On the global asymptotic stability of off-diagonally monotone dynamical systems

Global asymptotic stability of off-diagonally monotone nonlinear dynamical systems are studied. It is proved in this paper that if the nonlinear functions on the right hand sides are
M
-functions with negative signs then the systems are globally asymptotically stable without any more restrictive conditions whenever they have equilibrium points. To prove this the concept of weak
M
-functions is introduced. It is also shown that the condition obtained is insensible to delays even when delays are time varying, provided they are inserted off-diagonally.