Existence of analytic solutions of an iterative functional equation

Abstract: This paper is concern analytic solutions of an iterative functional equation of the formByconstructing a convergent power series solution of an auxiliary equationanalytic solutions of the original equation are obtained. We discuss not only these α appeared in the auxiliary equation at the hyperbolic case 0<∣α∣≠1 and resonance, i.e., at a root of the unity, but also those α near resonance (i.e., near a root of the unity) under Brjuno condition. [Copyright &y& Elsevier]