Advanced Search

We study existence of analytic solutions of a second-order iterative functional differential equation $x^{\text{'}\text{'}}\left(z\right)={\sum }_{j=0}^k{\sum }_{t=1}^{\infty }{C}_{t,j}\left(z\right){\left(x^{\left[j\right]}\left(z\right)\right)}^t+G\left(z\right)$ in the complex field ℂ. By constructing an invertible analytic solution y(z) of an auxiliary equation of the form $\alpha ²y^{\text{'}\text{'}}\left(\alpha z\right)y^{\text{'}}\left(z\right)=\alpha y^{\text{'}}\left(\alpha z\right)y^{\text{'}\text{'}}\left(z\right)+\left[y^{\text{'}}\left(z\right)\right]³[{\sum }_{j=0}^k{\sum }_{t=1}^{\infty }{C}_{t,j}\left(y\left(z\right)\right){\left(y\left({\alpha }^{j}z\right)\right)}^t+G\left(y\left(z\right)\right)]$ invertible analytic solutions of the form $y\left(\alpha y^{-1}\left(z\right)\right)$ for the original equation are obtained. Besides the hyperbolic case 0 < |α| < 1, we focus on α on the unit circle S¹, i.e., |α|=1. We discuss not only those α at resonance, i.e. at a root of unity, but also near resonance under the...

This work deals with Feigenbaum’s functional equation ⎧ $h\left(g\left(x\right)\right)=g^p\left(h\left(x\right)\right)$, ⎨ ⎩ g(0) = 1, -1 ≤ g(x) ≤ 1, x∈[-1,1] where p ≥ 2 is an integer, $g^p$ is the p-fold iteration of g, and h is a strictly monotone odd continuous function on [-1,1] with h(0) = 0 and |h(x)| < |x| (x ∈ [-1,1], x ≠ 0). Using a constructive method, we discuss the existence of continuous unimodal even solutions of the above equation.