Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

Gro Hovhannisyan

doi:10.1155/2013/946453

Abstract

We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinson's asymptotic theorem. Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theorem we prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.

Highlights

Consider an ordinary nonautonomous differential equation of the nth order nL (u (t)) = ∑aj (t) u(j) (t) j=0 (1)= 0, t > t0, an = 1, with complex-valued continuous variable coefficients aj(t), j = 0, . . . n − 1.A solution of (1) is said to be oscillatory if it has an infinite sequence of zeros in (t0, ∞) and nonoscillatory otherwise
Oscillation theorems for ordinary differential equation of the nth order in the case of real variable coefficients have been studied in many papers
Theorem 2 shows that the oscillations of the solutions could be produced by the complex conjugate phase functions, which one can see from the following example

Summary

Introduction

= 0, t > t0, an = 1, with complex-valued continuous variable coefficients aj(t), j = 0, . Equation (1) is said to be nonoscillatory if all nontrivial solutions are nonoscillatory. Oscillation theorems for ordinary differential equation of the nth order in the case of real variable coefficients have been studied in many papers (see [1, 2] and references therein). To the best of the author’s knowledge the oscillations of the solutions of nonautonomous nth order equations with complex coefficients have not been studied yet (except [3]). Let Ck(t0, ∞) be the set of k times differentiable functions on (t0, ∞) and L1(t0, ∞) the set of Lebesgue absolutely integrable functions on (t0, ∞). To consider the case of complex coefficients we are using asymptotic solutions of (1) in Euler form u(t) = e∫tt0 f(s)ds.

Main Theorems

Proofs