ALGEBRAIC MODEL OF DIFFERENCE EQUATIONS AND FUNCTIONAL EQUATIONS

Abstract

We will deal with the theory of Abel functional equations in the space of strictly monotonic functions \(S\). The Abel functional equation model reduces under specialization to a linear functional or to a linear difference equation. Definitions, structure, and general theory for Abel functional equations on \(S\) appear. The approach duplicates a rich body of known definitions, results and properties for classical functional and difference equations.

The setting for the algebraic model is in the space \(S\) of strictly monotonic real functions \(f\) defined on the interval \(J=(-\infty,\infty)\). It is required that \(f\) map \(J\) one-to-one onto an interval \((a,b)\), where \(a\) and \(b\) are extended real numbers.

The model equation is expressed in terms of iteration of a function \(\Phi\) in \(S\). The iteration process uses a canonical function in \(S\), which is an arbitrarily chosen increasing function \(X\in S\).

A method is presented for solving the new model equation. This method can be applied to solve, in particular, some classical linear functional and difference equations.