Existence, uniqueness and stability of stochastic neutral functional differential equations of Sobolev-type

In this paper, we are mainly concerned with a class of stochastic neutral functional differential equations of Sobolev-type with Poisson jumps. Under two different sets of conditions, we establish the existence of the mild solution by applying the Leray-Schauder alternative theory and the Sadakovskii´s fixed point theorem, respectively. Furthermore, we use the Bihari´s inequality to prove the Osgood type uniqueness. Also, the mean square exponential stability is investigated by applying the gronwall inequality. Finally, two examples are given to illustrate the theory results.