Analytic solutions for a class of differential equation with delays depending on state

In this paper, we study existence of analytic solutions of a class of the differential equation with state-dependent delaysx′(z)=∑j=0k∑t=1∞Ct,j(z)(x[j](z))t+G(z)in the complex field CC. By constructing a convergent power series solution y(z) of an auxiliary equation of the formαy′(αz)=y′(z)∑j=0k∑t=1∞Ct,j(y(z))(y(αjz))t+G(y(z))with α=G(0)∈C⧹{0}α=G(0)∈C⧹{0}, invertible analytic solutions of the form y(αy−1(z)) for the original equation are obtained.