Subnormal Solutions of Second-Order Nonhomogeneous Linear Differential Equations with Periodic Coefficients

Zhi-Bo Huang,Zong-Xuan Chen,Qian Li

doi:10.1155/2009/416273

Abstract

We obtain the representations of the subnormal solutions of nonhomogeneous linear differential equation , where and are polynomials in such that and are not all constants, . We partly resolve the question raised by G. G. Gundersen and E. M. Steinbart in 1994.

Highlights

We use the standard notations from Nevanlinna theory in this paper see 1–3
The study of the properties of the solutions of a linear differential equation with periodic coefficients is one of the difficult aspects in the complex oscillation theory of differential equations. It is one of the important aspects since it relates to many special functions
Some important researches were done by different authors; see, for instance, 4–9

Summary

Introduction

We use the standard notations from Nevanlinna theory in this paper see 1–3. The study of the properties of the solutions of a linear differential equation with periodic coefficients is one of the difficult aspects in the complex oscillation theory of differential equations. Whether the conclusions of Theorem A and Theorem B can be generalized or not, Gundersen and Steinbart considered the second-order nonhomogeneous linear differential equations f P1 ez f P2 ez f R1 ez R2 e−z , 1.5 where P1 z , P2 z , R1 z , and R2 z are polynomials in z such that P1 z , P2 z are not both constants. They found the exact forms of all subnormal solutions of 1.5 , that is, what is mentioned in 6, Theorem 2.2, Theorem 2.3 and Theorem 2.4. I If deg P1 > deg P2 and deg P1 > deg R1, f z must have the form f z eβz g1 ez g2 e−z , 1.8 where β is a constant, g1 z and g2 z are polynomials in z. ii If deg P1 > deg P2 and deg P1 ≤ deg R1, f z must have the form f z eβz g1 ez g2 e−z c1zg[3] e−z c2g4 e−z g0 ez , 1.9 where β is a constant, c1 and c2 are constants that may or may not be equal to zero, g0 z may be equal to zero or may be a polynomial in z, g1 z , g2 z , g3 z , and g4 z are polynomials in z with deg{g3} ≥ 1

Lemmas for the Proof

Proof of Theorem