JFSJournal of Function Spaces2314-88882314-8896Hindawi10.1155/2020/85983238598323Research ArticleVariational Method to p-Laplacian Fractional Dirichlet Problem with Instantaneous and Noninstantaneous ImpulsesChenYiruhttps://orcid.org/0000-0001-8408-3859GuHaiboMaLinaZhangXinguangSchool of Mathematics SciencesXinjiang Normal UniversityUrumqiXinjiang 830017Chinaxjnu.edu.cn202021720202020100420200106202021720202020Copyright © 2020 Yiru Chen et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, a research has been done about the existence of solutions to the Dirichlet boundary value problem for p-Laplacian fractional differential equations which include instantaneous and noninstantaneous impulses. Based on the critical point principle and variational method, we provide the equivalence between the classical and weak solutions of the problem, and the existence results of classical solution for our equations are established. Finally, an example is given to illustrate the major result.

Outstanding Young Science and technology personnel Training program of XinjiangNational Natural Science Foundation of China11961069Scientific Research Programs of Colleges in XinjiangXJEDU2018Y033Natural Science Foundation of Xinjiang Province2019D01A71
1. Introduction

Fractional calculus involves arbitrary order derivatives and integration, so it plays a very important role in various fields such as physical engineering, medical image processing, mathematics, chemical engineering, and electricity. For this reason, many scholars did research on the theory of fractional differential equations continuously and have made enormous achievements; readers who are interested in these kinds of researches can refer to relevant literature (see  and the references therein). Of course, in the science field, an impulsive phenomenon has also been spread widely when dealing with practical problems, and it has become a very effective tool for describing changes in sudden discontinuous jumps. In addition, from the perspective of the duration of the change, impulses can be divided into instantaneous impulses and noninstantaneous impulses. The difference between them is that the duration of the instantaneous impulse continuous change is relatively short compared to the duration of the entire process, and the noninstantaneous impulse change is to keep moving from any fixed point and at a certain time interval. In , Agarwal et al. provide a more detailed introduction to these two impulsive differential equations. We also noticed that some experts combine the theory of fractional calculus, of noncompactness, with Sadovskii’s fixed-point theorem and even more excellent methods have obtained more and better properties for fractional order equations with impulsive conditions, which greatly promotes its development (see, for example, , and the references therein).

For a long time, many scholars have conducted in-depth research on instantaneous impulsive differential equations. By using fixed-point theorems, critical point theorems, and variational methods, they obtained the existence of solutions (see ). But in many cases, instantaneous impulses cannot describe the development of certain dynamics. In 2013, Hernández and O’Regan first proposed the concept of noninstantaneous impulsive differential equations (see ). Since then, the existence of solutions to noninstantaneous impulsive problems has been gradually expanded by using some methods such as fixed-point theory and analytical semigroup theory, but the variational structure of general noninstantaneous impulsive differential equations has not received widespread attention. Among them, the existence of solutions for second-order differential equations with instantaneous and noninstantaneous impulses was investigated. Tian and Zhang obtained the existence results through the principle of variation (see ). In addition, it is worth noting that by using the critical point theory and variational method, Zhao et al. proved the existence and multiplicity of nontrivial solutions to the problem of nonlinear nontransient impulsive differential equations (see ). At the same time, some scholars have obtained the existence results of solutions for noninstantaneous impulsive differential equations by variational methods (see ).

In , Zhang and Liu generalized linear fractional differential equations to nonlinearities on the interval tsi,ti+1 and removed the restriction DtTα10CDtαu0=constant. By using a variational method, they considered the following Dirichlet problems with instantaneous and noninstantaneous impulse differential equations: (1)DtTα0CDtαut=fit,ut,tsi,ti+1,i=0,1,2,,n,ΔtDTα10CDtαuti=Iiuti,i=1,2,,n,tDTα10CDtαut=tDTα10CDtαuti+,tti,si,i=1,2,,n,tDTα10CDtαusi=tDTα10CDtαusi+,i=1,2,,n,u0=uT=0.

In , Zhou et al. discussed the existence of solutions for fractional differential equations of p-Laplacian with instantaneous and noninstantaneous impulses. The innovation was that when p=2, problem (2) can be regarded as problem (1), and when α=1, it can be simplified to a more general integer order case. Finally, they got the classical solution and prove that the weak solution was equivalent to the classical solution: (2)DtTαϕP0CDtαut+gtuP2ut=fit,ut,tsi,ti+1,i=0,1,2,,n,ΔtDTα1ϕP0CDtαuti=Iiuti,i=1,2,,n,tDTα1ϕP0CDtαut=tDTα1ϕP0CDtαuti+,tti,si,i=1,2,,n,tDTα1ϕP0CDtαusi=tDTα1ϕP0CDtαusi+,i=1,2,,n,u0=uT=0.

Motivated by the above-mentioned work, the paper focuses on the existence of solutions for the following p-Laplacian fractional differential equations with instantaneous and noninstantaneous impulses, and if DxFit,ututi+1=fit,ut, the following problem reduces to (2): (3)DtTαϕP0CDtαut+λtup2ut=DxFit,ututi+1,tsi,ti+1,i=0,1,2,,N,ΔtDTα1ϕP0CDtαuti=Iiuti,i=1,2,,N,tDTα1ϕP0CDtαut=tDTα1ϕP0CDtαuti+,tti,si,i=1,2,,N,tDTα1ϕP0CDtαusi=tDTα1ϕP0CDtαusi+,i=1,2,,N,u0=uT=0,where α1/p,1,p2,DtTα, and 0CDTα denote the right Riemann-Liouville fractional derivatives and left Caputo fractional derivatives. 0=s0<t0<s1<t1<<tN=T,IiCR,R, and there exists i1,2,,N such that Iuti0.fiCsi,ti+1×R,R,ϕp:RR is the p-Laplacian function defined as ϕps=sp2ss0,ϕp0=0.λL0,T, essinfλt0,Tt>Γα+1/Tαp. The nonlinear functions DxFit,x are the derivatives of Fit,x, for every i=0,1,2,,N. To the best of our knowledge, the instantaneous impulses start abruptly at point ti and the noninstantaneous impulses continue during the intervals ti,si, and (4)ΔtDTα1ϕp0CDtαuti=tDTα1ϕp0CDtαuti+tDTα1ϕp0CDtαuti,tDTα1ϕp0CDtαuti±=limtti±tDTα1ϕp0CDtαut,tDTα1ϕp0CDtαusi±=limtsi±tDTα1ϕp0CDtαut.

The rest of the paper is organized as follows. In Section 2, some basic knowledge and lemmas used in the latter are presented. In Section 3, we first give the equivalent form of solution of problem (3), and secondly, we establish the equivalence of the classical solution and the weak solution by using the critical point principle for the variational structure of problem (3) and finally prove the existence of solution. An illustrative example is given to show the practical usefulness of the analytical results in Section 4.

2. Preliminaries

In this section, we introduce some important definitions, lemmas, and theorems that are important to use later. For the definitions of the left and right fractional integrals and derivatives, we can refer to References [7, 23].

Definition 1 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

Let α1/p,1,p2,, the fractional space (5)E0α,p=u:0,TRU,0CDtαuLp0,T,R,u0=uTis defined by the closure of C00,T,R with the norm (6)uα,p=0Tλtutpdt+0T0cDtαutpdt1/p.

Definition 2 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

Let 0<α1, the fractional derivative space E0α,p is a reflexive as well as separable Banach space.

Remark 3 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

Let p=2, we define E0α=E0α,2. It is obvious that the fractional derivative space E0α is the space of functions uL20,T,R having an α-order Caputo fractional derivative 0cDtαuL20,T,R and u0=uT.

Proposition 4 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

Let 0<α1 and uACa,b, then for ta,b,(7)acDtαut=aDtαutuaΓ1αtaα,tcDbαut=tDbαutubΓ1αbtα.

Proposition 5 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

Let α0,1 and p1,+, for any uE0α,p, we have (8)uLpTαΓα+10T0CDtαutpdt1/p.

Moreover, if α>1/p and 1/p+1/q=1, then (9)uΩ0T0CDtαutpdt1/q,where Ω=Tα1/p/Γαα1q+11/q.

Proposition 6 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

Let α1/p,1, the space E0α,p is compactly embedded in C0,T,R.

Proposition 7 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

Let α0,1, and u,vLpa,b: (10)abtDbαutvtdt=abaDtαvtutdt.

Definition 8.

A function (11)uuAC0,T: siti+1utp+0CDtαutpdt<+,i=0,1,2,,Nis a classical solution of problem (3) if u satisfies the conditions of problem (3) and the boundary condition u0=uT=0 holds.

Lemma 9.

Let X be a reflexive Banach space and let φ:X,+ be weakly lower semicontinuous on X. If φ has a bounded minimizing sequence, then φ has a minimum on X.

Lemma 10.

If φ,+ is coercive, then φ has a bounded minimizing sequence.

Lemma 11 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

For uE0α,p, we define the norm (12)uα=0Tutpdt+0T0CDtαutpdt1/pis equivalent to the norm uα,p.

3. Main Results

In this section, we further discuss the variational structure of problem (3) and define the functional φ to prove that the critical point of φ is the weak solution of problem (3). In addition, we also give the equivalence between weak solution and the classical solution of this problem. Finally, the existence of classical solution for problem (3) is given by Theorem 16.

For each i=0,1,2,,N, the nonlinear functions Fi satisfy the following assumptions:

(H1).

Fit,x is measurable in t for every xR and continuously differentiable in x for a.e. tsi,ti+1, and there exist functions k1CR+,R+, k2L1si,ti+1,R+ such that (13)Fit,xk1xk2t,DxFit,xk1xk2t.

(H2).

There exist constants δ0,2,a>0, and the functions b0, b1L1si,ti,R+ such that (14)Fit,xax2+b0txδ+b1t,tsi,ti+1.

(H3).

There exist constants ci, di>0, σi0,1 (i=1,2,,N) such that Iiuci+diuσi, for any uR.

Lemma 12.

A function uE0α,p,vE0α,p, the following form is an equivalent form of the problem (3): (15)0Tϕp0CDtαut0CDtαvtdt=i=0Nsiti+1DxFit,ututi+1vtidti=0Nsiti+1λtutp2utvtdti=1NIiutivti.

Proof.

For vE0α,p, by (10) have (16)0TϕpDtα0CutDtα0Cvtdt=0TDTα1tϕpDtα0Cutvtdt=i=0Nsiti+1DTα1tϕpDtα0Cutvtdt+i=1NtisiDTα1tϕpDtα0Cutvtdt=DTα1tϕpDtα0Cut1vt1s0t1ddtDTα1tϕpDtα0Cutvtdt+i=1NDTα1tϕpDtα0Cutvttisii=1NtisiddtDTα1tϕpDtα0Cutvtdt+i=1N1DTα1tϕpDtα0Cutvtsiti+1DTα1tϕpDtα0CusN+vsNi=0N1siti+1ddtDTα1tϕpDtα0CutvtdtsNTddtDTα1tϕpDtα0Cutvtdt=0TDTαtϕpDtα0Cutvtdt+i=1NDTα1tϕpDtα0CusiDTα1tϕpDtα0Cusi+vsi+i=1NDTα1tϕpDtα0CutiDTα1tϕpDtα0Cuti+vti.

We have (17)0TtDTαϕp0CDtαutvtdt=0Tϕp0CDtαut0CDtαvtdt+i=1NIiutivti.

As for problem (3), we have (18)0TtDTαϕpD0Ctαutvtdt=i=0Nsiti+1tDTαϕpD0Ctαutvtdt+i=1NtisitDTαϕpD0Ctαutvtdt=i=0Nsiti+1DxFit,ututi+1vtidti=0Nsiti+1λtutp2utvtdti=1NtisiddttDTα1ϕpD0Ctαutvtdt=i=0Nsiti+1DxFit,ututi+1vtidti=0Nsiti+1λtutp2utvtdt.

Above all, problem (3) has the following equivalent form: (19)0TϕpD0CtαutD0Ctαvtdt=i=0Nsiti+1DxFit,ututi+1vtidti=0Nsiti+1λtutp2utvtdti=1NIiutivti.

Definition 13.

A function uE0α,p is a weak solution of problem (3) if (15) holds for all vE0α,p.

Define the function φ:E0α,pR as (20)φu=1pupi=0Nϕiu+i=1N0utiIisds,where ϕiu=siti+1Fit,ututi+1dt.

From (9), we have (21)ututi+12uTα1/pΓαα1q+11/quα=2Ωuα.

Under condition (H1), for a.e. tsi,ti+1, we have (22)limε01εFit,ututi+1+εvtFit,ututi+1=DxFit,ututi+1vt.

From (21), for all γ0,1, (23)ututi+1+γεvt2Ωuα+v2Ωuα+Ωvα.

According to the mean value theorem and condition (H1), for some γ0,1, (24)1εFit,ututi+1+εvtFit,ututi+1=DxFit,ututi+1+γεvtvtΩvαmaxω0,2Ωuα+Ωvαk1ωk2t.

By the Lebesgue dominated convergence theorem, it is easy to show that ϕi has a directional derivative at each point u: (25)ϕiu,v=siti+1DxFit,ututi+1vtdt,ϕiu,vsiti+1DxFit,ututi+1vtdtΩvαmaxω0,2Ωuα+Ωvαk1ωsiti+1k2tdt.

Thus, ϕiE0α,p, let ukuinE0α,p, then uk converges uniformly to u for all t0,T, we have (26)ϕiunϕiuαΩvαsiti+1DxFit,untunti+1DxFit,ututi+1dt.

Therefore, ϕi is continuous from E0α,p into E0α,p, φC1E0α,p,R and (27)φu,v=0Tϕp0CDtαut0CDtαvtdt+i=1NIiutivtii=0Nsiti+1DxFit,ututi+1vtidt+i=0Nsiti+1λtutp2utvtdt.

This yields that the critical points of φ are weak solutions of problem (3).

Lemma 14.

The functional φ:E0α,pR is a weak lower semicontinuous.

Proof.

Let sequence ukk=1 in E0α,p be weakly convergent to u in E0α,p, then ulimkinfuk.

From Proposition 6, sequence ukk=1 is convergent uniformly to u in C0,T,R; thus, we have (28)limkinfφuk=limkinf1qupi=0Nsiti+1Fit,uktukti+1+i=1N0uktiIisds1qupi=0Nsiti+1Fit,ututi+1+i=1N0uktiIisds=φu.

Lemma 15.

uE0α,p is a weak solution of the problem (3), if and only if u is a classical solution of the problem (3).

Proof.

From the definition of classical solutions, without loss of generality, if u is a classical solution of problem (3), then it is a weak solution of (3). Conversely, let uE0α,p be a weak solution of (3); thus, u0=uT=0 and (15) holds. Now, we prove that u is a classical solution of the problem (3). We take a test function vC0siti+1 satisfying vt=0, where t0,siti+1,T, i=0,1,2,,N. Substituting vt into (15) and by Proposition 7, we have (29)siti+1ϕp0CDtαut0CDtαvtdt+siti+1λtutp2utvtdt=siti+1DxFit,ututi+1vtidt,(30)siti+1ϕp0CDtαut0CDtαvtdt+siti+1tDTαϕp0CDtαutvtdt<+,that is to say, (31)tDTαϕp0CDtαut+λtutp2ut=DxFit,ututi+1,tsi,ti+1,i=0,1,,N.

Since uE0α,pC0,T, we have (32)siti+1utp+0CDtαutpdt<+.

Because of DxFit,ututi+1Csi,ti+1×R,R, by (31), we have (33)tDTαϕp0CDtαutACsi,ti+1.

Therefore, (34)tDTα1ϕp0CDtαusi+=limtsi+tDTα1ϕp0CDtαut,tDTα1ϕp0CDtαuti+1=limtti+1tDTα1ϕp0CDtαut.In view of (15) and (31), we have (35)0Tϕp0CDtαut0CDtαvtdt+i=1NIiutivti+i=1Nsiti+1ddttDTα1ϕp0CDtαutvtdt=0,that is, (36)i=1NtDTα1ϕp0CDtαusi+vsi+i=0NtDTα1ϕp0CDtαuti+1vti+1=i=1Nttisiϕp0CDtαut0CDtαvtdt+i=1NIiutivti.

Without loss of generality, we take the test function vC0ti,si such that vt=0, where t0,tisi,T,i=1,2,,N. Substituting vt into (30), we obtain tDTα1ϕp0CDtαut,tti,si, i=1,2,3N, is a constant, a.e. (37)tDTα1ϕp0CDtαuti+=tDTα1ϕp0CDtαusi=tDTα1ϕp0CDtαut,tti,si,i=1,2,N.

By using (36) and (37), we have (38)i=1NtDTα1ϕp0CDtαusi+vsi+i=0NtDTα1ϕp0CDtαuti+1vti+1=i=1NtDTα1ϕp0CDtαuti+vsi+i=1NIiutivti.

Hence, (39)tDTα1ϕp0CDtαuti+=tDTα1ϕp0CDtαutiIiutivti=tDTα1ϕp0CDtαuti+vsii=1NtDTα1ϕp0CDtαusi+vsi+.

Then, combining with (37), we have (40)tDTα1ϕp0CDtαuti+tDTα1ϕp0CDtαuti=Iiuti,tDTα1ϕp0CDtαusi+=tDTα1ϕp0CDtαusi.

Therefore, u is a classical solution of the problem (3). The proof is completed.

Theorem 16.

Suppose that (H2) and (H3) hold, then problem (3) has at least one classical solution.

Proof.

From Proposition 5, there exist constants A2,A3 such that (41)φu=1pupi=0Nsiti+1Fit,ututi+1dt+i=1N0utiIis1pupi=0Nsiti+1b1tdti=1Nc1ui=1Ndiσi+1ui=0Nsiti+1aututi+12dtsiti+1b0tututi+1δdt1pupa4Ω2uα2i=0Nti+1si2ΩδuαδA0A1uA2uασi+1A3,where (42)A0=i=0Nsiti+1b0tdt,A1=i=0Nsiti+1b1tdt.

Since p>2, we get limuα+φu=+, then φ is coercive. From Lemma 10 and Lemma 14 that φ satisfied all the conditions of Lemma 9, so φ has a minimum on E0α,p which is a critical point of φ. Since Iiuti0i=1,2,,N, then uconstant (u is a critical point of φ).

Therefore, problem (3) has at least one classical solution. The proof is completed.

4. Example

Considering the following p-Laplacian fractional differential equations with instantaneous and noninstantaneous impulses, (43)tDT3/4ϕp0CDT3/4ut+up2ut=DxFit,ututi+1,tsi,ti+1,i=0,1,2,,N,ΔtDT1/4ϕp0CDt3/4uti=Iiuti,i=1,2,,N,tDT1/4ϕp0CDt3/4ut=tDTα1ϕp0CDtαuti+,tti,si,i=0,1,2,,N,tDT1/4ϕp0CDt3/4usi=tDTα1ϕp0CDtausi+,i=1,2,,N,u0=uT=0,where α=3/4, T=2, λt=1, Iiuti=1/3+1/2sinu2/3, DxFit,x=1/9ututi+123ututi+121+t, we can find a=1/3, b0=31+t, b1=21+t, δ=1, ci=di=1, σi=2/3i=1,2,,N. It is easy to check that (H2) and (H3) hold. Then, by Theorem 16, problem (3) has at least one classical solution.

5. Conclusions

In this paper, we use the variational method to discuss the existence of solutions for Dirichlet boundary value problems with instantaneous and noninstantaneous fractional impulsive differential equations, and the existence results of classical solution for our equations are shown. In addition to extending the linear differential operator to a nonlinear differential operator, it is more important to use the p-Laplacian operator; as far as the author knows, there is not much work to study the solution of the generalized p-Laplacian impulsive fractional system using the variational method. Compared with (2), (3) contains the existence results of (2) when DxFit,ututi+1fit,ut. Without loss of generality, if p=2, α=1 it evolves into an integer order differential equation which has a certain auxiliary role in solving nonlinear similar problems in the future. At the same time, we also noticed the existence of multiple solutions or the existence of nontrivial solutions to the problem, so we will pay special attention in the following work. Overall, our work summarizes and supplements some of the results in the literature.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interest.

Authors’ Contributions

All authors read and approved the final manuscript.

Acknowledgments

This work is supported by the Natural Science Foundation of Xinjiang (2019D01A71), Scientific Research Programs of Colleges in Xinjiang (XJEDU2018Y033), National Natural Science Foundation of China (11961069), Outstanding Young Science and technology personnel Training program of Xinjiang.

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