MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2020/23275802327580Research ArticleMultiple Positive Solutions for Fractional Three-Point Boundary Value Problem with p-Laplacian OperatorLiDong1LiuYang2https://orcid.org/0000-0002-6223-908XWangChunli3LiuJia-Bao1Department of MathematicsCollege of Science Jiamusi UniversityJiamusiHeilongjiang 154007China2School of Mathematics and StatisticsHefei Normal UniversityHefeiAnhui 230061Chinahftc.edu.cn3Institute of Information Technology of GUETGuilinGuangxi 540004China2020972020202023052020200620209720202020Copyright © 2020 Dong Li 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, we investigate the existence of multiple positive solutions or at least one positive solution for fractional three-point boundary value problem with p-Laplacian operator. Our approach relies on the fixed point theorem on cones. The results obtained in this paper essentially improve and generalize some well-known results.

2019 Project of Foundational Research Ability Enhancement for Young and Middle-Aged University Faculties of Guangxi2019KY1046Natural Science Foundation of Anhui Province2008085QA08Scientific Research Projects of Institute of Information Technology of GUETB201911Department of Education, Heilongjiang Province12543079
1. Introduction

Nowadays, fractional calculus has been adapted to numerous fields, such as engineering, mechanics, physics, chemistry, and biology. Many essays and mongraphs studying various issues in fractional calculus have been researched (see ). In particular, fractional differential equations have been found to be a powerful tool in modeling various phenomena in many areas of science and engineering such as physics, fluid mechanics, and heat conduction. More details about research achievement on fractional differential equations and their applications are shown in .

Recently, fractional differential equations have gained considerable attention (see  and the references therein). Fractional differential equations and differential equations with p-Laplacian operators have attracted much attention from many mathematicians. As a result, meaningful research results have been drawn . Fractional-order boundary value problems involving classical, multipoint, high-order, and integral boundary conditions have extensively been studied by many researchers and a variety of results can be found in recent literature on the topic .

In , Chai studied the boundary value problems of fractional differential equations with p-Laplacian operator as follows:(1)D0+βϕpD0+αut+ft,ut=0,0<t<1,u0=0,u1+σD0+γu1=0,D0+αu0=0,where ϕps=sp2s,p>1,fC0,1×R+,R+, α1,2,β0,1,γ0,1, αγ1, σ>0, and D0+α, D0+β, D0+γ are the standard Riemann–Liouville derivatives. Some existence results of positive solutions are obtained by using the monotone iterative method.

In , by using Krasnosel'skii’s fixed point theorem, Tian et al. obtained the existence of positive solutions for a boundary value problem of fractional differential equations with p-Laplacian operator as follows:(2)D0+βϕpD0+αut=ft,ut,0<t<1,u0=u0=u1=D0+αu0=0,D0+αu1=λD0+αuη,where ϕps=sp2s,p>1,fC0,1×R+,R+, α2,3,β1,2, η0,1, λ0,, and D0+α, D0+β are the Riemann–Liouville fractional derivatives.

In , by using the monotone iterative method, Tian et al. obtained the existence of positive solutions for a boundary value problem of fractional differential equations with p-Laplacian operator as follows:(3)DγϕpDαut=ft,ut,0<t<1,u0=Dαu0=0,Dβu1=aDβuξ,Dαu1=bDαuη,where ϕps=sp2s,p>1,α1,2,0<βα1, ξ,η0,1, a,b0, and 1aξαβ1>0, 1bp1ηγ1>0, fC0,1×R+,R+, and Dα is the Riemann–Liouville fractional derivative.

In , by means of the p-Laplacian operator, Han et al. obtained the existence of positive solutions for the boundary value problem of fractional differential equation as follows:(4)D0+βϕD0+αut=λfut,0<t<1,u0=u0=u1=0,ϕD0+αu0=ϕD0+αu1=0,where ϕs=sp2s,p>1,α2,3,β1,2,f:0,+0,+ is continuous, and D0+α, D0+β are the Riemann–Liouville fractional derivatives.

Based on the above research, this paper analyzed the following fractional three-point boundary value problem with the p-Laplacian operator:(5)D0+βϕD0+αut=ft,ut,0<t<1,u0=u0=0,D0+μu1=δD0+μuη,ϕD0+αu0=ϕD0+αu1=0,where ϕs=sp2s,p>1,α2,3,β1,2,μ1,α1,fC0,1×R+,R+, δ0,0<η<1, and Δ=1δηαμ1>0.

The aim is to establish some existence and multiplicity results of positive solutions for BVP (5). This paper is organized as follows. In Section 2, some properties of Green’s function will be given, which are needed later. In Section 3, the existence of multiplicity results of positive solutions of BVP will be discussed (5).

2. Preliminary Knowledge and LemmasLemma 1.

(see ). Assume that Da+αL1a,b with a fractional derivative of order α>0. Then,(6)Ia+β1Da+β1ut=ut+c1taα1+c2taα2++cntaαn,for some ciR,i=1,2,,n, where n is the smallest integer greater than or equal to α.

Lemma 2.

If yC0,1, then the fractional boundary value problem is(7)D0+αut+yt=0,0<t<1,u0=u0=0,D0+μu1=δD0+μuη.

The unique solution is ut=01Gt,sysds, where Gt,s=G1t,s+δ/Δtα1G2η,s,(8)G1t,s=1Γαtα11sαμ1tsα1,st,tα11sαμ1,ts,G2η,s=1Γα1sαμ1ηαμ1ηsαμ1,sη,1sαμ1ηαμ1,sη.

Proof.

The general solution to the problem (7) is(9)ut=0ttsα1Γαysds+C1tα1+C2tα2+C3tα3.

From the boundary value condition of (7), C2=C3=0,(10)C1=1Δ011sαμ1Γαysdsδ0ηηsαμ1Γαysds.

Therefore,(11)ut=0ttsα1Γαysds+01tα11sαμ1Γαysds+δtα1Δ01ηαμ11sαμ1Γαysdsδtα1Δ0ηηsαμ1Γαysds=01G1t,sysds+δΔtα101G2η,sysds=01Gt,sysds.

Lemma 3.

If wC0,1, then the fractional boundary value problem is(12)D0+βϕD0+αut=wt,0<t<1,u0=u0=0,ϕD0+αu0=ϕD0+αu1=0.

The unique solution is ut=01Gt,sϕ101Hs,τwτdτds, where(13)Ht,s=1Γβtβ11sβ2tsβ1,st,tβ11sβ2,ts,

Proof.

Problem (12) is equivalent to(14)ϕD0+αut=0ttτβ1Γβwτdτ+C1tβ1+C2tβ2.

From the boundary value condition of (12), C2=0 and C1=011τβ2/Γβwτdτ; then,(15)ϕD0+αut=0ttτβ1Γβwτdτ01tβ11τβ2Γβwτdτ=01Ht,τwτdτ.

Therefore,(16)D0+αut+ϕ101Ht,τwτdτ=0.

Based on Lemma 2,(17)ut=01Gt,sϕ101Hs,τwτdτds.

Lemma 4.

The properties of Gt,s and Ht,s are

0Gt,sG1,s, for t,s0,1×0,1

Gt,s1/4α1G1,s, for t,sI×0,1=1/4,3/4×0,1

(see ). 0Ht,sHs,s, for t,s0,1×0,1

(see ). Ht,s1/4β1H1,s, for t,sI×0,1=1/4,3/4×0,1

Proof.

For ts, it is easy to show that /tG1t,s0; for ts,

(18)tG1t,s=α1tα21sαμ1tsα2Γαα1tsμ1tαμ11sαμ1tsαμ1Γα0.

So, for t,s0,1×0,1,

(19)tG1t,s=tG1t,s+α1δΔtα2G2η,s0.

Then, for t,s0,1×0,1, 0Gt,sG1,s.

For ts, it is easy to show that G1t,s/G11,s=tα1; for ts,

(20)G1t,sG11,s=tα11sαμ1tsα11sαμ11sα1tα11sαμ1tα11sα11sαμ11sα1=tα1.and for t,sI×0,1=1/4,3/4×0,1,(21)Gt,s=G1t,s+δΔtα1G2η,stα1G1t,s+δΔtα1G2η,s14α1G1,s.

Lemma 5.

(see ). Let E=E,. be a Banach space and let KE be a cone in E. Assume Ω1 and Ω2 are open subsets of E with 0Ω1 and Ω1¯Ω2 and let T:KΩ2¯\Ω1K be a continuous and completely continuous. In addition, suppose either

Tuu, uKΩ1, and Tuu, uKΩ2, or

Tuu, uKΩ1, and Tuu, uKΩ2

Lemma 6.

(see ). Let K be a cone in a real Banach space E, Kr=xk:x<r,ψ be nonnegative continuous concave functional on K such that ψxx,xK¯r, and(22)Kψ,d,e=xK:dψx,xe.

Suppose T:K¯rK¯r is completely continuous and there exist constants 0<c<d<er such that

xKψ,d,eψx>d and ψTx>d for xKψ,d,e

Tx<c for xc

ψTx>d for xKψ,d,r with Tx>e

Then, T has at least three fixed points x1,x2, and x3 with x1<c, d<ψx2, and c<x3 with ψx3<d.

3. Main Results

When E=0,1, any uE, u=max0t1ut, then E is a real Banach space. KE is a cone, which can be defined as K=uE:mint0,1ut0,mintIut1/4α1u. Defining the operator T:EE, for any uE,(23)Tut=01Gt,sϕ101Hs,τfτ,uτdτds,and for convenience, the following notation is introduced:(24)M=01G1,sϕ101Hτ,τdτds1,N=IG1,sϕ1IH1,τdτds1.

Theorem 1.

If there are two positive numbers 0<r1<r2 such that the following conditions hold:

B1ft,uϕr1N4α+β2, for t,u0,1×0,r1

B2ft,uϕr2M, for t,u0,1×0,r2

then the fractional three-point boundary value problem (5) has at least one positive solution u and r1ur2.

Proof.

From the continuity of G, H, f, it can be concluded that T:KK is continuous. For t,sI×0,1, uK, by Lemma 4, we have(25)mintITut=mintI01Gt,sϕ101Hs,τfτ,uτdτds14α101G1,sϕ101Hs,τfτ,uτdτds14α1Tu.

It means that TKK. Therefore, the Arzela–Ascoli theorem can prove that the operator T:KK is completely continuous.

Let Ω1=uK:uλ1, for uΩ1. From Lemma 4 and B1, we can conclude that(26)Tut=01Gt,sϕ101Hs,τfτ,uτdτdsr1N4α+β201Gt,sϕ101Hs,τdτdsr1N4α+β2I14α1G1,sϕ1I14β1H1,τdτds=r1NIG1,sϕ1IH1,τdτdsr1.

Then, when uΩ1, Tuu.

Let Ω2=uK:uλ2, for uΩ2. Then, we also can conclude from Lemma 4 and B2 that(27)Tut=01Gt,sϕ101Hs,τfτ,uτdτdsr2M01G1,sϕ1abHτ,τdτds=r2.

Therefore, for uΩ2, Tuu. In summary, by Lemma 5, the fractional three-point boundary value problem (5) has at least one positive solution u and r1ur2.

Theorem 2.

If there exist positive real numbers 0<c<d<1/4α1r such that the following conditions hold:

B3ft,u<ϕrM, for t,u0,1×0,r

B4ft,u>ϕdN/1/4α+β2, for t,uI×d,d/1/4α1

B5ft,u<ϕcM, for t,u0,1×0,c

then the fractional three-point boundary value problem (5) has at least three positive solutions u1, u2, and u3 with

(28)max0t1u1t<c,d<mintIu2t<r,c<max0t1u1t<r,mintIu3t<d.

Proof.

Firstly, if uKr¯, then we may assert that T:Kr¯Kr¯ is a completely continuous operator. To see this, suppose uKr¯; then, ur. It follows from Lemma 4 and (B3) that(29)Tut=max0t1Tut=max0t101Gt,sϕ101Hs,τfτ,uτdτdsr2M01G1,sϕ1abHτ,τdτds=r2.

Therefore, T:Kr¯Kr¯. This together with Lemma 5 implies that T:Kr¯Kr¯ is a completely continuous operator. In the same way, if uKc¯, then assumption (B5) yields Tu<c. Hence, condition (ii) of Lemma 6 is satisfied.

To check condition (i) of Lemma 6, we let ut=d/1/4α1 for t0,1. It is easy to verify that ut=d/1/4α1Kψ,d,d/1/4α1 and ψu=d/1/4α1>d, and so(30)uKψ,d,d1/4α1ψu>d.

Thus, for all uKψ,d,d/1/4α1, we have that dutd/1/4α1 for tI and TuK.

From Lemma 4 and (B4), one has(31)ψTut=mintITut=mintI01Gt,sϕ101Hs,τfτ,uτdτdsI14α1G1,sϕ101Hs,τfτ,uτdτdsI14α1Gt,sϕ1IHs,τfτ,uτdτdsdN1/4α+β2I14α1G1,sϕ1I14β1H1,τdτds=dNIG1,sϕ1IH1,τdτds=d.

This shows that condition (i) of Lemma 6 holds.

Secondly, we verify that (iii) of Lemma 6 is satisfied. By Lemma 6, we have(32)mintITut=mintI01Gt,sϕ101Hs,τfτ,uτdτds14α101G1,sϕ101Hs,τfτ,uτdτds14α1Tu>d,for uKψ,d,r with Tu>d/1/4α1, which shows that condition (iii) of Lemma 6 holds.

To sum up, all the conditions of Lemma 6 are satisfied; from Lemma 6, it follows that there exist three positive solutions u1, u2, and u3 with(33)max0t1u1t<c,d<mintIu2t<r,c<max0t1u1t<r,mintIu3t<d.

4. Conclusion

The existence of solutions to three-point boundary value problems of fractional differential equations with the p-Laplacian operators is discussed by using the fixed point exponential theorem and fixed point theorem of cone compression and cone tension. By extending the existence of solutions to boundary value problems, we have obtained the sufficient condition that the boundary value problem has multiple positive solutions or at least one positive solution.

Data Availability

The data used to support the findings of the study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by 2019 Project of Foundational Research Ability Enhancement for Young and Middle-Aged University Faculties of Guangxi (2019KY1046), Nature and Science Foundation of Anhui (2008085QA08), Scientific Research Projects of Institute of Information Technology of GUET (B201911), and Science and Technology Research Project of Heilongjiang Provincial Department of Education (12543079).