About coincidence points theorems on 2-step Carnot groups with 1-dimensional centre equipped with Box-quasimetrics

Грешнов, А. В.; Potapov, Vladimir N.

doi:10.3934/math.2023313

MATH

2023

DOI: 10.3934/math.2023313

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About coincidence points theorems on 2-step Carnot groups with 1-dimensional centre equipped with Box-quasimetrics

А. В. Грешнов¹,

Vladimir N. Potapov

Abstract: <abstract>For some class of 2-step Carnot groups $ D_n $ with 1-dimensional centre we find the exact values of the constants in $ (1, q_2) $-generalized triangle inequality for their $ \text{Box} $-quasimetrics $ \rho_{\text{Box}_{D_n}} $. Using this result we get the best version of the Coincidence Points Theorem of $ \alpha $-covering and $ \beta $-Lipschitz mappings defined on $ (D_n, \rho_{\text{Box}_{D_n}}) $.</abstract>

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Cited by 4 publications

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Optimal estimates of the number of links of basis horizontal broken lines for 2-step Carnot groups with horizontal distribution of corank 1

Greshnov,

Zhukov

2024

Russian Universities Reports. Mathematics

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For a 2-step Carnot group D_n, "dim" D_n=n+1, with horizontal distribution of corank 1, we proved that the minimal number N_(X_(D_n ) ) such that any two points u,v∈D_n can be joined by some basis horizontal k-broken line (i.e. a broken line consisting of k links) L_k^(X_(D_n ) ) (u,v), k≤N_(X_(D_n ) ), does not exeed n+2. The examples of D_n such that N_(X_(D_n ) )=n+i, i=1,2. were found. Here X_(D_n )={X_1,…,X_n} is the set of left invariant basis horizontal vector fields of the Lie algebra of the group D_n, and every link of L_k^(X_(D_n ) ) (u,v) has the form "exp"(asX_i)(w), s∈[0,s_0], a=const.

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Optimal estimates of the number of links of basis horizontal broken lines for 2-step Carnot groups with horizontal distribution of corank 1

Greshnov,

Zhukov

2024

Russian Universities Reports. Mathematics

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Fixed point theorems for $ (\alpha, \psi) $-rational type contractions in Jleli-Samet generalized metric spaces

Dumitrescu¹,

Pitea²

2023

MATH

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<abstract>The aim of this article is to present some results regarding $ (\alpha, \psi) $-rational type contractions in the setting of the generalized metric spaces introduced by Jleli and Samet. By the nature of these types of contractions which use also comparison functions, new fixed point theorems are established. Already known facts appear as consequences of our outcomes. Examples and comments point out the applicability of our approach.</abstract>

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Fixed point results for almost ($ \zeta -\theta _{\rho } $)-contractions on quasi metric spaces and an application

Güngör,

Altun

2024

MATH

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<abstract>This research paper investigated fixed point results for almost ($ \zeta-\theta _{\rho } $)-contractions in the context of quasi-metric spaces. The study focused on a specific class of ($ \zeta -\theta _{\rho } $)-contractions, which exhibit a more relaxed form of contractive property than classical contractions. The research not only established the existence of fixed points under the almost ($ \zeta -\theta _{\rho } $)-contraction framework but also provided sufficient conditions for the convergence of fixed point sequences. The proposed theorems and proofs contributed to the advancement of the theory of fixed points in quasi-metric spaces, shedding light on the intricate interplay between contraction-type mappings and the underlying space's quasi-metric structure. Furthermore, an application of these results was presented, highlighting the practical significance of the established theory. The application demonstrated how the theory of almost ($ \zeta -\theta _{\rho } $)-contractions in quasi-metric spaces can be utilized to solve real-world problems.</abstract>

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About coincidence points theorems on 2-step Carnot groups with 1-dimensional centre equipped with Box-quasimetrics

Cited by 4 publications

References 36 publications

Optimal estimates of the number of links of basis horizontal broken lines for 2-step Carnot groups with horizontal distribution of corank 1

Optimal estimates of the number of links of basis horizontal broken lines for 2-step Carnot groups with horizontal distribution of corank 1

Fixed point theorems for $ (\alpha, \psi) $-rational type contractions in Jleli-Samet generalized metric spaces

Fixed point results for almost ($ \zeta -\theta _{\rho } $)-contractions on quasi metric spaces and an application

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