Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

Analytic solutions of a second-order iterative functional differential equation near resonance

100%

Zhao H., Si J.

Annales Polonici Mathematici

2009

tom 96

nr 3

209-226

We study existence of analytic solutions of a second-order iterative functional differential equation $x''(z) = ∑_{j=0}^{k}∑_{t=1}^{∞}C_{t,j}(z)(x^{[j]}(z))^{t} + G(z)$ in the complex field ℂ. By constructing an invertible analytic solution y(z) of an auxiliary equation of the form $α²y''(αz)y'(z) = αy'(αz)y''(z) + [y'(z)]³[∑_{j=0}^{k}∑_{t=1}^{∞}C_{t,j}(y(z))(y(α^{j}z))^{t} + G(y(z))]$ invertible analytic solutions of the form $y(αy^{-1}(z))$ 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 Brjuno condition.

Solutions for the p-order Feigenbaum's functional equation $h(g(x)) = g^{p}(h(x))$

100%

Zhang M., Si J.

Annales Polonici Mathematici

2014

tom 111

nr 2

183-195

This work deals with Feigenbaum's functional equation ⎧ $h(g(x)) = g^p(h(x))$, ⎨ ⎩ 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.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Si J.

1 Zhang M.

1 Zhao H.

1 2014

1 2009

Wyniki wyszukiwania

Analytic solutions of a second-order iterative functional differential equation near resonance

Solutions for the p-order Feigenbaum's functional equation $h(g(x)) = g^{p}(h(x))$