Analytic Solutions of an Iterative Functional Differential Equation near Resonance

Abstract

We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown function $x^{\prime \prime }(z)+c{x}^{\prime }(z)=x\left(az+bx(z)\right)$ in the complex field $ℂ$. By reducing the equation with the Schröder transformation to the another functional differential equation without iteration of the unknown function ${\lambda }^2{g}^{\prime \prime }(\lambda z)g^{\prime }(z)-\lambda g^{\prime }(\lambda z)g^{\prime \prime }(z)$ + $c{\left(g^{\prime }\left(z\right)\right)}^2\left(\lambda g^{\prime }\left(\lambda z\right)-a{g}^{\prime }(z)\right)$ = ${\left(g^{\prime }\left(z\right)\right)}^3\left(g\left({\lambda }^{2}z\right)-ag\left(\lambda z\right)\right)$, we get its local invertible analytic solutions.