Global existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditions

The aim of this paper is to discuss the global existence, uniqueness and asymptotic stability of mild solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. A sufficient condition is given for judging the relative compactness of a class of abstract continuous family of functions on infinite intervals. With the aid of this criteria the compactness of the solution operator for the problem studied on the half line is obtained. The theorems proved in this paper improve and extend some related results in this direction. Discussions are based on stochastic analysis theory, analytic semigroup theory, relevant fixed point theory and the well known Gronwall-Bellman type inequality. An example to illustrate the feasibility of our main results is also given.