Representations of general solutions to some classes of nonlinear difference equations

Stevo Stević,Witold Kosmala,Bratislav Iričanin

doi:10.1186/s13662-019-2013-8

Abstract

Representations of general solutions to three related classes of nonlinear difference equations in terms of specially chosen solutions to linear difference equations with constant coefficients are given. Our results considerably extend some results in the literature and give theoretical explanations for them.

Highlights

Throughout the paper N, N0, Z, R, C denote the sets of natural, nonnegative, integer, real and complex numbers, respectively, whereas for k, l ∈ Z, the notation j = k, l denotes the set of all integers j such that k ≤ j ≤ l.No doubt that solvability of difference equations, as one of the basic and oldest topics in the theory of difference equations, is if not the most interesting certainly one of such topics
Further important results were obtained in the second half of the eighteenth century by several mathematicians, predominately by Lagrange and Laplace
What was known about difference equations up to the end of the nineteenth century was more or less summarized in books [4, 5]

Summary

Introduction

Using (26) in (25), we obtain that the representation of solutions to equation (15) given in (22) holds in this case. The following statements hold: (a) Assume that b = 0, and let (sn)n∈N be the solution to equation (15) satisfying initial condition (28). (b) Assume that b = 0, and let (sn)n∈N0 be the solution to equation (15) satisfying initial conditions (20).

Results

Conclusion