TY - JOUR
T1 - On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysangâ€™s Conjecture
TT -
JF - IJMSI
JO - IJMSI
VL - 13
IS - 1
UR - http://ijmsi.ir/article-1-775-en.html
Y1 - 2018
SP - 139
EP - 151
KW - Golden ratio
KW - Fibonacci functional equation
KW - Horadam functional equation
KW - convergence.
N2 - 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.
M3
ER -