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

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1 Zada et al. Advances in Difference Equations 217) 217:192 DOI /s R E S E A R C H Open Access Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems Akbar Zada 1, Peiguang Wang 2, Dhaou Lassoued 3,4,5 and Tongxing Li 6,7* * Correspondence: litongx27@163.com 6 LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 2765, P.R. China 7 School of Information Science and Engineering, Linyi University, Linyi, Shandong 2765, P.R. China Full list of author information is available at the end of the article Abstract We prove that the system θt)= t)θt), 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 φt)= t)φt)+e iγ t ξt), t R +, φ) = θ as the approximate solutions of θt)= t)θt), where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with ξ) =, and t) is a 2-periodic square matrix of order l. MSC: 34D2 Keywords: Hyers-Ulam stability; uniform exponential stability; nonautonomous system; periodic system 1 Introduction The stability theory is an important branch of the qualitative theory of differential equations. In 194, 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] relaxedthecondition forthe boundofthenormof Cauchy difference f x + y) f x) fy).tothebestofourknowledge,papersbyobłoza [5, 6] published in the late 199s 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)=yx), which was then extended to the Banach space-valued linear differential equation of the form y x)=λyx) 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 θt) = t)θt), t R to its dichotomy for dichotomy in autonomous case; see, e.g., [27, 28]). We conclude by mentioning that Barbu et al. [1] provedthathyers- The Authors) 217. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Zada et al. Advances in Difference Equations 217) 217:192 Page 2 of 7 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 θ n+1 = n θ n, n Z +, 1.1) where Z + is the set of all nonnegative integers and n )isanω-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 θt)= t)θt), t R +,where t) isasquarematrixoforderl. We proved that the 2-periodic system θt)= t)θt)ishyers- 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, 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, LZ +, C l )is the space of all C l -valued bounded functions with sup norm, and let W 2 R +, C l )bethe set of all continuous, bounded, and 2-periodic vectorial functions f with the property that f ) =. Let H be a square matrix of order l 1 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. Lemma 2.1 If H n < for any n Z +, then λ 1 for any λ ϒ. Proof Suppose to the contrary that λ > 1. 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 +,andthus H n H n θ / θ = λ n as n. Therefore, λ 1. The proof is complete. Lemma 2.2 If P j= Hj < for any P Z +, then 1 does not belong to ϒ. Proof If 1 ϒ, thenhθ = θ for some nonzero vector θ in C l and H k θ = θ for all k = 1,2,...,P. Therefore, we conclude that P sup H j I + H + + H P )θ) = sup sup P Z + θ θ P Z + j= sup P Z +,θ P θ θ =, and so 1 does not belong to ϒ. This completestheproof. Let S beasquarematrixoforderl 1 which has complex entries. We have the following two corollaries.

3 Zada et al. Advances in Difference Equations 217) 217:192 Page 3 of 7 Corollary 2.3 If P j= eiγ S) 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 2.2, 1 is not an eigenvalue of e iγ S,andthusI e iγ S is an invertible matrix or e iγ e iγ I S)isaninvertiblematrix,i.e., e iγ is not an eigenvalue of S. The proof is complete. Corollary 2.4 If P j= eiγ S) j < for any γ R and any P Z +, then λ <1for any eigenvalue λ of S. Proof By virtue of I e iγ S ) P = I e iγ S ) I + e iγ S + + e iγ S ) P 1) for any P Z + and any γ R, we deduce that e iγ S ) P 1+ I e iγ S ) I + e iγ S + + e iγ S ) P 1) S ) K. It follows from Lemmas 2.1 and 2.2 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,wehave, for any eigenvalue λ of S, λ < 1. This completes the proof. Definition 2.5 Let ɛ be a positive real number. If there exists a constant L suchthat, for every differentiable function φ satisfying the relation φt) t)φt) ɛ for any t R +, there exists an exact solution θt)of θt)= t)θt)suchthat φt) θt) Lɛ, 2.1) then the system θt)= t)θt)issaidtobehyers-ulamstable. Remark 2.6 If φt) is an approximate solution of θt) = t)θt), then φt) t)φt). Hence, letting g be an error function, then φt) is the exact solution of φt)= t)φt)+gt). On the basis of Remark2.6, Definition 2.5 can be modified as follows. Definition 2.7 Let ɛ be a positive real number. If there exists a constant L suchthat, for every differentiable function φ satisfying gt) ɛ for any t R +, there exists an exact solution θt)of θt)= t)θt)suchthat2.1) holds, then the system θt)= t)θt)issaid to be Hyers-Ulam stable. 3 Main results Let us consider the time dependent 2-periodic system θt)= t)θt), θt) C l and t R +, t) ) where t +2)= t) for all t R +.

4 Zada et al. Advances in Difference Equations 217) 217:192 Page 4 of 7 Definition 3.1 Let Bt) 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 Bt)B 1 s) Me αt s) for all t s. It follows from [11]thatsystem t)) is uniformly exponentially stable if and only if the spectrum of the matrix B2) lies inside of the circle of radius one. Consider now the Cauchy problem φt)= t)φt)+e iγ t ξt), t R +, ) t), γ, θ φ) = θ. The solution of the Cauchy problem t), γ, θ )isgivenby φt)=bt)b 1 )θ + Bt)B 1 s)e iγ s ξs) ds. For I := [, 2] and i {1, 2}, we define the functions π i : I C by t, t <1, π 1 t)= and π 2 t)=t2 t). 3.1) 2 t, 1 t 2, Let us denote by M i the set {ξ W 2R +, C l ):ξt)=bt)π i t), i {1, 2}}. Wearenowina position to state our main results. Theorem 3.2 Let the exact solution φt) of the Cauchy problem t), γ, θ ) be an approximate solution of system t)) with the error term e iγ t ξt), where γ R and ξ W 2R +, C l ). Then the following two statements hold. 1) If system t)) is uniformly exponentially stable, then system t)) is Hyers-Ulam stable. 2) If M := M 1 M 2, ξ M W 2R +, C l ), and system t)) is Hyers-Ulam stable, then system t)) is uniformly exponentially stable. Proof 1) Let ɛ >andφt) be the approximate solution of t)) such that sup t R+ φt) t)φt) = sup t R+ e iγ t ξt), φ) = θ,andsup t R+ ξt) ɛ, andletθt) be the exact solution of t)). Then sup φt) θt) = sup Bt)B 1 )θ + t R + = t R + Bt)B 1 s)e iγ s ξs) ds Bt)B 1 s)e iγ s ξs) ds Bt)B 1 )θ Me αt s) ξs) ds = Me αt Bt)B 1 s) ξs) ds e αs ξs) ds Me αt e αs ɛ ds = ɛ M 1 e αt ) M α α ɛ = Lɛ,

5 Zada et al. Advances in Difference Equations 217) 217:192 Page 5 of 7 where M >,α >,andl := M/α. Hence,sup t R+ φt) θt) Lɛ, which implies that system t)) is Hyers-Ulam stable. 2) The proof of the second part is more tricky. Let a C l and ξ 1 W 2R +, C l )besuch that Bs)sa, ifs [, 1), ξ 1 s)= Bs)2 s)a, ifs [1, 2]. Then we have, for each s R +, ξ 1 s)=bs)π 1 s)a, whereπ 1 is defined by 3.1). Thus, for any positive integer n 1, φ ξ1 2n)= 2n B2n)B 1 τ)e iγτ ξ 1 τ) dτ = 2k+2 k= 2k Let τ =2k + s.weknowthatb 1 2k + s)=b 1 2k)B 1 s), and so Define φ ξ1 2n)= 1 k= B2n)B 1 2k + s)e 2iγ k e iγ s ξ 1 s) ds = e 2iγ k B2n 2k)a 2 k= A 1 := R\{2kπ : k Z} and C 1 γ ):= e iγ s π 1 s) ds. 2 e iγ s π 1 s) ds. B2n)B 1 τ)e iγτ ξ 1 τ) dτ. It is not difficult to verify that C 1 γ ) foranyγ A 1, and hence φ ξ1 2n) C 1 γ ) ) 1 = e 2iγ k B2n 2k)a for all γ A ) k= Again, let ξ 2 W 2R +, C l )begivenon[,2]suchthatξ 2 s)=bs)π 2 s)a, whereπ 2 is defined as in 3.1). With a similar approach to above, we have φ ξ2 2n) C 2 γ ) ) 1 = e 2iγ k B2n 2k)a, γ A 2 := {2kπ : k Z}, 3.3) where C 2 γ ):= 2 k= e iγ s π 2 s) ds. By virtue of the fact that system t)) is Hyers-Ulam stable, we conclude that φ ξ1 and φ ξ2 are bounded functions, i.e., there exist two positive constants K 1 and K 2 such that φξ1 2n) K1 and φξ2 2n) K2 for all n =1,2,...

6 Zada et al. Advances in Difference Equations 217) 217:192 Page 6 of 7 It follows from 3.2)and3.3)that e 2iγ k B2n 2k)a K C 1 := R 1, ifγ A 1, 3.4) k= and e 2iγ k B2n 2k)a K 2 C 2 := R 2, ifγ A 2, 3.5) k= respectively. Hence, by virtue of 3.4)and3.5), we have, for any γ A 1 A 2 = R and each a C l, e 2iγ k B2n 2k)a R 1 + R ) k= Let n k = j.then e 2iγ k B2n 2k)a = e 2iγ n k= n e 2iγ j B2j)a. j=1 From 3.6), we obtain n e 2iγ j B2) ) j L <. j=1 Thus, using S = B2) in Corollary 2.4, wededucethatthespectrumofb2) lies in the interior of the circle of radius one, i.e., system t)) is uniformly exponentially stable. This completes the proof. Corollary 3.3 Let the exact solution φt) of the Cauchy problem t), γ, θ ) be an approximate solution of system t)) with the error term e iγ t ξt), where γ R, ξ M W 2 R +, C l ), and M := M 1 M 2. Then system t)) is uniformly exponentially stable if and only if it is Hyers-Ulam stable. Acknowledgements The authors express their sincere gratitude to the editors for useful comments that helped to improve the presentation of the results and accentuate important details. This research is supported by NNSF of P.R. China Grant Nos , , and ), CPSF Grant No. 215M58291), NSF of Shandong Province Grant No. ZR216JL21), DSRF of Linyi University Grant No. LYDX215BS1), and the AMEP of Linyi University, P.R. China. Competing interests The authors declare that they have no competing interests. Authors contributions All four authors contributed equally to this work. They all read and approved the final version of the manuscript. Author details 1 Department of Mathematics, University of Peshawar, Peshawar, 25, Pakistan. 2 College of Electronic and Information Engineering, Hebei University, Baoding, Hebei 712, P.R. China. 3 Département de Mathématiques, Faculté des Sciences de Gabès, Université de Gabès, Cité Erriadh, Gabès 672, Tunisia. 4 Laboratoire Équations aux dérivées partielles LEDP-LR3ES4, Faculté des Sciences de Tunis, Université de Tunis El Manar, Tunis, 292, Tunisia. 5 Laboratoire SAMM EA4543, Université Paris 1 Panthéon-Sorbonne, Centre P. M. F., 9 rue de Tolbiac, Paris Cedex 13, 75634, France. 6 LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 2765, P.R. China. 7 School of Information Science and Engineering, Linyi University, Linyi, Shandong 2765, P.R. China.

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