# Multiple Positive Solutions for Fractional Three-Point Boundary Value Problem withp-Laplacian Operator

Authors: Dong Li; Yang Liu; Chunli Wang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we investigate the existence of multiple positive solutions or at least one positive solution for fractional three-point boundary value problem withp-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.


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## Body

## 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 [1–6]). 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 [7–10].Recently, fractional differential equations have gained considerable attention (see [11–15] 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 [16–20]. 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 [21–25].In [15], 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=sp−2s,p>1,f∈C0,1×R+,R+, α∈1,2,β∈0,1,γ∈0,1, α−γ≥1, 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 [16], 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=u′0=u1=D0+αu0=0,D0+αu1=λD0+αuη,where ϕps=sp−2s,p>1,f∈C0,1×R+,R+, α∈2,3,β∈1,2, η∈0,1, λ∈0,∞, and D0+α, D0+β are the Riemann–Liouville fractional derivatives.In [17], 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=sp−2s,p>1,α∈1,2,0<β≤α−1, ξ,η∈0,1, a,b∈0,∞ and 1−aξα−β−1>0, 1−bp−1ηγ−1>0, f∈C0,1×R+,R+, and Dα is the Riemann–Liouville fractional derivative.In [18], 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=u′0=u′1=0,ϕD0+αu0=ϕD0+αu1′=0,where ϕs=sp−2s,p>1,α∈2,3,β∈1,2,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 thep-Laplacian operator:(5)D0+βϕD0+αut=ft,ut,0<t<1,u0=u′0=0,D0+μu1=δD0+μuη,ϕD0+αu0=ϕD0+αu1′=0,where ϕs=sp−2s,p>1,α∈2,3,β∈1,2,μ∈1,α−1,f∈C0,1×R+,R+, δ≥0,0<η<1, and Δ=1−δηα−μ−1>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 Lemmas

Lemma 1.
(see [14]). Assume that Da+α∈L1a,b with a fractional derivative of order α>0. Then,(6)Ia+β1Da+β1ut=ut+c1t−aα−1+c2t−aα−2+⋯+cnt−aα−n,for some ci∈R,i=1,2,…,n, where n is the smallest integer greater than or equal to α.Lemma 2.
Ify∈C0,1, then the fractional boundary value problem is(7)D0+αut+yt=0,0<t<1,u0=u′0=0,D0+μu1=δD0+μuη.The unique solution isut=∫01Gt,sysds, where Gt,s=G1t,s+δ/Δtα−1G2η,s,(8)G1t,s=1Γαtα−11−sα−μ−1−t−sα−1,s≤t,tα−11−sα−μ−1,t≤s,G2η,s=1Γα1−sα−μ−1ηα−μ−1−η−sα−μ−1,s≤η,1−sα−μ−1ηα−μ−1,s≤η.Proof.
The general solution to the problem (7) is(9)ut=−∫0tt−sα−1Γαysds+C1tα−1+C2tα−2+C3tα−3.
From the boundary value condition of (7), C2=C3=0,(10)C1=1Δ∫011−sα−μ−1Γαysds−δ∫0ηη−sα−μ−1Γαysds.
Therefore,(11)ut=−∫0tt−sα−1Γαysds+∫01tα−11−sα−μ−1Γαysds+δtα−1Δ∫01ηα−μ−11−sα−μ−1Γαysds−δtα−1Δ∫0ηη−sα−μ−1Γαysds=∫01G1t,sysds+δΔtα−1∫01G2η,sysds=∫01Gt,sysds.Lemma 3.
Ifw∈C0,1, then the fractional boundary value problem is(12)D0+βϕD0+αut=wt,0<t<1,u0=u′0=0,ϕD0+αu0=ϕD0+αu1=0.The unique solution isut=∫01Gt,sϕ−1∫01Hs,τwτdτds, where(13)Ht,s=1Γβtβ−11−sβ−2−t−sβ−1,s≤t,tβ−11−sβ−2,t≤s,Proof.
Problem (12) is equivalent to(14)ϕD0+αut=∫0tt−τβ−1Γβwτdτ+C1tβ−1+C2tβ−2.
From the boundary value condition of (12), C2=0 and C1=−∫011−τβ−2/Γβwτdτ; then,(15)ϕD0+αut=∫0tt−τβ−1Γβwτdτ−∫01tβ−11−τβ−2Γβwτdτ=−∫01Ht,τwτdτ.
Therefore,(16)D0+αut+ϕ−1∫01Ht,τwτdτ=0.
Based on Lemma2,(17)ut=∫01Gt,sϕ−1∫01Hs,τwτdτds.Lemma 4.
The properties ofGt,s and Ht,s are(i)
0≤Gt,s≤G1,s, for t,s∈0,1×0,1(ii)
Gt,s≥1/4α−1G1,s, for t,s∈I×0,1=1/4,3/4×0,1(iii)
(see [13]). 0≤Ht,s≤Hs,s, for t,s∈0,1×0,1(iv)
(see [13]). Ht,s≥1/4β−1H1,s, for t,s∈I×0,1=1/4,3/4×0,1Proof.
(i)
Fort≤s, it is easy to show that ∂/∂tG1t,s≥0; for t≥s,(18)∂∂tG1t,s=α−1tα−21−sα−μ−1−t−sα−2Γα≥α−1t−sμ−1tα−μ−11−sα−μ−1−t−sα−μ−1Γα≥0.So, fort,s∈0,1×0,1,(19)∂∂tG1t,s=∂∂tG1t,s+α−1δΔtα−2G2η,s≥0.Then, fort,s∈0,1×0,1, 0≤Gt,s≤G1,s.(ii)
Fort≤s, it is easy to show that G1t,s/G11,s=tα−1; for t≥s,(20)G1t,sG11,s=tα−11−sα−μ−1−t−sα−11−sα−μ−1−1−sα−1≥tα−11−sα−μ−1−tα−11−sα−11−sα−μ−1−1−sα−1=tα−1.and for t,s∈I×0,1=1/4,3/4×0,1,(21)Gt,s=G1t,s+δΔtα−1G2η,s≥tα−1G1t,s+δΔtα−1G2η,s≥14α−1G1,s.Lemma 5.
(see [20]). Let E=E,. be a Banach space and let K∈E be a cone in E. Assume Ω1 and Ω2 are open subsets of E with 0∈Ω1 and Ω1¯⊂Ω2 and let T:K∩Ω2¯\Ω1⟶K be a continuous and completely continuous. In addition, suppose either(1)
Tu≤u, u∈K∩∂Ω1, and Tu≥u, u∈K∩∂Ω2, or(2)
Tu≥u, u∈K∩∂Ω1, and Tu≤u, u∈K∩∂Ω2Lemma 6.
(see [16]). Let K be a cone in a real Banach space E, Kr=x∈k:x<r,ψ be nonnegative continuous concave functional on K such that ψx≤x,∀x∈K¯r, and(22)Kψ,d,e=x∈K:d≤ψx,x≤e.SupposeT:K¯r⟶K¯r is completely continuous and there exist constants 0<c<d<e≤r such that(i)
x∈Kψ,d,eψx>d≠∅ and ψTx>d for x∈Kψ,d,e(ii)
Tx<c for x≤c(iii)
ψTx>d for x∈Kψ,d,r with Tx>eThen,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

WhenE=0,1, any u∈E, u=max0≤t≤1ut, then E is a real Banach space. K∈E is a cone, which can be defined as K=u∈E:mint∈0,1ut≥0,mint∈Iut≥1/4α−1u. Defining the operator T:E⟶E, for any u∈E,(23)Tut=∫01Gt,sϕ−1∫01Hs,τfτ,uτdτds,and for convenience, the following notation is introduced:(24)M=∫01G1,sϕ−1∫01Hτ,τdτds−1,N=∫IG1,sϕ−1∫IH1,τdτds−1.Theorem 1.
If there are two positive numbers0<r1<r2 such that the following conditions hold:B1ft,u≥ϕr1N4α+β−2, for t,u∈0,1×0,r1B2ft,u≤ϕr2M, for t,u∈0,1×0,r2then the fractional three-point boundary value problem (5) has at least one positive solution u and r1≤u≤r2.Proof.
From the continuity ofG, H, f, it can be concluded that T:K⟶K is continuous. For t,s∈I×0,1, u∈K, by Lemma 4, we have(25)mint∈ITut=mint∈I∫01Gt,sϕ−1∫01Hs,τfτ,uτdτds≥14α−1∫01G1,sϕ−1∫01Hs,τfτ,uτdτds≥14α−1Tu.
It means thatTK⊂K. Therefore, the Arzela–Ascoli theorem can prove that the operator T:K⟶K is completely continuous.
LetΩ1=u∈K:u≤λ1, for u∈∂Ω1. From Lemma 4 and B1, we can conclude that(26)Tut=∫01Gt,sϕ−1∫01Hs,τfτ,uτdτds≥r1N4α+β−2∫01Gt,sϕ−1∫01Hs,τdτds≥r1N4α+β−2∫I14α−1G1,sϕ−1∫I14β−1H1,τdτds=r1N∫IG1,sϕ−1∫IH1,τdτdsr1.
Then, whenu∈∂Ω1, Tu≥u.
LetΩ2=u∈K:u≤λ2, for u∈∂Ω2. Then, we also can conclude from Lemma 4 and B2 that(27)Tut=∫01Gt,sϕ−1∫01Hs,τfτ,uτdτds≤r2M∫01G1,sϕ−1∫abHτ,τdτds=r2.
Therefore, foru∈∂Ω2, Tu≤u. In summary, by Lemma 5, the fractional three-point boundary value problem (5) has at least one positive solution u and r1≤u≤r2.Theorem 2.
If there exist positive real numbers0<c<d<1/4α−1r such that the following conditions hold:B3ft,u<ϕrM, for t,u∈0,1×0,rB4ft,u>ϕdN/1/4α+β−2, for t,u∈I×d,d/1/4α−1B5ft,u<ϕcM, for t,u∈0,1×0,cthen the fractional three-point boundary value problem (5) has at least three positive solutions u1, u2, and u3 with(28)max0≤t≤1u1t<c,d<mint∈Iu2t<r,c<max0≤t≤1u1t<r,mint∈Iu3t<d.Proof.
Firstly, ifu∈Kr¯, then we may assert that T:Kr¯⟶Kr¯ is a completely continuous operator. To see this, suppose u∈Kr¯; then, u≤r. It follows from Lemma 4 and (B3) that(29)Tut=max0≤t≤1Tut=max0≤t≤1∫01Gt,sϕ−1∫01Hs,τfτ,uτdτds≤r2M∫01G1,sϕ−1∫abHτ,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 u∈Kc¯, then assumption (B5) yields Tu<c. Hence, condition (ii) of Lemma 6 is satisfied.
To check condition (i) of Lemma6, we let ut=d/1/4α−1 for t∈0,1. It is easy to verify that ut=d/1/4α−1∈Kψ,d,d/1/4α−1 and ψu=d/1/4α−1>d, and so(30)u∈Kψ,d,d1/4α−1ψu>d≠∅.
Thus, for allu∈Kψ,d,d/1/4α−1, we have that d≤ut≤d/1/4α−1 for t∈I and Tu∈K.
From Lemma4 and (B4), one has(31)ψTut=mint∈ITut=mint∈I∫01Gt,sϕ−1∫01Hs,τfτ,uτdτds≥∫I14α−1G1,sϕ−1∫01Hs,τfτ,uτdτds≥∫I14α−1Gt,sϕ−1∫IHs,τfτ,uτdτds≥dN1/4α+β−2∫I14α−1G1,sϕ−1∫I14β−1H1,τdτds=dN∫IG1,sϕ−1∫IH1,τdτds=d.
This shows that condition (i) of Lemma6 holds.
Secondly, we verify that (iii) of Lemma6 is satisfied. By Lemma 6, we have(32)mint∈ITut=mint∈I∫01Gt,sϕ−1∫01Hs,τfτ,uτdτds≥14α−1∫01G1,sϕ−1∫01Hs,τfτ,uτdτds≥14α−1Tu>d,for u∈Kψ,d,r with Tu>d/1/4α−1, which shows that condition (iii) of Lemma 6 holds.
To sum up, all the conditions of Lemma6 are satisfied; from Lemma 6, it follows that there exist three positive solutions u1, u2, and u3 with(33)max0≤t≤1u1t<c,d<mint∈Iu2t<r,c<max0≤t≤1u1t<r,mint∈Iu3t<d.


## 4. Conclusion

The existence of solutions to three-point boundary value problems of fractional differential equations with thep-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.



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