# Variational Method top-Laplacian Fractional Dirichlet Problem with Instantaneous and Noninstantaneous Impulses

Authors: Yiru Chen; Haibo Gu; Lina Ma
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, a research has been done about the existence of solutions to the Dirichlet boundary value problem forp-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.


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## Body

## 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 [1–6] 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 [7], 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, [8–15], 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 [16–18]). 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 [19]). 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 [20]). 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 [21]). At the same time, some scholars have obtained the existence results of solutions for noninstantaneous impulsive differential equations by variational methods (see [22–24]).In [25], Zhang and Liu generalized linear fractional differential equations to nonlinearities on the interval t∈si,ti+1 and removed the restriction DtTα−10CDtαu0=constant. By using a variational method, they considered the following Dirichlet problems with instantaneous and noninstantaneous impulse differential equations:
(1)DtTα0CDtαut=fit,ut,t∈si,ti+1,i=0,1,2,⋯,n,ΔtDTα−10CDtαuti=Iiuti,i=1,2,⋯,n,tDTα−10CDtαut=tDTα−10CDtαuti+,t∈ti,si,i=1,2,⋯,n,tDTα−10CDtαusi−=tDTα−10CDtαusi+,i=1,2,⋯,n,u0=uT=0.In [26], 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+gtuP−2ut=fit,ut,t∈si,ti+1,i=0,1,2,⋯,n,ΔtDTα−1ϕP0CDtαuti=Iiuti,i=1,2,⋯,n,tDTα−1ϕP0CDtαut=tDTα−1ϕP0CDtαuti+,t∈ti,si,i=1,2,⋯,n,tDTα−1ϕP0CDtαusi−=tDTα−1ϕP0CDtαusi+,i=1,2,⋯,n,u0=uT=0.Motivated by the above-mentioned work, the paper focuses on the existence of solutions for the followingp-Laplacian fractional differential equations with instantaneous and noninstantaneous impulses, and if DxFit,ut−uti+1=fit,ut, the following problem reduces to (2):
(3)DtTαϕP0CDtαut+λtup−2ut=DxFit,ut−uti+1,t∈si,ti+1,i=0,1,2,⋯,N,ΔtDTα−1ϕP0CDtαuti=Iiuti,i=1,2,⋯,N,tDTα−1ϕP0CDtαut=tDTα−1ϕP0CDtαuti+,t∈ti,si,i=1,2,⋯,N,tDTα−1ϕP0CDtαusi−=tDTα−1ϕP0CDtαusi+,i=1,2,⋯,N,u0=uT=0,where α∈1/p,1,p≥2,DtTα, and 0CDTα denote the right Riemann-Liouville fractional derivatives and left Caputo fractional derivatives. 0=s0<t0<s1<t1<⋯<tN=T,Ii∈CR,R, and there exists i∈1,2,⋯,N such that Iuti≠0.fi∈Csi,ti+1×R,R,ϕp:R⟶R is the p-Laplacian function defined as ϕps=sp−2ss≠0,ϕp0=0.λ∈L∞0,T, essinfλt∈0,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)ΔtDTα−1ϕp0CDtαuti=tDTα−1ϕp0CDtαuti+−tDTα−1ϕp0CDtαuti−,tDTα−1ϕp0CDtαuti±=limt→ti±tDTα−1ϕp0CDtαut,tDTα−1ϕp0CDtαusi±=limt→si±tDTα−1ϕp0CDtαut.The rest of the paper is organized as follows. In Section2, 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 [26]).
Letα∈1/p,1,p∈2,∞, the fractional space
(5)E0α,p=u:0,T⟶RU,0CDtαu∈Lp0,T,R,u0=uTis defined by the closure of C0∞0,T,R with the norm
(6)uα,p=∫0Tλtutpdt+∫0T0cDtαutpdt1/p.Definition 2 (see [26]).
Let0<α≤1, the fractional derivative space E0α,p is a reflexive as well as separable Banach space.Remark 3 (see [25]).
Letp=2, we define E0α=E0α,2. It is obvious that the fractional derivative space E0α is the space of functions u∈L20,T,R having an α-order Caputo fractional derivative 0cDtαu∈L20,T,R and u0=uT.Proposition 4 (see [26]).
Let0<α≤1 and u∈ACa,b, then for t∈a,b,(7)acDtαut=aDtαut−uaΓ1−αt−a−α,tcDbαut=tDbαut−ubΓ1−αb−t−α.Proposition 5 (see [26]).
Letα∈0,1 and p∈1,+∞, for any u∈E0α,p, we have
(8)uLp≤TαΓα+1∫0T0CDtαutpdt1/p.
Moreover, ifα>1/p and 1/p+1/q=1, then
(9)u∞≤Ω∫0T0CDtαutpdt1/q,where Ω=Tα−1/p/Γαα−1q+11/q.Proposition 6 (see [26]).
Letα∈1/p,1, the space E0α,p is compactly embedded in C0,T,R.Proposition 7 (see [26]).
Letα∈0,1, and u,v∈Lpa,b:
(10)∫abtDbαutvtdt=∫abaDtαvtutdt.Definition 8.
A function(11)u∈u∈AC0,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.
LetX 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.
Ifφ⟶−∞,+∞ is coercive, then φ has a bounded minimizing sequence.Lemma 11 (see [26]).
Foru∈E0α,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 eachi=0,1,2,⋯,N, the nonlinear functions Fi satisfy the following assumptions:(H1).
Fit,x is measurable in t for every x∈R and continuously differentiable in x for a.e. t∈si,ti+1, and there exist functions k1∈CR+,R+, k2∈L1si,ti+1,R+ such that
(13)Fit,x≤k1xk2t,∣DxFit,x∣≤k1xk2t.(H2).
There exist constantsδ∈0,2,a>0, and the functions b0, b1∈L1si,ti,R+ such that
(14)Fit,x≤ax2+b0txδ+b1t,t∈si,ti+1.(H3).
There exist constantsci, di>0, σi∈0,1 (i=1,2,⋯,N) such that Iiu≤ci+diuσi, for any u∈R.Lemma 12.
A functionu∈E0α,p,∀v∈E0α,p, the following form is an equivalent form of the problem (3):
(15)∫0Tϕp0CDtαut0CDtαvtdt=∑i=0N∫siti+1DxFit,ut−uti+1vtidt−∑i=0N∫siti+1λtutp−2utvtdt−∑i=1NIiutivti.Proof.
Forv∈E0α,p, by (10) have
(16)∫0TϕpDtα0CutDtα0Cvtdt=∫0TDTα−1tϕpDtα0Cutv′tdt=∑i=0N∫siti+1DTα−1tϕpDtα0Cutv′tdt+∑i=1N∫tisiDTα−1tϕpDtα0Cutv′tdt=DTα−1tϕpDtα0Cut1−vt1−∫s0t1ddtDTα−1tϕpDtα0Cutvtdt+∑i=1NDTα−1tϕpDtα0Cutvttisi−∑i=1N∫tisiddtDTα−1tϕpDtα0Cutvtdt+∑i=1N−1DTα−1tϕpDtα0Cutvtsiti+1−DTα−1tϕpDtα0CusN+vsN−∑i=0N−1∫siti+1ddtDTα−1tϕpDtα0Cutvtdt−∫sNTddtDTα−1tϕpDtα0Cutvtdt=∫0TDTαtϕpDtα0Cutvtdt+∑i=1NDTα−1tϕpDtα0Cusi−−DTα−1tϕpDtα0Cusi+vsi+∑i=1NDTα−1tϕpDtα0Cuti−−DTα−1tϕpDtα0Cuti+vti.
We have(17)∫0TtDTαϕp0CDtαutvtdt=∫0Tϕp0CDtαut0CDtαvtdt+∑i=1NIiutivti.
As for problem (3), we have
(18)∫0TtDTαϕpD0Ctαutvtdt=∑i=0N∫siti+1tDTαϕpD0Ctαutvtdt+∑i=1N∫tisitDTαϕpD0Ctαutvtdt=∑i=0N∫siti+1DxFit,ut−uti+1vtidt−∑i=0N∫siti+1λtutp−2utvtdt−∑i=1N∫tisiddttDTα−1ϕpD0Ctαutvtdt=∑i=0N∫siti+1DxFit,ut−uti+1vtidt−∑i=0N∫siti+1λtutp−2utvtdt.
Above all, problem (3) has the following equivalent form:
(19)∫0TϕpD0CtαutD0Ctαvtdt=∑i=0N∫siti+1DxFit,ut−uti+1vtidt−∑i=0N∫siti+1λtutp−2utvtdt−∑i=1NIiutivti.Definition 13.
A functionu∈E0α,p is a weak solution of problem (3) if (15) holds for all v∈E0α,p.Define the functionφ:E0α,p⟶R as
(20)φu=1pup−∑i=0Nϕiu+∑i=1N∫0utiIisds,where ϕiu=∫siti+1Fit,ut−uti+1dt.From (9), we have
(21)ut−uti+1≤2u∞≤Tα−1/pΓαα−1q+11/quα=2Ωuα.Under condition (H1), for a.e. t∈si,ti+1, we have
(22)limε→01εFit,ut−uti+1+εvt−Fit,ut−uti+1=DxFit,ut−uti+1vt.From (21), for all γ∈0,1,
(23)ut−uti+1+γεvt≤2Ωuα+v∞≤2Ωuα+Ωvα.According to the mean value theorem and condition (H1), for some γ∈0,1,
(24)1εFit,ut−uti+1+εvt−Fit,ut−uti+1=DxFit,ut−uti+1+γεvtvt≤Ωvαmaxω∈0,2Ωuα+Ωvαk1ωk2t.By the Lebesgue dominated convergence theorem, it is easy to show thatϕi has a directional derivative at each point u:
(25)ϕi′u,v=∫siti+1DxFit,ut−uti+1vtdt,ϕi′u,v≤∫siti+1DxFit,ut−uti+1vtdt≤Ωvαmaxω∈0,2Ωuα+Ωvαk1ω∫siti+1k2tdt.Thus,ϕi∈E0α,p∗, let uk⟶uinE0α,p, then uk converges uniformly to u for all t∈0,T, we have
(26)ϕi′un−ϕi′uα≤Ωvα∫siti+1DxFit,unt−unti+1−DxFit,ut−uti+1dt.Therefore,ϕi′ is continuous from E0α,p into E0α,p∗, φ∈C1E0α,p,R and
(27)φ′u,v=∫0Tϕp0CDtαut0CDtαvtdt+∑i=1NIiutivti−∑i=0N∫siti+1DxFit,ut−uti+1vtidt+∑i=0N∫siti+1λtutp−2utvtdt.This yields that the critical points ofφ are weak solutions of problem (3).Lemma 14.
The functionalφ:E0α,p⟶R is a weak lower semicontinuous.Proof.
Let sequenceukk=1∞ in E0α,p be weakly convergent to u in E0α,p, then u≤limk→∞infuk.
From Proposition6, sequence ukk=1∞ is convergent uniformly to u in C0,T,R; thus, we have
(28)limk→∞infφuk=limk→∞inf1qup−∑i=0N∫siti+1Fit,ukt−ukti+1+∑i=1N∫0uktiIisds≥1qup−∑i=0N∫siti+1Fit,ut−uti+1+∑i=1N∫0uktiIisds=φu.Lemma 15.
u∈E0α,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, ifu is a classical solution of problem (3), then it is a weak solution of (3). Conversely, let u∈E0α,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 v∈C0∞siti+1 satisfying vt=0, where t∈0,si∪ti+1,T, i=0,1,2,⋯,N. Substituting vt into (15) and by Proposition 7, we have
(29)∫siti+1ϕp0CDtαut0CDtαvtdt+∫siti+1λtutp−2utvtdt=∫siti+1DxFit,ut−uti+1vtidt,(30)∫siti+1ϕp0CDtαut0CDtαvtdt+∫siti+1tDTαϕp0CDtαutvtdt<+∞,that is to say,
(31)tDTαϕp0CDtαut+λtutp−2ut=DxFit,ut−uti+1,t∈si,ti+1,i=0,1,⋯,N.
Sinceu∈E0α,p⊂C0,T, we have
(32)∫siti+1utp+0CDtαutpdt<+∞.
Because ofDxFit,ut−uti+1∈Csi,ti+1×R,R, by (31), we have
(33)tDTαϕp0CDtαut∈ACsi,ti+1.
Therefore,(34)tDTα−1ϕp0CDtαusi+=limt→si+tDTα−1ϕp0CDtαut,tDTα−1ϕp0CDtαuti+1−=limt→ti+1−tDTα−1ϕp0CDtαut.In view of (15) and (31), we have
(35)∫0Tϕp0CDtαut0CDtαvtdt+∑i=1NIiutivti+∑i=1N∫siti+1ddttDTα−1ϕp0CDtαutvtdt=0,that is,
(36)∑i=1NtDTα−1ϕp0CDtαusi+vsi+−∑i=0NtDTα−1ϕp0CDtαuti+1−vti+1−=∑i=1Nt∫tisiϕp0CDtαut0CDtαvtdt+∑i=1NIiutivti.
Without loss of generality, we take the test functionv∈C0∞ti,si such that vt=0, where t∈0,ti∪si,T,i=1,2,⋯,N. Substituting vt into (30), we obtain tDTα−1ϕp0CDtαut,t∈ti,si, i=1,2,3⋯N, is a constant, a.e.
(37)tDTα−1ϕp0CDtαuti+=tDTα−1ϕp0CDtαusi−=tDTα−1ϕp0CDtαut,t∈ti,si,i=1,2,⋯N.
By using (36) and (37), we have
(38)∑i=1NtDTα−1ϕp0CDtαusi+vsi+−∑i=0NtDTα−1ϕp0CDtαuti+1−vti+1−=∑i=1NtDTα−1ϕp0CDtαuti+vsi+∑i=1NIiutivti.
Hence,(39)tDTα−1ϕp0CDtαuti+=tDTα−1ϕp0CDtαuti−−Iiutivti=tDTα−1ϕp0CDtαuti+vsi−∑i=1NtDTα−1ϕp0CDtαusi+vsi+.
Then, combining with (37), we have
(40)tDTα−1ϕp0CDtαuti+−tDTα−1ϕp0CDtαuti−=Iiuti,tDTα−1ϕp0CDtαusi+=tDTα−1ϕp0CDtαusi−.
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 Proposition5, there exist constants A2,A3 such that
(41)φu=1pup−∑i=0N∫siti+1Fit,ut−uti+1dt+∑i=1N∫0utiIis≥1pup−∑i=0N∫siti+1b1tdt−∑i=1Nc1u∞−∑i=1Ndiσi+1u∞−∑i=0N∫siti+1aut−uti+12dt−∫siti+1b0tut−uti+1δdt≥1pup−a4Ω2uα2∑i=0Nti+1−si−2ΩδuαδA0−A1−uA2−uασi+1A3,where
(42)A0=∑i=0N∫siti+1b0tdt,A1=∑i=0N∫siti+1b1tdt.
Sincep>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 Iiuti≠0i=1,2,⋯,N, then u≠constant (u is a critical point of φ).
Therefore, problem (3) has at least one classical solution. The proof is completed.


## 4. Example

Considering the followingp-Laplacian fractional differential equations with instantaneous and noninstantaneous impulses,
(43)tDT3/4ϕp0CDT3/4ut+up−2ut=DxFit,ut−uti+1,t∈si,ti+1,i=0,1,2,⋯,N,ΔtDT−1/4ϕp0CDt3/4uti=Iiuti,i=1,2,⋯,N,tDT−1/4ϕp0CDt3/4ut=tDTα−1ϕp0CDtαuti+,t∈ti,si,i=0,1,2,⋯,N,tDT−1/4ϕp0CDt3/4usi−=tDTα−1ϕp0CDtausi+,i=1,2,⋯,N,u0=uT=0,where α=3/4, T=2, λt=1, Iiuti=1/3+1/2sinu2/3, DxFit,x=1/9ut−uti+12−3ut−uti+1−21+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 thep-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,ut−uti+1≡fit,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.



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