Upper and Lower Continuity of Soft Multifunctions

Akdağ, Metin; Erol, Fethullah

doi:10.12785/amis/080624

Cited by 6 publications

(6 citation statements)

References 34 publications

(56 reference statements)

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“…Ça g  man and et al redefined the operations of the soft sets and constructed a uni-int decision making method by using these new operations [7]. Then Akda g  and Erol [3,11,12] introduced the concept of soft multifunction and studied their properties. Many researcher studied on soft set theory [6,9,10] etc.…”

Section: Introductionmentioning

confidence: 99%

Upper and Lower Soft Contra Continuous Multifunctions

Akdağ

2016

CFD

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Abstract. In this paper, we introduce and study upper (lower) soft contra continuous multifunctions. Some characterizations and several properties concerning upper (lower) soft contra continuous multifunctions are obtained..The relationships between upper (lower) soft continuous multifunction and some known concepts are also discussed.. Keywords:Multifunction, contra-continuous multifunction, soft continuous multifunction, soft topological spaces. Esnek Küme Değerli Fonksiyonların Alttan ve Üstten Contra SüreklilikleriÖzet. Bu makalede, öncelikle esnek küme değerli dönüşümlerin alttan ve üstten kontra sürekliliği tanımlanmıştır. Sonra bu sürekliliğin bazı karakterizasyonları verilerek çeşitli özellikleri incelenmiştir. Sonunda ise esnek sürekli küme değerli dönüşüm ile arasındaki ilişki araştırılmıştır.Anahtar Kelimeler: Küme-değerli fonksiyon, contra-sürekli küme-değerli fonksiyon, esnek süreklilik, esnek küme düğerli fonksiyon, esnek topolojik uzay.

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Section: Introductionmentioning

confidence: 99%

Upper and Lower Soft Contra Continuous Multifunctions

Akdağ

2016

CFD

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“…Ç agman and et al redefined the operations of the soft sets and constructed a uni-int decision making method by using these new operations [7]. Later on, Akdag and Erol [3,11] introduced the concept of soft multifunction and studied their properties. Many researcher studied on soft set theory (see [9,10]).…”

Section: Introductionmentioning

confidence: 99%

Remarks on hyperspaces of soft sets

Erol

Akdağ

2015

J. of Adv. Stud. in Topol.

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“…It is well known today, that the notion of multifunction is playing a very important role in general topology, upper and lower continuity have been extensively studied on multifunctions F : (X, τ ) → (Y, σ). Currently using the notion of topological ideal introduced by Kuratowski [24], different types of upper and lower continuity in multifunction F : (X, τ, I) → (Y, σ) have been studied and characterized [1], [14], [15], [34], [37], [33], [30], [31], [10], [28], [13]. Kasahara in [21], introduced the concept of an operator on a topology τ on a set X as a map α : τ → P (X) such that U ⊆ α(U ) for all U ∈ τ .…”

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confidence: 99%

“…The notion of upper (resp. lower)-I-continuous multifunction [1], when we choose α = identity operator, β = I-interior operator, δ = identity operator, θ = identity operator, and I = {∅}.…”

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confidence: 99%

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Upper and lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions

et al. 2021

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Let $(X, \tau)$ and $(Y, \sigma)$ be topological spaces in which no separation axioms are assumed, unless explicitly stated and if $\mathcal{I}$ is an ideal on $X$.Given a multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$, $\alpha,\beta$ operators on $(X, \tau)$, $\theta,\delta$ operators on $(Y, \sigma)$ and $\mathcal{I}$ a proper ideal on $X$. We introduce and study upper and lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions.A multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$ is said to be: {1)} upper-$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous if $\alpha(F^{+}(\delta(V)))\setminus \beta(F^{+}(\theta(V)))\in \mathcal{I}$ for each open subset $V$ of $Y$;\{2)} lower-$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous if$\alpha(F^{-}(\delta(V)))\setminus \beta(F^{-}(\theta(V)))\in \mathcal{I}$ for each open subset $V$ of $Y$;\ {3)} $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous if it is upper-\ %$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuousand lower-$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous. In particular, the following statements are proved in the article (Theorem 2):Let $\alpha,\beta$ be operators on $(X, \tau)$ and $\theta, \theta^{*}, \delta$ operators on $(Y, \sigma)$: \noi\ \ {1.} The multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$ is upper $(\alpha,\beta,\theta\cap \theta^{*},\delta,\mathcal{I})$-continuous if and only if it is both upper $(\alpha,\beta,\theta,\delta,\mathcal{I})$-continuous and upper $(\alpha,\beta,\theta^{*},\delta,\mathcal{I})$-continuous. \noi\ \ {2.} The multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$ is lower $(\alpha,\beta,\theta\cap \theta^{*},\delta,\mathcal{I})$-continuous if and only if it is both lower $(\alpha,\beta,\theta,\delta,\mathcal{I})$-continuous and lower $(\alpha,\beta,\theta^{*},\delta,\mathcal{I})$-continuous,provided that $\beta(A\cap B) =\beta(A)\cap \beta(B)$ for any subset $A,B$ of $X$.

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Upper and Lower Continuity of Soft Multifunctions

Abstract: In this paper, we introduced the upper and lower α-continuous soft multifunctions. Also we obtain some characterizations and several properties concerning upper and lower α-continuous soft multifunctions.

Cited by 6 publications

References 34 publications

Upper and Lower Soft Contra Continuous Multifunctions

Upper and Lower Soft Contra Continuous Multifunctions

Remarks on hyperspaces of soft sets

Upper and lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions

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