Stability trichotomy, path stability, and relative stability for positive nonlinear difference equations of higher order

Abstract

For nonlinear difference equations of higher order which satisfy certain positivity conditions, in particular an inequalilty 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 s...