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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Periodic impulsive fractional differential equations

Michal FečkanORCID iD: https://orcid.org/0000-0002-7385-6737
  • Corresponding author
  • Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48, Bratislava; and Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia
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  • De Gruyter OnlineGoogle Scholar
/ Jin Rong WangORCID iD: https://orcid.org/0000-0002-6642-1946
Published Online: 2017-06-04 | DOI: https://doi.org/10.1515/anona-2017-0015

Abstract

This paper deals with the existence of periodic solutions of fractional differential equations with periodic impulses. The first part of the paper is devoted to the uniqueness, existence and asymptotic stability results for periodic solutions of impulsive fractional differential equations with varying lower limits for standard nonlinear cases as well as for cases of weak nonlinearities, equidistant and periodically shifted impulses. We also apply our result to an impulsive fractional Lorenz system. The second part extends the study to periodic impulsive fractional differential equations with fixed lower limit. We show that in general, there are no solutions with long periodic boundary value conditions for the case of bounded nonlinearities.

Keywords: Fractional differential equations; impulses; periodic solutions; existence result

MSC 2010: 34A08; 34A37; 34C25

1 Introduction

It is well known that fractional differential equations (FDE) have no nonconstant periodic solutions [7]. 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 [16]. 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 [12].

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 [11]. 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 0={0,1,,}. We consider

{Dtk,tqcx(t)=f(t,x(t)),t(tk,tk+1),k0,x(tk+)=x(tk-)+Δk(x(tk-)),k,x(0)=x0,(2.1)

where q(0,1), Dtk,tqcx(t) is the generalized Caputo fractional derivative with lower limit at tk. We suppose the following conditions:

  • (i)

    f:×mm is continuous and T-periodic in t.

  • (ii)

    There are constants K>0, Lk>0 such that

    f(t,x)-f(t,y)Kx-yandx+Δk(x)-y-Δk(y)Lkx-y

    for any t, x,ym and any k.

  • (iii)

    There is N such that T=tN+1, tk+N+1=tk+T and Δk+N+1=Δk for any k.

It is well known [8] that under assumptions (i) and (ii), (2.1) has a unique solution on +. So we can consider the Poincaré mapping

P(x0)=x(T-)+ΔN+1(x(T-)).

Clearly fixed points of P determine T-periodic solutions of (2.1) (see [4, Lemma 2.2]).

Lemma 2.1.

Under assumptions (i) and (ii), there holds

P(x)-P(y)Θx-yfor all x, for all ym

for Θ=k=0NLk+1Eq(K(tk+1-tk)q), where Eq is the Mittag-Leffler function (see [8, p. 40]).

Proof.

On all intervals (tk,tk+1), k=0,1,,N, equation (2.1) is equivalent to

x(t)=x(tk+)+1Γ(q)tkt(t-s)q-1f(s,x(s))𝑑s.(2.2)

Consider two solutions x and y of (2.1) with x(0)=x0 and y(0)=y0, respectively. Then by (2.2), we get

x(t)-y(t)x(tk+)-y(tk+)+1Γ(q)tkt(t-s)q-1f(s,x(s))-f(s,y(s))𝑑sx(tk+)-y(tk+)+KΓ(q)tkt(t-s)q-1x(s)-y(s)𝑑s.(2.3)

Applying a Gronwall fractional inequality [17, Corollary 2] to (2.3), we obtain

x(t)-y(t)x(tk+)-y(tk+)Eq(K(t-tk)q),t(tk,tk+1).(2.4)

Then using (2.4) with (ii), we get

x(tk+1+)-y(tk+1+)Lk+1Eq(K(tk+1-tk)q)x(tk+)-y(tk+),k=0,1,,N,

which implies

P(x0)-P(y0)k=0NLk+1Eq(K(tk+1-tk)q)x0-y0,(2.5)

as desired. ∎

Now we can prove the following result.

Theorem 2.2.

Suppose (i)(iii) are satisfied. If Θ<1, then (2.1) has a unique T-periodic solution, which is in addition asymptotically stable.

Proof.

Inequality (2.5) implies that P:mm is a contraction, so applying the Banach fixed point theorem yields that P has a unique fixed point x0. Moreover, since

Pn(x0)-Pn(y0)Θnx0-y0

for any y0m, 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

{Dtk,tqcx(t)=εf(t,x(t)),t(tk,tk+1),k0,x(tk+)=x(tk-)+εΔk(x(tk-)),k,x(0)=x0,(2.6)

where ε is a small parameter. Then (2.6) has a unique solution x(ε,t) and the Poincaré mapping is given by

P(ε,x0)=x(ε,T-)+εΔN+1(x(ε,T-)).

We suppose that

  • (C1)

    f and Δk are C2-smooth.

Then P(ε,x0) is C2-smooth as well. Moreover, we have

x(ε,t)=x0+εv(t)+O(ε2),

and thus

{P(ε,x0)=x0+εM(x0)+O(ε2),M(x0)=v(T-)+ΔN+1(x0).(2.7)

Note that v(t) solves

{Dtk,tqcv(t)=f(t,x0),t(tk,tk+1),k=0,1,,N,v(tk+)=v(tk-)+Δk(x0),k=1,2,,N+1,v(0)=0,

so we easily derive

v(T-)=k=1NΔk(x0)+1Γ(q)k=0Ntktk+1(tk+1-s)q-1f(s,x0)𝑑s.

Consequently, (2.7) gives

M(x0)=k=1N+1Δk(x0)+1Γ(q)k=0Ntktk+1(tk+1-s)q-1f(s,x0)𝑑s.

Now we are ready to prove the following result.

Theorem 2.3.

Suppose assumptions (i) and (C1). If there is a simple zero x0Rm of M, i.e., M(x0)=0 and detDM(x0)0, then (2.6) has a unique T-periodic solution located near x0 for any ε0 small. Moreover, if σ(DM(x0))<0 then it is asymptotically stable, and if σ(DM(x0))(0,), then it is unstable. If M(x0)0 for any x0Rm, then (2.6) has no T-periodic solution for ε0 small.

Proof.

To find a T-periodic solution of (2.6), we need to solve P(ε,x0)=x0, which by (2.7) is equivalent to

M(x0)+O(ε)=0.(2.8)

If there is a simple zero x0 of M, then (2.8) can be solved by the implicit function theorem to get its solution x0(ε) with x0(0)=x0. Moreover, DP(ε,x0(ε))=1+εDM(x0)+O(ε2). Thus stability and instability results follow directly by the known arguments (see [10]), so we omit details. ∎

2.3 Systems with equidistant and periodically shifted impulses

We consider

{Dkhqcx(t)=f(t,x(t)),t(kh,(k+1)h),k0,x(kh+)=x(kh-)+Δk,k,x(0)=x0,(2.9)

where h>0 and q(0,1). We consider the norm x=maxi=1,,m|xi| for x=(x1,,xm)m. We suppose the following conditions:

  • (I)

    f:×mm is continuous, locally Lipschitz in x and T-periodic in t, where T=(N+1)h for some N.

  • (II)

    There is a constant M>0 such that f(t,x)M for any t,x.

  • (III)

    The Δkm satisfy Δk+N+1=Δk for any k.

We are looking for T-periodic solutions of (2.9) on +. It is well known [8] that under above assumptions, (2.9) has a unique solution x(x0,t) on +, which is continuous in x0m, t+{khk} and left continuous in t at impulsive points {khk}. So we can consider the Poincaré mapping

P(x0)=x(x0,T+).

Clearly fixed points of P determine T-periodic solutions of (2.9).

Lemma 2.4.

Under assumptions (I)(III), there holds

Pr(x0)=x0+k=1r(N+1)Δk+1Γ(q)k=0rNkh(k+1)h((k+1)h-s)q-1f(s,x(x0,s))𝑑s(2.10)

and

x(x0,t)-x0k=1NΔk+M(N+1)hqΓ(q+1)(2.11)

for tI and rN, where Pr is the rth iteration of P.

Proof.

On all intervals (kh,(k+1)h), k0, equation (2.9) is equivalent to

x(x0,t)=x(kh+)+1Γ(q)kht(t-s)q-1f(s,x(x0,s))𝑑s,

which implies

x(x0,t)=x0+k=1nΔk+1Γ(q)k=0n-1kh(k+1)h((k+1)h-s)q-1f(s,x(x0,s))𝑑s+1Γ(q)nht(t-s)q-1f(s,x(x0,s))𝑑s

for t(nh,(n+1)h). This implies (2.10) since Pr(x0)=x(x0,rT+). Moreover, we have

|x(x0,t)-x0|k=1NΔk+MΓ(q)k=0Nkh(k+1)h((k+1)h-s)q-1𝑑s=k=1NΔk+M(N+1)hqΓ(q+1)

for tI. This implies (2.11). ∎

Now we can prove the following result.

Theorem 2.5.

Suppose assumptions (I)(III). If

k=1N+1ΔkN+1>MhqΓ(q+1),(2.12)

then (2.9) has no rT-periodic solution for any rN.

Proof.

We need to solve Pr(x0)=x0, which is equivalent to

-k=1r(N+1)Δk=1Γ(q)k=0rNkh(k+1)h((k+1)h-s)q-1f(s,x(x0,s))𝑑s.(2.13)

This gives

rk=1N+1Δk1Γ(q)k=0rNkh(k+1)h((k+1)h-s)q-1|f(s,x(x0,s))|𝑑sMr(N+1)hqΓ(q+1).

This contradicts (2.12). ∎

Example 2.6.

We consider the Lorenz system

f(x1,x2,x3)=(a(x2-x1),x1(b-x3)-x2,x1x2-cx3),(2.14)

which for q=0.995, a=10, b=120, c=83, 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 [3].

Chaotic attractor of (2.14).
Figure 1

Chaotic attractor of (2.14).

Take N=1 and tk=kh, k0. Let

𝒞={(x1,x2,x3)3:-50x150,-100x2100, 50x3200},

which embeds Lorenz’s attractor. To apply Theorem 2.5, we need to find

M=maxx𝒞f(x)=maxx𝒞max{|a(x2-x1)|,|x1(b-x3)-x2|,|x1x2-cx3|}533.333.

Let Δk=(Δk1,Δk2,Δk3). For Δ11=Δ21=Δ¯1, Δ12=0, Δ22=Δ¯2, and Δ31=Δ32=Δ¯3, condition (2.12) has the form

max{|Δ¯1|,|Δ¯2|2,|Δ¯3|}>MhqΓ(q+1)2.19767

for h=0.004. Note that T=(N+1)h=0.008. Summarizing, we obtain the following proposition.

Now we present existence results. Let f(t,x)=(f1(t,x),,fm(t,x)), Δk=(Δk1,,Δkm) and set

x^i=(x1,,xi-1,xi+1,,xm)m-1

for any i=1,,m. Now we are ready to prove the following result.

Theorem 2.8.

Suppose (I)(III) and, in addition, there are constants f±iR, i=1,,m, such that either

lim infxifi(t,x)f+i>f-ilim supxi-fi(t,x)(C1)

uniformly with respect to tI and x^iRm-1 for all i=1,,m, or

lim supxifi(t,x)f+i<f-ilim infxi-fi(t,x)(C2)

uniformly with respect to tI and x^iRm-1 for all i=1,,m. If

-k=1N+1ΔkiN+1hqΓ(q+1)(f-i,f+i)(2.15)

for all i=1,,m, then (2.9) has a T-periodic solution.

Proof.

We need to solve (2.13). For simplicity, we set

F(x0)=1Γ(q)k=0Nkh(k+1)h((k+1)h-s)q-1f(s,x(x0,s))𝑑s.

So we need to solve

-k=1N+1Δk=F(x0).(2.16)

Let F(x0)=(F1(x0),,Fm(x0)). By (2.11), there holds limx0i±xi(x0,t)=± uniformly with respect to tI and x^0im-1, where x0=(x01,,x0m) and x(x0,t)=(x1(x0,t),,xm(x0,t)). So by the first possibility (C1), for any ε>0, there is mε>0 such that

fi(s,x(x0,s))>f+i-εfor all x0i>mε,fi(s,x(x0,s))<f-i+εfor all x0i<-mε

uniformly with respect to sI and x^0im-1 for all i=1,,m. This implies

Fi(x0)1Γ(q)k=0Nkh(k+1)h((k+1)h-s)q-1(f+i-ε)𝑑s=(f+i-ε)(N+1)hqΓ(q+1)

for any x0imε and x^0im-1, x^0imε, while

Fi(x0)(f-i+ε)(N+1)hqΓ(q+1)

for any x0i-mε, x^0im-1, x^0imε, and for all i=1,,m. Further, let us take G:mm and H:[0,1]×mm given by

Gi(x0)=(f+i-ε)(N+1)hq2Γ(q+1)mε(x0i+mε)-(f-i+ε)(N+1)hq2Γ(q+1)mε(x0i-mε),H(λ,x0)=(1-λ)F(x0)+λG(x0).

Note that H(0,x0)=F(x0) and H(1,x0)=G(x0). It is easy to check that by fixing ε>0 sufficiently small, (2.15) implies

Hi(λ,x0)-k=1N+1Δki(2.17)

for any x0i=±mε, x^0im-1, x^0imε. Similarly, we verify that (2.17) holds also in the second possibility (C2). Consequently, we derive

Hi(λ,x0)-k=1N+1Δki

for any λ[0,1] and x0Ω for Ω={x0mx0mε}, where Ω is the border of Ω. This gives

deg(F,Ω,-k=1N+1Δk)=deg(G,Ω,-k=1N+1Δk),

where deg is the Brouwer topological degree. On the other hand, a linear equation G(x0)=-k=1N+1Δk has the only solution x¯0, which by (2.15) is located in Ω. Moreover, the linearization DG(x¯0) is a nonsingular matrix, so deg(G,Ω,-k=1N+1Δk)=±1, and thus deg(F,Ω,-k=1N+1Δk)=±10. 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.

Theorem 2.9.

Suppose (I)(III) and, in addition, there are constants f±iR, i=1,,m and a permutation σ:{1,,m}{1,,m} such that

𝑒𝑖𝑡ℎ𝑒𝑟lim infxifσ(i)(t,x)>f+σ(i)>f-σ(i)>lim supxifσ(i)(t,x)𝑜𝑟lim supxifσ(i)(t,x)<f+σ(i)<f-σ(i)<lim infxifσ(i)(t,x)

uniformly with respect to tI and x^iRm-1 for all i=1,,m. If (2.15) holds for all i=1,,m, then (2.9) has a T-periodic solution.

Clearly, either (C1) or (C2) implies (C3). Of course, assumptions (2.12) and (2.15) are complementary since

(f-i,f+i)(-M,M)

for all i=1,,m. Now we consider concrete examples.

Example 2.10.

Consider a system

fi(t,x)=Aitanhxi+Bicostcosxisin(k=1mxk)

for Ai>Bi>0 and all i=1,,m. Then we take

f+i=Ai-Bi,f-i=-Ai+Bi,M=maxi=1,,m{Ai+Bi},T=2π,h=2πN+1.

Hence, assumption (2.12) has the form

k=1N+1ΔkN+1>maxi=1,,m{Ai+Bi}hqΓ(q+1),

and assumption (2.15) has the form

-k=1N+1ΔkiN+1hqΓ(q+1)(-Ai+Bi,Ai-Bi)

for all i=1,,m, respectively. Consequently, Theorem 2.8 can be applied.

2.4 Systems with small equidistant and shifted impulses

We consider

{Dkhqcx(t)=f(x(t)),t(kh,(k+1)h),k0,x(kh+)=x(kh-)+Δ¯hq,k,x(0)=x0,(2.18)

where h>0, q(0,1), Δ¯m, and f:mm is Lipschitz. Under above assumptions [8], equation (2.18) has a unique solution x(x0,t) on +, which is continuous in x0m, t+{khk} and left continuous in t at impulsive points {khk}. So we can consider the Poincaré mapping

Ph(x0)=x(x0,h+).

On all intervals (kh,(k+1)h), k0, equation (2.18) is equivalent to

x(x0,t)=x(kh+)+1Γ(q)kht(t-s)q-1f(x(x0,s))𝑑s=x(kh+)+1Γ(q)0t-kh(t-kh-s)q-1f(x(x(kh+),s))𝑑s.(2.19)

Hence

x((k+1)h+)=Ph(x(kh+))(2.20)

and

Ph(x0)=x(x0,h+)=x0+Δ¯hq+1Γ(q)0h(h-s)q-1f(x(x0,s))𝑑s.(2.21)

Inserting

x(x0,t)=x0+hqy(x0,t),t[0,h],

into (2.19), we get

y(x0,t)=1Γ(q)hq0t(t-s)q-1f(x0+hqy(x0,s))𝑑s=1Γ(q+1)f(x0)+1Γ(q)hq0t(t-s)q-1(f(x0+hqy(x0,s))-f(x0))𝑑s=1Γ(q+1)f(x0)+O(hq)

since

0t(t-s)q-1(f(x0+hqy(x0,s))-f(x0))𝑑s0t(t-s)q-1f(x0+hqy(x0,s))-f(x0)𝑑shqmaxt[0,h]y(x0,t)Lloctqqh2qmaxt[0,h]y(x0,t)Llocq,

where Lloc is a local Lipschitz constant of f. Then

x(x0,t)=x0+hqΓ(q+1)f(x0)+O(h2q),t[0,h],

and (2.21) gives

Ph(x0)=x0+Δ¯hq+hqΓ(q+1)f(x0)+O(h2q).

Hence (2.20) becomes

x((k+1)h+)=x(kh+)+hq(Δ¯+1Γ(q+1)f(x(kh+)))+O(h2q).(2.22)

We note that (2.19) implies

x(x0,t)-x(kh+)=O(hq)(2.23)

for t[kh,(k+1)h]. We see that (2.22) is leading to its approximation

z((k+1)h+)=z(kh+)+hq(Δ¯+1Γ(q+1)f(z(kh+))),

which is the Euler numerical approximation of

z(t)=Δ¯+1Γ(q+1)f(z(t)).(2.24)

Using (2.23) and the known results about the Euler approximation method [6], we arrive at the following result.

Theorem 2.11.

Let z(t) be a solution of (2.24) with z(0)=x0 on [0,T]. Then the solution x(x0,t) of (2.18) exists on [0,T] and satisfies

x(x0,t)=z(thq-1)+O(hq)

for t[0,Th1-q]. If z(t) is a stable (hyperbolic) periodic solution of (2.24), then there is a stable (hyperbolic) invariant curve of Poincaré mapping Ph of (2.18) in a O(hq) neighborhood of z(t). Of course, h is sufficiently small.

Now we extend (2.18) for periodic impulses of the form

{Dkhqcx(t)=f(x(t)),t(kh,(k+1)h),k0,x(kh+)=x(kh-)+Δ¯khq,k,x(0)=x0,(2.25)

where the Δ¯km satisfy Δ¯k+N+1=Δ¯k for any k and some N. So we can consider the Poincaré mapping

Ph(x0)=x(x0,(N+1)h+),

which has a form

Ph=PN+1,hP1,h(2.26)

for

Pk,h(x0)=Δ¯khq+x(x0,h).

We already know

Pk,h(x0)=Δ¯khq+x(x0,h)=x0+Δ¯khq+hqΓ(q+1)f(x0)+O(h2q).

Then (2.26) implies

Ph(x0)=x0+hqk=1N+1Δ¯k+(N+1)hqΓ(q+1)f(x0)+O(h2q).

On the other hand, the ODE

z(t)=k=1N+1Δ¯kN+1+1Γ(q+1)f(z(t))(2.27)

has the Euler numerical approximation

x0+hq(k=1N+1Δ¯kN+1+1Γ(q+1)f(x0))

with the step size hq, and its N+1 iteration has a form

x0+hqk=1N+1Δ¯k+(N+1)hqΓ(q+1)f(x0)+O(h2q),

the same one as of Ph. Hence instead of (2.24) we have (2.27), and Theorem 2.11 is directly extended to (2.25) with (2.27).

3 FDE with Caputo derivatives with fixed lower limits

Let T=(N+1)h for a N and h>0. We consider

{D0qcx(t)=f(t,x(t)),t[0,T],x(kh+)=x(kh-)+Δk,k{1,,N},x(0)=x0,(3.1)

where q(0,1), D0qcx(t) is the generalized Caputo fractional derivative with lower limit at 0. We suppose the following conditions

  • (ci)

    f:[0,T]×mm is continuous and locally Lipschitz in x uniformly for t[0,T].

  • (cii)

    There is a constant M>0 such that |f(t,x)|K for any t[0,T] and xm.

We are looking for solutions of (3.1) satisfying x(0)=x(T). Following [16], by a solution of (3.1) we mean a function x(t) which is continuous in x0m, t[0,T]{khk=1,,N} and left continuous in t at impulsive points kh, and satisfying

x(t)=x0+i=1kΔi+1Γ(q)0t(t-s)q-1f(s,x(s))𝑑s,t(tk,tk+1].(3.2)

It is known that under above assumptions, (3.2) has a unique solution x(x0,t) on [0,T]

Now we can prove the following result.

Theorem 3.1.

Suppose assumptions (ci)(cii). If

k=1NΔk(N+1)q>MhqΓ(q+1),(3.3)

then (3.1) has no solution satisfying x(0)=x(T). Note that T=(N+1)h for some NN.

Proof.

We need to solve

x(0)=x(T)=x0+i=1NΔi+1Γ(q)0T(T-s)q-1f(s,x(s))𝑑s,

which is equivalent to

-k=1NΔk=1Γ(q)0(N+1)h((N+1)h-s)q-1f(s,x(x0,s))𝑑s.

This gives

k=1NΔk1Γ(q)0(N+1)h((N+1)h-s)q-1|f(s,x(x0,s))|𝑑sM(N+1)qhqΓ(q+1).

This contradicts (3.3). ∎

Remark 3.2.

Note that (3.3) is equivalent to

k=1NΔkTq>MΓ(q+1).(3.4)

Now we suppose that

  • (ciii)

    Δk+p=Δk for all k and some p.

Then we have the following corollary.

Corollary 3.3.

Suppose assumptions (ci)(ciii). Then there hold the following assertions:

  • (i)

    If

    k=1pΔk(p+1)q>MhqΓ(q+1),(3.5)

    then ( 3.1 ) has no solution satisfying x(0)=x(T) with T=(rp+1)h for any r.

  • (ii)

    If

    k=1pΔk0,(3.6)

    then there is no solution of ( 3.1 ) satisfying x(0)=x(T) with T=(rp+1)h for any r such that

    r>M(p+1)qhqk=1pΔkΓ(q+1)q-1.(3.7)

Proof.

We need to verify (3.3) for N=rp. First, we note that

k=1NΔk(N+1)q=k=1rpΔk(rp+1)qr1-qk=1pΔk(p+1)q(3.8)

since r1. Assuming (3.5), we have that (3.8) implies (3.3). This proves (i). Next, assuming (3.6) and (3.7), we obtain that (3.8) implies (3.3). This proves (ii). ∎

Of course, (ii) implies (i). Next, supposing h>0 as a small variable, we consider

{D0qcx(t)=f(x(t)),t(kh,(k+1)h),k0,x(kh+)=x(kh-)+Δ¯khq,k,x(0)=x0,(3.9)

where q(0,1), f:mm satisfies (ci), (cii) and the Δ¯km satisfy

  • (civ)

    Δ¯k+p=Δ¯k for all k and some p.

Then condition (3.5) becomes

k=1pΔ¯k(p+1)q>MΓ(q+1),

condition (3.6) becomes

k=1pΔ¯k0,

and (3.7) becomes

r>M(p+1)qk=1pΔ¯kΓ(q+1)q-1.(3.10)

Summarizing, we have the following result.

Corollary 3.4.

Under assumptions (ci), (cii) and (civ), there is no solution of (3.9) satisfying x(0)=x(T) for any T=(rp+1)h with rN satisfying (3.10).

Corollary 3.4 states that under assumptions (ci), (cii) and (civ), (3.9) has no ”periodic” solutions with large periods.

Example 3.5.

We consider system (2.14) for a=10, b=120 and c=83. Motivated by [9, Theorem 2], for the sphere

𝒮={(x1,x2,x3)3:x12+x22+(x3-a-b)2R2}

with

R=(a+b)c4(c-1)=134.263,

we find

M=maxx𝒮f(x)=maxx𝒮2max{|a(x2-x1)|,|x1(b-x3)-x2|,|x1x2-cx3|}8077.26.

By Corollary 3.3, for x1, Δ¯1 applied at 3kh, for x2, Δ¯2 applied at kh, and for x3, Δ¯3 applied at 2kh, now condition (3.5) has the form

max{|Δ¯1|,3|Δ¯2|,|Δ¯3|}>(p+1)qMhqΓ(q+1)132.214

for h=0.004 and q=0.995.

For the cuboid from Example 2.6,

𝒞={(x1,x2,x3)3:-50x150,-100x2100, 50x3200},

we have

M=maxx𝒞f(x)533.333.

Now condition (3.5) has the form

max{|Δ¯1|,3|Δ¯2|,|Δ¯3|}>(p+1)qMhqΓ(q+1)8.72997

again for h=0.004 and q=0.995.

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.

Theorem 3.6.

Suppose (ci), (cii) and (C3). If

-k=1NΔkiTq1Γ(q+1)(f-i,f+i)(3.11)

holds for all i=1,,m, then (3.9) has a solution satisfying x(0)=x(T).

Again, assumptions (3.4) and (3.11) are complementary. More related achievements are given in [16] and the references therein.

4 Conclusions

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

{Dtk,tqcx(t)=Ax(t)+f(t),t(tk,tk+1),k=0,1,,N,x(tk+)=(𝕀+Bk)x(tk-)+Δk,k=1,2,,N+1,x(0)=x0,(4.1)

where A,Bk:mm are matrices, Δk, 𝕀:mm is the unit matrix, and then consider a semilinear case

{Dtk,tqcx(t)=Ax(t)+f(t,x(t)),t(tk,tk+1),k=0,1,,N,x(tk+)=(𝕀+Bk)x(tk-)+Δk(x(tk-)),k=1,2,,N+1,x(0)=x0(4.2)

for achieving more specific results similar to the ones for impulsive ODEs [11] 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 C0-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 [13] to consider linear and semilinear differential equations with periodic noninstantaneous impulses as follows:

{x(t)=Ax(t),t(sk,tk+1],k=0,1,2,,x(ti+)=(𝕀+Bk)x(tk-)+Bkx(tk-),k=1,2,,x(t)=(𝕀+Bk)x(tk-)+Bkx(tk-),t(tk,sk],k=1,2,,x(sk+)=x(sk-),k=1,2,,(4.3)

and

{x(t)=Ax(t)+f(t,x(t)),t(sk,tk+1],k=0,1,2,,x(ti+)=(𝕀+Bk)x(ti-)+Bkx(tk-),k=1,2,,x(t)=(𝕀+Bk)x(tk-)+Bkx(tk-),t(tk,sk],k=1,2,,x(sk+)=x(sk-),k=1,2,,(4.4)

respectively, where tk acts as an impulsive point and sk acts as a junction point satisfying the conditions t0=s0<t1<s1<t2<<tk<sk<tk+1, tk and periodicity conditions tk+N+1=tk+T and sk+N+1=sk+T. 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.

Acknowledgements

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

Received: 2017-01-28

Revised: 2017-03-07

Accepted: 2017-03-09

Published Online: 2017-06-04


Funding Source: Ministerstvo školstva, vedy, výskumu a športu Slovenskej republiky

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 ([2015]7640) and Outstanding Scientific and Graduate ZDKC ([2015]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.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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