On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang's Conjecture

TL;DR

This paper derives a closed-form solution for a specific nonlinear difference equation and uses it to prove a case of Sroysang's conjecture related to the asymptotic behavior of functions satisfying a functional equation, providing new insights and proofs.

Contribution

The paper provides a clear derivation of the closed-form solution for a nonlinear difference equation and offers a novel proof for a case of Sroysang's conjecture, enhancing understanding of the conjecture's validity.

Findings

Explicit closed-form solution for the difference equation.

Verification of the limit involving the golden ratio for functions satisfying the functional equation.

Alternative proof approach for the remaining case of the conjecture.

Abstract

The purpose of this paper is twofold. First, we derive theoretically, using appropriate transformation on $x_n$, the closed-form solution of the nonlinear difference equation \[ x_{n+1} = \frac{1}{\pm 1 + x_n},\qquad n\in \mathbb{N}_0. \] We mention that the solution form of this equation was already obtained by Tollu et al. in 2013, but through induction principle, and one of our purpose is to clearly explain how was the formula appeared in such structure. After that, with the solution form of the above equation at hand, we prove a case of Sroysang's conjecture (2013); i.e., given a fixed positive integer $k$, we verify the validity of the following claim: \[ \lim_{x \rightarrow \infty}\left\{ \frac{f(x+k)}{f(x)}\right\}= \phi, \] where $\phi=(1+\sqrt{5})/2$ denotes the well-known golden ratio and the real valued function $f$ on $\mathbb{R}$ satisfies the functional equation $f(x+2k)=f(x+k) + f(x)$ for every $x\in \mathbb{R}$. We complete the proof of the conjecture by giving out an entirely different approach for the other case.