Abstract: This paper extended the method of perturbing cone-valued Lyapunov functions for a system of difference equations. Some properties of $(h_0,h)$-stability and $(h_0,h)$-asymptotical stability were discussed under much weaker assumptions.

Key words: uniformly $\phi_0$-stable, $(h_0, h)$-stable, $(h_0, h)$-asymptotically stable, \phi_0$-stable, uniformly $\phi_0$-stable, $(h_0, h)$-stable, $(h_0, h)$-asymptotically stable