DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 137890 10.1155/2013/137890 137890 Research Article Eigenvalue of Fractional Differential Equations with p-Laplacian Operator Wu Wenquan Zhou Xiangbing Xu Fuyi Department of Mathematics Aba Teachers College Wenchuan Sichuan 623002 China 2013 25 3 2013 2013 27 12 2012 19 02 2013 2013 Copyright © 2013 Wenquan Wu and Xiangbing Zhou. 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.

We investigate the existence of positive solutions for the fractional order eigenvalue problem with p-Laplacian operator -𝒟tβ(φp(𝒟tαx))(t)=λf(t,x(t)),  t(0,1),  x(0)=0,  𝒟tαx(0)=0,  𝒟tγx(1)=j=1m-2aj𝒟tγx(ξj), where 𝒟tβ,  𝒟tα,  𝒟tγ are the standard Riemann-Liouville derivatives and p-Laplacian operator is defined as φp(s)=|s|p-2s,  p>1.f:(0,1)×(0,+)[0,+) is continuous and f can be singular at t=0,1 and x=0. By constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of fractional differential equation is established.

1. Introduction

Differential equations of fractional order have been recently proved to be valuable tools in the modeling of many phenomena arising from science and engineering, such as viscoelasticity, electrochemistry, control, porous media, and electromagnetism. For detail, see the monographs of Kilbas et al. , Miller and Ross , and Podlubny  and the papers  and the references therein.

In , the authors investigated the nonlinear nonlocal boundary value problem: (1)𝒟tβ(φp(𝒟tαx))(t)+f(t,x(t))=0,t(0,1),x(0)=0,𝒟tαx(0)  =0,x(1)  =ax(ξ), where 0<α2,0<β1,0a1,0<ξ<1. By using Krasnoselskii's fixed point theorem and the Leggett-Williams theorem, some sufficient conditions for the existence of positive solutions to the above BVP are obtained. In , by using the upper and lower solutions method, under suitable monotone conditions, the authors investigated the existence of positive solutions to the following nonlocal problem: (2)𝒟tβ(φp(𝒟tαx))(t)+f(t,x(t))=0,t(0,1),x(0)=0,𝒟tαx(0)=0,x(1)=ax(ξ),𝒟tαx(1)=b𝒟tαx(η), where 0<α,β2,0a,b1,0<ξ,η<1. Recently, by means of the fixed point theorem on cones, Chai  investigated two-point boundary value problem of fractional differential equation with p-Laplacian operator: (3)𝒟tβ(φp(𝒟tαx))(t)+f(t,x(t))=0,t(0,1),x(0)=0,  𝒟tαx(0)=0,  x(1)+a𝒟tγx(1)=0. Some existence and multiplicity results of positive solutions are obtained.

As far as we know, no result has been obtained for the existence of positive solution for the fractional order eigenvalue problem with p-Laplacian operator: (4)-𝒟tβ(φp(𝒟tαx))(t)=λf(t,x(t)),t(0,1),x(0)=0,  𝒟tαx(0)=0,𝒟tγx(1)=j=1m-2aj𝒟tγx(ξj), where 𝒟tβ, 𝒟tα, 𝒟tγ are the standard Riemann-Liouville derivatives with 1<α2,0<β1,0<γ1,0α-γ-1,0<ξ1<ξ2<<ξp-2<1,aj[0,+) with c=j=1m-2ajξjα-γ-1<1, p-Laplacian operator is defined as φp(s)=|s|p-2s,p>1,f can be singular at t=0,1, and x=0. In order to obtain the existence of positive solutions of the fractional order eigenvalue problem (4), we will apply the upper and lower solutions method associated with the Schauder's fixed point theorem. It is worth emphasizing that the problem (4) not only includes the well-known Sturm-Liouville boundary value problems and the nonlocal boundary value problems as special case, but also f can be singular at t=0,1 and x=0.

The organization of this paper is as follows. In Section 2, we present some necessary definitions and preliminary results that will be used to prove our main results. In Section 3, we put forward and prove our main results. Finally, we will give an example to demonstrate our main results.

2. Preliminaries and Lemmas

In this section, we introduce some preliminary facts which are used throughout this paper.

Definition 1 (see [<xref ref-type="bibr" rid="B1">1</xref>–<xref ref-type="bibr" rid="B3">3</xref>]).

The Riemann-Liouville fractional integral of order α>0 of a function  x:(0,+) is given by (5)Iαx(t)=1Γ(α)0t(t-s)α-1x(s)ds provided that the right-hand side is pointwise defined on (0,+).

Definition 2 (see [<xref ref-type="bibr" rid="B1">1</xref>–<xref ref-type="bibr" rid="B3">3</xref>]).

The Riemann-Liouville fractional derivative of order α>0 of a function x:(0,+) is given by (6)𝒟tαx(t)=1Γ(n-α)(ddt)n0t(t-s)n-α-1x(s)ds, where n=[α]+1 and [α] denotes the integer part of number α, provided that the right-hand side is pointwise defined on (0,+).

Proposition 3 (see [<xref ref-type="bibr" rid="B1">1</xref>–<xref ref-type="bibr" rid="B3">3</xref>]).

(1) If xL1(0,1), ν>σ>0, then (7)IνIσx(t)=Iν+σx(t),𝒟tσIνx(t)=Iν-σx(t),𝒟tσIσx(t)=x(t).

(2) If ν>0, σ>0, then (8)𝒟tνtσ-1=Γ(σ)Γ(σ-ν)tσ-ν-1.

Proposition 4 (see [<xref ref-type="bibr" rid="B1">1</xref>–<xref ref-type="bibr" rid="B3">3</xref>]).

Let α>0, and f(x) is integrable, then (9)Iα𝒟tαf(x)  =f(x)  +c1xα-1+c2xα-2++cnxα-n, where ci(i=1,2,,n) and n is the smallest integer greater than or equal to α.

Definition 5.

A continuous function ψ(t) is called a lower solution of the BVP (4), if it satisfies (10)-𝒟tβ(φp(𝒟tαψ))(t)λf(t,ψ(t)),t(0,1),    ψ(0)0,𝒟tγψ(1)j=1m-2aj𝒟tγψ(ξj),  𝒟tαψ(0)0.

Definition 6.

A continuous function ϕ(t) is called an upper solution of the BVP (4), if it satisfies (11)-𝒟tβ(φp(𝒟tαϕ))(t)λf(t,ϕ(t)),      t(0,1),ϕ(0)0,𝒟tγϕ(1)j=1m-2aj𝒟tγϕ(ξj),𝒟tαϕ(0)0.

For forthcoming analysis, we first consider the following linear fractional differential equation: (12)𝒟tαx(t)+h(t)=0,t(0,1),x(0)=0,  𝒟tγx(1)=j=1m-2aj𝒟tγx(ξj).

Lemma 7 (see [<xref ref-type="bibr" rid="B15">15</xref>]).

If 1<α2 and hL1[0,1], then the boundary value problem (12) has the unique solution (13)x(t)=01G(t,s)h(s)ds, where (14)G(t,s)=g1(t,s)+tα-11-j=1m-2ajξjα-γ-1j=1m-2ajg2(ξj,s) is the Green function of the boundary value problem (12) and (15)g1(t,s)={tα-1  (1-s)α-γ-1-  (t-s)α-1Γ(α),  0st1,tα-1(1-s)α-γ-1Γ(α),0ts1,g2(t,s)={(t(1-s))α-γ-1-(t-s)α-γ-1Γ(α),0st1,(t(1-s))α-γ-1Γ(α),  0ts1.

Lemma 8.

The Green function G(t,s) in Lemma 7 has the following properties:

G(t,s) is continuous on [0,1]×[0,1];

G(t,s)>0 for any s,t(0,1);

tα-1σ1(s)G(t,s)tα-1σ2(s), for t,s[0,1], where (16)σ1(s)=j=1m-2ajg2(ξj,s)1-j=1m-2ajξjα-γ-1,  σ2(s)=(1-s)α-γ-1Γ(α)+σ1(s).

Let q>1 satisfy the relation 1/q+1/p=1, where p is given by (4). To study BVP (4), we first consider the associated linear BVP: (17)𝒟tβ(φp(𝒟tαx))(t)+h(t)=0,t(0,1),x(0)=0,𝒟tαx(0)=0,𝒟tγx(1)=j=1m-2aj𝒟tγx(ξj), for hL1[0,1] and h0. For convenience, let (18)b=(Γ(β))1-q, then we have the following lemma.

Lemma 9.

The associated linear BVP (17) has the unique positive solution (19)x(t)=b01G(t,s)(0s(s-τ)β-1h(τ)dτ)q-1ds.

Proof.

In fact, let w=𝒟tαx,v=φp(w). By Proposition 4, the solution of initial value problem (20)𝒟tβv(t)+h(t)=0,t(0,1),v(0)=0 is given by v(t)=C1tβ-1-Iβh(t),  t[0,1]. From the relations v(0)=0,0<β1, it follows that C1=0, and so (21)v(t)=-Iβh(t),t[0,1]. Noting that 𝒟tαx=w,w=φp-1(v), it follows from (21) that the solution of (17) satisfies (22)𝒟tαx(t)=φp-1(-Iβh(t)),t(0,1),x(0)=0,𝒟tγx(1)=j=1m-2aj𝒟tγx(ξj).By Lemma 7, the solution of (22) can be written as (23)x(t)=-01G(t,s)φp-1(-Iβh(s))ds,      t[0,1]. Since h(s)0, s[0,1], we have φp-1(-Iβh(s))=-(Iβh(s))q-1, s[0,1], which implies that the solution of (22) is given by (24)x(t)=b01G(t,s)(0s(s-τ)β-1h(τ)dτ)q-1ds,01Ga(0s(s-τ)β-1h(τ)dτ)q-1t[0,1].

The following lemma is a straightforward conclusion of Lemma 9.

Lemma 10.

If xC([0,1],R) satisfies (25)x(0)=0,𝒟tγx(1)  =j=1m-2aj𝒟tγx(ξj), and -𝒟tαx(t)0 for any t(0,1), then x(t)0, for t[0,1].

3. Main Results

Set (26)e(t)=tα-1.

We present the following two assumptions.

( H 1 ) f : ( ( 0,1 ) × ( 0 , ) [ 0 , + ) ) is continuous and decreasing in x.

( H 2 ) For any μ>0,f(t,μ)0, and (27)0<01σ2(s)(0s(s-τ)β-1f(τ,μe(τ))dτ)q-1ds<+.

Let E=C[0,1], and (28)P={yE:thereexistpositivenumbers0<lx<1,Lx>1  suchthat  lxe(t)x(t)Lxe(t),a0<lx<1,Lx>1  suchthat  lxe(t)xdgt[0,1]}. Clearly, e(t)P, so P is nonempty. For any xP, define an operator T by (29)(Tλx)(t)=λb01G(t,s)×(0s(s-τ)β-1f(τ,x(τ))dτ)q-1ds,×(0s(s-τ)β-1(τ,x(τ))p)t[0,1].

Theorem 11.

Suppose conditions (H1) and (H2) hold. In addition, if the following condition (S1) holds:

( S 1 ) (30) lim κ + κ 1 / ( q - 1 ) f ( t , κ x ) = + , for (t,x)(0,1)×(0,)uniformly holds. Then there exists a constant λ*>0 such that the BVP (4) has at least one positive solution w for any λ(λ*,+), and there exists one positive constant n>1 such that (31)e(t)w(t)ne(t),t[0,1].

Proof.

The proof is divided into four steps.

Step 1. We show that Tλ is well defined on P and Tλ(P)P, and Tλ is decreasing in x.

In fact, for any xP, by the definition of P, there exists two positive numbers 0<lx<1,  Lx>1 such that lxe(t)x(t)Lxe(t) for any t[0,1]. It follows from Lemma 8 and (H1)-(H2) that (32)(Tλx)(t)=λb01G(t,s)(0s(s-τ)β-1f(τ,x(τ))dτ)q-1dsλbe(t)01σ2(s)×(0s(s-τ)β-1f(τ,lxe(τ))dτ)q-1ds<+.

Now take c=maxt[0,1]x(t), by (H2), for any s(0,1), f(s,c)0. Thus by the continuity of f(t,x) and Lemma 8 and (32), we have (33)(Tλx)(t)λbe(t)01σ1(s)×(0s(s-τ)β-1f(τ,x(τ))dτ)q-1dsλbe(t)01σ1(s)(0s(s-τ)β-1f(τ,c)dτ)q-1ds>0,          t(0,1).

Take (34)lx=min{1,λb01σ1(s)(0s(s-τ)β-1f(τ,c)dτ)q-1ds},Lx=max{(0s(s-τ)β-1f(τ,lxe(τ))dτ)1,λb×01σ2(s)(0s(s-τ)β-1f(τ,lxe(τ))dτ)q-1ds}, then by (32) and (33), (35)lxe(t)(Tλx)(t)Lxe(t), which implies that Tλ is well defined and Tλ(P)P. And the operator Tλ is decreasing in x from (H1). Moreover, by direct computations, we also have (36)-𝒟tβ(φp(𝒟tα(Tλx)))(t)=λf(t,x(t)),    t(0,1),(Tλx)(0)=0,𝒟tα(Tλx)(0)=0,𝒟tγ(Tλx)(1)=j=1m-2aj𝒟tγ(Tλx)(ξj).

Step 2. In this step, we will focus on lower and upper solutions of the fractional boundary value problem (4).

By Lemma 8, we have (37)b01G(t,s)(0s(s-τ)β-1f(τ,e(τ))dτ)q-1dsae(t)b01σ1(s)(0s(s-τ)β-1f(τ,e(τ))dτ)q-1ds,e(t)b01(0s(s-τ)β-1f(τ,e(τ))dτ)q-1t[0,1]. Let (38)μ=(b01σ1(s)(0s(s-τ)β-1f(τ,e(τ))dτ)q-1ds)-1; it follows from (37) that (39)μb01G(t,s)(0s(s-τ)β-1f(τ,e(τ))dτ)q-1dse(t),  (0s(s-τ)β-1(τ,e(τ))dτ)q-1t[0,1].

On the other hand, take (40)ν(t)=b01G(t,s)(0s(s-τ)β-1f(τ,e(τ))dτ)q-1ds; then by monotonicity of f in x and (37)–(40), for any λ>μ, we have (41)01G(t,s)(0s(s-τ)β-1f(τ,λν(τ))dτ)q-1ds01G(t,s)(0s(s-τ)β-1f(τ,μν(τ))dτ)q-1ds01σ2(s)(0s(s-τ)β-1f(τ,e(τ))dτ)q-1ds<+. From (S1), we have (42)limκ+κ1/(q-1)f(t,κx)=+, uniformly on (t,x)(0,1)×(0,). Thus there exists large enough λ*>μ>0, such that, for any t(0,1), (43)λ*1/(q-1)f(s,λ*e(s))βb(01σ1(s)sβ(q-1)ds)-1/(q-1), which yields (44)λ*b01G(t,s)(0s(s-τ)β-1f(τ,λ*e(τ))dτ)q-1dsβ(01σ1(s)sβ(q-1)ds)-1/(q-1)a×01G(t,s)(0s(s-τ)β-1dτ)q-1dsβ(01σ1(s)sβ(q-1)ds)-1/(q-1)ai×01σ1(s)(0s(s-τ)β-1dτ)q-1dse(t)=e(t),×01σ1(s)(0s(s-τ)β-1dτ)q-1ghjhgt[0,1]. Letting (45)ϕ(t)=λ*b01G(t,s)(0s(s-τ)β-1f(τ,e(τ))dτ)q-1ds=λ*ν(t)=Tλ*e,ψ(t)=λ*b01G(t,s)(0s(s-τ)β-1f(τ,λ*ν(τ))dτ)q-1ds  =Tλ*ϕ, and by Lemma 9, (39), (44), and (45), one has (46)ϕ(t)=λ*b01G(t,s)(0s(s-τ)β-1f(τ,e(τ))dτ)q-1dse(t),ϕ(0)=0,  𝒟tϕϕ(0)=0,𝒟tγ(ϕx)(1)=j=1m-2aj𝒟tγϕ(ξj),(47)ψ(t)=λ*b01G(t,s)(0s(s-τ)β-1f(τ,λ*ν(τ))dτ)q-1dse(t),ψ(0)=0,𝒟tϕψ(0)=0,𝒟tγ(ψx)(1)=j=1m-2aj𝒟tγψ(ξj).

By Step  1 and (46), (47), we know ϕ(t),ψ(t)P. And it follows from (45)–(47) that (48)e(t)ψ(t)ϕ(t),t[0,1]. Consequently, it follows from (44)–(48) that (49)𝒟tβ(φp(𝒟tαψ))(t)+λ*f(t,ψ(t))=𝒟tβ(φp(𝒟tα(Tλ*ϕ)))(t)+λ*f(t,ψ(t))=-λ*f(t,ϕ(t))+λ*f(t,ψ(t))0,𝒟tβ(φp(𝒟tαϕ))(t)+λ*f(t,ϕ(t))=𝒟tβ(φp(𝒟tα(Tλ*e)))(t)+λ*f(t,ϕ(t))=-λ*f(t,e(t))+λ*f(t,ϕ(t))0; that is, ϕ(t) and ψ(t) are a couple of lower and upper solutions of fractional boundary value problem (4) by (46)–(49), respectively.

Step 3. Let (50)F(t,x)={f(t,ψ(t)),x<ψ(t),f(t,x(t)),ψ(t)xϕ(t),f(t,ϕ(t)),  x>ϕ(t). It follows from (H1) and (46) that F:(0,1)×[0,+)[0,+) is continuous.

We will show that the fractional boundary value problem (51)-𝒟tβ(φp(𝒟tαx))(t)=λ*F(t,x(t)),      t(0,1),x(0)=0,  𝒟tαx(0)=0,𝒟tγx(1)=j=1m-2aj𝒟tγx(ξj) has a positive solution.

To see this, we consider the operator Aλ*:C[0,1]C[0,1] defined as follows: (52)(Aλ*x)(t)=λ*b01G(t,s)×(0s(s-τ)β-1F(τ,x(τ))dτ)q-1ds,0s(s-τ)β-1τ,x(τ)sdggdsit[0,1]. Obviously, a fixed point of the operator Aλ* is a solution of the BVP (51). Noting that ϕP, then there exists a constant 0<lϕ<1 such that ϕ(t)lϕe(t),  t[0,1]. Thus for all xE, it follows from Lemma 8, (50), and (H2) that (53)(Aλ*x)(t)=λ*b01σ2(s)(0s(s-τ)β-1F(τ,x(τ))dτ)q-1ds=λ*b01σ2(s)(0s(s-τ)β-1f(τ,ϕ(τ))dτ)q-1ds=λ*b01σ2(s)(0s(s-τ)β-1f(τ,lϕe(τ))dτ)q-1ds=<+, which implies that the operator Aλ* is uniformly bounded.

From the uniform continuity of G(t,s) and the Lebesgue dominated convergence theorem, we easily obtain that A is equicontinuous. Thus by the means of the Arzela-Ascoli theorem, we have that Aλ*:EE is completely continuous. The Schauder fixed point theorem implies that Aλ* has at least a fixed point w such that w=Aλ*w.

Step 4. We will prove that the boundary value problem (4) has at least one positive solution.

In fact, we only need to prove that (54)ψ(t)w(t)ϕ(t),          t[0,1]. By (46), (47) and noticing that w is fixed point of Aλ*, we know that (55)ϕ(0)=0,𝒟tγϕ(1)=j=1m-2aj𝒟tγϕ(ξj),𝒟tαϕ(0)  =0,w(0)=0,𝒟tγw(1)=j=1m-2aj𝒟tγw(ξj),𝒟tαw(0)=0. Notice that the definition of F and the function f(t,x) is nonincreasing in x, we obtain (56)f(t,ϕ(t))F(t,x(t))f(t,ψ(t)),      xE. So by (48) and (56), (57)f(t,ϕ(t))F(t,x(t))f(t,e(t)),      xE. Thus one has by (57) (58)𝒟tβ(φp(𝒟tαϕ))(t)-𝒟tβ(φp(𝒟tαw))(t)=𝒟tβ(φp(𝒟tαϕ)-φp(𝒟tαw))(t)=-λ*f(t,e(t))+λ*F(t,w(t))0,=-λ*f(t,e(t))+λ*dfgdt[0,1]. Let z(t)=φp(𝒟tαϕ(t))-φp(𝒟tαw(t)); then (59)𝒟tβz(t)0,t[0,1], and (55) implies that z(0)=0. It follows from (21) that (60)z(t)0, and then (61)φp(𝒟tαϕ(t))-φp(𝒟tαw(t))0. Notice that φp is monotone increasing; we have (62)𝒟tαϕ(t)𝒟tαw(t),that  is,𝒟tα(ϕ-w)(t)0. It follows from Lemma 10 and (55) that (63)ϕ(t)-w(t)0. Thus we have w(t)ϕ(t) on [0,1]. By the same way, we also have w(t)ψ(t) on [0,1]. So (64)ψ(t)w(t)ϕ(t),t[0,1]. Consequently, F(t,w(t))=f(t,w(t)),  t[0,1]. Then w(t) is a positive solution of the problem (4).

Finally, by (48) and (64) and ϕP, we have (65)e(t  )ψ(t)w(t)ϕ(t)lϕe(t)=ne(t), where (66)n=lϕ>1.

In the end of this work we also remark the above results to the problem (4) with which f(t,x) is nonsingular at x=0 and t=0,1; that is, we have the following result.

Theorem 12.

If f(t,x):[0,1]×[0,+)[0,+) is continuous, decreasing in x and f(t,μ)0, for any μ>0, then the boundary value problem (4) has at least one positive solution w(t) for any λ>0, and there exists a constant n>1 such that (67)e(t)w(t)ne(t).

Proof.

The proof is similar to Theorem 11; we omit it here.

Example 13.

Consider the following boundary value problem: (68)-𝒟t4/3(φ2(𝒟t3/2x))(t)=1t1/2x(t),t(0,1),x(0)=0,𝒟t1/6x(1)=18𝒟t1/6x(14)+13𝒟t1/6x(34),𝒟t3/2x(0)=0.

Let α=3/2, β=4/3, γ=1/6, p=2, and (69)f(t,x)=1xt1/2. Firstly, (70)c=j=1p-2ajξjα-γ-1=18(14)1/3+13(34)1/3=0.3816<1. And, it is easy to check that (H1) holds. For any μ>0, f(t,μ)0 and (71)0<01σ2(s)(0s(s-τ)β-1f(τ,μe(τ))dτ)q-1ds=01σ2(s)0s(s-τ)1/3τ-1/2μ-1/2dτds=01s5/6σ2(s)01(1-τ)1/3τ-1/2μ-1/2dτds=μ-1/201s5/6σ2(s)dsB(43,12)<+, which implies that (H2) holds.

On the other hand, (72)limκ+κ1/(q-1)f(t,κx)=limκ+κ1κxt1/2=+. Thus (S1) also holds.

By Theorem 11, the boundary value problem (68) has at least one positive solution.

Acknowledgments

This work was supported by the Natural Sciences of Education and the Science Office Bureau of Sichuan Province of China, under Grants nos. 10ZC060, 2010JY0J41.

Kilbas A. A. Srivastava H. M. Trujillo J. J. Theory and Applications of Fractional Differential Equations 2006 204 Amsterdam, The Netherlands Elsevier North-Holland Mathematics Studies MR2218073 10.1016/S0304-0208(06)80001-0 ZBL1206.26007 Miller K. S. Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 New York, NY, USA Wiley MR1219954 ZBL0943.82582 Podlubny I. Fractional Differential Equations 1999 198 New York, NY, USA Academic Press Mathematics in Science and Engineering MR1658022 ZBL1056.93542 Babakhani A. Daftardar-Gejji V. Existence of positive solutions of nonlinear fractional differential equations Journal of Mathematical Analysis and Applications 2003 278 2 434 442 10.1016/S0022-247X(02)00716-3 MR1974017 ZBL1027.34003 Delbosco D. Rodino L. Existence and uniqueness for a nonlinear fractional differential equation Journal of Mathematical Analysis and Applications 1996 204 2 609 625 10.1006/jmaa.1996.0456 MR1421467 ZBL0881.34005 El-Sayed A. M. A. Nonlinear functional-differential equations of arbitrary orders Nonlinear Analysis: Theory, Methods and Applications A 1998 33 2 181 186 10.1016/S0362-546X(97)00525-7 MR1621105 ZBL0934.34055 Lakshmikantham V. Theory of fractional functional differential equations Nonlinear Analysis: Theory, Methods and Applications A 2008 69 10 3337 3343 10.1016/j.na.2007.09.025 MR2450543 ZBL1162.34344 Zhang S. Existence of positive solution for some class of nonlinear fractional differential equations Journal of Mathematical Analysis and Applications 2003 278 1 136 148 10.1016/S0022-247X(02)00583-8 MR1963470 ZBL1026.34008 Zhang X. Han Y. Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations Applied Mathematics Letters 2012 25 3 555 560 10.1016/j.aml.2011.09.058 MR2856032 ZBL1244.34009 Zhang X. Liu L. Wu Y. The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives Applied Mathematics and Computation 2012 218 17 8526 8536 10.1016/j.amc.2012.02.014 MR2921344 ZBL1254.34016 Bai Z. Lv H. Positive solutions for boundary value problem of nonlinear fractional differential equation Journal of Mathematical Analysis and Applications 2005 311 2 495 505 10.1016/j.jmaa.2005.02.052 MR2168413 ZBL1079.34048 Zhang X. Liu L. Wu Y. The uniqueness of positive solution for a singular fractional differential system involving derivatives Communications in Nonlinear Science and Numerical Simulation 2013 18 1400 1409 Kaufmann E. R. Mboumi E. Positive solutions of a boundary value problem for a nonlinear fractional differential equation Electronic Journal of Qualitative Theory of Differential Equations 2008 3 1 11 MR2369417 ZBL1183.34007 Bai C. Triple positive solutions for a boundary value problem of nonlinear fractional differential equation Electronic Journal of Qualitative Theory of Differential Equations 2008 24 1 10 MR2425095 ZBL1183.34005 Li C. F. Luo X. N. Zhou Y. Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations Computers and Mathematics with Applications 2010 59 3 1363 1375 10.1016/j.camwa.2009.06.029 MR2579500 ZBL1189.34014 Wang J. Xiang H. Liu Z. Positive solutions for three-point boundary value problems of nonlinear fractional differential equations with p-Laplacian Far East Journal of Applied Mathematics 2009 37 1 33 47 MR2583953 Wang J. Xiang H. Upper and lower solutions method for a class of singular fractional boundary value problems with p-Laplacian operator Abstract and Applied Analysis 2010 2010 12 971824 10.1155/2010/971824 MR2720032 Chai G. Positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator Boundary Value Problems 2012 2012, article 18 10.1186/1687-2770-2012-18 MR2904637 Zhang X. Liu L. Wiwatanapataphee B. Wu Y. Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives Abstract and Applied Analysis 2012 2012 16 512127 10.1155/2012/512127 MR2922960 ZBL1242.34015 Zhou Y. Existence and uniqueness of solutions for a system of fractional differential equations Fractional Calculus and Applied Analysis 2009 12 2 195 204 MR2498366 Zhang X. Liu L. Wu Y. Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives Applied Mathematics and Computation 2012 219 4 1420 1433 10.1016/j.amc.2012.07.046 MR2983853 Zhou Y. Existence and uniqueness of fractional functional differential equations with unbounded delay International Journal of Dynamical Systems and Differential Equations 2008 1 4 239 244 10.1504/IJDSDE.2008.022988 MR2502297 ZBL1175.34081 Zhang X. Liu L. Wu Y. Lu Y. The iterative solutions of nonlinear fractional differential equations Applied Mathematics and Computation 2013 219 9 4680 4691 10.1016/j.amc.2012.10.082 MR3001516