Abstract. In the present paper we give some characterizations for the stability and instability of the evolution families by using the Lyapunov function, in Banach spaces. These results are a generalization of those obtained by N.U. Ahmed in  for the case of $C_0$-semigroups.
AMS Subject Classification
(1991): 34D05, 34D09, 47D06, 93D20
Lyapunov operator inequality,
Received August 31, 2012, and in revised form April 25, 2014. (Registered under 66/2012.)