We study the existence of periodic solutions of a first order nonlinear impulsive differential system with piecewise constant arguments.
In the past two decades, the theory of impulsive differential equations has been developed very rapidly. Such equations consist of differential equations with impulse effects and emerge in modelling of real-world problems observed in engineering, physics and biology, etc. The books [1–3] are good sources for the study of impulsive differential equations and their applications. In addition to these, there exist many papers that investigate the behaviour of solutions of impulsive differential equations [4–8].
Since the early 1980s, differential equations with piecewise constant arguments have attracted great deal of attention of researchers in science. Differential systems with piecewise constant arguments appear in diverse areas such as engineering, physics and mathematics. The work  covers a systematical study on mathematical models with piecewise constant arguments. Differential equations with piecewise constant arguments are closely related to difference and differential equations. Therefore, they are stated as hybrid dynamical systems . The qualitative works on oscillation, periodicity and convergence of solutions of differential equations with piecewise constant arguments have been done by works [11–19]. Also, Wiener’s book  is a distinguished source with respect to this area.
Moreover, in , Seifert has taken into consideration the scalar equation
which shows a continuous dynamical system and proved that this equation has a periodic solution with period 2.
So, we have been motivated consider the impulsive differential system with piecewise constant arguments
where is a real constant, is a continuously differentiable function, and i. e., and are right continuous at and [.] denotes the greatest integer function.
(Carvalho’s method, ) If is a positive integer and is a periodic sequence of period then the following hold true:
If is odd and then(3)
If is even and then(4)
For example, if then
The components are continuous for with the possible exception of the points ,
is right continuous and has left-hand limits at the points
exists for every with the possible exception of the points where one-sided derivates exist,
satisfies system (1) on each interval
and satisfy, respectively, (2) at
We prove the following results:
and for satisfies the difference system
system (1), in the interval can be reduced to the ordinary differential equations system
Solving system (8), we get
Replacing by we obtain (6).
Before applying the impulse condition at , we also find the solution of system (1) in the interval as
where and are real constants. Also, we note that under the same conditions the difference system (7) has a unique solution.
From (6), in the interval we have
Hence, the proof is complete.
Assume that is a sufficiently small real constant. If is an odd function and there is a number such that
For it is and also Hence, (15) reduces to the system
Now, let be sufficiently small.
Again, we should establish a solution of system (7) as in the form of (13). To fulfill this, we use the Implicit Function Theorem to show that there exists a such that there are functions and which are continuous for and Putting (13) into (7), we find
Again, if this system holds for and , then it will be satisfied for all Substituting and into (18), respectively, we obtain the system
Since is odd, the Jacobian determinant of system (19) at is
From (10), we obtain that at
So, for sufficiently small there is a such that there exist functions that are continuous on and form a solution of system (19) such that
Hence, the proof is complete.
In this case, Theorem 3, 4 and 5 are still valid for
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Lafci, M., Bereketoglu, H. On a certain impulsive differential system with piecewise constant arguments. Math Sci 8, 121 (2014). https://doi.org/10.1007/s40096-014-0121-x
- Carvalho’s method
- Periodic solution
- Impulsive differential system
- Piecewise constant arguments