Volume 69 , issue 2 ( 2017 ) | back |

Fixed points of generalized $TAC$-contractive mappings in $b$-metric spaces | 75$-$88 |

**Abstract**

We introduce generalized $TAC$-contractive mappings in $b$-metric spaces and we prove some new fixed point results for this class of mappings. We provide examples in support of our results. Our results extend the results of [S. Chandok, K. Tas and A. H. Ansari, Some fixed point results for $TAC$-type contractive mappings, J. Function Spaces, Vol. 2016, Article ID 1907676, 6 pages] from the metric space setting to $b$-metric spaces and generalize a result of [D. Djorić, Common fixed point for generalized $(\psi,\varphi)$-weak contractions, Appl. Math. Lett. 22 (2009) 1896--1900].

**Keywords:** $b$-metric space; cyclic $(\alpha,\beta)$-admissible mapping; generalized $TAC$-contractive
mapping; fixed point.

**MSC:** 47H10, 54H25

On the contact and co-contact of higher order | 89$-$100 |

**Abstract**

The unifying methodologies are based on the construction of `bridges' connecting distinct mathematical theories with each other. The purpose of this paper is to study the relationship between the geo\-me\-tric and algebraic formulation of completely integrable systems of order $k$ and dimension $n$ over a differentiable manifold, in terms of contact $C^{k,n}M$ and co-contact $(C^{k,n}M)^0$ of higher order, as seen in [A. Morimoto, {Prolongation of Geometric Structures}, Math. Inst. Nagoya University, Nagoya, (1969)], to establish an equivalence between both formulations.

**Keywords:** geometric structures on manifolds; differential systems; contact theory; co-contact of higher order

**MSC:** 53C15, 53B25

Cubic symmetric graphs of order $6p^{3}$ | 101$-$117 |

**Abstract**

A graph is called $s$-regular if its automorphism group acts regularly on the set of its $s$-arcs. In this paper, we classify all connected cubic $s$-regular graphs of order $6p^3$ for each $s\geq1$ and all primes $p$.

**Keywords:** symmetric graphs; $s$-regular graphs; regular coverings.

**MSC:** 05C10, 05C25

On relative Gorenstein homological dimensions with respect to a dualizing module | 118$-$125 |

**Abstract**

Let $R$ be a commutative Noetherian ring. The aim of this paper is studying the properties of relative Gorenstein modules with respect to a dualizing module. It is shown that every quotient of an injective module is $G_{C}$-injective, where $C$ is a dualizing $R$-module with $id_{R}(C) \leq 1$. We also prove that if $C$ is a dualizing module for a local integral domain, then every $G_{C}$-injective $R$-module is divisible. In addition, we give a characterization of dualizing modules via relative Gorenstein homological dimensions with respect to a semidualizing module.

**Keywords:** semidualizing; dualizing; $C$-injective; $G_{C}$-injective.

**MSC:** 13D05, 13D45, 18G20

Towards Cantor intersection theorem and Baire category theorem in partial metric spaces | 126$-$132 |

**Abstract**

In this paper we consider a suitable definition of convergence and introduce star closed sets that enable us to establish a variant of Cantor intersection theorem as well as Baire category theorem in partial metric spaces.

**Keywords:** partial metric space; star closed sets; second category.

**MSC:** 54E50, 54E52

On generalizations of Boehmian space and Hartley transform | 133$-$143 |

**Abstract**

Boehmians are quotients of sequences which are constructed by using a set of axioms. In particular, one of these axioms states that the set $S$ from which the denominator sequences are formed should be a commutative semigroup with respect to a binary operation. In this paper, we introduce a generalization of abstract Boehmian space, called generalized Boehmian space or $G$-Boehmian space, in which $S$ is not necessarily a commutative semigroup. Next, we provide an example of a $G$-Boehmian space and we discuss an extension of the Hartley transform on it.

**Keywords:** Bohemians; convolution; Hartley transform.

**MSC:** 44A15, 44A35, 44A40

A note on convergence of double sequences in a topological space | 144$-$152 |

**Abstract**

In this paper we have shown that a double sequence in a topological space satisfies certain conditions which in turn are capable to generate a topology on a nonempty set. Also we have used the idea of $I$-convergence of double sequences to study the idea of $I$-sequential compactness in the sense of double sequences [A.K. Banerjee, A. Banerjee, A note on $I$-convergence and $I^{*}$-convergence of sequences and nets in a topological space, Mat. Vesnik 67, 3 (2015), 212--221].

**Keywords:** double sequence; $d$-limit space; ${I}$-convergence; ${I}$-limit point; ${I}$-cluster point; ${I}$-sequential compactness.

**MSC:** 54A20, 40A35, 40A05