Uniqueness and Asymptotic Behavior of Positive Solutions for a Fractional-Order Integral Boundary Value Problem

Abstract

We study a model arising from porous media, electromagnetic, and signal processing of wireless communication system −𝒟_𝐭^𝛼𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥^′(𝑡),𝑥”(𝑡),…,𝑥^(𝑛−2)(𝑡)),??0<𝑡<1,?? 𝑥(0)=𝑥^′(0)=⋯=𝑥^(𝑛−2)(0)=0,?? 𝑥^(𝑛−2)(1)=∫¹₀𝑥^(𝑛−2)(𝑠)𝑑𝐴(𝑠), where 𝑛−1<𝛼≤𝑛,?? 𝑛∈ℕ and 𝑛≥2, 𝒟_𝐭^𝛼 is the standard Riemann-Liouville derivative, ∫¹₀𝑥(𝑠)𝑑𝐴(𝑠) is linear functionals given by Riemann-Stieltjes integrals, 𝐴 is a function of bounded variation, and 𝑑𝐴 can be a changing-sign measure. The existence, uniqueness, and asymptotic behavior of positive solutions to the singular nonlocal integral boundary value problem for fractional differential equation are obtained. Our analysis relies on Schauder's fixed-point theorem and upper and lower solution method.

1. Introduction

Recently, fractional-order models have proved to be more accurate than integer order models, that is, there are more degrees of freedom in the fractional-order models.Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic, and signal processing of wireless communication system. Especially, in application of the digital signal processing, the fractional digital signal processing can greatly improve the high frequency components of signal, enhance an intermediate frequency component of signal, and reserve nonlinear low frequency signal. According to this analysis, fractional differential equation applied to the edge information extraction will get higher signal to noise ratio than that of the traditional method based on one- or-two order differential equation.

Many applications on digital signal processing were found recently, for example, Hartley et al. [1] derived a closed-loop control electromagnetic and signal processing system of equations with the Chua resistor 𝒟_𝐭^𝛼𝑥(𝑡)=𝑎𝒟_𝐭^𝛼−1𝑦(𝑡)(𝑡)−𝑏−𝑥74𝑥(𝑡)−𝑥³(𝑡),𝑑𝑦(𝑡)(𝑡)(𝑡).𝑑𝑡=𝑥−𝑦 (1.1) Caputo [2] presented a fractional model system to describe the relationship between the electric field 𝐸 and the current density 𝐷 in the study of a electrochemical polarization medium, 𝛾𝒟_𝐭^𝜈𝐷(𝑡)+𝛼𝐷(𝑡)=𝜎𝐸(𝑡)+𝛿𝒟_𝐭^𝜈𝐸(𝑡), (1.2) where 𝛾, 𝛼, 𝜎, 𝛿 are constants and 𝜈 is a real number. Anastasio [3] believed that signal processing of communication system of vestibule visual reflex effect is fractional, which can be described by the following model 𝜏₁𝑟^′(𝑡)+𝑟(𝑡)=𝜏₁𝜏₂𝒟_𝐭^𝛼+1𝑣(𝑡)+𝜏₁𝒟_𝐭^𝛼𝑣(𝑡), (1.3) where 𝑟(𝑡) is the vestibular reflex nerve discharge rate, 𝑣(𝑡) is the head of the rotational angular velocity, 𝛼 is fraction.

Motivated by systems (1.1)–(1.3) and their application background in electromagnetic and signal processing of wireless communication system, in this paper, we consider the existence, uniqueness, and asymptotic behavior of positive solutions for the higher nonlocal fractional differential equation −𝒟_𝐭^𝛼𝑥(𝑡)=𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡),𝑥^′′(𝑡),…,𝑥^(𝑛−2)(𝑡)𝑥(0),0<𝑡<1,=𝑥^′(0)=⋯=𝑥^(𝑛−2)(0)=0,𝑥^(𝑛−2)(1)=¹₀𝑥^(𝑛−2)(𝑠)𝑑𝐴(𝑠), (1.4) where 𝑛−1<𝛼≤𝑛,??𝑛∈ℕ and 𝑛≥2, 𝒟_𝐭^𝛼 is the standard Riemann-Liouville derivative, ∫¹₀𝑥(𝑠)𝑑𝐴(𝑠) is linear functionals given by Riemann-Stieltjes integrals, 𝐴 is a function of bounded variation, and 𝑑𝐴 can be a changing-sign measure, and 𝑓∶(0,1)×(0,+∞)^𝑛−1→[0,+∞) is continuous, 𝑓 may be singular at 𝑥_𝑖=0 and 𝑡=0,1.

The nonlocal integral-boundary value problems represent a class of interesting and important problems arising in physical, biological, and chemical processes and have attracted the attention of Khan [4], Gallardo [5], Karakostas and Tsamatos [6], Ahmad et al. [7], Feng et al. [8], Corduneanu [9], and Agarwal and O’Regan [10].

When 𝛼 is an integer, a lot of work has been done dealing with nonlocal 3-point boundary value problems, see [11, 12]. In [12], Eloe and Ahmad studied the following 𝑛th-order nonlocal differential equation 𝑢^(𝑛)(𝑡)(𝑡)𝑓(𝑢)𝑢(0)+𝑎=0,0<𝑡<1,=𝑢^′(0)=⋯=𝑢^(𝑛−2)(0)=0,𝑢(1)=𝜂𝑢(𝜉), (1.5) where 0<𝜉<1, 0<𝜂𝜉^𝑛−1<1. By applying the fixed point theorem in cones, the authors prove the existence of at least one positive solution when 𝑎 and 𝑓 is continuous and 𝑓 is either sublinear or superlinear. Recently, Hao et al. [13] studied the existence of positive solutions for the BVP (1.5) with integer order 𝑛, and nonlinear term is replaced by 𝑎(𝑡)𝑓(𝑡,𝑥(𝑡)), where 𝑎 can be singular at 𝑡=0,1, 𝑓 can be singular at 𝑥=0 and no singularity at 𝑡=0,1.

If 𝛼 is fractional, Yuan [14] studied the (𝑛−1,1)-type conjugate boundary value problem 𝒟_𝐭^𝛼𝑢(𝑡)((𝑡𝑢(0)+𝑓𝑡,𝑢))=0,0<𝑡<1,𝑛−1<𝛼≤𝑛,𝑛≥3,=𝑢^′(0)=⋯=𝑢^(𝑛−2)(0)=0,𝑢(1)=0, (1.6) where 𝑓 is continuous and semipositone, 𝒟_𝐭^𝛼 is the standard Riemann-Liouville derivative. By giving properties of Green’s function and using the Guo-Krasnosel’skii fixed-point theorem on cones, the existence of multiple positive solutions were obtained. More recently, Zhang [15] considered the following BVP whose nonlinear term and boundary condition contain derivatives of unknown functions 𝒟_𝐭^𝛼𝑥(𝑡)+𝑞(𝑡)𝑓𝑥,𝑥^′,…,𝑥^(𝑛−2)𝑥(0)=0,0<𝑡<1,𝑛−1<𝛼≤𝑛,=𝑥^′(0)=⋯=𝑥^(𝑛−2)(0)=𝑥^(𝑛−2)(1)=0, (1.7) where 𝒟_𝐭^𝛼 is the standard Riemann-Liouville fractional derivative of order 𝛼, 𝑞 may be singular at 𝑡=0 and 𝑓 may be singular at 𝑥=0,??𝑥^′=0,…,𝑥^(𝑛−2)=0, by using fixed-point theorem of the mixed monotone operator, the unique existence result of positive solution to problem (1.7) was established. Other some recent results, see [16–20]

Motivated by the results mentioned above, in this paper, we study the existence, uniqueness, and asymptotic behavior of positive solutions for the BVP (1.4) where the nonlinear terms and boundary conditions all involve derivatives of unknown functions and with Riemann-Stieltjes integral boundary condition, moreover 𝑓 may be singular at 𝑥_𝑖=0 and 𝑡=0,1. Our main tool relies on Schauder’s fixed-point theorem and upper and lower solution method.

2. Preliminaries and Lemmas

In this section, we present here the necessary definitions from fractional calculus theory. These definitions can be found in the recent literatures.

Definition 2.1 (see [21, 22]). The Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑥∶(0,+∞)→ℝ is given by the following: 𝐼^𝛼𝑥(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑥(𝑠)𝑑𝑠 (2.1) provided that the right-hand side is pointwise defined on (0,+∞).

Definition 2.2 (see [21, 22]). The Riemann-Liouville fractional derivative of order 𝛼>0 of a function 𝑥∶(0,+∞)→ℝ is given by the following: 𝒟_𝐭^𝛼𝑥(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑥(𝑠)𝑑𝑠, (2.2) where 𝑛=[𝛼]+1, [𝛼] denotes the integer part of number 𝛼, provided that the right-hand side is pointwise defined on (0,+∞).

Proposition 2.3 (see [21, 22]). (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). (2.3) (2) If 𝛼>0,𝜎>0, then 𝒟_𝐭^𝛼𝑡^𝜎−1=Γ(𝜎)Γ()𝑡𝜎−𝛼^{𝜎−𝛼−1}. (2.4)

Proposition 2.4 (see [21, 22]). 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, (2.5) where 𝑐_𝑖∈ℝ(𝑖=1,2,…,𝑛), 𝑛 is the smallest integer greater than or equal to 𝛼.

Our discussion is based on the assumption −1<𝛼−𝑛≤0 in this paper.

Let 𝑥(𝑡)=𝐼^𝑛−2𝑦(𝑡),𝑦(𝑡)∈𝐶[0,1], by standard discuss, we easily reduce order the BVP (1.4) to the following equivalent BVP −𝒟_𝐭^{𝛼−𝑛+2}𝑦(𝑡)=𝑓𝑡,𝐼^𝑛−2𝑦(𝑡),𝐼^𝑛−3𝑦(𝑡),…,𝐼¹𝑦(𝑡)(𝑡,𝑦(0)(1)=,𝑦)=0,𝑦¹₀𝑦(𝑠)𝑑𝐴(𝑠). (2.6)

In fact, let 𝑥(𝑡)=𝐼^𝑛−2𝑦(𝑡),𝑦(𝑡)∈𝐶[0,1], then 𝒟_𝐭^𝛼𝑥(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝐼^𝑛−𝛼𝑥(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝐼^𝑛−𝛼𝐼^𝑛−2𝑦(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝐼^{2𝑛−𝛼−2}𝑦(𝑡)=𝒟_𝐭^{𝛼−𝑛+2}𝑦(𝑡),𝑥^′(𝑡)=𝒟_𝐭¹𝐼^𝑛−2𝑦(𝑡)=𝐼^𝑛−3𝑦(𝑡),𝑥^′′(𝑡)=𝒟_𝐭²𝐼^𝑛−2𝑦(𝑡)=𝐼^𝑛−4𝑦(𝑡),⋮𝑥^(𝑛−3)(𝑡)=𝒟_𝐭^𝑛−3𝐼^𝑛−2𝑦(𝑡)=𝐼¹𝑦(𝑡),𝑥^(𝑛−2)(𝑡)=𝒟_𝐭^𝑛−2𝐼^𝑛−2𝑦(𝑡)=𝑦(𝑡). (2.7) By (2.7), we have 𝑥^(𝑛−2)(0)=𝑦(0)=0 and 𝑦(1)=¹₀𝑦(𝑠)𝑑𝐴(𝑠). (2.8) Moreover −𝒟_𝐭^{𝛼−𝑛+2}𝑦(𝑡)=𝑓(𝑡,𝐼^𝑛−2𝑦(𝑡),𝐼^𝑛−3𝑦(𝑡),…,𝐼¹𝑦(𝑡),𝑦(𝑡)). Thus, (1.4) is transformed into (2.6).

On the other hand, if 𝑦∈𝐶([0,1],[0,+∞)) is a solution for problem (2.6). Then, from Proposition 2.3 and (2.7), one has −𝒟_𝐭^𝛼𝑥(𝑡)=−𝒟_𝐭^{𝛼−𝑛+2}𝑦(𝑡)=𝑓𝑡,𝐼^𝑛−2𝑦(𝑡),𝐼^𝑛−3𝑦(𝑡),…,𝐼¹𝑦(𝑡)(𝑡(𝑡),𝑦)=𝑓𝑡,𝑥,𝑥^′(𝑡),𝑥^′′(𝑡),…,𝑥^(𝑛−2)(𝑡),0<𝑡<1. (2.9) Notice for any 𝛼>0, 𝐼^𝛼𝑦(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠, (2.10) which implies that 𝐼^𝛼𝑦(0)=0. So 𝑥(0)=0, and from (2.7), for 𝑖=1,2,…,𝑛−2, we have 𝑥^(𝑖)(0)=0, and 𝑥^(𝑛−2)(1)=¹₀𝑥^(𝑛−2)(𝑠)𝑑𝐴(𝑠). (2.11) Consequently, the BVP (2.6) is transformed into the BVP (1.4).

Applying Propositions 2.3 and 2.4, by standard discuss, we have the following Lemma.

Lemma 2.5. Given 𝑦∈𝐿¹(0,1), then the problem 𝒟_𝐭^{𝛼−𝑛+2}𝑦(𝑡)(𝑡)𝑦(0)(1)+ℎ=0,0<𝑡<1,=0,𝑦=0, (2.12) has the unique solution 𝑦(𝑡)=¹₀𝐺(𝑡,𝑠)ℎ(𝑠)𝑑𝑠, (2.13) where 𝐺(𝑡,𝑠) is the Green function of the BVP (2.12) and is given by the following: 𝐺()=𝑡𝑡,𝑠^{𝛼−𝑛+1}(1−𝑠)^{𝛼−𝑛+1}−(𝑡−𝑠)^{𝛼−𝑛+1}Γ()𝑡𝛼−𝑛+2,0≤𝑠≤𝑡≤1,_{𝛼−𝑛+1}(1−𝑠)^{𝛼−𝑛+1}Γ(𝛼−𝑛+2),0≤𝑡≤𝑠≤1. (2.14)

By Proposition 2.4, the unique solution of the problem 𝒟_𝐭^{𝛼−𝑛+2}𝑦(𝑡)𝑦(0)(1)=0,0<𝑡<1,=0,𝑦=1, (2.15) is 𝑡^{𝛼−𝑛+1}. Let 𝒞=¹₀𝑡^{𝛼−𝑛+1}𝑑𝐴(𝑡), (2.16) and define 𝒢_𝐴(𝑠)=¹₀𝐺(𝑡,𝑠)𝑑𝐴(𝑡). (2.17) Then the Green function for the nonlocal BVP (2.6) is (the detail see [23] or [16]) 𝐾()=𝑡𝑡,𝑠^{𝛼−𝑛+1}𝒢1−𝒞^𝐴(𝑠)+𝐺(𝑡,𝑠). (2.18)

Throughout paper, we always assume the following holds.

(𝐇𝟎)𝐴 is a function of bounded variation such that 𝒢_𝐴(𝑠)≥0 for 𝑠∈[0,1] and 0≤𝒞<1, where 𝒞 is defined by (2.16).

Lemma 2.6. Suppose (𝐇𝟎) holds, then the Green function defined by (2.18) satisfies (1)𝐾(𝑡,𝑠)>0, for all 𝑡,𝑠∈(0,1).(2)𝑡^{𝛼−𝑛+1}𝒢1−𝒞^𝐴(𝑠)≤𝐾(𝑡,𝑠)≤ℋ(𝑠)𝑡^{𝛼−𝑛+1}, (2.19)where ℋ(𝑠)=(1−𝑠)^{𝛼−𝑛+1}Γ()+𝒢𝛼−𝑛+2^𝐴(𝑠).1−𝒞 (2.20)

Proof. (1) is obvious. For (2), by (2.14) and (2.18), we have 𝐻()=𝑡𝑡,𝑠^{𝛼−𝑛+1}𝒢1−𝒞^𝐴(𝑠)()≤𝑡+𝐺𝑡,𝑠^{𝛼−𝑛+1}(1−𝑠)^{𝛼−𝑛+1}Γ()+𝑡𝛼−𝑛+2^{𝛼−𝑛+1}𝒢1−𝒞^𝐴(𝑠)=ℋ(𝑠)𝑡^{𝛼−𝑛+1},𝐻()=𝑡𝑡,𝑠^{𝛼−𝑛+1}𝒢1−𝒞^𝐴(𝑠)()≥𝑡+𝐺𝑡,𝑠^{𝛼−𝑛+1}𝒢1−𝒞^𝐴(𝑠). (2.21)

Definition 2.7. A continuous function 𝜓(𝑡) is called a lower solution of the BVP (2.6), if it satisfies −𝒟_𝐭^{𝛼−𝑛+2}𝜓(𝑡)(𝑡)≤𝑓𝑡,𝐼^𝑛−2𝜓(𝑡),𝐼^𝑛−3𝜓(𝑡),…,𝐼¹𝜓(𝑡)(𝑡,𝜓(0)(1)≥,𝜓)≥0,𝜓¹₀𝜓(𝑠)𝑑𝐴(𝑠). (2.22)

Definition 2.8. A continuous function 𝜙(𝑡) is called a upper solution of the BVP (2.6), if it satisfies −𝒟_𝐭^{𝛼−𝑛+2}𝜙(𝑡)(𝑡)≥𝑓𝑡,𝐼^𝑛−2𝜙(𝑡),𝐼^𝑛−3𝜙(𝑡),…,𝐼¹𝜙(𝑡)(𝑡,𝜙(0)(1)≤,𝜙)≤0,𝜙¹₀𝜙(𝑠)𝑑𝐴(𝑠). (2.23)
It follows from Lemma 2.5, we have the following maximum principle.

Lemma 2.9 (maximum principle). If 𝑦∈𝐶([0,1],ℝ) satisfies 𝑦(0)=0,𝑦(1)=¹₀𝑦(𝑠)𝑑𝐴(𝑠), (2.24) and 𝒟_𝐭^{𝛼−𝑛+2}𝑦(𝑡)≤0 for any 𝑡∈(0,1). Then 𝑦(𝑡)≥0,𝑡∈[0,1]. (2.25)

3. Main Results

Let 𝐸=𝐶[0,1], and 𝑃=𝑦∈𝐸∶thereexistpositivenumbers0<𝑙_𝑦<1,𝐿_𝑦>1suchthat𝑙_𝑦𝑡^{𝛼−𝑛+1}≤𝑦(𝑡)≤𝐿_𝑦𝑡^{𝛼−𝑛+1},𝑡∈[0,1]} (3.1) Clearly, 𝑡^{𝛼−𝑛+1}∈𝑃, so 𝑃 is nonempty. For any 𝑦∈𝑃, define an operator 𝑇 by the following: 𝑇𝑦(𝑡)=¹₀𝐾(𝑡,𝑠)𝑓𝑠,𝐼^𝑛−2𝑦(𝑠),𝐼^𝑛−3𝑦(𝑠),…,𝐼¹𝑦(𝑠),𝑦(𝑠)𝑑𝑠. (3.2)

Let 𝜅_𝑖(𝑡)=Γ(𝛼−𝑛+2)Γ()𝑡𝛼−𝑖^{𝛼−𝑖−1},𝑖=0,1,2,…,𝑛−2, (3.3) then 𝜅₀(𝑡)=𝐼^𝑛−2𝑠^{𝛼−𝑛+1},𝜅₁(𝑡)=𝐼^𝑛−3𝑠^{𝛼−𝑛+1},…,𝜅_𝑛−3(𝑡)=𝐼¹𝑠^{𝛼−𝑛+1},𝜅_𝑛−2(𝑡)=𝑡^{𝛼−𝑛+1}. (3.4)

The conditions imposed on 𝑓 are the following.?(H1)??𝑓∈𝐶((0,1)×(0,∞)^𝑛−1,[0,+∞)), and 𝑓(𝑡,𝑥₀,𝑥₁,𝑥₂,…,𝑥_𝑛−2) is decreasing in 𝑥_𝑖>0 for 𝑖=0,1,2,…,𝑛−2;?(H2)??for any 𝜆_𝑖>0, 𝑓(𝑡,𝜆₀𝑡^𝑛−2,𝜆₁𝑡^𝑛−3,…,𝜆_𝑛−3𝑡,𝜆_𝑛−2)≢0,??𝑡∈(0,1), and 0<¹₀ℋ(𝑠)𝑓𝑠,𝜆₀𝜅₀(𝑠),𝜆₁𝜅₁(𝑠),𝜆₂𝜅₂(𝑠),…,𝜆_𝑛−2𝜅_𝑛−2(𝑠)𝑑𝑠<+∞. (3.5)

Lemma 3.1. Suppose (𝐇𝟎)−(𝐇𝟐) hold, then 𝑇 is well defined and 𝑇(𝑃)⊂𝑃.

Proof. For any 𝑦∈𝑃, by the definition of 𝑃, there exists two positive numbers 0<𝑙_𝑦<1,𝐿_𝑦>1 such that 𝑙_𝑦𝑡^{𝛼−𝑛+1}≤𝑦(𝑡)≤𝐿_𝑦𝑡^{𝛼−𝑛+1} for any 𝑡∈[0,1]. It follows from (2.19) and (𝐇𝟏)-(𝐇𝟐) that 𝑇𝑦(𝑡)=¹₀𝐾(𝑡,𝑠)𝑓𝑠,𝐼^𝑛−2𝑦(𝑠),𝐼^𝑛−3𝑦(𝑠),…,𝐼¹𝑦(𝑠)(𝑠)≤,𝑦𝑑𝑠₁₀ℋ(𝑠)𝑓𝑠,𝐼^𝑛−2𝑙_𝑦𝑠^{𝛼−𝑛+1},…,𝐼¹𝑙_𝑦𝑠^{𝛼−𝑛+1},𝑙_𝑦𝑠^{𝛼−𝑛+1}=𝑑𝑠₁₀ℋ(𝑠)𝑓𝑠,𝑙_𝑦𝜅₀(𝑠),𝑙_𝑦𝜅₁(𝑠),…,𝑙_𝑦𝜅_𝑛−2(𝑠)𝑑𝑠<+∞. (3.6)
Now take 𝐶=max_𝑡∈[0,1]𝑦(𝑡), then for any 𝑠∈(0,1), 𝒢_𝐴(𝑠)𝑓𝐶𝑠,()!𝑠𝑛−2^𝑛−2,𝐶()!𝑠𝑛−3^𝑛−3𝐶,…,1!𝑠,𝐶≢0, (3.7) by (𝐇𝟐). Thus by the continuity of 𝑓(𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛−2), we have ¹₀𝒢_𝐴(𝑠)𝑓𝐶𝑠,()!𝑠𝑛−2^𝑛−2,𝐶()!𝑠𝑛−3^𝑛−3𝐶,…,1!𝑠,𝐶𝑑𝑠>0. (3.8) This yields ¹₀𝒢_𝐴(𝑠)𝑓𝑠,𝐼^𝑛−2𝐶,𝐼^𝑛−3𝐶,…,𝐼¹=𝐶,𝐶𝑑𝑠₁₀𝒢_𝐴(𝑠)𝑓𝐶𝑠,()!𝑠𝑛−2^𝑛−2,𝐶()!𝑠𝑛−3^𝑛−3𝐶,…,1!𝑠,𝐶𝑑𝑠>0. (3.9) By (2.19) and (3.6)–(3.9), we have 𝑇𝑦(𝑡)=¹₀𝐾(𝑡,𝑠)𝑓𝑠,𝐼^𝑛−2𝑦(𝑠),𝐼^𝑛−3𝑦(𝑠),…,𝐼¹𝑦(𝑠)(𝑠)≥𝑡,𝑦𝑑𝑠_{𝛼−𝑛+1}1−𝒞¹₀𝒢_𝐴(𝑠)𝑓𝑠,𝐼^𝑛−2𝐶,𝐼^𝑛−3𝐶,…,𝐼¹𝐶,𝐶𝑑𝑠≥𝑙^′_𝑦𝑡^{𝛼−𝑛+1}, (3.10) where 𝑙^′_𝑦1=min1,1−𝒞¹₀𝒢_𝐴(𝑠)𝑓𝑠,𝐼^𝑛−2𝐶,𝐼^𝑛−3𝐶,…,𝐼¹𝐶,𝐶𝑑𝑠. (3.11) On the other hand, it follows from (2.19) that 𝑇𝑦(𝑡)=¹₀𝐾(𝑡,𝑠)𝑓𝑠,𝐼^𝑛−2𝑦(𝑠),𝐼^𝑛−3𝑦(𝑠),…,𝐼¹𝑦(𝑠),𝑦(𝑠)𝑑𝑠≤𝑡^{𝛼−𝑛+11}₀ℋ(𝑠)𝑓𝑠,𝑙_𝑦𝜅₀(𝑠),𝑙_𝑦𝜅₁(𝑠),…,𝑙_𝑦𝜅_𝑛−2(𝑠)𝑑𝑠,≤𝐿^′_𝑦𝑡^{𝛼−𝑛+1}, (3.12) where 𝐿^′_𝑦=max1,¹₀ℋ(𝑠)𝑓𝑠,𝑙_𝑦𝜅₀(𝑠),𝑙_𝑦𝜅₁(𝑠),…,𝑙_𝑦𝜅_𝑛−2(𝑠)𝑑𝑠. (3.13) It follows from (3.6)–(3.12) that 𝑇 is well defined and 𝑇(𝑃)⊂𝑃.

Take 𝐿=max1,¹₀ℋ(𝑠)𝑓𝑠,𝜅₀(𝑠),𝜅₁(𝑠),…,𝜅_𝑛−2(𝑠),1𝑑𝑠𝑙=min1,1−𝒞¹₀𝒢_𝐴(𝑠)𝑓𝑠,𝜅₀(𝑠),𝜅₁(𝑠),…,𝜅_𝑛−2(𝑠)𝑑𝑠. (3.14) By (H2) and Weierstrass distinguishing method, we know 𝑎(𝑡)=¹₀𝐾(𝑡,𝑠)𝑓𝑠,𝜅₀(𝑠),𝜅₁(𝑠),…,𝜅_𝑛−2(𝑠)𝑑𝑠 (3.15) is a continuous function on [0,1], that is, 𝑎(𝑡)∈𝐸.

Theorem 3.2 (existence). Suppose (𝐇𝟎)−(𝐇𝟐) hold, and if ¹₀𝒢_𝐴(𝑠)𝑓𝐿𝑠,𝑙𝜅₀(𝑠),𝐿𝑙𝜅₁(𝑠)𝐿,…,𝑙𝜅_𝑛−2(𝑠),𝐿𝑙𝜅_𝑛−1(𝑠)𝑑𝑠≥1−𝒞. (3.16) Then the BVP (1.4) has at least a positive solution 𝑤(𝑡).

Proof. By (2.18) and (3.2), we have −𝒟_𝐭^{𝛼−𝑛+2}𝑇𝑦(𝑡)=𝑓𝑡,𝐼^𝑛−2𝑦(𝑡),𝐼^𝑛−3𝑦(𝑡),…,𝐼¹𝑦(𝑡)(𝑡,(0)(1)=,𝑦)𝑇𝑦=0,𝑇𝑦¹₀𝑇𝑦(𝑠)𝑑𝐴(𝑠). (3.17) It follows from (2.19) that 𝑙𝑡^{𝛼−𝑛+1}≤𝑎(𝑡)≤𝐿𝑡^{𝛼−𝑛+1}, (3.18) which implies that 𝑎(𝑡)∈𝑃. Let 𝑏(𝑡)=𝑎(𝑡)𝑙, (3.19) then we also have 𝑡^{𝛼−𝑛+1}(𝑡)≤𝐿≤𝑏𝑙𝑡^{𝛼−𝑛+1}. (3.20)
Thus according to the fact that the operator 𝑇 is nonincreasing relative to 𝑦, Lemma 3.1 and (3.20), we have 𝑇𝑏(𝑡)≤𝑇𝑡^{𝛼−𝑛+1}(𝑡)≤𝑎(𝑡)=𝑎𝑙=𝑏(𝑡). (3.21) and 𝑇𝑏(𝑡)∈𝑃. By (2.19), (3.12), and (3.20), one has (𝑡)≥1𝑇𝑏𝑡1−𝒞^{𝛼−𝑛+11}₀𝒢_𝐴(𝑠)𝑓𝑠,𝐼^𝑛−2𝑏(𝑠),𝐼^𝑛−3𝑏(𝑠),…,𝐼¹𝑏(𝑠)(𝑠)≥1,𝑏𝑑𝑠𝑡1−𝒞^{𝛼−𝑛+11}₀𝒢_𝐴(𝑠)𝑓𝐿𝑠,𝑙𝜅₀(𝑠),𝐿𝑙𝜅₁(𝑠)𝐿,…,𝑙𝜅_𝑛−2(𝑠)𝑑𝑠≥𝑡^{𝛼−𝑛+1}. (3.22)
Thus, by (3.20), (3.21) and the operator 𝑇 is nonincreasing relative to 𝑦, we have 𝒟_𝐭^{𝛼−𝑛+2}𝑏(𝑡)+𝑓𝑡,𝐼^𝑛−2𝑏(𝑡),𝐼^𝑛−3𝑏(𝑡),…,𝐼¹𝑏(𝑡)(𝑡1,𝑏)=−𝑙𝑓𝑠,𝐼^𝑛−2𝑡^{𝛼−𝑛+1},…,𝐼¹𝑡^{𝛼−𝑛+1},𝑡^{𝛼−𝑛+1}+𝑓𝑡,𝐼^𝑛−2𝑏(𝑡),…,𝐼¹𝑏(𝑡)(𝑡)1,𝑏≤−𝑙𝑓𝑠,𝐼^𝑛−2𝑡^{𝛼−𝑛+1},…,𝐼¹𝑡^{𝛼−𝑛+1},𝑡^{𝛼−𝑛+1}+𝑓𝑠,𝐼^𝑛−2𝑡^{𝛼−𝑛+1},…,𝐼¹𝑡^{𝛼−𝑛+1},𝑡^{𝛼−𝑛+1}𝒟≤0,_{𝐭𝛼−𝑛+2}𝑇𝑏(𝑡)+𝑓𝑡,𝐼^𝑛−2𝑇𝑏(𝑡),𝐼^𝑛−3𝑇𝑏(𝑡),…,𝐼¹𝑇𝑏(𝑡),𝑇𝑏(𝑡)=−𝑓𝑡,𝐼^𝑛−2𝑏(𝑡),𝐼^𝑛−3𝑏(𝑡),…,𝐼¹𝑏(𝑡),𝑏(𝑡)+𝑓𝑡,𝐼^𝑛−2𝑇𝑏(𝑡),𝐼^𝑛−3𝑇𝑏(𝑡),…,𝐼¹(𝑡)(𝑡)[].𝑇𝑏,𝑇𝑏≥0,𝑡∈0,1 (3.23) However (3.17) implies that 𝑏(𝑡), 𝑇𝑏(𝑡) satisfy boundary conditions of BVP (2.6). Let 𝜓(𝑡)(𝑡)(𝑡)=1=𝑇𝑏,𝜙𝑙𝑇𝑡^{𝛼−𝑛+1}=1𝑙𝑎(𝑡)=𝑏(𝑡), (3.24) then (3.21), (3.23) imply that 𝜓(𝑡),𝜙(𝑡) are lower and upper solution of BVP (2.6), respectively, and 𝜓(𝑡),𝜙(𝑡)∈𝑃.
Define the function 𝐹 and the operator 𝐴 in 𝐸 by the following: 𝐹=𝑓𝑡,𝑦𝑡,𝐼^𝑛−2𝜓(𝑡),𝐼^𝑛−3𝜓(𝑡),…,𝐼¹𝜓(𝑡)(𝑡(𝑡),𝑓,𝜓),𝑦<𝜓𝑡,𝐼^𝑛−2𝑦(𝑡),𝐼^𝑛−3𝑦(𝑡),…,𝐼¹𝑦(𝑡)(𝑡(𝑡)(𝑡),𝑓,𝑦),𝜓≤𝑦≤𝜙𝑡,𝐼^𝑛−2𝜙(𝑡),𝐼^𝑛−3𝜙(𝑡),…,𝐼¹𝜙(𝑡)(𝑡)(𝑡),(𝑡)=,𝜙,𝑦>𝜙𝐵𝑦¹₀𝐾(𝑡,𝑠)𝐹𝑠,𝑦(𝑠)𝑑𝑠,∀𝑦∈𝐸. (3.25)(3.26) Clearly, 𝐹∶[0,1]×[0,+∞)→[0,+∞) is continuous by (3.25). Consider the following boundary value problem −𝒟_𝐭^{𝛼−𝑛+2}𝑦(𝑡)𝑦(0)(1)==𝐹𝑡,𝑦,0<𝑡<1,=0,𝑦¹₀𝑦(𝑠)𝑑𝐴(𝑠). (3.27) Obviously, a fixed point of the operator 𝐵 is a solution of the BVP (3.27).
For all 𝑦∈𝐸, it follows from Lemma 2.6 and (3.22) that 𝐵𝑦(𝑡)≤¹₀ℋ(𝑠)𝐹𝑠,𝑦(𝑠)𝑑𝑠≤¹₀ℋ(𝑠)𝑓𝑠,𝐼^𝑛−2𝜓(𝑠),𝐼^𝑛−3𝜓(𝑠),…,𝐼¹𝜓(𝑠)(𝑠)≤,𝜓𝑑𝑠₁₀ℋ(𝑠)𝑓𝑠,𝐼^𝑛−2𝑠^{𝛼−𝑛+1},…,𝐼¹𝑠^{𝛼−𝑛+1},𝑠^{𝛼−𝑛+1}=𝑑𝑠₁₀ℋ(𝑠)𝑓𝑠,𝜅₀(𝑠),𝜅₁(𝑠),…,𝜅_𝑛−2(𝑠)𝑑𝑠<+∞. (3.28) So 𝐵 is bounded. It is easy to see 𝐵∶𝐸→𝐸 is continuous from the continuity of 𝐹 and 𝐾.
Let Ω⊂𝐸 be bounded, that is, there exists a positive constant 𝑁>0 such that ||𝑦||≤𝑁, for all 𝑦∈Ω. Let 𝐿=max_{0≤𝑡≤1,0≤𝑦≤𝑁}|𝐹(𝑡,𝑦)|+1, and since 𝐾(𝑡,𝑠) is uniformly continuous on [0,1]×[0,1], then for any 𝜀>0 and 𝑠∈[0,1], there exists 𝛿>0 such that 𝐾𝑡₁,𝑠−𝐾𝑡₂<𝜀,𝑠𝐿, (3.29) for |𝑡₁−𝑡₂|<𝛿. Then 𝐵𝑦𝑡₁−𝐵𝑦𝑡₂≤¹₀𝐾𝑡₁,𝑠−𝐾𝑡₂,𝑠𝐹𝑠,𝑦(𝑠)𝑑𝑠<𝜀. (3.30) This implies that 𝐵(Ω) is equicontinuous.
By the means of the Arzela-Ascoli theorem, we have 𝐵∶𝐸→𝐸 is completely continuous. Thus, by using Schauder’s fixed-point theorem, 𝐵 has at least a fixed-point 𝑤 such that 𝑤=𝐵𝑤.
Now we prove 𝜓(𝑡)≤𝑤(𝑡)≤𝜙(𝑡),𝑡∈[0,1]. (3.31)
Let 𝑧(𝑡)=𝜙(𝑡)−𝑤(𝑡),??𝑡∈[0,1]. By 𝜙(𝑡) is upper solution of BVP (2.6) and 𝑤 is fixed point of 𝐵, we know 𝑤(0)=0,𝑤(1)=¹₀𝑤(𝑠)𝑑𝐴(𝑠). (3.32) From the definition of 𝐹 and (3.22), we obtain 𝑓𝑡,𝐼^𝑛−2𝜙(𝑡),𝐼^𝑛−3𝜙(𝑡),…,𝐼¹𝜙(𝑡),𝜙(𝑡)≤𝐹𝑡,𝑦(𝑡)≤𝑓𝑡,𝐼^𝑛−2𝜓(𝑡),𝐼^𝑛−3𝜓(𝑡),…,𝐼¹𝜓(𝑡),𝜓(𝑡)≤𝑓𝑡,𝐼^𝑛−2𝑡^{𝛼−𝑛+1},…,𝐼¹𝑡^{𝛼−𝑛+1},𝑡^{𝛼−𝑛+1},𝑡∈[0,1]. (3.33) Thus (3.16) and (3.33) imply 𝒟_𝐭^{𝛼−𝑛+2}𝑧(𝑡)=𝒟_𝐭^{𝛼−𝑛+2}𝜙(𝑡)−𝒟_𝐭^{𝛼−𝑛+2}𝑤(𝑡)1=−𝑙𝑓𝑡,𝐼^𝑛−2𝑡^{𝛼−𝑛+1},…,𝐼¹𝑡^{𝛼−𝑛+1},𝑡^{𝛼−𝑛+1}((𝑡1+𝐹𝑡,𝑤))≤−𝑙𝑓𝑡,𝐼^𝑛−2𝑡^{𝛼−𝑛+1},…,𝐼¹𝑡^{𝛼−𝑛+1},𝑡^{𝛼−𝑛+1}+𝑓𝑡,𝐼^𝑛−2𝑡^{𝛼−𝑛+1},…,𝐼¹𝑡^{𝛼−𝑛+1},𝑡^{𝛼−𝑛+1}≤0,∀𝑡∈[0,1]. (3.34) By (3.32),??(3.34), and Lemma 2.9, we know 𝑧(𝑡)≥0 which implies 𝑤(𝑡)≤𝜙(𝑡) on [0,1]. By the same way, it is easy to prove 𝑤(𝑡)≥𝜓(𝑡) on [0,1]. So we obtain 𝜓(𝑡)≤𝑤(𝑡)≤𝜙(𝑡),𝑡∈[0,1]. (3.35) Consequently, 𝐹(𝑡,𝑤(𝑡))=𝑓(𝑡,𝐼^𝑛−2𝑤(𝑡),𝐼^𝑛−3𝑤(𝑡),…,𝐼¹𝑤(𝑡),𝑤(𝑡)),??𝑡∈[0,1]. Then 𝑤(𝑡) is a positive solution of the BVP (2.6), which implies that 𝑥(𝑡)=𝐼^𝑛−2𝑤(𝑡) is a positive solution of the BVP (1.4).

Theorem 3.3 (asymptotic behavior). Suppose the conditions of Theorem 3.2 are satisfied. Then there exist two constants ℬ₁,ℬ₂ such that the positive solution 𝑥(𝑡) of the BVP (1.4) satisfies ℬ₁𝑡^𝛼−1≤𝑥(𝑡)≤ℬ₂𝑡^𝛼−1,lim_𝑡→0+𝑥(𝑡)𝑡^𝛼−1=0, (3.36) where ℬ₁=Γ(𝛼−𝑛+2)Γ(𝛼),ℬ₂=𝐿ℬ₁. (3.37)

Proof. By (3.22) and (3.35), we know 𝑤(𝑡)≥𝜓(𝑡)≥𝑡^{𝛼−𝑛+1}. (3.38) On the other hand, it follows from (3.38) and (2.19) that 𝑤(𝑡)=¹₀𝐾(𝑡,𝑠)𝑓𝑠,𝐼^𝑛−2𝑤(𝑠),𝐼^𝑛−3𝑤(𝑠),…,𝐼¹𝑤(𝑠),𝑤(𝑠)𝑑𝑠≤𝑡^{𝛼−𝑛+11}₀ℋ(𝑠)𝑓𝑠,𝐼^𝑛−2𝑠^{𝛼−𝑛+1},…,𝐼¹𝑠^{𝛼−𝑛+1},𝑠^{𝛼−𝑛+1}𝑑𝑠=𝑡^{𝛼−𝑛+11}₀ℋ(𝑠)𝑓𝑠,𝜅₀(𝑠),𝜅₁(𝑠),…,𝜅_𝑛−2(𝑠),𝜅_𝑛−1(𝑠)𝑑𝑠≤𝐿𝑡^{𝛼−𝑛+1}. (3.39) Then 𝑡^{𝛼−𝑛+1}≤𝑤(𝑡)≤𝐿𝑡^{𝛼−𝑛+1}. (3.40)
Since 𝐼^𝑛−2𝑡^{𝛼−𝑛+1}=1Γ(𝑛−2)^𝑡₀(𝑡−𝑠)^𝑛−3𝑠^{𝛼−𝑛+1}Γ()𝑑𝑠=𝛼−𝑛+2Γ(𝛼)𝑡^𝛼−1, (3.41) so it follows from (3.40) that Γ(𝛼−𝑛+2)Γ(𝛼)𝑡^𝛼−1≤𝐼^𝑛−2𝑤(𝑡)(𝑡)Γ()=𝑥≤𝐿𝛼−𝑛+2Γ(𝛼)𝑡^𝛼−1, (3.42) That is, ℬ₁𝑡^𝛼−1≤𝑥(𝑡)≤ℬ₂𝑡^𝛼−1. (3.43)
In the end, from 𝑛−1<𝛼≤𝑛 and L’Hospital Rule, we have Lim_𝑡→0+𝑥(𝑡)𝑡^𝛼−1=lim_𝑡→0+𝑥^(𝑛−2)(𝑡)(𝛼−1)(𝛼−2)⋯(𝛼−𝑛+2)𝑡^{𝛼−𝑛+1}=lim_𝑡→0+𝑥^(𝑛−1)(𝑡)(𝛼−1)(𝛼−2)⋯(𝛼−𝑛+1)𝑡^𝛼−𝑛=0. (3.44)

Theorem 3.4 (uniqueness). Suppose the conditions of Theorem 3.2 are satisfied and 𝛼=𝑛. Then the positive solution 𝑥(𝑡) of the BVP (1.4) is unique.

Proof. We only need the BVP (2.6) has unique solution. Suppose that 𝑤₁,𝑤₂ are 𝐶[0,1] positive solutions to the BVP (2.6). We may assume, without loss of generality, that there exists 𝑡^∗∈(0,1) such that 𝑤₂(𝑡^∗)−𝑤₁(𝑡^∗)=max{𝑤₂(𝑡)−𝑤₁(𝑡)}>0. Let 𝑎=inf𝑡₁∣0≤𝑡₁<𝑡^∗,𝑤₂(𝑡)≥𝑤₁(𝑡),𝑡∈𝑡₁,𝑡^∗;𝑏=inf𝑡₂∣𝑡^∗<𝑡₂≤1,𝑤₂(𝑡)≥𝑤₁(𝑡),𝑡∈𝑡^∗,𝑡₂;𝑧(𝑡)=𝑤₂(𝑡)−𝑤₁(𝑡),𝑡∈[0,1]. (3.45) Evidently, 𝑡^∗∈(𝑎,𝑏),𝑤₂(𝑡)≥𝑤₁(𝑡),𝑓𝑡,𝐼^𝑛−2𝑤₂(𝑡),𝐼^𝑛−3𝑤₂(𝑡),…,𝐼¹𝑤₂(𝑡),𝑤₂(𝑡)≤𝑓𝑡,𝐼^𝑛−2𝑤₁(𝑡),𝐼^𝑛−3𝑤₁(𝑡),…,𝐼¹𝑤₁(𝑡),𝑤₂(𝑡),𝑡∈[𝑎,𝑏]. (3.46) Thus 𝒟_𝐭^{𝛼−𝑛+2}𝑧(𝑡)=𝑧^′′(𝑡)≥0,𝑡∈[𝑎,𝑏]. (3.47)
By the boundary conditions of the BVP (2.6), it is easy to check that there exist the following two possible cases: (1)𝑧(𝑎)(𝑏)(2)𝑧(𝑎)(𝑏)=0,𝑧=0;=0,𝑧>0; (3.48)

Case 1. From 𝑧^′′(𝑡)≥0 and 𝑧(𝑎)=0,𝑧(𝑏)=0, we get that 𝑧(𝑡)≤0,𝑡∈[𝑎,𝑏], which implies a contradiction with 𝑤₂(𝑡^∗)>𝑤₁(𝑡^∗).

Case 2. In this case we have 𝑏=1 and 𝑧^′(𝑡^∗)=0. Since 𝑧^′(𝑡) is increasing on [𝑎,𝑏], we also have 𝑧^′(𝑡) is increasing on 𝑡∈[𝑎,𝑡^∗], that is, 𝑧^′(𝑡)≤𝑧^′(𝑡^∗)=0 for any 𝑡∈[𝑎,𝑡^∗]. Thus 𝑧(𝑡) is nonincreasing on 𝑡∈[𝑎,𝑡^∗], which implies that 𝑧(𝑡^∗)≤𝑧(𝑎)=0, this is a contradiction with 𝑤₂(𝑡^∗)>𝑤₁(𝑡^∗).
Therefore the BVP (2.6) has unique solution, and the BVP (1.4) also has unique solution.

Remark 3.5. We only get the uniqueness of positive solution for the BVP (1.4) as 𝛼 is integer, but it remains unknown as to whether an analogous unique result holds for the case 𝑛−1<𝛼<𝑛. We believe the result should also hold and try to study this problem, but it is a pity that we failed to find the effective method to solve this problem.

Remark 3.6. The BVP (1.4) allows the nonlinearity 𝑓(𝑡,𝑥₀,𝑥₂,…,𝑥_𝑛−2) has singularity at 𝑡=0,1 and 𝑥_𝑖=0,??𝑖=0,1,…,𝑛−2. If 𝑓(𝑡,𝑥₀,𝑥₂,…,𝑥_𝑛−2) is continuous at 𝑥_𝑖=0,??𝑖=0,1,…,𝑛−2, that is
𝑓∈𝐶((0,1)×[0,∞)^𝑛−1,[0,+∞)), and 𝑓(𝑡,𝑥₀,𝑥₁,𝑥₂,…,𝑥_𝑛−2) is decreasing in 𝑥_𝑖>0 for 𝑖=0,1,2,…,𝑛−2;0<¹₀ℋ(𝑠)𝑓(𝑠,0,0,…,0)𝑑𝑠<+∞. (3.49) then the BVP (1.4) has at least a positive solution 𝑥(𝑡), and there exists a constant ℬ₃>0 such that 0≤𝑥(𝑡)≤ℬ₃𝑡^𝛼−1. (3.50) If 𝛼=𝑛, the positive solution of the BVP (1.4) also is unique.

Proof. In fact, in Theorem 3.2 the set 𝑃 is replaced by the following: 𝑃₁={𝑥∈𝐸∶𝑥(𝑡)≥0,𝑡∈[0,1]} (3.51) and (3.21) can be replaced by the following: 0≤𝜙(𝑡)=𝑇(0),0≤𝜓(𝑡)=𝑇𝜙(𝑡)≤𝑇(0)=𝜙(𝑡). (3.52) Clearly 𝜙(𝑡),𝜓(t)∈𝑃₁, and 𝒟_𝐭^{𝛼−𝑛+2}𝜙(𝑡)+𝑓𝑡,𝐼^𝑛−2𝜙(𝑡),𝐼^𝑛−3𝜙(𝑡),…,𝐼¹𝜙(𝑡)(𝑡(),𝜙)=−𝑓𝑡,0,0,…,0+𝑓𝑡,𝐼^𝑛−2𝑇(0),𝐼^𝑛−3𝑇(0),…,𝐼¹𝑇(0)(0)𝒟,𝑇≤0,_{𝐭𝛼−𝑛+2}𝜓(𝑡)+𝑓𝑡,𝐼^𝑛−2𝜓(𝑡),𝐼^𝑛−3𝜓(𝑡),…,𝐼¹𝜓(𝑡),𝜓(𝑡)=−𝑓𝑡,𝐼^𝑛−2𝜙(𝑡),𝐼^𝑛−3𝜙(𝑡),…,I¹𝜙(𝑡),𝜙(𝑡)+𝑓𝑡,𝐼^𝑛−2𝜓(𝑡),𝐼^𝑛−3𝜓(𝑡),…,𝐼¹𝜓(𝑡)(𝑡)[].,𝜓≥0,𝑡∈0,1 (3.53) Thus the rest of proof is similar to those of Theorem 3.2.
On the other hand, by Lemma 2.6 and (𝐆𝟐), the positive solution of the BVP (2.6) satisfies 0≤𝑤(𝑡)≤𝜙(𝑡)=𝑇0=¹₀𝐾()𝑓()≤𝑡,𝑠𝑠,0,0,…,0𝑑𝑠₁₀ℋ(𝑠)𝑓(𝑠,0,0,…,0)𝑑𝑠𝑡^{𝛼−𝑛+1}, (3.54) and then 0≤𝐼^𝑛−2𝑤(𝑡)=𝑥(𝑡)≤ℬ₃𝑡^𝛼−1, (3.55) where ℬ₃=Γ(𝛼−𝑛+2)Γ(𝛼)¹₀ℋ(𝑠)𝑓(𝑠,0,0,…,0)𝑑𝑠. (3.56) Moreover if 𝛼=𝑛, the positive solution of the BVP (1.4) also is unique.

Example 3.7. Consider the existence of positive solutions for the nonlinear fractional differential equation −𝒟_𝐭^7/2𝑥(𝑡)=10𝑡^−1/4𝑥^′−1/3+𝑥^−1/8𝑥(0),0<𝑡<1,=𝑥^′(0)=0,𝑥^′(1)=¹₀𝑥^′(𝑠)𝑑𝐴(𝑠), (3.57) where 𝐴(𝑡)=10,𝑡∈0,2,321,𝑡∈2,34,31,𝑡∈4,1. (3.58) Then the BVP (3.58) is equivalent to the following 4-point BVP with coefficients of both signs −𝒟_𝐭^7/2𝑥(𝑡)=10𝑡^−1/4𝑥^′−1/3+𝑥^−1/8𝑥(0),0<𝑡<1,=𝑥^′(0)=0,𝑥^′(1)=32𝑥^′12−12𝑥^′34, (3.59)
Conclusion: The BVP (3.59) has at least a positive solution 𝑥(𝑡) such that 25𝑡^5/2≤𝑥(𝑡)≤14.886𝑡^5/2. (3.60)

Proof. Clearly, 0≤𝒞=¹₀𝑡^5/2𝑑𝐴(𝑡)=1−^3/4_1/232𝑑𝑡^5/2+¹_3/4𝑑𝑡^5/2𝐺()=𝐺≈0.0217<1,𝑡,𝑠¹()=𝑡𝑡,𝑠^3/2(1−𝑠)^3/2Γ()𝐺5/2,0≤𝑡≤𝑠≤1,₂()=𝑡𝑡,𝑠^3/2(1−𝑠)^3/2−(𝑡−𝑠)^3/2Γ(5/2),0≤𝑠≤𝑡≤1. (3.61) Thus, 𝒢_𝐴(𝑠)=32𝐺₂12−1,𝑠2𝐺₂341,𝑠,0≤𝑠<2,32𝐺₁12−1,𝑠2𝐺₂34,1,𝑠23≤𝑠<4,32𝐺₁12−1,𝑠2𝐺₁34,3,𝑠4=()√≤𝑠≤1,3/22−(3/4)√3(1−𝑠)^3/2+2((3/4)−𝑠)^3/2−6((1/2)−𝑠)^3/23√𝜋1,0≤𝑠<2,(3/2)√2−(3/4)√3(1−𝑠)^3/2+2((3/4)−𝑠)^3/23√𝜋,123≤𝑠<4,(3/2)√2−(3/4)√3(1−𝑠)^3/23√𝜋,34ℋ(𝑠)=()≤𝑠≤1,1−𝑠^3/2Γ()+𝒢5/2^𝐴(𝑠).0.9783 (3.62) Clearly, (𝐇𝟎) and (𝐇𝟏) hold.
On the other hand, for any 𝜆_𝑖>0, 𝑖=0,1, 𝑓(𝑡,𝜆₀𝑡,𝜆₁)=𝜆₀^−1/8𝑡^−3/8+𝜆₁^−1/3𝑡^−1/4≢0, 𝑡∈(0,1), and 𝜅₀(𝑠)=25𝑠^5/2,𝜅₁(𝑠)=𝑠^3/2,0<¹₀ℋ(𝑠)𝑓𝑠,𝜆₀𝜅₀(𝑠),𝜆₁𝜅₁(𝑠)𝑑𝑠=10¹₀(1−𝑠)^3/2Γ()+𝒢5/2^𝐴(𝑠)20.97835𝜆₀^−1/8𝑠^−9/16+𝜆₁^−1/3𝑠^−3/4𝑑𝑠<+∞. (3.63) Thus (𝐇𝟐) is satisfied.
Now we compute 𝐿,𝑙. Since ¹₀𝒢_𝐴(𝑠)𝑓𝑠,𝜅₀(𝑠),𝜅₁(𝑠)𝑑𝑠=10¹₀𝒢_𝐴(𝑠2)5^−1/8𝑠^−9/16+𝑠^−3/4𝑑𝑠=10₀^1/2(3/2)√2−(3/4)√3(1−𝑠)^3/2+2((3/4)−𝑠)^3/2−6((1/2)−𝑠)^3/23√𝜋×25^−1/8𝑠^−9/16+𝑠^−3/4𝑑𝑠+10^3/4_1/2(3/2)√2−(3/4)√3(1−𝑠)^3/2+2((3/4)−𝑠)^3/23√𝜋25^−1/8𝑠^−9/16+𝑠^−3/4+10¹_3/4(3/2)√2−(3/4)√3(1−𝑠)^3/23√𝜋25^−1/8𝑠^−9/16+𝑠^−3/4𝑑𝑠=2.600,₁₀ℋ(𝑠)𝑓𝑠,𝜅₀(𝑠),𝜅₁(𝑠)𝑑𝑠=10¹₀ℋ(𝑠2)5^−1/8𝑠^−9/16+𝑠^−3/4𝑑𝑠=10¹₀(1−𝑠)^3/2Γ()+𝒢5/2^𝐴(𝑠)20.97835^−1/8𝑠^−9/16+𝑠^−3/4𝑑𝑠=37.215. (3.64) Thus 𝐿=max1,¹₀ℋ(𝑠)𝑓𝑠,𝜅₀(𝑠),𝜅₁(𝑠){}1𝑑𝑠=max1,37.215=37.215,𝑙=min1,1−𝒞¹₀𝒢_𝐴(𝑠)𝑓𝑠,𝜅₀(𝑠),𝜅₁(𝑠)𝑑𝑠=min{1,2.658}=2.658. (3.65)
So ¹₀𝒢_𝐴(𝑠)𝑓𝐿𝑠,𝑙𝜅₀(𝑠),𝐿𝑙𝜅₁(𝑠)𝑑𝑠=10¹₀𝒢_𝐴(𝑠2)14×5^−1/8𝑠^−9/16+14^−1/3𝑠^−3/4𝑑𝑠=4.540≥1−𝒞=0.9783, (3.66) which implies that (3.16) holds. Then the BVP (1.4) has at least a positive solution 𝑥(𝑡), and there exist two constants ℬ₁=Γ(𝛼−𝑛+2)Γ(𝛼)=25,ℬ₂=𝐿ℬ₁=14.886, (3.67) such that 25𝑡^5/2≤𝑥(𝑡)≤14.886𝑡^5/2. (3.68)

Acknowledgment

This work is supported by the National Natural Science Foundation of China (61201143) and the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2010091), the National Science Foundation for Postdoctoral Scientists of China (Grant no. 2012M510956), and the Postdoctoral Funds of Heilongjiang Province (Grant no. LBHZ11128).