The existence of multiple positive periodic solutions for functional differential equations

Abstract

The existence of multiple positive solutions for the integral equation $x(t)={\int }_t^{t+\omega }G(t\text{,}s)b(s)f(s\text{,}x(s-{\tau }_1(s))\text{,}\dots \text{,}x(s-{\tau }_n(s)))\mathit{ds}$ is studied by using fixed point index theory. Using these results we obtain new results on the existence of multiple positive periodic solutions for some first order periodic functional differential equations.

Introduction

Functional differential equations have been extensively studied by many authors by using the coincidence degree theory of Mawhin, the fixed point theorem in cones, Banach’s contraction mapping principle, method of lower and upper solutions, the Leggett–Williams fixed point theorem and so on. We refer the reader to [1], [2], [3], [4], [5], [6] and references cited therein.

In [2], Sun et al. studied functional differential equation

$$z^{\prime }(t)=-a(t)f(t\text{,}z(t))z(t)+g(t\text{,}z(t-\tau (t)))$$

under the following assumptions:$(H_1)a\in C(R\text{,}{R}^{+})\text{,}a(t)$ is a T-periodic function for t;$(H_2)f(t\text{,}x)\in C([0\text{,}+\infty )\times [0\text{,}+\infty )\text{,}[0\text{,}+\infty ))$, and f is a bounded function;$(H_3)g\in C(R\times [0\text{,}+\infty )\text{,}[0\text{,}+\infty ))\text{,}\tau \in C(R\text{,}R)\text{,}\tau (t)\text{,}g(t\text{,}·)$ are T-periodic function with respect to t; where $T>0$ is a constant. They obtained the existence of positive periodic solutions (see [2]). In [3], Zhang et al. investigated the existence of positive periodic solutions for the scalar functional differential equation

$$y^{\prime }(t)=-a(t)y(t)+f(t\text{,}y(t-{\tau }_1(t))\text{,}\dots \text{,}y(t-{\tau }_n(t)))$$

under the following assumptions:$(H_4)a\text{,}{\tau }_i\in C(R\text{,}{R}^{+})\text{,}a(t)\not\equiv 0\text{,}a(t)\text{,}{\tau }_i(t)$ are $\omega$-periodic functions, $i=1\text{,}2\text{,}\dots \text{,}n$; $(H_5)f\in C(R\times R_{+}^n\text{,}{R}_{+})$ is $\omega$-periodic with respect to the first variable. They got a few theorems about the existence of positive periodic solutions of (1.2).

These types of equation have been proposed as models for a variety physiological processes and conditions including production of blood cells, respiration, and cardiac arrhythmias.

In this paper, we obtain a few theorems of existence of positive periodic solutions for integral equation

$$x(t)={\int }_t^{t+\omega }G(t\text{,}s)b(s)f(s\text{,}x(s-{\tau }_1(s))\text{,}\dots \text{,}x(s-{\tau }_n(s)))\mathit{ds}\text{.}$$

What is interesting is that $b(t)$ may be singular at any point of R. Using these results, we obtain new results on the existence of multiple positive periodic solutions for functional differential Eqs. (1.1), (1.2). Our results different from those of [2], [3] and extend the results of [2], [3]. Particularly, we do not need any continuous assumption on the terms $a(t)\text{,}f(t\text{,}{x}_1\text{,}\dots \text{,}{x}_n)$ and $g(t\text{,}x)$, which is essential for the technique used in [2], [3].

In this paper, we will always suppose the following conditions are satisfied:

• $b(t)\in L^1[0\text{,}\omega ]\text{,}b(t)⩾0$, a.e. $t\in [0\text{,}1]$. $b(t)$ is a $\omega$-periodic function and ${\int }_0^{\omega }b(t)\mathit{dt}>0\text{;}$

• $G(t\text{,}s):R^2\to R_{+}$ is measurable and $G(t+\omega \text{,}s+\omega )=G(t\text{,}s)$. For $t\in R\text{,}s\in [t\text{,}t+2\omega ]\text{,}0<l⩽G(t\text{,}s)⩽L$. $G(t\text{,}s)$ is monotone with respect to t. For every $\tau \in R$, we have

  $$\underset{t\to \tau }{\lim }|G(t\text{,}s)-G(\tau \text{,}s)|=0\text{,a.e.}s\in R\text{;}$$

• $f:R\times R_{+}^n\to R_{+}$ maps bounded subset of $R\times R_{+}^n$ into bounded subset of $R_{+}$, f is $\omega$-periodic with respect to the first variable and $f(t\text{,}{x}_1\text{,}\dots \text{,}{x}_n)$ satisfies Caratheodory conditions;

• ${\tau }_i\in C(R\text{,}R)$ is $\omega -$periodic, $i=1\text{,}2\text{,}\dots \text{,}n$.