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. (1)

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

  2. (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.