On a Nevanlinna type result for solutions of nonautonomous equations y′=a( t) F₁( x,y), x′=b( t)F₂( x,y )

Abstract

Oscillation problems (investigation of zeros) were widely studied for solutions of ordinary differential equation (ODE). In this article, we transfer oscillation problems to the case of solutions

of nonautonomous system of equations

. By analogy with the theory of oscillation for one ODE, we consider number n(t ₁,t ₂,0) of zeros τ _i in (t ₁,t ₂) of the solutions, that is the number of those points τ _i , where x(τ _i ) =0 and y(τ _i )=0. It turns out that the above bounds for n(t ₁,t ₂,0) can be given in terms of a( t) , b( t) , F ₁, F ₂, t ₁ and t ₂. Also by analogy with the concept of a-points in the complex analysis, we consider values a:=(a′,a″) in the ( x, y )-plane and define a-points of the solutions as those points τ _i , where

and

. Denoting by n(t ₁,t ₂, a) the number of a-points in (t ₁,t ₂) of the solutions we give above bounds for the sum

, where a ₁,a ₂,…,a_q is a given totality of pairwise different points. Thus we obtain for the solutions of the above equation an analog of the second fundamental theorem in the Nevanlinna value distribution theory; the last one also considers a similar sum for the number of a-points of meromorphic functions. As an immediate application we obtain below bounds for the periods of periodic solutions.

Keywords:

• Oscillations
• ODE
• Periodic solutions
• Nevanlinna theory
• AMS 2000 Mathematics Subject Classifications: 30D30
• 30C99
• 34A26
• 34A34
• 34A99
• 34C10
• 34C99

Notes

Dedicated to Heinrich Begehr on his 65th birthday.

Additional information

Notes on contributors

G. A. Barsegian

Dedicated to Heinrich Begehr on his 65th birthday.