# A note on stability of impulsive differential equations

YuMei Liao12 and JinRong Wang12*

Author Affiliations

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

2 Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, P.R. China

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Boundary Value Problems 2014, 2014:67  doi:10.1186/1687-2770-2014-67

 Received: 5 December 2013 Accepted: 5 March 2014 Published: 24 March 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

### Abstract

In this note, we study a new class of ordinary differential equations with non-instantaneous impulses. Both existence and generalized Ulam-Hyers-Rassias stability results are established. Finally, an example is given to illustrate our theoretical results.

##### Keywords:
impulsive differential equations; non-instantaneous impulses; stability

### 1 Introduction

Many evolution processes studied in applied sciences are represented by differential equations. However, the situation is quite different in many modeled phenomena which have a sudden change in their states such as population dynamics, biotechnology processes, chemistry, engineering, medicine and so on. One of the mathematical models about such processes can be formulated by the following impulsive differential equations:

(1)

where the function and impulsive conditions , . We set and . The fixed time sequence is increasing, i.e., . and represent the right and left limits of at , respectively. Here, the impulsive conditions are the combination of the traditional initial value problems and the short-term perturbations whose duration can be negligible in comparison with the duration of such a process.

However, the above short-term perturbations could not show the dynamic change of evolution processes completely in pharmacotherapy. As we know, the introduction of the drugs in the bloodstream and the consequent absorption for the body are a gradual and continuous process. Thus, we have to use a new model to describe such an evolution process. In fact, the above situation has fallen in a new impulsive action, which starts at an arbitrary fixed point and keeps active on a finite time interval. To achieve this aim, Hernández and O’Regan [1] introduced a new class of abstract semilinear impulsive differential equations with non-instantaneous impulses. Then, the concept of mild solutions and existence results are presented. Next, Pierri et al.[2] continued the work and developed the results in [1] and obtained new existence results in a fractional power space.

In 1940, the famous stability of functional equations was firstly offered by Ulam at Wisconsin University and concerned approximate homomorphisms. Thereafter, Ulam’s stability problem [3] has attracted many famous researchers, one can refer to the interesting monographs of Hyers [4,5], Rassias [6], Jung [7], Cădariu [8], and an important survey of Brillouët-Belluot et al.[9] via the recent special issue on Ulam-type stability edited by Brzdȩk et al.[10]. For the recent Ulam’s stability concepts and results on ordinary differential equations (with impulses), one can see [11,12] and reference therein.

Motivated by [1,2,11,12], we introduce a new Ulam-type stability concept for the following semilinear differential equations with non-instantaneous impulses:

(2)

where are pre-fixed numbers, is continuous, and is continuous for all .

The novelty of our paper is considering a new type of equation (2), then presenting a generalized Ulam-Hyers-Rassias stability definition and finding reasonable conditions on equation (2) to show that equation (2) is generalized Ulam-Hyers-Rassias stable.

In Section 2, we introduce a new Ulam-type stability concept for equation (2) (see Definition 2.2). In Section 3, we mainly prove a generalized Ulam-Hyers-Rassias stability result for equation (2) on a compact interval. Finally, an example is given to illustrate our theoretical results.

### 2 Preliminaries

Throughout this paper, let be the Banach space of all continuous functions from J into ℝ with the norm for . We introduce the Banach space := {, , and there exist and , , with } with the norm . Meanwhile, we set with . Clearly, endowed with the norm is also a Banach space.

By virtue of the concept about the solutions in [1], we can introduce the following definition.

Definition 2.1 A function is called a classical solution of the problem

(3)

if x satisfies

Next, we adopt the idea in [12] and introduce a new Ulam-type stability concept for equation (2). Set . Let and . We consider the following inequality:

(4)

Definition 2.2 Equation (2) is generalized Ulam-Hyers-Rassias stable with respect to if there exists such that for each solution of inequality (4), there exists a solution of equation (2) with

Remark 2.3 Definition 2.2 has practical meaning in the following sense. Consider an evolution process with not sudden changes of states but acting on an interval, which can be modeled by equation (2). Assume that we can measure the state of the process at any time to get a function . Putting this into equation (2), in general, we do not expect to get a precise solution of equation (2). All what is required is to get a function which satisfies the suitable approximation inequality (4). Our result of Section 3 will guarantee that there is a solution of inequality (4) close to the measured output and closeness is defined in the sense of generalized Ulam-Hyers-Rassias stability. This technique is quite useful in many applications such as numerical analysis, optimization, biology and economics, where it is quite difficult to find the exact solution.

Remark 2.4 A function is a solution of inequality (4) if and only if there is and a sequence , (which depend on y) such that

(i) , and , ;

(ii) , , ;

(iii) , , .

Remark 2.5 If is a solution of inequality (4), then y is a solution of the following integral inequality:

(5)

In fact, by Remark 2.4 we get

(6)

Clearly, the solution of equation (6) is given by

For each , , we get

Proceeding as above, we derive that

In order to deal with Ulam-type stability, we need the following result (see Theorem 16.4, [13]).

Lemma 2.6Let the following inequality hold:

where, ais nondecreasing and, , .

Then, for, the following inequality is valid:

where.

### 3 Main results

We introduce the following assumptions:

(H1) .

(H2) There exists a positive constant such that

() is strongly measurable for the first variable and is continuous for the second variable. There exists a positive constant and a nondecreasing function such that

(H3) and there are positive constants , , such that

(H4) There exists a constant and a nondecreasing function such that

Concerning the existence results for the solutions about problem (3), one can repeat the same procedure in Theorems 2.1 and 2.2 of Hernández and O’Regan [1] to derive the following results. So we omit the proof here.

Theorem 3.1Assume that (H1), (H2) and (H3) are satisfied. Then problem (3) has the unique solutionprovided that

(7)

Theorem 3.2Assume that () and (H3) are satisfied, the functionsare bounded. Then problem (3) has at least one solutionprovided that

Now, we discuss the stability of equation (2) by using the concept of generalized Ulam-Hyers-Rassias in the above section.

Theorem 3.3Assume that (H1), (H2), (H3) and (H4) are satisfied. Then equation (2) is generalized Ulam-Hyers-Rassias stable with respect toprovided that (7) holds.

Proof Let be a solution of inequality (4). Denote by x the unique solution of the impulsive Cauchy problem

(8)

Then we get

Keeping in mind (5), for each , , we have

and for each , , we have

and for each , we have

Hence, for each , , we get

Thus, by Lemma 2.6, we have

(9)

for each , .

Further, for each , , we have

which yields that

(10)

Moreover, for each , we have

By Gronwall’s inequality, we obtain

(11)

Summarizing, we combine (9), (10) and (11) and derive that

for all , which implies that equation (2) is generalized Ulam-Hyers-Rassias stable with respect to . The proof is completed. □

### 4 Example

Let and . Denote with for and with for . We set and .

Consider

(12)

and

(13)

Let be a solution of inequality (13). Then there exist and such that

(14)

For , integrating (14) from 0 to t, we have

For , we have

After checking the conditions in Theorem 3.1, we find that

(15)

has a unique solution. Let us take the solution x of problem (15) given by

For , we have

For , we have

which yields that

Summarizing, we have

which yields that equation (12) is generalized Ulam-Hyers-Rassias stable with respect to .

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

This work was carried out in collaboration between all authors. JRW raised these interesting problems in this research. YML and JRW proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, read and approved the manuscript.

### Acknowledgements

The authors thank the referees for their careful reading of the manuscript and insightful comments, which helped to improve the quality of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improving the presentation of the paper. This work is supported by Project of Guizhou Normal College (12YB023), Doctor Project of Guizhou Normal College (13BS010), Guizhou Province Education Planning Project (2013A062), Key Project on the Reforms of Teaching Contents and Course System and Key Support Subject (Applied Mathematics) of Guizhou Normal College.

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