About estimates of stability of contraction~mappings on~the~first Heisenberg group in the fixed point theorem

A. V. Greshnov

Vestnik rossijskih universitetov. Matematika, Tome 30 (2025) no. 149, pp. 15-27

Résumé

On a symmetric $(1,q_2)$-quasimetric space $(\Bbb H^1_{\alpha},\mathrm{Box}_{\Bbb H^1_{\alpha}}),$ where $\mathrm{Box}_{\Bbb H^1_{\alpha}}$ is the $\mathrm{Box}$-quasimetic of the first Heisenberg group $\Bbb H^1_{\alpha},$ we studied a constant $\mathrm{L}_{\Phi}$ in the estimate $\mathrm{Box}_{\Bbb H^1_{\alpha}}(u,\xi)\leq\frac{\mathrm{L}_{\Phi}\mathrm{Box}_{\Bbb H^1_{\alpha}}\big(u,\Phi(u)\big)}{1-\varepsilon}$ of stability of the $\varepsilon$-contracting mapping $\Phi$ with respect to the identity mapping; here $\xi$ is a fixed point of the mapping $\Phi$ and $u$ is an arbitrary point of $\Bbb H^1_{\alpha}.$ In the paper, we got that $\mathrm{L}_{\Phi}=1$ when the mapping $\Phi$ is the composition of the left translation and the homogeneous dilation subgroup. Examples of the contracting mappings $\Phi$ on the first Heisenberg group such that $\mathrm{L}_{\Phi}$ is not less then $C\sqrt{q_2}$ were found; here positive constant $C$ does not depend on the choice of point $u\in\Bbb H^1_{\alpha}.$

Détail

Mots-clés : $(q_1,q_2)$-quasimetric, $\mathrm{Box}$-quasimetric
Keywords: canonical Carnot group, contraction mapping, estimates of stability, fixed point