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.
Keywords:
impulsive differential equations; noninstantaneous impulses; stability1 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
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
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
By virtue of the concept about the solutions in [1], we can introduce the following definition.
Definition 2.1 A function
if x satisfies
Next, we adopt the idea in [12] and introduce a new Ulamtype stability concept for equation (2). Set
Definition 2.2 Equation (2) is generalized UlamHyersRassias stable with respect to
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
Remark 2.4 A function
(i)
(ii)
(iii)
Remark 2.5 If
In fact, by Remark 2.4 we get
Clearly, the solution of equation (6) is given by
For each
Proceeding as above, we derive that
In order to deal with Ulamtype stability, we need the following result (see Theorem 16.4, [13]).
Lemma 2.6Let the following inequality hold:
where
Then, for
where
3 Main results
We introduce the following assumptions:
(H_{1})
(H_{2}) There exists a positive constant
(
(H_{3})
(H_{4}) There exists a constant
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 (H_{1}), (H_{2}) and (H_{3}) are satisfied. Then problem (3) has the unique solution
Theorem 3.2Assume that (
Now, we discuss the stability of equation (2) by using the concept of generalized UlamHyersRassias in the above section.
Theorem 3.3Assume that (H_{1}), (H_{2}), (H_{3}) and (H_{4}) are satisfied. Then equation (2) is generalized UlamHyersRassias stable with respect to
Proof Let
Then we get
Keeping in mind (5), for each
and for each
and for each
Hence, for each
Thus, by Lemma 2.6, we have
for each
Further, for each
which yields that
Moreover, for each
By Gronwall’s inequality, we obtain
Summarizing, we combine (9), (10) and (11) and derive that
for all
4 Example
Let
Consider
and
Let
For
For
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
For
which yields that
Summarizing, we have
which yields that equation (12) is generalized UlamHyersRassias 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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