JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi 10.1155/2017/3187492 3187492 Research Article Positive Solutions of Fractional Differential Equations with p-Laplacian Tian Yuansheng 1 http://orcid.org/0000-0003-0667-6442 Sun Sujing 2 http://orcid.org/0000-0002-5131-9252 Bai Zhanbing 2 Liu Lishan 1 College of Mathematics and Finance Xiangnan University Chenzhou 423000 China xnu.edu.cn 2 College of Mathematics and System Science Shandong University of Science and Technology Qingdao 266590 China sdust.edu.cn 2017 14112017 2017 21 06 2017 05 10 2017 23 10 2017 14112017 2017 Copyright © 2017 Yuansheng Tian 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.

The multiplicity of positive solution for a new class of four-point boundary value problem of fractional differential equations with p-Laplacian operator is investigated. By the use of the Leggett-Williams fixed-point theorem, the multiplicity results of positive solution are obtained. An example is given to illustrate the main results.

National Natural Science Foundation of China 11571207 Education Department of Hunan Province 16A198 Natural Science Foundation of Hunan Province 2015JJ6101 Construct Program of the Key Discipline in Hunan Province
1. Introduction

In recent years, boundary value problems of nonlinear fractional differential equations have been studied extensively (see  and the references therein). By the use of some fixed-point theorem, the existence results of positive solutions are obtained for singular factional problem [2, 4, 24, 25], impulsive fractional problem [3, 20], nonlocal problem [1, 12, 14, 16, 22, 23], numerical solution problem [17, 18], initial value problem [19, 20], Dirichlet value problem [6, 9, 15], iterative solution problem , and so on. Cui  considered the following boundary values problems: (1)D0+αxt+ptft,xt+qxt=0,0<t<1,u0=u0=u1=0, where 2<α3 is a real number and D0+α is the standard Riemann-Liouville differentiation. Under the assumption that f(t,x) is a Lipschitz continuous function, by the use of u0-positive operator, they studied the uniqueness results for the fractional differential equation.

On the other hand, because of the wide mathematical and physical background, the existence of positive solutions for nonlinear integer-order boundary values problems with p-Laplacian operator has received wide attention (see [8, 12, 13, 21, 2429]). For example, Su et al.  considered the following four-point boundary values problems with p-Laplacian operator: (2)ϕput+atfut=0,0<t<1,αϕpu0-βϕpuξ=0,γϕpu1-δϕpuη=0,where ϕps=sp-2s,p>1. Liu et al. , Dong et al. , and Zhang et al.  studied p-Laplacian boundary value problems with fractional derivative. By using the fixed-point index theory, they obtained the existence of positive solutions.

In this paper, we investigate the multiplicity of positive solution for a new class of four-point boundary value problem of fractional differential equations with p-Laplacian operator: (3)D0+γϕpD0+αut=ft,ut,0<t<1,u0=D0+αu0=0,D0+βu1=λuξ,D0+αu1=μD0+αuη, where α,β,γR,1<α,γ2,β>0,1+βα and ξ,η(0,1),λ,μ[0,+),(1-β)Γ(α)λΓ(α-β)ξα-2,1-μp-1ηγ-20 and ϕps=sp-2s,p>1,D0+α is the standard Riemann-Liouville differentiation and fC([0,1]×[0,+),[0,+)). By using the Leggett-Williams fixed-point theorem on a cone, the multiplicity results of positive solution are obtained.

2. Preliminaries Lemma 1 (see [<xref ref-type="bibr" rid="B12">10</xref>]).

Assume that xC(0,1)L(0,1) with a fractional derivative of order α>0 that belongs to C(0,1)L(0,1). Then (4)I0+αD0+αxt=xt+c1tα-1+c2tα-2++cNtα-N,ciR,i=1,2,,N,where N is the smallest integer greater than or equal to α.

Lemma 2.

If hC[0,1],ϕp(s)=sp-2s,p>1,ϕq=(ϕp)-1,1/p+1/q=1,α,β,γR,1<α,γ2,β>0,1+βα and ξ,η(0,1),λ,μ[0,+),MλΓ(α-β)ξα-1,Nμp-1ηγ-1, then the problem (5)D0+γϕpD0+αut=ht,0<t<1,u0=D0+αu0=0,D0+βu1=λuξ,D0+αu1=μD0+αuη has a unique solution (6)ut=01Gt,sϕq01Ks,τhτdτds, where (7)Gt,s=Γαtα-11-sα-β-1-λΓα-βtα-1ξ-sα-1ΓαΓα-M-t-sα-1Γα,0st1,sξ,Γαtα-11-sα-β-1-Γα-Mt-sα-1ΓαΓα-M,0<ξst1,Γαtα-11-sα-β-1-λΓα-βtα-1ξ-sα-1ΓαΓα-M,0tsξ<1,tα-11-sα-β-1Γα-M,0ts1,ξs.Ks,τ=s1-τγ-1-μp-1sη-τγ-1-1-Ns-τγ-11-NΓγ,0τs1,τη,s1-τγ-1-1-Ns-τγ-11-NΓγ,0<ητs1,s1-τγ-1-μp-1sη-τγ-11-NΓγ,0sτη<1,s1-τγ-11-NΓγ,0sτ1,ητ.

Proof.

Suppose u is a solution of the problem (5). By Lemma 1, there is (8)ϕpD0+αut=Iγht+d1tγ-1+d2tγ-2, for some d1,d2R. Taking into account the fact that D0+αu(0)=0, we have d2=0 and (9)ϕpD0+αut=1Γγ0tt-τγ-1hτdτ+d1tγ-1. Thus, (10)ϕpD0+αu1=1Γγ011-τγ-1hτdτ+d1,ϕpD0+αuη=1Γγ0ηη-τγ-1hτdτ+d1ηγ-1. By D0+αu(1)=μD0+αu(η), (10), it holds that(11)d1=-011-τγ-1Γγ1-Nhτdτ+0ημp-1η-τγ-1Γγ1-Nhτdτ. So, (12)ϕpD0+αut=0tt-τγ-1Γγhτdτ-01tγ-11-τγ-1Γγ1-Nhτdτ+0ημp-1tγ-1η-τγ-1Γγ1-Nhτdτ=-01Kt,τhτdτ,(13)D0+αut+ϕq01Kt,τhτdτ=0. Applying Lemma 1, we can reduce (13) to an equivalent integral equation (14)ut=-I0+αϕq01Kt,τhτdτ+c1tα-1+c2tα-2, for some c1,c2R. With the condition u(0)=0, there is c2=0. Consequently, (15)ut=-I0+αϕq01Kt,τhτdτ+c1tα-1. By (15), one has (16)D0+βut=-D0+βI0+αϕq01Kt,τhτdτ+c1D0+βtα-1=-I0+α-βϕq01Kt,τhτdτ+c1ΓαΓα-βtα-β-1. So, (17)Dβu1=-011-sα-β-1Γα-βϕq01Ks,τhτdτds+c1ΓαΓα-β,uξ=-0ξξ-sα-1Γαϕq01Ks,τhτdτds+c1ξα-1. By Dβu(1)=λu(ξ), (17), we have (18)c1=Γα-βΓα-M011-sα-β-1Γα-βϕq01Ks,τhτdτds-λ0ξξ-sα-1Γαϕq01Ks,τhτdτds. So, the unique solution of problem (5) is (19)ut=-0tt-sα-1Γαϕq01Ks,τhτdτds+Γα-βtα-1Γα-M011-sα-β-1Γα-βϕq01Ks,τhτdτds-λ0ξξ-sα-1Γαϕq01Ks,τhτdτds=01Gt,sϕq01Ks,τhτdτds. The proof is completed.

Lemma 3.

Suppose that (1-β)Γ(α)λΓ(α-β)ξα-2,1-μp-1ηγ-20. The functions G(t,s) and K(t,s) satisfy the following:

G(t,s),K(t,s)C([0,1]×[0,1]);G(t,s)>0,K(t,s)>0 for t,s(0,1),

G(t,s)G(s,s),K(t,s)K(s,s) for t,s[0,1],

There exists a positive function ψ(s)C(0,1) such that (20)minξt1Gt,sψsmax0t1Gt,s=ψsGs,s,s0,1.

Proof.

The Proof of the Statement (1). From definitions, it is clear that G(t,s),K(t,s)C([0,1]×[0,1]). By (1-β)Γ(α)λΓ(α-β)ξα-2,1-μp-1ηγ-20, we have (21)Γα1-βΓαλΓα-βξα-2>λΓα-βξα-1=M,1-N=1-μp-1ηγ-1>1-μp-1ηγ-20.

If 0<st<1,sξ, let(22)gt,s=tα-11-sα-β-1-t-sα-1Γα. It is obvious that g(t,s)>0 for 0<st<1. Hence, we have (23)Gt,s=Γαtα-11-sα-β-1-Γα-Mt-sα-1-λΓα-βtα-1ξ-sα-1ΓαΓα-M=1Γα1+MΓα-Mtα-11-sα-β-1-t-sα-1Γα-λΓα-βtα-1ξ-sα-1ΓαΓα-M=tα-11-sα-β-1-t-sα-1Γα+λΓα-βtα-1ξα-11-sα-β-1-ξ-sα-1ΓαΓα-M=gt,s+λΓα-βtα-1Γα-Mgξ,s>0.

By using the analogous argument, it holds that G(t,s)>0 for other situations. Hence, G(t,s)>0 for t,s(0,1).

If 0<st<1,sη, let (24)kt,s=t1-sγ-1-t-sγ-1Γγ. It is obvious that k(t,s)>0 for 0<st<1. Hence, we have (25)Kt,s=t1-sγ-1-μp-1tη-sγ-1-1-Nt-sγ-11-NΓγ=1+N1-Nt1-sγ-1Γγ-t-sγ-1Γγ-μp-1tη-sγ-11-NΓγ=t1-sγ-1-t-sγ-1Γγ+μp-1tγ-1ηγ-11-sγ-1-η-sγ-11-NΓγ=kt,s+μp-1tγ-11-Nkη,s>0. Similarly, it holds that

K(t,s)>0 for 0<ηst<1 or 0<tsη<1 or 0<ts<1,ηs.

Hence, K(t,s)>0 for t,s(0,1).

The Proof of the Statement (2). Let (26)g1t,s=Γαtα-11-sα-β-1-λΓα-βtα-1ξ-sα-1ΓαΓα-M-t-sα-1Γα,0st1,sξ,g2t,s=Γαtα-11-sα-β-1-Γα-Mt-sα-1ΓαΓα-M,0<ξst1,g3t,s=Γαtα-11-sα-β-1-λΓα-βtα-1ξ-sα-1ΓαΓα-M,0tsξ<1,g4t,s=tα-11-sα-β-1Γα-M,0ts1,ξs.

It is easy to check that g3(t,s) and g4(t,s) are increasing with respect to t on [0,s]. We will show that g1(t,s) and g2(t,s) are decreasing with respect to t on [s,1]. For 0st1,sξ, let (27)h1t,s=ΓαΓα-Mg1t,s; then we have (28)h1t,s=Γαtα-11-sα-β-1-Γα-Mt-sα-1-λΓα-βtα-1ξ-sα-1,h1t,st=α-1tα-2Γα1-sα-β-1-Γα-M1-stα-2-λΓα-βξ-sα-1α-1tα-2Γα1-sα-β-1-Γα-M1-sα-2-λΓα-βξ-sα-1=α-1tα-21-sα-2Γα1-s1-β-Γα-M-λΓα-βξ-sα-11-s2-αα-1tα-21-sα-2Γα1-1-βs-Γα-M-λΓα-βξ-sα-11-s2-α=α-1tα-21-sα-2M-Γα1-βsλΓα-βξ-sα-11-s2-α.

For 0sξ, let (29)ωs=M-Γα1-βs-λΓα-βξ-sα-11-s2-α=λΓα-βξα-1-Γα1-βs-λΓα-βξ-sα-11-s2-α. We have (30)ω0=λΓα-βξα-1-λΓα-βξα-1=0,ωξ=λΓα-βξα-1-1-βξΓα=ξλΓα-βξα-2-1-βΓα0,ωs=β-1Γα+λΓα-βξ-sα-11-s2-αα-11-s+ξ-s2-α1-sξ-s.

If there exists s(0,ξ) such that ω(s)=0, then (31)λΓα-βξ-sα-11-s2-α=1-βΓα1-sξ-sα-11-s+ξ-s2-α. Therefore, we have (32)ωs=λΓα-βξα-1-1-βΓαs+1-sξ-sα-11-s+ξ-s2-α=λΓα-βξα-1-1-βΓα×sα-11-s+ξ-s2-α+1-sξ-sα-11-s+ξ-s2-α=λΓα-βξα-1-1-βΓαξα-11-s-ξ-sα-11-sα-11-s+ξ-s2-α+ξξ-s2-α-ξ-s22-α+1-sξ-sα-11-s+ξ-s2-α=λΓα-βξα-1-1-βΓαξ+ξ-s2-α1-s-ξ-s22-αα-11-s+ξ-s2-α=λΓα-βξα-1-1-βΓαξ+ξ-s2-α1-ξα-11-s+ξ-s2-α=λΓα-βξα-1-1-βΓαξ-1-βΓαξ-s2-α1-ξα-11-s+ξ-s2-αλΓα-βξα-2-1-βΓαξ0. In fact, taking into account the fact that λΓ(α-β)ξα-2-(1-β)Γ(α)0 and 2-α0, one has (33)-1-βΓαξ-s2-α1-ξα-11-s+ξ-s2-α0. So, we have max0sξω(s)0, which implies that h1(t,s)/t0. Hence g1(t,s) is decreasing with respect to t on [s,1].

For 0<ξst1, let h2(t,s)=Γ(α)(Γ(α)-M)g2(t,s); then we have (34)h2t,s=Γαtα-11-sα-β-1-Γα-Mt-sα-1,(35)h2t,st=α-1tα-2Γα1-sα-β-1-Γα-M1-stα-2α-1tα-2Γα1-sα-β-1-Γα-M1-sα-2=α-1tα-21-sα-2Γα1-s1-β-Γα-Mα-1tα-21-sα-2Γα1-1-βs-Γα-Mα-1tα-21-sα-2λΓα-βξα-1-1-βξΓα=α-1tα-21-sα-2ξλΓα-βξα-2-1-βΓα0, which implies that g2(t,s) is decreasing with respect to t on [s,1].

We can conclude that G(t,s) is increasing with respect to t for ts and G(t,s) is decreasing with respect to t for ts. Hence, G(t,s)G(s,s) for t,s[0,1].

By using the analogous method, we can conclude that K(t,s) is increasing with respect to t for ts and K(t,s) is decreasing with respect to t for ts. Hence, K(t,s)K(s,s) for t,s[0,1].

The Proof of the Statement (3). Taking into account the definition of G(t,s), there is (36)minξt1Gt,s=minξt1g1t,s,0sξ,minminξt1g2t,s,minξt1g4t,s,ξs1,=g11,s,0sξ,φs,ξs1, where φ(s)=min{g2(1,s),g4(ξ,s)}. It is easy to see that φ(s) is continuous on [ξ,1] and φ(s)>0 for all s[ξ,1). Let (37)ψs=g11,sGs,s,0<sξ,φsGs,s,ξs<1, where (38)Gs,s=Γαsα-11-sα-β-1-λΓα-βsα-1ξ-sα-1ΓαΓα-M,0sξ,sα-11-sα-β-1Γα-M,ξs1. Then (39)minξt1Gt,sψsmax0t1Gt,s=ψsGs,s,s0,1. The proof is completed.

Lemma 4 (see [<xref ref-type="bibr" rid="B13">11</xref>]).

Let P be a cone in a real Banach space E,Pc={uPuc},θ be a nonnegative continuous concave functional on a cone P such that θ(u)u, for all uP¯c, and P(θ,b,d)={uPbθ(u),ud}. Suppose T:P¯cP¯c is completely continuous and there exist constants 0<a<b<dc such that

{uP(θ,b,d)θ(u)>b} and θ(Tu)>b for uP(θ,b,d);

Tu<a for ua;

θ(Tu)>b for uP(θ,b,c) with Tu>d.

Then T has at least three fixed points u1,u2,u3 satisfying

u1<a,b<θ(u2) and u3>a with θ(u3)<b.

3. Main Result

Let E=C[0,1] be a Banach space with u=max0t1ut. Define the cone PE by P={uEu(t)0,0t1}. Let the nonnegative continuous concave functional θ on the P be defined by (40)θu=minξt1ut.

Lemma 5.

Let T:PE be an operator defined by (41)Tut=01Gt,sϕq01Ks,τfτ,uτdτds. Then T:PP is completely continuous.

Proof.

T : P P is continuous in view of nonnegativity and continuity of G(t,s),K(t,s) and f(t,u). Furthermore it is easy to see that by the Arzela-Ascoli theorem and Lebesgue dominated convergence theorem T:PP is completely continuous.

For convenience, we introduce the following notations: (42)N=01Gs,sϕq01Ks,τdτds-1,L=ξ1ψsGs,sϕq01Ks,τdτds-1.

Theorem 6.

Suppose fC([0,1]×[0,+),[0,+)) and there exist constants 0<a<b<c such that

f(t,u)<ϕp(Na), for (t,u)[0,1]×[0,a];

f(t,u)ϕp(Lb), for (t,u)[ξ,1]×[b,c];

f(t,u)ϕp(Nc), for (t,u)[0,1]×[0,c].

Then problem (3) possesses at least three positive solutions u1,u2, and u3 with (43)max0t1u1t<a,b<minξt1u2t<max0t1u2tc,a<max0t1u3tc,minξt1u3t<b.

Proof.

In order to apply Lemma 4, we divide the proof into four steps.

Step  1. If uP¯c, then uc. Assumption (B3) implies f(t,u(t))ϕp(Nc) for 0t1. Consequently, (44)Tu=max0t101Gt,sϕq01Ks,τfτ,uτdτds01Gs,sϕq01Ks,τϕpNcdτdsNc01Gs,sϕq01Ks,τdτds=c. Hence T:P¯cP¯c.

Step  2. We claim that the condition (A1) of Lemma 4 is satisfied. Let u(t)=(b+c)/2,0t1. we can easily see that u(t)=(b+c)/2P(θ,b,c),θ(u)=θ((b+c)/2)>b; consequently, {uP(θ,b,c)θ(u)>b}. Hence, if uP(θ,b,c), then bu(t)c for ξt1. From assumption (B2), we have (45)ft,utϕpLb,ξt1. So (46)θTu=minξt1Tut=minξt101Gt,sϕq01Ks,τfτ,uτdτdsminξt101Gt,sϕq01Ks,τϕpLbdτds>ξ1ψsGs,sϕq01Ks,τϕpLbdτds=Lbξ1ψsGs,sϕq01Ks,τdτds=b. Then, we have (47)θTu>b,uPθ,b,c. This implies that condition (A1) of Lemma 4 is satisfied.

Step  3. We now prove the condition (A2) of Lemma 4 is satisfied. If uP¯a, then ua. Assumption (B1) implies f(t,u(t))<ϕp(Na) for 0t1. Thus (48)Tu=max0t101Gt,sϕq01Ks,τfτ,uτdτds<01Gs,sϕq01Ks,τϕpNadτds=Na01Gs,sϕq01Ks,τdτds=a. Hence, the condition (A2) of Lemma 4 is also satisfied.

Step  4. Finally, we prove that the condition (A3) of Lemma 4 is satisfied. If uP(θ,b,c), by Step  2, we have θ(Tu)>b. Hence, the condition (A3) of Lemma 4 is also satisfied.

By Lemma 4, problem (3) has at least three positive solutions u1,u2, and u3 with (49)max0t1u1t<a,b<minξt1u2t<max0t1u2tc,a<max0t1u3tc,minξt1u3t<b.The proof is complete.

Example 7.

Consider the following boundary value problem: (50)D3/2ϕ3/2D3/2ut=ft,ut,0<t<1,u0=D3/2u0=0,D1/2u1=14u12,D3/2u1=18D3/2u14, where (51)ft,u=18t2+2u2,u<1,137+18t2+17u,u1. Choose p=α=γ=3/2,β=ξ=1/2,η=λ=1/4,μ=1/8. It is easy to check that (1-β)Γ(α)-λΓ(α-β)ξα-2>0 and 1-μp-1ηγ-2>0; by simple computation, we have (52)N1.3788,L3.9213.

Let a=1/4,b=1,c=7; one can check that the function f(t,u) satisfies

f(t,u)=1/8t2+2u20.25<ϕ3/2(Na)0.5871, for (t,u)[0,1]×[0,1/4];

f(t,u)=13/7+1/8t2+1/7u>2.0312>ϕ3/2(Lb)1.9802, for (t,u)[1/2,1]×[1,7];

f(t,u)=13/7+1/8t2+1/7u<2.9822<ϕ3/2(Nc)3.1067, for (t,u)[0,1]×[0,7].

That is to say that all the conditions of Theorem 6 hold. Thus Theorem 6 implies the problem (50) has at least three positive solutions u1,u2, and u3 such that (53)max0t1u1t<14,1<min1/2t1u2t<max0t1u2t7,14<max0t1u3t7,min1/2t1u3t<1.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The Project is supported by NSFC (11571207), the Scientific Research Foundation of Hunan Provincial Education Department (16A198), the Hunan Provincial Natural Science Foundation of China (2015JJ6101), and the Construct Program of the Key Discipline in Hunan Province.

Ahmad B. Sivasundaram S. On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order Applied Mathematics and Computation 2010 217 2 480 487 MR2678559 10.1016/j.amc.2010.05.080 2-s2.0-77955713929 Bai Z. Chen Y. Lian H. Sun S. On the existence of blow up solutions for a class of fractional differential equations Fractional Calculus and Applied Analysis 2014 17 4 1175 1187 2-s2.0-84989211590 10.2478/s13540-014-0220-2 Zbl1312.34007 Bai Z. Dong X. Yin C. Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions Boundary Value Problems 2016 2016 1, article no. 63 2-s2.0-84961256181 10.1186/s13661-016-0573-z Bai Z. Qiu T. Existence of positive solution for singular fractional differential equation Applied Mathematics and Computation 2009 215 7 2761 2767 MR2563488 10.1016/j.amc.2009.09.017 2-s2.0-70350714213 Bai Z. Zhang S. Sun S. Yin C. Monotone iterative method for fractional differential equations Electronic Journal of Differential Equations 2016 Paper No. 6, 8 MR3466477 Zhou Y. Chen F. Luo X. Existence results for nonlinear fractional difference equation Advances in Difference Equations 2011 2011 2-s2.0-79952236374 10.1155/2011/713201 713201 Cui Y. Uniqueness of solution for boundary value problems for fractional differential equations Applied Mathematics Letters 2016 51 48 54 10.1016/j.aml.2015.07.002 MR3396346 Dong X. Bai Z. Zhang S. Positive solutions to boundary value problems of p-Laplacian with fractional derivative Boundary Value Problems 2017 2017 1, article no. 5 2-s2.0-85008315211 10.1186/s13661-016-0735-z Zbl1357.34012 Jiao F. Zhou Y. Existence results for fractional boundary value problem via critical point theory International Journal of Bifurcation and Chaos 2012 22 4 2-s2.0-84861207614 10.1142/S0218127412500861 1250086 Kilbas A. A. Srivastava H. M. Trujillo J. J. Theory and Applications of Fractional Differential Equations 2006 Elsevier New York, NY, USA MR2218073 Zbl1092.45003 Leggett R. W. Williams L. R. Multiple positive fixed points of nonlinear operators on ordered Banach spaces Indiana University Mathematics Journal 1979 28 4 673 688 10.1512/iumj.1979.28.28046 MR542951 Zbl0421.47033 Liu X. Jia M. Ge W. The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator Applied Mathematics Letters 2017 65 56 62 10.1016/j.aml.2016.10.001 MR3575172 Liu X. Jia M. Ge W. Multiple solutions of a p-Laplacian model involving a fractional derivative Advances in Difference Equations 2013 2013, article no. 126 10.1186/1687-1847-2013-126 2-s2.0-84893948671 Qi T. Liu Y. Cui Y. Existence of solutions for a class of coupled fractional differential systems with nonlocal boundary conditions Journal of Function Spaces 2017 Art. ID 6703860, 9 MR3672060 Zbl06781956 Song Q. Dong X. Bai Z. Chen B. Existence for fractional Dirichlet boundary value problem under barrier strip conditions Journal of Nonlinear Sciences and Applications. JNSA 2017 10 7 3592 3598 MR3680301 10.22436/jnsa.010.07.19 Tian Y. Positive solutions to m-point boundary value problem of fractional differential equation Acta Mathematicae Applicatae Sinica 2013 29 3 661 672 10.1007/s10255-013-0242-2 MR3129832 Wang Z. A numerical method for delayed fractional-order differential equations Journal of Applied Mathematics 2013 2013 7 10.1155/2013/256071 256071 MR3056227 Zbl1266.65118 Wang Z. Huang X. Zhou J. A numerical method for delayed fractional-order differential equations: based on G-L definition Applied Mathematics & Information Sciences 2013 7 2L 525 529 10.12785/amis/072L22 MR3056028 Wei Z. Li Q. Che J. Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative Journal of Mathematical Analysis and Applications 2010 367 1 260 272 MR2600395 10.1016/j.jmaa.2010.01.023 2-s2.0-77049084247 Wang J. Feckan M. Zhou Y. Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions Bulletin des Sciences mathématiques 2017 10.1016/j.bulsci.2017.07.007 Zbl06790451 Zhang X. Ge W. Impulsive boundary value problems involving the one-dimensional p-Laplacian Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal 2009 70 4 1692 1701 10.1016/j.na.2008.02.052 MR2483590 Zou Y. Liu L. Cui Y. The existence of solutions for four-point coupled boundary value problems of fractional differential equations at resonance Abstract and Applied Analysis 2014 2014 2-s2.0-84899417467 10.1155/2014/314083 314083 Zou Y. Cui Y. Existence results for a functional boundary value problem of fractional differential equations Advances in Difference Equations 2013 2013, article no. 233 2-s2.0-84893522615 10.1186/1687-1847-2013-233 Tian Y. Chen A. Ge W. Multiple positive solutions to multipoint one-dimensional p-Laplacian boundary value problem with impulsive effects Czechoslovak Mathematical Journal 2011 61(136) 1 127 144 10.1007/s10587-011-0002-5 MR2782764 Zhang X. Liu L. Wu Y. Cui Y. Entire blow-up solutions for a quasilinear p-Laplacian Schrödinger equation with a non-square diffusion term Applied Mathematics Letters 2017 74 85 93 MR3677846 Chen C. Song H. Yang H. Liouville type theorems for stable solutions of p-Laplace equation in RN Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal 2017 160 44 52 10.1016/j.na.2017.05.004 MR3667674 Cui Y. Sun J. A generalization of Mahadevan's version of the Krein-Rutman theorem and applications to p-Laplacian boundary value problems Abstract and Applied Analysis 2012 2012 305279 10.1155/2012/305279 2-s2.0-84866173612 Su H. Wei Z. Wang B. The existence of positive solutions for a nonlinear four-point singular boundary value problem with a p-Laplacian operator Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal 2007 66 10 2204 2217 10.1016/j.na.2006.03.009 MR2311023 Wang Y. Hou C. Existence of multiple positive solutions for one-dimensional p-Laplacian Journal of Mathematical Analysis and Applications 2006 315 1 144 153 10.1016/j.jmaa.2005.09.085 MR2196536