Solvability of a third-order system of nonlinear difference equations via a generalized Fibonacci sequence

Hamioud, Hamida and Dekkar, Imane and Touafek, Nouressadat (2024) Solvability of a third-order system of nonlinear difference equations via a generalized Fibonacci sequence. Miskolc Mathematical Notes, 25 (1). pp. 271-285. ISSN 1787-2413

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Abstract

In this paper, we solve in closed-form the following third-order system of nonlinear difference equations x_{n+1}=(y_ny_{n-1}x^p_{n-1})/(x_{n}(a_n y^q_{n-2}+b_n y_n y_{n-1})), y_{n+1}=(x_n x_{n-1}y^q_{n-1})/(y_{n}(c_n x^p_{n-2}+d_n x_n x_{n-1})), p, q ∈ ℕ, n ∈ ℕ_0 where the initial values x−i, y−i, i = 0, 1, 2 and the parameters (a_n)n∈ℕ_0, (b_n)n∈ℕ0, (cn)n∈ℕ_0, (dn)n∈ℕ_0 are non-zero real numbers. The form of the solutions of the one dimensional case of our system and a more general system defined by one to one functions are also presented.

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