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 $$w^{\prime }(z)=w(z)\left[\frac{P(z,w(z))}{Q(z,w(z))}+\sum _{j=1}^k{b}_j(z)w(z-c_j)\right]$$ with distinct delays $c_1,\dots ,c_k\in \mathbb{C}\setminus \{0\},$ 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 ${\mathrm{limsup}}_{r\to \infty }\frac{\log T(r,w)}{r}=0,$ then $$de{g}_w(P/Q)\le k+2.$$ Furthermore, if $Q(z,w(z))$ has at least one nonzero root, then $de{g}_w(P)=de{g}_w(Q)+1\le k+2;$ if all roots of $Q(z,w(z))$ are nonzero, then $de{g}_w(P)=de{g}_w(Q)+1\le k+1;$ if $de{g}_w(Q)=0,$ then $de{g}_w(P)\le 1.$\par In particular, whenever $de{g}_w(Q)=0$ and $de{g}_w(P)\le 1$ 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 $de{g}_w(P/Q)=0$ ) satisfies that $N(r,w)$ and $T(r,w)$ have the same growth category. Some examples support our results.

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References in corpus (1)

• Logarithmic difference lemma in several complex variables and partial difference equations