We investigate the boundary value problems of impulsive fractional order differential equations. First, we obtain the existence of at least one solution by the minimization result of Mawhin and Willem. Then by the variational methods and a very recent critical points theorem of Bonanno and Marano, the existence results of at least triple solutions are established. At last, two examples are offered to demonstrate the application of our main results.
1. Introduction
Fractional differential equations have been an area of great interest recently. This is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, and engineering; see [1–3] and the references therein. Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attentions. But for almost all the papers above, the main methods are some fixed theorems, the coincidence degree theory, and the monotone iterative methods.
On the other hand, critical point theory and the variational methods have been very useful in dealing with the existence and multiplicity of solutions for integer order differential equations with some boundary conditions. We refer the readers to the books (or surveys) of Mawhin and Willem [4] and Rabinowitz [5] and [6, 7] and the references therein. But until now, there are few works that deal with the fractional differential equations via the variational methods; [8, 9] are the pioneer in the use of the variational methods to fractional models. By using the variational methods, Jiao and Zhou [8] first considered the following fractional boundary value problems:(1)DtTαD0tαut=∇Ft,ut,a.e.t∈0,T,u0=uT=0,where α∈(0,1) and D0tα and DTαt are the left and right Riemann-Liouville fractional derivatives, respectively. F:[0,T]×RN→R (with N≥1) is a suitable given function and ∇F(t,x) is the gradient of F with respect to x. The problem in [8] and the related problems were further considered by critical point theory and the variational methods in [10–12].
By means of critical point theory, Jiao and Zhou [9] considered the following fractional boundary value problems:(2)-12ddtDt-β0+DtT-βu′t=∇Ft,ut,a.e.t∈0,T,u0=uT=0,where β∈[0,1) and D0t-β and DT-βt are the left and right Riemann-Liouville fractional derivatives gradient of F with respect to x.
The above problem arises from the phenomena of advection dispersion and was first investigated by Ervin and Roop in [13]. The authors in [14–17] further studied the existence and multiplicity of solutions for the above problem or related problems by critical point theory.
The impulsive differential equations arise from the real world problems to describe the dynamics of processes in which sudden, discontinuous jumps occur. Such processes are naturally seen in biology, physics, engineering, and so forth. Due to their significance, many authors have established the solvability of impulsive differential equations. There have been many different approaches to the study of the existence of solutions to impulsive fractional differential equations, such as topological degree theory, fixed point theory, upper and lower solutions method, and monotone iterative technique.
To the best of our knowledge, the fractional boundary value problems with impulses using variational methods and critical point theory have received considerably less attention [18–20].
In [20], the authors investigated the following fractional differential equations with impulses:(3)DTαtD0ctαut+atut=ft,ut,t≠tj,a.e.t∈0,T,ΔDTα-1tD0ctαutj=Ijutj,j=1,2,…,n,u0=uT=0.By using critical point theory and variational methods, the authors give some criteria of the existence of solutions.
Motivated by the work above, we consider the following problem (4) of impulsive fractional differential equations: (4)-12ddtDt-β0+DT-βtu′t=λ∇Ft,ut,t≠tk,a.e.t∈0,T,ΔDtαutk=λIkutk,tk∈0,T,k=1,2,…,l,u0=uT=0,where λ∈(0,+∞), β∈[0,1), α=1-β/2∈(1/2,1], 0Dt-β, DT-βt are the left and right Riemann-Liouville fractional integrals of order β, Dtα0c and DTαtc are the left and right Caputo fractional derivative of order α, respectively, 0=t0<t1<t2<⋯<tl<tl+1=1, F∈[0,T]×R→R a given function satisfying some assumptions and ∇F(t,x) is the gradient of F at x, Ik∈C([0,T],R), and (5)Dtαut=12Dtα-10Dtα0cu-DTα-1tDTαtcut,ΔDtαutk=12Dtα-10Dtα0cu-DTα-1tDTαtcutk+-12Dtα-10Dtα0cu-DTα-1tDTαtcutk-,12Dtα-10Dtα0cu-DTα-1tDTαtcutk+=limt→tk+12Dtα-10Dtα0cu-DTα-1tDTαtcut,12Dtα-10Dtα0cu-DTα-1tDTαtcutk-=limt→tk-12Dtα-10Dtα0cu-DTα-1tDTαutct,fork=1,…,l.
In this paper, the existence results of at least one solution or triple solutions of problem (4) are established. The rest of this paper is organized as follows. In Section 2, some definitions and lemmas which are essential to prove our main results are stated. In Section 3, we give the main results. At last, two examples are offered to demonstrate the application of our main results.
2. Preliminaries
At first, we present the necessary definitions for the fractional calculus theory and several lemmas which are used further in this paper.
Definition 1 (see [3]).
Let f be a function defined by [a,b]. The left and right Riemann-Liouville fractional integrals of order α for function f denoted by Dat-αf(t) and Db-αtf(t) function, respectively, are defined by(6)Dt-αaft=1Γα∫att-sα-1fsds,Db-αtft=1Γα∫tbs-tα-1fsds,fort∈a,b,α>0.
The left and right Riemann-Liouville fractional derivatives of order α for function f denoted by Dtαaf(t) and Dbαtf(t) function, respectively, are defined by(7)Dtαaft=dndtnDtα-nαft=1Γn-αdndtn∫att-sn-α-1fsds,Dbαtft=-1ndndtnDbα-ntft=-1nΓn-αdndtn∫tbs-tn-α-1fsds,fort∈a,b,n-1≤α<n,n∈N,provided that the right-hand side integral is pointwise defined on [a,b].
Lemma 2 (see [3]).
The left and right Riemann-Liouville fractional integral operators have the property of a semigroup; that is,(8)Dt-γ1aDt-γ2aft=Dt-γ1-γ2aft,Db-γ1tDb-γ2tft=Db-γ1-γ2tft,∀γ1,γ2>0in any point t∈[a,b] for continuous function f and for almost every point in [a,b] if the function f∈L1([a,b],RN).
Definition 3 (see [3]).
If α∈(n-1,n) and f∈ACn([a,b],R), then the left and right Caputo fractional derivatives of order α of a function f denoted by Dtαacf(t) and Dbαtcft function, respectively, are defined by(9)Dtαacft=Dtα-nadndtnft=1Γn-α∫att-sn-α-1fnsds,Dbαtcft=-1nDbα-ntdndtnft=-1nΓn-α∫tbs-tn-α-1fnsds,respectively, where t∈[a,b].
In view of Definition 1 and Lemma 2, we can easily transfer problem (4) to the following problem:(10)ddt12Dtα-10Dtα0cut-12DTα-1tDTαtcut+λ∇Ft,ut=0,t≠tk, a.e.,t∈0,T,ΔDtαutk=λIkutk,tk∈0,T,k=1,2,…,l,u0=uT=0,where β∈[0,1), α=1-β/2∈(1/2,1], (Dtαu)(t)=(1/2){Dtα-10(Dtα0cu)-DTα-1t(DTαtcu)}(t).
Then problem (4) is equivalent to problem (10). Therefore, a solution of problem (10) corresponds to a solution of BVP (4).
In order to establish a variational structure which enables us to reduce the existence of solution of problem (10) to existence of the critical point of corresponding functional, we construct the following appropriate function spaces.
Let us recall that, for any fixed t∈[0,T] and 1≤p≤∞,(11)u∞=maxt∈0,Tut,uLp=∫0Tuspds1/p.
Let 0<α≤1, and we define the fractional derivative spaces E0α by the closure of C0∞([0,J]) with respect to the weighted norm uα=(∫0TDtα0cut2dt+∫0Tu(t)2dt)1/2, ∀u∈E0α, where (12)C0∞0,J=u∈C∞0,J:u0=uT.
Clearly, the fractional derivative space E0α is the space of functions u∈L2([0,T]∖{t1,t2,…,tl} having α-order Caputo left and right fractional derivatives and Riemann-Liouville left and right fractional derivatives, Dtα0c, DTαtc, Dtα0ut, DTαt∈L2([0,T]∖{t1,t2,…,tl} and u(0)=u(T)=0.
Lemma 4 (see [9]).
Let 0<α≤1 and 1<p<∞. For all u∈E0α, one has(13)uLp≤TαΓα+1Dtαu0cLp.Moreover, if α>1/p and 1/p+1/q=1, then (14)u∞≤Tα-1/pΓαα-1q+11/qDtαu0cLp.
Then we can conclude that uα=(∫0TDtα0cu(t)2dt+∫0Tu(t)2dt)1/2 is equivalent to uα=(∫0TDtα0cu(t)2dt)1/2, ∀u∈E0α. In the following, we will consider the fractional derivative spaces E0α with respect to the norm uα=(∫0TDtα0cu(t)2dt)1/2. Obviously, E0α is a reflexive and separable Banach space with the norm uα=(∫0TDtα0cu(t)2dt)1/2.
Definition 5.
A function (15)u∈u∈AC0,T:∫tjtj+1Dtα0cut2+ut21/2<∞,j=0,…,lis called a classic solution of problem (10) if
exist and satisfy the impulsive condition ΔDtαu(tk)=Ikutk and the boundary condition u(0)=u(T)=0 holds,
u satisfies (10) a.e. on t∈0,T∖t1,t2,…,tl.
Definition 6.
A function u∈E0α is called a weak solution of problem (10) if (17)∫0T-12Dtα0cutDTαtcvt+DTαtcutDtα0cvtdt+λ∑j=1lIjutjvtj-λ∫0T∇Ft,utvtdt=0,for all v(t)∈E0α.
Similar to the proof of Lemma 2.1 in [18], we have the following Lemma 7.
Lemma 7.
The function u∈E0α is weak solution of (10), if and only if u is a classical solution of (10).
Lemma 8 (see [9]).
Let 1/2<α≤1 and the sequence {uk} converges weakly to u in E0α; then uk→u in C([0,T],R); that is, u-u∞→0, as k→∞.
Lemma 9 (see [9]).
Let 1/2<α≤1. For any u∈E0α, one has(18)cosπαuα2≤-∫0TDtα0cut,DTαtcutdt≤1cosπαuα2.
The proofs of the main results in this paper are based on the following critical point theorems.
Theorem 10 (see [4, Theorem 1.1]).
If Φ is weakly lower semicontinuous (WLSC) on a reflexive Banach space X and has a bounded minimizing sequence, then Φ has a minimum on X.
Theorem 11 (see [21]).
Let X be a reflexive real Banach space, let Φ:X→R be a sequentially weakly lower semicontinuous, coercive, and continuously Gateaux differentiable functional whose Gateaux derivative admits a continuous inverse on X∗, and let Ψ:X→R be a sequentially weakly upper semicontinuous and continuously Gateaux differentiable functional whose Gateaux derivative is compact. Assume that there exist r∈R and u1∈X with 0<r<Φ(u1), such that
supu∈Φ-1-∞,rΨu<r(Ψu1/Φu1),
for every λ∈Λr≔Φ(u1)/Ψ(u1),r/supu∈Φ-1([-∞,r])Ψ(u), the functional Φ-λΨ is coercive.
Then, for each λ∈Λr the functional Φ-λΨ has at least three distinct critical points in X.
3. Main Results
For convenience, we give the following hypothesis (H1).
(H1)λ is a positive real parameter, β∈[0,1), α=1-β/2∈(1/2,1], F:[0,T]×R→R is a function such that F(·,x) is continuous in [0,T] for every x∈R, F(t,·) is a C1 function in R for any t∈[0,T], F(t,0)=0, and ∇F(t,x) is the gradient of F at x,Ij∈C([0,T],R+), j=1,…,l.
Theorem 12.
Let (H1) hold. If λ<1, the following assumption (H2) is satisfied.
(H2) There exists a positive constant a1<Γ2(α+1)cos(πα)/2T2α such that (19)limsupx→∞Ft,xx2<a1,uniformly for x∈R, t∈[0,T].
Then (4) possesses at least one solution.
Proof.
We define(20)φu=∫0T-12Dtα0cutDTαtcutdt-λ∫0TFt,utdt+λ∑j=1l∫0utjIjsds. Then, for all v(t)∈E0α, we know(21)φ′uv=∫0T-12Dtα0cutDTαtcvt+DTαtcutDtα0cvtdt+λ∑j=1lIjutjvtj-λ∫0T∇Ft,utvtdt,which shows that a critical point of the functional φ is a weak solution of problem (10).
Our aim is to apply Theorem 10 to problem (10).
We begin by proving that φ is weakly lower semicontinuous. Since X is a separable and reflexive real Banach space, we assume that {uk}⊂X converges weakly to u in ⊂X. By Lemma 8, we can obtain that uk→u uniformly in C([0,T],R), as k→∞; that is, (22)uk-u∞⟶0,ask⟶∞,liminfk→∞ukα≥uα.Together with (H1), one has (23)liminfk→∞φuk=liminfk→∞∫0T-12Dtα0cuktDTαtcuktdt-λ∫0TFt,uktdt+λ∑j=1l∫0uktjIjsds≥∫0T-12Dtα0cutDTαtcutdt-λ∫0TFt,utdt+λ∑j=1l∫0utjIjsds=φu.
Then it implies that φ is weakly lower semicontinuous.
Now, we are in the position of showing that the functional φ is coercive.
From (H2), we know that there exists a positive constant a2 large enough such that (24)Ft,u<a1u2,foru>a2,t∈0,T.
On the other hand, from the continuity of F(t,u), we concluded that F(t,u) is bounded for u≤a2, t∈[0,T]. Then there exists a constant b1>0 such that(25)Ft,u<b1,for u≤a2, t∈[0,T].
Hence, for all (t,u)∈[0,T]×R, we can get (26)Ft,u<a1u2+b1.
Then it follows from (H1) and Lemmas 4 and 9 that(27)φu=∫0T-12Dtα0cutDTαtcutdt-λ∫0TFt,utdt+λ∑j=1l∫0utjIjsds≥12cosπαuα2-∫0Ta1u2+b1dt≥12cosπα-a1T2αΓ2α+1uα2-b1T.
In view of a1<Γ2(α+1)cos(πα)/2T2α, we have (28)φu⟶∞,asuα⟶∞.
Then we know that φ is coercive. Thus, by virtue of Theorem 10, the functional φ has a minimum, which is a critical point of φ. It follows that the boundary value problem (10) has one weak solution. By virtue of Lemma 7, we can deduce that the boundary value problem (10) has one solution which implies that the boundary value problem (4) possesses at least one solution.
Remark 13.
If the asymptotically quadratic case in (H2) becomes the subquadratic case, that is, (29)limsupx→∞Ft,x<exe1,1<e1<2,e>0,then we can get the similar result.
For u∈E0α, we define the functional Φ,Ψ:E0α→R as follows: (31)Φu=∫0T-12Dtα0cutDtcTαutdt,(32)Ψu=∫0TFt,utdt-∑j=1l∫0utjIjsds.
Theorem 14.
Let (H1) hold. Suppose that there exist a constant r>0 and a function ω such that ωα2>2r/cos(πα) and the following assumptions H3-(H4) are satisfied:
(H3)supξ∈Ω(Mr)Ψ(ξ)/r<2cos(πα)Ψ(ω)/ωα2;
(H4)limξ→+∞Ft,ξ/ξ2<cos(πα)supu∈Φ-1([-∞,r])Ψ(u)/2Lr, where Ω(Mr)={ξ∈R:(1/2)ξ2≤Mr}.
Then, for each (33)λ∈Λ≔ωα22cosπαΨω,rsupu∈Φ-1-∞,rΨu,the boundary value problem (4) has at least three distinct solutions in E0α.
Proof.
Obviously E0α is compactly embedded in C0([0,T],R). From (31), (32), and (H1), it is well known that Φ,Ψ are well defined and are Gateaux differentiable functional whose Gateaux derivative at the point u∈E0α is the functionals Φ′(u),Ψ′(u)∈E0α∗, given by(34)Φ′ux=∫0T-12D0ctαutDTαtcxt+DTαtcutDtα0cxtdt,Ψ′ux=∫0T∇Ft,utxtdt-∑j=1lIjutjxtjfor every x∈E0α.
For un, u∈E0α, un→u in E0α as n→∞, by Lemma 8, we know that {un} converges uniformly to u in C([0,T],R). Hence(35)limsupn→∞Ψun≤∫0Tlimsupn→∞Ft,undt-∑j=1l∫0limsupn→∞untjIjsds=∫0TFt,udt-∑j=1l∫0utjIjsds=Ψu,which implies that Ψ is sequentially weak upper semicontinuous.
By (H1), one can get F(t,un)→F(t,u) as n→∞. By the Lebesgue control convergence theorem, Ψ′(un)→Ψ′(u) strongly, which implies that Ψ′:E0α→E0α is strongly continuous on E0α; that is, Ψ′ is a compact operator.
From (H1), Lemma 9, and (31), it is also easy to verify that Φ:E0α→R is sequentially weakly lower semicontinuous, coercive, and its derivative Φ′ admits a continuous inverse on (E0α)∗. By the condition ωα2>2r/cos(πα), (31), and Lemma 9, we have (36)Φω=-12∫0TDtα0cωtDTαtcωtdt>12cosπαωα2>r>0and Φ(0)=0.
Then, in order to apply Theorem 11, we only need to show that (i) and (ii) of Theorem 11 hold.
From Lemma 9, (14), and (31), one has(37)Φ-1-∞,r=u∈E0α:Φu≤r=u∈E0α:-12∫0TDtα0cutDTαtcutdt≤r⊆u∈E0α:12cosπαuα2≤r⊆u∈E0α:cosπαΓ2α2α-12T2α-1u∞2≤r⊆u∈E0α:ut22≤Mr,∀t∈0,T,and it follows that (38)supu∈Φ-1-∞,rΨu=supu∈Φ-1-∞,r∫0TFt,utdt-∑j=1l∫0utjIjsds≤supu∈ΩMr∫0TFt,utdt-∑j=1l∫0utjIjsds,From (31), (32), Lemma 9, and (H3), we obtain(39)supu∈Φ-1-∞,rΨur≤supu∈ΩMr∫0TFt,utdt-∑j=1l∫0utjIjsdsr<2cosπα∫0TFt,ωtdt-∑j=1l∫0ωtjIjsdsωα2≤ΨωΦω.Then (i) of Theorem 11 holds.
Furthermore, according to (H4) there exist two constants k,ε∈R with (40)k<cosπαsupu∈Φ-1-∞,rΨur,such that (41)Ft,ξ≤k2Lξ2+ε,fort∈0,T,ξ∈R.
Next we consider the functional Φ(u)-λΨ(u). From (41), one has(42)Φu-λΨu=-12∫0TDtα0cutDTαtcutdt-λ∫0TFt,utdt-∑j=1l∫0utjIjsds≥-12∫0TDtα0cutDTαtcutdt-λ∫0TFt,utdt≥12cosπαuα2-λk2LuL22-λTε.Obviously, if k≤0, it follows that Φ(u)-λΨ(u)→+∞, as uα→+∞.
On the other hand, if k>0, from (13) and (30), then we have(43)Φu-λΨu≥12cosπαuα2-λk2LuL22-λTε≥12cosπαuα2-λk2LT2αΓ2α+1uα2-λTε=12cosπα-λk2uα2-λTε.
For λ∈ωα2/2cosπαΨ(ω),r/supu∈Φ-1([-∞,r])Ψ(u)⊆Φ(ω)/Ψ(ω),r/supu∈Φ-1([-∞,r])Ψ(u), by (40), we can deduce(44)Φu-λΨu≥12cosπα-λk2uα2-λTε≥12cosπα1-λ2supu∈Φ-1-∞,rΨuruα2-λTε≥14cosπαuα2-λTε⟶+∞,uα⟶+∞.
Hence the functional Φ(u)-λΨ(u) is coercive. So condition (ii) of Theorem 11 holds. Then, by virtue of Theorem 11, we can conclude that the equation Φ′(u)-λΨ′(u)=0 has at least three distinct solutions. That is, the boundary value problem (10) has at least three distinct weak solutions. As a consequence of Lemma 7, we deduce that the boundary value problem (10) has at least three distinct solutions which implies that problem (4) possesses at least three distinct solutions.
Finally, we give two examples to illustrate the usefulness of our main result. Consider the following impulsive system of fractional differential equations.
Example 15.
Consider(45)-12ddtDt-0.50+D1-0.5tu′t=λ∇Ft,ut,t≠tk, a.e.,t∈0,1,ΔDtαutk=λ2,k=1,…,l,u0=u1=0,where 0=t0<t1<t2<⋯<tl<tl+1=1. From (45), we know β=0.5, α=0.75, I(u)=1/2, T=1. Let F(t,u(t))=u2(t)/18; we can easily verify that all the conditions of hypothesis (H1) are satisfied.
Choose a1=1/16; it follows that(46)Γ2α+1cosπα2T2α=24Γ232≈0.277621>a1=116.It is also easy to see (47)limsupu→∞Ft,uu2=118<a1=116,which implies that condition (H2) holds.
Then problem (45) satisfies all the conditions in Theorem 12. In view of Theorem 12, problem (45) has at least one solution for 0<λ<1.
Example 16.
Consider(48)-12ddtDt-0.50+D1-0.5tu′t=λ∇Ft,ut,t≠tk, a.e.,t∈0,1,ΔDtαutk=λ10-6,k=1,…,l,u0=u1=0,where 0=t0<t1<t2<⋯<tl<tl+1=1. From (48), we can see that β=0.5, α=0.75, I(u)=10-6, T=1. Let (49)Ft,ut=1+t2u4,u≤1,1+t3u-u2,u>1.Then it is easy to verify that assumption (H1) holds.
We choose ω(t)=t(1-t)Γ(1.25), r=0.001. By a direct calculation, we can obtain(50)M=T2α-1Γ2α2α-1cosπα≈1.50177,L=2T2αΓ2α+1≈2.3677,ωα2=∫01Dt0.750cωt2dt≈0.158>0.0042=2rcosπα,supξ∈ΩMr∫0TFt,ξdt-∑j=1l∫0ξtjIjsdsr≈0.26<5.733≈2cosπαΨωωα2,limξ→+∞Ft,ξξ2=-1<0<cosπαsupu∈Φ-1-∞,r∫0TFt,utdt-∑j=1l∫0utjIjsds2Lr.
Hence all the assumptions of Theorem 14 are satisfied. Then our results can be applied to problem (48), which shows that problem (48) possesses at least three distinct solutions in E0α, for λ∈0.17443,38.461538.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work is supported by National Natural Science Foundation of China (nos. 11001274, 11101126, and 11261010), China Postdoctoral Science Foundation (no. 20110491249), Youth Science Foundation of Henan University of Science and Technology (no. 2012QN010), and Innovative Natural Science Foundation of Henan University of Science and Technology (no. 2013ZCX020).
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