arXiv subjects

Ruslan Salimov

Publications and source records attributed to Ruslan Salimov.

At least 19 recordsLinked to original sources

arXiv2026

Arithmetic-geometric mean, additive, and multiplicative contractions: New generalizations of the Banach contraction principle

We introduce new contraction conditions based on classical inequality between arithmetic and geometric means. By incorporating an auxiliary semimetric $\delta$, we define arithmetic-geometric mean, multiplicative-type, and additive-type contractions. Connections between these types of contractions are found. Fixed point theorems are proved in the case of continuity of the above mentioned contractions. Under suitable regularity conditions on $\delta$ (such as being d-regular, strongly d-regular, or d-lower bounded) we obtain constructive corollaries. Various examples demonstrating our results are constructed. It is shown that with certain caveats fixed point theorem for additive-type mappings is equivalent to the fixed point theorem for perturbed metric spaces, which were recently introduced by M. Jleli and B. Samet.

math.GN↗

arXiv2025

On quasisymmetric mappings between ultrametric spaces

In 1980 P. Tukia and J. Väisälä in seminal paper [P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn., Ser. A I, Math. 5, 97--114 (1980)] extended a concept of quasisymmetric mapping known from the theory of quasiconformal mappings to the case of general metric spaces. They also found an estimation for the ratio of diameters of two subsets which are images of two bounded subsets of a metric space under a quasisymmetric mapping. We improve this estimation for the case of ultrametric spaces. It was also shown that the image of an ultrametric space under an $η$-quasisymmetric mapping with $η(1)=1$ is again an ultrametric space. In the case of finite ultrametric spaces it is proved that such mappings are ball-preserving.

math.GN↗

arXiv2025

On generalizations of some fixed point theorems in semimetric spaces with triangle functions

In the present paper, we prove generalizations of Banach, Kannan, Chatterjea, Ćirić-Reich-Rus fixed point theorems, as well as of the fixed point theorem for mappings contracting perimeters of triangles. We consider corresponding mappings in semimetric spaces with triangle functions introduced by M. Bessenyei and Z. Páles. Such an approach allows us to derive corollaries for various types of semimetric spaces including metric spaces, ultrametric spaces, b-metric spaces etc. The significance of these generalized theorems extends across multiple disciplines, including optimization, mathematical modeling, and computer science. They may serve to establish stability conditions, demonstrate the existence of optimal solutions, and improve algorithm design.

math.GN↗

arXiv2024

On asymptotics of ring $Q$-homeomorphisms with respect to $p$-modulus near the origin

We consider the class of ring $Q$-homeomorphisms with respect to $p$-modulus in $\mathbb{R}^{n}$ with $p > n$, and obtain lower bounds for limsups of the distance distortions under such mappings. These estimates can be treated as Hölder's continuity of the inverses near the origin. The sharpness is illustrated by example

math.CV↗

arXiv2024

On quasisymmetric mappings in semimetric spaces

The class of quasisymmetric mappings on the real axis was first introduced by A. Beurling and L. V. Ahlfors in 1956. In 1980 P. Tukia and J. Väisälä considered these mappings between general metric spaces. In our paper we generalize the concept of quasisymmetric mappings to the case of general semimetric spaces and study some properties of these mappings. In particular, conditions under which quasisymmetric mappings preserve triangle functions, Ptolemy's inequality and the relation ``to lie between'' are found. Considering quasisymmetric mappings between semimetric spaces with different triangle functions we have found a new estimation for the ratio of diameters of two subsets, which are images of two bounded subsets. This result generalizes the well-known Tukia-Väisälä inequality. Moreover, we study connections between quasisymmetric mappings and weak similarities which are a special class of mappings between semimetric spaces.

math.GN↗

arXiv2024

Asymptotic behavior of solutions of the nonlinear Beltrami equation with the Jacobian

We investigate the asymptotic behavior at infinity of regular homeomorphic solutions of the nonlinear Beltrami equation with the Jacobian on the right-hand side. The sharpness of the above bounds is illustrated by several examples.

math.AP↗

arXiv2024

On three-point generalizations of Banach and Edelstein fixed point theorems

Let $X$ be a metric space. Recently in~[1] it was considered a new type of mappings $T\colon X\to X$ which can be characterized as mappings contracting perimeters of triangles. These mappings are defined by the condition based on the mapping of three points of the space instead of two, as it is adopted in many fixed-point theorems. In the present paper we consider so-called $(F,G)$-contracting mappings, which form a more general class of mappings than mappings contracting perimeters of triangles. The fixed-point theorem for these mappings is proved. We prove also a fixed-point theorem for mappings contracting perimeters of triangles in the sense of Edelstein.

math.GN↗

arXiv2024

Refined geometric characterizations of weak $p$-quasiconformal mappings

In this paper we consider refined geometric characterizations of weak $p$-quasiconformal mappings $φ:Ω\to\widetildeΩ$, where $Ω$ and $\widetildeΩ$ are domains in $\mathbb R^n$. We prove that mappings with the bounded on the set $Ω\setminus S$, where a set $S$ has $σ$-finite $(n-1)$-measure, geometric $p$-dilatation, are $W^1_{p,\loc}$-- mappings and generate bounded composition operators on Sobolev spaces.

math.AP↗

arXiv2023

Capacity inequalities and Lipschitz continuity of mappings

In this paper we consider topological mappings defined by $p$-capacity inequalities in domains of $\mathbb R^n$. In the case $p=n-1$ we prove the Lipschitz continuity of such mappings, that extends the result by F.~W.~Gehring.

math.AP↗

arXiv2022

Hölder and Lipschitz continuity in Orlicz-Sobolev classes, distortion and harmonic mappings

In this article, we consider the Hölder continuity of injective maps in Orlicz-Sobolev classes defined on the unit ball. Under certain conditions on the growth of dilatations, we obtain the Hölder continuity of the indicated class of mappings. In particular, under certain special restrictions, we show that Lipschitz continuity of mappings holds. We also consider Hölder and Lipschitz continuity of harmonic mappings and in particular of harmonic mappings in Orlicz-Sobolev classes. In addition in planar case, we show in some situations that the map is bi-Lipschitzian if Beltrami coefficient is Hölder continuous.

math.CV↗

arXiv2022

On the global behavior of solutions of the Beltrami equations

In this paper, the estimate for growth of homeomorphic solutions of the Beltrami equation at infinity is obtained, provided that the dilatation quotient has a global finite mean oscillation.

math.AP↗

arXiv2020

On the Asymptotic Behavior of Ring $Q$-homeomorphisms with Respect to $P$-modulus

We study the behavior at infinity of ring $Q$-homeomorphisms with respect to $p$-modulus for $p>n$.

math.CV↗

arXiv2018

Asymptotic dilation of regular homeomorphisms

We study the asymptotic behavior of the ratio $|f(z)|/|z|$ as $z\to 0$ for mappings differentiable a.e. in the unit disc with non-degenerated Jacobian. The main tools involve the length-area functionals and angular dilatations depending on some real number $p.$ The results are applied to homeomorphic solutions of a nonlinear Beltrami equation. The estimates are illustrated by examples.

math.CV↗

arXiv2016

Lower bounds for areas of images of discs

In this article we consider Q-homeomorphisms with respect to the p-modulus on the complex plane with p>2. It is obtained a lower area estimate for image of discs under such mappings. We solved the extremal problem about minimization of the area functional of images of discs.

math.CA↗

arXiv2014

On some local properties of space generalized quasiisometries

For some class of mappings, which are generalization of space quasiisometries, an upper estimate for a measure of image of a ball is obtained. As consequence, it is obtained one analog of Schwartz lemma for mappings mentioned above. Results of the paper are applicable in Sobolev and Orlicz--Sobolev classes.

math.CV↗

arXiv2014

On Vaisala inequality for angular dilatation of mappings and some applications

We study some type of mappings with finite distortion $f:D\rightarrow D^{\prime},$ $D, D^{\prime}\subset{\Bbb R}^n,$ $n\ge 2,$ which admit branch points. It is proved some inequality playing essential role at investigation of some problems of plane and space mappings. As application, it is investigated a problem of removable singularities for open discrete mappings with finite length distortion.

math.CV↗

arXiv2013

Equicontinuity and normality of mappings with integrally bounded $p$-moduli

We consider the generic discrete open mappings in ${\mathbb R}^n$ under which the perturbation of extremal lengths of curve collections is controlled integrally via $\int Q(x)η^p(|x-x_0|) dm(x)$ with $n-1<p<n$, where $Q$ is a measurable function on ${\mathbb R}^n$ and $\int\limits_{r_1}^{r_2} η(r) dr \ge 1$ for any $η$ on a given interval $[r_1,r_2].$ We proved that the family of all open discrete mappings of above type is normal under appropriate restrictions on the majorant $Q.$

math.CV↗

arXiv2013

Holder continuity for the ring $Q$-homeomorphisms with respect to $p$-modulus

It is founded the sufficient condition of Holder continuity of the ring $Q$-homeomorphisms in $\mathbb{R}^n, n\geq 2$ with respect to $p$-modulus at $n-1<p<n$.

math.CV↗