On the decomposition problem of stability for Volterra integrodifferential equations

In this paper, we discuss the problem of stability of Volterra integrodifferential equationwith the decompositionwhere andin which is an matrix of functionB continuous forin which is an matrix of functionscontinuous for 0≤S≤ t<∞.According to the decomposition theory of large scale system and with the help of Liapunovfunctional, we give a criterion for concluding that the zero solution of (2) (i.e. large scale system(1)) is uniformly asymptotically stable.We also discuss the large scale system with the decompositionand give a criterion for determining that the solutions of (4) (i.e. large scale system (3)) areuniformly bounded and uniformly ultimately bounded.Those criteria are of simple forms, easily checked and applied.     

Runge-Kutta theory for Volterra integrodifferential equations

Summary The present paper develops the theory of general Runge-Kutta methods for Volterra integrodifferential equations. The local order is characterized in terms of the coefficients of the method. We investigate the global convergence of mixed and extended Runge-Kutta methods and give results on asymptotic error expansions. In a further section we construct examples of methods up to order 4.     

Second order linear Volterra integrodifferential equations

In this paper we study mild and classical solutions of the second order linear Volterra integrodifferential equation $$(VE^f )\left\{ {\begin{array}{*{20}c} {u''(t) = Au(t) + {\text{ }}\int_0^t {B(t - s)u(s)ds + f(t){\text{ }}for{\text{ }}t \in [0,T]} } \\ {u(0) = x{\text{ }}and{\text{ }}u'(0) = y,} \\ \end{array} } \right.$$ whereA is a closed linear operator whose domainD(A) is not necessarily dense in a Banach spaceX, and {B(t)|t≥0} is a family of bounded linear operators from the Banach space,D(A) endowed with the graph norm intoX. We also give two examples to illustrate the abstract results.     

Stability properties for partial Volterra integrodifferential equations

Summary Stability properties of the solutions of a Volterra's population equation including infinite delay and diffusion terms are studied via a linearized stability argument, which leads to investigating on operational characteristic equation. For a specific class of delay kernels time periodic solutions are shown to appear as the delay is increased.     

Stability of solutions of Volterra integrodifferential equations

Systems with past memory (or after-effect), the state of which is given by nonlinear Volterra- type integrodifferential equations with small perturbations, are investigated. The equations depend on functionals in integral form and, in particular, on analytic functionals represented by Fréchet series. The integral kernels can allow for singularities with Abel’s kernel. The stability under persistent disturbances, and the structure of the general solution, are investigated in the neighborhood of zero for an equation with holomorphic nonlinearity assuming asymptotic stability of the trivial solution of the linearized unperturbed equation. Stability in the critical cases (in Lyapunov’s sense) of a single zero root and of pairs of pure imaginary roots for the unperturbed equation is analyzed.     

Unique existence of periodic solutions of neutral Volterra integrodifferential equations

This paper deals with the uniqueness and existence of periodic solutions of neutral Volterra integrodifferential equations (1) and (2). Some new unique existence criteria are obtained.The project is supported by the National Natural Science Foundation of China.     

Local existence for nonlinear Volterra integrodifferential equations with infinite delay

In this paper, we investigate the local existence and uniqueness of solutions to integrodifferential equations with infinite delay, which are more general than those in previous studies. We assume that the linear part of the equation is nondensely defined and satisfies a Hille–Yosida condition. Moreover, the continuity of solutions with respect to initial conditions is also studied. In order to illustrate our abstract results, we conclude this work with an example.     

Natural Continuous Extensions of Runge-Kutta methods for Volterra integrodifferential equations

Summary In this paper we deal with a very general class of Runge-Kutta methods for the numerical solution of Volterra integrodifferential equations. Our main contribution is the development of the theory of Natural Continuous Extensions (NCEs), i.e. piecewice polynomial functions which interpolate the values given by the RK-method at the mesh points. The particular features of these NCEs allow us to construct tail approximations which are quite efficient since they require a minimal number of kernel evaluations.     

On three-particle integrodifferential equations

Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 81, No. 1, pp. 86–93, October, 1989.     

Stability analysis of multilag and modified multilag methods for Volterra integrodifferential equations

Stability analysis of multilag and modified multilag methodsfor Volterra integrodifferential equations is presented, withrespect to the nonconvolution test equation where , , µ, and are real parameters. The applicationof these methods to this test equation leads to difference equationswith variable coefficients which are of Poincar type. Usingthe extension of the Perron theorem, the conditions under whichthe solutions to such equations are bounded are derived. Asa consequence, a complete characterization of stability regionsof multilag and modified multilag methods with respect to theabove nonconvolution test equation is obtained.     

Mild solutions of functional semilinear evolution Volterra integrodifferential equations on an unbounded interval

In this paper, we study the existence of mild solutions for initial value problems for semilinear Volterra integrodifferential equations in a Banach space. The arguments are based on the concept of measure of noncompactness in Fréchet space and the Tikhonov fixed point theorem.     

Controllability of stochastic Volterra integrodifferential systems

In this paper, sufficient conditions for the controllability of stochastic integrodifferential systems in Banach spaces are established. The results are obtained by using a fixed point theorem. An example is provided to illustrate the theory.