Positive periodic solutions for an iterative differential equation related to a discrete derivatives sequence

Received: 13.II.2016, Accepted: 27.IV.2016

Abstract. In this paper, we use Krasnoselskii’s fixed point theorem to study the existence and uniqueness of periodic solutions of a class of iterative differential equation
$f'(t)=\frac{1}{K(f_1(t))^{a_1}(f_2(t))^{a_2}\cdots (f_n(t))^{a_n}},$ where $K\in\mathbb{R}\setminus\{0\}, a_i\in\mathbf{R}, f_i(t)$ denotes $i$th iterate of $f(t), i=1,2,\ldots,n.$ The above equation is closely related to a discrete derivatives sequence $F'(m)$ (see [Y.-F.S. Petermann, Jean-LucRemy, Ilan Vardi, Discrete derivative of sequences, Adv. in Appl. Math. 27(2001) 562-584]).

Keywords: Iterative differential equation, Discrete derivatives sequence, Periodic solutions, Fixed point theorem

Mathematics Subject Classification (2010): 39B12, 39B82

Author:
Hou Yu Zhao, School of Mathematics, Chongqing Normal University, Chongqing, P.R.China, and Department of Pure Mathematics, University of Waterloo, Waterloo, Canada