It is well known that fractional differential equations (FDE) have no nonconstant periodic solutions . Then asymptotically periodic solutions are shown [1, 2, 15]. This paper is devoted to this problem. We first study impulsive FDE (IFDE) with Caputo derivatives when the lower limits are periodically varying at each impulses. We show that such kind of IFDE do may possess periodic solutions. We present several particular cases of IFDE along with nonexistence results. For completeness of the paper, we also discuss IFDE when the lower limit of the Caputo derivative is fixed, which is introduced in [5, 14]. Then there is also a problem on the definition of solutions for such kind of IFDE; this is mentioned in . Here we just study periodic boundary value conditions on finite intervals. For more development, methods, and the current research on impulsive and functional fractional differential equations, we refer to .
The paper is organized as follows: Section 2 is devoted to the study of IFDE with varying lower limits for Caputo derivatives. First, in Section 2.1, we deal with general IFDE to show uniqueness, existence and asymptotic stability results for periodic solutions. In Section 2.2, we study IFDE with weak nonlinearities leading to the averaging theory for IFDE like in . Section 2.3 is devoted to IFDE with equidistant and periodically shifted impulses. This specific form allows us to obtain a nonexistence result for periodic solutions which is applied to an impulsive fractional Lorenz system. Then we obtain Landesman–Lazer-type existence results for periodic solutions. Section 2.4 continues the investigation of IFDE with equidistant and periodically shifted impulses, but all are small. The corresponding limiting ODE is derived and, by using known results from numerical dynamics, the relationship between dynamics of the limiting ODE and original IFDE is studied. For completeness of the paper, Section 3 deals with IFDE with fixed lower limit of Caputo derivative. We extend some results of Section 2 to that case. Several concrete examples are given in the paper for illustration of theoretical results. Section 4 briefly outlines further possible ways for studies.
2 FDE with Caputo derivatives with varying lower limits
2.1 General impulses
Let . We consider
where , is the generalized Caputo fractional derivative with lower limit at . We suppose the following conditions:
is continuous and T-periodic in t.
There are constants , such that
for any , and any .
There is such that , and for any .
Under assumptions (i) and (ii), there holds
for , where is the Mittag-Leffler function (see [8, p. 40]).
On all intervals , , equation (2.1) is equivalent to
Then using (2.4) with (ii), we get
as desired. ∎
Now we can prove the following result.
Suppose (i)–(iii) are satisfied. If , then (2.1) has a unique T-periodic solution, which is in addition asymptotically stable.
Inequality (2.5) implies that is a contraction, so applying the Banach fixed point theorem yields that P has a unique fixed point . Moreover, since
for any , we see that the corresponding periodic solution is asymptotically stable. ∎
2.2 Systems with weak nonlinearities
Now we consider (2.1) with small nonlinearities of the form
where ε is a small parameter. Then (2.6) has a unique solution and the Poincaré mapping is given by
We suppose that
f and are -smooth.
Then is -smooth as well. Moreover, we have
Note that solves
so we easily derive
Consequently, (2.7) gives
Now we are ready to prove the following result.
Suppose assumptions (i) and (C1). If there is a simple zero of M, i.e., and , then (2.6) has a unique T-periodic solution located near for any small. Moreover, if then it is asymptotically stable, and if , then it is unstable. If for any , then (2.6) has no T-periodic solution for small.
If there is a simple zero of M, then (2.8) can be solved by the implicit function theorem to get its solution with . Moreover, . Thus stability and instability results follow directly by the known arguments (see ), so we omit details. ∎
2.3 Systems with equidistant and periodically shifted impulses
where and . We consider the norm for . We suppose the following conditions:
is continuous, locally Lipschitz in x and T-periodic in t, where for some .
There is a constant such that for any .
The satisfy for any .
We are looking for T-periodic solutions of (2.9) on . It is well known  that under above assumptions, (2.9) has a unique solution on , which is continuous in , and left continuous in t at impulsive points . So we can consider the Poincaré mapping
Clearly fixed points of P determine T-periodic solutions of (2.9).
Under assumptions (I)–(III), there holds
for and , where is the rth iteration of P.
On all intervals , , equation (2.9) is equivalent to
for . This implies (2.10) since . Moreover, we have
for . This implies (2.11). ∎
Now we can prove the following result.
Suppose assumptions (I)–(III). If
then (2.9) has no rT-periodic solution for any .
We need to solve , which is equivalent to
This contradicts (2.12). ∎
We consider the Lorenz system
which for , , , , and without impulses presents a chaotic attractor (see Fig. 1). The utilized numerical scheme to integrate Lorenz’s system is the predictor-corrector Adams–Bashforth–Moulton method for FDEs .
Take and , . Let
which embeds Lorenz’s attractor. To apply Theorem 2.5, we need to find
Let . For , , , and , condition (2.12) has the form
for . Note that . Summarizing, we obtain the following proposition.
Now we present existence results. Let , and set
for any . Now we are ready to prove the following result.
Suppose (I)–(III) and, in addition, there are constants , , such that either
uniformly with respect to and for all , or
uniformly with respect to and for all . If
for all , then (2.9) has a T-periodic solution.
We need to solve (2.13). For simplicity, we set
So we need to solve
uniformly with respect to and for all . This implies
for any and , , while
for any , , , and for all . Further, let us take and given by
Note that and . It is easy to check that by fixing sufficiently small, (2.15) implies
for any and for , where is the border of Ω. This gives
where is the Brouwer topological degree. On the other hand, a linear equation has the only solution , which by (2.15) is located in Ω. Moreover, the linearization is a nonsingular matrix, so , and thus . Summarizing, we see that (2.16) has a solution in Ω, and so (2.13) is solvable. ∎
Following the proof of Theorem 2.8, we have a modified result.
Suppose (I)–(III) and, in addition, there are constants , and a permutation such that
for all . Now we consider concrete examples.
2.4 Systems with small equidistant and shifted impulses
where , , , and is Lipschitz. Under above assumptions , equation (2.18) has a unique solution on , which is continuous in , and left continuous in t at impulsive points . So we can consider the Poincaré mapping
On all intervals , , equation (2.18) is equivalent to
into (2.19), we get
where is a local Lipschitz constant of f. Then
and (2.21) gives
Hence (2.20) becomes
We note that (2.19) implies
for . We see that (2.22) is leading to its approximation
which is the Euler numerical approximation of
for . If is a stable (hyperbolic) periodic solution of (2.24), then there is a stable (hyperbolic) invariant curve of Poincaré mapping of (2.18) in a neighborhood of . Of course, h is sufficiently small.
Now we extend (2.18) for periodic impulses of the form
where the satisfy for any and some . So we can consider the Poincaré mapping
which has a form
We already know
Then (2.26) implies
On the other hand, the ODE
has the Euler numerical approximation
with the step size , and its iteration has a form
3 FDE with Caputo derivatives with fixed lower limits
Let for a and . We consider
where , is the generalized Caputo fractional derivative with lower limit at 0. We suppose the following conditions
is continuous and locally Lipschitz in x uniformly for .
There is a constant such that for any and .
It is known that under above assumptions, (3.2) has a unique solution on
Now we can prove the following result.
Suppose assumptions (ci)–(cii). If
then (3.1) has no solution satisfying . Note that for some .
We need to solve
which is equivalent to
This contradicts (3.3). ∎
Note that (3.3) is equivalent to
Now we suppose that
for all and some .
Then we have the following corollary.
Suppose assumptions (ci)–(ciii). Then there hold the following assertions:
We need to verify (3.3) for . First, we note that
Of course, (ii) implies (i). Next, supposing as a small variable, we consider
where , satisfies (ci), (cii) and the satisfy
for all and some .
Then condition (3.5) becomes
condition (3.6) becomes
and (3.7) becomes
Summarizing, we have the following result.
for and .
For the cuboid from Example 2.6,
Now condition (3.5) has the form
again for and .
Finally, for the completeness of the paper, we present the following existence result, whose proof is similar to Theorem 2.9, so we omit it.
Suppose (ci), (cii) and (C3). If
holds for all , then (3.9) has a solution satisfying .
In this paper, fractional differential equations with periodic impulses are investigated. We establish new criteria for the uniqueness, existence and asymptotic stability of periodic solutions for impulsive fractional differential equations with either varying or fixed lower limits.
To conclude this paper, we roughly outline a possible further progress on this topic. Firstly, one can consider a nonhomogeneous linear case
where are matrices, , is the unit matrix, and then consider a semilinear case
for achieving more specific results similar to the ones for impulsive ODEs  in the finite-dimensional case. Of course, we can extend (4.1) and (4.2) to an infinite-dimensional space, that is, we consider the associated impulsive fractional evolution equation by setting A as a generator of a -semigroup on an infinite-dimensional Banach space, and then using the theory of semigroups with nonlinear functional analysis. In addition, the investigation of the existence of almost periodic solutions for (4.1) and (4.2) may be more interesting.
Secondly, one can also extend our recent results in  to consider linear and semilinear differential equations with periodic noninstantaneous impulses as follows:
respectively, where acts as an impulsive point and acts as a junction point satisfying the conditions , and periodicity conditions and . In addition, one can study the associated fractional order and infinite-dimensional cases for (4.3) and (4.4). The issues on the existence and stability of almost periodic solutions would be another interesting branch.
Finally, one can consider controllability and iterative learning control for the above equations with impulsive periodic controls arising from some real problems in engineering.
We thank Professor Marius Danca for a motivating discussion during the preparation of this paper. The authors are grateful to the referees for their careful reading of the manuscript and valuable comments, and to the editor for his help.
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About the article
Published Online: 2017-06-04
Award identifier / Grant number: APVV-14-0378
Award identifier / Grant number: 2/0153/16
Award identifier / Grant number: 1/0078/17
Funding Source: National Natural Science Foundation of China
Award identifier / Grant number: 11661016
Funding Source: Science and Technology Foundation of Guizhou Province
Award identifier / Grant number: (2016)4006
The first author is supported by the Slovak Research and Development Agency under the contract No. APVV-14-0378 and by the Slovak Grant Agency VEGA No. 2/0153/16 and No. 1/0078/17. The second author is supported by National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Unite Foundation of Guizhou Province (7640) and Outstanding Scientific and Graduate ZDKC (003).
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 482–496, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0015.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0