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

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

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

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

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

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

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

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

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