Eigenvalue of Fractional Differential Equations with 𝑝-Laplacian Operator

Abstract

We investigate the existence of positive solutions for the fractional order eigenvalue problem with 𝑝-Laplacian operator −𝒟_𝑡^𝛽(𝜑_𝑝(𝒟_𝑡^𝛼𝑥))(𝑡)=𝜆𝑓(𝑡,𝑥(𝑡)),𝑡∈(0,1),𝑥(0)=0,𝒟_𝑡^𝛼𝑥(0)=0,𝒟_𝑡^𝛾𝑥(1)=∑^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝑥(𝜉_𝑗), where 𝒟_𝑡^𝛽,𝒟_𝑡^𝛼,𝒟_𝑡^𝛾 are the standard Riemann-Liouville derivatives and 𝑝-Laplacian operator is defined as 𝜑_𝑝(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. [1], Miller and Ross [2], and Podlubny [3] and the papers [4–23] and the references therein.

In [16], the authors investigated the nonlinear nonlocal boundary value problem: 𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝑥(𝑡)((𝑡(),𝑥(0)+𝑓𝑡,𝑥))=0,𝑡∈0,1=0,𝒟_𝑡^𝛼𝑥(0)=0,𝑥(1)=𝑎𝑥(𝜉), (1) where 0<𝛼≤2,0<𝛽≤1,0≤𝑎≤1,0<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 [17], 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: 𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝑥(𝑡)((𝑡(),𝑥(0)+𝑓𝑡,𝑥))=0,𝑡∈0,1=0,𝒟_𝑡^𝛼𝑥(0)(1)(𝜉),𝒟=0,𝑥=𝑎𝑥_𝑡𝛼𝑥(1)=𝑏𝒟_𝑡^𝛼𝑥𝜂, (2) where 0<𝛼,𝛽≤2,0≤𝑎,𝑏≤1,0<𝜉,𝜂<1. Recently, by means of the fixed point theorem on cones, Chai [18] investigated two-point boundary value problem of fractional differential equation with 𝑝-Laplacian operator: 𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝑥(𝑡)((𝑡(),𝑥(0)+𝑓𝑡,𝑥))=0,𝑡∈0,1=0,𝒟_𝑡^𝛼𝑥(0)=0,𝑥(1)+𝑎𝒟_𝑡^𝛾𝑥(1)=0. (3) 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: −𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝑥(𝑡)((𝑡(),𝑥(0)=𝜆𝑓𝑡,𝑥)),𝑡∈0,1=0,𝒟_𝑡^𝛼𝑥(0)𝒟=0,_𝑡𝛾𝑥(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝑥𝜉_𝑗, (4) where 𝒟_𝑡^𝛽, 𝒟_𝑡^𝛼, 𝒟_𝑡^𝛾 are the standard Riemann-Liouville derivatives with 1<𝛼≤2,0<𝛽≤1,0<𝛾≤1,0≤𝛼−𝛾−1,0<𝜉₁<𝜉₂<⋯<𝜉_𝑝−2<1,𝑎_𝑗∈[0,+∞) with 𝑐=∑^𝑚−2_𝑗=1𝑎_𝑗𝜉_𝑗^{𝛼−𝛾−1}<1, 𝑝-Laplacian operator is defined as 𝜑_𝑝(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 [1–3]). The Riemann-Liouville fractional integral of order 𝛼>0 of a function𝑥∶(0,+∞)→ℝ is given by 𝐼^𝛼𝑥(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑥(𝑠)𝑑𝑠 (5) provided that the right-hand side is pointwise defined on (0,+∞).

Definition 2 (see [1–3]). The Riemann-Liouville fractional derivative of order 𝛼>0 of a function 𝑥∶(0,+∞)→ℝ is given by 𝒟_𝑡^𝛼𝑥(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑥(𝑠)𝑑𝑠, (6) 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 [1–3]). (1) If 𝑥∈𝐿¹(0,1), 𝜈>𝜎>0, then 𝐼^𝜈𝐼^𝜎𝑥(𝑡)=𝐼^𝜈+𝜎𝑥(𝑡),𝒟_𝑡^𝜎𝐼^𝜈𝑥(𝑡)=𝐼^𝜈−𝜎𝑥(𝑡),𝒟_𝑡^𝜎𝐼^𝜎𝑥(𝑡)=𝑥(𝑡). (7)
(2) If 𝜈>0, 𝜎>0, then 𝒟_𝑡^𝜈𝑡^𝜎−1=Γ(𝜎)Γ()𝑡𝜎−𝜈^{𝜎−𝜈−1}. (8)

Proposition 4 (see [1–3]). Let 𝛼>0, and 𝑓(𝑥) is integrable, then 𝐼^𝛼𝒟_𝑡^𝛼𝑓(𝑥)=𝑓(𝑥)+𝑐₁𝑥^𝛼−1+𝑐₂𝑥^𝛼−2+⋯+𝑐_𝑛𝑥^𝛼−𝑛, (9) where 𝑐_𝑖∈ℝ(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 −𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝜓(𝑡)(𝑡)(),𝜓(0)≤𝜆𝑓𝑡,𝜓,𝑡∈0,1≥0,𝒟_𝑡^𝛾𝜓(1)≥^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝜓𝜉_𝑗,𝒟_𝑡^𝛼𝜓(0)≥0. (10)

Definition 6. A continuous function 𝜙(𝑡) is called an upper solution of the BVP (4), if it satisfies −𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝜙(𝑡)(𝑡)(),𝜙(0)≥𝜆𝑓𝑡,𝜙,𝑡∈0,1≤0,𝒟_𝑡^𝛾𝜙(1)≤^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝜙𝜉_𝑗,𝒟_𝑡^𝛼𝜙(0)≤0. (11)

For forthcoming analysis, we first consider the following linear fractional differential equation: 𝒟_𝑡^𝛼𝑥(𝑡)(𝑡)(),𝑥(0)+ℎ=0,𝑡∈0,1=0,𝒟_𝑡^𝛾𝑥(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝑥𝜉_𝑗. (12)

Lemma 7 (see [15]). If 1<𝛼≤2 and ℎ∈𝐿¹[0,1], then the boundary value problem (12) has the unique solution 𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)ℎ(𝑠)𝑑𝑠, (13) where 𝐺(𝑡,𝑠)=𝑔₁()+𝑡𝑡,𝑠^𝛼−11−∑^𝑚−2_𝑗=1𝑎_𝑗𝜉_𝑗^{𝛼−𝛾−1𝑚−2}_𝑗=1𝑎_𝑗𝑔₂𝜉_𝑗,𝑠 (14) is the Green function of the boundary value problem (12) and 𝑔₁()=𝑡𝑡,𝑠^𝛼−1(1−𝑠)^{𝛼−𝛾−1}−(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑡,0≤𝑠≤𝑡≤1,_𝛼−1(1−𝑠)^{𝛼−𝛾−1}Γ(𝛼)𝑔,0≤𝑡≤𝑠≤1,₂()=(𝑡(𝑡,𝑠1−𝑠))^{𝛼−𝛾−1}−(𝑡−𝑠)^{𝛼−𝛾−1}Γ(𝛼)(𝑡(,0≤𝑠≤𝑡≤1,1−𝑠))^{𝛼−𝛾−1}Γ(𝛼),0≤𝑡≤𝑠≤1. (15)

Lemma 8. The Green function 𝐺(𝑡,𝑠) in Lemma 7 has the following properties:(i)𝐺(𝑡,𝑠) is continuous on [0,1]×[0,1];(ii)𝐺(𝑡,𝑠)>0 for any 𝑠,𝑡∈(0,1);(iii)𝑡^𝛼−1𝜎₁(𝑠)≤𝐺(𝑡,𝑠)≤𝑡^𝛼−1𝜎₂(𝑠), for 𝑡,𝑠∈[0,1], where 𝜎₁(𝑠)=∑^𝑚−2_𝑗=1𝑎_𝑗𝑔₂𝜉_𝑗,𝑠1−∑^𝑚−2_𝑗=1𝑎_𝑗𝜉_𝑗^{𝛼−𝛾−1},𝜎₂(𝑠)=(1−𝑠)^{𝛼−𝛾−1}Γ(𝛼)+𝜎₁(𝑠). (16)

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: 𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝑥(𝑡)(𝑡)(),𝑥(0)+ℎ=0,𝑡∈0,1=0,𝒟_𝑡^𝛼𝑥(0)𝒟=0,_𝑡𝛾𝑥(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝑥𝜉_𝑗, (17) for ℎ∈𝐿¹[0,1] and ℎ≥0. For convenience, let 𝑏=Γ𝛽^1−𝑞, (18) then we have the following lemma.

Lemma 9. The associated linear BVP (17) has the unique positive solution 𝑥(𝑡)=𝑏¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑑𝜏^𝑞−1𝑑𝑠. (19)

Proof. In fact, let 𝑤=𝒟_𝑡^𝛼𝑥,𝑣=𝜑_𝑝(𝑤). By Proposition 4, the solution of initial value problem 𝒟_𝑡^𝛽𝑣(𝑡)(𝑡)(),𝑣(0)+ℎ=0,𝑡∈0,1=0 (20) is given by 𝑣(𝑡)=𝐶₁𝑡^𝛽−1−𝐼^𝛽ℎ(𝑡),𝑡∈[0,1]. From the relations 𝑣(0)=0,0<𝛽≤1, it follows that 𝐶₁=0, and so 𝑣(𝑡)=−𝐼^𝛽ℎ(𝑡),𝑡∈[0,1]. (21) Noting that 𝒟_𝑡^𝛼𝑥=𝑤,𝑤=𝜑_𝑝⁻¹(𝑣), it follows from (21) that the solution of (17) satisfies 𝒟_𝑡^𝛼𝑥(𝑡)=𝜑_𝑝⁻¹−𝐼^𝛽ℎ(𝑡)(),𝑥(0),𝑡∈0,1=0,𝒟_𝑡^𝛾𝑥(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝑥𝜉_𝑗. (22)By Lemma 7, the solution of (22) can be written as 𝑥(𝑡)=−¹₀𝐺(𝑡,𝑠)𝜑_𝑝⁻¹−𝐼^𝛽ℎ(𝑠)𝑑𝑠,𝑡∈[0,1]. (23) Since ℎ(𝑠)≥0, 𝑠∈[0,1], we have 𝜑_𝑝⁻¹(−𝐼^𝛽ℎ(𝑠))=−(𝐼^𝛽ℎ(𝑠))^𝑞−1, 𝑠∈[0,1], which implies that the solution of (22) is given by 𝑥(𝑡)=𝑏¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑑𝜏^𝑞−1[].𝑑𝑠,𝑡∈0,1 (24)

The following lemma is a straightforward conclusion of Lemma 9.

Lemma 10. If 𝑥∈𝐶([0,1],𝑅) satisfies 𝑥(0)=0,𝒟_𝑡^𝛾𝑥(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝑥𝜉_𝑗, (25) and −𝒟_𝑡^𝛼𝑥(𝑡)≥0 for any 𝑡∈(0,1), then 𝑥(𝑡)≥0, for 𝑡∈[0,1].

3. Main Results

Set 𝑒(𝑡)=𝑡^𝛼−1. (26)

We present the following two assumptions.

(H1)𝑓∶((0,1)×(0,∞)→[0,+∞)) is continuous and decreasing in 𝑥.

(H2) For any 𝜇>0,𝑓(𝑡,𝜇)≢0, and 0<¹₀𝜎₂(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓𝜏,𝜇𝑒(𝜏)𝑑𝜏^𝑞−1𝑑𝑠<+∞. (27)

Let 𝐸=𝐶[0,1], and 𝑃=𝑦∈𝐸∶thereexistpositivenumbers0<𝑙_𝑥<1,𝐿_𝑥>1suchthat𝑙_𝑥𝑒(𝑡)≤𝑥(𝑡)≤𝐿_𝑥𝑒(𝑡),𝑡∈[0,1]}. (28) Clearly, 𝑒(𝑡)∈𝑃, so 𝑃 is nonempty. For any 𝑥∈𝑃, define an operator 𝑇 by 𝑇_𝜆𝑥(𝑡)=𝜆𝑏¹₀𝐺()×𝑡,𝑠_𝑠0(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1[].𝑑𝑠,𝑡∈0,1 (29)

Theorem 11. Suppose conditions (H1) and (H2) hold. In addition, if the following condition (S1) holds:
(S1)lim_𝜅→+∞𝜅^1/(𝑞−1)𝑓(𝑡,𝜅𝑥)=+∞, (30) 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 𝑒(𝑡)≤𝑤(𝑡)≤𝑛𝑒(𝑡),𝑡∈[0,1]. (31)

Proof. The proof is divided into four steps.
Step 1. We show that 𝑇_𝜆 is well defined on 𝑃 and 𝑇_𝜆(𝑃)⊂𝑃, and 𝑇_𝜆 is decreasing in 𝑥.
In fact, for any 𝑥∈𝑃, by the definition of 𝑃, there exists two positive numbers 0<𝑙_𝑥<1,𝐿_𝑥>1 such that 𝑙_𝑥𝑒(𝑡)≤𝑥(𝑡)≤𝐿_𝑥𝑒(𝑡) for any 𝑡∈[0,1]. It follows from Lemma 8 and (H1)-(H2) that 𝑇_𝜆𝑥(𝑡)=𝜆𝑏¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1(𝑡)𝑑𝑠≤𝜆𝑏𝑒¹₀𝜎₂(𝑠)×^𝑠₀(𝑠−𝜏)^𝛽−1𝑓𝜏,𝑙_𝑥𝑒(𝜏)𝑑𝜏^𝑞−1𝑑𝑠<+∞. (32)
Now take 𝑐=max_𝑡∈[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 𝑇_𝜆𝑥(𝑡)≥𝜆𝑏𝑒(𝑡)¹₀𝜎₁(𝑠)×^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1(𝑡)𝑑𝑠≥𝜆𝑏𝑒¹₀𝜎₁(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑐)𝑑𝜏^𝑞−1().𝑑𝑠>0,𝑡∈0,1 (33)
Take 𝑙^′_𝑥=min1,𝜆𝑏¹₀𝜎₁(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑐)𝑑𝜏^𝑞−1,𝐿𝑑𝑠_′𝑥×=max1,𝜆𝑏₁₀𝜎₂(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓𝜏,𝑙_𝑥𝑒(𝜏)𝑑𝜏^𝑞−1𝑑𝑠, (34) then by (32) and (33), 𝑙^′_𝑥𝑒(𝑡)≤𝑇_𝜆𝑥(𝑡)≤𝐿^′_𝑥𝑒(𝑡), (35) which implies that 𝑇_𝜆 is well defined and 𝑇_𝜆(𝑃)⊂𝑃. And the operator 𝑇_𝜆 is decreasing in 𝑥 from (H1). Moreover, by direct computations, we also have −𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝑇_𝜆𝑥(𝑡)((𝑡(),𝑇=𝜆𝑓𝑡,𝑥)),𝑡∈0,1_𝜆𝑥(0)=0,𝒟_𝑡^𝛼𝑇_𝜆𝑥(0)𝒟=0,_𝑡𝛾𝑇_𝜆𝑥(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝑇_𝜆𝑥𝜉_𝑗. (36)
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 𝑏¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑒(𝜏))𝑑𝜏^𝑞−1(𝑡)𝑏𝑑𝑠≥𝑒¹₀𝜎₁(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑒(𝜏))𝑑𝜏^𝑞−1[].𝑑𝑠,∀𝑡∈0,1 (37) Let 𝜇=𝑏¹₀𝜎₁(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑒(𝜏))𝑑𝜏^𝑞−1𝑑𝑠⁻¹; (38) it follows from (37) that 𝜇𝑏¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑒(𝜏))𝑑𝜏^𝑞−1(𝑡),[].𝑑𝑠≥𝑒∀𝑡∈0,1 (39)
On the other hand, take 𝜈(𝑡)=𝑏¹₀𝐺(𝑡,𝑠)^s₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑒(𝜏))𝑑𝜏^𝑞−1𝑑𝑠; (40) then by monotonicity of 𝑓 in 𝑥 and (37)–(40), for any 𝜆>𝜇, we have ¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝜆𝜈(𝜏))𝑑𝜏^𝑞−1≤𝑑𝑠₁₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓𝜏,𝜇𝜈(𝜏)𝑑𝜏^𝑞−1≤𝑑𝑠₁₀𝜎₂(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑒(𝜏))𝑑𝜏^𝑞−1𝑑𝑠<+∞. (41) From (S1), we have lim_𝜅→+∞𝜅^1/(𝑞−1)𝑓(𝑡,𝜅𝑥)=+∞, (42) uniformly on (𝑡,𝑥)∈(0,1)×(0,∞). Thus there exists large enough 𝜆^∗>𝜇>0, such that, for any 𝑡∈(0,1), 𝜆^{∗1/(𝑞−1)}𝑓𝑠,𝜆^∗𝑒(s)≥𝛽𝑏¹₀𝜎₁(𝑠)𝑠^{𝛽(𝑞−1)}𝑑𝑠^{−1/(𝑞−1)}, (43) which yields 𝜆^∗𝑏¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓𝜏,𝜆^∗𝑒(𝜏)𝑑𝜏^𝑞−1𝑑𝑠≥𝛽¹₀𝜎₁(𝑠)𝑠^{𝛽(𝑞−1)}𝑑𝑠^{−1/(𝑞−1)}×¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑑𝜏^𝑞−1𝑑𝑠≥𝛽¹₀𝜎₁(𝑠)𝑠^{𝛽(𝑞−1)}𝑑𝑠^{−1/(𝑞−1)}×¹₀𝜎₁(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑑𝜏^𝑞−1(𝑡)(𝑡),[].𝑑𝑠𝑒=𝑒∀𝑡∈0,1 (44) Letting 𝜙(𝑡)=𝜆^∗𝑏¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑒(𝜏))𝑑𝜏^𝑞−1𝑑𝑠=𝜆^∗𝜈(𝑡)=𝑇_𝜆^∗𝜓(𝑡)𝑒,=𝜆^∗𝑏¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓𝜏,𝜆^∗𝜈(𝜏)𝑑𝜏^𝑞−1𝑑𝑠=𝑇_𝜆^∗𝜙, (45) and by Lemma 9, (39), (44), and (45), one has 𝜙(𝑡)=𝜆^∗𝑏¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓(𝜏,𝑒(𝜏))𝑑𝜏^𝑞−1(𝑡),𝜙(0)𝑑𝑠≥𝑒=0,𝒟_𝑡^𝜙𝜙(0)𝒟=0,_𝑡𝛾𝜙𝑥(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝜙𝜉_𝑗,𝜓(𝑡)=𝜆^∗𝑏¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓𝜏,𝜆^∗𝜈(𝜏)𝑑𝜏^𝑞−1(𝑡),𝜓(0)𝑑𝑠≥𝑒=0,𝒟_𝑡^𝜙𝜓(0)𝒟=0,_𝑡𝛾𝜓𝑥(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝜓𝜉_𝑗. (46)(47)
By Step  1 and (46), (47), we know 𝜙(𝑡),𝜓(𝑡)∈𝑃. And it follows from (45)–(47) that 𝑒(𝑡)≤𝜓(𝑡)≤𝜙(𝑡),∀𝑡∈[0,1]. (48) Consequently, it follows from (44)–(48) that 𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝜓(𝑡)+𝜆^∗𝑓𝑡,𝜓(𝑡)=𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝑇_𝜆^∗𝜙(𝑡)+𝜆^∗𝑓𝑡,𝜓(𝑡)=−𝜆^∗𝑓𝑡,𝜙(𝑡)+𝜆^∗𝑓(𝑡)𝒟𝑡,𝜓≥0,_𝑡𝛽𝜑_𝑝𝒟_𝑡^𝛼𝜙(𝑡)+𝜆^∗𝑓𝑡,𝜙(𝑡)=𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝑇_𝜆^∗𝑒(𝑡)+𝜆^∗𝑓𝑡,𝜙(𝑡)=−𝜆^∗𝑓(𝑡,𝑒(𝑡))+𝜆^∗𝑓𝑡,𝜙(𝑡)≤0; (49) 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 𝐹()=𝑓(𝑡)(𝑡),𝑓((𝑡(𝑡)(𝑡),𝑓(𝑡)(𝑡).𝑡,𝑥𝑡,𝜓,𝑥<𝜓𝑡,𝑥)),𝜓≤𝑥≤𝜙𝑡,𝜙,𝑥>𝜙 (50) It follows from (H1) and (46) that 𝐹∶(0,1)×[0,+∞)→[0,+∞) is continuous.
We will show that the fractional boundary value problem −𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝑥(𝑡)=𝜆^∗𝐹((𝑡(),𝑥(0)𝑡,𝑥)),𝑡∈0,1=0,𝒟_𝑡^𝛼𝑥(0)𝒟=0,_𝑡𝛾𝑥(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝑥𝜉_𝑗 (51) has a positive solution.
To see this, we consider the operator 𝐴_𝜆^∗∶𝐶[0,1]→𝐶[0,1] defined as follows: 𝐴_𝜆^∗𝑥(𝑡)=𝜆^∗𝑏¹₀𝐺()×𝑡,𝑠_𝑠0(𝑠−𝜏)^𝛽−1𝐹(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1[].𝑑𝑠,𝑡∈0,1 (52) Obviously, a fixed point of the operator 𝐴_𝜆^∗ is a solution of the BVP (51). Noting that 𝜙∈𝑃, then there exists a constant 0<𝑙_𝜙<1 such that 𝜙(𝑡)≥𝑙_𝜙𝑒(𝑡),𝑡∈[0,1]. Thus for all 𝑥∈𝐸, it follows from Lemma 8, (50), and (H2) that 𝐴_𝜆^∗𝑥(𝑡)≤𝜆^∗𝑏¹₀𝜎₂(𝑠)^s₀(𝑠−𝜏)^𝛽−1𝐹(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1𝑑𝑠≤𝜆^∗𝑏¹₀𝜎₂(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓𝜏,𝜙(𝜏)𝑑𝜏^𝑞−1𝑑𝑠≤𝜆^∗𝑏¹₀𝜎₂(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓𝜏,𝑙_𝜙𝑒(𝜏)𝑑𝜏^𝑞−1𝑑𝑠<+∞, (53) which implies that the operator 𝐴_𝜆^∗ 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 𝐴_𝜆^∗∶𝐸→𝐸 is completely continuous. The Schauder fixed point theorem implies that 𝐴_𝜆^∗ has at least a fixed point 𝑤 such that 𝑤=𝐴_𝜆^∗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 𝜓(𝑡)≤𝑤(𝑡)≤𝜙(𝑡),𝑡∈[0,1]. (54) By (46), (47) and noticing that 𝑤 is fixed point of 𝐴_𝜆^∗, we know that 𝜙(0)=0,𝒟_𝑡^𝛾𝜙(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝜙𝜉_𝑗,𝒟_𝑡^𝛼𝜙(0)𝑤(0)=0,=0,𝒟_𝑡^𝛾𝑤(1)=^𝑚−2_𝑗=1𝑎_𝑗𝒟_𝑡^𝛾𝑤𝜉_𝑗,𝒟_𝑡^𝛼𝑤(0)=0. (55) Notice that the definition of 𝐹 and the function 𝑓(𝑡,𝑥) is nonincreasing in 𝑥, we obtain 𝑓𝑡,𝜙(𝑡)≤𝐹(𝑡,𝑥(𝑡))≤𝑓𝑡,𝜓(𝑡),∀𝑥∈𝐸. (56) So by (48) and (56), 𝑓𝑡,𝜙(𝑡)≤𝐹(𝑡,𝑥(𝑡))≤𝑓(𝑡,𝑒(𝑡)),∀𝑥∈𝐸. (57) Thus one has by (57) 𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝜙(𝑡)−𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝑤(𝑡)=𝒟_𝑡^𝛽𝜑_𝑝𝒟_𝑡^𝛼𝜙−𝜑_𝑝𝒟_𝑡^𝛼𝑤(𝑡)=−𝜆^∗𝑓(𝑡,𝑒(𝑡))+𝜆^∗𝐹((𝑡[].𝑡,𝑤))≤0,∀𝑡∈0,1 (58) Let 𝑧(𝑡)=𝜑_𝑝(𝒟_𝑡^𝛼𝜙(𝑡))−𝜑_𝑝(𝒟_𝑡^𝛼𝑤(𝑡)); then 𝒟_𝑡^𝛽𝑧(𝑡)≤0,𝑡∈[0,1], (59) and (55) implies that 𝑧(0)=0. It follows from (21) that 𝑧(𝑡)≤0, (60) and then 𝜑_𝑝𝒟_𝑡^𝛼𝜙(𝑡)−𝜑_𝑝𝒟_𝑡^𝛼𝑤(𝑡)≤0. (61) Notice that 𝜑_𝑝 is monotone increasing; we have 𝒟_𝑡^𝛼𝜙(𝑡)≤𝒟_𝑡^𝛼𝑤(𝑡),thatis,𝒟_𝑡^𝛼𝜙−𝑤(𝑡)≤0. (62) It follows from Lemma 10 and (55) that 𝜙(𝑡)−𝑤(𝑡)≥0. (63) 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 𝜓(𝑡)≤𝑤(𝑡)≤𝜙(𝑡),𝑡∈[0,1]. (64) 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 𝜙∈𝑃, we have 𝑒(𝑡)≤𝜓(𝑡)≤𝑤(𝑡)≤𝜙(𝑡)≤𝑙_𝜙𝑒(𝑡)=𝑛𝑒(𝑡), (65) where 𝑛=𝑙_𝜙>1. (66)

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 𝑒(𝑡)≤𝑤(𝑡)≤𝑛𝑒(𝑡). (67)

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

Example 13. Consider the following boundary value problem: −𝒟_𝑡^4/3𝜑₂𝒟_𝑡^3/2𝑥(𝑡)=1𝑡^1/2𝑥(𝑡)(),𝑥(0),𝑡∈0,1=0,𝒟_𝑡^1/6𝑥(1)=18𝒟_𝑡^1/6𝑥14+13𝒟_𝑡^1/6𝑥34,𝒟_𝑡^3/2𝑥(0)=0. (68)

Let 𝛼=3/2, 𝛽=4/3, 𝛾=1/6, 𝑝=2, and 𝑓()=1𝑡,𝑥𝑥𝑡^1/2. (69) Firstly, 𝑐=^𝑝−2_𝑗=1𝑎_𝑗𝜉_𝑗^{𝛼−𝛾−1}=1814^1/3+1334^1/3=0.3816<1. (70) And, it is easy to check that (H1) holds. For any 𝜇>0, 𝑓(𝑡,𝜇)≢0 and 0<¹₀𝜎₂(𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑓𝜏,𝜇𝑒(𝜏)𝑑𝜏^𝑞−1=𝑑𝑠₁₀𝜎₂(𝑠)^𝑠₀(𝑠−𝜏)^1/3𝜏^−1/2𝜇^−1/2𝑑𝜏𝑑𝑠=¹₀𝑠^5/6𝜎₂(𝑠)¹₀(1−𝜏)^1/3𝜏^−1/2𝜇^−1/2𝑑𝜏𝑑𝑠=𝜇^−1/21₀𝑠^5/6𝜎₂(𝑠)4𝑑𝑠𝐵3,12<+∞, (71) which implies that (H2) holds.

On the other hand, lim_𝜅→+∞𝜅^1/(𝑞−1)𝑓(𝑡,𝜅𝑥)=lim_𝜅→+∞𝜅1𝜅𝑥𝑡^1/2=+∞. (72) 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.