Multiple Positive Solutions of a Singular Semipositone Integral Boundary Value Problem for Fractional 𝑞-Derivatives Equation

Abstract

By using the fixed point index theorem, this paper investigates a class of singular semipositone integral boundary value problem for fractional 𝑞-derivatives equations and obtains sufficient conditions for the existence of at least two and at least three positive solutions. Further, an example is given to illustrate the applications of our main results.

1. Introduction

Studies on 𝑞-difference equations appeared already at the beginning of the 20th century in intensive works especially by Jackson [1], Carmichael [2], and other authors such as Poincare, Picard and, Ramanujan [3]. Up to date, it has evolved into a multidisciplinary subject, for example, see [4–7] and the references therein. For some recent work on 𝑞-difference equations, we refer the reader to the papers [8–21], and the basic definitions and properties of 𝑞-difference calculus can be found in the book [3, 22]. On the other hand, fractional differential equations have gained importance due to their numerous applications in many fields of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, and probability [23]. Many researchers studied the existence of solutions to fractional boundary value problems, for example, [24–35] and the references therein.

The fractional 𝑞-difference calculus had its origin in the works by Al-Salam [36] and Agarwal [37]. More recently, perhaps due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional 𝑞-difference calculus were made, specifically, 𝑞-analogues of the integral and differential fractional operators properties such as the Mittag-Leffler function, the 𝑞-Laplace transform, and 𝑞-Taylor’s formula [3, 13, 22, 38], just to mention some.

However, the theory of boundary value problems for nonlinear 𝑞-difference equations is still in the initial stage and many aspects of this theory need to be explored.

In [17], Ferreira considered a Dirichlet type nonlinear 𝑞-difference boundary value problem as follows: 𝐷^𝛼_𝑞𝑢(𝑡)(𝑢(𝑡𝑢(0)(1)+𝑓))=0,0<𝑡<1,1<𝛼≤2,=𝑢=0. (1) By applying a fixed point theorem in cones, sufficient conditions for the existence of nontrivial solutions were enunciated.

In other paper, Ferreira [18] studied the existence of positive solutions to nonlinear 𝑞-difference boundary value problem as follows: 𝐷^𝛼_𝑞𝑢(𝑡)(𝑢(𝑡𝑢(0)+𝑓))=0,0<𝑡<1,2<𝛼≤3,=𝐷_𝑞𝑢(0)=0,𝐷_𝑞𝑢(1)=𝛽≥0. (2)

By using a fixed point theorem in a cone, El-Shahed and Al-Askar [19] were concerned with the existence of positive solutions to nonlinear 𝑞-difference equation: _𝐶𝐷^𝛼_𝑞𝑢(𝑡)(𝑡)𝑓(𝑢(𝑡𝑢(0)+𝑎))=0,0<𝑡≤1,2<𝛼≤3,=𝐷²_𝑞𝑢(0)=0,𝑎𝐷_𝑞𝑢(1)+𝑏𝐷²_𝑞𝑢(1)=0, (3) where 𝑎,𝑏≥0 and _𝐶𝐷^𝛼_𝑞 is the fractional 𝑞-derivatives of the Caputo type.

Recently, Liang and Zhang [20] discussed the following nonlinear 𝑞-fractional three-point boundary value problem: 𝐷^𝛼_𝑞𝑢(𝑡)((𝑡𝑢(0)=𝐷+𝑓𝑡,𝑢))=0,0<𝑡<1,2<𝛼≤3,_𝑞𝑢(0)=0,𝐷_𝑞𝑢(1)=𝛽𝐷_𝑞𝑢𝜂. (4) By using a fixed point theorem in partially ordered sets, the authors obtained sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions to the above boundary value problem.

In [21], Graef and Kong investigated the following boundary value problem with fractional 𝑞-derivatives: 𝐷^𝛼_𝑞𝑢(𝑡)((𝑡𝐷+𝑓𝑡,𝑢))=0,0<𝑡<1,𝑛−1<𝛼≤𝑛,𝑛∈𝐍,_𝑖𝑞𝑢(0)=0,𝑖=0,…,𝑛−2,𝑏𝐷_𝑞𝑢(1)=^𝑚_𝑗=1𝑎_𝑗𝐷_𝑞𝑢𝑡_𝑗+𝜆, (5) where 𝜆≥0 is a parameter, and the uniqueness, existence, and nonexistence of positive solutions are considered in terms of different ranges of 𝜆.

Furthermore, Ahmad et al. [11] studied the following nonlinear fractional 𝑞-difference equation with nonlocal boundary conditions _𝐶𝐷^𝛼_𝑞𝑢(𝑡)((𝑡𝑎=𝑓𝑡,𝑢)),0≤𝑡≤1,1<𝛼≤2,₁𝑢(0)−𝑏₁𝐷_𝑞𝑢(0)=𝑐₁𝑢𝜂₁,𝑎₂𝑢(1)+𝑏₂𝐷_𝑞𝑢(1)=𝑐₂𝑢𝜂₂, (6) where _𝐶𝐷^𝛼_𝑞 is the fractional 𝑞-derivative of the Caputo type, and 𝑎_𝑖,𝑏_𝑖,𝑐_𝑖,𝜂_𝑖∈𝐑. The existence of solutions for the problem is shown by applying some well-known tools of fixed point theory such as Banach’s contraction principle, Krasnoselskii’s fixed point theorem, and the Leray-Schauder nonlinear alternative.

It is known that the fractional 𝑞-derivative of Riemann-Liouville type played an important role in the development of the theory of fractional 𝑞-derivatives and 𝑞-integrals and for its applications in pure mathematics, and is a useful tool in the description of nonconservative models. The details can be found in [22].

Since 𝑞-calculus has a tremendous potential for applications [3, 22], we find it pertinent to investigate problems in this field. Motivated by the papers [20, 21, 30], we will deal with the following integral boundary value problem of nonlinear fractional 𝑞-derivatives equation: 𝐷^𝛼_𝑞𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(0,1),2<𝛼≤3, (7) subject to the boundary conditions 𝑢(0)=𝐷_𝑞𝑢(0)=0,𝑢(1)=𝜇¹₀𝑢(𝑠)𝑑_𝑞𝑠, (8) where 𝑞∈(0,1), 𝜇 is parameter with 0<𝜇<[𝛼]_𝑞, 𝐷^𝛼_𝑞 is the 𝑞-derivative of Riemann-Liouville type of order 𝛼,𝑓∶[0,1]×𝐑⁺₀→𝐑 is continuous and semipositone and may be singular at 𝑢=0, in which 𝐑⁺₀=(0,+∞),𝐑=(−∞,+∞). In the present work, we investigate the existence of positive solutions for fractional 𝑞-derivatives integral boundary value problem (7) and (8) involving the Riemann-Liouville’s fractional derivative, which is different from [11]. We gave the corresponding Green’s function of the boundary value problem (7) and (8), gave some properties of Green’s function, and constructed a cone by properties of Green’s function. Moreover the existence of at least two and three positive solutions to the boundary value problem (7) and (8) is enunciated.

2. Preliminaries on 𝑞-Calculus and Lemmas

For the convenience of the reader, below we recall some known facts on fractional 𝑞-calculus. The presentation here can be found in, for example, [1, 3, 12, 19, 21, 22].

Let 𝑞∈(0,1) and define [𝑎]_𝑞=1−𝑞^𝑎1−𝑞,𝑎∈𝐑. (9) The 𝑞-analogue of the power function (𝑎−𝑏)^𝑛 with 𝑛∈𝐍₀∶={0,1,2,…} is (𝑎−𝑏)⁽⁰⁾=1,(𝑎−𝑏)^(𝑛)=^𝑛−1_𝑘=0𝑎−𝑏𝑞^𝑘,𝑛∈𝐍,𝑎,𝑏∈𝐑. (10) More generally, if 𝛾∈𝐑, then (𝑎−𝑏)^(𝛾)=𝑎^𝛾∞_𝑘=0𝑎−𝑏𝑞^𝑘𝑎−𝑏𝑞^𝛾+𝑘,𝑎≠0. (11) Clearly, if 𝑏=0 then 𝑎^(𝛾)=𝑎^𝛾. The 𝑞-gamma function is defined by Γ_𝑞(𝑥)=1−𝑞^(𝑥−1)1−𝑞^𝑥−1,𝑥∈𝐑⧵{0,−1,−2,…}, (12) and satisfies Γ_𝑞(𝑥+1)=[𝑥]_𝑞Γ_𝑞(𝑥).

The 𝑞-derivative of a function 𝑓 is defined by 𝐷_𝑞𝑓(𝑥)=𝑓𝑞𝑥−𝑓(𝑥)𝑥,𝐷𝑞−1_𝑞𝑓(0)=lim_𝑥→0𝐷_𝑞𝑓(𝑥), (13) and the 𝑞-derivatives of higher order by 𝐷⁰_𝑞𝑓(𝑥)(𝑥),𝐷=𝑓_𝑛𝑞𝑓(𝑥)=𝐷_𝑞𝐷_𝑞^𝑛−1𝑓(𝑥),𝑛∈𝐍. (14) The 𝑞-integral of a function 𝑓 defined in the interval [0,𝑏] is given by 𝐼_𝑞𝑓(𝑥)=^𝑥₀𝑓(𝑠)𝑑_𝑞𝑠=𝑥1−𝑞^∞_𝑘=0𝑓𝑥𝑞^𝑘𝑞^𝑘,𝑥∈[0,𝑏]. (15) If 𝑎∈[0,𝑏] and 𝑓 is defined in the interval [0,𝑏], then its integral from 𝑎 to 𝑏 is defined by ^𝑏_𝑎𝑓(𝑠)𝑑_𝑞𝑠=^𝑏₀𝑓(𝑠)𝑑_𝑞𝑠−^𝑎₀𝑓(𝑠)𝑑_𝑞𝑠. (16) Similar to that for derivatives, an operator 𝐼^𝑛_𝑞 is given by 𝐼⁰_𝑞𝑓(𝑥)(𝑥),𝐼=𝑓_𝑛𝑞𝑓(𝑥)=𝐼_𝑞𝐼_𝑞^𝑛−1𝑓(𝑥),𝑛∈𝐍. (17) The fundamental theorem of calculus applies to these operators 𝐼_𝑞 and 𝐷_𝑞, that is, 𝐷_𝑞𝐼_𝑞𝑓(𝑥)=𝑓(𝑥), (18) and if 𝑓 is continuous at 𝑥=0, then 𝐼_𝑞𝐷_𝑞𝑓(𝑥)=𝑓(𝑥)−𝑓(0). (19) The following formulas will be used later, namely, the integration by parts formula: ^𝑥₀𝑓(𝑠)𝐷_𝑞𝑔(𝑠)𝑑_𝑞𝑠=𝑓(𝑠)𝑔(𝑠)^𝑠=𝑥_𝑠=0−^𝑥₀𝐷_𝑞𝑓(𝑠)𝑔𝑞𝑠𝑑_𝑞[𝑎(𝑠,𝑡−𝑠)]^(𝛾)=𝑎^𝛾(𝑡−𝑠)^(𝛾),_𝑡𝐷_𝑞(𝑡−𝑠)^(𝛾)=𝛾_𝑞(𝑡−𝑠)^(𝛾−1),_𝑠𝐷_𝑞(𝑡−𝑠)^(𝛾)=𝛾_𝑞𝑡−𝑞𝑠^(𝛾−1),_𝑥𝐷_𝑞^𝑥₀𝑓(𝑥,𝑠)𝑑_𝑞𝑠(𝑥)=^𝑥_0𝑥𝐷_𝑞𝑓(𝑥,𝑠)𝑑_𝑞𝑠+𝑓𝑞𝑥,𝑥, (20)(21)(22) where _𝑡𝐷_𝑞 denotes the derivative with respect to the variable 𝑡.

Definition 1. Let 𝛼≥0 and 𝑓 be a function defined on [0,1]. The fractional 𝑞-integral of Riemann-Liouville type is (𝐼⁰_𝑞𝑓)(𝑥)=𝑓(𝑥) and 𝐼^𝛼_𝑞𝑓(𝑥)=1Γ_𝑞(𝛼)^𝑥₀𝑥−𝑞𝑠^(𝛼−1)𝑓(𝑠)𝑑_𝑞𝑠,𝛼>0,𝑥∈[0,1]. (23)
The fractional 𝑞-derivative of order 𝛼≥0 is defined by (𝐷⁰_𝑞𝑓)(𝑥)=𝑓(𝑥) and (𝐷^𝛼_𝑞𝑓)(𝑥)=(𝐷^𝑘_𝑞𝐼_𝑞^𝑘−𝛼𝑓)(𝑥) for 𝛼>0, where 𝑘 is the smallest integer greater than or equal to 𝛼.

Remark 2. Let 𝛼≥0 and 𝑢,𝑣 be two functions defined on [0,1], then 𝐷^𝛼_𝑞(𝑢(𝑡)+𝑣(𝑡))=𝐷^𝛼_𝑞𝑢(𝑡)+𝐷^𝛼_𝑞𝑣(𝑡).

Lemma 3. Assume that 𝛾≥0 and 𝑎≤𝑏≤𝑡, then (𝑡−𝑎)^(𝛾)≥(𝑡−𝑏)^(𝛾).

Lemma 4. Let 𝛼,𝛽≥0 and 𝑓 be a function defined in [0,1]. Then, the following formulas hold: (1)(𝐼^𝛽_𝑞𝐼^𝛼_𝑞𝑓)(𝑥)=(𝐼_𝑞^𝛼+𝛽𝑓)(𝑥), (2)(𝐷^𝛼_𝑞𝐼^𝛼_𝑞𝑓)(𝑥)=𝑓(𝑥).

Lemma 5 (see [17]). Let 𝛼>0 and 𝑛 be a positive integer. Then, the following equality holds: 𝐼^𝛼_𝑞𝐷^𝑛_𝑞𝑓(𝑥)=𝐷^𝑛_𝑞𝐼^𝛼_𝑞𝑓(𝑥)−^𝑛−1_𝑘=0𝑥^{𝛼−𝑛+𝑘}Γ_𝑞()𝐷𝛼+𝑘−𝑛+1^𝑘𝑞𝑓(0). (24)

Lemma 6. Let 𝑦∈𝐶[0,1],𝑞∈(0,1),2<𝛼≤3,1<𝜇<[𝛼]_𝑞. Then the unique solution of the equation 𝐷^𝛼_𝑞𝑢(𝑡)+𝑦(𝑡)=0,𝑡∈(0,1), (25) subject to BC (8) is given by 𝑢(𝑡)=¹₀𝐺𝑡,𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠, (26) where 𝐺()=𝑡𝑡,𝑠^𝛼−1(1−𝑠)^(𝛼−1)[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼−1𝑠−[𝛼]_𝑞−𝜇(𝑡−𝑠)^(𝛼−1)×[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⁻¹𝑡,0≤𝑠≤𝑡≤1,_𝛼−1(1−𝑠)^(𝛼−1)[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼−1𝑠×[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⁻¹,0≤𝑡≤𝑠≤1. (27)

Proof. Let us put 𝑛=3. In view of Definition 1 and Lemma 4, we see that 𝐷^𝛼_𝑞𝑢(𝑡)=−𝑦(𝑡)⟺𝐼^𝛼_𝑞𝐷³_𝑞𝐼_𝑞^3−𝛼𝑢(𝑡)=−𝐼^𝛼_𝑞𝑦(𝑡). (28) Then, it follows from Lemma 5 that the solution 𝑢(𝑡) of (25) and BC (8) is given by 𝑢(𝑡)=𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+𝑐₃𝑡^𝛼−3−^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠, (29) for some constants 𝑐₁,𝑐₂,𝑐₃∈𝐑. Since 𝑢(0)=0, we have 𝑐₃=0. Differentiating both sides of (29) and with the help of (20) and (22), we obtain 𝐷_𝑞𝑢(𝑡)=[𝛼−1]_𝑞𝑐₁𝑡^𝛼−2+[𝛼−2]_𝑞𝑐₂𝑡^𝛼−3−^𝑡₀[𝛼−1]_𝑞𝑡−𝑞𝑠^(𝛼−2)Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠. (30) Then by the boundary conditions (𝐷_𝑞𝑢)(0)=0, we get 𝑐₂=0. Thus, (29) reduces to 𝑢(𝑡)=𝑐₁𝑡^𝛼−1−^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠. (31) Using the boundary condition 𝑢(1)=𝜇∫¹₀𝑢(𝑠)𝑑_𝑞𝑠, we get 𝑐₁=𝜇¹₀𝑢(𝑠)𝑑_𝑞𝑠+¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠. (32) Hence, we have 𝑢(𝑡)=𝑡^𝛼−1𝜇¹₀𝑢(𝑠)𝑑_𝑞𝑠+¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠−^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠. (33) Integrate the above equation (33) from 0 to 1, and using (11), (19) and (20), we obtain ¹₀𝑢(𝑡)𝑑_𝑞𝑡=¹₀𝑡^𝛼−1𝜇¹₀𝑢(𝑠)𝑑_𝑞𝑠𝑑_𝑞𝑡+¹₀𝑡^𝛼−11₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠𝑑_𝑞𝑡−¹₀^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠𝑑_𝑞𝑡=1−𝑞𝜇1−𝑞^𝛼1₀𝑢(𝑠)𝑑_𝑞𝑠+1−𝑞1−𝑞^𝛼1₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠−¹₀1−𝑞𝑠^(𝛼)[𝛼]_𝑞Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠=𝜇[𝛼]_𝑞¹₀𝑢(𝑠)𝑑_𝑞𝑠+¹₀1−𝑞𝑠^(𝛼−1)𝑞^𝛼𝑠[𝛼]_𝑞Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠, (34) then ¹₀𝑢(𝑡)𝑑_𝑞𝑞𝑡=^𝛼[𝛼]_𝑞−𝜇¹₀1−𝑞𝑠^(𝛼−1)𝑠Γ_𝑞(𝛼)𝑦(𝑠)𝑑_𝑞𝑠. (35) Combining this with (29) and (31) yields 𝑢(𝑡)=𝜇𝑞^𝛼[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)¹⁰𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠𝑦(𝑠)𝑑_𝑞𝑠+1Γ_𝑞(𝛼)¹₀𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑦(𝑠)𝑑_𝑞𝑠−1Γ_𝑞(𝛼)^𝑡₀𝑡−𝑞𝑠^(𝛼−1)𝑦(𝑠)𝑑_𝑞𝑠=¹₀[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼𝑠[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⋅𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑦(𝑠)𝑑_𝑞𝑠−1Γ_𝑞(𝛼)^𝑡₀𝑡−𝑞𝑠^(𝛼−1)𝑦(𝑠)𝑑_𝑞𝑠=¹₀𝐺𝑡,𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠. (36) This completes the proof of the lemma.

Remark 7. For the special case where 𝜇=0, Lemmas 6 has been obtained by Ferreira [18].

Lemma 8. The function 𝐺(𝑡,𝑠) defined by (27) satisfies the following conditions: (i)𝐺(𝑡,𝑠) is a continuous function on (𝑡,𝑠)∈[0,1]×[0,1], and 𝐺(𝑡,𝑞𝑠)≥0, for (𝑡,𝑠)∈[0,1]×[0,1];(ii)𝜇𝑞^𝛼𝑡^𝛼−1𝜌(𝑠)≤([𝛼]_𝑞−𝜇)Γ_𝑞(𝛼)𝐺(𝑡,𝑞𝑠)≤𝜆𝑡^𝛼−1,([𝛼]_𝑞−𝜇)Γ_𝑞(𝛼)𝐺(𝑡,𝑞𝑠)≤𝜆𝜌(𝑠), for (𝑡,𝑠)∈[0,1]×[0,1], where 𝜌(𝑠)=1−𝑞𝑠^(𝛼−1)]𝑠,𝜆=max[𝛼−1_𝑞[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼,𝑞^𝛼−1[𝛼]_𝑞. (37)

Proof. The continuity of 𝐺 is easily checked. On the other hand, when 0≤𝑞𝑠≤𝑡≤1, in view of Lemma 3, we have [𝛼]_𝑞−𝜇Γ_𝑞(𝛼)𝐺𝑡,𝑞𝑠=𝑡^𝛼−11−𝑞𝑠^(𝛼−1)[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼𝑠−[𝛼]_𝑞−𝜇𝑡−𝑞𝑠^(𝛼−1)=𝑡^𝛼−11−𝑞𝑠^(𝛼−1)[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼𝑠−[𝛼]_𝑞−𝜇𝑡^𝛼−1𝑠1−𝑞𝑡^(𝛼−1)≥𝑡^𝛼−11−𝑞𝑠^(𝛼−1)[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼𝑠−[𝛼]_𝑞−𝜇𝑡^𝛼−11−𝑞𝑠^(𝛼−1)=𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠=𝜇𝑞^𝛼𝑡^𝛼−1𝜌(𝑠)≥0. (38)
Further, since we have 𝐺𝑡,𝑞𝑠=[𝛼]_𝑞−𝜇𝑡^𝛼−11−𝑞𝑠^(𝛼−1)−𝑡−𝑞𝑠^(𝛼−1)+𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠×[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⁻¹=[𝛼]_𝑞−𝜇𝑡−𝑡𝑞𝑠^(𝛼−1)−𝑡−𝑞𝑠^(𝛼−1)+𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠×[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⁻¹=[𝛼]_𝑞−𝜇^{𝑡−𝑡𝑞𝑠}_{𝑡−𝑞𝑠}𝐷_𝑞𝑥^(𝛼−1)𝑑_𝑞𝑥+𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠×[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⁻¹≤[𝛼−1]_𝑞[𝛼]_𝑞−𝜇𝑡−𝑡𝑞𝑠^(𝛼−2)×𝑡−𝑡𝑞𝑠−𝑡−𝑞𝑠+𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠×[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⁻¹=𝑞[𝛼−1]_𝑞[𝛼]_𝑞−𝜇𝑡^𝛼−21−𝑞𝑠^(𝛼−2)𝑠(1−𝑡)+𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠×[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⁻¹≤𝑞[𝛼−1]_𝑞[𝛼]_𝑞−𝜇𝑡^𝛼−21−𝑞𝑠^(𝛼−2)𝑠1−𝑞𝑠+𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠×[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⁻¹≤𝑞[𝛼−1]_𝑞[𝛼]_𝑞−𝜇𝑡^𝛼−21−𝑞𝑠^(𝛼−2)𝑠1−𝑞^𝛼𝑠+𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠×[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⁻¹=𝑞[𝛼−1]_𝑞[𝛼]_𝑞−𝜇𝑡^𝛼−21−𝑞𝑠^(𝛼−1)𝑠+𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠×[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)⁻¹, (39) we get [𝛼]_𝑞−𝜇Γ_𝑞(𝛼)𝐺≤𝑞[]𝑡,𝑞𝑠𝛼−1_𝑞[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼1−𝑞𝑠^(𝛼−1)𝑠(𝑠),[𝛼]≤𝜆𝜌_𝑞−𝜇Γ_𝑞(𝛼)𝐺≤[]𝑡,𝑞𝑠𝛼−1_𝑞[𝛼]_𝑞−𝜇𝑡^𝛼−11−𝑞𝑠^(𝛼−1)+𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠≤[𝛼−1]_𝑞[𝛼]_𝑞−𝜇1−𝑞𝑠^(𝛼−1)+𝜇𝑞^𝛼1−𝑞𝑠^(𝛼−1)𝑠𝑡^𝛼−1≤[𝛼−1]_𝑞[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼𝑡^𝛼−1≤𝜆𝑡^𝛼−1. (40) When 0≤𝑡≤𝑞𝑠≤1, since 𝜇<[𝛼]_𝑞, we have [𝛼]_𝑞−𝜇Γ_𝑞(𝛼)𝐺𝑡,𝑞𝑠=𝑡^𝛼−11−𝑞𝑠^(𝛼−1)[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼𝑠≥𝜇𝑞^𝛼𝑡^𝛼−11−𝑞𝑠^(𝛼−1)𝑠=𝜇𝑞^𝛼𝑡^𝛼−1𝜌(𝑠),[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)𝐺𝑡,𝑞𝑠=𝑡^𝛼−11−𝑞𝑠^(𝛼−1)[𝛼]_𝑞−𝜇+𝜇𝑞^𝛼𝑠≤𝑞^𝛼−1[𝛼]_𝑞𝑠^𝛼−11−𝑞𝑠^(𝛼−1)≤𝜆𝜌(𝑠). (41) This completes the proof of the lemma.

Remark 9. If we let 0<𝜏<1, then min_{[]𝑡∈𝜏,1}𝐺𝑡,𝑞𝑠≥𝜇𝑞^𝛼𝜏^𝛼−1[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)𝜌(𝑠),for𝑠∈[0,1]. (42) According to [16], we may take 𝜏=𝑞^𝑛,𝑛∈𝐍.

Lemma 10. Let 𝑝∈𝐶[0,1]. Then the boundary value problem 𝐷^𝛼_𝑞𝑢(𝑡)(𝑡)()𝑢(0)=𝐷+𝑝=0,𝑡∈0,1,2<𝛼≤3,_𝑞𝑢(0)=0,𝑢(1)=𝜇¹₀𝑢(𝑠)𝑑_𝑞𝑠, (43) has a unique solution 𝑤(𝑡)=∫¹₀𝐺(𝑡,𝑞𝑠)𝑝(𝑠)𝑑_𝑞𝑠 with 𝑤(𝑡)≤𝜆[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)𝑡^𝛼−11₀𝑝(𝑠)𝑑_𝑞𝑠=𝜆₀𝑡^𝛼−11₀𝑝(𝑠)𝑑_𝑞𝑠, (44) where 𝜆₀=𝜆/([𝛼]_𝑞−𝜇)Γ_𝑞(𝛼).
From [39, Theorem  2.3.1], one has the following definition. Let 𝑄 be a retract of real Banach space 𝐸, Ω be a relatively bounded open subset of 𝑄,𝑇∶Ω→𝑄 be completely continuous operator. The integer 𝑖(𝑇,Ω,𝑄) be defined by 𝑖(𝑇,Ω,𝑄)=deg𝐼−𝑇⋅𝜒,𝐵(𝜃,𝑟)∩𝜒⁻¹(Ω),𝜃, (45) where 𝜒∶𝐸→𝑄 is an arbitrary retraction and 𝑟>0 such that 𝐵(𝜃,𝑟)⊃Ω. Then the integer 𝑖(𝑇,Ω,𝑄) is called the fixed point index of 𝑇 on Ω with respect to 𝑄.

Lemma 11 (see [39]). Let 𝑃 be a cone in a Banach space 𝐸. Let Ω be an open bounded subset of 𝐸 with Ω_𝑟=Ω∩𝑃≠𝜙 and Ω_𝑟≠𝑃. Assume that 𝑇∶Ω_𝑟→𝑃 is a compact map such that 𝑢≠𝑇𝑢 for 𝑢∈𝜕Ω_𝑟. Then (i)if 𝑇(Ω_𝑟)⊂Ω_𝑟 for 𝑢∈Ω_𝑟, then 𝑖(𝑇,Ω_𝑟,𝑃)=1;(ii)if there exists 𝑢₀∈𝑃⧵{𝜃} such that 𝑢≠𝑇𝑢+𝜆𝑢₀ for all 𝑢∈𝜕Ω_𝑟 and 𝜆>0, then 𝑖(𝑇,Ω_𝑟,𝑃)=0.

3. The Main Results

In order to abbreviate our discussion, we give the following assumptions.(𝐻0)  There exists 𝑝(𝑡)∈𝐂[0,1], such that 𝜙(𝑡)ℎ₀(𝑢)≤𝑓(𝑡,𝑢)+𝑝(𝑡)≤𝜑(𝑡)𝑔(𝑢)+ℎ(𝑢) (46)for all (𝑡,𝑢)∈[0,1]×𝐑⁺₀, where 𝜙, 𝜑∈𝐂([0,1],𝐑⁺₀), 𝑔∈𝐂(𝐑⁺₀,𝐑⁺₀), and 𝑔(𝑢) is nonincreasing with respect to 𝑢;ℎ₀, ℎ∈(𝐑⁺,𝐑⁺) and ℎ₀(𝑢), ℎ(𝑢) are nondecreasing with respect to 𝑢, where 𝐑⁺=[0,+∞).(𝐻1)  0<∫¹₀𝜑(𝑠)𝜌(𝑠)𝑑_𝑞𝑠<+∞.

Let 𝑋=𝐶[0,1] be the Banach space endowed with morm ‖𝑢‖=max_𝑡∈[0,1]|𝑢(𝑡)|, and define the cone Ω∈𝑋 by Ω=𝑢∈𝑋∶𝑢(𝑡)≥𝜇𝑞^𝛼𝑡^𝛼−1𝜆‖𝑢‖_𝑋,𝑞∈(0,1),2<𝛼≤3. (47) By a positive solution of BVP (7) and (8), we mean a function 𝑢∈𝐶[0,1] such that 𝑢(𝑡) satisfies (7) and (8) and 𝑢(𝑡)>0 on (0,1).

Setting 𝐹(𝑡,𝑢)=𝑓(𝑡,𝑢)+𝑝(𝑡) and for any 𝑢∈Ω, 𝑚∈𝑁, we consider the following singular nonlinear boundary value problem: 𝐷^𝛼_𝑞𝑢(𝑡)+𝐹𝑡,[𝑢(𝑡)]^∗+1𝑚(),𝑢(0)=𝐷=0,𝑡∈0,1_𝑞𝑢(0)=0,𝑢(1)=𝜇¹₀𝑢(𝑠)𝑑_𝑞𝑠, (48) where [𝑢(𝑡)]^∗=max{𝑢(𝑡)−𝑤(𝑡),0}.

According to Remark 2, we can see that if 𝑢(𝑡)≥𝑤(𝑡) for 𝑡∈[0,1] is a positive solution of BVP (48), then 𝑢−𝑤 is a positive solution of BVP (7) and (8).

Lemma 12. For any 𝑚∈𝑁, let 𝑇_𝑚∶Ω→Ω be the operator defined by 𝑇_𝑚𝑢(𝑡)=¹₀𝐺𝑡,𝑞𝑠𝐹𝑠,[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠. (49) Then 𝑇_𝑚∶Ω→Ω is completely continuous.

Proof. For any 𝑢∈Ω,𝑚∈𝑁, Lemma 8 implies that (𝑇_𝑚𝑢)(𝑡)≥0 on [0,1], and 𝑇_𝑚𝑢(𝑡)≥𝜇𝑞^𝛼𝑡^𝛼−1[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)¹⁰𝜌(𝑠)𝐹𝑠,[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠. (50) On the other hand, 𝑇_𝑚𝑢_𝑋=max_0≤𝑡≤1𝑇_𝑚𝑢(𝑡)≤𝜆[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)¹⁰𝜌(𝑠)𝐹𝑠,[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠. (51) Then (𝑇_𝑚𝑢)(𝑡)≥(𝜇𝑞^𝛼𝑡^𝛼−1/𝜆)‖𝑇_𝑚𝑢‖_𝑋, which leads to 𝑇_𝑚(Ω)⊂Ω. Thus 𝑇_𝑚∶Ω→Ω.
It follows from the nonnegativeness and continuity of 𝐺(𝑡,𝑠) and 𝐹 that the operator 𝑇_𝑚 is continuous. Suppose 𝐵⊂Ω is any bounded set; then, for any 𝑢∈𝐵, there is a constant number 𝑀>0 such that ‖𝑢‖_𝑋≤𝑀. Let 𝐿𝑚⁻¹,𝑀,𝜆₀=𝜆₀𝑔1𝑚+ℎ(𝑀+1). (52) for all 𝑢∈𝐵, by Lemma 8, we have 𝑇_𝑚𝑢(𝑡)≤¹₀𝐺𝑡,𝑞𝑠𝐹𝑠,[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠≤𝜆₀¹₀𝜌(𝑠)𝐹𝑠,[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠≤𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝑔[𝑢(𝑠)]^∗+1𝑚+ℎ[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠≤𝐿𝑚⁻¹,𝑀,𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝑑_𝑞𝑠<+∞. (53) Hence, 𝑇_𝑚(𝐵) is bounded.
On the other hand, for any 𝜀>0, according to (𝐻1), there is a constant 𝛿>0 such that ^𝛿₀𝜑(𝑠)𝜌(𝑠)𝑑_𝑞𝜀𝑠<6𝐿𝑚⁻¹,𝑀,𝜆_011−𝛿,𝜑(𝑠)𝜌(𝑠)𝑑_𝑞𝜀𝑠<6𝐿𝑚⁻¹,𝑀,𝜆₀. (54) From the property of continuity of 𝐺, there exists 𝜂 with 0<𝜂<𝛿 such that for any 𝑡₁,𝑡₂∈[0,1], and 𝑠∈[0,1], when 𝑡₂−𝑡₁<𝜂 we have 𝐺𝑡₂,𝑞𝑠−𝐺𝑡₁<𝜀,𝑞𝑠3𝜑₀𝐿𝑚⁻¹,𝑀,𝜆₀,𝑇_𝑚𝑢𝑡₂−𝑇_𝑚𝑢𝑡₁≤¹₀𝐺𝑡₂,𝑞𝑠−𝐺𝑡₁,𝑞𝑠𝐹𝑠,[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠≤𝐿𝑚⁻¹,𝑀,𝜆₀¹₀𝐺𝑡₂,𝑞𝑠−𝐺𝑡₁,𝑞𝑠𝜑(𝑠)𝑑_𝑞𝑠=2𝐿𝑚⁻¹,𝑀,𝜆₀^𝛿₀𝜑(𝑠)𝐺𝑠,𝑞𝑠𝑑_𝑞𝑠+¹_1−𝛿𝜑(𝑠)𝐺𝑠,𝑞𝑠𝑑_𝑞𝑠+𝐿𝑚⁻¹,𝑀,𝜆_0𝛿^1−𝛿𝐺𝑡₂,𝑞𝑠−𝐺𝑡₁,𝑞𝑠𝜑(𝑠)𝑑_𝑞𝑠<2𝜀3+𝐿𝑚⁻¹,𝑀,𝜆₀𝜑₀×_𝛿^1−𝛿𝐺𝑡₂,𝑞𝑠−𝐺𝑡₁,𝑞𝑠𝑑_𝑞𝑠<𝜀, (55) where 𝜑₀=max{𝜑(𝑡)∶𝛿≤𝑡≤1−𝛿}. By means of the Arzela-Ascoli Theorem, 𝑇_𝑚∶Ω→Ω is completely continuous.

Theorem 13. Suppose (𝐻0) and (𝐻1) hold. In addition, assume that the following conditions are satisfied. (𝐶1) There exists a constant 𝑟>2𝜆²𝑘₀/𝜇𝑞^𝛼 such that 𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝑔𝜇𝑞^𝛼𝑠^𝛼−1𝑟()𝑑2𝜆+ℎ𝑟+1_𝑞𝑠<𝑟, (56) where 𝑘₀=∫¹₀𝑝(𝑠)𝑑_𝑞𝑠/([𝛼]_𝑞−𝜇)Γ_𝑞(𝛼).(𝐶2) There exist constants 𝜉₁,𝜉₂ with 𝜉₂>𝜉₁>𝑟 such that 𝜇^∗ℎ₀12𝜉_𝑖¹_𝜏𝜌(𝑠)𝜙(𝑠)𝑑_𝑞𝑠>𝜉_𝑖,𝑖=1,2, (57) where 𝜇^∗=𝜇𝑞^𝛼𝜏^𝛼−1/([𝛼]_𝑞−𝜇)Γ_𝑞(𝛼),𝜏=𝑞^𝑛,𝑛∈𝐍.(𝐶3)  lim_𝑢→+∞ℎ(𝑢)𝑢=0. (58)
Then BVP (7) and (8) has at least two positive solutions 𝑢^∗,𝑢^∗∗ with 𝑟≤‖𝑢^∗‖_𝑋≤𝜉₁<𝜉₂≤‖𝑢^∗∗‖_𝑋.

Proof. First, we prove that 𝑖𝑇_𝑚,Ω₀,Ω=1, (59) where Ω₀={𝑢∈Ω∶‖𝑢‖_𝑋<𝑟}.
To see this, let 𝑢∈Ω∩𝜕Ω₀. Then ‖𝑢‖_𝑋=𝑟 and 𝑢(𝑡)≥(𝜇𝑞^𝛼/𝜆)𝑡^𝛼−1𝑟 for 𝑡∈[0,1]. Now for 𝑡∈(0,1), we get 𝑢(𝑡)(𝑡)(𝑡)−𝜆−𝑤≥𝑢[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)𝑡^𝛼−11₀𝑝(𝑠)𝑑_𝑞𝑠≥𝑢(𝑡)−𝜇𝑞^𝛼𝑡2𝜆^𝛼−11𝑟≥2𝑢(𝑡)≥𝜇𝑞^𝛼𝑡^𝛼−1‖𝑢‖2𝜆^𝑋=𝜇𝑞^𝛼𝑡^𝛼−12𝜆𝑟. (60) So, for any 𝑢∈Ω∩𝜕Ω₀,𝑡∈(0,1), we get 𝜇𝑞^𝛼𝑡^𝛼−1(𝑡)(𝑡)2𝜆𝑟≤𝑢−𝑤≤𝑟. (61) It follows from (𝐶1), (61) and Lemma 8 that, for any 𝑢∈Ω∩𝜕Ω₀, 𝑇_𝑚𝑢(𝑡)=¹₀𝐺𝑡,𝑞𝑠𝐹𝑠,[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠≤𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝑔[𝑢(𝑠)]^∗+1𝑚+ℎ[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠≤𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝑔𝜇𝑞^𝛼𝑠^𝛼−1𝑟()𝑑2𝜆+ℎ𝑟+1_𝑞=‖𝑢‖𝑠<𝑟_𝑋. (62) This together with (56) yields 𝑇_𝑚(Ω₀)⊂Ω₀. From the (i) of Lemma 11, (59) is satisfied.
Let us choose 𝜖>0 such that 𝜖𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝑑_𝑞𝑠<1. (63) Then for the above 𝜖, according to (𝐶2) and (𝐶3), there exists 𝑅>𝜉₂>0 such that, for any 𝑢≥𝑅, ℎ(𝑢)≤𝜖𝑢. (64)
Take 𝑅₁=𝑅+𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)×𝑔𝜇𝑞^𝛼𝑠^𝛼−1𝑅()𝑑2𝜆+ℎ𝑅+1+𝜖_𝑞𝑠×1−𝜖𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝑑_𝑞𝑠⁻¹, (65) then 𝑅₁>𝑅>𝜉₂.
Now let Ω₁={𝑢∈Ω∶‖𝑢‖_𝑋<𝑅₁} and 𝜕Ω₁={𝑢∈Ω∶‖𝑢‖_𝑋=𝑅₁}. Then, for any 𝑢∈Ω∩𝜕Ω₁, we have 𝑇_𝑚𝑢(𝑡)=¹₀𝐺𝑡,𝑞𝑠𝐹𝑠,[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠≤𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)×𝑔[𝑢(𝑠)]^∗+1𝑚+ℎ[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠≤𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)×𝑔[𝑢(𝑠)]^∗+1𝑚+max_{[]𝑢∈0,𝑅}ℎ[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠+𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝜖‖𝑢‖_𝑋+1𝑑_𝑞𝑠≤𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)×𝑔𝜇𝑞^𝛼𝑠^𝛼−1𝑅2𝜆¹+ℎ(𝑅+1)𝑑_𝑞𝑠+𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝜖𝑅₁+1𝑑_𝑞𝑠≤𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)×𝑔𝜇𝑞^𝛼𝑠^𝛼−1𝑅()𝑑2𝜆+ℎ𝑅+1+𝜖_𝑞𝑠+𝜖𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝑑_𝑞𝑠⋅𝑅₁<𝑅₁=‖𝑢‖_𝑋. (66) This together with (56) yields 𝑇_𝑚(Ω₁)⊂Ω₁, which implies that 𝑖𝑇_𝑚,Ω₁,Ω=1. (67)
In the following, let Ω₁₀=𝑢∈Ω∶‖𝑢‖<𝑅₁,min_{[]𝑡∈𝜏,1}𝑢(𝑡)>𝜉₁,Ω₁₁=𝑢∈Ω∶‖𝑢‖<𝑅₁,min_{[]𝑡∈𝜏,1}𝑢(𝑡)>𝜉₂. (68) It is easy to see that Ω_𝑖,Ω_1𝑖,(𝑖=0,1) are bounded sets and satisfy Ω₀⊂Ω₁,Ω₁₀⊂Ω₁,Ω₁₁⊂Ω₁₀,Ω₁₀∩Ω₀=∅. (69) For any 𝑢∈Ω∩𝜕Ω₁₀, by Remark 9 and (61), we have 𝑢(𝑡)(𝑡)≥1−𝑤2𝑢(𝑡)≥12𝜉₁,forany𝑡∈[𝜏,1]. (70) Thus min_{[]𝑡∈𝜏,1}𝑇_𝑚𝑢(𝑡)=min_{[]𝑡∈𝜏,1}¹₀𝐺𝑡,𝑞𝑠𝐹𝑠,[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠≥min_{[]𝑡∈𝜏,1}¹₀𝐺𝑡,𝑞𝑠𝜙(𝑠)ℎ₀[𝑢(𝑠)]^∗+1𝑚𝑑_𝑞𝑠≥𝜇^∗ℎ₀12𝜉₁¹_𝜏𝜌(𝑠)𝜙(𝑠)𝑑_𝑞𝑠>𝜉₁. (71) This means that 𝑇_𝑚(Ω₁₀)⊂Ω₁₀, for 𝑢∈Ω₁₀. Thus, it follows from the (i) of Lemma 11 that 𝑖𝑇_𝑚,Ω₁₀,Ω=1. (72) Similarly, we can prove that, for any 𝑢∈Ω₁₁, 𝑖𝑇_𝑚,Ω₁₁,Ω=1. (73) Thus, using (59), (67), and (72), we obtain 𝑖𝑇_𝑚,Ω₁⧵Ω₀∩Ω₁₀𝑇,Ω=𝑖_𝑚,Ω₁,Ω−𝑖𝑇_𝑚,Ω₀𝑇,Ω−𝑖_𝑚,Ω₁₀,Ω=−1 (74) which implies that 𝑇_𝑚 has at least one fixed point 𝑢_𝑚1∈Ω₁⧵(Ω₀∩Ω₁₀) and satisfies 𝑇_𝑚𝑢_𝑚1(𝑡)=¹₀𝐺𝑡,𝑞𝑠𝐹𝑠,𝑢_𝑚1(𝑠)^∗+1𝑚𝑑_𝑞𝑠,𝑡∈[0,1], (75) and 𝑟≤‖𝑢_𝑚1‖_𝑋≤𝜉₁.
Obviously, 𝐹(𝑡,[𝑢_𝑚1(𝑡)]^∗+1/𝑚) is continuous for any 𝑢_𝑚1∈𝐶([0,1],𝐑⁺),𝑡∈(0,1). Also, it can be seen that 𝑢_𝑚1 has uniform lower and upper bounds. This directly comes from 𝑢_𝑚1∈Ω₁⧵(Ω₀∩Ω₁₀). Hence, in order to pass the solution 𝑢_𝑚1 of the problem (48) to that of the original problem (7) and (8), we need the following fact: 𝑢_𝑚1^∞_𝑚=1isanequicontinuousfamilyon[0,1]. (76)
As in the proof of Lemma 12, we can prove that the sequence {𝑢_𝑚1}^∞_𝑚=1 is equicontinuous on [0,1]. Now the Arzela-Ascoli Theorem guarantees that the sequence {𝑢_𝑚1}^∞_𝑚=1 has a subsequence {𝑢_𝑚𝑘,1}^∞_𝑘=1, converging uniformly on [0,1] to 𝑢₁∈𝑋. As for (61) and the fact 𝑟≤‖𝑢_𝑚1‖_𝑋≤𝜉₁, we obtain that 𝜇𝑞^𝛼𝑡^𝛼−12𝜆𝑟≤𝑢₁(𝑡)−𝑤(𝑡)≤𝜉₁,∀𝑡∈[0,1]. (77) Moreover, 𝑢_𝑚𝑘,1 satisfies the following integral equation: 𝑢_𝑚𝑘,1=¹₀𝐺𝑡,𝑞𝑠𝐹𝑠,𝑢_𝑚𝑘,1(𝑠)^∗+1𝑚_𝑘𝑑_𝑞𝑠. (78) Letting 𝑘→+∞, we have 𝑢₁=¹₀𝐺𝑡,𝑞𝑠𝐹𝑠,𝑢₁(𝑠)^∗𝑑_𝑞𝑠. (79) Let 𝑢^∗(𝑡)=𝑢₁(𝑡)−𝑤(𝑡), then 𝑢^∗ is a positive solution of BVP (7) and (8).
From (73), 𝑇_𝑚 has at least one fixed point 𝑢_𝑚2∈Ω₁₁. Similar to (76), there exists a subsequence {𝑢_𝑚𝑘,2}^∞_𝑘=1⊆{𝑢_𝑚2}^∞_𝑚=1 such that lim_𝑘→+∞𝑢_𝑚𝑘,2(𝑡)=𝑢₂(𝑡)∈𝑋, and 𝜉₂≤‖𝑢₂‖_𝑋≤𝑅₁. Let 𝑢^∗∗(𝑡)=𝑢₂(𝑡)−𝑤(𝑡), then 𝑢^∗∗ is also a positive solution of BVP (7) and (8).
Since 𝜉₁<𝜉₂, we have 𝑢^∗≠𝑢^∗∗. This implies that 𝑢^∗,𝑢^∗∗ are two different positive solutions of BVP (7) and (8).

Theorem 14. Suppose (𝐻0), (𝐻1), (𝐶1), and (𝐶2) hold. In addition, assume that the following conditions are satisfied. (𝐶4)  There exists 𝑅₁ with 𝑅₁>𝜉₂(𝜉₂ as given in (𝐶2)) such that 𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝑔𝜇𝑞^𝛼𝑠^𝛼−1𝑅2𝜆¹+ℎ𝑅₁+1𝑑_𝑞𝑠<𝑅₁. (80)(𝐶5)   lim_𝑢→+∞ℎ₀(𝑢)𝑢=+∞. (81)Then BVP (7) and (8) has at least three different positive solutions.

Proof. It can be seen that condition (𝐶4) is equivalent to condition (𝐶3). As a consequence we obtain that the BVP (7) and (8) has at least two different positive solutions 𝑢₁, 𝑢₂ with 𝑟≤‖𝑢₁‖_𝑋<‖𝑢₂‖_𝑋<𝑅₁.
On the other hand, choose a real number 𝑀^∗>0 such that 𝑀^∗≥2𝜆𝜇^∗𝜇𝑞^𝛼𝜏^𝛼−1∫¹_𝜏𝜌(𝑠)𝜙(𝑠)𝑑_𝑞𝑠. (82)
By (𝐶5), there exists 𝑅^∗₁>𝑅₁ such that ℎ₀(𝑢)≥𝑀^∗𝑢,𝑢≥𝑅^∗₁. (83) Choose 𝑅₂>max𝑅₁,2𝜆𝑅^∗₁𝜇𝑞^𝛼𝜏^𝛼−1, (84) and let Ω₂={𝑢∈Ω∶‖𝑢‖_𝑋<𝑅₂}.
In the following, we will prove that 𝑢≠𝑇_𝑚𝑢+𝜈ℏ,ℏ∈Ω⧵{𝜃},∀𝑢∈𝜕Ω₂,𝜈∈[0,1],𝑚∈𝐍. (85) Suppose that (85) is false; then there exists 𝜈₀∈[0,1],𝑢₀∈𝜕Ω₂, such that 𝑢₀=𝑇_𝑚𝑢₀+𝜈₀ℏ. (86)
For 𝑢₀∈𝜕Ω₂ and for any 𝑡∈[𝜏,1], we have 𝑢₀(𝑡)−𝑤(𝑡)≥𝑢₀(𝑡)−𝜆[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)𝑡^𝛼−11₀𝑝(𝑠)𝑑_𝑞𝑠≥𝜆1−²∫¹₀𝜌(𝑠)𝑑_𝑞𝑠𝜇𝑞^𝛼[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)𝑅₂𝑢₀(𝑡)≥12𝑢₀(𝑡)≥𝜇𝑞^𝛼𝑡^𝛼−1𝑢2𝜆^0𝑋≥𝜇𝑞^𝛼𝜏^𝛼−1𝑅2𝜆²>𝑅^∗₁>0. (87) It follows from (83) and (85) that we have 𝑅₂=𝑢_0𝑋=𝑇_𝑚𝑢₀+𝜈₀ℏ_𝑋≥¹₀𝐺𝑡,𝑞𝑠𝐹𝑠,𝑢₀(𝑠)^∗+1𝑚𝑑_𝑞𝑠>𝜇^∗1_𝜏𝜌(𝑠)𝜙(𝑠)ℎ₀𝑢₀(𝑠)^∗+1𝑚𝑑_𝑞𝑠≥𝜇^∗1_𝜏𝜌(𝑠)𝜙(𝑠)𝑑_𝑞𝑠⋅𝑀^∗𝑢₀(𝑠)^∗+1𝑚≥𝑀^∗𝜇^∗𝜇𝑞^𝛼𝜏^𝛼−1∫¹_𝜏𝜌(𝑠)𝜙(𝑠)𝑑_𝑞𝑠2𝜆⋅𝑅₂≥𝑅₂, (88) which is a contradiction. Hence, (85) is ture; from the (ii) of Lemma 11, we get 𝑖𝑇_𝑚,Ω₂,Ω=0. (89) Combining this with (67) yields 𝑖𝑇_𝑚,Ω₂⧵Ω₁,Ω=𝑖𝑇_𝑚,Ω₂,Ω−𝑖𝑇_𝑚,Ω₁,Ω=−1. (90) This implies that 𝑇_𝑚 has at least one fixed point 𝑢_𝑚3∈Ω₂⧵Ω₁ with 𝑅₁<‖𝑢_𝑚3‖_𝑋≤𝑅₂. Similar to (76), there exists a subsequence {𝑢_𝑚𝑘,3}^∞_𝑘=1⊆{𝑢_𝑚3}^∞_𝑚=1 such that lim_𝑘→+∞𝑢_𝑚𝑘,3(𝑡)=𝑢^∗₃(𝑡)∈𝑋, and 𝑅₁<‖𝑢^∗₃‖_𝑋≤𝑅₂. Let 𝑢₃(𝑡)=𝑢^∗₃(𝑡)−𝑤(𝑡), then 𝑢₃ is also a positive solution of BVP (7) and (8). So the proof is complete.

By the induction method, we can obtain the following multiplicity results for BVP (7) and (8).

Corollary 15. Suppose (𝐻0), (𝐻1), (𝐶1), and (𝐶5) hold. In addition, there exist constants 𝑟_𝑖, 𝜉_𝑖𝑘, (𝑖=1,…,𝑚;𝑘=1,2) with 𝑟<𝜉₁₁<𝜉₁₂<𝑟₁<𝜉₂₁<𝜉₂₂<𝑟₂<⋯<𝜉_𝑚1<𝜉_𝑚2<𝑟_𝑚 such that 𝜆¹₀𝜌(𝑠)𝜑(𝑠)𝑔𝜇𝑞^𝛼𝑠^𝛼−1𝑟2𝜆^𝑖+ℎ𝑟_𝑖+1𝑑_𝑞𝑠<𝑟_𝑖𝜇,𝑖=1,2,…,𝑚,_∗ℎ₀12𝜉_𝑖𝑘¹_𝜏𝜌(𝑠)𝜙(𝑠)𝑑_𝑞𝑠>𝜉_𝑖𝑘,𝑖=1,2,…,𝑚;𝑘=1,2, (91) where 𝜇^∗=𝜇𝑞^𝛼𝜏^𝛼−1/([𝛼]_𝑞−𝜇)Γ_𝑞(𝛼).
Then BVP (7) and (8) has at least 2𝑚+1 different positive solutions.

Now we present an example to illustrate our main results.

Example 16. Consider the following problem: 𝐷^2.5_0.5𝑢(𝑡)+𝑡110(𝑢()𝑢(0)=𝐷20𝑢+ℎ)−401+𝑡=0,0<𝑡<1,_0.5𝑢(0)=0,𝑢(1)=1.5¹₀𝑢(𝑠)𝑑_0.5𝑠, (92) where ℎ(𝑢)=18𝑢,0≤𝑢≤60,𝑢490³,60≤𝑢≤900,30𝑢²,900≤𝑢≤1600,3.2×10⁵√𝑢,𝑢≥1600. (93) Then BVP (91) has at least two positive solutions.

Proof. In this case, 𝛼=2.5, 𝑞=0.5, 𝜇=1.5, and 𝑓()=𝑡𝑡,𝑢110(𝑢().20𝑢+ℎ)−401+𝑡 (94) Let 𝑝(𝑡)()(𝑡)=𝑡=401+𝑡,𝜙(𝑡)=√10,𝜑𝑡,𝑔(𝑢)=11020𝑢,ℎ₀(𝑢)=78ℎ(𝑢). (95) It is easy to see that the assumptions (𝐻0), (𝐻1) hold.
Take 𝑛=2, then 𝜏=𝑞^𝑛=0.25. By calculation, we get 𝜆≈0.7335,𝜆₀≈1.3913,𝜇^∗𝑘≈0.0524,₀=∫¹₀𝑝(𝑠)𝑑_𝑞𝑠[𝛼]_𝑞−𝜇Γ_𝑞(𝛼)≈∫¹₀40(1+𝑠)𝑑_0.5𝑠0.5272≈12.6454. (96) Set 𝑟=52, then 𝑟=52>2𝜆²𝑘₀𝜇𝑞^𝛼≈51.3086,Δ∶=𝜆₀¹₀𝜌(𝑠)𝜑(𝑠)𝑔𝜇𝑞^𝛼𝑠^𝛼−1𝑟()𝑑2𝜆+ℎ𝑟+1_𝑞𝑠=𝜆₀10¹₀11−2𝑠^(3/2)𝑠^3/2×𝜆10𝜇𝑞^𝛼𝑠^3/2𝑟+ℎ(𝑟+1)𝑑_𝑞𝑠≤𝜆₀𝜆10520𝜇𝑞^𝛼()×+ℎ𝑟+1₁₀11−2𝑠^(3/2)𝑑_𝑞𝑠≈35.8280<𝑟=52, (97) which implies that the assumption (𝐶1) holds.
On the other hand, take 𝜉₁=1×10³, 𝜉₂=1.1×10³, then 𝜇^∗ℎ₀12𝜉₁¹_𝜏𝜌(𝑠)𝜙(𝑠)𝑑_𝑞𝑠=𝜇^∗ℎ10⁰12𝜉₁¹_1/411−2𝑠^(3/2)𝑠^3/2𝑑_𝑞𝑠≥𝜇^∗ℎ160⁰12𝜉₁¹_1/411−2𝑠^(3/2)𝑑_𝑞𝑠≈1.1681×10³>1×10³=𝜉₁,𝜇^∗ℎ₀12𝜉₂¹_𝜏𝜌(𝑠)𝜙(𝑠)𝑑_𝑞𝑠≥1.4135×10³>1.1×10³=𝜉₂ (98) which implies that the assumption (𝐶2) holds.
Finally, we have lim_𝑢→+∞ℎ(𝑢)𝑢=lim_𝑢→+∞3.2×10⁵√𝑢𝑢=lim_𝑢→+∞3.2×10⁵√𝑢=0. (99) Thus (𝐶3) also holds. It follows from Theorem 13 that the BVP (91) has at least two positive solutions 𝑢^∗,𝑢^∗∗ with 52≤‖𝑢^∗‖≤1×10³<1.1×10³≤‖𝑢^∗∗‖.

Acknowledgments

The authors would like to express their thanks to the editor of the journal and the anonymous referees for their carefully reading of the paper. The authors are supported by the National Natural Science Foundation of China (11271372, and 11201138) is also supported by Hunan Provincial Natural Science Foundation of China (12JJ2004), and the Scientific Research Fund of Hunan Provincial Education Department (12B034).