Positive Solutions to a Fractional-Order Two-Point Boundary Value Problem with 𝑝-Laplacian Operator

Abstract

This paper systematically investigates positive solutions to a kind of two-point boundary value problem (BVP) for nonlinear fractional differential equations with 𝑝-Laplacian operator and presents a number of new results. First, the considered BVP is converted to an operator equation by using the property of the Caputo derivative. Second, based on the operator equation and some fixed point theorems, several sufficient conditions are presented for the nonexistence, the uniqueness, and the multiplicity of positive solutions. Finally, several illustrative examples are given to support the obtained new results. The study of illustrative examples shows that the obtained results are effective.

1. Introduction

Fractional differential equation has recently attracted many scholars’ interest due to its wide applications [1–3] in engineering, technology, biology, chemical process, and so on. The first issue for the theory of fractional differential equations is the existence of solutions to kinds of boundary value problems (BVPs), which has been studied recently by many scholars, and lots of excellent results have been obtained [4–17] by means of fixed point theorems, upper and lower solutions technique, and so forth.

As an important branch of BVPs, 𝑝-Laplacian equation was firstly introduced in [18] to model the following turbulent flow in a porous medium: 𝜑_𝑝𝑢^′(𝑡)^′=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡), (1) where Φ_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1, Φ_𝑝⁻¹=Φ_𝑞, and 1/𝑝+1/𝑞=1. Then, it was investigated in both integer-order BVPs [19, 20] and fractional BVPs [21–25]. In [21], Chen and Liu considered the antiperiodic boundary value problem of fractional differential equation with 𝑝-Laplacian operator and obtained the existence of one solution by using Schaefer’s fixed point theorem under certain nonlinear growth conditions. Han et al. [22] investigated a class of fractional boundary value problem with 𝑝-Laplacian operator and boundary parameter and presented several existence results for a positive solution in terms of the boundary parameter. It is noted that although there exist several results on the existence of one solution to fractional 𝑝-Laplacian BVPs, there are, to our best knowledge, relatively few results on the nonexistence, the uniqueness, and the multiplicity of positive solutions to fractional 𝑝-Laplacian BVPs.

In this paper, we study the following two-point boundary value problem of nonlinear fractional differential equations with 𝑝-Laplacian operator: 𝐷^𝛽₀₊Φ_𝑝𝐷^𝛼₀₊𝑥(𝑡)((𝑡[],𝑥(0)(1)=𝑓𝑡,𝑥)),𝑡∈0,1=𝛾𝑥,𝐷^𝛼₀₊𝑥(0)=𝜂𝐷^𝛼₀₊𝑥(1), (2) where Φ_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1, Φ_𝑝⁻¹=Φ_𝑞, 1/𝑝+1/𝑞=1, 0<𝛼, 𝛽≤1, 1<𝛼+𝛽≤2, 0<𝛾, 𝜂<1, 𝑓∶[0,1]×[0,+∞)→[0,+∞), and 𝐷^𝛼₀₊ is the Caputo derivative. We first convert BVP (2) into an equivalent operator equation by using the property of the Caputo derivative and then present some sufficient conditions for the nonexistence, the uniqueness, and the multiplicity of positive solutions to this problem. Finally, we give several illustrative examples to support our new results.

The main contributions of this paper are as follows. (i) We systematically study the nonexistence, the uniqueness, and the multiplicity of positive solutions to BVP (2) and propose some techniques to deal with fractional 𝑝-Laplacian BVPs, which enriches this academic area. (ii) We present a sufficient condition for the existence of three positive solutions to BVP (2) by introducing a free constant to construct a proper concave functional. The main feature of this free constant is that one can weaken the conditions by regulating the free constant (see Remark 12).

Throughout this paper, we consider BVP (2) in the real Banach space 𝐸=𝐶[0,1] with the norm ‖𝑥‖=max_𝑡∈[0,1]|𝑥(𝑡)|. Let 𝑃={𝑥∈𝐸∶𝑥(𝑡)≥0, for all 𝑡∈[0,1]}. Then, 𝑃 is a normal solid cone of 𝐸 with 𝑃^∘={𝑥∈𝐸∶𝑥(𝑡)>0, for all 𝑡∈[0,1]}. A solution 𝑥(𝑡)∈𝐸 is said to be a positive solution to BVP (2), if 𝑥∈𝑃 and 𝑥(𝑡)≢0. We will study the existence of positive solutions to BVP (2) in 𝑃.

The rest of this paper is structured as follows. Section 2 contains some preliminaries on the Caputo derivative. Section 3 investigates the nonexistence, the uniqueness, and the multiplicity of positive solutions to BVP (2) and presents the main results of this paper. In Section 4, three illustrative examples are worked out to support our obtained results.

2. Preliminaries

In this section, we give some necessary preliminaries on the Caputo derivative, which will be used in the sequel. For details, please refer to [1–3] and the references therein.

Definition 1 (see [3]). The Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑦∶(0,+∞)→ℝ is given by 𝐼^𝛼₀₊𝑦(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠, (3) provided that the right side is pointwise defined on (0,+∞).

Definition 2 (see [3]). The Caputo fractional derivative of order 𝛼>0 of a continuous function 𝑦∶(0,+∞)→ℝ is given by 𝐷^𝛼₀₊𝑦(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑦^(𝑛)(𝑠)(𝑡−𝑠)^{𝛼−𝑛+1}𝑑𝑠, (4) where 𝑛 is the smallest integer greater than or equal to 𝛼, provided that the right side is pointwise defined on (0,+∞).

One can easily obtain the following property from the definition of the Caputo derivative.

Proposition 3 (see [3]). Let 𝛼>0. Assume that 𝑦, 𝐷^𝛼₀₊𝑦∈𝐿(0,1). Then, the following equality holds: 𝐼^𝛼₀₊𝐷^𝛼₀₊𝑦(𝑡)=𝑦(𝑡)+𝐶₀+𝐶₁𝑡+⋯+𝐶_𝑛−1𝑡^𝑛−1, (5) for some 𝐶_𝑖∈ℝ, 𝑖=0,1,…,𝑛−1, where 𝑛 is the smallest integer greater than or equal to 𝛼.

3. Main Results

In this section, we first convert BVP (2) into an equivalent operator equation and then present some new results on the nonexistence, the uniqueness, and the multiplicity of positive solutions to BVP (2).

Firstly, we convert BVP (2) into an equivalent operator equation.

Theorem 4. Given that ℎ∈𝐶[0,1], the unique solution of 𝐷^𝛽₀₊Φ_𝑝𝐷^𝛼₀₊𝑥(𝑡)(𝑡)[],𝑥(0)(1)=ℎ,𝑡∈0,1=𝛾𝑥,𝐷^𝛼₀₊𝑥(0)=𝜂𝐷^𝛼₀₊𝑥(1) (6) is 𝑥(𝑡)=𝐼^𝛼₀₊Φ_𝑞𝐼^𝛽₀₊ℎ(𝑡)(𝑡)(𝑡)=1+𝐴ℎ+𝐵ℎΓ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)(𝑠(𝑡),𝑑𝜏+𝐴ℎ)𝑑𝑠+𝐵ℎ (7) where (𝑡)=Φ𝐴ℎ^𝑝𝜂1−Φ_𝑝𝜂𝐼^𝛽₀₊ℎ(𝑡)_𝑡=1=Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝑠)^𝛽−1ℎ(𝑠)[],(𝑡)=𝛾𝑑𝑠,∀𝑡∈0,1𝐵ℎ𝐼1−𝛾^𝛼0+Φ_𝑞𝐼^𝛽₀₊ℎ(𝑡)+𝐴ℎ(𝑡)_𝑡=1=𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)(𝑠)[].𝑑𝜏+𝐴ℎ𝑑𝑠,∀𝑡∈0,1 (8)

Proof. Assume that 𝑥(𝑡) satisfies (6). Then, from Proposition 3, we have Φ_𝑝𝐷^𝛼₀₊𝑥(𝑡)=𝐼^𝛽₀₊ℎ(𝑡)+𝑐₀,𝑐₀∈ℝ. (9)
From the boundary value condition 𝐷^𝛼₀₊𝑥(0)=𝜂𝐷^𝛼₀₊𝑥(1), one can see that 𝑐₀=Φ_𝑝𝜂1−Φ_𝑝𝜂𝐼^𝛽₀₊ℎ(𝑡)_𝑡=1=𝐴ℎ(𝑡). (10) Thus, we have 𝑥(𝑡)=𝐼^𝛼₀₊Φ_𝑞𝐼^𝛽₀₊ℎ(𝑡)+𝐴ℎ(𝑡)+𝑐₁,𝑐₁∈ℝ, (11) which together with the boundary value condition 𝑥(0)=𝛾𝑥(1) yields that 𝑐₁=𝛾𝐼1−𝛾^𝛼0+Φ_𝑞𝐼^𝛽₀₊ℎ(𝑡)+𝐴ℎ(𝑡)_𝑡=1=𝐵ℎ(𝑡). (12)
The proof is completed.

Lemma 5. Consider that 𝑥(𝑡)∈𝐸 is a solution to BVP (2), if and only if 𝑥(𝑡)=𝑇𝑥(𝑡), where 𝑇𝑥(𝑡)=𝐼^𝛼₀₊Φ_𝑞𝐼^𝛽₀₊(𝑡)(𝑡)(𝑡)=1𝑁𝑥+𝐴𝑁𝑥+𝐵𝑁𝑥Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓((𝜏+𝛾𝜏,𝑥))𝑑𝜏𝑑𝑠11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓(𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠, (13) and 𝑁𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)).

Proof. The necessity is obvious. Next, we prove the sufficiency.
In fact, if 𝑥(𝑡)=𝑇𝑥(𝑡), a straightforward calculation shows that 𝑥(0)=𝛾𝑥(1).
Since 𝐷^𝛼₀₊𝑥(𝑡)=Φ_𝑞𝐼^𝛽₀₊𝑁𝑥(𝑡)+𝐴𝑁𝑥(𝑡)=Φ_𝑞1Γ𝛽^𝑡₀(𝑡−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓(𝜏,𝑥(𝜏))𝑑𝜏, (14) we have 𝐷^𝛼₀₊𝑥(0)=𝜂𝐷^𝛼₀₊𝑥(1), and 𝐷^𝛽₀₊Φ_𝑝𝐷^𝛼₀₊𝑥(𝑡)=𝐷^𝛽₀₊Φ_𝑝Φ_𝑞𝐼^𝛽₀₊𝑁𝑥(𝑡)+𝐴𝑁𝑥(𝑡)=𝐷^𝛽₀₊𝐼^𝛽₀₊𝑓((𝑡(𝑡)((𝑡.𝑡,𝑥))+𝐴𝑁𝑥=𝑓𝑡,𝑥)) (15)
Therefore, 𝑥(𝑡) is a solution to BVP (2).

In the following, we consider the nonexistence of positive solutions to BVP (2) and present the following result.

Theorem 6. Consider BVP (2). Assume that 𝑓(𝑡,𝑥)≤𝑚₁𝑥^𝑝−1,∀(𝑡,𝑥)∈[0,1]×[0,+∞), (16) where 𝑚₁<(1−𝛾)Γ(𝛼+1)(1−Φ_𝑝(𝜂))^𝑞−1(Γ(𝛽+1))^𝑞−1 is a positive constant. Then, BVP (2) has no positive solution.

Proof. In fact, if BVP (2) has a positive solution 𝑥(𝑡)∈𝑃 with 𝑥(𝑡)≢0. Then, one can see from (16) that |𝑥(𝑡)|=|𝑇𝑥(𝑡)|=𝐼^𝛼₀₊Φ_𝑞𝐼^𝛽₀₊(𝑡)(𝑡)(𝑡)=1𝑁𝑥+𝐴𝑁𝑥+𝐵𝑁𝑥Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓((𝜏+𝛾𝜏,𝑥))𝑑𝜏𝑑𝑠11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓(𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠≤𝑚₁‖𝑥‖1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1+Φ𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑑𝜏𝑑𝑠+𝑚₁‖𝑥‖𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1+Φ𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑑𝜏𝑑𝑠=𝑚₁‖𝑥‖1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝑠𝛽+1^𝛽+Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1𝑑𝑠+𝑚₁‖𝑥‖𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝑠𝛽+1^𝛽+Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1𝑑𝑠≤𝑚₁‖𝑥‖1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1𝑑𝑠+𝑚₁‖𝑥‖𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1𝑑𝑠=𝑚₁‖𝑥‖1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝑑𝑠Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1+𝑚₁‖𝑥‖𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1𝑑𝑠Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1=𝑚₁‖𝑥‖1Γ()𝑡𝛼+1^𝛼Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1+𝑚₁‖𝑥‖𝛾11−𝛾Γ(𝛼)Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1≤𝑚₁‖𝑥‖×1Γ(𝛼+1)1−Φ_𝑝𝜂^𝑞−1Γ𝛽+1^𝑞−1+𝑚₁‖𝑥‖×𝛾1−𝛾Γ(𝛼+1)1−Φ_𝑝𝜂^𝑞−1Γ𝛽+1^𝑞−1≤𝑚₁‖𝑥‖×11−𝛾Γ(𝛼+1)1−Φ_𝑝𝜂^𝑞−1Γ𝛽+1^𝑞−1<‖𝑥‖,∀𝑡∈[0,1], (17) which is a contradiction. Hence, BVP (2) has no positive solution.

Next, we study the existence of a unique positive solution to BVP (2). To this end, we need the following fixed point theorem [26].

Definition 7. Let 𝑃 be a solid cone in a real Banach space 𝐸, 𝑇∶𝑃^∘→𝑃^∘ an operator, and 0≤𝜃<1. Then, 𝑇 is called a 𝜃-concave operator if 𝑇(𝑘𝑢)≥𝑘^𝜃𝑇𝑢,∀0<𝑘<1,𝑢∈𝑃^∘. (18)

Lemma 8. Assume that 𝑃 is a normal solid cone in a real Banach space 𝐸, 0≤𝜃<1, and 𝑇∶𝑃^∘→𝑃^∘ is a 𝜃-concave increasing operator. Then 𝑇 has only one fixed point in 𝑃^∘.

Based on Lemma 8, we have the following result.

Theorem 9. Consider BVP (2). Assume that for all 𝑡∈[0,1], 𝑓∶[0,1]×(0,+∞)→(0,+∞) is increasing in 𝑥, and there exists a constant 0≤𝜃<1 such that 𝑓(𝑡,𝑘𝑥)≥𝑘^{(𝑝−1)𝜃}𝑓(),∀()∈[]×()𝑡,𝑥𝑡,𝑥0,10,+∞,∀0<𝑘<1. (19) Then, BVP (2) has a unique positive solution.

Proof. Since 𝑓∶[0,1]×(0,+∞)→(0,+∞), one can see that 𝑇∶𝑃^∘→𝑃^∘.
For any 𝑥₁,𝑥₂∈𝑃^∘ with 𝑥₁≤𝑥₂, from the monotonicity of 𝑓 and 𝑥^𝑞−1, we have 𝑇𝑥₂(𝑡)−𝑇𝑥₁(𝑡)≥0,∀𝑡∈[0,1]. (20) Thus, 𝑇 is increasing in 𝑃^∘.
Next, we prove that 𝑇 is a 𝜃-concave operator.
In fact, from (19), for any 0<𝑘<1, 𝑥∈𝑃^∘, it is easy to see that 𝑇(𝑡)=1𝑘𝑥)(Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑘𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓((𝜏+𝛾𝜏,𝑘𝑥))𝑑𝜏𝑑𝑠11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑘𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓(𝜏,𝑘𝑥(𝜏))𝑑𝜏𝑑𝑠≥𝑘^𝜃1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓(𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠+𝑘^𝜃𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓(𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠=𝑘^𝜃𝑇(𝑥)(𝑡), (21) which implies that 𝑇 is a 𝜃-concave operator.
By Lemma 8, BVP (2) has a unique positive solution.

Finally, we investigate the multiplicity of positive solutions to BVP (2).

We first recall the famous Leggett-Williams fixed point theorem [26].

Let 𝐸=(𝐸,‖⋅‖) be a Banach space and 𝑃⊂𝐸 be a cone on 𝐸. A continuous mapping 𝜔∶𝑃→[0,+∞) is said to be a concave nonnegative continuous functional on 𝑃, if 𝜔 satisfies 𝜔(𝜆𝑥+(1−𝜆)𝑦)≥𝜆𝜔(𝑥)+(1−𝜆)𝜔(𝑦) for all 𝑥,𝑦∈𝑃 and 𝜆∈[0,1].

Let 𝑎, 𝑏, 𝑑>0 be constants. Define 𝑃_𝑑={𝑥∈𝑃∶‖𝑥‖<𝑑}, 𝑃_𝑑={𝑥∈𝑃∶‖𝑥‖≤𝑑}, and 𝑃(𝜔,𝑎,𝑏)={𝑥∈𝑃∶𝜔(𝑥)≥𝑎,‖𝑥‖≤𝑏}.

Lemma 10. Let 𝐸=(𝐸,‖⋅‖) be a Banach space, 𝑃⊂𝐸 a cone of 𝐸, and 𝑐>0 a constant. Suppose that there exists a concave nonnegative continuous functional 𝜔 on 𝑃 with 𝜔(𝑥)≤‖𝑥‖ for all 𝑥∈𝑃_𝑐. Let 𝑇∶𝑃_𝑐→𝑃_𝑐 be a completely continuous operator. Assume that there are numbers 𝑎, 𝑏, and 𝑑 with 0<𝑑<𝑎<𝑏≤𝑐, such that(i){𝑥∈𝑃(𝜔,𝑎,𝑏)∶𝜔(𝑥)>𝑎}≠∅ and 𝜔(𝑇𝑥)>𝑎 for all 𝑥∈𝑃(𝜔,𝑎,𝑏);(ii)‖𝑇𝑥‖<𝑑 for all 𝑥∈𝑃_𝑑;(iii)𝜔(𝑇𝑥)>𝑎 for all 𝑥∈𝑃(𝜔,𝑎,𝑐) with ‖𝑇𝑥‖>𝑏.
Then, 𝑇 has at least three fixed points 𝑥₁, 𝑥₂, and 𝑥₃ in 𝑃_𝑐. Furthermore, 𝑥₁∈𝑃_𝑎; 𝑥₂∈{𝑥∈𝑃(𝜔,𝑎,𝑐)∶𝜔(𝑥)>𝑎}; 𝑥₃∈𝑃_𝑐⧵(𝑃(𝜔,𝑏,𝑐)∪𝑃_𝑎).

Now, based on Lemma 10, we consider the existence of three positive solutions to BVP (2).

Let 𝜉∈[0,1) be a constant. Define 𝜔(𝑥)=min_{𝑡∈[𝜉,1]}𝑥(𝑡), for all 𝑥∈𝑃. Then, one can easily see that 𝜔 is a concave non-negative continuous functional. We have the following result.

Theorem 11. Consider BVP (2). Suppose that 𝑓∈𝐶([0,1]×[0,+∞)→[0,+∞)). Assume that there exist positive constants 𝑎 and 𝑐 with 𝑎<𝜂(1−𝜉)^{𝛽(𝑞−1)}[(1−𝛾)𝜉^𝛼+𝛾(1−𝜉)^𝛼]𝑐, such that (H1) there exists a constant 0<𝑚₂≤(1−𝛾)Γ(𝛼+1)(Γ(𝛽+1))^𝑞−1(1−Φ_𝑝(𝜂))^𝑞−1, such that 𝑓(𝑡,𝑥)≤𝑚₂𝑐^𝑝−1,∀(𝑡,𝑥)∈[0,1]×[0,𝑐]; (22) (H2) there exists a constant (1−𝛾)Γ(𝛼+1) (Γ(𝛽+1))^𝑞−1(1−Φ_𝑝(𝜂))^𝑞−1 / 𝜂(1−𝜉)^{𝛽(𝑞−1)} [(1−𝛾)𝜉^𝛼+𝛾(1−𝜉)^𝛼]≤𝑚₃<𝑚₂𝑐/𝑎, such that 𝑓(𝑡,𝑥)>𝑚₃𝑎^𝑝−1,∀(𝑡,𝑥)∈𝜉,1×[𝑎,𝑐]; (23) (H3) there exists a constant 0<𝑚₄<(1−𝛾)^𝑝−1(Γ(𝛼+1))^𝑝−1Γ(𝛽+1)(1−Φ_𝑝(𝜂)), such that lim_𝑥→0+sup_[]𝑡∈0,1𝑓(𝑡,𝑥)𝑥^𝑝−1=𝑚₄. (24)Then, BVP (2) has at least three positive solutions.

Proof. Let us divide the proof into 4 steps.
Step 1. By (H1) and Lemma 5, for any 𝑥∈𝑃_𝑐, we have |𝑥(𝑡)|=|𝑇𝑥(𝑡)|=𝐼^𝛼₀₊Φ_𝑞𝐼^𝛽₀₊(𝑡)(𝑡)(𝑡)=1𝑁𝑥+𝐴𝑁𝑥+𝐵𝑁𝑥Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓((𝜏+𝛾𝜏,𝑥))𝑑𝜏𝑑𝑠11−𝛾Γ(𝛼)₁₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓(𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠≤𝑚₂𝑐1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1+Φ𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑑𝜏𝑑𝑠+𝑚₂𝑐𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1+Φ𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽¹₀(1−𝜏)^𝛽−1𝑑𝜏𝑑𝑠=𝑚₂𝑐1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝑠𝛽+1^𝛽+Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1𝑑𝑠+𝑚₂𝑐𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝑠𝛽+1^𝛽+Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1𝑑𝑠≤𝑚₂𝑐1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1𝑑𝑠+𝑚₂𝑐𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1𝑑𝑠=𝑚₂𝑐1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑑𝑠Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1+𝑚₂𝑐𝛾11−𝛾Γ(𝛼)¹₀(1−𝑠)^𝛼−1𝑑𝑠Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1=𝑚₂𝑐1Γ()𝑡𝛼+1^𝛼Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1+𝑚₂𝑐𝛾11−𝛾Γ(𝛼)Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽+1≤𝑚₂𝑐×1Γ(𝛼+1)1−Φ_𝑝𝜂^𝑞−1Γ𝛽+1^𝑞−1+𝑚₂𝑐×𝛾1−𝛾Γ(𝛼+1)1−Φ_𝑝𝜂^𝑞−1Γ𝛽+1^𝑞−1≤𝑚₂𝑐×11−𝛾Γ(𝛼+1)1−Φ_𝑝𝜂^𝑞−1Γ𝛽+1^𝑞−1≤𝑐,∀𝑡∈[0,1], (25)
which implies that ‖𝑇𝑥‖≤𝑐. Thus, 𝑇∶𝑃_𝑐→𝑃_𝑐.
Next, let us show that 𝑇∶𝑃_𝑐→𝑃_𝑐 is completely continuous.
By the continuity of 𝑓 and 𝑢^𝑞−1, we can get that 𝑇∶𝑃_𝑐→𝑃_𝑐 is continuous and 𝑇(𝑃_𝑐) is bounded. Moreover, there exists a constant 𝑀>0 such that |𝐼^𝛽₀₊𝑁𝑥(𝑡)+𝐴𝑁𝑥(𝑡)|≤𝑀, for all 𝑥∈𝑃_𝑐, 𝑡∈[0,1].
In addition, for any 𝑡₁,𝑡₂∈[0,1] and 𝑥∈𝑃_𝑐, we have 𝑇𝑥𝑡₂−𝑇𝑥𝑡₁=𝐼^𝛼₀₊Φ_𝑞𝐼^𝛽₀₊𝑁𝑥(𝑡)+𝐴𝑁𝑥(𝑡)_𝑡=𝑡2−𝐼^𝛼₀₊Φ_𝑞𝐼^𝛽₀₊𝑁𝑥(𝑡)+𝐴𝑁𝑥(𝑡)_𝑡=𝑡2=1Γ(𝛼)^𝑡2₀𝑡₂−𝑠^𝛼−1Φ_𝑞𝐼^𝛽₀₊(𝑠)(𝑠)−𝑁𝑥+𝐴𝑁𝑥𝑑𝑠_𝑡10𝑡₁−𝑠^𝛼−1Φ_𝑞𝐼^𝛽₀₊(𝑠)(𝑠)=1𝑁𝑥+𝐴𝑁𝑥𝑑𝑠Γ(𝛼)^𝑡1₀𝑡₂−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1Φ_𝑞𝐼^𝛽₀₊(𝑠)(𝑠)+𝑁𝑥+𝐴𝑁𝑥𝑑𝑠_𝑡2𝑡1𝑡₂−𝑠^𝛼−1Φ_𝑞𝐼^𝛽₀₊(𝑠)(𝑠)≤𝑀𝑁𝑥+𝐴𝑁𝑥𝑑𝑠_𝑞−1Γ(𝛼)^𝑡1₀𝑡₁−𝑠^𝛼−1−𝑡₂−𝑠^𝛼−1+𝑑𝑠_𝑡2𝑡1𝑡₂−𝑠^𝛼−1=𝑀𝑑𝑠_𝑞−1Γ()𝑡𝛼+1^𝛼1−𝑡^𝛼₂+2𝑡₂−𝑡₁^𝛼𝑡⟶0,₂−𝑡₁⟶0. (26)
Thus, the Arzela-Ascoli theorem guarantees that 𝑇∶𝑃_𝑐→𝑃_𝑐 is compact. Therefore, 𝑇∶𝑃_𝑐→𝑃_𝑐 is completely continuous.
Step 2. Choose a constant 𝑏=𝑐. Let 𝑥₀(𝑡)=(𝑎+𝑏)/2, for all 𝑡∈[0,1]. Then, 𝜔(𝑥₀)=(𝑎+𝑏)/2>𝑎 and ‖𝑥₀‖=(𝑎+𝑏)/2<𝑏. Thus, 𝑥₀∈{𝑥∈𝑃(𝜔,𝑎,𝑏)∶𝜔(𝑥)>𝑎}≠∅.
Now, let us prove that 𝜔(𝑇𝑥)>𝑎 holds for all 𝑥∈𝑃(𝜔,𝑎,𝑏). In fact, 𝑥∈𝑃(𝜔,𝑎,𝑏) implies that 𝑎≤𝑥(𝑡)≤𝑏, for all 𝑡∈[𝜉,1]. For any 𝑡∈[𝜉,1], one can obtain from (H2) that 𝑇𝑥(𝑡)≥𝐼^𝛼₀₊Φ_𝑞((𝑡(𝑡)≥1𝐴𝑁𝑥))+𝐵𝑁𝑥Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹_𝜉(1−𝜏)^𝛽−1𝑓((𝜏+𝛾𝜏,𝑥))𝑑𝜏𝑑𝑠11−𝛾Γ(𝛼)×¹_𝜉(1−𝑠)^𝛼−1Φ_𝑞Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹_𝜉(1−𝜏)^𝛽−1𝑓((𝜏>1𝜏,𝑥))𝑑𝜏𝑑𝑠Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹_𝜉(1−𝜏)^𝛽−1𝑚₃𝑎^𝑝−1+𝛾𝑑𝜏𝑑𝑠11−𝛾Γ(𝛼)×¹_𝜉(1−𝑠)^𝛼−1Φ_𝑞Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹_𝜉(1−𝜏)^𝛽−1𝑚₃𝑎^𝑝−1=𝑚𝑑𝜏𝑑𝑠₃𝑎Γ()𝑡𝛼+1^𝛼Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ()𝛽+11−𝜉^𝛽𝑞−1+𝛾𝑚1−𝛾³𝑎Γ()()𝛼+11−𝜉^𝛼×Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ()𝛽+11−𝜉^𝛽𝑞−1≥𝑚₃𝑎Γ()𝜉𝛼+1^𝛼Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ()𝛽+11−𝜉^𝛽𝑞−1+𝛾𝑚1−𝛾³𝑎Γ()()𝛼+11−𝜉^𝛼×Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ()𝛽+11−𝜉^𝛽𝑞−1=Φ_𝑝𝜂^𝑞−1(1−𝜉)^{𝛽(𝑞−1)}1−𝛾𝜉^𝛼+𝛾(1−𝜉)^𝛼1−𝛾Γ(𝛼+1)Γ𝛽+1^𝑞−11−Φ_𝑝𝜂^𝑞−1×𝑚₃𝑎≥𝑎. (27) Hence, condition (i) of Lemma 10 holds.
Step 3. It is easy to see from (H3) that for all 𝑡∈[0,1], for all 0 < 𝜀 < (1−𝛾)^𝑝−1(Γ(𝛼+1))^𝑝−1Γ(𝛽+1)(1−Φ_𝑝(𝜂))−𝑚₄, there exists 𝛿>0, such that 𝑓(𝑡,𝑥)<𝑚₄+𝜀𝑥^𝑝−1,0≤𝑥<𝛿. (28) Let 0<𝑑<min{𝛿,𝑎}. Now, we prove that ‖𝑇𝑥‖<𝑑 for all 𝑥∈𝑃_𝑑={𝑥∈𝑃∶‖𝑥‖≤𝑑}. As a matter of fact, for all 𝑥∈𝑃_𝑑, one can see that |𝑥(𝑡)|=|𝑇𝑥(𝑡)|=𝐼^𝛼₀₊Φ_𝑞𝐼^𝛽₀₊(𝑡)(𝑡)(𝑡)=1𝑁𝑥+𝐴𝑁𝑥+𝐵𝑁𝑥Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓((𝜏+𝛾𝜏,𝑥))𝑑𝜏𝑑𝑠11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1𝑓((𝜏+Φ𝜏,𝑥))𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1𝑓((𝜏≤𝑚𝜏,𝑥))𝑑𝜏𝑑𝑠₄+𝜀^𝑞−1‖𝑥‖1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1+Φ𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1+𝑚𝑑𝜏𝑑𝑠₄+𝜀^𝑞−1‖𝑥‖𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝛽^𝑠₀(𝑠−𝜏)^𝛽−1+Φ𝑑𝜏_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽×¹₀(1−𝜏)^𝛽−1=𝑚𝑑𝜏𝑑𝑠₄+𝜀^𝑞−1‖𝑥‖1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ𝑠𝛽+1^𝛽+Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ+𝑚𝛽+1𝑑𝑠₄+𝜀^𝑞−1‖𝑥‖𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ𝑠𝛽+1^𝛽+Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ≤𝑚𝛽+1𝑑𝑠₄+𝜀^𝑞−1‖𝑥‖1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ+𝑚𝛽+1𝑑𝑠₄+𝜀^𝑞−1‖𝑥‖𝛾11−𝛾Γ(𝛼)×¹₀(1−𝑠)^𝛼−1Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ=𝑚𝛽+1𝑑𝑠₄+𝜀^𝑞−1‖𝑥‖1Γ()𝑡𝛼+1^𝛼Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ+𝑚𝛽+1₄+𝜀^𝑞−1‖𝑥‖𝛾11−𝛾Γ(𝛼)Φ_𝑞1Γ+Φ𝛽+1^𝑝𝜂1−Φ_𝑝𝜂1Γ≤𝑚𝛽+1₄+𝜀^𝑞−1‖𝑥‖×1Γ(𝛼+1)1−Φ_𝑝𝜂^𝑞−1Γ𝛽+1^𝑞−1+𝑚₄+𝜀^𝑞−1‖𝑥‖×𝛾1−𝛾Γ(𝛼+1)1−Φ_𝑝𝜂^𝑞−1Γ𝛽+1^𝑞−1≤𝑚₄+𝜀^𝑞−11×𝑑1−𝛾Γ(𝛼+1)1−Φ_𝑝𝜂^𝑞−1Γ𝛽+1^𝑞−1<𝑑,∀𝑡∈[0,1], (29) which implies that ‖𝑇𝑥‖<𝑑. Thus, ‖𝑇𝑥‖<𝑑, for all 𝑥∈𝑃_𝑑.
Step 4. Let us prove that 𝜔(𝑇𝑥)>𝑎 holds for all 𝑥∈𝑃(𝜔,𝑎,𝑐) with ‖𝑇𝑥‖>𝑏.
For 𝑥∈𝑃(𝜔,𝑎,𝑐) with ‖𝑇𝑥‖>𝑏, we have 𝑎≤𝑥(𝑡)≤𝑐, for all 𝑡∈[𝜉,1]. From (H2), one can see that 𝑇𝑥(𝑡)≥𝐼^𝛼₀₊Φ_𝑞((𝑡(𝑡)≥1𝐴𝑁𝑥))+𝐵𝑁𝑥Γ(𝛼)×^𝑡_𝜉(𝑡−𝑠)^𝛼−1Φ_𝑞Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽¹_𝜉(1−𝜏)^𝛽−1𝑓((𝜏+𝛾𝜏,𝑥))𝑑𝜏𝑑𝑠11−𝛾Γ(𝛼)×¹_𝜉(1−𝑠)^𝛼−1Φ_𝑞Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽¹_𝜉(1−𝜏)^𝛽−1𝑓((𝜏>1𝜏,𝑥))𝑑𝜏𝑑𝑠Γ(𝛼)×^𝑡_𝜉(𝑡−𝑠)^𝛼−1Φ_𝑞Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽¹_𝜉(1−𝜏)^𝛽−1𝑚₃𝑎^𝑝−1+𝛾𝑑𝜏𝑑𝑠11−𝛾Γ(𝛼)×¹_𝜉(1−𝑠)^𝛼−1Φ_𝑞Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ𝛽¹_𝜉(1−𝜏)^𝛽−1𝑚₃𝑎^𝑝−1=𝑚𝑑𝜏𝑑𝑠₃𝑎Γ()𝜉𝛼+1^𝛼Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ()𝛽+11−𝜉^𝛽𝑞−1+𝛾𝑚1−𝛾³𝑎Γ()()𝛼+11−𝜉^𝛼×Φ_𝑝𝜂1−Φ_𝑝𝜂1Γ()𝛽+11−𝜉^𝛽𝑞−1=Φ_𝑝𝜂^𝑞−1(1−𝜉)^{𝛽(𝑞−1)}1−𝛾𝜉^𝛼+𝛾(1−𝜉)^𝛼1−𝛾Γ(𝛼+1)Γ𝛽+1^𝑞−11−Φ_𝑝𝜂^𝑞−1×𝑚₃𝑎≥𝑎,∀𝑡∈𝜉,1. (30) Therefore, condition (iii) of Lemma 10 is satisfied.
To sum up, all conditions of Lemma 10 hold. By Lemma 10, BVP (2) has at least three positive solutions.

Remark 12. It is noted that (H2) is dependent of the selection on 𝜉. Specifically, to obtain a weaker condition, one can choose 𝜉^∗∈[0,1) which maximizes 𝑘(𝜉)=𝜉^𝛼(1−𝜉)^{𝛽(𝑞−1)}1−𝛾+𝛾(1−𝜉)^{𝛼+𝛽(𝑞−1)}. (31)

4. Illustrative Examples

In this section, we give three illustrative examples to support our new results.

Example 1. Consider the following BVP: 𝐷^1/2₀₊Φ_3/2𝐷^2/3₀₊𝑥(𝑡)=1+𝑡²𝑥(𝑡)[],𝑥(0)=120,𝑡∈0,12𝑥(1),𝐷^2/3₀₊𝑥(0)=12𝐷^2/3₀₊𝑥(1). (32)
A simple calculation shows that (1−𝛾)Γ(𝛼+1)(1−Φ_𝑝(𝜂))^𝑞−1(Γ(𝛽+1))^𝑞−1≈0.0301.
Set 𝑚₁=1/100<0.0301, then one can see that 𝑓(𝑡,𝑥)=1+𝑡²√𝑥≤20𝑥()∈[]×[],100,∀𝑡,𝑥0,10,∞ (33) which implies that (16) holds. By Theorem 6, BVP (32) has no positive solution.

Example 2. Consider the following BVP: 𝐷^𝛽₀₊Φ₂𝐷^𝛼₀₊𝑥(𝑡)=1+𝑡²⁴𝑥(𝑡)[],𝑥(0)(1),𝑡∈0,1=𝛾𝑥,𝐷^𝛼₀₊𝑥(0)=𝜂𝐷^𝛼₀₊𝑥(1), (34) where 0<𝛼, 𝛽≤1, 1<𝛼+𝛽≤2, and 0<𝛾, 𝜂<1 are arbitrary.

Set 𝜃=1/4; then it is easy to see that 𝑓(𝑡,𝑘𝑥)=1+𝑡√²⁴𝑘𝑥=𝑘^1/4𝑓(),∀()∈[]×()𝑡,𝑥𝑡,𝑥0,10,+∞,∀0<𝑘<1, (35) Thus, (19) is satisfied. By Theorem 9, BVP (34) has a unique positive solution.

Let 𝛼=𝛽=1 and 𝛾=𝜂=1/2; we can draw the graph of the unique positive solution to BVP (34) by using MATLAB, shown in Figure 1.

Figure 1 

The unique positive solution to BVP (34) with 𝛼=𝛽=1 and 𝛾=𝜂=1/2.

Example 3. Consider the following BVP: 𝐷^2/3₀₊Φ₃𝐷^1/2₀₊𝑥(𝑡)((𝑡[],𝑥(0)=1=𝑓𝑡,𝑥)),𝑡∈0,13𝑥(1),𝐷^1/2₀₊𝑥(0)=34𝐷^1/2₀₊𝑥(1), (36) where 𝑓()=𝑡,𝑥1+𝑡²𝑥²,()∈[]×140𝑡,𝑥0,10,2,159980𝑥−799,()∈[]×180𝑡,𝑥0,12,()∈[]×[).,110,𝑡,𝑥0,11,+∞ (37)

Set 𝑎=1, 𝑐=50, 𝜉=0, 𝑚₂=1/5, 𝑚₃=2, and 𝑚₄=1/20. By a straightforward calculation, one can easily see that (H1)–(H3) hold. By Theorem 11, BVP (36) has at least three positive solutions.

Acknowledgments

The authors would like to thank the anonymous reviewers for their constructive comments and suggestions which improved the quality of the paper. This work is supported by the National Natural Science Foundation of China (G61174036), the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education, and the Graduate Independent Innovation Foundation of Shandong University (yzc10064).