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

On a symmetric (1,q_2) -quasimetric space (H_α^1,〖"Box" 〗_(H_α^1 ) ), where 〖"Box" 〗_(H_α^1 ) is the "Box" -quasimetic of the first Heisenberg group H_α^1, we studied a constant "L" _Φ in the estimate 〖"Box" 〗_(H_α^1 ) (u,ξ)≤("L" _Φ 〖"Box" 〗_(H_α^1 ) (u,Φ(u)))/(1-ε) of stability of the ε -contracting mapping Φ with respect to the identity mapping; here ξ is a fixed point of the mapping Φ and u is an arbitrary point of H_α^1. In the paper, we got that "L" _Φ=1 when the mapping Φ is the composition of the left translation and the homogeneous dilation subgroup. Examples of the contracting mappings Φ on the first Heisenberg group such that "L" _Φ is not less then C√(q_2 ) were found; here positive constant C does not depend on the choice of point u∈H_α^1.