Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems

Abstract

We prove that the system $\dot{\theta}(t) =\Lambda(t)\theta(t)$, $t\in\mathbb{R}_{+}$, is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions; we take the exact solutions of the Cauchy problem $\dot{\phi}(t)=\Lambda(t)\phi(t)+e^{i\gamma t}\xi(t)$, $t\in\mathbb{R}_{+}$, $\phi(0)=\theta_{0}$ as the approximate solutions of $\dot{\theta}(t)=\Lambda(t)\theta(t)$, where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with $\xi(0)=0$, and $\Lambda(t)$ is a 2-periodic square matrix of order l.

1 Introduction

The stability theory is an important branch of the qualitative theory of differential equations. In 1940, Ulam [1] queried a problem regarding the stability of differential equations for homomorphism as follows: when can an approximate homomorphism from a group $G_{1}$ to a metric group $G_{2}$ be approximated by an exact homomorphism?

Hyers [2] brilliantly gave a partial answer to this question assuming that $G_{1}$ and $G_{2}$ are Banach spaces. Later on, Aoki [3] and Rassias [4] extended and improved the results obtained in [2]. In particular, Rassias [4] relaxed the condition for the bound of the norm of Cauchy difference $f(x+y)-f(x)-f(y)$. To the best of our knowledge, papers by Obłoza [5, 6] published in the late 1990s were among the first contributions dealing with the Hyers-Ulam stability of differential equations.

Since then, many authors have studied the Hyers-Ulam stability of various classes of differential equations. Properties of solutions to different classes of equations were explored by using a wide spectrum of approaches; see, e.g., [7–26] and the references cited therein. Alsina and Ger [7] proved Hyers-Ulam stability of a first-order differential equation $y'(x) = y(x)$, which was then extended to the Banach space-valued linear differential equation of the form $y'(x)=\lambda y(x)$ by Takahasi et al. [24]. Zada et al. [26] generalized the concept of Hyers-Ulam stability of the nonautonomous w-periodic linear differential matrix system $\dot{\theta}(t)=\Lambda(t)\theta(t)$, $t\in\mathbb{R}$ to its dichotomy (for dichotomy in autonomous case; see, e.g., [27, 28]). We conclude by mentioning that Barbu et al. [10] proved that Hyers-Ulam stability and the exponential dichotomy of linear differential periodic systems are equivalent.

Very recently, Li and Zada [19] gave connections between Hyers-Ulam stability and uniform exponential stability of the first-order linear discrete system

$$ \theta_{n+1}=\Lambda_{n}\theta_{n}, \quad n\in\mathbb{Z}_{+}, $$

(1.1)

where $\mathbb{Z}_{+}$ is the set of all nonnegative integers and $(\Lambda_{n})$ is an ω-periodic sequence of bounded linear operators on Banach spaces. They proved that system (1.1) is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions. The natural question now is: is it possible to extend the results of [19] to continuous nonautonomous systems over Banach spaces? The purpose of this paper is to develop a new method and give an affirmative answer to this question in finite dimensional spaces. We consider the first-order linear nonautonomous system $\dot{\theta}(t)=\Lambda(t)\theta(t)$, $t\in\mathbb{R}_{+}$, where $\Lambda(t)$ is a square matrix of order l. We proved that the 2-periodic system $\dot{\theta}(t)=\Lambda (t)\theta(t)$ is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions. Our result can be extended to any q-periodic system, because we choose 2 as the period in our approach.

2 Notation and preliminaries

Throughout the paper, $\mathbb{R}$ is the set of all real numbers, $\mathbb{R}_{+}$ denotes the set of all nonnegative real numbers, $\mathbb{Z_{+}}$ stands for the set of all nonnegative integers, $\mathbb{C}^{l}$ denotes the l-dimensional space of all l-tuples complex numbers, $\Vert \cdot \Vert$ is the norm on $\mathbb{C}^{l}$, $\mathcal{L}(\mathbb{Z_{+}}, \mathbb{C}^{l})$ is the space of all $\mathbb{C}^{l}$-valued bounded functions with ‘sup’ norm, and let $\mathcal{W}_{0}^{2}(\mathbb{R_{+}},\mathbb{C}^{l})$ be the set of all continuous, bounded, and 2-periodic vectorial functions f with the property that $f(0)=0$.

Let $\mathcal{H}$ be a square matrix of order $l \geq1$ which has complex entries and let ϒ denote the spectrum of $\mathcal{H}$, i.e., $\Upsilon:=\{\lambda: \lambda \text{ is an eigenvalue of } \mathcal{H}\}$. We have the following lemmas.

Lemma 2.1

If $\Vert \mathcal{H}^{n} \Vert < \infty$ for any $n\in\mathbb {Z}_{+}$, then $\vert \lambda \vert \leq1$ for any $\lambda\in\Upsilon$.

Proof

Suppose to the contrary that $\vert \lambda \vert > 1$. By the definition of eigenvalue, there exists a nonzero vector $\theta\in\mathbb{C}^{l}$ such that $\mathcal{H}\theta= \lambda\theta$, which implies that $\mathcal{H}^{n}\theta= \lambda^{n}\theta$ for any $n\in\mathbb{Z} _{+}$, and thus $\Vert \mathcal{H}^{n} \Vert \geq{ \Vert \mathcal{H}^{n}\theta \Vert }/{ \Vert \theta \Vert } = { \vert \lambda \vert }^{n} \rightarrow\infty$ as $n \rightarrow\infty$. Therefore, $\vert \lambda \vert \leq1$. The proof is complete. □

Lemma 2.2

If $\Vert \sum_{j=0}^{P}{\mathcal{H}}^{j} \Vert < \infty$ for any $P\in\mathbb{Z}_{+}$, then 1 does not belong to ϒ.

Proof

If $1 \in\Upsilon$, then $\mathcal{H}\theta=\theta$ for some nonzero vector θ in $\mathbb{C}^{l}$ and $\mathcal{H}^{k} \theta=\theta$ for all $k= 1,2,\ldots, P$. Therefore, we conclude that

$$ \sup_{P\in\mathbb{Z}_{+}} \Biggl\Vert \sum_{j=0}^{P}{ \mathcal{H}} ^{j} \Biggr\Vert = \sup_{P\in\mathbb{Z}_{+}} \sup _{\theta\neq0}\frac{ \Vert (I + \mathcal{H} + \cdots+ \mathcal{H}^{P})(\theta) \Vert }{ \Vert \theta \Vert } \geq \sup_{ P\in\mathbb{Z}_{+}, \theta\neq0} \frac{P \Vert \theta \Vert }{ \Vert \theta \Vert }= \infty, $$

and so 1 does not belong to ϒ. This completes the proof. □

Let $\mathcal{S}$ be a square matrix of order $l \geq1$ which has complex entries. We have the following two corollaries.

Corollary 2.3

If $\Vert \sum_{j=0}^{P}{(e^{i\gamma}\mathcal{S})}^{j} \Vert < \infty$ for any $\gamma\in\mathbb{R}$ and any $P\in\mathbb{Z}_{+}$, then $e^{-i\gamma}$ is not an eigenvalue of $\mathcal{S}$.

Proof

Let $\mathcal{H} = e^{i\gamma}\mathcal{S}$. By virtue of Lemma 2.2, 1 is not an eigenvalue of $e^{i\gamma}\mathcal{S}$, and thus $I-e^{i\gamma}\mathcal{S}$ is an invertible matrix or $e^{i\gamma}(e^{-i\gamma}I - \mathcal{S})$ is an invertible matrix, i.e., $e^{-i\gamma}$ is not an eigenvalue of $\mathcal{S}$. The proof is complete. □

Corollary 2.4

If $\Vert \sum_{j=0}^{P}{(e^{i\gamma}\mathcal{S})}^{j} \Vert < \infty$ for any $\gamma\in\mathbb{R}$ and any $P\in\mathbb{Z}_{+}$, then $\vert \lambda \vert <1$ for any eigenvalue λ of $\mathcal{S}$.

Proof

By virtue of

$$ I - \bigl(e^{i\gamma}\mathcal{S} \bigr)^{P}= \bigl(I - e^{i\gamma}\mathcal{S} \bigr) \bigl(I + e ^{i\gamma}\mathcal{S} + \cdots + \bigl(e^{i\gamma}\mathcal{S} \bigr)^{P-1} \bigr) \quad \text{for any } P \in\mathbb{Z}_{+} \text{ and any } \gamma\in\mathbb{R}, $$

we deduce that

$$ \bigl\Vert \bigl(e^{i\gamma}\mathcal{S} \bigr)^{P} \bigr\Vert \leq1 + \bigl\Vert \bigl(I - e^{i\gamma} \mathcal{S} \bigr) \bigr\Vert \bigl\Vert \bigl(I + e^{i\gamma}\mathcal{S} + \cdots+ \bigl(e^{i\gamma } \mathcal{S} \bigr)^{P-1} \bigr) \bigr\Vert \leq1 + \bigl( 1 + \Vert \mathcal{S} \Vert \bigr) K. $$

It follows from Lemmas 2.1 and 2.2 that the absolute value of each eigenvalue λ of $e^{i\gamma}\mathcal{S}$ is less than or equal to one and $e^{-i\gamma}$ is in the resolvent set of $\mathcal{S}$, respectively. Thus, we have, for any eigenvalue λ of $\mathcal{S}$, $\vert \lambda \vert <1$. This completes the proof. □

Definition 2.5

Let ϵ be a positive real number. If there exists a constant $L\geq0$ such that, for every differentiable function ϕ satisfying the relation $\Vert \dot{\phi}(t)-\Lambda(t)\phi(t) \Vert \leq \epsilon$ for any $t\in\mathbb{R}_{+}$, there exists an exact solution $\theta(t)$ of $\dot{\theta}(t)=\Lambda(t)\theta(t)$ such that

$$ \bigl\Vert \phi(t)-\theta(t) \bigr\Vert \leq L\epsilon, $$

(2.1)

then the system $\dot{\theta}(t) = \Lambda(t)\theta(t)$ is said to be Hyers-Ulam stable.

Remark 2.6

If $\phi(t)$ is an approximate solution of $\dot{\theta}(t)=\Lambda (t)\theta(t)$, then $\dot{\phi}(t)\approx\Lambda(t)\phi(t)$. Hence, letting g be an error function, then $\phi(t)$ is the exact solution of $\dot{\phi}(t)=\Lambda(t)\phi(t)+g(t)$.

On the basis of Remark 2.6, Definition 2.5 can be modified as follows.

Definition 2.7

Let ϵ be a positive real number. If there exists a constant $L\geq0$ such that, for every differentiable function ϕ satisfying $\Vert g(t) \Vert \leq\epsilon$ for any $t\in \mathbb{R}_{+}$, there exists an exact solution $\theta(t)$ of $\dot{\theta}(t)=\Lambda(t) \theta(t)$ such that (2.1) holds, then the system $\dot{\theta}(t)=\Lambda(t)\theta(t)$ is said to be Hyers-Ulam stable.

3 Main results

Let us consider the time dependent 2-periodic system

$$ \dot{\theta} {(t)} = \Lambda(t)\theta{(t)}, \quad\theta(t) \in \mathbb{C}^{l} \text{ and } t\in\mathbb{R}_{+}, \hspace{160.5pt} \bigl(\Lambda(t) \bigr) $$

where $\Lambda(t+2)= \Lambda(t)$ for all $t \in\mathbb{R}_{+}$.

Definition 3.1

Let $\mathcal{B}(t)$ be the fundamental solution matrix of $(\Lambda (t))$. The system $(\Lambda(t))$ is said to be uniformly exponentially stable if there exist two positive constants M and α such that

$$ \bigl\Vert \mathcal{B}(t){\mathcal{B}}^{-1}(s) \bigr\Vert \leq Me^{-\alpha(t-s)} \quad\mbox{for all } t \geq s. $$

It follows from [11] that system $(\Lambda(t))$ is uniformly exponentially stable if and only if the spectrum of the matrix $\mathcal{B}(2)$ lies inside of the circle of radius one.

Consider now the Cauchy problem

$$ \textstyle\begin{cases} \dot{\phi}(t) = \Lambda(t)\phi (t)+e^{i\gamma t}\xi(t), \quad t\in\mathbb{R_{+}}, \\ \phi(0) = \theta_{0}. \end{cases}\displaystyle \hspace{134.5pt} \bigl( \Lambda(t), \gamma, \theta_{0} \bigr) $$

The solution of the Cauchy problem $(\Lambda(t), \gamma, \theta_{0})$ is given by

$$ \phi(t) =\mathcal{B}(t)\mathcal{B}^{-1}(0)\theta_{0}+ \int_{0}^{t} \mathcal{B}(t){\mathcal{B}}^{-1}(s)e^{i\gamma s} \xi(s) \,ds. $$

For $I:=[0,2]$ and $i\in\{1,2\}$, we define the functions $\pi_{i}: I \rightarrow\mathbb{C}$ by

$$ \pi_{1}(t) = \textstyle\begin{cases} t, & 0\leq t< 1, \\ 2-t, & 1\leq t\leq2, \end{cases}\displaystyle \quad \text{and} \quad\pi_{2}(t)=t(2-t). $$

(3.1)

Let us denote by $\mathcal{M}_{i}$ the set $\{\xi\in\mathcal{W}_{0} ^{2}({ \mathbb{R}}_{+}, \mathbb{C}^{l}): \xi(t)=\mathcal{B}(t)\pi _{i}(t), i\in\{1,2\}\}$. We are now in a position to state our main results.

Theorem 3.2

Let the exact solution $\phi(t)$ of the Cauchy problem $(\Lambda(t), \gamma, \theta_{0})$ be an approximate solution of system $(\Lambda (t))$ with the error term $e^{i\gamma t}\xi(t)$, where $\gamma \in\mathbb{R}$ and $\xi\in\mathcal{W}_{0}^{2}({ \mathbb{R}}_{+}, \mathbb{C}^{l})$. Then the following two statements hold.

• (1) If system $(\Lambda(t))$ is uniformly exponentially stable, then system $(\Lambda(t))$ is Hyers-Ulam stable.

• (2) If $\mathcal{M}:=\mathcal{M}_{1}\cup\mathcal{M}_{2}$, $\xi\in\mathcal{M}\subset\mathcal{W}_{0}^{2}({ \mathbb{R}}_{+}, \mathbb{C}^{l})$, and system $(\Lambda(t))$ is Hyers-Ulam stable, then system $(\Lambda(t))$ is uniformly exponentially stable.

Proof

(1) Let $\epsilon> 0$ and $\phi(t)$ be the approximate solution of $(\Lambda(t))$ such that $\sup_{t\in\mathbb{R}_{+}} \Vert \dot {\phi}(t)- \Lambda(t)\phi(t) \Vert =\sup_{t\in\mathbb{R}_{+}} \Vert e^{i\gamma t}\xi(t) \Vert$, $\phi(0)=\theta_{0}$, and $\sup_{t\in \mathbb{R}_{+}} \Vert \xi(t) \Vert \leq\epsilon$, and let $\theta(t)$ be the exact solution of $(\Lambda(t))$. Then

$$\begin{aligned} \sup_{t\in\mathbb{R}_{+}} \bigl\Vert \phi(t)-\theta(t) \bigr\Vert =& \sup _{t\in\mathbb{R}_{+}} \biggl\Vert \mathcal{B}(t)\mathcal{B}^{-1}(0) \theta_{0}+ \int_{0}^{t}\mathcal{B}(t){\mathcal{B}}^{-1}(s)e^{i\gamma s} \xi(s) \,ds-\mathcal{B}(t)\mathcal{B}^{-1}(0)\theta_{0} \biggr\Vert \\ =& \int_{0}^{t}\mathcal{B}(t){\mathcal{B}}^{-1}(s)e^{i\gamma s} \xi(s) \,ds \leq \int_{0}^{t} \bigl\Vert \mathcal{B}(t){ \mathcal{B}}^{-1}(s) \bigr\Vert \bigl\Vert \xi(s) \bigr\Vert \,ds \\ \leq& \int_{0}^{t}Me^{-\alpha(t-s)} \bigl\Vert \xi(s) \bigr\Vert \,ds = Me^{-\alpha t} \int_{0}^{t}e^{\alpha s} \bigl\Vert \xi(s) \bigr\Vert \,ds \\ \leq& Me^{-\alpha t} \int_{0}^{t}e^{\alpha s}\epsilon\,ds = \epsilon \frac{M}{\alpha} \bigl(1 - e^{-\alpha t} \bigr) \leq \frac{M}{\alpha} \epsilon= L\epsilon, \end{aligned}$$

where $M> 0$, $\alpha>0$, and $L:={M}/{\alpha}$. Hence, $\sup_{t\in\mathbb{R}_{+}} \Vert \phi(t)-\theta(t) \Vert \leq L\epsilon$, which implies that system $(\Lambda(t))$ is Hyers-Ulam stable.

(2) The proof of the second part is more tricky. Let $a\in \mathbb{C}^{l}$ and $\xi_{1}\in\mathcal{W}_{0}^{2}(\mathbb{R_{+}}, \mathbb{C}^{l})$ be such that

$$ \xi_{1}(s)= \textstyle\begin{cases} \mathcal{B}(s)s a, & \mbox{if } s\in[0,1), \\ \mathcal{B}(s)(2-s)a, & \mbox{if } s \in[1,2]. \end{cases} $$

Then we have, for each $s\in\mathbb{R}_{+}$, $\xi_{1}(s)= \mathcal{B}(s)\pi_{1}(s)a$, where $\pi_{1}$ is defined by (3.1). Thus, for any positive integer $n\geq1$,

$$ \phi_{\xi_{1} }(2n) = \int_{0}^{2n}\mathcal{B}(2n){\mathcal{B}}^{-1}( \tau)e^{i\gamma\tau}\xi_{1}(\tau) \,d\tau=\sum _{k=0}^{n-1} \int_{2k}^{2k+2}\mathcal{B}(2n){\mathcal{B}}^{-1}( \tau)e^{i\gamma \tau}\xi_{1}(\tau) \,d\tau. $$

Let $\tau=2k+s$. We know that $\mathcal{B}^{-1}(2k + s) = \mathcal{B}^{-1}(2k)\mathcal{B}^{-1}(s)$, and so

$$ \begin{aligned} \phi_{\xi_{1} }(2n)&=\sum_{k=0}^{n-1} \int_{0}^{1}\mathcal{B}(2n) {\mathcal{B}}^{-1}(2k + s)e^{2i\gamma k}e^{i\gamma s}\xi_{1}(s) \,ds \\ & = \sum _{k=0}^{n-1}e^{2i\gamma k}\mathcal{B}(2n-2k)a \int_{0}^{2}e ^{i\gamma s}\pi_{1}(s) \,ds. \end{aligned} $$

Define

$$ \mathcal{A}_{1}:=\mathbb{R} \backslash{\{ 2k \pi: k\in\mathbb{Z} \}} \quad\mbox{and} \quad\mathcal{C}_{1}(\gamma):= \int_{0}^{2}e^{i \gamma s}\pi_{1}(s) \,ds. $$

It is not difficult to verify that $\mathcal{C}_{1}(\gamma) \neq0$ for any $\gamma\in\mathcal{A}_{1}$, and hence

$$ \phi_{\xi_{1} }(2n) \bigl({\mathcal{C}_{1}(\gamma )} \bigr)^{-1}=\sum_{k=0} ^{n-1}e^{2i\gamma k}\mathcal{B}(2n-2k)a \quad\mbox{for all } \gamma \in \mathcal{A}_{1}. $$

(3.2)

Again, let $\xi_{2} \in\mathcal{W}_{0}^{2}(\mathbb{R_{+}}, \mathbb{C}^{l})$ be given on $[0,2]$ such that $\xi_{2}(s)= \mathcal{B} (s)\pi_{2}(s)a$, where $\pi_{2}$ is defined as in (3.1). With a similar approach to above, we have

$$ \phi_{\xi_{2} }(2n) \bigl(\mathcal{C}_{2}(\gamma) \bigr)^{-1}=\sum_{k=0} ^{n-1}e^{2i\gamma k} \mathcal{B}(2n-2k)a, \quad\gamma\in\mathcal{A} _{2}:={\{ 2k\pi: k \in \mathbb{Z}\}}, $$

(3.3)

where

$$ \mathcal{C}_{2}(\gamma) := \int_{0}^{2}e^{i\gamma s} \pi_{2}(s) \,ds. $$

By virtue of the fact that system $(\Lambda(t))$ is Hyers-Ulam stable, we conclude that $\phi_{\xi_{1} }$ and $\phi_{\xi_{2} }$ are bounded functions, i.e., there exist two positive constants $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$ such that

$$ \bigl\Vert \phi_{\xi_{1} }(2n) \bigr\Vert \leq\mathcal{K}_{1} \quad\mbox{and} \quad \bigl\Vert \phi_{\xi_{2} }(2n) \bigr\Vert \leq \mathcal{K}_{2} \quad\mbox{for all } n= 1,2,\ldots. $$

It follows from (3.2) and (3.3) that

$$ \Biggl\Vert \sum_{k=0}^{n-1} e^{2i\gamma k}\mathcal{B} {(2n-2k)}a \Biggr\Vert \leq\frac{\mathcal {K}}{ \vert \mathcal{C}_{1} \vert } := \mathcal{R}_{1}, \quad\text{if } \gamma\in\mathcal{A}_{1}, $$

(3.4)

and

$$ \Biggl\Vert \sum_{k=0}^{n-1} e^{2i\gamma k}\mathcal{B} {(2n-2k)}a \Biggr\Vert \leq\frac{\mathcal {K}_{2}}{ \vert \mathcal{C}_{2} \vert } := \mathcal{R}_{2}, \quad\text{if } \gamma\in\mathcal{A}_{2}, $$

(3.5)

respectively. Hence, by virtue of (3.4) and (3.5), we have, for any $\gamma\in\mathcal{A}_{1}\cup\mathcal{A}_{2}=\mathbb{R}$ and each $a \in\mathbb{C}^{l}$,

$$ \Biggl\Vert \sum_{k=0}^{n-1} e^{2i\gamma k}\mathcal{B} {(2n-2k)}a \Biggr\Vert \leq\mathcal{R}_{1} + \mathcal{R}_{2}. $$

(3.6)

Let $n-k=j$. Then

$$ \sum_{k=0}^{n-1} e^{2i\gamma k}\mathcal{B} {(2n-2k)}a=e^{2i \gamma n}\sum_{j=1}^{n} e^{-2i\gamma j}\mathcal{B} {(2j)}a. $$

From (3.6), we obtain

$$ \Biggl\Vert \sum_{j=1}^{n} e^{-2i\gamma j} \bigl(\mathcal{B} {(2)} \bigr)^{j} \Biggr\Vert \leq L < \infty. $$

Thus, using $\mathcal{S}=\mathcal{B}(2)$ in Corollary 2.4, we deduce that the spectrum of $\mathcal{B}(2)$ lies in the interior of the circle of radius one, i.e., system $(\Lambda(t))$ is uniformly exponentially stable. This completes the proof. □

Corollary 3.3

Let the exact solution $\phi(t)$ of the Cauchy problem $(\Lambda(t), \gamma, \theta_{0})$ be an approximate solution of system $(\Lambda (t))$ with the error term $e^{i\gamma t}\xi(t)$, where $\gamma \in\mathbb{R}$, $\xi\in\mathcal{M}\subset\mathcal{W}_{0}^{2}( { \mathbb{R}}_{+}, \mathbb{C}^{l})$, and $\mathcal{M}:=\mathcal{M} _{1}\cup\mathcal{M}_{2}$. Then system $(\Lambda(t))$ is uniformly exponentially stable if and only if it is Hyers-Ulam stable.