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A note on stability of impulsive differential equations
Boundary Value Problems volume 2014, Article number: 67 (2014)
Abstract
In this note, we study a new class of ordinary differential equations with noninstantaneous impulses. Both existence and generalized UlamHyersRassias stability results are established. Finally, an example is given to illustrate our theoretical results.
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:
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 shortterm perturbations whose duration can be negligible in comparison with the duration of such a process.
However, the above shortterm 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 noninstantaneous 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 BrillouëtBelluot et al. [9] via the recent special issue on Ulamtype 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 Ulamtype stability concept for the following semilinear differential equations with noninstantaneous impulses:
where $0={t}_{0}={s}_{0}<{t}_{1}\le {s}_{1}\le {t}_{2}<\cdots <{s}_{m1}\le {t}_{m}\le {s}_{m}\le {t}_{m+1}=T$ are prefixed 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 UlamHyersRassias stability definition and finding reasonable conditions on equation (2) to show that equation (2) is generalized UlamHyersRassias stable.
In Section 2, we introduce a new Ulamtype stability concept for equation (2) (see Definition 2.2). In Section 3, we mainly prove a generalized UlamHyersRassias 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
if x satisfies
Next, we adopt the idea in [12] and introduce a new Ulamtype stability concept for equation (2). Set $PC(J,{\mathbb{R}}_{+}):=\{x\in PC(J,\mathbb{R}):x(t)\ge 0\}$. Let $\psi \ge 0$ and $\phi \in PC(J,{\mathbb{R}}_{+})$. We consider the following inequality:
Definition 2.2 Equation (2) is generalized UlamHyersRassias 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 UlamHyersRassias 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 fact, by Remark 2.4 we get
Clearly, the solution of equation (6) is given by
For each $t\in ({s}_{i},{t}_{i+1}]$, $i=0,1,2,\dots ,m$, we get
Proceeding as above, we derive that
In order to deal with Ulamtype 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}) $f\in C(J\times \mathbb{R},\mathbb{R})$.
(H_{2}) There exists a positive constant ${L}_{f}$ such that
(${\mathrm{H}}_{2}^{\prime}$) $f:J\times \mathbb{R}\to \mathbb{R}$ is strongly measurable for the first variable and is continuous for the second variable. There exists a positive constant ${L}_{f}^{\prime}$ and a nondecreasing function ${W}_{f}\in C([0,\mathrm{\infty}),{\mathbb{R}}_{+})$ such that
(H_{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
(H_{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 UlamHyersRassias 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 UlamHyersRassias 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
Then we get
Keeping in mind (5), for each $t\in [{s}_{i},{t}_{i+1}]$, $i=1,2,\dots ,m$, we have
and for each $t\in ({t}_{i},{s}_{i}]$, $i=1,2,\dots ,m$, we have
and for each $t\in [0,{t}_{1}]$, we have
Hence, for each $t\in ({s}_{i},{t}_{i+1}]$, $i=1,2,\dots ,m$, we get
Thus, by Lemma 2.6, we have
for each $t\in ({s}_{i},{t}_{i+1}]$, $i=1,2,\dots ,m$.
Further, for each $t\in ({t}_{i},{s}_{i}]$, $i=1,2,\dots ,m$, we have
which yields that
Moreover, for each $t\in [0,{t}_{1}]$, we have
By Gronwall’s inequality, we obtain
Summarizing, we combine (9), (10) and (11) and derive that
for all $t\in J$, which implies that equation (2) is generalized UlamHyersRassias 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 ${g}_{1}(t,x(t))=\frac{x(t)}{(5e+{e}^{t})(2+x(t))}$ with ${L}_{{g}_{1}}=\frac{1}{10}$ for $t\in (1,2]$. We set $\phi (t)={e}^{t}$ and $\psi =1$.
Consider
and
Let $y\in P{C}^{1}([0,2],\mathbb{R})$ be a solution of inequality (13). Then there exist $G(\cdot )\in P{C}^{1}([0,2],\mathbb{R})$ and ${G}_{1}\in \mathbb{R}$ such that
For $t\in [0,1]$, integrating (14) from 0 to t, we have
For $t\in (1,2]$, we have
After checking the conditions in Theorem 3.1, we find that
has a unique solution. Let us take the solution x of problem (15) given by
For $t\in [0,1]$, we have
For $t\in (1,2]$, we have
which yields that
Summarizing, we have
which yields that equation (12) is generalized UlamHyersRassias stable with respect to $({e}^{t},1)$.
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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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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.
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Keywords
 impulsive differential equations
 noninstantaneous impulses
 stability