Abstract
In this paper, we study the existence of affine-periodic solutions of nonlinear impulsive differential equations. The affine-periodic solutions have the form \(x(t+T)=Qx(t)\) with some nonsingular matrix Q. We give a theorem on the existence of the affine-periodic solutions, respectively, depending on wether \(\operatorname{det}(I-Q)\) (I= identity matrix) is equal to 0 or not.
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1 Introduction
The periodicity is a very important property in the study of the impulsive differential equations [1, 2]. However, not all natural phenomena can be described alone by periodicity. Some differential equations often exhibit certain symmetries rather than periodicity. For example, consider the system
where \(f:R^{1}\times R^{n}\rightarrow R^{n}\) is continuous, and for some \(Q\in GL_{n}(R)\) (general linear group), satisfies the following affine symmetry:
We call it a \((Q,T)\)-affine-periodic system. For this \((Q,T)\)-affine-periodic system, we are concerned with the existence of \((Q,T)\)-affine-periodic solutions \(x(t)\) with
It should be pointed out that when \(Q=I\) (identity matrix) or \(Q=-I\), the solutions are just the pure periodic solutions or antiperiodic ones; when \(Q\in SO_{n}\) (special orthogonal group), the solutions correspond to the solutions with Q-rotating symmetry, particularly to some special quasi-periodic solutions. So the interest to particular kinds of periodic solutions that we are going to study is not purely theoretical. The antiperiodicity property or some quasi-periodicity property, which is obviously a particular case of affine-periodic solutions, has drawn wide attention from physicists and astronomers [3, 4].
Recently, these conceptions and existence results of the solutions have been introduced and proved by Li and his coauthors; see [5] for Levinson’s problem, [6] for Lyapunov function type theorems, [7] for averaging methods of affine-periodic solutions, and [8] for some dissipative dynamical systems. The aim of this paper is to touch such a topic for affine-periodic solutions of nonlinear impulsive differential equations.
The paper is organized as follows. We first change the affine-periodic solutions problem to the boundary value problem in Sect. 2. In Sect. 3, when \(\operatorname{det}(I-Q)\neq0\), we give an unique affine-periodic solution by using the Banach contraction mapping principle. Furthermore, via the topological degree theory, we prove the existence of affine-periodic solutions for nonlinear impulsive system when \(\operatorname{det}(I-Q)=0\) in Sect. 4. We give two examples by numerical simulation in Sect. 5.
2 Nonlinear impulsive differential system
In this paper, we investigate the following system:
The system satisfies the following hypotheses H:
-
(1)
\(f(\cdot)\in C(R\times R^{n},R^{n})\) and \(f(t+T,x)= Qf(t,Q^{-1}x)\) for some \(G\in SO_{n}(R)\).
-
(2)
\(I_{k}(\cdot)\in C(R^{n},R^{n})\), \(t_{k}< t_{k+1}\) (\(k \in Z\)).
-
(3)
There exists \(q\in N\) such that \(I_{k+q}(x)=Q I_{k}(Q^{-1}x)\) and \(t_{k+q}=t_{k}+T\) (\(k \in Z\)).
In system (2), the continuous part corresponds to a nonlinear \((Q,T)\)-affine-periodic system. The discrete component models the affine-periodic impulsive change of \(x(t)\).
Lemma 2.1
The existence of Q-affine-periodic solutions of equation (2) is equivalent to the existence of the boundary value problem (2) with \(x(T)=Qx(0)\).
Proof
Let \(x(t)\) be a solution of equation (2) defined on \(t\in[0,T]\). Then
is a Q-affine-periodic solution of (2). Indeed, if \(t\in(jT,jT+T]\) and \(t\neq t_{k}\), then \(t-jT \in(0,T]\), and
and if \(t_{k}\in(jT,jT+T]\), then \(t_{k-jq}=t_{k}-jT\in(0,T]\) and
Let \(x(t)\) be any solution of (2) with \(x(T)=Qx(0)\). Then \(x(t)\) has the form
Denote \(x(0)\) by \(x_{0}\). Then we have
□
3 Noncritial case
In this case, \((I-Q)^{-1}\) exists. Then
So, the existence of Q-affine-periodic solutions of equation (2) is equivalent to the existence of solutions of the following impulsive integral equation:
Let
and define the norm \(\|x\|=\sup_{t\in[0,T]}|x(t)|\). It is easy to see that X is a Banach space with norm \(\|x\|\). We also define the norm of the matrix \(\|X(t)\|=\|(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))\|= \max_{i=1,2,\ldots,n}\|x_{i}\|\). Then we have the following theorem.
Theorem 3.1
Let a function \(p\in L([0,T],R^{+})\) and nonnegative constants \(\alpha_{k}\) (\(k=1,2,\ldots,q\)) be such that
and
Then system (2) has an unique Q-affine-periodic solution.
Proof
Define
Then
So, if \((\int_{0}^{T}p(s)\,ds+\sum_{k=1}^{q}a_{k})<\frac{1}{\|(I-Q)^{-1} \|+1}\), then by the Banach contraction mapping principle system (2) has a unique Q-affine-periodic solution. □
4 Critial case
To investigate the existence of solutions of system (2), the following auxiliary equation is often considered:
Then we give the following existence theorem for (Q,T)-affine-periodic solutions by using the topological degree theory [6, 7, 9–11].
Theorem 4.1
Let \(D\subset R^{n}\) be a bounded open set. Assume that the following hypotheses hold for system (10):
-
(H1)
For each \(\lambda\in(0,1]\), every Q-affine-periodic solution \(x(t)\) of system (10) satisfies
$$ x(t)\notin\partial D \quad \textit{for all }t; $$ -
(H2)
the Brouwer degree,
$$ \operatorname{deg}\bigl(g,D\cap \operatorname{Ker}(I-Q),0 \bigr)\neq0 \quad \textit{if } \operatorname{Ker}(I-Q) \neq{0}, $$
where
with an orthogonal projection \(P:R^{n}\rightarrow \operatorname{Ker}(I-Q)\).
Then system (2) has at least one Q-affine-periodic solution \(x_{*}(t)\in D\) for all t.
Proof
Consider the auxiliary equation (10) with the boundary value condition \(x(T)=Qx(t)\), where \(\lambda\in(0,1]\). Let \(x(t)\) be any solution of (10) with \(x(T)=Qx(0)\). Then
In this case, \((I-Q)^{-1}\) does not exist. By coordinate transformation, without loss of generality, we can just let
where \((I-Q_{1})^{-1}\) exists. Here \(Q=Q_{1}\oplus I\).
Let \(P:R^{n}\rightarrow{ \operatorname{Ker}(I-Q)}\) be the orthogonal projection. Then
where \(x^{0}_{\ker }\in \operatorname{Ker}(I-Q)\), \(x^{0}_{\bot}\in \operatorname{Im}(I-Q)\) and \(x_{0}=x^{0}_{\ker }+x^{0}_{\bot}\).
Let \(L_{p}=(I-Q)|_{\operatorname{Im}(I-Q)}\). It is easy to see that \(L^{-1}_{p}\) exists. Thus equation (13) is equivalent to
Thus we have
For \(x\in\textrm{X}\) such that \(x(t)\in\overline{D}\) for all \(t\in[0,T]\), we define the operator \(\mathtt{T}(x^{0}_{\ker },x, \lambda)\) by
where \(\lambda\in[0,1]\). We claim that each fixed point x of T in X is a solution of (10) with \(x(T)=Qx(0)\).
In fact, if x is a fixed point of T, we have
Thus
By equation (16) we know that
According to \((I-Q)x^{0}_{\ker }=0\), we have
Since equation (15) holds, we have
Thus
Then
By equations (16) and (17), equation (11) holds. Thus,
Then,
This means that the fixed point x is a solution of (10) with \(x(T)=Qx(0)\).
Now, we need to prove the existence of the fixed point of T. Take a constant M such that \(M> \sup_{t\in[0,T],x\in\overline{D}}|f(t,x)|\), and let
Then, it is easy to make a retraction \(\alpha_{\lambda}:\textrm{X} \rightarrow\textrm{X}_{\lambda}\).
Define an operator \(\widehat{\mathtt{T}}(x^{0}_{\ker },x,\lambda)\) by
Since \(P:R^{n}\rightarrow \operatorname{Ker}(I-Q)\), it is easy to see that
Also,
Let us consider the homotopy
where \(\widetilde{D}=\{x\in X:x(t)\in D \text{ for all } t \in[0,T]\}\).
We claim that
Suppose, on the contrary, that there exists \((\widehat{x}^{0}_{\ker }, \widehat{x},\widehat{\lambda})\in\partial((D\cap \operatorname{Ker}(I-Q)\times \widetilde{D})\times[0,1]\) such that \((id-H)(\widehat{x}^{0}_{\ker }, \widehat{x},\widehat{\lambda})=0\). Since \(\widehat{x}^{0}_{\ker } \in\partial D\) is contradictory to (\(H_{1}\)) and since \(\partial(D \cap \operatorname{Ker}(I-Q))\subset\partial D\), we have that \(\widehat{x}^{0}_{ \ker }\notin\partial(D\cap \operatorname{Ker}(I-Q))\). In other words, \(\widehat{x}\in\partial D\). Then (21) can be proved as follows.
(i) When \(\widehat{\lambda}=0\), by the definition of the set \(\textrm{X}_{\lambda}\) we have
Hence \(\alpha_{0}\circ x(t)\equiv\alpha_{0}\circ x(t_{k+1})\) for all \(t\in(t_{k},t_{k+1}]\). Since \((id-H)(\widehat{x}^{0}_{\ker}, \widehat{x},0)=0\), we have
This means that \(\widehat{x}(t)\equiv\widehat{x}(0)\) for all \(t\in[0,T]\). Taking \(\widehat{x}(t)=p\), we have \(\alpha_{0}\circ \widehat{x}^{0}_{\ker }=\widehat{x}(t)=p\). Consequently,
and this is equivalent to \(g(p)=0\) by the definition of \(g(a)\). Notice that \(\widehat{x}\in\partial\widetilde{D}\) and \(\widetilde{D}=\{x \in D \text{ for all } t\in[0,T]\}\). Then there exists \(t_{0}\in[0,T]\) such that \(\widehat{x(t)}_{0}\in\partial D\). Since \(\widehat{x}(t)\equiv p\) for all \(t\in[0,T]\), we obtain that \(p\in\partial D\). Thus, we have \(p\in\partial D\) and \(g(p)=0\). It is contradictory to (\(H_{2}\)) because the Brouwer degree \(\operatorname{deg}(g,D,0) \neq0\).
(ii) When \(\widehat{\lambda}\in(0,1]\), as \(0=(id-H)(\widehat{x}^{0} _{\ker },\widehat{x},\widehat{\lambda})\), we have
Thus
and
Note that
By the definition of \(\textrm{X}_{\lambda}\) we obtain \(\widehat{x} \in\textrm{X}_{\widehat{\lambda}}\), which means that \(\alpha_{\widehat{\lambda}}\circ\widehat{x}=\widehat{x}\). Now we can rewrite equation (23) as
By a similar discussion of equation (16) we can prove that \(\widehat{x}(t)\) is a solution of equation (10). By hypothesis (\(H_{1}\)) we know that \(\widehat{x}(t)\notin\partial\widetilde{D}\) for any \(t\in[0,T]\). This is a contradiction to \(\widehat{x}\in\partial \widetilde{D}\).
By (i) and (ii) we obtain that
Therefore, by the homotopy invariance and the theory of Brouwer degree we have
This means that there exists \(\widehat{x}_{*}\in\widetilde{D}\) such that
Similarly to the proof in (ii), we get \(\widehat{x}_{*}\in\textrm{X} _{\lambda}\). Then
By equations (24) and (25) we obtain that \(\widehat{x}_{*}\) is a fixed point of T in X. Thus, \(\widehat{x}_{*}\) is a solution of system (2) with boundary value condition \(x(T)=Qx(0)\). □
5 Numerical simulation
Example 1
Consider the system
Set
In this example, \(Q=-I\). System (26) has an antiperiodic solution (see Fig. 1).
The antiperiodic solution of system (26). The green line is the trajectory of \(x(t)\) for \(t\in(0,1]\), the red line corresponds to the trajectory of \(Qx(t)\) for \(t\in(0,1]\)
Example 2
Consider the system
Set
Similarly to Example 1, system (26) has a \((Q,1)\)-affine-periodic solution., which is a quasi-periodic solution (see Fig. 2).
The \((Q,T)\)-affine-periodic solution of system (27). The black line is the trajectory of \(x(t)\) for \(t\in(0,1]\), the red line corresponds to the trajectory of \(Qx(t)\) for \(t\in(0,1]\), and the green line corresponds to the trajectory of \(Q^{2}x(t)\) for \(t\in(0,1]\). It is easy to see that \(x(t)\) is a quasi-periodic solution of system (27)
Abbreviations
- I :
-
identity matrix
- GL:
-
general linear group
- SO:
-
special orthogonal group
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The author expresses sincere thanks to the anonymous referees for their comments.
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The author is supported by the fund of the “Thirteen Five” Scientific and Technological Research Planning Project of the Department of Education of Jilin Province (JJKH20170301KJ).
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Wang, S. The existence of affine-periodic solutions for nonlinear impulsive differential equations. Bound Value Probl 2018, 113 (2018). https://doi.org/10.1186/s13661-018-1033-8
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DOI: https://doi.org/10.1186/s13661-018-1033-8