The existence and asymptotic behaviour of solutions to non-Lipschitz stochastic functional evolution equations driven by Poisson jumps

In this paper, we consider the existence and uniqueness of the energy solutions to the following non-Lipschitz stochastic functional evolution equation driven both by Brownian motion and by Poisson jumps {dX(t) = [A(t, X(t)) + f(t, X-t)]dt + g(t, X-t)dW(t) + integral(U) k(t,X-t,y)q(dtdy), t >= ), X-0 = (sic) is an element of D([-r, 0],H), where A(t, .) : V -> V* is a linear or nonlinear bounded operator, f : [0, infinity) x D([-r, 0], H) -> H, g : [0, infinity) x D([-r, 0], H) -> L-2(0)(K, H) and [0, infinity) x D([-r, 0], H) x F -> H are measurable functions. We also investigate the almost sure exponential stability of energy solutions by using the energy equality for this equation.