Wiman’s formula for a second order dynamic equation

Lynn Erbe,Alan Peterson,Gro Hovhannisyan

doi:10.1186/1687-1847-2014-61

Abstract

Abstract We derive Wiman’s asymptotic formula for the number of generalized zeros of (nontrivial) solutions of a second order dynamic equation on a time scale. The proof is based on the asymptotic representation of solutions via exponential functions on a time scale. By using the Jeffreys et al. approximation we prove Wiman’s formula for a dynamic equation on a time scale. Further we show that using the Hartman-Wintner approximation one can derive another version of Wiman’s formula. We also prove some new oscillation theorems and discuss the results by means of several examples. MSC:34E20, 34N05.

Highlights

Consider the equation u u(t) (t) + (wσ (t)w(t)) =, t ≥ t, ( . )on [t, t) T = T ∩ [t, t), where T is a time scale
In this paper under some restrictions on the graininess of the time scale and the asymptotic behavior of the coefficient w(t), we obtain an explicit Jeffreys, Wentzel, Kramers and Brillouin (JWKB) asymptotic representation of solutions of ( . ). Using this representation we prove the analogue of Wiman’s formula for ( . ) on [t, ∞) T, which is given by t)
First we recall some basic definitions and notation used in time scale analysis

Summary

Introduction

Let N(t , t) be the number of generalized zeros of solutions of ) is oscillatory on [t , ∞), and w (t) is a differentiable function such that lim w (t) = , Is given by the formula t ep(t, t ) = exp lim ln + qp(s) t q μ(s) q s where ln is the principal logarithmic function. Assume η : [t , ∞) T → R and w : [t , ∞) T → ( , ∞) are rd-continuous functions, w(t) > on [t , ∞) T and let i θ (t) := η(t) + w (t) , θ (t)

Results

Conclusion