ENTIRE SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS OF MALMQUIST TYPE

Ran-Ran Zhang,Zhi-Bo Huang

doi:10.11948/20190176

Abstract

The celebrated Malmquist theorem states that a differential equation, which admits a transcendental meromorphic solution, reduces into a Riccati differential equation. Motivated by the integrability of difference equations, this paper investigates the delay differential equations of form $ w(z+1)-w(z-1)+a(z)\frac{w'(z)}{w(z)} = R(z, w(z))(*), $ where $ R(z, w(z)) $ is an irreducible rational function in $ w(z) $ with rational coefficients and $ a(z) $ is a rational function. We characterize all reduced forms when the equation $ (*) $ admits a transcendental entire solution with hyper-order less than one. When we compare with the results obtained by Halburd and Korhonen[Proc. Amer. Math. Soc. 145, no.6 (2017)], we obtain the reduced forms without the assumptions that the denominator of rational function $ R(z, w(z)) $ has roots that are nonzero rational functions in $ z $. The value distribution and forms of transcendental entire solutions for the reduced delay differential equations are studied. The existence of finite iterated order entire solutions of the Kac-van Moerbeke delay differential equation is also detected.

Highlights

Introduction and resultsWe assume that the reader is familiar with the standard notations and basic results of the Nevanlinna theory, see e.g. [11]
In this paper, we investigate the delay differential equations of Malmquist type of form w(z
Let w be a meromorphic function in the complex plane

Summary

Introduction

Introduction and resultsWe assume that the reader is familiar with the standard notations and basic results of the Nevanlinna theory, see e.g. [11]. Theorem C has an assumption “Q(z, w) has roots that are nonzero rational functions of z” under which the equation (1.2) has no transcendental entire solutions with σ2(w) < 1. Theorem 1.3 Let a(z), a2(z), a1(z) and a0(z) ≡ 0 be rational functions, and let w(z) be a transcendental entire solution of the equation (1.4) with σ2(w) < 1.

Results

Conclusion