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

We prove that the system θ˙(t)=Λ(t)θ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot{\theta}(t) =\Lambda(t)\theta(t)$\end{document}, t∈R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\in\mathbb{R}_{+}$\end{document}, 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 ϕ˙(t)=Λ(t)ϕ(t)+eiγtξ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot{\phi}(t)=\Lambda(t)\phi(t)+e^{i\gamma t}\xi(t)$\end{document}, t∈R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\in\mathbb{R}_{+}$\end{document}, ϕ(0)=θ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi(0)=\theta_{0}$\end{document} as the approximate solutions of θ˙(t)=Λ(t)θ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot{\theta}(t)=\Lambda(t)\theta(t)$\end{document}, where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with ξ(0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\xi(0)=0$\end{document}, and Λ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Lambda(t)$\end{document} is a 2-periodic square matrix of order l.


Introduction
The stability theory is an important branch of the qualitative theory of differential equations. In , Ulam [] queried a problem regarding the stability of differential equations for homomorphism as follows: when can an approximate homomorphism from a group G  to a metric group G  be approximated by an exact homomorphism? Hyers [] brilliantly gave a partial answer to this question assuming that G  and G  are Banach spaces. Later on, Aoki [] and Rassias [] extended and improved the results obtained in []. In particular, Rassias [] 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 [, ] published in the late s 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., [-] and the references cited therein. Alsina and Ger [] 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) = λy(x) by Takahasi et al. []. Zada et al. [] generalized the concept of Hyers-Ulam stability of the nonautonomous w-periodic linear differential matrix systemθ(t) = (t)θ (t), t ∈ R to its dichotomy (for dichotomy in autonomous case; see, e.g., [, ]). We conclude by mentioning that Barbu et al. [] proved that Hyers-Ulam stability and the exponential dichotomy of linear differential periodic systems are equivalent.
Very recently, Li and Zada [] gave connections between Hyers-Ulam stability and uniform exponential stability of the first-order linear discrete system where Z + is the set of all nonnegative integers and ( n ) is an ω-periodic sequence of bounded linear operators on Banach spaces. They proved that system (.) 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 [] 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 systemθ (t) = (t)θ (t), t ∈ R + , where (t) is a square matrix of order l. We proved that the -periodic systemθ(t) = (t)θ (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  as the period in our approach.

Notation and preliminaries
Throughout the paper, R is the set of all real numbers, R + denotes the set of all nonnegative real numbers, Z + stands for the set of all nonnegative integers, C l denotes the l-dimensional space of all l-tuples complex numbers, · is the norm on C l , L(Z + , C l ) is the space of all C l -valued bounded functions with 'sup' norm, and let W   (R + , C l ) be the set of all continuous, bounded, and -periodic vectorial functions f with the property that f () = .
Let H be a square matrix of order l ≥  which has complex entries and let ϒ denote the spectrum of H, i.e., ϒ := {λ : λ is an eigenvalue of H}. We have the following lemmas.
Proof Suppose to the contrary that |λ| > . By the definition of eigenvalue, there exists a nonzero vector θ ∈ C l such that Hθ = λθ , which implies that H n θ = λ n θ for any n ∈ Z + , and thus H n ≥ H n θ / θ = |λ| n → ∞ as n → ∞. Therefore, |λ| ≤ . The proof is complete.
Proof If  ∈ ϒ, then Hθ = θ for some nonzero vector θ in C l and H k θ = θ for all k = , , . . . , P. Therefore, we conclude that and so  does not belong to ϒ. This completes the proof.
Let S be a square matrix of order l ≥  which has complex entries. We have the following two corollaries.
j < ∞ for any γ ∈ R and any P ∈ Z + , then e -iγ is not an eigenvalue of S.
Proof Let H = e iγ S. By virtue of Lemma .,  is not an eigenvalue of e iγ S, and thus I -e iγ S is an invertible matrix or e iγ (e -iγ I -S) is an invertible matrix, i.e., e -iγ is not an eigenvalue of S. The proof is complete.
Corollary . If P j= (e iγ S) j < ∞ for any γ ∈ R and any P ∈ Z + , then |λ| <  for any eigenvalue λ of S.
Proof By virtue of Ie iγ S P = Ie iγ S I + e iγ S + · · · + e iγ S P- for any P ∈ Z + and any γ ∈ R, we deduce that It follows from Lemmas . and . that the absolute value of each eigenvalue λ of e iγ S is less than or equal to one and e -iγ is in the resolvent set of S, respectively. Thus, we have, for any eigenvalue λ of S, |λ| < . This completes the proof.
Definition . Let be a positive real number. If there exists a constant L ≥  such that, for every differentiable function φ satisfying the relation φ (t) -(t)φ(t) ≤ for any t ∈ R + , there exists an exact solution θ (t) ofθ (t) = (t)θ (t) such that then the systemθ (t) = (t)θ (t) is said to be Hyers-Ulam stable.
Definition . Let be a positive real number. If there exists a constant L ≥  such that, for every differentiable function φ satisfying g(t) ≤ for any t ∈ R + , there exists an exact solution θ (t) ofθ (t) = (t)θ (t) such that (.) holds, then the systemθ (t) = (t)θ (t) is said to be Hyers-Ulam stable.

Main results
Let us consider the time dependent -periodic systeṁ where (t + ) = (t) for all t ∈ R + .

Definition . Let B(t) be the fundamental solution matrix of ( (t)). The system ( (t))
is said to be uniformly exponentially stable if there exist two positive constants M and α such that It follows from [] that system ( (t)) is uniformly exponentially stable if and only if the spectrum of the matrix B() lies inside of the circle of radius one.
Consider now the Cauchy problem The solution of the Cauchy problem ( (t), γ , θ  ) is given by For I := [, ] and i ∈ {, }, we define the functions π i : I → C by Let us denote by M i the set {ξ ∈ W   (R + , C l ) : ξ (t) = B(t)π i (t), i ∈ {, }}. We are now in a position to state our main results.
Theorem . Let the exact solution φ(t) of the Cauchy problem ( (t), γ , θ  ) be an approximate solution of system ( (t)) with the error term e iγ t ξ (t), where γ ∈ R and ξ ∈ W   (R + , C l ). Then the following two statements hold.
() The proof of the second part is more tricky. Let a ∈ C l and ξ  ∈ W   (R + , C l ) be such that Then we have, for each s ∈ R + , ξ  (s) = B(s)π  (s)a, where π  is defined by (.). Thus, for any positive integer n ≥ , It is not difficult to verify that C  (γ ) =  for any γ ∈ A  , and hence Again, let ξ  ∈ W   (R + , C l ) be given on [, ] such that ξ  (s) = B(s)π  (s)a, where π  is defined as in (.). With a similar approach to above, we have where C  (γ ) :=   e iγ s π  (s) ds.
By virtue of the fact that system ( (t)) is Hyers-Ulam stable, we conclude that φ ξ  and φ ξ  are bounded functions, i.e., there exist two positive constants K  and K  such that Thus, using S = B() in Corollary ., we deduce that the spectrum of B() lies in the interior of the circle of radius one, i.e., system ( (t)) is uniformly exponentially stable. This completes the proof.
Corollary . Let the exact solution φ(t) of the Cauchy problem ( (t), γ , θ  ) be an approximate solution of system ( (t)) with the error term e iγ t ξ (t), where γ ∈ R, ξ ∈ M ⊂ W   (R + , C l ), and M := M  ∪ M  . Then system ( (t)) is uniformly exponentially stable if and only if it is Hyers-Ulam stable.