Integrodifferential Equations In Hilbert Space Research Articles

Stochastic integrodifferential equations in Hilbert spaces with applications in electromagnetics

In this work we present some results on deterministic and stochastic integrodifferential equations in Hilbert spaces, motivated from and applied to problems arising from the mathematical modeling of electromagnetics fields in complex random media. We examine the mild, strong and classical well posedness for the Cauchy problem of the integrodifferential equation which describes Maxwell’s equations complemented with the general (and therefore nonlocal in time) linear constitutive relations describing such media, with either additive or multiplicative noise.

Approximate controllability of nonlinear stochastic impulsive integrodifferential systems in hilbert spaces

Many practical systems in physical and biological sciences have impulsive dynamical behaviours during the evolution process which can be modeled by impulsive differential equations. This paper studies the approximate controllability properties of nonlinear stochastic impulsive integrodifferential and neutral functional stochastic impulsive integrodifferential equations in Hilbert spaces. Assuming the conditions for the approximate controllability of these linear systems we obtain the sufficient conditions for the approximate controllability of these associated nonlinear stochastic impulsive integrodifferential systems in Hilbert spaces. The results are obtained by using the Nussbaum fixed-point theorem. Finally, two examples are presented to illustrate the utility of the proposed result.

Weak Solutions of Abstract Evolutionary Integro-Differential Equations in Hilbert Spaces

We prove existence and uniqueness of weak solutions to certain abstract evolutionary integro-differential equations in Hilbert spaces, including evolution equations of fractional order less than 1. Our results apply, e.g., to parabolic partial integro-differential equations in divergence form with merely bounded and measurable coefficients. AMS subject classification: 45N05, 45K05

On Some Fractional Stochastic Integrodifferential Equations in Hilbert Space

We study a class of fractional stochastic integrodifferential equations considered in a real Hilbert space. The existence and uniqueness of the Mild solutions of the considered problem is also studied. We also give an application for stochastic integropartial differential equations of fractional order.

Controllability of McKean–Vlasov Stochastic Integrodifferential Evolution Equation in Hilbert Spaces

Sufficient conditions for controllability of McKean–Vlasov stochastic integrodifferential evolution equation in Hilbert spaces are obtained by using Banach fixed point analysis approach. This paper is motivated primarily by the work of Ahmed and Ding (Stochastic Processes Appl. 1995; 60:65–85), which is extended to a more general class of McKean–Vlasov stochastic integrodifferential equation in Hilbert space for establishing controllability result.

ON SEMI-LINEAR SECOND ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS IN HILBERT SPACE

Existence, uniqueness and regularity of weak solutions of semilinear second order Volterra integro-differential equations in Hilbert space are established by the variational method. As an application we give a well-posedness result for the semilinear viscoelastic equations with long memory.

ON SEMI-LINEAR SECOND ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS IN HILBERT SPACE

Existence, uniqueness and regularity of weak solutions of semilinear second order Volterra integro-differential equations in Hilbert space are established by the variational method. As an application we give a well-posedness result for the semilinear viscoelastic equations with long memory.

A stability result for a class of nonlinear integrodifferential equations with L1kernels

We study second order nonlinear integro-differential equations in Hilbert spaces with weakly singular convolution kernels obtaining energy estimates for the solutions, uniform in $t$. Then we show that the solutions decay exponentially at $\infty $ in the

ON EXPONENTIAL ASYMPTOTIC STABILITY IN LINEAR VISCOELASTICITY

This paper establishes results concerning the exponential decay of strong solutions of a linear hyperbolic integrodifferential equation in Hilbert space. Rather than the more commonly used assumptions that the relaxation function is non-negative, decreasing and convex, dissipation is modelled by requiring that the dynamic viscosity be a positive function. This restriction has a firm thermodynamic basis. Frequency domain methods are employed.

Semilinear Integrodifferential Equations of Hyperbolic Type: Existence in the Large

The main object of this work is the existence of global solutions to semilinear integrodifferential equations in Hilbert spaces for absolutely continuous convolution kernels, where the nonlinear term is the gradient of a functional having superlinear growth at infinity. An application to the theory of viscoelasticity concludes the paper providing motivations for the abstract theory.

Asymptotic Behavior of a Class of Abstract Semilinear Integrodifferential Equations and Applications

A class of semilinear hyperbolic Volterra integrodifferential equations in Hilbert spaces is considered. The main results in this paper are the global existence of solutions to the initial problem of the equation for large data and an asymptotic estimation of the solution. An application to a system relating to three-dimensional viscoelastic dynamics is presented.

Uniform asymptotic stability via Liapunov-Razumikhin technique

The Liapunov-Razumikhin technique is applied to obtain the uniform asymptotic stability for linear integrodifferential equations in Hilbert spaces, \[ x ′ ( t ) = A [ x ( t ) + ∫ # t F ( t − s ) x ( s ) d s ] , t ≥ t 0 ≥ 0 ( # = 0 or − ∞ ) , x’(t) = A\left [ {x(t) + \int _\# ^t {F(t - s)x(s)\,ds} } \right ],\quad t \geq {t_0} \geq 0(\# = 0\;{\text {or}} - \infty ), \] which occur in viscoelasticity and in heat conduction for materials with memory.

Application of Rothe's method to nonlinear integro-differential equations in Hilbert spaces

Application of Rothe's method to nonlinear integro-differential equations in Hilbert spaces

Rothe′s method to semilinear hyperbolic integrodifferential equations

In this paper we consider an application of Rothe′s method to abstract semi‐linear hyperbolic integrodifferential equations in Hilbert spaces. With the aid of Rothe′s method we establish the existence of a unique strong solution.

Exponential stabilization of Volterra integrodifferential equations in Hilbert space

We consider exponential stabilization of an abstract linear Volterra integrodifferential equation in a Hilbert space u ′=−E 1Au(t)+E 2∫ 0 1k(t−s)Au(s)ds+f(t) by a feedback control f( t) = − C 0u( t) − C 1u′( t) with bounded linear operators C 0 and C 1 of finite rank. A is a positive semidefinite, self-adjoint unbounded operator, k is nonnegative, convex, and decreasing with exponential decay, k(0) being finite. We introduce a concept of essential growth rate for resolvent operators, similar to that for C 0 -semigroups and prove that the resolvent for the Volterra equation has the same essential growth as the solution semigroup to u′'(t)= − E 1 Au(t) − E 2k(0) E 1 u′(t) + f(t) . Stability can be checked by the location of poles, and it is shown that exponential stabilization is possible with a decay rate of at most E 2k(0) 2E 1 . We apply our results to two problems concerning stabilization of motion of viscoelastic bodies.

A linear Volterra integro-differential equation for viscoelastic rods and plates

It is proved that the resolvent kernel of a certain Volterra integrodifferential equation in Hilbert space is absolutely integrable on ( 0 , ∞ ) \left ( {0,\infty } \right ) . Weaker assumptions on the convolution kernel appearing in the integral term are used than in existing results. The equation arises in the linear theory of isotropic viscoelastic rods and plates.

Uniform $L^1 $ Behavior in Classes of Integrodifferential Equations with Completely Monotonic Kernels

We find conditions on a family $\mathfrak{a}$ of completely monotone, locally integrable, nonconstant functions a which enable us to write the solution $u(t;a)$ of $u'(t) + \int_0^t {a(t - s)u(s)ds = 0} $, $u(0) = 1$, as $u(t;a) = - \int_{ - \varepsilon }^0 {e^{\sigma t} } d\mu (\sigma ;a) + u_1 (t;a)$, where $\mu ( \cdot ;a)$ is a finite nonnegative measure on $[ - \varepsilon ,0]$ and $|u_1 (t;a)| \leqslant Qe^{ - \varepsilon t} $ with Q, $\varepsilon $ positive constants independent of $a \in \mathfrak{a}$. This formula is then utilized to give conditions on the collection $\mathfrak{a}$ which ensure that $\rho (t)\sup _{a \in \mathfrak{a}} |u(t;a)| \to 0(t \to \infty )$ and $\int_0^\infty {\rho (t)\sup _{a \in \mathfrak{a}} |u(t;a)|dt < \infty } $ , where $\rho $ is a given weight function. These results can be combined with a resolvent formula to investigate the asymptotic behavior as $t \to \infty $ of solutions of certain integrodifferential equations in Hilbert space.

A linear integro-differential equation for viscoelastic rods and plates

It is proved that the resolvent kernel of a certain integrodifferential equation in Hilbert space is absolutely integrable on ( 0 , ∞ ) \left ( {0,\infty } \right ) . The equation arises in the linear theory of isotropic viscoelastic rods and plates.

A Version of the Chain Rule and Integrodifferential Equations in Hilbert Spaces

For Hilbert-space valued functions u, subdifferentials $\partial \varphi $ and certain scalar kernels a, an estimate on $\int_0^t {\langle {\dot u,{d / ds}\,a * \partial \varphi (u)} \rangle } $ is derived and then applied to certain integrodifferential equations.

Lower bounds for solutions to a class of nonlinear integrodifferential equations in hilbert space

Growth estimates are obtained for solutions of the abstract nonlinear initial-value problem where H + is a Hilbert subspace of some real Hilbert space H, with topologically as well as algebraically, and with H _ being the dual of H + via the inner product of H and denoting the space of all bounded symmetric linear operators from H + into H _.