Existence of nonnegative mild solutions of stochastic evolution inclusions via weak topology

This paper addresses the existence of nonnegative mild solutions for stochastic evolution inclusions through a weak topology approach. Precisely, the study focuses on stochastic evolution inclusions characterized by multivalued nonlinearities and perturbed by a Q-Wiener process within a Hilbert space framework. The primary objective is to establish the existence of mild solutions under conditions that ensure sublinear growth for the involved functions and multivalued mappings. By means of the weak topology method, we establish the existence of at least one mild solution in an appropriate function space. Additionally, when the associated semigroup is positive and a specific sign condition is met, we show the existence of nonnegative solutions. The methodological approach involves approximating the problem via truncation on bounded intervals and applying fixed-point theorems in weak topology to extend local solutions to the entire half-line.