General Nondegenerate Solution of a System of Functional Equations

Bogdanova, R. A. , Mikhailichenko, G. G.
Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 1.

Abstract:
Systems of functional equations of the form \(f(\bar x,\bar y,\bar \xi,\bar \eta,\bar \mu,\bar \nu ) = \chi (g(x,y,\xi, \eta ),\mu,\nu)\) with six unknown functions \(\bar x\), \(\bar y\), \(\bar \xi\), \(\bar \eta\), \(\bar \mu\), \(\bar \nu\) arise when establishing the mutual embedding of two-metric phenomenologically symmetric geometries of two sets (TPhS GTS). When establishing an embedding of an additive TPhS GTS of rank \((2, 2)\) with a known vector function \(g(x,y,\xi,\eta )=({g^1},{g^1})=(x+\xi,y+\eta)\) into a dual GDM DFS of rank \((3, 2)\) with a known vector function \(f(x,y,\xi,\eta,\mu,\nu )=({f^1},{f^2})=(x\xi+\mu,x\eta+y\xi+\nu )\) the explicit form of the system of two functional equations is as follows: \(\overline x \overline \xi + \overline \mu = \chi^1(x + \xi,y + \eta,\mu,\nu )\), \(\overline x \overline\eta+\overline y\overline\xi+\overline\nu=\chi^2(x+\xi,y+\eta,\mu,\nu )\). This system of two functional equations is solvable because the expressions for the vector functions \(g\) and \(f\) in the system are known. To find a general nondegenerate solution to a given system of functional equations, it is necessary to develop a solving method, which is an interesting and meaningful mathematical problem. The basis of the method is the differentiation of one of the functional equations included in the system, followed by the transition to differential equations. Further, the solutions of the differential equations are substituted into the second functional equation of the original system of functional equations, from which, under appropriate restrictions, its general nondegenerate solution is found. This method can be developed and applied to other systems of functional equations of the same type that arise in the framework of the TPhS GTS embedding problem in order to find their general nondegenerate solution.

Keywords: geometry of two sets, sequential in rank embedding, system of functional equations, general nondegenerate solution of a system of functional equations, system of differential equations.

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