Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 425102, 6 pages

http://dx.doi.org/10.1155/2013/425102

Research Article

On the Stability of Nonautonomous Linear Impulsive Differential Equations

JinRong Wang1,2 and Xuezhu Li²

¹School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, China

²Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

Received 2 May 2013; Accepted 21 June 2013

Academic Editor: Gestur Ólafsson

Copyright © 2013 JinRong Wang and Xuezhu Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce two Ulam's type stability concepts for nonautonomous linear impulsive ordinary differential equations. Ulam-Hyers and Ulam-Hyers-Rassias stability results on compact and unbounded intervals are presented, respectively.

1. Introduction

During the past decades, the impulsive differential equations have attracted many authors since it is better to describe dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting and diseases than the corresponding differential equations without impulses. For the basic theory on the impulsive differential equations and impulsive controls, the reader can refer to the monographs of Baĭnov and Simeonov [1], Lakshmikantham et al. [2], Yang [3], and Benchohra et al. [4] and references therein. In particular, exponential, asymptotical, strong, weak and Lyapunov stability of all kinds of impulsive differential equations has been studied extensively in the previous monographs and references therein.

In addition to the previously mentioned stability theory, Ulam stability of functional equation, which was formulated by Ulam on a talk given to a conference at Wisconsin University in 1940, is one of the central subjects in the mathematical analysis area. Many researchers paid much attention to discuss the stability properties of all kinds of equations. In fact, Ulam’s type stability problems have been taken up by a large number of mathematicians, and the study of this area has grown to be one of the most important subjects in the mathematical analysis area. For the advanced contribution on such problems, we refer the reader to András and Kolumbán [5], András and Mészáros [6], Burger et al. [7], Cǎdariu [8], Castro and Ramos [9], Ciepliński [10], Cimpean and Popa [11], Hyers et al. [12], Hegyi and Jung [13], Jung [14, 15], Lungu and Popa [16], Miura et al. [17, 18], Moslehian and Rassias [19], Rassias [20, 21], Rus [22, 23], Takahasi et al. [24], and Wang et al. [25–27].

As far as we know, there are few results on Ulam’s type stability of nonautonomous impulsive differential equations. Motivated by recent works [23, 25, 27], we study Ulam’s type stability of nonautonomous linear impulsive differential equations:

where , , is -order real diagonal matrix, and and are -order bounded diagonal matrix and -dimensional bounded vector, respectively. Impulsive sequence satisfy , , and and represent the right and left limits of at , .

Firstly, we will modify the Ulam’s type stability concepts in [23] and introduce two Ulam’s type stability concepts for (1). Secondly, we pay attention to check the Ulam-Hyers and Ulam-Hyers-Rassias stability results on a compact and unbounded intervals, respectively.

2. Preliminaries

Let be the Banach space of all continuous functions from into with the norm for , where . Also, we use the Banach space , , and there exist and , , with with the norm . Denote . Set . It can be seen that endowed with the norm , is also a Banach space.

If , , , by , we mean that for .

It follows [3], we introduce the concept of piecewise continuous solutions.

Definition 1. By a , from solution of the following impulsive Cauchy problem

we mean that the function which satisfies

where is called impulsive evolution matrix which is given by

is the evolution matrix for the system and denotes the identity matrix.

If there exists , such that for any , then satisfy

By proceeding with the same elementary computation in Lemma of [28], we have

Next, we introduce two Ulam’s type stability definitions for (1) which can be regarded as the extension of the Ulam’s type stability concepts for ordinary differential equations in [23].

Let , , and be nondecreasing functions where . For , denote

We consider the following inequalities:

Definition 2. Equation (1) is Ulam-Hyers stable, if there exist constants , , such that for each and for each solution of inequality (8) there exists a solution of (1) with

Definition 3. Equation (1) is Ulam-Hyers-Rassias stable with respect to if there exist , such that for each and for each solution of inequality (9) there exists a solution of (1) with

3. Stability Results in the Case

We introduce the following assumptions.

().(),  .

Denote , and denote .

Now, we are ready to state our first Ulam-Hyers stable result on a compact interval.

Theorem 4. Assume that are satisfied. Then (1) is Ulam-Hyers stable.

Proof. Let be a solution of inequality (8). Define

Then we have

According to Definition 1, for each , we have

Suppose that be the unique solution of the impulsive Cauchy problem:

Then for each , we have

It follows that, for each , we can derive

Thus,

where

So (1) is Ulam-Hyers stable. The proof is completed.

In order to discuss Ulam-Hyers-Rassias stability, we need the following condition.

() There exist ,  , such that

where is nondecreasing.

Theorem 5. Assume that are satisfied. Then (1) is Ulam-Hyers-Rassias stable.

Proof. Let be a solution of inequality (9). For and defined in (12), we have

Let be the unique solution of (15). Hence, for each , it follows from that

Thus, we obtain

where

Thus, (1) is Ulam-Hyers-Rassias stable. The proof is completed.

4. Stability Results in the Case

In this section, we will present stability results on an unbounded interval.

We change () to the following strong condition.

() is continuous and uniformly bounded function on , . Thus, there exists such that for any .

Theorem 6. Assume that are satisfied. Then (1) is Ulam-Hyers stable.

Proof. Let be a solution of inequality (8). For and , defined in (12), we have

Let be the unique solution of (15). Hence, for each , we have

Thus, we obtain

where

Thus, (1) is Ulam-Hyers stable. The proof is completed.

Next, we suppose the following.

() There exist ,  , such that

where is nondecreasing.

We have

Theorem 7. Assume that and are satisfied. Then (1) is Ulam-Hyers-Rassias stable.

Proof. Let be a solution of inequality (9). For and defined in (12), we have

Let be the unique solution of (15). Hence for each , it follows from that

Thus, we obtain

where

Thus, (1) is Ulam-Hyers-Rassias stable. The proof is completed.

Acknowledgments

The authors thank the referee for valuable comments and suggestions which improved their paper. This work is supported by the National Natural Science Foundation of China (11201091), Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169) and Key Support Subject (Applied Mathematics) of Guizhou Normal College.

References

D. D. Baĭnov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66, Longman Scientific & Technical, New York, NY, USA, 1993.

V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific Publishing, Teaneck, NJ, USA, 1989.

T. Yang, Impulsive Control Theory, vol. 272 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2001.

M. Benchohra, J. Henderson, and S. K. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2, Hindawi Publishing Corporation, New York, NY, USA, 2006.

S. András and J. J. Kolumbán, “On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 82, pp. 1–11, 2013.

S. András and A. R. Mészáros, “Ulam-Hyers stability of dynamic equations on time scales via Picard operators,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4853–4864, 2013.

M. Burger, N. Ozawa, and A. Thom, “On Ulam stability,” Israel Journal of Mathematics, vol. 193, no. 1, pp. 109–129, 2013.

L. Cǎdariu, Stabilitatea Ulam-Hyers-Bourgin Pentru Ecuatii Functionale, Universitatii de Vest Timisoara, Timişara, Romania, 2007.

L. P. Castro and A. Ramos, “Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations,” Banach Journal of Mathematical Analysis, vol. 3, no. 1, pp. 36–43, 2009.

K. Ciepliński, “Applications of fixed point theorems to the Hyers-Ulam stability of functional equations—a survey,” Annals of Functional Analysis, vol. 3, no. 1, pp. 151–164, 2012.

D. S. Cimpean and D. Popa, “Hyers-Ulam stability of Euler's equation,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1539–1543, 2011.

D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, Mass, USA, 1998.

B. Hegyi and S.-M. Jung, “On the stability of Laplace's equation,” Applied Mathematics Letters, vol. 26, no. 5, pp. 549–552, 2013.

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, FL, USA, 2001.

S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order,” Applied Mathematics Letters, vol. 17, no. 10, pp. 1135–1140, 2004.

N. Lungu and D. Popa, “Hyers-Ulam stability of a first order partial differential equation,” Journal of Mathematical Analysis and Applications, vol. 385, no. 1, pp. 86–91, 2012.

T. Miura, S. Miyajima, and S.-E. Takahasi, “A characterization of Hyers-Ulam stability of first order linear differential operators,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 136–146, 2003.

T. Miura, S. Miyajima, and S.-E. Takahasi, “Hyers-Ulam stability of linear differential operator with constant coefficients,” Mathematische Nachrichten, vol. 258, pp. 90–96, 2003.

M. S. Moslehian and T. M. Rassias, “Stability of functional equations in non-Archimedean spaces,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 2, pp. 325–334, 2007.

T. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264–284, 2000.

T. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000.

I. A. Rus, “Ulam stability of ordinary differential equations,” Studia. Universitatis Babeş-Bolyai. Mathematica, vol. 54, no. 4, pp. 125–133, 2009.

I. A. Rus, “Ulam stabilities of ordinary differential equations in a Banach space,” Carpathian Journal of Mathematics, vol. 26, no. 1, pp. 103–107, 2010.

S.-E. Takahasi, T. Miura, and S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued differential equation ,” Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp. 309–315, 2002.

J. Wang, Y. Zhou, and M. Fečkan, “Nonlinear impulsive problems for fractional differential equations and Ulam stability,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3389–3405, 2012.

J. Wang and Y. Zhou, “Mittag-Leffler-Ulam stabilities of fractional evolution equations,” Applied Mathematics Letters, vol. 25, no. 4, pp. 723–728, 2012.

J. Wang, M. Fečkan, and Y. Zhou, “Ulam's type stability of impulsive ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 395, no. 1, pp. 258–264, 2012.

J. Wang, X. Xiang, W. Wei, and Q. Chen, “Existence and global asymptotical stability of periodic solution for the -periodic logistic system with time-varying generating operators and -periodic impulsive perturbations on Banach spaces,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 524945, 16 pages, 2008.