⊆ (τ1, τ2)-pInt(F +(Y − σ1σ2-Int(K))). Thus, X − F−(K) ⊆ (τ1, τ2)-pInt(F +(Y − σ1σ2-Int(K))) = (τ1, τ2)-pInt(X − F−(σ1σ2-Int(K))) = X − (τ1, τ2)-pCl(F −(σ1σ2-Int(K))) and hence (τ1, τ2)-pCl(F −(σ1σ2-Int(K))) ⊆ F−(K). C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 10 of 18 (2) ⇒ (3): Let B be any subset of Y . Then, (σ1, σ2)-sCl(B)) is (σ1, σ2)s-closed in Y , by (2) we have (τ1, τ2)-pCl(F −(σ1σ2-Int((σ1, σ2)-sCl(B)))) ⊆ F−((σ1, σ2)-sCl(B)). (3) ⇒ (4): Let B be any subset of Y . By (3), we have X − F+((σ1, σ2)-sInt(B)) = F−((σ1, σ2)-sCl(Y −B)) ⊇ (τ1, τ2)-pCl(F −(σ1σ2-Int((σ1, σ2)-sCl(Y −B)))) = (τ1, τ2)-pCl(F −(σ1σ2-Int(Y − (σ1, σ2)-sInt(B)))) = (τ1, τ2)-pCl(F −(Y − σ1σ2-Cl((σ1, σ2)-sInt(B)))) = (τ1, τ2)-pCl(X − F+(σ1σ2-Cl((σ1, σ2)-sInt(B)))) = X − (τ1, τ2)-pInt(F +(σ1σ2-Cl((σ1, σ2)-sInt(B)))) and hence F+((σ1, σ2)-sInt(B)) ⊆ (τ1, τ2)-pInt(F +(σ1σ2-Cl((σ1, σ2)-sInt(B)))). (4) ⇒ (1): Let V be any (σ1, σ2)s-open set of Y . Then, V = (σ1, σ2)-sInt(V ) and by (4), F+(V ) ⊆ (τ1, τ2)-pInt(F +(σ1σ2-Cl(V ))). By Theorem 5, F is upper almost contra- (τ1, τ2)p-continuous. Theorem 8. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost contra-(τ1, τ2)p-continuous; (2) (τ1, τ2)-pCl(F +(σ1σ2-Int(K))) ⊆ F+(K) for every (σ1, σ2)s-closed set K of Y ; (3) (τ1, τ2)-pCl(F +(σ1σ2-Int((σ1, σ2)-sCl(B)))) ⊆ F+((σ1, σ2)-sCl(B)) for every subset B of Y ; (4) F−((σ1, σ2)-sInt(B)) ⊆ (τ1, τ2)-pInt(F −(σ1σ2-Cl((σ1, σ2)-sInt(B)))) for every subset B of Y . Proof. The proof is similar to that of Theorem 7. Lemma 5. [80] For a bitopological space (X, τ1, τ2), the following properties hold: (1) α(τ1, τ2)-Cl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)β-open set V of X; (2) (τ1, τ2)-pCl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)s-open set V of X; (3) (τ1, τ2)-sCl(V ) = τ1τ2-Int(τ1τ2-Cl(V )) for every (τ1, τ2)p-open set V of X. Theorem 9. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost contra-(τ1, τ2)p-continuous; (2) F−(V ) is (τ1, τ2)p-open in X for every s(σ1, σ2)θ-open set V of Y ; (3) F+(K) is (τ1, τ2)p-closed in X for every s(σ1, σ2)θ-closed set K of Y ; C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 11 of 18 (4) (τ1, τ2)-pCl(F +(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+((σ1, σ2)-sCl(B)) for every subset B of Y ; (5) (τ1, τ2)-pCl(F +(B)) ⊆ F+(s(σ1, σ2)θ-Cl(B)) for every subset B of Y ; (6) F ((τ1, τ2)-pCl(A)) ⊆ s(σ1, σ2)θ-Cl(F (A)) for every subset A of X. Proof. (1) ⇒ (2): Let V be any s(σ1, σ2)θ-open set of Y . There exists a family of (σ1, σ2)r-closed sets {Kγ | γ ∈ ∇} such that V = ∪{Kγ | γ ∈ ∇}. It follows from Theorem 5 that F−(V ) = ∪{F−(Kγ) | γ ∈ ∇} is (τ1, τ2)p-open in X. (2) ⇒ (3): The proof is obvious. (3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Int(σ1σ2-Cl(B)) is (σ1, σ2)r-open and hence σ1σ2-Int(σ1σ2-Cl(B)) is s(σ1, σ2)θ-closed in Y . By (3), F+(σ1σ2-Int(σ1σ2-Cl(B))) is (τ1, τ2)p-closed in X. Thus, (τ1, τ2)-pCl(F +(σ1σ2-Int(σ1σ2-Cl(B)))) = F+(σ1σ2-Int(σ1σ2-Cl(B))) ⊆ F+((σ1, σ2)-sCl(B)). (4) ⇒ (5): Let B be any subset of Y . For any (σ1, σ2)r-open set V of Y with B ⊆ V , by (4) and Lemma 5 we have (τ1, τ2)-pCl(F +(B)) ⊆ (τ1, τ2)-pCl(F +(V )) = (τ1, τ2)-pCl(F +(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+((σ1, σ2)-sCl(V )) = F+(V ). Thus, (τ1, τ2)-pCl(F +(B)) ⊆ F+(∩{V | V is (σ1, σ2)r-open in Y and B ⊆ V }) = F+(s(σ1, σ2)θ-Cl(B)). (5) ⇒ (1): Let V be any (σ1, σ2)s-open set of Y . By (5), we have X − (τ1, τ2)-pInt(F −(σ1σ2-Cl(V ))) = (τ1, τ2)-pCl(F +(Y − σ1σ2-Cl(V ))) ⊆ F+((σ1, σ2)-sCl(Y − σ1σ2-Cl(V ))) = F+(Y − σ1σ2-Cl(V )) = X − F−(σ1σ2-Cl(V )) and hence F−(V ) ⊆ F−(σ1σ2-Cl(V )) ⊆ (τ1, τ2)-pInt(F −(σ1σ2-Cl(V ))). By Theorem 6, F is lower almost contra-(τ1, τ2)p-continuous. (5) ⇒ (6): Let A be any subset of X and B = F (A). Then, A ⊆ F+(B) and by (5), (τ1, τ2)-pCl(A) ⊆ (τ1, τ2)-pCl(F +(B)) ⊆ F+(s(σ1, σ2)θ-Cl(B)). Thus, F ((τ1, τ2)-pCl(A)) ⊆ F (F+(s(σ1, σ2)θ-Cl(B))) C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 12 of 18 ⊆ s(σ1, σ2)θ-Cl(B) = s(σ1, σ2)θ-Cl(F (A)). (6) ⇒ (5): Let B be any subset of Y . By (6), we have F ((τ1, τ2)-pCl(F +(B))) ⊆ s(σ1, σ2)θ-Cl(F (F+(B))) ⊆ s(σ1, σ2)θ-Cl(B) and hence (τ1, τ2)-pCl(F +(B)) ⊆ F+(s(σ1, σ2)θ-Cl(B)). Theorem 10. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost contra-(τ1, τ2)p-continuous; (2) F+(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)β-open set V of Y ; (3) F+(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)s-open set V of Y ; (4) F−(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X for every (σ1, σ2)p-open set V of Y . Proof. (1) ⇒ (2): Let V be any (σ1, σ2)β-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r- closed in Y , by Theorem 5 we have F+(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X. (2) ⇒ (3): This is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open. (3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . This implies that Y − σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-closed and (σ1, σ2)s-open in Y . By (3), we have X − F−(σ1σ2-Int(σ1σ2-Cl(V ))) = F+(Y − σ1σ2-Int(σ1σ2-Cl(V ))) = F+(σ1σ2-Cl(Y − σ1σ2-Int(σ1σ2-Cl(V )))) is (τ1, τ2)p-open in X. Thus, F−(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X. (4) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y . Then, V is (σ1, σ2)p-open in Y and by (4), F−(V ) = F−(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X. Thus by Theorem 5, F is upper almost contra-(τ1, τ2)p-continuous. Theorem 11. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost contra-(τ1, τ2)p-continuous; (2) F−(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)β-open set V of Y ; (3) F−(σ1σ2-Cl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)s-open set V of Y ; C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 13 of 18 (4) F+(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X for every (σ1, σ2)p-open set V of Y . Proof. The proof is similar to that of Theorem 10. Corollary 1. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost contra-(τ1, τ2)p-continuous; (2) F+(α(σ1, σ2)-Cl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)β-open set V of Y ; (3) F+((σ1, σ2)-pCl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)s-open set V of Y ; (4) F−((σ1, σ2)-sCl(V )) is (τ1, τ2)p-closed in X for every (σ1, σ2)p-open set V of Y . Corollary 2. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost contra-(τ1, τ2)p-continuous; (2) F−(α(σ1, σ2)-Cl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)β-open set V of Y ; (3) F−((σ1, σ2)-pCl(V )) is (τ1, τ2)p-open in X for every (σ1, σ2)s-open set V of Y ; (4) F+((σ1, σ2)-sCl(V )) is (τ1, τ2)p-closed in X for every (σ1, σ2)p-open set V of Y . Theorem 12. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost contra-(τ1, τ2)p-continuous; (2) (τ1, τ2)-pCl(F −(V )) ⊆ F−(σ1σ2-Int(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) (τ1, τ2)-pCl(F −(V )) ⊆ F−((σ1, σ2)-sCl(V )) for every σ1σ2-open set V of Y . Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y . Then, σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-open in Y . Thus by (1), F−(σ1σ2-Int(σ1σ2-Cl(V ))) is (τ1, τ2)p-closed in X. Since V ⊆ σ1σ2-Int(σ1σ2-Cl(V )), we have F−(V ) ⊆ F−(σ1σ2-Int(σ1σ2-Cl(V ))) and hence (τ1, τ2)-pCl(F −(V )) ⊆ F−(σ1σ2-Int(σ1σ2-Cl(V ))). (2) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y . By (2), we have (τ1, τ2)-pCl(F −(V )) ⊆ F−(σ1σ2-Int(σ1σ2-Cl(V ))) = F−(V ) and hence F−(V ) is (τ1, τ2)p-closed in X, by Theorem 5 we have F is upper almost contra-(τ1, τ2)p-continuous. (2) ⇔ (3): It follows from Lemma 5. C. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (2) (2025), 6009 14 of 18 Theorem 13. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost contra-(τ1, τ2)p-continuous; (2) (τ1, τ2)-pCl(F +(V )) ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ; (3) (τ1, τ2)-pCl(F +(V )) ⊆ F+((σ1, σ2)-sCl(V )) for every σ1σ2-open set V of Y . Proof. The proof is similar to that of Theorem 12. Theorem 14. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction. If (τ1, τ2)-pCl(F −(B)) ⊆ F−(σ1σ2-ker(B)) for every subset B of Y , then F is upper almost contra-(τ1, τ2)p-continuous. Proof. Let V be any (σ1, σ2)r-open set of Y . By Lemma 1, we have (τ1, τ2)-pCl(F −(V )) ⊆ F−(σ1σ2-ker(V )) = F−(V ) and hence F−(V ) is (τ1, τ2)p-closed in X. Thus by Theorem 5, F is upper almost contra- (τ1, τ2)p-continuous. Theorem 15. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction. If F ((τ1, τ2)-pCl(A)) ⊆ σ1σ2-ker(F (A)) for every subset A of X, then F is lower almost contra-(τ1, τ2)p-continuous. Proof. Let V be any (σ1, σ2)r-open set of Y . This implies that F ((τ1, τ2)-pCl(F +(V ))) ⊆ σ1σ2-ker(V ) and hence (τ1, τ2)-pCl(F +(V )) ⊆ F+(σ1σ2-ker(V )). By Lemma 1, we have (τ1, τ2)-pCl(F +(V )) ⊆ F+(σ1σ2-ker(V )) = F+(V ) and so F+(V ) is τ1τ2-closed in X. By Theorem 6, F is lower almost contra-(τ1, τ2)p- continuous. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] C. Boonpok and J. Khampakdee. 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