On solutions of certain nonlinear delay-differential equations

Abstract

The properties of solutions of two classes Tumura-Clunie type delay-differential equations are studied by applying the Nevanlinna theory. On the one hand, the growth and structure of entire solutions of the equationfn(z)+ωfn−1(z)f(k)(z)+q(z)eQ(z)f(z+c)=p1eλ1z+p2eλ2z are characterized, which improves some previous results in [2,6], where n≥3, k≥1 are integers, ω,c,λ1,λ2,p1, p2 are nonzero constants, and q≢0, Q are polynomials with Q is not a constant. On the other hand, the structure and the exponent of convergence of zeros of meromorphic solutions of the equationfn(z)+ωfn−1(z)f′(z)+P(z)f(k)(z+c)=H0(z)+H1(z)eω1zq+⋯+Hm(z)eωmzq are described, which extends some recent results in [12], where n≥2,m≥1,q≥1,k≥0 are integers, c,ω are nonzero complex numbers, ω1,…,ωm are distinct nonzero complex numbers, and P,H0,…,Hm are entire functions of order less than q with PH1⋯Hm≢0. Some examples are given to show these results.