Existence of periodic solutions for differential equations with multiple delays under dichotomy condition
- Parinya Sa Ngiamsunthorn^{1, 2}Email author
https://doi.org/10.1186/s13662-015-0598-0
© Sa Ngiamsunthorn 2015
Received: 6 April 2015
Accepted: 10 August 2015
Published: 21 August 2015
Abstract
Using Krasnoselskii’s fixed point theorem and dichotomy theory, we prove the existence of periodic solutions for differential equations with multiple delays of the form \(x'(t)+ cx'(t-\tau) = A(t)x(t) +f(t, x(t-\alpha_{1}(t)),\ldots, x(t-\alpha_{m}(t)))\), where the parameter \(c \ll1\) is a small perturbation for a delayed forced term. Moreover, we discuss the convergence of these solutions to a solution of the unperturbed problem as \(c \rightarrow0\).
Keywords
MSC
1 Introduction
Delay differential equations are of interest in many areas of applications, such as population dynamics, drug administration, automatic control, laser optics, neural networks, economics and others (see for example [1–5]). There are significant theoretical researches on delay differential equation addressing many aspects of the dynamics, for example, stability of equilibria, existence of periodic solutions, complicated behavior, and chaos. Several methods were developed to obtain periodic solutions of autonomous delay differential equations, both for equations with time-invariant delay and with state-dependent delay. For references, see [6–8].
The paper is organized as follows. In the next section, some preliminary results on integrable dichotomy based on the result of [13], and the frameworks of our problem are introduced. Section 3 is devoted to establishing some criteria for the existence of periodic solutions of system (1.2). Finally, in Section 4, we discuss the convergence of these solutions to a solution of unperturbed problem as \(c \rightarrow0\).
2 Preliminaries and frameworks
The concept of an exponential dichotomy has been extensively used when studying bounded solutions of differential equations. Several results on the existence and uniqueness of bounded solutions, periodic solutions and almost periodic solutions of both linear and nonlinear differential equations are obtained under the assumption that the associated homogeneous linear equation satisfies the exponential dichotomy condition. However, there are similar results on the existence and uniqueness of bounded solutions under more general conditions such as the \((h,k)\)-dichotomy, integrable dichotomy, and integrable \((h,k)\)-dichotomy.
2.1 Periodic solutions for linear differential systems
Definition 2.1
A special case of an integrable dichotomy includes the following class of integrable \((h,k)\)- dichotomies.
Definition 2.2
Remark
If the differential system (1.1) has an integrable \((h,k)\)-dichotomy with projection P, then (1.1) has an integrable dichotomy for which \(\Phi(t)P\Phi^{-1}(t)\) is bounded.
We review the following result about bounded solutions of linear differential system (1.1).
Proposition 2.1
([13])
Suppose that a linear differential system (1.1) has an integrable dichotomy. Then \(x(t) = 0\) is the unique bounded solution of (1.1).
Theorem 2.1
([13])
In addition, if the differential operator \(A(t)\) and the non-homogeneous term \(f(t)\) are T-periodic, we can obtain periodic solutions of (2.4).
2.2 Frameworks
In this paper, we will investigate the existence of a periodic solution of a delay differential equation of the form (1.2), where τ and c are constants with \(|c|\ll1\) sufficiently small perturbation and \(\alpha_{i}(t)\), \(i=1,2,\ldots,m\), are real continuous functions on \(\mathbb{R}\) with period \(T>0\).
We assume the following conditions.
Assumption 1
We assume that \(A(t)\) is an \(N \times N\) real continuous matrix function defined on \(\mathbb{R}\) and T-periodic, that is, \(A(t+T) = A(t)\) for all \(t \in\mathbb{R}\).
Assumption 2
System (1.1) has an integrable dichotomy for which \(\Phi(t)P\Phi^{-1}(t)\) is bounded.
Here, \(\Phi(t)\) is the fundamental matrix solution of (1.1). Hence, the associated Green matrix \(G (t,s)\) given by (2.1) satisfies (2.2) for some positive constant μ.
In addition, we impose the following condition on f.
Assumption 3
- (i)
\(f(t+T,u_{1},\ldots,u_{m}) =f(t,u_{1},\ldots,u_{m})\) for all \((t,u_{1},\ldots,u_{m}) \in\mathbb{R} \times\mathbb{R}^{N} \times\cdots\times \mathbb{R}^{N}\).
- (ii)There exists a positive constant \(r < \frac {1}{m(1+L\mu+\mu)}\) such thatfor every \(u_{1}, u_{2},\ldots, u_{m},v_{1},v_{2},\ldots,v_{m} \in\mathbb{R}^{N}\) and \(t\in \mathbb{R}\),$$\bigl|f(t,u_{1},\ldots,u_{m}) -f(t,v_{1}, \ldots,v_{m}) \bigr| < r \bigl(|u_{1}-v_{1}|+|u_{2}-v_{2}|+ \cdots+|u_{m}-v_{m}| \bigr), $$
We prove the existence of a periodic solution of (1.2) under an integrable dichotomy condition.
3 Existence of periodic solutions
In this section, we prove our main result on the existence of periodic solutions to system (1.2).
Theorem 3.1
Suppose that Assumptions 1, 2, and 3 are satisfied. For every \(|c| \ll1\) sufficiently small, there exists at least a T-periodic solution of system (1.2).
To establish the existence result, we will apply Krasnoselskii’s fixed point theorem [18] as stated below.
Theorem 3.2
- (i)
\(\Gamma_{1} x + \Gamma_{2} y \in K\) for all \(x,y \in K\);
- (ii)
\(\Gamma_{1}\) is a contraction on K;
- (iii)
\(\Gamma_{2}\) is completely continuous on K.
Lemma 3.1
The operators V and W defined above are operators from \(BC^{1}(\mathbb{R}, \mathbb{R}^{N})\) into itself, that is, \(V,W: BC^{1}(\mathbb{R}, \mathbb{R}^{N}) \rightarrow BC^{1}(\mathbb{R}, \mathbb{R}^{N})\).
Proof
It is clear that if \(V+W\) has a fixed point, then the fixed point is a periodic solution of (1.2). Hence, we will turn to the problem of establishing a fixed point of the operator \(V+W\).
Lemma 3.2
The operator \(V: BC^{1}(\mathbb{R}, \mathbb{R}^{N}) \rightarrow BC^{1}(\mathbb{R}, \mathbb{R}^{N})\) defined by (3.1) is a contraction.
Proof
The result is clear since \(\|Vu\| = |c| \|u\|\) for \(u \in BC^{1}(\mathbb{R}, \mathbb{R}^{N})\) and \(|c| < 1\). □
Lemma 3.3
There exists \(M>0\) such that, for any \(v,w \in K_{M}\), we have \(Vv + Ww \in K_{M}\) whenever \(|c| \ll1\) is sufficiently small.
Proof
We next prove that W is a completely continuous operator on \(K_{M}\), which is a consequence of the following lemma.
Lemma 3.4
The set \(W (B_{M})\) is relatively compact in \(BC^{1}(\mathbb{R}, \mathbb{R}^{N})\).
Proof
It follows from the above lemmas and Krasnoselskii’s fixed point theorem (Theorem 3.2) that there exists a T-periodic solution of (1.2). Hence, we have proved Theorem 3.1.
Remark
- (1)
The result in Theorem 3.1 can be applied to the case when \(c=0\), however, we indeed obtain a unique periodic solution (see Lemma 4.1).
- (2)We may consider an alternative approach to establish the existence of periodic solutions of (1.2) by using the transformation \(y(t) = x(t) + c x(t - \tau)\). Hence, system (1.2) can be written aswhere \(g(t) := f(t, x(t - \alpha_{1}(t)),\ldots,x(t-\alpha_{m}(t))) -A(t) c x(t- \tau)\). By Theorem 2.1, we know that the solution of the above system satisfies the integral equation$$y'(t) = A(t)y(t) + g(t), $$Hence, the problem is reduced to showing the existence of periodic solution of the following delay integral equation$$y(t) = \int_{-\infty}^{\infty} G(t,s) g(s)\,ds. $$$$x(t) = -c x(t- \tau) \int_{-\infty}^{\infty} G(t,s) \bigl[f \bigl(s, x\bigl(s - \alpha _{1}(s)\bigr),\ldots,x\bigl(s- \alpha_{m}(s)\bigr)\bigr) -A(s) c x(s- \tau)\bigr]\,ds. $$
4 Continuity of periodic solutions in a neighborhood of \(c=0\)
Lemma 4.1
Suppose that Assumptions 1, 2, and 3 are satisfied. Then the T-periodic solution of (4.1) exists and is unique.
Proof
Theorem 4.1
Suppose that Assumptions 1, 2, and 3 are satisfied. For \(|c| \ll1\) sufficiently small, a sequence of T-periodic solutions \(u_{c}\) of system (1.2) converges to the T-periodic solution of system (4.1) as \(c \rightarrow0\).
Proof
Declarations
Acknowledgements
This project was supported by KMUTT Research Fund 2014 and the Theoretical and Computational Science Center (TaCS). We would like to thank anonymous referees and editor for valuable comments and suggestions to improve this work.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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