Nondegenerate canonical solutions of a certain system of functional equations

V. A. Kyrov   ; G. G. Mikhailichenko

Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 1, pp. 44-53

Résumé

The problem of embedding a non-additive two-metric phenomenologically symmetric geometry of rank $ (2,2)$ with the function $g (x, y, \xi, \eta) = (g^{1}, g^{2}) $ into a two-metric phenomenologically symmetric geometry of rank $ (3,2)$ with the function $f (x, y,\xi,\eta,\mu, \nu) = (f^{1}, f^{2})$ leads to the existence problem of nondegenerate solutions for corresponding system $f(\bar{x},\bar{y},\bar{\xi}, \bar{\eta}, \bar{\mu}, \bar{\nu})= \chi(g (x, y, \xi, \eta), \mu, \nu)$ of two functional equations. This system is solvable since the functions $g$ and $f$ are previously known and hence the system takes an explicit form: $\bar{x} \bar{\xi} + \bar{y}\bar{\mu}=\chi^{1}((x+\xi)y,(x+\xi)\eta,\mu,\nu),$ $\bar{x} \bar{\eta}+\bar{y}\bar{\nu}=\chi^{2}((x + \xi) y,(x+\xi)\eta,\mu,\nu).$ It is difficult to find a general solution to such a system. However, one can first find a canonical solution associated with the Jordan form of second-order matrices, since their number is small, and then determine the general solution using an appropriate transformation of matrices and vectors. This reformulation of the main problem makes it simpler and mathematically more interesting. In the process of searching for canonical solutions of the original system of functional equations, we first differentiate with respect to the variables $x$ and $\xi$, as a result, we obtain a system of differential equations with a matrix of coefficients $A$ of general form: $\left(\begin{array}{c}{\bar{x}_{x}}{\bar{y}_{x}}\end{array}\right) =A\left(\begin{array}{c}{\bar{x}} {\bar{y}}\end{array}\right)$. The matrix $A$ can be reduced to Jordan form and then the system of differential equations with such a Jordan matrix is solved. Further, with the solutions of the system of differential equations, we return to the original system of functional equations, from which additional constraints are found. As a result, nondegenerate canonical solutions of the original system of functional equations are obtained. These canonical solutions are then used to write down the general solutions of the original system.