Donald A. Lutz

Exponentially asymptotically constant systems of difference equations with an application to hyperbolic equilibria

In a recent paper, Agarwal and Pituk have considered scalar linear difference equations whose coefficients are asymptotically constant and whose corresponding perturbations are exponentially small. Using the method of generating functions and results from complex analysis, they derived an asymptotic representation for solutions, which was then applied to study the asymptotic behaviour of solutions of certain nonlinear autonomous scalar difference equations near a hyperbolic equilibrium. Here, we first show using standard matrix analysis how their results can be extended to systems of linear difference equations and that the error estimates can be made more precise. Our method also can be extended to weakly nonlinear systems. That result can in turn be used to analyze solutions of some autonomous nonlinear difference systems near an equilibrium.

On the remainders of the asymptotic expansion of solutions of differential equations near irregular singular points

We consider systems of linear differential equations with an irregular singular point of Poincare rank 1 at infinity. It is well known that there is a fundamental system of formal solution vectors and, for each halfplane, a fundamental system of actual solution vectors having the formal ones as asymptotic expansions. These asymptotic expansions (in the sense of Poincare) describe the behavior of the actual solutions as theindependent variable zgrows indefinitely, but give no precise error bounds for a given z, if the asymptotic series are truncated after Nterms. In this paper we show that for large values of |z| the best choice of Nis proportional to |z| and that the resulting error terms are exponentially small.

Asymptotic integration of differential and difference equations

Introduction, Notation, and Background.- Asymptotic Integration for Differential Systems.- Asymptotics for Solutions of Difference Systems.- Conditioning Transformations for Differential Systems.- Conditioning Transformations for Difference Systems.- Perturbations of Jordan Differential Systems.- Perturbations of Jordan Difference Systems.- Applications to Classes of Scalar Linear Differential Equations.- Applications to Classes of Scalar Linear Difference Equations.- Asymptotics for Dynamic Equations on Time Scales.

A Hartman–Wintner theorem for dynamic equations on time scales

Classical results of the Hartman–Wintner type lead to the asymptotic integration of certain types of linear, non-autonomous systems of differential or difference equations. Here, we show how to extend such results to linear dynamic equations on time scales having bounded graininess, using both averaged as well as pointwise dichotomy conditions.

Perturbations of Jordan Differential Systems

Whereas in Chaps. 2 and 4, we studied the asymptotic behavior of solutions of perturbations of diagonal systems of differential equations, we are now interested in the asymptotic behavior of solutions of systems of the form

Conditioning Transformations for Difference Systems

\displaystyle{ y^{{\prime}} = \left [J(t) + R(t)\right ]y(t)t \geq t_{ 0}, }

Applications to Classes of Scalar Linear Differential Equations

(6.1) where J(t) is now in Jordan form and R(t) is again a perturbation. Early results on perturbations of constant Jordan blocks include works by Dunkel [50] and Hartman–Wintner [73]. The focus here is an approach, developed by Coppel and Eastham, to reduce perturbed Jordan systems to a situation where Levinson’s fundamental theorem can be applied.

Asymptotics for Dynamic Equations on Time Scales

This book is concerned with the problem of determining the asymptotic behavior of solutions of non-autonomous systems of linear differential and linear difference equations. It has been observed from the work by Poincare and Perron that there is a very close and symbiotic relationship between many results for differential and difference equations, and we wish to further demonstrate this by treating the asymptotic theories here in parallel. In Chaps. 2, 4, 6, and 8 we will discuss topics related to asymptotic behavior of solutions of differential equations, and in Chaps. 3, 5, 7, and 9 some corresponding results for difference equations. In Chap. 10 we will show how some of these results can be simultaneously treated within the framework of so-called dynamic equations on time scales.