Dynamics and global stability of higher order nonlinear difference equation (Q2792522)

This paper is concerned with the difference equation NEWLINE\[NEWLINE x_{n+1}=ax_{n-l}+bx_{n-k}+\frac{cx_{n-s}+dx_{n-t}}{ex_{n-s}+fx_{n-t}} NEWLINE\]NEWLINE under the initial conditions that \(x_{-r},x_{-r+1},\dots,x_{-1},x_{0}\) are positive, where the parameters \(a,b,c,d,e,f\) are positive numbers and \(r=\max \{l,k,s,t\}\).NEWLINENEWLINEUnder the condition that \(a+b<1\), it is shown that every solution is bounded and that NEWLINE\[NEWLINE \overline{x}=\frac{c+d}{(1-a-b)(e+f)} NEWLINE\]NEWLINE is the unique positive equilibrium point which is locally asymptotically stable if we assume further that NEWLINE\[NEWLINE 2\left| cf-de\right| <\left( e+f\right) \left( c+d\right); NEWLINE\]NEWLINE and is a global attractor if we assume either \(cf-de\geq 0\) and \(d\geq c\), or \(de-cf\geq 0\) and \(c\geq d\).NEWLINENEWLINENecessary and sufficient conditions are also found such that the above equation has a periodic solution of prime period \(2\).