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

Authors : Merve Kara

Pages : 126-136

Doi:10.47000/tjmcs.1366596

View : 45 | Download : 41

Publication Date : 2024-06-30

Article Type : Research Paper

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