Volume 73 , issue 4 ( 2021 ) | back |

AN EXTENSION OF THE CONCEPT OF $\boldsymbol\gamma$-CONTINUITY FOR MULTIFUNCTIONS | 223--242 |

**Abstract**

A function $f:(X,\tau) \rightarrow (Y,\tau^{\ast})$ between topological spaces is called $\gamma$-continuous if $f^{-1}(W) \subset Cl(Int(f^{-1}(W))) \cup Int(Cl(f^{-1}(W)))$ for each open $W \subset Y$, where $Cl$ (resp. $Int$ ) denotes the closure (resp. interior) operator on X. When we use the other possible operators obtained by multiple composing $Cl$ and $Int$, then this condition boils down to the definitions of known types of generalized continuity. The case of multifunctions is quite different. The appropriate condition have two forms: $F^{+}(W) \subset Cl(Int(F^{+}(W))) \cup Int(Cl(F^{+}(W)))$ called $u.\gamma .c.$ or, $F^{-}(W) \subset Cl(Int(F^{-}(W))) \cup Int(Cl(F^{-}(W)))$ called $l.\gamma .c.$, where F$^{+}$(W) = $\left\{x \in X: F(x) \subset W \right\}$ and F$^{-}$(W) = $\left\{x \in X: F(x) \cap W \neq\emptyset\right\}$. So, one can consider the simultaneous use of the two different inverse images namely, $F^{+}(W)$ and $F^{-}(W)$. We will show that in this case the usage of all possible multiple compositions of $Cl$ and $Int$ leads to the new different types of continuity for multifunctions, which together with the previous defined types of continuity forms a collection which is complete in a certain topological sense.

**Keywords:** Multifunction; upper semi continuity; quasi-continuity; $\gamma$-continuity.

**MSC:** 54C05, 54C08, 54C60, 54A05, 58C07

ON JET LIKE BUNDLES OF VECTOR BUNDLES | 243--255 |

**Abstract**

We describe completely the so called jet like functors of a vector bundle $E$ over an $m$-dimensional manifold $M$, i.e.\ bundles $FE$ over $M$ canonically depending on $E$ such that $F(E_1\times_M E_2)=FE_1\times_MFE_2$ for any vector bundles $E_1$ and $E_2$ over $M$. Then we study how a linear vector field on $E$ can induce canonically a vector field on $FE$.

**Keywords:** Bundle functor; gauge bundle functor; natural transformation; (gauge) natural operator; vector bundle;
module bundle; jet.

**MSC:** 58A05, 58A20, 58A32

CONICS FROM THE ADJOINT REPRESENTATION OF $SU(2)$ | 256--267 |

**Abstract**

The aim of this paper is to introduce and study the class of conics provided by the symmetric matrices of the adjoint representation of the Lie group $SU(2)=S^3$. This class depends on three real parameters as components of a point of sphere $S^2$ and various relationships between these parameters give special subclasses of conics. A symmetric matrix inspired by one giving by Barning as Pythagorean triple preserving matrix and associated hyperbola are carefully analyzed. We extend this latter hyperbola to a class of hyperbolas with integral coefficients. A complex approach is also included.

**Keywords:** Conic; adjoint representation of $SU(2)$; complex variable.

**MSC:** 11D09, 51N20, 30C10, 22E47

AN INTRODUCTION TO $\mathfrak{U}$-METRIC SPACE AND NON-LINEAR CONTRACTION WITH APPLICATION TO THE STABILITY OF FIXED POINT EQUATION | 268--281 |

**Abstract**

In this paper, we introduce the notion of $U$-metric space of $n$-tuples which generalizes several known metric-type spaces. We study topological properties of such newly constructed spaces and prove Cantor's intersection-like theorem. Banach contraction principle theorem is proved in this space and we apply the theorem to obtain the stability of a fixed point equation.

**Keywords:** $\mathfrak{U}$-metric space; Cantor's intersection like theorem; fixed point; stability of fixed point equation.

**MSC:** 47H10, 54H25

CONFORMAL YAMABE SOLITON AND $*$-YAMABE SOLITON WITH TORSE FORMING POTENTIAL VECTOR FIELD | 282--292 |

**Abstract**

The goal of this paper is to study conformal Yamabe soliton and $*$-Yamabe soliton, whose potential vector field is torse forming. Here, we have characterized conformal Yamabe soliton admitting potential vector field as torse forming with respect to Riemannian connection, semi-symmetric metric connection and projective semi-symmetric connection on Riemannian manifold. We have also shown the nature of $*$-Yamabe soliton with torse forming vector field on Riemannian manifold admitting Riemannian connection. Lastly we have developed an example to corroborate some theorems regarding Riemannian connection on Riemannian manifold.

**Keywords:** Yamabe soliton; conformal Yamabe soliton; $*$-Yamabe soliton; torse-forming vector field; torqued vector field;
semisymmetric metric connection; projective semisymmetric connection.

**MSC:** 53C21, 53C25, 53B50, 35Q51