Exact and explicit analytic solutions of an extended Jabotinsky functional differential equation

In this paper, an extended Jabotinsky-type functional iterative differential equation is investigated in the complex field C for the existence of local analytic solutions. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. Furthermore, the exact and explicit analytic solution of the original equation is investigated for the first time. We discuss not only the general case, but also critical cases as well, especially for α a unit root, and in case (H4) we deal with the equation under the Brjuno condition, which is weaker than the Diophantine condition. Such equations are important in both applications and the theory of iteration.