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

*Correspondence: litongx2007@163.com 6LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P.R. China 7School of Information Science and Engineering, Linyi University, Linyi, Shandong 276005, P.R. China Full list of author information is available at the end of the article Abstract We prove that the system θ̇ (t) = (t)θ (t), t ∈ 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 φ̇(t) = (t)φ(t) + eiγ tξ (t), t ∈R+, φ(0) = θ0 as the approximate solutions of θ̇ (t) = (t)θ (t), where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with ξ (0) = 0, and (t) 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.