Existence and uniqueness of positive periodic solutions for a first-order functional differential equation

Chen Yang,Chengbo Zhai,Mengru Hao

doi:10.1186/s13662-014-0341-2

Abstract

In this article, we establish sufficient conditions for the existence and uniqueness of positive periodic solutions for a class of first-order functional differential equations. Our analysis relies on some fixed point theorems for mixed monotone operators. Our results can not only guarantee the existence of unique positive periodic solutions, but also be applied to construct an iterative scheme for approximating them. Some examples are given to illustrate our main results.

Highlights

This article will investigate the existence and uniqueness of positive periodic solutions for the following first-order functional differential equation:y (t) = –δ(t)y(t) + f t, y t – τ (t), y t – τ (t) + g t, y t – τ (t), ( . )where T >, δ, τ : R → R are continuous T-periodic functions and δ(t) > for t ∈ R, f : R → R and g : R → R.Functional differential equations with periodic delays appear in a number of ecological, economical, control and physiological models
Most of the authors have investigated the existence of positive periodic solutions for functional differential equations
In most of the existing works, in order to establish the existence of positive periodic solutions, a key condition is that the

Summary

Introduction

This article will investigate the existence and uniqueness of positive periodic solutions for the following first-order functional differential equation:. Few papers can be found in the literature on the existence and uniqueness of positive periodic solutions for four-point fractional differential equations. Motivated by the works [ , ], in our paper, we will use three fixed point theorems for mixed monotone operators to study the existence and uniqueness of positive periodic solutions for problem Our results can guarantee the existence of unique positive periodic solutions, and be applied to construct an iterative scheme for approximating them With this context in mind, the outline of this paper is as follows. We need the following fixed point theorems for mixed monotone operators, which were given and proved in [ , ]. ( ) there exist u , v ∈ Ph and r ∈ ( , ) such that rv ≤ u < v , u ≤ A(u , v ) + Bu ≤ A(v , u ) + Bv ≤ v ;

In our considerations we will work in the Banach space

And thus