Oscillation of solutions of second-order linear differential equations and corresponding difference equations

In this paper, we present conditions ensuring that solutions of linear second-order differential equations oscillate, provided solutions of corresponding difference equations oscillate. We also establish the converse result, namely, when oscillation of solutions of difference equations implies oscillation of solutions of corresponding differential equations.     

Dissipativity of Volterra functional differential equations

This paper is concerned with the dissipativity of Volterra functional differential equations in a Hilbert space. A sufficient condition for dissipativity of one class of such equations is obtained. This result is applied to delay differential equations and integro-differential equations to obtain dissipativity results that are more general and deeper than related results in the previous literature.     

Dissipativity of delay functional differential equations with bounded lag

Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if θ=1. One numerical example is given to confirm our theoretical result.     

Dissipativity of Runge-Kutta methods for Volterra functional differential equations

This paper is concerned with the numerical dissipativity of nonlinear Volterra functional differential equations (VFDEs). We give some dissipativity results of Runge-Kutta methods when they are applied to VFDEs. These results provide unified theoretical foundation for the numerical dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice. Numerical examples are given to confirm our theoretical results.     

Some differences between difference equations and differential equations

In this expository paper we discuss some instances in which analogues which might be expected between the behavior of solutions of differential equations and difference equations fail to hold, focussing particularly on questions related to boundedness and oscillation.     

On nonlinear difference and differential equations

A generalization of the logarithmic norm to nonlinear operators, the Dahlquist constant is introduced as a useful tool for the estimation and analysis of error propagation in general nonlinear first-order ODE's. It is a counterpart to the Lipschitz constant which has similar applications to difference equations. While Lipschitz constants can also be used for ODE's, estimates based on the Dahlquist constant always give sharper results.The analogy between difference and differential equations is investigated, and some existence and uniqueness results for nonlinear (algebraic) equations are given. We finally apply the formalism to the implicit Euler method, deriving a rigorous global error bound for stiff nonlinear problems.Dedicated to my teacher and friend, Professor Germund Dahlquist, on the occasion of his 60th birthday.     

Periodic solutions of certain hybrid delay-differential equations and their corresponding difference equations

In this paper we use a method due to Carvalho [L.A.V. Carvalho, On a method to investigate bifurcation of periodic solution in retarded differential equations, J. Difference Equ. Appl. 4 (1998) 17–27] to obtain conditions for the existence of nonconstant periodic solutions of certain systems of hybrid delay-differential equations. We first deal with a scalar equation of Lotka-Valterra type, followed by a system of 2 equations in 2 unknowns that could model the interactions of 2 identical neurons. It will be seen that such solutions are determined by solutions of corresponding difference equations. Another paper in which this method is used is by Cook and Ladeira [K.L. Cook, L.A.C. Laderia, Applying Carvalho’s method to find periodic solutions of difference equations, J. Difference Equ. Appl. 2 (1996) 105–115. [2]].     

Periodic solutions of certain hybrid delay-differential equations and their corresponding difference equations

In this paper we use a method due to Carvalho (A method to investigate bifurcation of periodic solution in retarded differential equations, J. Differ. Equ. Appl. 4 (1998), pp. 17–27) to obtain conditions for the existence of nonconstant periodic solutions of certain systems of hybrid delay-differential equations. We first deal with a scalar equation of Lotka–Valterra type; then a system of two equations in two unknowns that could model the interactions of two identical neurons. It will be seen that such solutions are determined by solutions of corresponding difference equations. Another paper in which this method is used is by Cooke and Ladeira (Applying Carvalho's method to find periodic solutions of difference equations, J. Differ. Equ. Appl. 2 (1996), pp. 105–115).

We first state Carvalho's result.     

Interior and boundary difference equations for hyperbolic differential equations

Interior and boundary difference equations are derived for several hyperbolic partial differential equations by means of an integral method. The method is applied to a simple transport equation, to waves in a compressible, isentropic fluid, and to surface waves in shallow water. Boundary conditions treated are (a) a perfectly reflecting boundary, (b) an open boundary with outgoing waves and a specified incoming wave, and (c) a partially reflecting boundary. For open boundaries, the major assumption for the algorithms to be valid is that outgoing waves can be defined, an assumption equivalent to the most general statement of Sommerfeld's radiation condition. The difference equations obtained are conservative, second-order accurate, two time-level, explicit, and stable (for one-dimensional, time-dependent problems) for cΔt/Δx ? 1 where c is the wave speed, Δt is the temporal grid size, and Δx is the spatial grid size. Numerical calculations demonstrate the excellent accuracy of the procedure.     

The geometry of differential difference equations

To each maximal commuting subalgebra h of gl_m( )is associated a system of differential difference equations, generalizing several known systems. Starting from a Grassmann manifold, solutions are constructed, their properties are discussed and the relation with other systems is given. Finally it is shown how to express these solutions in τ-functions.     

Stability of difference approximations to differential equations

Consider the differential equation (1) ? = f(x) in a Banach space and let x¹ be an equilibrium. The basic question treated is the following: if x¹ is asymptotically stable and if (2) x_{k + 1} = x_k + h?(x_k, h) is a one-step method, with fixed step size h, for integrating (1), then does the sequence x_k converge to x¹? It is shown that uniform asymptotic stability of (1) implies stability of (2) and that exponential asymptotic stability of (1) implies asymptotic stability of (2).     

On the strong stability of differential equations and difference schemes

Paper deals with the construction of a priori estimates for the solution of Cauchy problem for an abstract linear differential equation in Hilbert space under perturbations of the initial condition, right-hand side, and operators of the problem. It is shown that a priori estimates of strong stability can be obtained directly on the basis of various a priori estimates for the solution of the Cauchy problem. The perturbations of the operators of the problem are estimated in the corresponding operator norms. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monotony of solutions of some difference and differential equations

In this contribution motivated by some analysis of the first author concerning bounds of topological entropy it is shown that a well known sufficient condition for a difference and differential equation with constant real coefficients to possess strictly monotone solution appears to be also necessary. Transparent proofs of adequate generalizations to Banach space analogs are presented.     

A note on the oscillatory property for nonlinear difference equations and differential equations

Criteria are given to determine the oscillatory property of solutions of the nonlinear difference equation: Δ^du_n + ∑_{i = 1}^mp_inf_i(u_n, Δu_n,…,Δ^{d ? 1}u_n) = 0, n = 0, 1, 2,…, where d is an arbitrary integer, generalizing results that have been obtained by B. Szmanda (J. Math. Anal. Appl.79 (1981), 90–95) for d = 2. Analogous results are given for the differential equation: u^(d) + ∑_{i = 1}^mp_i(t)f_i(u, u′,…, u^{(d ? 1)}) = 0, t ? t₀, which coincide with the criteria given by 2., 3., 599–602) and 4., 5., 6., 715–719) for the case m = 1.     

On stability of linear periodic differential difference equations

The paper considers the equation

where the operator-valued bounded functions a_j and b_j are 2π-periodic, and the operator-valued kernels m and n are 2π-periodic with respect to the first argument. The connection between the input-output stability of the equation and the invertibility of a family of operators acting on the space of periodic functions is investigated.     

Periodic solutions of non linear differential difference equations

In this paper we will be interested in the behaviour of long chains of coupled gravitational pendula. We will prove existence and uniqueness of periodic solutions for such chains under periodic forcing and will prove that under some smoothness assumptions the chain behaves as an uncoupled one. We will also analyse a more general class of differential difference equations and prove existence and unicity results for periodic solutions.Research partially supported by AFOSR under U.R.I, contract F49620-86-C-0131 to Northeastern University.