Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations

This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results

Existence and asymptotic behavior of solutions for neutral stochastic partial integrodifferential equations with infinite delays

In this work we study the existence, uniqueness and asymptotic behavior of mild solutions for neutral stochastic partial integrodifferential equations with infinite delays. To prove the results, we use the theory of resolvent operators as developed by R. Grimmer [12] R. Grimmer. Resolvent operators for integral equations in a banach space. Transactions of the American Mathematical Society, 273(1): 333-349, 1982, as well as a version of the fixed point principle. We establish sufficient conditions ensuring that the mild solutions are exponentially stable in pth-moment. An example is provided to illustrate the abstract results.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Controllability of nonlocal impulsive stochastic quasilinear integrodifferential systems

Sufficient conditions for controllability of nonlocal impulsive stochastic quasilinear integrodifferential systems in Hilbert spaces are established. The results are obtained by using evolution operator, semigroup theory and fixed point technique. As an application, an example is provided to illustrate the obtained result

Global existence and controllability to a stochastic integro-differential equation

In this paper, we are focused upon the global uniqueness results for a stochastic integro-differential equation in Fréchet spaces. The main results are proved by using the resolvent operators combined with a nonlinear alternative of Leray-Schauder type in Fréchet spaces due to Frigon and Granas. As an application, a controllability result with one parameter is given to illustrate the theory

General existence results for abstract McKean-Vlasov stochastic equations with variable delay

Results concerning the global existence and uniqueness of mild solutions for a class of first-order abstract stochastic integro-differential equations with variable delay in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time t, but also on the corresponding probability distribution at time t are established. The classical Lipschitz is replaced by a weaker so-called Caratheodory condition under which we still maintain uniqueness. The time-dependent case is discussed, as well as an extension of the theory to the case of a nonlocal initial condition. Two examples illustrating the applicability of the general theory are provided