Positive Solutions for a Fractional Boundary Value Problem with Changing Sign Nonlinearity

Abstract

We discuss the existence of positive solutions to the following fractional m-point boundary value problem with changing sign nonlinearity 𝐷^𝛼₀₊𝑢(𝑡)+𝜆𝑓(𝑡,𝑢(𝑡))=0,0<𝑡<1,𝑢(0)=0,𝐷^𝛽₀₊𝑢(1)=∑^𝑚−2_𝑖=1𝜂_𝑖𝐷^𝛽₀₊𝑢(𝜉_𝑖), where 𝜆 is a positive parameter, 1<𝛼≤2, 0<𝛽<𝛼−1, 0<𝜉₁<⋯<𝜉_𝑚−2<1 with ∑^𝑚−2_𝑖=1𝜂_𝑖𝜉_𝑖^{𝛼−𝛽−1}<1, 𝐷^𝛼₀₊ is the standard Riemann-Liouville derivative, 𝑓 and may be singular at 𝑡=0 and/or 𝑡=1 and also may change sign. The work improves and generalizes some previous results.

1. Introduction

In this paper, we consider the following fractional differential equation with 𝑚-point boundary conditions: 𝐷^𝛼₀₊𝑢(𝑡)+𝜆𝑓(𝑡,𝑢(𝑡))=0,0<𝑡<1,𝑢(0)=0,𝐷^𝛽₀₊𝑢(1)=^𝑚−2_𝑖=1𝜂_𝑖𝐷^𝛽₀₊𝑢𝜉_𝑖, (1.1) where 1<𝛼≤2, 𝜆>0 is a parameter, 0<𝛽<𝛼−1, 0<𝜉₁<⋯<𝜉_𝑚−2<1 with ∑^𝑚−2_𝑖=1𝜂_𝑖𝜉_𝑖^{𝛼−𝛽−1}<1, 𝐷^𝛼₀₊ is the standard Riemann-Liouville derivative, and 𝑓∈𝐶((0,1)×[0,+∞)→(−∞,+∞)) may be singular at 𝑡=0 and/or 𝑡=1 and also may change sign. In this paper, by a positive solution to (1.1), we mean a function 𝑢∈𝐶[0,1] which is positive on (0,1] and satisfies (1.1).

In recent years, great efforts have been made worldwide to study the existence of solutions for nonlinear fractional differential equations by using nonlinear analysis methods [1–24]. Fractional-order multipoint boundary value problems (BVP) have particularly attracted a great deal of attention (see, e.g., [13–19]). In [10], the authors discussed some properties of the Green function for the Direchlet-type BVP of nonlinear fractional differential equations 𝐷^𝛼₀₊𝑢(𝑡)((𝑡𝑢(0)(1)+𝑓𝑡,𝑢))=0,0<𝑡<1,=0,𝑢=0, (1.2) where 1<𝛼<2, 𝐷^𝛼₀₊ is the standard Riemann-Liouville derivative and 𝑓∶𝐶([0,1]×[0,+∞)→[0,+∞)) is continuous. By using the Krasnosel'skii fixed-point theorem, the existence of positive solutions was obtained under some suitable conditions on 𝑓.

In [14], the authors investigated the existence and multiplicity of positive solutions by using some fixed-point theorems for the fractional differential equation given by 𝐷^𝛼₀₊𝑢(𝑡)((𝑡𝑢(0)+𝑓𝑡,𝑢))=0,0<𝑡<1,=0,𝐷^𝛽₀₊𝑢(1)=𝑎𝐷^𝛽₀₊𝑢(𝜉), (1.3) where 1<𝛼≤2, 0≤𝛽≤1, 0<𝜉<1, 0≤𝑎≤1 with 𝑎𝜉^{𝛼−𝛽−2}<1−𝛽, 0≤𝛼−𝛽−1, 𝑓 is nonnegative.

It should be noted that in most of the works in literature the nonlinearity needs to be nonnegative in order to establish positive solutions. As far as we know, semipositone fractional nonlocal boundary value problems with 1<𝛼≤2 have been seldom studied due to the difficulties in finding and analyzing the corresponding Green function.

In [23], the authors investigated the following fractional differential equation with three-point boundary conditions: 𝐷^𝛼₀₊𝑢(𝑡)((𝑡(𝑡)𝑢(0)+𝑓𝑡,𝑢))+𝑒=0,0<𝑡<1,=0,𝐷^𝛽₀₊𝑢(1)=𝑎𝐷^𝛽₀₊𝑢(𝜉), (1.4) where 1<𝛼≤2, 0<𝛽≤1, 0<𝜉<1, 0≤𝑎≤1,0≤𝛼−𝛽−1, 𝑒(𝑡)∈𝐿[0,1], and 𝑓 satisfies the Caratheodory conditions. The authors obtained the properties of the Green function for (1.4) as follows: 𝛽𝑡^𝛼−1𝑠(1−𝑠)^{𝛼−𝛽−1}Γ(𝛼)()≤𝑡≤𝐺𝑡,𝑠^𝛼−1(1−𝑠)^{𝛼−𝛽−1}Γ(𝛼)1−𝑎𝜉^{𝛼−𝛽−1}. (1.5) By using the Schauder fixed-point theorem, the authors obtained the existence of positive solution of (1.4) with the following assumptions:(𝐴₁) for each 𝐿>0, there exists a function 𝜙_𝐿≻0 such that 𝑓(𝑡,𝑡^𝛼−1𝑥)≥𝜙_𝐿(𝑡) for a.e. 𝑡∈(0,1), for all 𝑥∈(0,𝐿];(𝐴₂) there exist 𝑔(𝑥),ℎ(𝑥), and 𝑘(𝑡)≻0, such that 0≤𝑓(𝑡,𝑥)≤𝑘(𝑡)𝑔(𝑥)+ℎ(𝑥),∀𝑥∈(0,∞),a.e.𝑡∈(0,1), (1.6)here 𝑔∶(0,∞)→[0,∞) is continuous and nonincreasing, ℎ∶[0,∞)→[0,∞) is continuous, and ℎ/𝑔 is nondecreasing;(𝐴₃) There exist two positive constants 𝑅>𝑟>0 such that 𝑅>Φ_𝑅1+𝛾_∗≥𝑟>0,₁₀𝑘(𝑠)𝑔𝑟𝑠^𝛼−1ℎ(𝑅)𝑑𝑠<+∞,𝑅≥1+𝑔(𝑅)¹₀(1−𝑠)^{𝛼−𝛽−1}Γ(𝛼)1−𝑎𝜉^{𝛼−𝛽−1}𝑘(𝑠)𝑔𝑟𝑠^𝛼−1𝑑𝑠+𝛾_∗. (1.7)Here Φ_𝑅1=¹₀𝛽𝑠(1−𝑠)^{𝛼−𝛽−1}Γ(𝛼)𝜙_𝑅(𝑠)𝑑𝑠. (1.8) The assumptions on nonlinearity are not suitable for frequently used conditions, such as superlinear or some sublinear. For instance, 𝑓(𝑡,𝑥)=𝑥^𝜇, 𝜇>0, obviously, 𝑓 does not satisfy (𝐴₁).

Inspired by the previous work, the aim of this paper is to establish conditions for the existence of positive solutions of the more general BVP (1.1). Our work presented in this paper has the following new features. Firstly, we consider few cases of 1<𝛼≤2 which has been studied before, and in dealing with the difficulties related to the Green function for this case, some new properties of the Green function have been discovered. Secondly, the BVP (1.1) possesses singularity; that is, 𝑓 may be singular at 𝑡=0 and/or 𝑡=1. Thirdly, the nonlinearity 𝑓 may change sign and may be unbounded from below. Finally, we impose weaker positivity conditions on the nonlocal boundary term; that is, some of the coefficients 𝜂_𝑖 may be negative.

The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas that are to be used to prove our main results. We also discover some new positive properties of the corresponding Green function. In Section 3, we discuss the existence of positive solutions of the semipositone BVP (1.1). In Section 4, we give an example to demonstrate the application of our theoretical results.

2. Basic Definitions and Preliminaries

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions can also be found in the recent literature.

Definition 2.1. The fractional integral of order 𝛼>0 of a function 𝑢∶(0,+∞)→𝑅 is given by 𝐼^𝛼₀₊𝑢(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑢(𝑠)𝑑𝑠, (2.1) provided that the right-hand side is pointwisely defined on (0,+∞).

Definition 2.2. The Riemann-Liouville fractional derivative of order 𝛼>0 of a function 𝑢∶(0,+∞)→𝑅 is given by 𝐷^𝛼₀₊𝑢(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑢(𝑠)𝑑𝑠, (2.2) where 𝑛=[𝛼]+1 and [𝛼] denotes the integer part of the number 𝛼, provided that the right-hand side is pointwisely defined on (0,+∞).

Lemma 2.3 (see [3]). Let 𝛼>0. Then the following equality holds for 𝑢∈𝐿(0,1), 𝐷^𝛼₀₊𝑢∈𝐿(0,1); 𝐼^𝛼₀₊𝐷^𝛼₀₊𝑢(𝑡)=𝑢(𝑡)+𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋅⋅⋅+𝑐_𝑛𝑡^𝛼−𝑛, (2.3) where 𝑐_𝑖∈𝑅, 𝑖=1,2,…,𝑛, 𝑛−1<𝛼≤𝑛.

Set 𝐺₀()=1𝑡,𝑠Γ(𝛼)𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}𝑡,0≤𝑡≤𝑠≤1,_𝛼−1(1−𝑠)^{𝛼−𝛽−1}−(𝑡−𝑠)^𝛼−1𝑝(𝑠),0≤𝑠≤𝑡≤1,=1−_{𝑠≤𝜉𝑖}𝜂_𝑖𝜉_𝑖−𝑠1−𝑠^{𝛼−𝛽−1},𝐺(𝑡,𝑠)=𝐺₀(𝑡,𝑠)+𝑞(𝑠)𝑡^𝛼−1, (2.4)(2.5)(2.6) where 𝑞(𝑠)=𝑝(𝑠)−𝑝(0)Γ(𝛼)𝑝(0)(1−𝑠)^{𝛼−𝛽−1},𝑝(0)=1−^𝑚−2_𝑖=1𝜂_𝑖𝜉_𝑖^{𝛼−𝛽−1}. (2.7) For convenience in presentation, we here list the assumption to be used throughout the paper.(𝐻₁)𝑝(0)>0, 𝑞(𝑠)≥0 on [0,1].

Remark 2.4. If 𝜂_𝑖=0(𝑖=1,…,𝑚−2), we have 𝑝(0)=1 and 𝑞(𝑠)≡0. If 𝜂_𝑖≥0(𝑖=1,…,𝑚−2) and ∑^𝑚−2_𝑖=1𝜂_𝑖𝜉_𝑖^{𝛼−𝛽−1}<1, we have 𝑞(𝑠)≥0 on [0,1].

Lemma 2.5 (see [14]). Assume that 𝑔(𝑡)∈𝐿[0,1] and 𝛼>𝛽>0. Then 𝐷^𝛽₀₊^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠)Γ(𝛼)𝑑𝑠=Γ𝛼−𝛽^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}𝑔(𝑠)𝑑𝑠. (2.8)

Lemma 2.6. Assume (𝐻₁) holds, and 𝑦(𝑡)∈𝐿[0,1]. Then the unique solution of the problem 𝐷^𝛼₀₊𝑢(𝑡)(𝑡)𝑢(0)+𝑦=0,0<𝑡<1,=0,𝐷^𝛽₀₊𝑢(1)=^𝑚−2_𝑖=1𝜂_𝑖𝐷^𝛽₀₊𝑢𝜉_𝑖 (2.9) is 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠, (2.10) where 𝐺(𝑡,𝑠) is the Green function of the boundary value problem (2.9).

Proof. From Lemma 2.3, the solution of (2.9) is 𝑢(𝑡)=−𝐼^𝛼₀₊𝑦(𝑡)+𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2. (2.11) Consequently, 𝑢(𝑡)1=−Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠+𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2. (2.12) From 𝑢(0)=0, we have 𝑐₂=0.
By Lemma 2.5, we have 𝐷^𝛽₀₊𝑢(𝑡)1=−Γ𝛼−𝛽^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}𝑦(𝑠)𝑐𝑑𝑠+¹Γ(𝛼)Γ𝑡𝛼−𝛽^{𝛼−𝛽−1}. (2.13) Therefore, 𝐷^𝛽₀₊𝑢(1)1=−Γ𝛼−𝛽¹₀(1−𝑠)^{𝛼−𝛽−1}𝑦(𝑠)𝑐𝑑𝑠+¹Γ(𝛼)Γ,𝐷𝛼−𝛽_𝛽0+𝑢𝜉_𝑖1=−Γ𝛼−𝛽^𝜉𝑖₀𝜉_𝑖−𝑠^{𝛼−𝛽−1}𝑦(𝑠)𝑐𝑑𝑠+¹Γ(𝛼)Γ𝜉𝛼−𝛽^{𝑖𝛼−𝛽−1}. (2.14) By 𝐷^𝛽₀₊𝑢(1)=∑^𝑚−2_𝑖=1𝜂_𝑖𝐷^𝛽₀₊𝑢(𝜉_𝑖), we have 𝑐₁=∫¹₀(1−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠−∑^𝑚−2_𝑖=1𝜂_𝑖∫^𝜉𝑖₀𝜉_𝑖−𝑠^{𝛼−𝛽−1}𝑦(𝑠)𝑑𝑠Γ(𝛼)𝑝(0)=∫¹₀(1−𝑠)^{𝛼−𝛽−1}𝑝(𝑠)𝑦(𝑠)𝑑𝑠Γ(𝛼)𝑝(0). (2.15) Therefore, the solution of (2.9) is 𝑢(𝑡)=𝑐₁𝑡^𝛼−1−1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑦(𝑠)=𝑑𝑠₁₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠. (2.16)

Lemma 2.7. The function 𝐺₀(𝑡,𝑠) has the following properties:(1)𝐺₀(𝑡,𝑠)>0, for 𝑡,𝑠∈(0,1);(2)Γ(𝛼)𝐺₀(𝑡,𝑠)≤𝑡^𝛼−1, for 𝑡,𝑠∈[0,1];(3)𝛽𝑡^𝛼−1ℎ(𝑠)≤Γ(𝛼)𝐺₀(𝑡,𝑠)≤ℎ(𝑠)𝑡^𝛼−2,  for  𝑡,𝑠∈(0,1),where ℎ(𝑠)=𝑠(1−𝑠)^{𝛼−𝛽−1}. (2.17)

Proof. (1) When 0<𝑡≤𝑠<1, it is clear that 𝐺₀()=1𝑡,𝑠Γ(𝛼)𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}>0. (2.18) When 0<𝑠≤𝑡<1, we have 𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}−(𝑡−𝑠)^𝛼−1≥𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}−𝑡^𝛼−1(1−𝑠)^𝛼−1=𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}1−(1−𝑠)^𝛽>0. (2.19)
(2) By (2.4), for any 𝑡,𝑠∈[0,1], we have Γ(𝛼)𝐺₀(𝑡,𝑠)≤𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}≤𝑡^𝛼−1. (2.20)
In the following, we will prove (3).(i) When 0<𝑠≤𝑡<1, noticing that 0<𝛽<𝛼−1≤1, we have 𝜕𝑡𝜕𝛽^𝛼−2𝑠(1−𝑠)^{𝛼−𝛽−1}−𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}=𝑡^𝛼−2(1−𝑠)^{𝛼−𝛽−1}(𝑡−𝑠)ln(1−𝑠)≤0. (2.21)Therefore, 𝑡^𝛼−2𝑠(1−𝑠)^{𝛼−𝛽−1}−𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}−(𝑡−𝑠)^𝛼−1≥𝑡^𝛼−2𝑠−𝑡^𝛼−1+(𝑡−𝑠)^𝛼−1=−𝑡^𝛼−2(𝑡−𝑠)+(𝑡−𝑠)^𝛼−1≥0, (2.22) which implies Γ(𝛼)𝐺₀(𝑡,𝑠)≤ℎ(𝑠)𝑡^𝛼−2. (2.23)
On the other hand, we have 𝑑()𝑑𝑠𝛽𝑠+1−𝑠^𝛽≤0,𝑠∈[0,1). (2.24) Therefore, 𝛽𝑠+(1−𝑠)^𝛽≤1, which implies 1−(1−𝑠)^𝛽≥𝛽𝑠. (2.25) Then Γ(𝛼)𝐺₀(𝑡,𝑠)=𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}−(𝑡−𝑠)^𝛼−1≥𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}−(𝑡−𝑠)^𝛽(𝑡−𝑡𝑠)^{𝛼−𝛽−1}=𝑠1−1−𝑡^𝛽𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}≥1−(1−𝑠)^𝛽𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}≥𝛽𝑡^𝛼−1ℎ(𝑠). (2.26)(ii) When 0<𝑡≤𝑠<1, we have Γ(𝛼)𝐺₀(𝑡,𝑠)=𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}=𝑡^𝛼−2𝑡(1−𝑠)^{𝛼−𝛽−1}≤𝑡^𝛼−2𝑠(1−𝑠)^{𝛼−𝛽−1}=ℎ(𝑠)𝑡^𝛼−2, (2.27)On the other hand, clearly we have Γ(𝛼)𝐺₀(𝑡,𝑠)=𝑡^𝛼−1(1−𝑠)^{𝛼−𝛽−1}≥𝛽𝑡^𝛼−1ℎ(𝑠). (2.28) The inequalities (2.23)–(2.28) imply that (3) holds.

By Lemma 2.7, we have the following results.

Lemma 2.8. Assume (𝐻₁) holds, then the Green function defined by (2.6) satisfies(1)𝐺(𝑡,𝑠)>0, for all 𝑡,𝑠∈(0,1);(2)𝐺(𝑡,𝑠)≤𝑡^𝛼−1((1/(Γ(𝛼)))+𝑞(𝑠)), for all 𝑡,𝑠∈[0,1];(3)𝛽𝑡^𝛼−1Φ(𝑠)≤𝐺(𝑡,𝑠)≤𝑡^𝛼−2Φ(𝑠), for all 𝑡,𝑠∈(0,1),where Φ(𝑠)=ℎ(𝑠)Γ(𝛼)+𝑞(𝑠). (2.29)

Lemma 2.9. Assume (𝐻₁) holds, then the function 𝐺^∗(𝑡,𝑠)=∶𝑡^2−𝛼𝐺(𝑡,𝑠) satisfies(1)𝐺^∗(𝑡,𝑠)>0, for all 𝑡,𝑠∈(0,1);(2)𝐺^∗(𝑡,𝑠)≤𝑡((1/(Γ(𝛼)))+𝑞(𝑠)), for all 𝑡,𝑠∈[0,1];(3)𝛽𝑡Φ(𝑠)≤𝐺^∗(𝑡,𝑠)≤Φ(𝑠), for all 𝑡,𝑠∈[0,1].

For convenience, we list here four more assumptions to be used later:(𝐻₂)𝑓∈𝐶((0,1)×[0,+∞),(−∞,+∞)) satisfies 𝑓(𝑡,𝑥)≥−𝑟(𝑡),𝑓𝑡,𝑡^𝛼−2𝑥≤𝑧(𝑡)𝑔(𝑥),𝑡∈(0,1),𝑥∈[0,+∞), (2.30)where 𝑟,𝑧∈𝐶((0,1),[0,+∞)), 𝑔∈𝐶([0,+∞),[0,+∞)).(𝐻₃)∫¹₀𝑟(𝑠)𝑑𝑠<+∞, 0<∫¹₀𝑧(𝑠)𝑑𝑠<+∞.(𝐻₄) There exists [𝑎,𝑏]⊂(0,1) such that liminf_𝑥→+∞min_{𝑡∈[𝑎,𝑏]}𝑓(𝑡,𝑥)𝑥=+∞. (2.31)(𝐻₅) There exists [𝑐,𝑑]⊂(0,1) such that liminf_𝑥→+∞min_{[]𝑡∈𝑐,𝑑}𝑓(𝑡,𝑥)=+∞,lim_𝑥→+∞𝑔(𝑥)𝑥=0. (2.32)

Remark 2.10. The second limit of (𝐻₅) implies lim_𝑢→+∞𝑔^∗(𝑢)𝑢=0, (2.33) where 𝑔^∗(𝑢)=max_{𝑥∈[0,𝑢]}𝑔(𝑥). (2.34)

Proof. By lim_𝑢→+∞(𝑔(𝑢)/𝑢)=0, for any 𝜖>0, there exists 𝑁₁>0, such that for any 𝑢>𝑁₁ we have 0≤𝑔(𝑢)<𝜖𝑢. (2.35) Let 𝑁=max{𝑁₁,((𝑔^∗(𝑁₁))/𝜖)}, for any 𝑢>𝑁 we have 0≤𝑔^∗(𝑢)<𝜖𝑢+𝑔^∗𝑁₁<2𝜖𝑢. (2.36) Therefore, lim_𝑢→+∞((𝑔^∗(𝑢))/𝑢)=0.

Lemma 2.11. Assume (𝐻₁) holds and 𝑟(𝑡)∈𝐶(0,1)∩𝐿[0,1] is nonnegative, then the BVP 𝐷^𝛼₀₊𝑢(𝑡)+𝑟(𝑡)=0,0<𝑡<1,𝑢(0)=0,𝐷^𝛽₀₊𝑢(1)=^𝑚−2_𝑖=1𝜂_𝑖𝐷^𝛽₀₊𝑢𝜉_𝑖 (2.37) has a unique solution 𝜔(𝑡)=∫¹₀𝐺(𝑡,𝑠)𝑟(𝑠)𝑑𝑠 with 𝜔(𝑡)≤𝑘𝑡^𝛼−1, where 𝑘=¹₀1Γ(𝛼)+𝑞(𝑠)𝑟(𝑠)𝑑𝑠,𝑡∈[0,1]. (2.38)

Proof. By Lemma 2.6, 𝜔(𝑡)=∫¹₀𝐺(𝑡,𝑠)𝑟(𝑠)𝑑𝑠 is the unique solution of (2.37). By (2) of Lemma 2.8, we have 𝜔(𝑡)=¹₀𝐺(𝑡,𝑠)𝑟(𝑠)𝑑𝑠≤𝑡^𝛼−11₀1Γ(𝛼)+𝑞(𝑠)𝑟(𝑠)𝑑𝑠. (2.39)

Let 𝐸=𝐶[0,1] be endowed with the maximum norm ‖𝑢‖=max_0≤𝑡≤1|𝑢(𝑡)| and define a cone 𝑃 by 𝑃=𝑢(𝑡)∈𝐸∶thereexists𝑙_𝑢>0suchthat𝛽𝑡‖𝑢‖≤𝑢(𝑡)≤𝑙_𝑢𝑡, (2.40) and then set 𝐵_𝑟={𝑢(𝑡)∈𝐸∶‖𝑢‖<𝑟}, 𝑃_𝑟=𝑃∩𝐵_𝑟, 𝜕𝑃_𝑟=𝑃∩𝜕𝐵_𝑟.

Next we consider the following singular nonlinear BVP: 𝐷^𝛼₀₊𝑢(𝑡)+𝜆𝑓𝑡,[𝑢(𝑡)−𝜆𝜔(𝑡)]⁺+𝑟(𝑡)=0,0<𝑡<1,𝑢(0)=0,𝐷^𝛽₀₊𝑢(1)=^𝑚−2_𝑖=1𝜂_𝑖𝐷^𝛽₀₊𝑢𝜉_𝑖, (2.41) where 𝜆>0, [𝑣(𝑡)]⁺=max{𝑣(𝑡),0}, 𝜔(𝑡) is defined in Lemma 2.11.

Let 𝑇𝑢(𝑡)=𝜆¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)−𝜆𝜔(𝑠)⁺+𝑟(𝑠)𝑑𝑠. (2.42)

Clearly, if 𝑢(𝑡)∈𝑃 is a fixed point of 𝑇, then 𝑦(𝑡)=𝑡^𝛼−2𝑢(𝑡) is a positive solution of (2.41).

Lemma 2.12. Suppose that (𝐻₁)–(𝐻₃) hold. Then 𝑇∶𝑃→𝑃 is a completely continuous operator.

Proof. It is clear that 𝑇 is well defined on 𝑃. For any 𝑢∈𝑃, Lemma 2.9 implies 𝑇𝑢(𝑡)≥𝛽𝑡𝜆¹₀Φ(𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)−𝜆𝜔(𝑠)⁺+𝑟(𝑠)𝑑𝑠. (2.43) On the other hand, 𝑇𝑢(𝑡)≤𝜆¹₀Φ(𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)−𝜆𝜔(𝑠)⁺+𝑟(𝑠)𝑑𝑠. (2.44) Therefore, 𝑇𝑢(𝑡)≥𝛽𝑡‖𝑇𝑢‖. Noticing that 𝑇𝑢(𝑡)≤𝜆𝑡¹₀1Γ(𝛼)+𝑞(𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)−𝜆𝜔(𝑠)⁺+𝑟(𝑠)𝑑𝑠, (2.45) we have 𝑇∶𝑃→𝑃.
Using the Ascoli-Arzela theorem, we can then get that 𝑇∶𝑃→𝑃 is a completely continuous operator.  

Lemma 2.13 (see [25]). Let 𝐸 be a real Banach space and let 𝑃⊂𝐸 be a cone. Assume that Ω₁ and Ω₂ are two-bounded open subsets of 𝐸 with 𝜃∈Ω₁,Ω₁⊂Ω₂, 𝑇∶𝑃∩(Ω₂⧵Ω₁)→𝑃 a completely continuous operator such that either(1)‖𝑇𝑢‖≤‖𝑢‖, 𝑢∈𝑃∩𝜕Ω₁ and ‖𝑇𝑢‖≥‖𝑢‖, 𝑢∈𝑃∩𝜕Ω₂, or(2)‖𝑇𝑢‖≥‖𝑢‖, 𝑢∈𝑃∩𝜕Ω₁ and ‖𝑇𝑢‖≤‖𝑢‖, 𝑢∈𝑃∩𝜕Ω₂.Then 𝑇 has a fixed point in 𝑃∩(Ω₂⧵Ω₁).

3. Existence of Positive Solutions

Theorem 3.1. Suppose that (𝐻₁)–(𝐻₄) hold. Then there exists 𝜆^∗>0 such that the BVP (1.1) has at least one positive solution for any 𝜆∈(0,𝜆^∗).

Proof . Choose 𝑟₁>𝑘𝛽⁻¹. Let 𝜆^∗𝑟=min1,¹𝑔^∗𝑟₁+1∫¹₀Φ(𝑠)(𝑧(𝑠)+𝑟(𝑠))𝑑𝑠, (3.1) where 𝑔^∗(𝑟)=max_{𝑥∈[0,𝑟]}𝑔(𝑥). (3.2) In the following of the proof, we suppose 𝜆∈(0,𝜆^∗).
For any 𝑢∈𝜕𝑃_𝑟1, noticing 𝑢(𝑡)≥𝛽𝑡𝑟₁ and Lemma 2.11, we have 𝑡^𝛼−2𝑢(𝑡)−𝜆𝜔(𝑡)≥𝛽𝑟₁−𝜆𝑘𝑡^𝛼−1≥𝛽𝑟₁−𝑘𝑡^𝛼−1𝑟≥0,₁≥𝑢(𝑡)−𝜆𝑡^2−𝛼𝜔(𝑡)≥𝛽𝑟₁−𝑘𝑡≥0. (3.3)(3.4) Therefore, 𝑇𝑢(𝑡)=𝜆¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)−𝜆𝜔(𝑠)⁺+𝑟(𝑠)𝑑𝑠≤𝜆¹₀Φ(𝑠)𝑧(𝑠)𝑔𝑢(𝑠)−𝜆𝑠^2−𝛼𝜔(𝑠)⁺(𝑠)𝑔+𝑟𝑑𝑠≤𝜆^∗𝑟₁+1¹₀Φ(𝑠)(𝑧(𝑠)+𝑟(𝑠))𝑑𝑠<𝜆^∗𝑔^∗𝑟₁+1¹₀Φ(𝑠)(𝑧(𝑠)+𝑟(𝑠))𝑑𝑠≤𝑟₁. (3.5) Thus, ‖𝑇𝑢‖≤‖𝑢‖,∀𝑢∈𝜕𝑃_𝑟1. (3.6)
Now choose a real number 2𝐿>𝜆𝛽²∫^𝑏_𝑎Φ(𝑠)𝑠^𝛼−1.𝑑𝑠 (3.7) By (𝐻₄), there exists a constant 𝑁>0 such that 𝑓(𝑡,𝑥)>𝐿𝑥,forany𝑡∈[𝑎,𝑏],𝑥≥𝑁. (3.8) Let 𝑟₂=𝑟₁+2𝑘𝛽+2𝑁𝛽𝑎^𝛼−1. (3.9) Then for any 𝑢∈𝜕𝑃_𝑟2, we have 𝑡^𝛼−2𝑢(𝑡)−𝜆𝜔(𝑡)≥𝛽𝑟₂−𝑘𝑡^𝛼−1≥𝛽𝑟₂2𝑡^𝛼−1,∀𝑡∈(0,1]. (3.10) Thus, for any 𝑡∈[𝑎,𝑏], we have 𝑡^𝛼−2𝑢(𝑡)−𝜆𝜔(𝑡)>𝑁. Hence, we get ‖𝑇𝑢‖=max_[]𝑡∈0,1𝜆¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)−𝜆𝜔(𝑠)⁺+𝑟(𝑠)𝑑𝑠≥max_[]𝑡∈0,1𝜆^𝑏_𝑎𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)−𝜆𝜔(𝑠)𝑑𝑠≥max_[]𝑡∈0,1𝜆𝐿^𝑏_𝑎𝐺^∗(𝑡,𝑠)𝑠^𝛼−2𝑢(𝑠)−𝜆𝜔(𝑠)𝑑𝑠≥max_[]𝑡∈0,1𝜆𝐿^𝑏_𝑎𝐺^∗(𝑡,𝑠)𝛽𝑟₂2𝑠^𝛼−1𝑑𝑠≥max_[]𝑡∈0,1𝜆𝐿𝛽²𝑟₂2𝑡^𝑏_𝑎Φ(𝑠)𝑠^𝛼−1=𝑑𝑠𝜆𝐿𝛽²𝑟₂2^𝑏_𝑎Φ(𝑠)𝑠^𝛼−1𝑑𝑠≥𝑟₂. (3.11) Therefore, ‖𝑇𝑢‖≥‖𝑢‖,∀𝑢∈𝜕𝑃_𝑟2. (3.12) By Lemma 2.13, 𝑇 has a fixed point 𝑢∈𝑃 such that 𝑟₁≤‖𝑢‖≤𝑟₂. Let 𝑢(𝑡)=𝑡^𝛼−2𝑢(𝑡)−𝜆𝜔(𝑡). Since ‖𝑢‖≥𝑟₁, by (3.3) we have 𝑢(𝑡)≥0 on (0,1] and lim_𝑡→0+𝑡^𝛼−2𝑢(𝑡)=0. Notice that 𝜔(𝑡) is the solution of (2.37) and 𝑡^𝛼−2𝑢(𝑡) is the solution of (2.41). Thus, 𝑢(𝑡) is a positive solution of the BVP (1.1).

Theorem 3.2. Suppose that (𝐻₁)–(𝐻₃) and (𝐻₅) hold. Then there exists 𝜆^∗>0 such that the BVP (1.1) has at least one positive solution for any 𝜆∈(𝜆^∗,+∞).

Proof. By the first limit of (𝐻₅), there exists 𝑁>0 such that 𝑓(𝑡,𝑥)≥2𝑘𝛽²∫^𝑑_𝑐Φ(𝑠)[]𝑑𝑠,forany𝑡∈𝑐,𝑑,𝑥≥𝑁. (3.13)
Let 𝜆^∗=𝑁𝑘𝑐^𝛼−1. (3.14) In the following part of the proof, we suppose 𝜆>𝜆^∗.
Let 𝑅₁=2𝜆𝑘𝛽. (3.15) Then for any 𝑢∈𝜕𝑃_𝑅1, we have 𝑡^𝛼−2𝑢(𝑡)−𝜆𝜔(𝑡)≥𝛽𝑅₁−𝜆𝑘𝑡^𝛼−1=𝜆𝑘𝑡^𝛼−1≥𝜆^∗𝑘𝑡^𝛼−1,∀𝑡∈(0,1]. (3.16) Therefore, 𝑡^𝛼−2𝑢(𝑡)−𝜆𝜔(𝑡)≥𝑁, for any 𝑡∈[𝑐,𝑑] and 𝑢∈𝜕𝑃_𝑅1. Then 𝑇𝑢(𝑡)=𝜆¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)−𝜆𝜔(𝑠)⁺+𝑟(𝑠)𝑑𝑠≥𝜆^𝑑_𝑐𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)−𝜆𝜔(𝑠)⁺≥𝑑𝑠2𝜆𝑘𝛽²∫^𝑑_𝑐Φ(𝑠)𝑑𝑠^𝑑𝑐𝐺^∗()≥𝑡,𝑠𝑑𝑠2𝜆𝑘𝑡𝛽∫^𝑑_𝑐Φ(𝑠)𝑑𝑠^𝑑𝑐Φ(𝑠)𝑑𝑠=𝑅₁𝑡. (3.17) This implies ‖𝑇𝑢‖≥‖𝑢‖,∀𝑢∈𝜕𝑃_𝑅1. (3.18)
On the other hand, 𝑔(𝑥) is continuous on [0,+∞), and thus from the second limit of (𝐻₅), we have lim_𝑥→+∞𝑔^∗(𝑥)𝑥=0, (3.19) where 𝑔^∗(𝑥) is defined by (3.2). For 𝜖=2𝜆¹₀Φ(𝑠)𝑧(𝑠)𝑑𝑠⁻¹, (3.20) there exists 𝑋₀>0 such that 𝑔(𝑢)≤𝜀𝑥 for any 𝑥≥𝑋₀ and 𝑢∈[0,𝑥].
Let 𝑅₂=𝑋₀+𝑅₁+2𝜆¹₀Φ(𝑠)𝑟(𝑠)𝑑𝑠. (3.21) For any 𝑢∈𝜕𝑃_𝑅2, by (3.16) we can get 𝑅₂≥𝑢(𝑡)−𝜆𝑡^2−𝛼𝜔(𝑡)≥0, for all 𝑡∈[0,1]. Therefore, ‖𝑇𝑢‖≤𝜆¹₀Φ(𝑠)𝑧(𝑠)𝑔𝑢(𝑠)−𝜆𝑠^2−𝛼𝜔(𝑠)⁺+𝑟(𝑠)𝑑𝑠≤𝜆𝜀𝑅₂¹₀Φ(𝑠)𝑧(𝑠)𝑑𝑠+𝜆¹₀Φ(𝑠)𝑟(𝑠)≤𝑅𝑑𝑠₂2+𝑅₂2=𝑅₂. (3.22) Thus, ‖𝑇𝑢‖≤‖𝑢‖,∀𝑢∈𝜕𝑃_𝑅2. (3.23) By Lemma 2.13, 𝑇 has a fixed point 𝑢∈𝑃 such that 𝑅₁≤‖𝑢‖≤𝑅₂. Let 𝑢(𝑡)=𝑡^𝛼−2𝑢(𝑡)−𝜆𝜔(𝑡). Since ‖𝑢‖≥𝑅₁, by (3.16) we have 𝑢(𝑡)≥0 on (0,1] and lim_𝑡→0+𝑡^𝛼−2𝑢(𝑡)=0. Notice that 𝜔(𝑡) is a solution of (2.37) and 𝑡^𝛼−2𝑢(𝑡) is a solution of (2.41). Thus, 𝑢(𝑡) is a positive solution of the BVP (1.1).

By the proof of Theorem 3.2, we have the following corollary.

Corollary 3.3. The conclusion of Theorem 3.2 is valid if (𝐻₅) is replaced by (𝐻^∗₅). There exist [𝑐,𝑑]⊂(0,1) and 𝑁>0 such that for any 𝑡∈[𝑐,𝑑] and 𝑥≥𝑁, 𝑓(𝑡,𝑥)≥2𝑘𝛽²∫^𝑑_𝑐Φ(𝑠),𝑑𝑠lim_𝑥→+∞𝑔(𝑥)𝑥=0. (3.24)

4. Example

Example 4.1 (a 4-point BVP with coefficients of both signs). Consider the following problem: 𝐷^7/4₀₊𝑢(𝑡)((𝑡()(0)𝐷+𝜆𝑓𝑡,𝑢))=0,𝑡∈0,1,𝑢=0,_1/40+𝑢(1)=𝐷^1/4₀₊𝑢14−12𝐷^1/4₀₊𝑢49, (4.1) where 𝑓(𝑡,𝑥)=𝑥²+ln𝑡. (4.2)
We have 𝐺₀()=1𝑡,𝑠Γ()𝑡7/4^3/4(1−𝑠)^1/2𝑡,0≤𝑡≤𝑠≤1,_3/4(1−𝑠)^1/2−(𝑡−𝑠)^3/4𝑝(𝑠)=(),0≤𝑠≤𝑡≤1,1−1/4−𝑠1−𝑠^1/2−12(4/9)−𝑠1−𝑠^1/21,0≤𝑠≤4,11−2(4/9)−𝑠1−𝑠^1/2,144<𝑠≤9,41,9<𝑠≤1. (4.3)(4.4) By direct calculations, we have 𝑝(0)=(1/6) and 𝑞(𝑠)≥0, which implies that (𝐻₁) holds.
Let 𝑟(𝑡)=−ln𝑡, 𝑧(𝑡)=𝑡^−1/2, 𝑔(𝑥)=𝑥². It is easy to see that (𝐻₂) and (𝐻₃) hold. Moreover, liminf_𝑥→+∞min_{(),()]𝑡∈[1/43/4}𝑓(𝑡,𝑥)𝑥=+∞. (4.5) Therefore, the assumptions of Theorem 3.1 are satisfied. Thus, Theorem 3.1 ensures that there exists 𝜆^∗>0 such that the BVP (4.1) has at least one positive solution for any 𝜆∈(0,𝜆^∗).

Remark 4.2. Noticing that 𝜆𝑥² does not satisfy (𝐴₁), therefore, the work in the present paper improves and generalizes the main results of [23].

Acknowledgments

The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 11101237) and the Natural Science Foundation of Shandong Province of China (ZR2011AQ008, ZR2011AL018). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.