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# A note on stability of impulsive differential equations

- YuMei Liao
^{1, 2}and - JinRong Wang
^{1, 2}Email author

**2014**:67

https://doi.org/10.1186/1687-2770-2014-67

© Liao and Wang; licensee Springer. 2014

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

## 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

where the function $f:J\times \mathbb{R}\to \mathbb{R}$ and impulsive conditions ${I}_{k}:\mathbb{R}\to \mathbb{R}$, $k=1,2,\dots ,m$. We set ${t}_{0}=0$ and ${t}_{m+1}=T$. The fixed time sequence ${\{{t}_{k}\}}_{k=1,2,\dots ,m}$ is increasing, *i.e.*, ${t}_{k}<{t}_{k+1}$. $x({t}_{k}^{+})={lim}_{\u03f5\to {0}^{+}}x({t}_{k}+\u03f5)$ and $x({t}_{k}^{-})={lim}_{\u03f5\to {0}^{-}}x({t}_{k}+\u03f5)$ represent the right and left limits of $x(t)$ at $t={t}_{k}$, 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.

where $0={t}_{0}={s}_{0}<{t}_{1}\le {s}_{1}\le {t}_{2}<\cdots <{s}_{m-1}\le {t}_{m}\le {s}_{m}\le {t}_{m+1}=T$ are pre-fixed numbers, $f:[0,T]\times \mathbb{R}\to \mathbb{R}$ is continuous, and ${g}_{i}:[{t}_{i},{s}_{i}]\times \mathbb{R}\to \mathbb{R}$ is continuous for all $i=1,2,\dots ,m$.

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 $C(J,\mathbb{R})$ be the Banach space of all continuous functions from *J* into ℝ with the norm ${\parallel x\parallel}_{C}:=sup\{|x(t)|:t\in J\}$ for $x\in C(J,\mathbb{R})$. We introduce the Banach space $PC(J,\mathbb{R})$ := {$x:J\to \mathbb{R}:x\in C(({t}_{k},{t}_{k+1}],\mathbb{R})$, $k=0,1,\dots ,m$, and there exist $x({t}_{k}^{-})$ and $x({t}_{k}^{+})$, $k=1,\dots ,m$, with $x({t}_{k}^{-})=x({t}_{k})$} with the norm ${\parallel x\parallel}_{PC}:=sup\{|x(t)|:t\in J\}$. Meanwhile, we set $P{C}^{1}(J,\mathbb{R}):=\{x\in PC(J,\mathbb{R}):{x}^{\prime}\in PC(J,\mathbb{R})\}$ with ${\parallel x\parallel}_{P{C}^{1}}:=max\{{\parallel x\parallel}_{PC},{\parallel {x}^{\prime}\parallel}_{PC}\}$. Clearly, $P{C}^{1}(J,\mathbb{R})$ endowed with the norm ${\parallel \cdot \parallel}_{P{C}^{1}}$ 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 $x\in P{C}^{1}(J,\mathbb{R})$ is called a classical solution of the problem

*x*satisfies

**Definition 2.2**Equation (2) is generalized Ulam-Hyers-Rassias stable with respect to $(\phi ,\psi )$ if there exists ${c}_{f,{g}_{i},\phi ,m}>0$ such that for each solution $y\in P{C}^{1}(J,\mathbb{R})$ of inequality (4), there exists a solution $x\in P{C}^{1}(J,\mathbb{R})$ 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 $x(\cdot )$. Putting this $x(\cdot )$ 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 $y(\cdot )$ of inequality (4) close to the measured output $x(\cdot )$ 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 $y\in P{C}^{1}(J,\mathbb{R})$ is a solution of inequality (4) if and only if there is $G\in PC(J,\mathbb{R})$ and a sequence ${G}_{i}$, $i=1,2,\dots ,m$ (which depend on

*y*) such that

- (i)
$|G(t)|\le \phi (t)$, $t\in J$ and $|{G}_{i}|\le \psi $, $i=1,2,\dots ,m$;

- (ii)
${y}^{\prime}(t)=f(t,y(t))+G(t)$, $t\in ({s}_{i},{t}_{i+1}]$, $i=0,1,2,\dots ,m$;

- (iii)
$y(t)={g}_{i}(t,y(t))+{G}_{i}$, $t\in ({t}_{i},{s}_{i}]$, $i=1,2,\dots ,m$.

**Remark 2.5**If $y\in P{C}^{1}(J,\mathbb{R})$ is a solution of inequality (4), then

*y*is a solution of the following integral inequality:

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

**Lemma 2.6**

*Let the following inequality hold*:

*where* $u,a,b\in PC({\mathbb{R}}_{+},{\mathbb{R}}_{+}):=\{x\in PC({\mathbb{R}}_{+},\mathbb{R}):x(t)\ge 0\}$, *a* *is nondecreasing and* $b(t)>0$, ${\beta}_{k}>0$, $k=1,\dots ,m$.

*Then*,

*for*$t\in {\mathbb{R}}_{+}$,

*the following inequality is valid*:

*where* $\beta =max\{{\beta}_{k}:k=1,\dots ,m\}$.

## 3 Main results

We introduce the following assumptions:

(H_{1}) $f\in C(J\times \mathbb{R},\mathbb{R})$.

_{2}) There exists a positive constant ${L}_{f}$ such that

_{3}) ${g}_{i}\in C([{t}_{i},{s}_{i}]\times \mathbb{R},\mathbb{R})$ and there are positive constants ${L}_{{g}_{i}}$, $i=1,2,\dots ,m$, such that

_{4}) There exists a constant ${c}_{\phi}>0$ and a nondecreasing function $\phi \in PC(J,{\mathbb{R}}_{+})$ 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 3.1**

*Assume that*(H

_{1}), (H

_{2})

*and*(H

_{3})

*are satisfied*.

*Then problem*(3)

*has the unique solution*$x\in P{C}^{1}(J,\mathbb{R})$

*provided that*

**Theorem 3.2**

*Assume that*(${\mathrm{H}}_{2}^{\prime}$)

*and*(H

_{3})

*are satisfied*,

*the functions*${g}_{i}(\cdot \phantom{\rule{0.2em}{0ex}},0)$

*are bounded*.

*Then problem*(3)

*has at least one solution*$x\in P{C}^{1}(J,\mathbb{R})$

*provided that*

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

**Theorem 3.3** *Assume that* (H_{1}), (H_{2}), (H_{3}) *and* (H_{4}) *are satisfied*. *Then equation* (2) *is generalized Ulam*-*Hyers*-*Rassias stable with respect to* $(\phi ,\psi )$ *provided that* (7) *holds*.

*Proof*Let $y\in P{C}^{1}(J,\mathbb{R})$ be a solution of inequality (4). Denote by

*x*the unique solution of the impulsive Cauchy problem

for each $t\in ({s}_{i},{t}_{i+1}]$, $i=1,2,\dots ,m$.

for all $t\in J$, which implies that equation (2) is generalized Ulam-Hyers-Rassias stable with respect to $(\phi ,\psi )$. The proof is completed. □

## 4 Example

Let $J=[0,2]$ and $0={t}_{0}={s}_{0}<{t}_{1}=1<{s}_{1}=2$. Denote $f(t,x(t))=\frac{|x(t)|}{(1+9{e}^{t})(1+|x(t)|)}$ with ${L}_{f}=\frac{1}{10}$ for $t\in (0,1]$ and ${g}_{1}(t,x(t))=\frac{|x(t)|}{(5-e+{e}^{t})(2+|x(t)|)}$ with ${L}_{{g}_{1}}=\frac{1}{10}$ for $t\in (1,2]$. We set $\phi (t)={e}^{t}$ and $\psi =1$.

*t*, we have

*x*of problem (15) given by

which yields that equation (12) is generalized Ulam-Hyers-Rassias stable with respect to $({e}^{t},1)$.

## Declarations

### 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.

## Authors’ Affiliations

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## Copyright

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