دوره 13، شماره 1 - ( 2-1397 )                   جلد 13 شماره 1 صفحات 139-151 | برگشت به فهرست نسخه ها

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Rabago J F. On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang’s Conjecture. IJMSI. 2018; 13 (1) :139-151
URL: http://ijmsi.ir/article-1-775-fa.html
On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang’s Conjecture. مجله علوم ریاضی و انفورماتیک ایرانیان. 1397; 13 (1) :139-151

URL: http://ijmsi.ir/article-1-775-fa.html


چکیده:  

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.

نوع مطالعه: پژوهشي | موضوع مقاله: تخصصي

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