Datko-type theorems concerning asymptotic behaviour of exponential type in mean

Abstract

In this paper, we study the concept of exponential (in)stability in mean for stochastic skew-evolution semiflows, in which the exponential (in)stability in the classical sense is replaced by an average with respect to a probability measure. Our paper consists of three major results. The first is to obtain Datko-type characterizations for the exponential stability in mean of stochastic skew-evolution semiflows. Next, we acquire Datko-type characterizations for the exponential instability in mean by extending the stability techniques. The last is to extend Lyapunov-type equations to the case of exponential (in)stability in mean.