Analytic invariant curves for an iterative equation related to mosquito model

In this paper, we study the existence of analytic invariant curves of a iterative equation f ( f ( x ) ) = ( af ( x ) + bx e − x)e − f ( x )which from mosquito model. By constructing an invertible analytic solution g ( x ) of an auxiliary equation of the form g (α 2 x )e g ( x ) + g ( αx )= ag ( αx )e g ( x )+ bg ( x )invertible analytic solutions of the form g ( αg  − 1 ( x )) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < |  α  | < 1, we focus on those α on the unit circle S 1 , that is, |  α  | = 1. We discuss not only those α at resonance, that is at a root of the unity, but also those α near resonance under the Brjuno condition. Copyright © 2013 John Wiley & Sons, Ltd.