Entire solutions of delay differential equations of Malmquist type

In this paper, we investigate the delay differential equations of Malmquist type of form \begin{equation*} w(z+1)-w(z-1)+a(z)\frac{w'(z)}{w(z)}=R(z, w(z)),~~~~~~~~~~~~~~(*) \end{equation*} where $R(z, w(z))$ is an irreducible rational function in $w(z)$ with rational coefficients and $a(z)$ is a rational function. We characterize all reduced forms when the equation $(*)$ admits a transcendental entire solutions with hyper-order less than one. When we compare with the results obtained by Halburd and Korhonen[Proc.Amer.Math.Soc., forcoming],we obtain the reduced forms without the assumptions that the denominator of rational function $R(z,w(z))$ has roots that are nonzero rational functions in $z$. The growth order and value distribution of transcendental entire solutions for the reduced forms are also investigated.