Representation of solutions of a solvable nonlinear difference equation of second order
We present a representation of welldefined solutions to the following nonlinear secondorder difference equation xn+1 = a + b xn c xnxn−1 , n ∈ N0, where parameters a, b, c, and initial values x−1 and x0 are complex numbers such that c 6= 0, in terms of the parameters, initial values, and a special...
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Format:  Serial 
Published: 
2018

Series:  Electronic journal of qualitative theory of differential equations

Kulcsszavak:  Másodrendű differenciálegyenlet 
doi:  10.14232/ejqtde.2018.1.95 
Online Access:  http://acta.bibl.uszeged.hu/56907 
Summary:  We present a representation of welldefined solutions to the following nonlinear secondorder difference equation xn+1 = a + b xn c xnxn−1 , n ∈ N0, where parameters a, b, c, and initial values x−1 and x0 are complex numbers such that c 6= 0, in terms of the parameters, initial values, and a special solution to a thirdorder homogeneous linear difference equation with constant coefficients associated to the nonlinear difference equation, generalizing a recent result in the literature, completing the proof therein by using an essentially constructive method, and giving some theoretical explanations related to the method for solving the difference equation. We also give a more concrete representation of the solutions to the nonlinear difference equation by calculating the special solution to the thirdorder homogeneous linear difference equation in terms of the zeros of the characteristic polynomial associated to the linear difference equation. 

Physical Description:  118 
ISSN:  14173875 