Meromorphic solutions of a first order differential equations with delays

Abstract

The main purpose of this paper is to study meromorphic solutions of the first order differential equations with delays \begin{equation*} w(z+1)-w(z-1)+a(z)\left(\frac{w^{\prime }(z)}{w(z)}\right)^k=R(z,w(z)) \end{equation*} and \begin{equation*} w(z+1)+a(z)\left(\frac{w^{\prime }(z)}{w(z)}\right)^k=R(z,w(z)), \end{equation*} where $k$ is a positive integer, $a(z)$ is a rational function, $R(z,w)$ is rational in $w$ with rational coefficients. Some necessary conditions on the degree of $R(z, w)$ are obtained for the equation to admit a transcendental meromorphic solution of minimal hypertype. These are extensions of some previous results due to Halburd, Korhonen, Liu and others. Some examples are given to support our conclusions.