| Volume 10, Number 5, 2020, Pages 1720-1740 DOI：10.11948/20190176 |
| Entire solutions of delay differential equations of Malmquist type |
| Ran-Ran Zhang,Zhi-Bo Huang |
| Keywords:Delay differential equation, entire solution, Nevanlinna theory. |
| 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. |
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