Stability Trichotomy, Path Stability, and Relative Stability for Positive Nonlinear Difference Equations of Higher Order

For nonlinear difference equations of higher order which satisfy certain positivity conditions, in particular an inequality involving the partial derivatives of the defining function, three kinds of global stability are considered. In case the difference equation is autonomous, conditions for stability trichotomy are given. A difference equation is said to possess stability trichotomy if either all (non-zero) solutions are unbounded or all solutions converge to zero or all (non-zero) solutions converge to a positive equilibrium. In case of a non-autonomous difference equation it is shown that under certain conditions the solutions, even if highly erratic, exhibit path stability. Path stability means that a solution when being disturbed comes finally back to its original behavior. For an autonomous and homogeneous difference equation with possibly unbounded solutions. conditions are specified under which the solutions are relative stable, which means that for any (non-zero) solution the ratio of any two successive values converges to the same value, the growth factor, irrespective of initial values. To prove these results, the given difference equation is considered as a discrete dynamical system to which methods from the recently developed theory of positive discrete dynamical systems are applied. The positivity assumptions made for the difference equation lead thereby to contractivity properties of the dynamical system with respect to a particular metric. The stability results obtained are applied to explore the behavior of various special nonlinear difference equations.