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

The Domains of Admissible Parameters of Box-quasimetrics for Canonical Heisenberg Groups and their Generalizations

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.

<abstract><p>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.</p></abstract>

Box-Quasimetrics and Horizontal Joinability on Cartan Groups

<abstract><p>The purpose of this paper is to present a study of $ \alpha $-$ \eta $-type generalized $ F $-proximal contraction mappings in the framework of modular metric spaces and to prove some best proximity point theorems for these types of mappings. Some examples are given to show the validity of our results. We also apply our results to establish the existence of solutions for a certain type of non-linear integral equation.</p></abstract>

Coincidence, fixed points in symmetric <inline-formula><tex-math id="M1">$ (q_1,q_2) $</tex-math></inline-formula>-quasi-metric spaces and sensitivity analysis of generalized equations

<abstract><p>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.</p></abstract>