Asymptotic behavior and oscillation of functional differential equations

Abstract

Asymptotic relations between the solutions of a linear autonomous functional differential equation and the solutions of the corresponding perturbed equation are established. In the scalar case, it is shown that the existence of a nonoscillatory solution of the perturbed equation often implies the existence of a real eigenvalue of the limiting equation. The proofs are based on a recent Perron type theorem for functional differential equations.

Asymptotic Expansions for Higher-Order Scalar Difference Equations

“…This result is in fact a consequence of the asymptotic expansion of the solutions of Poincaré difference equation established in Theorem 2.3. Our results may be viewed as the discrete analogs of similar qualitative results known for ordinary and functional differential equations (see [4, Chapter 13, Theorem 4.5], [5,Proposition 7.2], or [6,Theorem 3.1]). The simple short proof presented below is based on the inversion formula for the z-transform and the residue theorem.…”

Asymptotically exponential solutions in nonlinear integral and differential equations

“…The study of asymptotic properties of different classes of integral and differential equations is an active research area, see, e.g., [6][7][8]12,14,15,17,[24][25][26]28] and the references therein. Most of the work in this direction has been done for linear equations, and guarantees only pure exponential growth/decay of the solutions.…”

A Perron-type theorem for nonautonomous delay equations

“…Earlier results were obtained by Perron [9], Lettenmeyer [7], and Hartman and Wintner [6]. Corresponding results for perturbations of autonomous delay equations = L were obtained by Pituk [10,11] (for values in C and finite delay) and Matsui, Matsunaga and Murakami [8] (for values in a Banach space and infinite delay). Related results for perturbations of autonomous difference equations were first obtained by Coffman [3].…”