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

Abstract

In this paper, we solve in closed-form the following third-order system of nonlinear difference equations xn+1 = ynyn−1xn−1p xn(anyn−2q + bnynyn−1),yn+1 = xnxn−1yn−1q yn(cnxn−2p + dnxnxn−1),p,q ∈ ℕ,n ∈ ℕ0 where the initial values x−i,y−i,i = 0,1,2 and the parameters (an)n∈ℕ0,(bn)n∈ℕ0,(cn)n∈ℕ0, (dn)n∈ℕ0 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.