Meromorphic solutions of delay differential equations related to logistic type and generalizations

Abstract

Let $\{b_{j}\}_{j=1}^{k}$ be meromorphic functions, and let $w$ be admissible meromorphic solutions of delay differential equation $$w'(z)=w(z)\left[\frac{P(z, w(z))}{Q(z,w(z))}+\sum_{j=1}^{k}b_{j}(z)w(z-c_{j})\right]$$ with distinct delays $c_{1}, \ldots, c_{k}\in\mathbb{C}\setminus\{0\},$ where the two nonzero polynomials $P(z, w(z))$ and $Q(z, w(z))$ in $w$ with meromorphic coefficients are prime each other. We obtain that if $\limsup_{r\rightarrow\infty}\frac{\log T(r, w)}{r}=0,$ then $$deg_{w}(P/Q)\leq k+2.$$ Furthermore, if $Q(z, w(z))$ has at least one nonzero root, then $deg_{w}(P)=deg_{w}(Q)+1\leq k+2;$ if all roots of $Q(z, w(z))$ are nonzero, then $deg_{w}(P)=deg_{w}(Q)+1\leq k+1;$ if $deg_{w}(Q)=0,$ then $deg_{w}(P)\leq 1.$\par In particular, whenever $deg_{w}(Q)=0$ and $deg_{w}(P)\leq 1$ and without the growth condition, any admissible meromorphic solution of the above delay differential equation (called Lenhart-Travis' type logistic delay differential equation) with reduced form can not be an entire function $w$ satisfying $\overline{N}(r, \frac{1}{w})=O(N(r, \frac{1}{w}));$ while if all coefficients are rational functions, then the condition $\overline{N}(r, \frac{1}{w})=O(N(r, \frac{1}{w}))$ can be omitted. Furthermore, any admissible meromorphic solution of the logistic delay differential equation (that is, for the simplest special case where $k=1$ and $deg_{w}(P/Q)=0$ ) satisfies that $N(r,w)$ and $T(r, w)$ have the same growth category. Some examples support our results.