On the Exact Forms of Meromorphic Solutions of Certain Non-linear Delay-Differential Equations

Abstract

In this paper, we consider transcendental meromorphic solutions f of finite order $\rho$ and few poles in the sense that $S_{\lambda }(r,f):=O(r^{\lambda +\varepsilon })$, where $\lambda <\rho$ and $\varepsilon \in (0,\rho -\lambda )$, of the delay-differential equation

$$\begin{aligned} f^n+L(z,f)=p_1(z)e^{\alpha _{1}(z)}+p_2(z)e^{\alpha _{2}(z)}, \end{aligned}$$

where $n\ge 2$ is an integer, L(z, f) is a linear delay-differential polynomial with coefficients of growth $S_{\lambda }(r,f)$. In addition, $p_1(z)$, $p_2(z)$ are non-zero small functions of f in the sense $S_{\lambda }(r,f)$ and $\alpha _{1}(z)$, $\alpha _{2}(z)$ are non-constant polynomials. In fact, we give the exact forms of all possible meromorphic solutions of the above equation and we improve some recent results.