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

1.   Introduction

Consider a set $X$ consisting of at least two points. Function $\rho_X :X\times X\to\Bbb R^+$, $\rho_X(x, y) = 0\Leftrightarrow x = 0$, is called $(q_1, q_2)$-quasimetric if the following $(q_1, q_2)$-generalized triangle inequality takes place:

where $q_1, q_2$ are some positive numbers. Pair $(X, \rho_X)$ is called $(q_1, q_2)$-quasimetric space ^{[1,2,3,4,5,6,7,8,9]}. The expression $\rho_X(x, y)$ denotes a $(q_1, q_2)$-quasi-distance exactly from the point $x$ to the point $y$. If $q_1 = q_2 = 1$, then $(X, \rho_X)$ is a quasimetric space ^[11].

If for a $(q_1, q_2)$-quasimetric $\rho_X$ the following condition holds

for some $q_0 > 0$ then we refer to a $(q_1, q_2)$-quasimetric space $(X, \rho_X)$ as a $q_0$-symmetric one; for the case when $q_0 = 1$, we use the notion of symmetric $(q_1, q_2)$-quasimetric space. The $(q_2, q_1)$-quasimetric $\overline{\rho}_X(x, y) = \rho_X(y, x)$ is said to be conjugate to $\rho_X(x, y)$. If $\rho_X$ is symmetric then $\overline{\rho}_X$ is symmetric too.

The class of $(q_1, q_2)$-quasimetric spaces is suffciently wide; it includes quasimetric spaces, b-metric spaces introduced by Bakhtin in 1989, Carnot-Carathéodory spaces with Box-quasimetrics, $L_p$-spaces with $p\in(0, 1)$, etc. (see ^[10]).

Definition 1.1. For a $(q_1, q_2)$-quasimetric space $(X, \rho_X)$ we denote by $R = R(\rho_X)$ the set of points $(q'_1, q'_2)\in\Bbb R^2$, such that for $\rho_X$ $(q'_1, q'_2)$-generalized triangle inequality holds.

The next Property 1.2 follows directly from the Definition 1.1.

Property 1.2 (^[1,2]). $1^0$ The set $R = R (d)$ is convex and closed, and, moreover, $R\subseteq\{(x, y)\in\Bbb R^2\mid x\geq1, \, y\geq1\}$;

$2^0$ The condition $(1, 1)\in R$ is equivalent to the fact that $\rho_X$ is a quasimetric;

$3^0$ If $(q_1, q_2)$-quasimetric is symmetric, then the set $R$ is symmetric with respect to the bisector of the right upper coordinate angle of the Euclidean plane.

If $q'\in R$ and $\tilde q\geq q'$ (in the sense that $\tilde q_1\geq q'_1$, $\tilde q_2\geq q'_2$), then $\tilde q\in R$. By considering the support lines at the boundary points of the closed convex set $R$ we obtain that $R$ has extreme points. (Recall that a point $x_0\in A$ is called an extreme point of a set $A$, if there are no points $x_1, x_2\in A$, such that $x_0\in (x_1, x_2)$, that is, $x_0 = tx_1+(1-t)x_2$ for some $0 < t < 1$.) We easily see that each extreme point of $R$ is a Pareto optimal point of $R$ (in the sense of minimization of components), but not conversely. Extreme points of $R$ are said to be extreme for $\rho_X$. A point $q\in R$ is said to be best for $\rho_X$ if $q\leq q'$ for all $q'\in R$. See the examples of $(q_1, q_2)$-quasimetric spaces with the best points $q = (q^0_1, q^0_2)$ such that $q^0_1+q^0_2 > 2$, in ^[1,4,5,6].

An important special case of symmetric $(q_1, q_2)$-quasimetric spaces are the symmetric $(1, q_2)$- quasimetric spaces ^[1]; these include Carnot groups and more general equiregular Carnot-Carathéodory spaces $(M, \rho_{\text{Box}_M})$, equipped by $\text{Box}$-quasimetrics $\rho_{\text{Box}_M}$ ^{[6,7,8,9,10,11,12,13,14,15]}. Moreover, in the general case, the constant $q_2$ does not equal $1$ ^[16]. $\text{Box}$-quasimetrics were introduced in ^[17]. $(1, q_2)$-generalized triangle inequality plays a crucial role in obtaining the divergence estimates of the equiregular Carnot-Carathéodory space $(M, \rho_{\text{Box}_M})$ from its nilpotent tangent cone, see ^[18,19].

Define the sets

A set $U\subset X$ is said to be open if, for every point $u\in U$ there is a number $r_u > 0$ such that ${B}^{o}_X(u, r_u)\subset U$. A set is said to be closed if its complement is open. The open sets defined in this way determine a topology on $X$.

A sequence of points $\{x_i\}\subset (X, \rho_X)$ converges to a point $x_0\in X$ (we write $x_i\to x_0$) if, for every $\varepsilon > 0$ ball ${B}^{o}_X(x_0, \varepsilon)$ contains all points $x_i$, starting with some of them. The point $x_0$ is called the limit of the sequence $\{x_i\}$. Clearly, this definition may equivalently be restated in the following form: A sequence $\{x_i\}$ converges to $x_0$, if $\lim\limits_{i\to \infty}\rho_X(x_0, x_i) = 0$.

A sequence $\{x_n\}$ in a $(q_1, q_2)$-quasimetric space $(X, \rho_X)$ is called a fundamental sequence or a Cauchy sequence, if for every $\varepsilon > 0$ there is an $N$ such that for all $n > m > N$ we have $\rho_X(x_m, x_n) < \varepsilon$. A $(q_1, q_2)$-quasimetric space $(X, \rho_X)$ is said to be complete if each of its fundamental sequences has a limit (possibly non-unique).

Consider a $(q_1, q_2)$-quasimetric space $(X, \rho_X)$ and a $(q'_1, q'_2)$-quasimetric space $(Y, \rho_Y)$. Let $\Psi, \Phi:X\to Y$ be mappings and $\alpha > \beta \geq 0$ be numbers.

Definition 1.3 (^[1,2,3]). A point $x \in X$ is called a coincidence point of the mappings $\Psi, \Phi$ if

Definition 1.4 (^[1,2,3]). A mapping $\Psi$ is said to be $\alpha$-covering if

Definition 1.5. A mapping $\Phi$ is said to be $\beta$-Lipschitz if

$\alpha$-covering of $\Psi$ means that for every $x_0\in X$, $y_1\in Y$ there is $x_1\in X$ such that $y_1 = \Psi(x_1)$, $\rho_X(x_0, x_1)\leq\frac{\rho_Y(\Psi(x_0), y_1)}{\alpha}$; hence the mapping $\Psi$ is surjective.

The Banach open mapping theorem provides a classical example of a covering mapping. Recall that the theorem states that if $X$ is a Banach space, $Y$ is a normed space, and $\psi$ is a linear, continuous, and surjective operator, then for some $\alpha > 0$ the operator $\psi$ is $\alpha$-covering. Covering mappings and their properties have been studied in detail since the middle of the 20th Century. One of the first papers devoted to this issue was the paper ^[20] by L. M. Graves. In this paper, the covering property of linear mappings in Banach spaces were used to derive conditions for smooth mappings to be locally covering. Subsequently, Milyutin ^[21] obtained a theorem that provides sufficient covering conditions.

Theorem 1.6 (^[21]). Let $X$ be a complete metric space, $Y$ be a linear metric space with a translation-invariant metric $\rho_Y$, $\psi:X\to Y$ be continuous and $\alpha$-covering, $\phi:X\to Y$ be $\beta$-Lipschitz, and $\beta < \alpha$. Then the mapping $\psi+\phi$ is $(\alpha-\beta)$-covering.

This result is commonly called the Milyutin theorem on Lipschitz perturbations of covering mappings. Subsequently, the covering property and its stability under perturbations was a subject of numerous studies (see, for instance, ^{[22,23,24,25]}). Another problem to which the covering mappings theory is applicable is the coincidence points problem. Sufficient conditions for the existence of coincidence points stated in terms of covering mappings were proved by Arutyunov in ^[26] on metric spaces. In ^[26] there were also obtained conditions for existence of coincidence points of set-valued mappings. In ^[27,28] the stability of coincidence points was investigated. The discussed above and some other results of covering mappings theory has applications in investigations of control systems ^[29], implicit differential equations (see ^[30,31]), Volterra equations (see ^[32]). The theory of coincidence points of both single-valued and set-valued mappings of metric spaces plays an important role in analysis (see ^[33]). This theory is applied to the study of inclusions (see ^[34]). We note the following recent interesting works on the theory of coincidence points ^[35,36,37].

In their recent papers ^[1,2,3,10], Arutyunov and Greshnov introduced $(q_1, q_2)$-quasimetric spaces and studied their properties; they studied covering mappings from one $(q_1, q_2)$-quasimetric space to another and obtained sufficient conditions for the existence of coincidence points of two mappings between such spaces provided that one of them is covering and the other satisfies the Lipschitz condition. These results were extended to multi-valued mappings. Also it was proved that the coincidence points are stable under small perturbations of the mappings. The development of the theory of coincidence points of mappings on $(q_1, q_2)$-quasimetric spaces initiated interest in the study of more general $f$-quasimetric spaces ^[4] and in generalizing Banach's fixed point theorem to such spaces ^[38].

Let's formulate the results from ^[1,2], we will deal with further.

As usual, $gph(F) = \{(x, y)\in X\times Y\mid y\in F(x)\}$ is the graph of the mapping $F:(X, \rho_X)\to(Y, \rho_Y)$. We say that a mapping $F$ closed if, for all sequences $\{x_i\} \subset X$ and $\{y_i\} \subset Y$ converging to points $x_0$ and $y_0$ respectively, such that $(x_i, y_i) \in gph(F)$ for all $i$, it holds that $(x_0, y_0) \in gph(F).$

Given a function $f:X\times X\to\Bbb R^+$ of two variables and a point $(x_1, x_2)\in X\times X$, we write $\lim\limits_{\overline {y\to x_1}}f(y, x_1)$ for its lower limit in the first variable at the point $(x_1, x_2)$. This limit is defined as the infimum of the lower limits $\inf \lim\limits_{\overline {y_i\to x_1}} f(y_i, x_2)$, where the infimum is taken over all sequences $\{y_i\}$ that converge to $x_1$. The lower limit $\lim\limits_{\overline {y\to x_2}}f(x_1, y)$ in the second variable is defined in a similar way.

Given any $u\in [0, 1)$ and any positive integer $n$, we write $S(u, n)$ for the sum of $n$ terms of the geometric progression $\sum\limits_{i = 0}^{n-1}u^i$ and, therefore, $S(u, n) = \frac{1-u^n}{1-u}$. We shall assume that $S(u, 0) = 0$ and $\beta^0 = 1$ for $\beta = 0$. For all $q_0, q_1, q_2\geq 1$ we put

and under the assumption that $q_0^2\beta < \alpha$, we put

Theorem 1.7 (On the existence of coincidence points ^[1,2]). Assume that the $(q_1, q_2)$-quasimetric space $(X, \rho_X)$ is complete. Let the mapping $\Psi$ be $\alpha$-covering and closed, while the mapping $\Phi$ be $\beta$-Lipschitz. Fix an arbitrary point $x_0\in X$. Then the mappings $\Psi$ and $\Phi$ have a coincidence point $\xi$, such that

If the space $(X, \rho_X)$ is $q_0$-symmetric, then $\xi$ satisfies the estimate

and if, additionally, $q_0^2\beta < \alpha$, then $\xi$ also satisfies the estimates

Let $X = Y$ and $\Psi$ be the identity mapping, i.e., $\Psi(x)\equiv x$. Then $\alpha = 1$, the condition $\beta < 1$ means that $\Phi$ is a contraction mapping, and the coincidence point becomes a fixed point.

Corollary 1.8 (Fixed-point theorem for a contraction mapping). A closed contraction mapping of a complete $(q_1, q_2)$-quasimetric space to itself has a fixed point, and this point is unique.

An extended version of Theorem 1.7 is proved in [1,Theorem 4.5].

Theorem 1.9 (^[1,2]). Let the space $(X, \rho_X)$ be complete, the mapping $\Psi$ be $\alpha$-covering and closed, and the mapping $\Phi$ be $\beta$-Lipschitz. Fix an arbitrary point $x_0\in X$.

$1^0$ Let $q_1 = 1$. Then the mappings $\Psi$ and $\Phi$ have a coincidence point $\xi$ such that

$2^0$ Let the space $(X, \rho_X)$ be $q_0$-symmetric, $q_0^2\beta < \alpha$, $q_2 = 1$. Then there exists a coincidence point $\xi$, such that

The bounds (1.1)–(1.6) from the Theorem 1.7 and Theorem 1.9 are the estimates of divergence of coincidence point of $\alpha$-covering and closed and $\beta$-Lipschitz mappings from an arbitrary point $x_0\in X$.

Examples showing that the bounds (1.3)–(1.6) are unimprovable have been found in ^[1]. So the problem of finding the optimal bounds in (1.3)–(1.6) is directly related to finding optimal values of the constants $q_1$, $q_2$. Let us consider $(1, q_2)$-quasimetric spaces, in particular, Carnot groups $G$ equipped with symmetric $\text{Box}$-quasimetric $\rho_{\text{Box}_G}$; from this point of view the problem of finding of minimal values of $q_2$ becomes relevant, see (1.5) and (1.6). Further, we will use the term exact value that implies such value of the constant $q_2$ that for every number $q'_2$, $q'_2 < q_2$, the $(1, q'_2)$-generalized triangle inequality does not hold for $\rho_{\text{Box}_G}$. Note that the exact values of the constant $q_2$ for the $(1, q_2)$-generalized triangle inequality of $\text{Box}$-quasimetrics were obtained: on the canonical Heisenberg groups $\Bbb H^n_{\alpha}$, $n\in \Bbb N$, and the canonical Engel group $\Bbb E_{\alpha, \beta}$ in ^[16], on some low-dimensional 2-step canonical Carnot groups in ^[39]. (See the definition of canonical finite-dimensional Lie group in ^[39,40].)

The aim of this paper is to find the exact value of the constant $q_2$ on some $(n+1)$-dimensional 2-step canonical Carnot group $D_n$ with the 1-dimensional centre of a special kind (see the Definition 2.2). The main result of our work is Theorem 3.7 where the exact values for $q_2$ on $D_n$ are obtained. Using Theorem 3.7 we prove Theorem 3.9 which is the best version of the Theorem 1.9. The exact estimates obtained in the Theorem 3.9 can be used in optimal control problems, in particular, to obtain theorems Milyutin type theorems in sub-riemannian geometry. The proof of the Theorem 3.7 is based on some special combinatorial theorems, with which we find the maximum of some special bilinear function $f(A, B)$, where $A, B$ belong to the set of all vertices of a standard unit $n$-dimensional cube (Theorem 1.7, Property 3.5); these results and methods of their proofs can be used in optimization problems of arbitrary functions over vertices of polytopes (see ^[41]).

2.   Preliminaries

In this section, we recall some basic definitions and notations which will be required in proving our main results.

A Lie algebra is called graduated ^[42], if it decomposes into a direct sum of vector subspaces $V = \bigoplus\limits_{i = 1}^rV_i$, and, moreover, $[V_i, V_k]\subset V_{i+k}$, if $i+k\leq r$, and $[V_i, V_k] = 0$, if $i+k > r$. Note that a graduated algebra is always nilpotent of degree $r$. $r$-step stratifed Lie algebra $V$ ^[43] is a Lie algebra nilpotent of degree $r$, that has a stratifcation, i. e.

An $r$-step Carnot algebra ^[43] is a graduated Lie algebra $V$, which has a stratifcation; a simply connected Lie group $G$, corresponding to an $r$-step Carnot algebra $V$, is called an $r$-step Carnot group. Let $N = \sum\limits_{i = 1}^{r}n_i$, $n_i = \dim V_i$, and the basis of the left-invariant vector fields $\{X_1, \ldots, X_N\}$ of the Carnot group $G$ is ordered such that the values of the first $n_1$ of them form at every point $u\in G$ the basis of the subspace $V_1(u)$, the values of the next $n_2$ of them form at every point $u\in G$ the basis of the subspace $V_2(u)$, and so on. We assign to every vector field $X_k$ a natural number $j = \deg X_k$, defined by the inclusion $X_k\in V_j$.

Definition 2.1 (^{[12,13,14,15,17]}). A $\text{Box}$-quasimetric $\rho_{\text{Box}_G}$ is defined as

where exp is standard exponential mapping.

Note that standard exponential mapping is identical on canonical Lie group. The Definition 2.1 implies that $\rho_{\text{Box}_G}$ satisfies the identity and symmetry axioms.

A canonical 2-step group $\Bbb D_n$ with the 1-dimensional centre is defined in the standard Euclidean space $\Bbb R^{n+1}$ with the coordinate system $(x_1, \ldots, x_n, t)$ and the coordinate frame $(O, e_1, \ldots, e_n, e)$ with the help of the following commutator table

the rest of possible commutators of $e_1, \ldots, e_{n+1}$ equal $0$. Suppose that $x = (x_1, \ldots, x_n, t)$, $x' = (x'_1, \ldots, x'_{n}, t')$. Using the Campbell-Hausdorff formula ^[44], with the help of (2.2) we obtain

The values of basis left-invariant vector fields $X_1, \ldots, X_n, T$ of the group $\Bbb D_n$ at every point $x = (x_1, \ldots, x_n, t)$ are defined as

If in (2.2) we put $n = 2m$, $m\in\Bbb N$, $\sum\limits_{i = 1}^{m-1}\alpha^2_{2i, 2i+1} = 0$ and $\alpha_{2j-1, 2j} = \alpha\neq0$, $j = 1, \ldots, m$, then we obtain a commutator table that defines the canonical Heisenberg group $\Bbb H^m_{\alpha}$ ^[16]. In particular, $\Bbb D_2 = \Bbb H^1_{\alpha}$.

According to (2.1), $(1, q_2)$-quasimetric $\rho_{\text{Box}_{\Bbb D_n}}$ is defined by the rule

Definition 2.2. Carnot group $D_n$ is such a group $\Bbb D_n$ for which the relations $\alpha_{ij} = 1$ $\forall i < j$ are fulfilled in (2.2).

Let us consider some basis $E_0 = \{e_1, \ldots, e_n\}$ in a $n$-dimensional vector space $\text{Vec}_n$.

Definition 2.3. We say that a basis $E' = \{e'_1, \ldots, e'_n\}$ is affine equivalent to basis $E_0$ on $\text{Vec}_n$ if

where

Let us consider skew-symmetric bilinear bracket function $[x, y]:\text{Vec}_n\times\text{Vec}_n\to\Bbb R$. Let

Since $[x, y]$ is skew-symmetric then $a_{ij} = -a_{ji}$, $a_{ii} = 0$ for all $i$.

Lemma 2.4. Basis $E_0 = \{e_1, \ldots, e_n\}$ is affine equivalent to some basis $E' = \{e'_1, \ldots, e'_n\}$ such that

Proof. Without loss of generality, we put $[e_1, e_2] = a_{12}$, where $a_{12} > 0$. Build the basis $\{e'_1, \ldots, e'_n\}$ step by step.

$1^0$ Consider all brackets $[e_1, e_i]$, $i > 2$. Suppose that there are numbers $a_{1i}\leq0$. Then, instead the basis $E_0 = \{e_1, \ldots, e_n\}$ let us consider the basis

And we have

if $A_1$ is a large enough number then $a_{12}A_1+a_{1i} > 0$.

$2^0$ Next, we consider the basis $E_1$, but in order to avoid inconvenience, we will use the notation $\{e_1, \ldots, e_n\}$ for $E_1$ and the symbols $a_{ij}$. So we have

Consider all brackets $[e_2, e_i]$, $i > 3$. Suppose that there are numbers $a_{2i}\leq0$. Then, instead the basis $E_1 = \{e_1, \ldots, e_n\}$ let us consider the basis

We have

if $|A_2|$ is a large enough number then $a_{2i}-A_2a_{12} > 0$.

$3^0$ Next, we consider the basis $E_2$, but in order to avoid inconvenience, we will use the notation $\{e_1, \ldots, e_n\}$ for $E_2$ and the symbols $a_{ij}$. We have

Consider all brackets $[e_3, e_i]$, $i > 3$. Suppose that there are numbers $a_{3i}\leq0$. Then instead the basis $E_2 = \{e_1, \ldots, e_n\}$ let us consider the basis

We have

if $A_3$ is a large enough number.

$4^0$ Next, we consider the basis $E_3$, but in order to avoid inconvenience, we will use the notation $\{e_1, \ldots, e_n\}$ for $E_3$ and the symbols $a_{ij}$. We have

Consider all brackets $[e_4, e_i]$, $i > 4$. Suppose that there are numbers $a_{4i}\leq0$. Then, instead the basis $E_3 = \{e_1, \ldots, e_n\}$ let us consider the basis

We have

if $A_4$ is a large enough number.

The next steps are obvious.

In some cases the basis $E_0 = \{e_1, \ldots, e_n\}$ is affine equivalent to such a basis $E' = \{e'_1, \ldots, e'_n\}$ that

Consider, for example, the 3-dimensional case

Taking into account Lemma 2.4, we can assume that $a_{ij} > 0$. Let

then

so

It is not difficult to see that vectors $e'_i = \frac{e_i}{x_i}$, $i = 1, 2, 3$, satisfy the identities $[e'_i, e'_j] = 1$, $i < j$, $i, j = 1, 2, 3$.

3.   Main results

Next, we consider the basis $\{e_1, \ldots, e_n\}$ of $\text{Vec}_n$ satisfying the table

Let $x = \sum\limits_{i = 1}^nx_ie_i$, $y = \sum\limits_{i = 1}^ny_ie_i$; then, using (3.1), we get

Let's find $\max\limits_{x, y\in \text{Vert}(n)}f(x, y)$, where $\text{Vert}(n)$ is the set of all vertices of unit $n$-dimensional cube in vector space $\text{Vec}_n$, i. e. all possible points whose coordinates consist only of numbers $\pm1$.

Let $A[n](x, y)$ is a $(n\times n)$-matrix consisting of elements

where $x, y\in \text{Vert}(n)$. Denote by $L(A)$ the number of $-1$ in $A[n](x, y)$.

Theorem 3.1. $1)$ $\min\limits_{x, y\in \mathit{\text{Vert}}(2k)}L(A[2k](x, y))\geq k^2-k$, $2)$ $\min\limits_{x, y\in \mathit{\text{Vert(2k+1)}}}L(A[2k+1](x, y))\geq k^2$.

Denote by $A'_i$ the $(n-1)\times(n-1)$-matrix that is obtained from the matrix $A[n](x, y)$ by deleting $i$-line and $i$-column. Denote by $A''_{ij}$ $(n-1)\times(n-1)$-matrix, that is obtained from the matrix $A[n](x, y)$ by deleting $i$-line and $j$-column.

Lemma 3.2. Let us consider the matrix $A = A[2k+2](x, y)$. Then

Proof. We have

where $l(i', j') = 0$ in the case when $(A'')_{i'j'}\neq-1$, and $l(i', j')$ is equal to the number of all matrices $A''_{ij}$ containing the element $(A'')_{i'j'}$ in the case when $(A'')_{i'j'} = -1$. It is not difficult to see that $l = l(i', j')$ does not depend on the choice of a pair $i', j'$. We have

where $N_k$ is the number of non-diagonal elements of a $(k\times k)$-matrix, and $C^2_{2k+2}$ is the number of ways to choose two pairs of $i$ and $j$ lines and columns in a $(2k+2)\times (2k+2)$-matrix. Then

hence Lemma 3.2 follows.

Lemma 3.3. Let us consider the matrix $A = A[2k+1](x, y)$. Then

Proof. The proof of Lemma 3.3 is similar to the proof of Lemma 3.2.

Proof of Theorem 3.1. $1)$ The proof is carried out using the method of mathematical induction. The cases $k = 1, 2, 3$ are clear. Suppose that for $k+1$ the Theorem 1.7 does not hold, i. e. there is a matrix $A = A[2k+2](x, y)$ such that

But then for matrix $A$ there will be a matrix $A''_{ij}$ for which the Theorem 1.7 does not hold too. Let $A''_{ij}$ be a matrix for which $L(A''_{ij})$ is minimal. Then using Lemma 3.2 we have

We have

Inequality

is equivalent to inequality

that is right for $k\geq3$. Then $L(A''_{ij}) < k^2-k$ but this is contradiction.

The proof of the point $2)$ is similar using Lemma 3.3.

Corollary 3.4.

Property 3.5.

Proof. Let $x, y\in \text{Vert}(2k)$. You can see that if $x_i = 1$, $i = 1, \ldots, 2k$, $\, \, y_j = 1$, $j = 1, \ldots, k$, $\, \, y_l = -1$, $l = k+1, \ldots, 2k$, then

Let $x, y\in \text{Vert}(2k+1)$. You can see that if $x_i = 1$, $i = 1, \ldots, 2k$, $\, \, y_j = 1$, $j = 1, \ldots, k$, $\, \, y_l = -1$, $l = k+1, \ldots, 2k+1$, then

Using some results from work ^[39], we find the exact value of the constant in the $(1, q_2)$-generalized triangle inequality for the canonical Carnot group $D_n$. Let

compare with (2.3).

Theorem 3.6 (^[39]). On canonical Carnot group $\Bbb D_n$ the following formula gives the exact value of the constant $q_2$ in the $(1, q_2)$-generalized triangle inequality

Using Property 3.5 and Theorem 3.6 we get the following:

Theorem 3.7. The exact value of the constant in the $(1, q_2)$-generalized triangle inequality for canonical Carnot group $D_n$ is defined by the formula $q_2 = \frac{M_n}2$, where

Lemma 3.8. Let $(X, \rho_X)$ be a symmetric $(1, q)$-quasimetric space. Then $(X, \rho_X)$ is $(q, 1)$-quasimetric space.

Proof. Obviously.

Consider a $(q'_1, q'_2)$-quasimetric space $(Y, \rho_Y)$. The following Theorem 3.9 follows from Lemma 3.8, Theorem 1.9 and Theorem 3.7.

Theorem 3.9. Let mapping $\Psi:(D_n, \rho_{\mathit{\text{Box}}_{D_n}})\to(Y, \rho_Y)$ be $\alpha$-covering and closed, and the mapping $\Phi:(D_n, \rho_{\mathit{\text{Box}}_{D_n}})\to(Y, \rho_Y)$ be $\beta$-Lipschitz. Fix an arbitrary point $x_0\in D_n$. The mappings $\Psi$ and $\Phi$ have a coincidence point $\xi$ such that

4.   Conclusions

In this paper, on some class of 2-step Carnot groups $D_n$ with 1-dimensional centre we found the exact values of the constants in $(1, q_2)$-generalized triangle inequality for their $\text{Box}$-quasimetrics $\rho_{\text{Box}_{D_n}}$. As a consequence, we obtained the best version of the Coincidence Points Theorem of $\alpha$-covering and $\beta$-Lipschitz mappings defined on $(D_n, \rho_{\text{Box}_{D_n}})$. The results obtained and the methods of their proofs can be applied in fixed point theory, optimal control theory, optimization problems, quasimetric analysis, sub-riemannian geometry.

Acknowledgments

The authors are thankful to Professor Arutyunov and Professor Storozhuk for helpful discussions and valuable comments. The authors would like to thank the anonymous referees for their comments that have helped us improve this article. The research of the first author has been carried out within the framework of a state assignment of the Ministry of Education and Science of the Russian Federation for the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (project no. FWNF-2022-0006). The research of the second author has been carried out within the framework of a state assignment of the Ministry of Education and Science of the Russian Federation for the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (project no. FWNF-2022- 0017).

Conflict of interest

The authors declare no conflicts of interest.