English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The purpose of this paper is twofold. First, we derive theoretically, usingappropriate transformation on $x_n$, the closed-form solution of the nonlineardifference 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 alreadyobtained by Tollu et al. in 2013, but through induction principle, and one ofour purpose is to clearly explain how was the formula appeared in suchstructure. After that, with the solution form of the above equation at hand, weprove a case of Sroysang's conjecture (2013); i.e., given a fixed positiveinteger $k$, 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 valuedfunction $f$ 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 bygiving out an entirely different approach for the other case.

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