Volume 70 , issue 2 ( 2018 ) | back |

Strongly $I$ and $I^*$-statistically pre-Cauchy double sequences in probabilistic metric spaces | 97$-$109 |

**Abstract**

In this paper we consider the notion of strongly $I$-statistically pre-Cauchy double sequences in probabilistic metric spaces in line of Das et.\ al.\ (\emph{On $I$-statistically pre-Cauchy sequences}, Taiwanese J. Math 18 (1) (2014), 115--126) and introduce the new concept of strongly $I^*$-statistically pre-Cauchy double sequences in probabilistic metric spaces. We mainly study interrelationship among strong $I$-statistical convergence, strong $I$-statistical pre-Cauchy condition and strong $I^*$-statistical pre-Cauchy condition for double sequences in probabilistic metric spaces and examine some basic properties of these notions.

**Keywords:** Probabilistic metric space; strong $I$-statistical convergence; strong $I$-statistical pre-Cauchy condition; strong $I^*$-statistical pre-Cauchy condition.

**MSC:** 54E70, 40B05

Groups of generalized isotopies and generalized $G$-spaces | 110$-$119 |

**Abstract**

The group of generalized isotopies of topological space is studied. A relationship of this group with the group of homeomorphisms is established in case of locally compact and locally connected space. Notions of generalized $G$-spaces and there equivariant maps are introduced. It is proved that a new category of generalized $G$-spaces is a natural extension of the category of $G$-spaces.

**Keywords:** Compact-open topology; group of homeomorphisms; isotopy; group of generalized isotopies; generalized $G$-space.

**MSC:** 54H15, 57S99

Some calculus of the composition of functions in Besov-type spaces | 120$-$133 |

**Abstract**

In the Besov-type spaces $B^{s,\tau}_{p,q}(R^n)$, we will prove that the composition operator $T_f: g \to f \circ g$ takes both $B^{s}_{\infty,q}(R^n)\cap B^{s,\tau}_{p,q}(R^n)$ and $W^1_{\infty}(R^n)\cap B^{s,\tau}_{p,q}(R^n)$ to $B^{s,\tau}_{p,q}(R^n)$, under some restrictions on $s, \tau, p,q$, and if the real function $f$ vanishes at the origin and belongs locally to $B^{s+1}_{\infty, q}({R})$.

**Keywords:** Besov spaces; Besov-type spaces; Littlewood-Paley decomposition; composition operator.

**MSC:** 46E35

Curvatures of tangent bundle of Finsler manifold with Cheeger-Gromoll metric | 134$-$146 |

**Abstract**

Let $(M,F)$ be a Finsler manifold and $G$ be the Cheeger-Gromoll metric induced by $F$ on the slit tangent bundle $\widetilde{TM}=TM\backslash 0$. In this paper, we will prove that the Finsler manifold $(M,F)$ is of scalar flag curvature $K=\alpha$ if and only if the unit horizontal Liouville vector field $\xi=\frac{y^i}{F}\frac{\delta}{\delta x^i}$ is a Killing vector field on the indicatrix bundle $IM$ where $\alpha: TM\rightarrow R$ is defined by $\alpha(x,y)=1+g_x(y,y)$. Also, we will calculate the scalar curvature of a tangent bundle equipped with Cheeger-Gromoll metric and obtain some conditions for the scalar curvature to be a positively homogeneous function of degree zero with respect to the fiber coordinates of $\widetilde{TM}$.

**Keywords:** Besov spaces; Besov-type spaces; Littlewood-Paley decomposition; composition operator.

**MSC:** 53C60, 53C07, 53C15

A study on elliptic PDE involving the $p$-harmonic and the $p$-biharmonic operators with steep potential well | 147$-$154 |

**Abstract**

In this paper, we give an existence result pertaining to a nontrivial solution to the problem $\Delta^2_p u -\Delta_p u + \lambda V(x)|u|^{p-2}u = f(x,u)\,,\,x\in R^N,\ u \in W^{2,p}(R^N)$, where $p>1$, $\lambda>0$, $V\in C(R^N,R^+)$, $f\in C(R^N \times R,R)$, $N>2p$. We also explore the problem in the limiting case of $\lambda \rightarrow \infty$.

**Keywords:** $p$-Laplacian; $p$-biharmonic; elliptic PDE; Sobolev space.

**MSC:** 35J35, 35J60, 35J92

A note on some operators acting on central Morrey spaces | 155$-$160 |

**Abstract**

We prove boundedness of maximal commutators and convolution operators with generalized Poisson kernels on central Morrey spaces.

**Keywords:** Morrey space; maximal commutator; Weinstein transform.

**MSC:** 42B35, 26D10, 44A35

Strong linear preservers of ut-Toeplitz weak majorization on $\mathbb{R}^{n}$ | 161$-$166 |

**Abstract**

Let $x, y \in {R}^{n}$, we say $x$ is ut-Toeplitz weak majorized by $y$ (written as $x\prec_ {uT}y$) if there exists an upper triangular substochastic Toeplitz matrix $A$ such that $x=Ay$. In this paper, we characterize all linear functions that strongly preserve $\prec_ {uT}$ on ${R}^n$.

**Keywords:** Substochastic matrix; ut-Toeplitz weak majorization; linear preserver.

**MSC:** 15A04, 15A21

Some remarks on TAC-contractive mappings in $b$-metric spaces | 167$-$175 |

**Abstract**

In this paper we complement and improve fixed point results established on TAC-contractive mappings in the framework of $b$-metric spaces. Our analysis starts by a recent paper of Babu and Dula [G. V. R. Babu and T. M. Dula, Fixed points of generalized TAC-contractive mappings in $b$-metric spaces, Matematički Vesnik 69 (2017), no. 2, 75--88].

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

**MSC:** 47H10, 54C30, 54H25

A note on $IA$-automorphisms of a finite $p$-group | 176$-$182 |

**Abstract**

Let $G$ be a finite group. An automorphism $\alpha$ of $G$ is called an $IA$-auto\-mor\-phism if $x^{-1}x^{\alpha}\in G'$ for all $x\in G$. The set of all $IA$-automorphisms of $G$ is denoted by $Aut^{G'}(G)$. A group $G$ is called semicomplete if and only if $Aut^{G'}(G)=Inn(G)$. In this paper, we obtain certain conditions on a finite $p$-group to be semicomplete.

**Keywords:** Automorphism group; $p$-group; semicomplete group.

**MSC:** 20D45, 20D15

Liouville theorem on conformal mappings of domains in multidimensional Euclidean and Pseudoeuclidean spaces | 183$-$188 |

**Abstract**

Everybody who attended a course in complex analysis, knows Riemann Theorem on conformal mappings, demonstrating conformal flexibility of domains in the two-dimensional plane (more generally, in a two-dimensional surface). In contrast to the plane case, domains in spaces of dimension greater than two are conformally rigid. This is the content of a (less popular) Liouville theorem, which appeared almost in the same time as the mentioned Riemann theorem. Here we present one of the possible proofs of this theorem together with a contemporary bibliography containing new approaches to this theorem together with its generalizations and extensions.

**Keywords:** Quasiconformal mapping; conformal rigidity.

**MSC:** 30C65