Functional integro‐differential stochastic evolution equations inHilbert space

Abstract

We investigate a class of abstract functional integro-differential stochastic evolution equations in a real separable Hilbert space. Global existence results concerning mild and periodic solutions are formulated under various growth and compactness conditions. Also, related convergence results are established and an example arising in the mathematical modeling of heat conduction is discussed to illustrate the abstract theory.

Existence Results for Stochastic Semilinear Differential Inclusions with Nonlocal Conditions

“…See for instance 9-12 and the references therein. So far, very few articles have been devoted to the study of stochastic differential inclusions with nonlocal conditions, see [13][14][15] and the references therein. Our objective is to contribute to the study of SDIns with nonlocal conditions.…”

On a Class of Measure-Dependent Stochastic Evolution Equations Driven by fBm

“…The purpose of this work is to study the class of abstract stochastic evolution equations obtained by accounting for more general nonlinear perturbations (in the sense of McKean-Vlasov equations, as described in [19]) in the mathematical description of phenomena involving an fBm. In particular, the existence and convergence results we present constitute generalizations of the theory governing standard models arising in the mathematical modeling of nonlinear diffusion processes [1,[16][17][18][19]22], communication networks [4], Sobolev-type equations arising in the study of consolidation of clay [8], shear in second-order fluids [23], and fluid flow through fissured rocks [24]. As a part of our general discussion, we establish an approximation result concerning the effect of the dependence of the nonlinearity on the probability law of the state process, as well as the noise arising from the stochastic integral, for a special case of (1.1) arising often in applications.…”

Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion

“…For the second way, we point out that accounting for certain types of nonlinearities in the mathematical modeling of some phenomena-for instance, nonlinear waves and the dynamic buckling of a hinged extensible beamrequires that the nonlinearities depend on the probability distribution of the solution process. This notion was first studied in the finite-dimensional setting [9,10], was initiated in the infinite-dimensional setting by Ahmed and Ding [11], and subsequently was studied by various authors [12,13].…”