On the stability of a class of splitting methods for integro-differential equations

The classical convection–diffusion–reaction equation has the unphysical property that if a sudden change in the dependent variable is made at any point, it will be felt instantly everywhere. This phenomena violate the principle of causality. Over the years, several authors have proposed modifications in an effort to overcome the propagation speed defect. The purpose of this paper is to study, from analytical and numerical point of view a modification to the classical model that take into account the memory effects. Besides the finite speed of propagation, we establish an energy estimate to the exact solution. We also present a numerical method which has the same qualitative property of the exact solution. Finally we illustrate the theoretical results with some numerical simulations.     

On the asymptotic stability of solutions of Volterra integro-differential equations

Nonlinear differential delay equations are investigated by means of their associated semigroups. Conditions are found for which solutions have nonexponential decay, but nevertheless behave asymptotically as an inverse power of t.     

On a certain system of equations with delay,occurring in aeroelasticity

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 54, pp. 123–130, 1990.     

Asymptotic stability of a class of integro-differential equations

We examine the asymptotic stability of the zero solution of the first-order linear equation x′(t) = Ax(t) + ∝₀^tB(t ? s) x(s) ds, where B(t) is integrable and does not change sign on [0, ∞). The results are applied to an examination of the stability of equilibrium of some nonlinear population models.     

On the existence of solutions of a nonlinear integro-differential system of transport equations

We consider the system of equations describing transport processes in inhomogeneous distributed media, such as those in nuclear reactors. For a given system of equations, a mixed problem is posed. Under certain conditions on the initial data, we prove the global solvability of the problem in the weak generalized sense by using the standard scheme of nonlinear functional analysis.     

On a family of singular integro-differential equations

Integro-differential systems in which the matrix multiplying the derivative of the unknown vector function is identically singular are analyzed. This analysis is based on the special properties of the matrix polynomials associated with the original system. An existence theorem is proved, and a numerical method for finding the solution is proposed and justified.     

Absolute stability of a stochastic integro-differential system

A stochastic integro-differential system (1) $\text{x}\text{?}\text{(t, ω) = A(ω) X(t, ω) + ∝}{}_0{}^{\mathrm{<mtext>t</mtext>}}\text{B(t ? s, ω) X(s, ω) ds + b(ω) Φ(σ(t, ω))}\text{subject to}\text{σ(t, ω) = 〈c(ω), X(t, ω)〉, t ? 0}$ is considered where the integrals are interpreted as Bochner integrals. Existence and absolute stability of random solutions of (1) are studied.     

Stability of a system of Volterra integro-differential equations

Using new and known forms of Lyapunov functionals, this paper proposes new stability criteria for a system of Volterra integro-differential equations.     

Asymptotic solution of a system of integro-differential equations

An asymptotic solution of a system of inegro-differential equations is constructed for the case where turning points are present. Vinnychenko Kirovograd Pedagogic Institute, Kirovograd. Translated from Ukrainskii Matematicheskii Zhurnal. Vol. 49, No. 12, pp. 1617–1623, December, 1997     

On the integro-differential equations with reflection

In this paper, by developing important properties on the composition of functions with reflection, using some exponential dichotomy properties and an application of the fixed-point theorem, several new sufficient conditions for the existence and the uniqueness of an pseudo almost automorphic solutions with measure for some general-type reflection integro-differential equations. We suppose that the nonlinear part is measure pseudo almost automorphic and in which we distinguish the two constant and variable cases for the Lipschitz coefficients of the functions associated with this part. It is assumed that the linear part of the equation considered admits an exponential dichotomy. Finally, an application is given on the very interesting model of Markus and Yamabe.     

On the stability in terms of two measures for perturbed impulsive integro-differential equations

This paper establishes several stability criteria for perturbed impulsive integro-differential equations with fixed moments of impulsive effect. By using a new comparison theorem, which connects the solutions of perturbed system and the unperturbed one, some sufficient conditions for the stability in terms of two measures are obtained for the perturbed system while unperturbed one dissatisfied which because of the effect of the perturbed terms.     

On the perturbation of Volterra integro-differential equations

In this work, we will prove that every solution of a perturbed Volterra integro-differential equation can be approximated by a solution of the Volterra integro-differential equation.     

Solvability of a nonlinear nonstationary system of equations

V. Kapsukas Vilnius State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 3, pp. 452–463, July–September, 1989.     

Floquet theory and stability of nonlinear integro-differential equations

Summary One of the classical topics in the qualitative theory of differential equations is the Floquet theory. It provides a means to represent solutions and helps in particular for stability analysis. In this paper first we shall study Floquet theory for integro-differential equations (IDE), and then employ it to address stability problems for linear and nonlinear equations.     

Exponential stability of nonlinear impulsive neutral integro-differential equations

In this paper, a nonlinear impulsive neutral integro-differential equation with time-varying delays is considered. By establishing a singular impulsive delay integro-differential inequality and transforming the n$n$-dimensional impulsive neutral integro-differential equation to a 2n$2n$-dimensional singular impulsive delay integro-differential equation, some sufficient conditions ensuring the global exponential stability in PC¹$P{C}^1$ of the zero solution of an impulsive neutral integro-differential equation are obtained. The results extend and improve the earlier publications. An example is also discussed to illustrate the efficiency of the obtained results.     

Boundedness of solutions of a system of integro-differential equations

Let b: [?1, 0] →$\text{R}$ be a nondecreasing, strictly convex C²-function with b(? 1) = 0, and let g: $\text{R}$ⁿ → $\text{R}$ⁿ be a locally Lipschitzian mapping, which is the gradient of a function G: $\text{R}$ⁿ → $\text{R}$. Consider the following vector-valued integro-differential equation of the Levin-Nohel type

$\text{x}\text{?}\text{(t)=?∝}{}_{\mathrm{?1}}{}^0\text{b(θ)g(x(t + θ))dθ}$

. (E) This equation is used in applications to model various viscoelastic phenomena. By LaSalle's invariance principle, every bounded solution x(t) goes to a connected set of zeros of g, as time t goes to infinity. It is the purpose of this paper to give several geometric criteria assuring the boundedness of solutions of (E) or some of its components.