ENTIRE SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS OF MALMQUIST TYPE

Ran-Ran Zhang; Zhi-Bo Huang

doi:10.11948/20190176

2020 Volume 10 Issue 5

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Ran-Ran Zhang, Zhi-Bo Huang. ENTIRE SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS OF MALMQUIST TYPE[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1720-1740. doi: 10.11948/20190176

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Ran-Ran Zhang, Zhi-Bo Huang. ENTIRE SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS OF MALMQUIST TYPE[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1720-1740. doi: 10.11948/20190176

ENTIRE SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS OF MALMQUIST TYPE

Ran-Ran Zhang¹,
Zhi-Bo Huang^2, ,

1.
Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

Corresponding author: Email: huangzhibo@scnu.edu.cn(Z. Huang)
Fund Project: The authors were supported by National Natural Science Foundation of China (11801093, 11871260), National Science Foundation of Guangdong (2018A030313508) and Characteristic Innovation Project (Natural Science) of Guangdong Province(2019KTSCX119)

Abstract

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.
- Delay differential equation /
- entire solution /
- Nevanlinna theory
MSC: 30D35, 34M05

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References

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