The boundedness and stability of neutral delay differential equations with damped stochastic perturbations via fixed point theory

In this paper, a class of neutral stochastic differential equation is studied. By using the fixed point method, some conditions are established for ensuring the equation is bounded, asymptotically stable and ΨγΨγ-stable in mean square. Then, with Borel–Cantelli lemma, several sufficient conditions for almost surely asymptotic stability and ΨγΨγ-stability are given. The conditions do not require the boundedness of delays, nor do they ask for the continuousness on the coefficient functions. The results of this paper are new and they extend and improve previously known results.