On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang's Conjecture
Author(s):
Article Type:
Research/Original Article (دارای رتبه معتبر)
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/(±1 x(n)), n ∈ N_0. The form of solution of this equation, however, was first obtained in [10] but through induction principle. Then, with the solution of the above equation at hand, we prove a case of Sroysangs conjecture (2013) [9] i.e., given a fixed positive integer k, we verify the validity of the following claim: lim x→∞ f(x k)/f(x) = φ, where φ = (1 √5)/2 denotes the well-known golden ratio and the real valued function f on R satisfies the functional equation f(x 2k) =f(x k) f(x) for every x ∈ R. We complete the proof of the conjecture by giving out an entirely different approach for the other case.
Language:
English
Published:
Iranian Journal of Mathematical Sciences and Informatics, Volume:13 Issue: 1, May 2018
Pages:
139 to 151
https://www.magiran.com/p1818359