Stochastic Integro-differential Equations Research Articles

On existence of solutions of a impulsive stochastic partial functional integro-differential equation with the measure of noncompactness

In this article, we study the existence of mild solutions for a new class of impulsive stochastic partial neutral functional integro-differential equations with infinite delay and non-instantaneous impulses in separable Hilbert spaces. The new results are obtained by using the Hausdorff measure of noncompactness, and the theory of analytic resolvent operators and fractional power of closed operators with the fixed point theorems. An example is also given to illustrate the obtained theory.

On classical solutions of linear stochastic integro-differential equations

We prove the existence of classical solutions to parabolic linear stochastic integro-differential equations with adapted coefficients using Feynman–Kac transformations, conditioning, and the interlacing of space-inverses of stochastic flows associated with the equations. The equations are forward and the derivation of existence does not use the “general theory” of SPDEs. Uniqueness is proved in the class of classical solutions with polynomial growth.

Sobolev-Type Fractional Stochastic Integrodifferential Equations with Nonlocal Conditions in Hilbert Space

We prove the existence of mild solution to Sobolev-type fractional stochastic integrodifferential equation with nonlocal conditions in Hilbert space. Also, we study the controllability of Sobolev-type fractional stochastic integrodifferential equations with impulsive conditions. We use uniformly continuous semigroups and fixed point technique for the main results. An example is provided to illustrate the theory.

The Solvability and Optimal Controls for Impulsive Fractional Stochastic Integro-Differential Equations via Resolvent Operators

In this manuscript, we investigate the solvability and optimal controls for impulsive fractional stochastic integro-differential equations in Hilbert space. Sufficient conditions are obtained for the existence of mild solution of the considered system by using analytic resolvent operators, the uniform continuity of the resolvent, and Leray---Schauder fixed point theorem. Then, the existence of optimal controls is discussed for the considered system. Finally, the obtained theoretical result is validated through an example.

Existence and stability solutions of nonlinear stochastic functional partial integrodifferential equations with infinite delay

In this paper, we are focused upon the results on existence, uniqueness and stability of mild solution for stochastic functional integrodifferential equations with non-Lipschitz condition. The theory of resolvent operator is utilised to exhibit the existence of these mild solutions. The results are obtained by using the method of successive approximation and Bihari's inequality. Finally, an example is provided for demonstration.

Sobolev type fractional stochastic integro-differential evolution

In this paper, we prove the existence of $\alpha$-mild solutions for a class of fractional stochastic integrodifferential evolution equations of sobolev type with fractional sobolev stochastic nonlocal conditions in a real separable Hilbert space. To establish our main results, we use the Banach contraction mapping principle, fractional calculus, stochastic analysis and an analytic semigroup of linear operators. An example is given to illustrate the feasibility of our abstract result.

Pseudo almost automorphic in distribution solutions of semilinear stochastic integro-differential equations by measure theory

This paper deals with the [Formula: see text]-pseudo almost automorphy in distribution for a stochastic process. Under suitable conditions, we establish the existence, uniqueness of [Formula: see text]-pseudo almost automorphic in distribution mild solutions for semilinear stochastic integro-differential equations.

Existence and stability results for a partial impulsive stochastic integro-differential equation with infinite delay

This article presents the result on existence and stability of mild solutions of stochastic partial differential equations with infinite delay in the phase space \(\mathcal {B}\) with non-lipschitz coefficients. We use the theory of resolvent operator devolopped in Grimmer (Trans Am Math Soc 273(1):333–349, 1982) to show the existence of mild solutions. An example is provided to illustrate the results of this work.

Neutral Stochastic Integrodifferential Equations Driven by a Fractional Brownian Motion with Impulsive Effects and Time-varying Delays

This paper deals with the existence,uniqueness and asymptotic behaviors of mild solutions to neutral stochastic delay functional integrodifferential equations with impulsive effects, perturbed by a fractional Brownian motion B H , with Hurst parameter \({H \in (\frac{1}{2},1)}\). We use the theory of resolvent operators developed in Grimmer (Trans Am Math Soc 273(1982):333–349, 2009) to show the existence of mild solutions. An example is provided to illustrate the results of this work.

Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion

In this paper, we establish the results on existence and uniqueness of mild solution of impulsive neutral stochastic integrodifferential equations driven by a fractional Brownian motion. Further, by using an impulsive integral inequality, some novel sufficient conditions are derived to ensure the exponential stability of mild solution in the mean square moment. The results are obtained by utilizing the fractional power of operators and the semigroup theory. Finally, an example is presented to demonstrate the effectiveness of the proposed result.

On a new class of impulsive stochastic partial neutral integro-differential equations

In this paper, we introduce a new class of impulsive stochastic partial neutral functional integro-differential equations with infinite delay and not instantaneous impulses in separable Hilbert spaces. We prove the existence of mild solutions for these equations in the -norm without the assumptions of compactness. The results are obtained using suitable fixed point theorems with the properties of analytic resolvent operators. Finally, an example is presented to illustrate the theory.

Existence and stability for impulsive neutral stochastic integrodifferential equations with infinite delays

This work presents result on the existence and stability of mild solutions for some stochastic partial differential equations with infinite delay with non-lipschitz coefficients. We use the theory of resolvent operator devolopped in R. Grimmer (1982)[8] to show the existence and stability of the mild solutions. An example is provided to illustrate the results of this work.

On degenerate linear stochastic evolution equations driven by jump processes

We prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equations driven by jump processes in a Hilbert scale using the variational framework of stochastic evolution equations and the method of vanishing viscosity. As an application of this result, we derive the existence and uniqueness of solutions of degenerate parabolic linear stochastic integro-differential equations (SIDEs) in the Sobolev scale. The SIDEs that we consider arise in the theory of non-linear filtering as the equations governing the conditional density of a degenerate jump–diffusion signal given a jump–diffusion observation, possibly with correlated noise.

On the asymptotic stability of impulsive neutral stochastic partial integrodifferential equations with variable delays and Poisson jumps

In this work, we study the existence and asymptotic stability in the \(p\)-th moment of mild solutions of nonlinear impulsive stochastic partial functional integrodifferential equations with delays. We suppose that the linear part possesses a resolvent operator in the sense given in Grimmer (Trans. Am. Math. Soc. 273(1), 333–349, 1982) and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is employed for achieving the required result. An example is provided to illustrate the results of this work.

Exponential Behavior of Solutions to Stochastic Integrodifferential Equations with Distributed Delays

In this work, we study the existence, uniqueness, and exponential asymptotic behavior of mild solutions to stochastic integrodifferential delay evolution equations. We assume that the non-delay part generates a C0-semigroup.

Approximate controllability of fractional stochastic integro-differential equations with infinite delay of order 1 < α < 2

Approximate controllability of fractional stochastic integro-differential equations with infinite delay of order 1 < α < 2

A parabolic Triebel–Lizorkin space estimate for the fractional Laplacian operator

In this paper we prove a parabolic Triebel-Lizorkin space estimate for the operator given by \[T^{\alpha}f(t,x) = \int_0^t \int_{{\mathbb R}^d} P^{\alpha}(t-s,x-y)f(s,y) dyds,\] where the kernel is \[P^{\alpha}(t,x) = \int_{{\mathbb R}^d} e^{2\pi ix\cdot\xi} e^{-t|\xi|^\alpha} d\xi.\] The operator $T^{\alpha}$ maps from $L^{p}F_{s}^{p,q}$ to $L^{p}F_{s+\alpha/p}^{p,q}$ continuously. It has an application to a class of stochastic integro-differential equations of the type $du = -(-\Delta)^{\alpha/2} u dt + f dX_t$.

Existence of mild solution result for fractional neutral stochastic integro-differential equations with nonlocal conditions and infinite delay

We investigate in this paper the existence of mild solutions for the fractional differential equations of neutral type with nonlocal conditions and infinite delay in Hilbert spaces by employing fractional calculus and Krasnoselski-Schaefer fixed point theorem. Finally an example is provided to illustrate the application of the obtained results.

EXISTENCE RESULTS FOR A NEW CLASS OF FRACTIONAL IMPULSIVE PARTIAL NEUTRAL STOCHASTIC INTEGRO-DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper, a new class of fractional impulsive partial neutral stochastic integro-differential equations with infinite delay is introduced.Under some dissipative conditions, we obtain the existence, uniqueness and continuous dependence of mild solutions for these equations. An application involving a fractional stochastic parabolic system with not instantaneous impulses is considered.

Stochastic stability of quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

A procedure for determining the asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises is proposed. For the case of resonance with α resonant relations, the averaged Ito^ stochastic differential equations (SDEs) for quasi-integrable and resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are first derived by using the stochastic averaging method. Then, the expression for the largest Lyapunov exponent is obtained by generalizing Khasminskii׳s procedure to the averaged Ito^ SDEs and using the property of the stochastic integro-differential equations (SIDEs). Finally, the stochastic stability of the original system is determined approximately by using the largest Lyapunov exponent. An example of two non-linear damping oscillators under parametric excitations of combined Gaussian and Poisson white noises is worked out to illustrate the application of the proposed procedure. The validity of the proposed procedure is verified by the good agreement between the analytical results and those from Monte Carlo simulation.