Solvability of some classes of nonlinear first-order difference equations by invariants and generalized invariants

Abstract

We introduce notion of a generalized invariant for difference equations, which naturally generalizes notion of an invariant for the equations. Some motivations, basic examples and methods for application of invariants in the theory of solvability of difference equations are given. By using an invariant, as well as, a generalized invariant it is shown solvability of two classes of nonlinear first-order difference equations of interest, for nonnegative initial values and parameters appearing therein, considerably extending and explaining some problems in the literature. It is also explained how these classes of difference equations can be naturally obtained from some linear second-order difference equations with constant coefficients.