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

Let {b(j)}(j=1)(k) be meromorphic functions, and let wbe admissible meromorphic solutions of delay differential equation omega'(z) = w(z) [P(z, w(z))/Q(z, w(z)) + Sigma(k)(j=1)b(j)(z)w(z-c(j))] with distinct delays c(1),..., c(k) is an element of C\{0}, where the two nonzero polynomials P( z, w(z)) and Q( z, w(z)) in wwith meromorphic coefficients are prime each other. We obtain that if limsup(r ->infinity) logT(r,w)/r= 0, then deg(w)(P/Q) <= k + 2. Furthermore, if Q(z, w(z)) has at least one nonzero root, then deg(w)(P) = deg(w)(Q) + 1 <= k+ 2; if all roots of Q( z, w(z)) are nonzero, then deg(w)(P) = deg(w)(Q) + 1 <= k+1; if deg(w)(Q) = 0, then deg(w)(P) <= 1. In particular, whenever deg(w)(Q) = 0 and deg(w)(P) <= 1and 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 wsatisfying (N) over bar (r, 1/w) = O(N(r, 1/w)); while if all coefficients are rational functions, then the condition (N) over bar( r, 1/w) = O(N( r, 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= 1and deg(w)( P/Q) = 0) satisfies that N( r, w) and T( r, w) have the same growth category. Some examples support our results. (c) 2021 Elsevier Masson SAS. All rights reserved.