On a General Non-Linear Difference Equation of Third-Order

Abstract

In this paper, we investigate the following general difference equations \begin{equation*} x_{n+1}=h^{-1}\left( h\left( x_{n}\right) \frac{Ah\left( x_{n-1}\right)+Bh\left( x_{n-2}\right) }{Ch\left( x_{n-1}\right)+Dh\left( x_{n-2}\right)}\right) ,\ n\in \mathbb{N}_{0}, \end{equation*} where the parameters $A, B, C, D$ and the initial values $x_{-\Phi}$, for $\Phi=\overline{0,2}$ are real numbers, $h$ is a continuous and strictly monotone function, $h\left( \mathbb{R}\right) =\mathbb{R}$, $h\left( 0\right) =0$. In addition, we obtain closed-form solutions of aforementioned difference equations. Finally, numerical applications are given.

Keywords

• Riccati difference equation
• solution
• closed form

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