On the rational recursive sequence \(x_{n+1}=Ax_{n}+Bx_{n-k}+\frac{\beta x_{n}+\gamma x_{n-k}}{cx_{n}+Dx_{n-k}}\)
The authors investigate some qualitative behavior of the solutions of the difference equation \[ x_{n+1}=Ax_{n}+Bx_{n-k}+\dfrac{\beta x_{n}+\gamma x_{n-k}}{Cx_{n}+Dx_{n-k} },\quad n=0,1,2,\dots, \] where the initial conditions \(x_{-k},\dots,x_{-1},x_{0}\) and the coefficients \(A,B,C,D,\beta,\gamma\) are positive real numbers, while \(k>0\) is an integer. Among other properties, they analyse the local and global stability, the boundedness, and the existence of periodic solutions. Finally, they consider some interesting examples.
scientific article; zbMATH DE number 5760431
scientific article