%0 Journal Article
%A Rabago, J. F.
%T On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysangâ€™s Conjecture
%J Iranian Journal of Mathematical Sciences and Informatics
%V 13
%N 1
%U http://ijmsi.ir/article-1-775-en.html
%R
%D 2018
%K Golden ratio, Fibonacci functional equation, Horadam functional equation, convergence.,
%X 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/(±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 Sroysang’s 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.
%> http://ijmsi.ir/article-1-775-en.pdf
%P 139-151
%& 139
%!
%9 Research paper
%L A-10-1534-1
%+ University of the Philippines Baguio
%G eng
%@ 1735-4463
%[ 2018