Positive almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument

Lequn Peng,Wentao Wang

doi:10.14232/ejqtde.2010.1.6

Abstract

Positive almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument

Highlights

Consider the following model for nonlinear Duffing equation with a deviating argument x′′(t) + cx′(t) − ax(t) + bxm(t − τ (t)) = p(t), (1.1)where τ (t) and p(t) are almost periodic functions on R, m > 1, a, b and c are constants
Some results on existence of the almost periodic solutions were obtained in the literature
A primary purpose of this paper is to study the problem of positive almost periodic solutions of (1.1)

Summary

Introduction

Without assuming (H0), we derive some sufficient conditions ensuring the existence of positive almost periodic solutions for Eq (1.1), which are new and complement to previously known results. From the theory of almost periodic functions in [8,9], it follows that for any ǫ > 0, it is possible to find a real number l = l(ǫ) > 0, for any interval with length l(ǫ), there exists a number δ = δ(ǫ) in this interval such that. Let C([−τ, 0], R) denote the space of continuous functions φ : [−τ, 0] → R with the supremum norm · It is known in [1−4] that for τ, Q1 and Q2 continuous, given a continuous initial function φ ∈ C([−τ, 0], R) and a number y0, there exists a solution of (1.4) on an interval [0, T ) satisfying the initial condition and satisfying (1.4) on [0, T ).

Preliminary Results

Main Results

An Example