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

C. Carpintero; E.  Rosas; J.  Sanabria; J. Vielma

doi:10.30970/ms.55.2.206-213

C. Carpintero Corporación Universitaria del Caribe-CECAR, Sincelejo, Colombia and, Departamento de Matemáticas, Universidad De Oriente, Cumaná, Venezuela
E. Rosas Departamento de Matemáticas, Universidad De Oriente, Cumaná, Venezuela and, Departamento de Ciencias Naturales y Exactas, Universidad de la Costa, Barranquilla, Colombia
J. Sanabria Departamento de Matemáticas, Universidad De Sucre, Sincelejo, Colombia
J. Vielma Escuela Superior Politécnica del Litoral, Facultad de Ciencias Naturales y Matemáticas, Guayaquil

DOI: https://doi.org/10.30970/ms.55.2.206-213

Keywords: (α, β, θ, δ, I)-continuous multifunctions, P-continuous multifunctions.

Abstract

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-\ %$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous
and 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 $X$.

References

M. Akdag, On upper and lower I-continuos multifunctions, Far East J. Math. Sci., 25 (2007), No1, 49–57.

A. Al-Omari, M.S.M. Noorani, Contra-I-continuous and almost I-continuous functions, Int. J. Math. Math. Sci., 9 (2007), 169–179.

C. Arivazhagi, N. Rajesh, On upper and lower weakly I-continuous multifunctions, Ital. J. Pure Appl. Math., 36 (2016), 899–912.

C. Arivazhagi, N. Rajesh, On upper and lower I-nontinuous multifunctions, Bol. Soc. Paran. Mat., 36 (2018), No2, 99–106.

C. Arivazhagi, N. Rajesh, Nearly I-continuous multifunctions, Bol. Soc. Paran. Mat., 37 (2019), No11, 33–38.

C. Arivazhagi, N. Rajesh, S. Shanthi, On upper and lower almost contra-I-continuous multifunctions, Int. J. Pure Appl. Math., 115 (2017), No4, 787–799.

C. Arivazhagi, N. Rajesh, On upper and lower almost I-continuous multifunctions, submitted.

C. Arivazhagi, N. Rajesh, On slightly I-continuous multifunctions, Journal of New Results in Science, 11 (2016), 17–23.

D. Carnahan, Locally nearly compact spaces, Boll. Un. mat. Ital., 4 (1972), No6, 143–153.

C. Carpintero, J. Pacheco, N. Rajesh, E. Rosas, S. Saranyasri, Properties of nearly ω-continuous multifunctions, Acta Univ. Sapientiae Math., 9 (2017), No1, 13–25.

C. Carpintero, E. Rosas, J. Moreno, More on upper and lower almost nearly I-continuous multifunctions, Int. J. Pure Appl. Math., 117 (2017), No3, 521–537.

C. Carpintero, N. Rajesn, E. Rosas, S. Saranyasri, On upper and lower contra ω-continuous multifunctions, Novi Sad J. Math., 44 (2014), No1, 143–151.

C. Carpintero, N. Rajesn, E. Rosas, S. Saranyasri, On upper and lower faintly ω-continuous multifunctions, Bol. Mat., 21 (2014), No1, 1–8.

E. Ekici, Nearly continuous multifunctions, Acta Math. Univ. Comenianae, 72 (2003), 229–235.

E. Ekici, Almost nearly continuous multifunctions, Acta Math. Univ. Comenianae, 73 (2004), 175–186.

E. Ekici, S. Jafari, V. Popa, On Almost contra-continuous multifunctions, Lobachevskii J. Math., 30 (2009), No2, 124–131.

E. Ekici, S. Jafari, T. Noiri, On upper and lower contra-continuous multifunctions, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), Tomul LIV, 2008, 75–85.

D.S. Jankovic, T.R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), No4, 295–310.

N. Levine, Semi open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36-41.

A. Kambir, I. J. Reilly, On almost l-continuous multifunctions, Hacet. J. Math. Stat., 35 (2005), No2, 181–188.

S. Kasahara, Operation compact spaces, Math. Japonica, 24 (1966), 97–105.

J. K. Kolii, C.P. Arya, Strongly and perfectly continuous multifunctions, Sci. Stud. Res. Ser. Math. Inform., 20 (2010), No1, 103–118.

J.K. Kohli, C.P. Arya, Upper and lower (almost) completely continuous multifunctions, preprint.

K. Kuratowski, Topology, Academic Press, New York, 1966.

A.S. Mashhour, M.E. Abd El-Monsef, El-Deep on precontinuous and weak precontinuous mappings, Proced. Phys. Soc. Egypt, 53 (1982), 47–53.

T. Noiri, V. Popa, On upper and lower M -continuos multifunctions, Filomat, 14 (2000), 73–86.

T. Noiri, V. Popa, Slightly m-continuos multifunctions, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 1 (2006), No4, 485–505.

T. Noiri, V. Popa, Almost weakly continuos multifunctions, Demonstrat. Math., 22 (1993), No2, 362–380.

V. Popa, Some properties of H-almost continuous multifunctions, Chaos Solitons Fractals, 12 (2000), 2387–2394.

E. Rosas, C. Carpintero, J. Moreno, Upper and lower (I, J) continuous multifunctions, Int. J. Pure Appl. Math., 117 (2017), No1, 87–97.

E. Rosas, C. Carpintero, J. Sanabria, Weakly (I, J) continuous multifunctions and contra (I, J) continuous multifunctions, Ital. J. Pure Appl. Math., 41 (2019), 547–556.

E. Rosas, C. Carpintero, J. Moreno, Upper and lower (I,J)-continuous multifunctions, Int. J. Pure Appl. Math., 117 (2017), No1.

E. Rosas, C. Carpintero, J. Sanabria, Almost contra (I, J)-continuous multifunctions, Novi Sad J. Math. on line March 28, 2019.

R.E. Simithson, Almost and weak continuity for multifunctions, Bull. Calcutta Math. Soc., 70 (1978), 383–390.

M. Stone, Applications of the theory of boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 374–381.

J. Vielma, E. Rosas, (α, β, θ, δ, I)-continuous mappings and their decomposition, Divulgaciones Matematicas, 12 (2004), No1, 53–64.

I. Zorlutuna, I-continuous multifunctions, Filomat, 27 (2013), No1, 155–162.