Positive Solutions of a Two-Point Boundary Value Problem for Singular Fractional Differential Equations in Banach Space

Abstract

This paper investigates the existence of positive solutions to a two-point boundary value problem (BVP) for singular fractional differential equations in Banach space and presents a number of new results. First, by constructing a novel cone and using the fixed point index theory, a sufficient condition is established for the existence of at least two positive solutions to the approximate problem of the considered singular BVP. Second, using Ascoli-Arzela theorem, a sufficient condition is obtained for the existence of at least two positive solutions to the considered singular BVP from the convergent subsequence of the approximate problem. Finally, an illustrative example is given to support the obtained new results.

1. Introduction

Fractional differential equations have been widely investigated recently due to its wide applications [1–3] in biology, physics, medicine, control theory, and so forth. As a matter of fact, fractional derivatives provide a more excellent tool for the description of memory and hereditary properties of various materials and processes than integer derivatives. As an important issue for the theory of fractional differential equations, the existence of positive solutions to kinds of boundary value problems (BVPs) has attracted many scholars’ attention, and lots of excellent results have been obtained [4–11] by means of fixed point theorems, upper and lower solutions technique, and so forth.

It is noted that as a special class of fractional differential equations, the singular fractional differential equations with kinds of boundary values have been studied in a series of recent works [7, 12, 13]. In [7], Jiang et al. studied a singular nonlinear semipositone fractional differential system with coupled boundary conditions and presented some sufficient conditions for the existence of a positive solution by using the fixed point theory in cone and constructing some available integral operators together with approximating technique. Zhang et al. [13] considered a class of two-point BVP for singular fractional differential equations with a negatively perturbed term and established some results on the multiplicity of positive solutions by using the approximating technique. In [12], Agarwal et al. investigated the existence of positive solutions for a two-point singular fractional boundary value problem and proposed some existence criteria by using sequential techniques. It should be pointed out that the nonlinearities of [7, 13] are singular at 𝑡=0,1, while the nonlinearity of [12] is singular at 𝑥=0. To our best knowledge, there are fewer results on two-point BVPs for singular fractional differential equations with the nonlinearity being singular at both 𝑡=0,1 and 𝑥=0. Motivated by this, we consider the following two-point BVP of singular fractional differential equations in Banach space: 𝐷^𝛽₀₊𝑥(𝑡)((𝑡𝑥(0)(1)+𝑓𝑡,𝑥))=𝜃,𝑡∈𝐽,=𝑥=𝜃, (1) where 1<𝛽≤2 is a real number, 𝐽=[0,1], 𝑓∶𝐽×𝐄→𝐄 is continuous, 𝜃 denotes the null element in the Banach space 𝐄 with the norm ‖⋅‖, 𝐷^𝛽₀₊ is the standard Riemann-Liouville fractional derivative, and 𝑓(𝑡,𝑥) may be singular at 𝑡=0,1 and 𝑥=𝜃. Firstly, we establish a sufficient condition for the existence of at least two positive solutions to the approximate problem of BVP (1) by constructing a novel cone and using the fixed point index theory. Secondly, using Ascoli-Arzela theorem, we obtained a sufficient condition for the existence of at least two positive solutions to BVP (1) from the convergent subsequence of the approximate problem. Finally, we give an illustrative example to support the obtained new results.

The main features of this paper are as follows. (i) A class of fractional-order two-point boundary value problems with the nonlinearity being singular at both 𝑡=0,1 and 𝑥=𝜃 is firstly studied in this paper, which generalizes the existing singular fractional differential equations [7, 12, 13] and has wider applications. (ii) A sequential-based method is proposed for singular fractional differential equations with the nonlinearity being singular at both 𝑡=0,1 and 𝑥=0, which enriches the theory of fractional differential equations.

The rest of this paper is organized as follows. Section 2 contains the definition of Riemann-Liouville fractional derivative and some notation. The main results are presented in Section 3, which is followed by an illustrative example in Section 4.

2. Preliminaries

We first recall some well-known results about Riemann-Liouville derivative. For details, please refer to [14, 15] and the references therein.

Definition 1. The Riemann-Liouville fractional integral of order 𝛽>0 of a function 𝑦∶(0,∞)→𝑅 is given by 𝐼^𝛽₀₊𝑦(𝑡)=1Γ𝛽^𝑡₀(𝑡−𝑠)^𝛽−1𝑦(𝑠)𝑑𝑠, (2) provided the right side is pointwise defined on (0,∞).

Definition 2. The Riemann-Liouville fractional derivative of order 𝛽>0 of a continuous function 𝑦∶(0,∞)→𝑅 is given by 𝐷^𝛽₀₊𝑦(𝑡)=1Γ𝑑𝑛−𝛽𝑑𝑡^𝑛𝑡₀𝑦(𝑠)(𝑡−𝑠)^{𝛽−𝑛+1}𝑑𝑠, (3) where 𝑛 is the smallest integer greater than or equal to 𝛽, provided that the right side is pointwise defined on (0,∞).
One can easily obtain the following properties from the definition of Riemann-Liouville derivative.

Proposition 3 (see [15]). Let 𝛽>0; if one assumes that 𝑢∈𝐶(0,1)∩𝐿(0,1), then, the fractional differential equation 𝐷^𝛽₀₊𝑢(𝑡)=0 has 𝑢(𝑡)=𝐶₁𝑡^𝛼−1+𝐶₂𝑡^𝛽−2+⋯+𝐶_𝑛𝑡^𝛽−𝑛, 𝐶_𝑖∈𝑅,𝑖=1,2,…,𝑛 as unique solutions, where 𝑛 is the smallest integer greater than or equal to 𝛽.

Proposition 4 (see [15]). Assume that 𝑢∈𝐶(0,1)∩𝐿(0,1) with a fractional derivative of order 𝛽>0 that belongs to 𝐶(0,1)∩𝐿(0,1). Then, 𝐼^𝛽₀₊𝐷^𝛽₀₊𝑢(𝑡)=𝑢(𝑡)+𝐶₁𝑡^𝛽−1+𝐶₂𝑡^𝛽−2+⋯+𝐶_𝑛𝑡^𝛽−𝑛, (4) for some 𝐶_𝑖∈𝑅, 𝑖=1,2,…,𝑛, where 𝑛 is the smallest integer greater than or equal to 𝛽.

The following lemmas will be used in the proof of the main results.

Lemma 5 (see [16]). If 𝑆⊂𝐶[𝐽,𝐄] is bounded and equicontinuous, then 𝛼_𝑐(𝑆)=sup_𝑡∈𝐽(𝛼𝑆(𝑡)), (5) where 𝛼(⋅) and 𝛼_𝑐(⋅) denote the Kuratowski noncompactness measure of bounded sets in 𝐄 and 𝐶[𝐽,𝐄], respectively, 𝑆(𝑡)={𝑥(𝑡)∶𝑥∈𝑆}(𝑡∈𝐽), and 𝐶[𝐽,𝐄] is the Banach space of all continuous functions 𝑥∶𝐽→𝐄 with the norm ‖𝑥‖_𝑐=max_𝑡∈𝐽‖𝑥(𝑡)‖.

Lemma 6 (see [16]). Let 𝑃 be a cone in 𝐄, and let 𝑃_𝑟={𝑥∈𝑃∶‖𝑥‖<𝑟}. Let 𝐹∶𝑃_𝑟→𝑃 be a strict set contraction. Assume that there exist a 𝑢₀∈𝑃 and 𝑢₀≠𝜃 such that 𝑥−𝐹𝑥≠𝜆𝑢₀ for any 𝑥∈𝜕𝑃_𝑟 and 𝜆≥0. Then 𝑖(𝐹,𝑃_𝑟,𝑃)=0.

Lemma 7 (see [16]). Let 𝑉={𝑥_𝑛}∈𝐿[𝐽,𝐄], and there exists a 𝑔∈𝐿[𝐽,𝑅⁺] such that ‖𝑥_𝑛(𝑡)‖≤𝑔(𝑡),𝑎.𝑒.𝑡∈𝐽 for all 𝑥_𝑛∈𝑉; then 𝛼({∫^𝑡_𝑎𝑥_𝑛(𝑠)𝑑𝑠∶𝑛∈𝑁})≤2∫^𝑡_𝑎𝛼(𝑉(𝑠))𝑑𝑠, 𝑡∈𝐽.

Lemma 8 (Ascoli-Arzela theorem [16]). 𝐻⊂𝐶[𝐽,𝐄] is relative compact if and only if 𝐻 is equicontinuous, and for any 𝑡∈𝐽, 𝐻(𝑡) is a relatively compact set in 𝐄.

Lemma 9 (see [17]). Let 𝐁⊆𝐄 be bounded open set. 𝐴∶𝑃∩𝐁→𝐁 is condensing. If there exists 1>𝜇>0, such that 𝐴𝑥=𝜇𝑥, where 𝑥∈𝑃⋂𝜕𝐁, then 𝑖(𝐴,𝑃⋂𝐁,𝑃)=1.

3. Main Results

Let 𝑃={𝑥∈𝐄∶𝑥(𝑡)≥𝜃,∀𝑡∈𝐽}. Then, one can see that 𝑃 is a normal solid cone of 𝐄. Define 𝑃_𝑑={𝑥∈𝑃∶‖𝑥‖<𝑑} and 𝑃_𝑑={𝑥∈𝑃∶‖𝑥‖≤𝑑}. Let 𝑃^∗ denote the dual cone of 𝑃. We consider BVP (1) in 𝐶[𝐽,𝐄]. 𝑥 is called a solution to BVP (1), if 𝑥 satisfies (1). In addition, we call 𝑥(𝑡) a positive solution to BVP (1), if 𝑥(𝑡)>𝜃,∀𝑡∈𝐽.

For convenience, let us list the following assumptions.(H1)𝑓∈𝐶[(0,1)×𝑃⧵{𝜃},𝑃] and 𝑓𝑡,𝑡^𝛽−2𝑥≤𝑘(𝑡)𝑞(𝑥),𝑡∈(0,1),𝑥∈𝑃⧵{𝜃}, (6) where 𝑘∶(0,1)→(0,+∞), ∫¹₀𝑠(1−𝑠)^𝛽−1𝑘(𝑠)𝑑𝑠<+∞, and 𝑞∶𝑃⧵{𝜃}→𝑃.(H2)1/Γ(𝛽)∫¹₀𝑠(1−𝑠)^𝛽−1𝑘(𝑠)𝑞[(𝛽−1)𝑠(1−𝑠)^𝛽−1𝑟₁,𝑅₁]𝑑𝑠<+∞ for all 𝑅₁>𝑟₁>0, and there exists 𝑅>0 such that 1/Γ(𝛽)∫¹₀𝑠(1−𝑠)^𝛽−1𝑘(𝑠)𝑞[(𝛽−1)𝑠(1−𝑠)^𝛽−1𝑅,𝑅+1]𝑑𝑠<𝑅, where 𝑞𝑟₁,𝑅₁∶=sup_{𝑥∈𝑃𝑅1⧵𝑃𝑟1}𝑞(𝑥)<+∞,𝑅₁>𝑟₁>0. (7) (H3)𝑓(𝑡,𝑡^𝛽−2𝑥) is uniformly continuous with respect to 𝑡 on [𝛾,1−𝛾]×𝑃_𝑅1⧵𝑃_𝑟1, where 𝛾∈(0,1/2) and 𝑅₁>𝑟₁>0.(H4) There exists a constant 𝐿≥0, such that 𝛼𝑓𝑡,𝑡^𝛽−2𝐷(𝐷),()≤𝐿𝛼∀𝑡∈0,1,𝐷⊆𝑃_𝑅1⧵𝑃_𝑟1, (8) where 𝐿<min{Γ(𝛽)𝛽^𝛽/2(𝛽−1)^𝛽−1,Γ(𝛽)𝛽(𝛽+1)/2}, and 𝑅₁>𝑟₁>0.(H5) There exists 𝜑^∗∈𝑃^∗ with ‖𝜑^∗‖=1 and 𝜑∈𝐿[0,1], such that lim_{‖𝑥‖→0}inf_𝑥∈𝑃𝜑^∗𝑓𝑡,𝑡^𝛽−2𝑥≥𝜑(𝑡), (9) uniformly in 𝑡∈(0,1), where 0<∫¹₀𝑠(1−𝑠)^𝛽−1𝜑(𝑠)𝑑𝑠<+∞.(H6) There exist 𝜓^∗∈𝑃^∗ with ‖𝜓^∗‖=1 and [𝑎,𝑏]⊆(0,1), such that lim_{‖𝑥‖→+∞}inf_𝑥∈𝑃𝜓^∗𝑓𝑡,𝑡^𝛽−2𝑥‖𝑥‖=+∞, (10) uniformly on 𝑡∈[𝑎,𝑏].

According to [18], BVP (1) is equivalent to 𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠, (11) where 𝐺()=[𝑡(𝑡,𝑠1−𝑠)]^𝛽−1−(𝑡−𝑠)^𝛽−1Γ𝛽[𝑡(,0≤𝑠≤𝑡≤1,1−𝑠)]^𝛽−1Γ𝛽,0≤𝑡≤𝑠≤1. (12)

Consider the operator 𝐴 associated with the singular boundary value problem (1), which is defined by (𝐴𝑥)(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠. (13) Set 𝐺^∗(𝑡,𝑠)=𝑡^2−𝛽𝐺(𝑡,𝑠). As in [19], we have 𝐺^∗(𝑡,𝑠)≥𝛽−1𝑡(1−𝑡)𝐺^∗(),𝐺𝜏,𝑠_∗()≤1𝑡,𝑠Γ𝛽𝑠(1−𝑠)^𝛽−1. (14)

Define 𝐴^∗𝑥(𝑡)=¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑡^𝛽−2𝑥(𝑠). (15) Then, one can see that 𝑥(𝑡) is a fixed point of the operator 𝐴^∗ if and only if 𝑡^𝛽−2𝑥(𝑡) is a solution of BVP (1).

Choose 𝑒∈𝑃^𝑜 with ‖𝑒‖=1. We consider the following approximate problem of (15): 𝐴^∗_𝑛𝑥(𝑡)=¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑡^𝛽−2𝑥(𝑠)+𝑒𝑛𝑑𝑠,𝑛∈𝑁. (16)

Denote 𝑄=𝑥∈𝐶[𝐽,𝑃]∶𝑥(𝑡)≥𝛽−1𝑡(1−𝑡)𝑥(𝑠),∀𝑡,𝑠∈𝐽. (17) It is easy to check that 𝑄 is a cone in 𝐶[𝐽,𝐄]. Let 𝑄_𝑟={𝑥∈𝑄∶‖𝑥‖<𝑟}. By (H1) and (H2), we can conclude that 𝐴^∗𝑥≤¹₀𝑠(1−𝑠)^𝛽−1Γ(𝛼)𝑘(𝑠)×𝑞𝛽−1𝑠(1−𝑠)^𝛽−1𝑟,𝑟𝑑𝑠<+∞,∀𝑥∈𝑄_𝑟, (18) which implies that the operator 𝐴^∗ is well defined. In addition, from the definition of 𝑄, we can prove that ‖𝑥(𝑡)‖≥𝛽−1𝑡(1−𝑡)‖𝑥‖_𝑐,∀𝑥∈𝑄,𝑡∈𝐽. (19) By (16), we have (𝐴^∗_𝑛𝑥)(𝑡)≥(𝛽−1)𝑡(1−𝑡)(𝐴^∗_𝑛𝑥)(𝑠),∀𝑡,𝑠∈𝐽. Thus 𝐴^∗_𝑛𝑄⊂𝑄.

In the following, we first investigate the existence of two positive solutions to the approximate problem (16) and then establish the existence criterion for the existence of two positive solutions to BVP (1) by using the sequential-based technique and Ascoli-Arzela theorem.

Lemma 10. Let (H1) and (H2) be satisfied. Then for any 𝑟>0, the operator 𝐴^∗_𝑛 is a continuous bounded operator from 𝑄_𝑟 into 𝑄.

Proof. By (H1), we can easily see that 𝐴^∗_𝑛 is bounded from 𝑄_𝑟 to 𝑄.
Next, we prove that 𝐴^∗_𝑛 is continuous.
Let ‖𝑥_𝑚−𝑥‖_𝑐→0 as 𝑚→+∞(𝑥_𝑚,𝑥∈𝑄_𝑟). From (H1), we have 𝑓𝑠,𝑥_𝑚(𝑠)+𝑒𝑛𝑠^𝛽−2≤𝑘(𝑠)𝑞𝑥_𝑚(𝑠)+𝑒𝑛(𝑠)𝑞1≤𝑘𝑛1,𝑟+𝑛. (20) Hence 𝐴^∗_𝑛𝑥_𝑚(𝑡)=¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑥_𝑚(𝑠)+𝑒𝑛𝑠^𝛽−2≤1𝑑𝑠Γ𝛽¹₀𝑠(1−𝑠)^𝛽−1𝑘(𝑠)𝑞1𝑛1,𝑟+𝑛𝑑𝑠<+∞, (21) which together with the dominated convergence theorem imply that lim_𝑚→+∞𝐴^∗_𝑛𝑥_𝑚(𝑡)=𝐴^∗_𝑛𝑥(𝑡). (22)
We now show that lim_𝑚→+∞𝐴^∗_𝑛𝑥_𝑚−𝐴^∗_𝑛𝑥_𝑐=0. (23) In fact, if (23) is not true, then there exist a positive number 𝜀₀ and a subsequence {𝑥_𝑚𝑖}⊂{𝑥_𝑚} such that 𝐴^∗_𝑛𝑥_𝑚𝑖−𝐴^∗_𝑛𝑥_𝑐≥𝜀₀. (24) Since {𝐴^∗_𝑛𝑥_𝑚} is relatively compact, there is a subsequence of {𝐴^∗_𝑛𝑥_𝑚𝑖} which converges to some 𝑦∈𝑄. Without loss of generality, we assume that {𝐴^∗_𝑛𝑥_𝑚𝑖} itself converges to 𝑦, that is, lim_𝑖→∞𝐴^∗_𝑛𝑥_𝑚𝑖−𝑦_𝑐=0. (25) By virtue of (22) and (25), we have 𝑦=𝐴^∗_𝑛𝑥, which contradicts with (24). Hence (23) holds, and the continuity of 𝐴^∗_𝑛 is proved.

Lemma 11. Let (H1)–(H4) be satisfied. Then for any 𝑟>0, the operator 𝐴^∗_𝑛 is a strict set contraction from 𝑄_𝑟 to 𝑄.

Proof. By virtue of Lemma 10, we know that 𝐴^∗_𝑛𝑆 is bounded and equicontinuous on 𝐽. Thus, from Lemma 5, one can see that 𝛼_𝑐𝐴^∗_𝑛𝑆=sup_𝑡∈𝐽𝛼𝐴^∗_𝑛𝑆(𝑡),𝑟>0,𝑆⊆𝑄_𝑟, (26) where 𝐴^∗_𝑛𝑆(𝑡)={(𝐴^∗_𝑛𝑥)(𝑡)∶𝑥∈𝑆},𝑡∈𝐽.
Set 𝐷_𝛿=_𝛿^1−𝛿𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2,1𝑑𝑠∶𝑥∈𝑆𝛿∈0,2. (27) Based on (H1) and (H2), we have _𝛿^1−𝛿𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2−𝑑𝑠₁₀𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2𝑑𝑠≤𝑐^𝛿₀𝑠(1−𝑠)^𝛽−1𝑘(𝑠)𝑑𝑠+₁^1−𝛿𝑠(1−𝑠)^𝛽−1𝑘(𝑠)1𝑑𝑠,𝑥∈𝑆,𝛿∈0,2, (28) where 𝑐=𝑞[1/𝑛,𝑟+(1/𝑛)]. Hence 𝑑_𝐻𝐷_𝛿,𝐴^∗_𝑛𝑆⟶0,𝛿⟶0⁺, (29) where 𝑑_𝐻(⋅,⋅) denotes the Hausdorff metrics, which implies that 𝛼𝐴^∗_𝑛𝑆=lim_𝛿→0+𝛼𝐷_𝛿. (30) Since _𝛿^1−𝛿𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛽−2𝑥(𝑠)+𝑒𝑛∈()×𝑑𝑠1−2𝛿co𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛽−2𝑥(𝑠)+𝑒𝑛∶𝑠∈[𝛿,1−𝛿], (31) by Lemmas 5 and 7, we have 𝛼𝐷_𝛿=𝛼_𝛿^1−𝛿𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛽−2𝑥(𝑠)+𝑒𝑛≤()𝑑𝑠,𝑥∈𝑆1−2𝛿×𝛼co𝐺^∗()𝑥(𝑠)+𝑒𝑡,𝑠×𝑓𝑠,𝑛𝑠^𝛽−2[]𝐺∶𝑠∈𝛿,1−𝛿,𝑥∈𝑆≤𝛼^∗()𝑥(𝑠)+𝑒𝑡,𝑠×𝑓𝑠,𝑛𝑠^𝛽−2[]≤1∶𝑠∈𝛿,1−𝛿,𝑥∈𝑆Γ𝛽𝛽−1^𝛽−1𝛽^𝛽𝑓𝑥(𝑠)+𝑒×𝛼𝑠,𝑛𝑠^𝛽−2[]≤1∶𝑠∈𝛿,1−𝛿,𝑥∈𝑆Γ𝛽𝐿𝛽−1^𝛽−1𝛽^𝛽𝛼𝑆𝐼_𝛿≤1Γ𝛽2𝐿𝛽−1^𝛽−1𝛽^𝛽𝛼_𝑐(𝑆), (32) where 𝐼_𝛿=[𝛿,1−𝛿]. By (H4), we can obtain 𝛼𝐴_𝑛𝑆=lim_𝛿→0+𝛼𝐷_𝛿≤1Γ𝛽𝛽−1^𝛽−1𝛽^𝛽2𝐿𝛼_𝑐(𝑆)<𝛼_𝑐(𝑆). (33) Consequently, the operator 𝐴_𝑛 is a strict set contraction from 𝑄_𝑟 into 𝑄.

Theorem 12. Let (H1)–(H6) hold. Then, there exists 𝑟∈(0,𝑅) such that the operator 𝐴^∗_𝑛 has two fixed points in 𝑄_𝑅⧵𝑄_𝑟 and 𝑄⧵𝑄_𝑅, respectively, for arbitrary sufficiently large positive integer 𝑛, where 𝑅 is given in (H2).

Proof. For any given 𝑡∈(0,1), we have ∫¹₀𝐺^∗(𝑡,𝑠)𝜑(𝑠)𝑑𝑠≤∫¹₀𝑠(1−𝑠)^𝛽−1𝜑(𝑠)𝑑𝑠. Since ∫¹₀𝐺^∗(𝑡,𝑠)𝜑(𝑠)𝑑𝑠>0, there exists a 𝜀^′>0 such that 𝑟^′=¹₀¹₀𝐺^∗(𝑡,𝑠)𝜑(𝑠)−𝜀^′𝑑𝑠𝑑𝑡>0. (34) Then by (H5), there exists 𝑟^′′∈(0,𝑅) such that 𝜑^∗𝑓𝑡,𝑡^𝛽−2𝑥(𝑡)≥𝜑(𝑡)−𝜀^′,∀𝑡∈(0,1) (35) holds for 𝑥∈𝑄 and ‖𝑥‖_𝑐≤𝑟^′′. Letting 0<𝑟<𝑙∶=min{𝑟^′,𝑟^′′}, we prove that 𝑥−𝐴^∗_𝑛𝑥≠𝜆𝑒 for any 𝑥∈𝜕𝑄_𝑟, 𝜆≥0 as 𝑛>1/(𝑙−𝑟). In fact, suppose that this is false. Then there exist 𝜆>0, 𝑥∈𝜕𝑄_𝑟 such that 𝑥−𝐴^∗_𝑛𝑥=𝜆𝑒. It is obvious that 𝑥(𝑡)=𝐴^∗_𝑛𝑥(𝑡)+𝜆𝑒≥𝐴^∗_𝑛𝑥𝑡)=₁₀(𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2𝑑𝑠. (36) Thus 𝜑^∗(𝑥(𝑡≥))₁₀𝐺^∗(𝑡,𝑠)𝜑^∗𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2≥𝑑𝑠₁₀𝐺^∗(𝑡,𝑠)𝜑(𝑠)−𝜀^′𝑑𝑠. (37) This implies that ¹₀𝜑^∗(𝑥(𝑡≥))𝑑𝑡₁₀₁₀𝐺^∗(𝑡,𝑠)𝜑(𝑠)−𝜀^′𝑑𝑠𝑑𝑡=𝑟^′>𝑟, (38) which is in contradiction with 𝜑^∗(𝑥(𝑡))≤‖𝑥(𝑡)‖≤‖𝑥‖_𝑐=𝑟,𝑡∈𝐽. (39) Combining with Lemma 6, we obtain that 𝑖𝐴^∗_𝑛,𝑄_𝑟,𝑄=0. (40)
Now, we show that 𝑖(𝐴^∗_𝑛,𝑄_𝑅,𝑄)=1. By Lemma 9, we need only to prove that 𝐴^∗_𝑛𝑥≠𝜆𝑥 for 𝑥∈𝜕𝑄_𝑅 and 𝜆≥1. In fact, if there exist 𝑥∈𝜕𝑄_𝑅 and some 𝜆≥1 such that 𝐴^∗_𝑛𝑥=𝜆𝑥, then 𝑥(𝑡)=1𝜆¹₀𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2≤𝑑𝑠₁₀𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2≤1𝑑𝑠Γ𝛽¹₀𝑠(1−𝑠)^𝛽−1𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2𝑑𝑠. (41) Therefore, by (H1) and (H2), we have 𝑅=‖𝑥‖_𝑐<1Γ𝛽¹₀𝑠(1−𝑠)^𝛽−1𝑘(𝑠𝑞𝑥(𝑠)+𝑒)𝑛<1𝑑𝑠Γ𝛽¹₀𝑠(1−𝑠)^𝛽−1𝑘(𝑠)×𝑞𝛽−1𝑠(1−𝑠)^𝛽−1𝑅,𝑅+1𝑑𝑠<𝑅, (42) which is a contradiction. Consequently, 𝑖𝐴^∗_𝑛,𝑄_𝑅,𝑄=1. (43)
In the following, we choose 𝑅^′>𝛽−1𝑎(1−𝑏)^𝑏_𝑎max_𝑡∈𝐽𝐺^∗(𝑡,𝑠)𝑑𝑠⁻¹. (44) By (H6), there exists 𝑀>𝑅 such that 𝜑^∗(𝑓(𝑡,𝑡^𝛽−2𝑥))≥𝑅^′‖𝑥‖ holds for 𝑥>𝑀.
Let 𝑅=max𝑀+1𝑎(),1𝛽−11−𝑏1+𝛽−1𝑅^′𝑎(1−𝑏)max_𝑡∈𝐽∫^𝑏_𝑎𝐺^∗().𝑡,𝑠𝑑𝑠−1 (45) We now claim that 𝑥−𝐴^∗_𝑛𝑥≠𝜆𝑒 for 𝑥∈𝜕𝑄_𝑅 and 𝜆≥0. As a matter of fact, if this is not true, then there exists 𝜆>0 such that 𝑥−𝐴^∗_𝑛𝑥=𝜆𝑒, that is, 𝑥(𝑡)>𝐴^∗_𝑛𝑥(𝑡), which implies that 𝑅≥𝜑^∗(𝑥(𝑡))≥𝜑^∗𝐴^∗_𝑛𝑥(𝑡≥₁₀)𝐺^∗(𝑡,𝑠)𝜑^∗𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2≥𝑑𝑠₁₀𝐺^∗(𝑡,𝑠)𝑅^′𝑎(𝑥‖)≥𝑎()𝑅𝛽−11−𝑏)(‖−1𝑑𝑠𝛽−11−𝑏^′𝑅−1^𝑏_𝑎𝐺^∗()>𝑡,𝑠𝑑𝑠𝑅, (46) which is a contradiction. Therefore, by Lemma 6, we can obtain that 𝑖𝐴^∗_𝑛,𝑄_𝑅,𝑄=0. (47) This together with (40), (43), and (47) implies that 𝑖𝐴^∗_𝑛,𝑄_𝑅⧵𝑄_𝑟,𝑄=𝑖𝐴^∗_𝑛,𝑄_𝑅,𝑄−𝑖𝐴^∗_𝑛,𝑄_𝑟𝑖𝐴,𝑄=1,_∗𝑛,𝑄_𝑅⧵𝑄_𝑅,𝑄=𝑖𝐴^∗_𝑛,𝑄_𝑅,𝑄−𝑖𝐴^∗_𝑛,𝑄_𝑅,𝑄=−1. (48) Thus, there exist 𝑥∈𝑄_𝑅⧵𝑄_𝑟 and 𝑦∈𝑄_𝑅⧵𝑄_𝑅 such that 𝐴^∗_𝑛𝑥=𝑥 and 𝐴^∗_𝑛𝑦=𝑦. This completes the proof.

Theorem 13. Let (H1)–(H6) be satisfied. Then BVP (1) has at least two positive solutions in 𝐶[𝐽,𝐄].

Proof. From Theorem 12, there exists an integer 𝑛₀>0, such that the operator 𝐴^∗_𝑛, 𝑛≥𝑛₀ has two fixed points in 𝑥_𝑛∈𝑄_𝑅⧵𝑄_𝑟 and 𝑦_𝑛∈𝑄⧵𝑄_𝑅. Denote 𝐷∶={𝑥_𝑛∶𝑛≥𝑛₀}. Obviously 𝐷 is uniformly bounded. Next, we show that 𝐷 is equicontinuous.
Firstly, we prove that lim_𝑡→0+¹₀𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2𝑑𝑠=0,lim_𝑡→1−¹₀𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2𝑑𝑠=0. (49)(50)
On one hand, by the absolute continuity of integration [19], for any 𝜀>0, there exists 𝛿₁>0 such that ∫^𝑡₀(1/Γ(𝛽))𝑠(1−𝑠)^𝛽−1𝑓(𝑠,(𝑥(𝑠)+𝑒/𝑛)𝑠^𝛽−2)𝑑𝑠<𝜀/2,∀0<𝑡≤𝛿₁.
On the other hand, from (H2), we set 𝑀∶=¹_𝛿1𝑠(1−𝑠)^𝛽−1Γ𝛽𝑘(𝑠)𝑞𝛽−1𝑠(1−𝑠)^𝛽−1𝑟,𝑅𝑑𝑠, (51) and 𝛿=min{𝛿₁,𝛿₁𝜀/𝑀}. For 0<𝑡<𝛿, we have ¹₀𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2=𝑑𝑠_𝛿10𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2+𝑑𝑠_1𝛿1𝐺^∗()𝑓𝑥(𝑠)+𝑒𝑡,𝑠𝑠,𝑛𝑠^𝛽−2<𝜀𝑑𝑠2+¹_𝛿1𝑡(1−𝑠)^𝛽−1Γ𝛽𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2≤𝜀𝑑𝑠2+𝑡𝛿₁¹_𝛿1𝑠(1−𝑠)^𝛽−1Γ𝛽𝑘(𝑠)𝑞𝑥(𝑠)+𝑒𝑛≤𝜀𝑑𝑠2+𝑡𝛿₁¹_𝛿1𝑠(1−𝑠)^𝛽−1Γ𝛽𝑘(𝑠)𝑞𝛽−1𝑠(1−𝑠)^𝛽−1𝑟,𝑅𝑑𝑠≤𝜀, (52) which implies that (49) holds. Similarly, one can prove that (50) holds.
Next, we show that 𝐷 is equicontinuous for 𝑡∈[𝛾,1−𝛾], 0<𝛾<1/2.
From (H3), we define 𝑀=max_{𝑡∈[𝛾,1−𝛾]}𝑓𝑥(𝑡)+𝑒𝑡,𝑛𝑡^𝛽−2. (53)
By the absolute continuity of integration, for 𝛾∈(0,1/2) and the previous 𝜀>0, there exists 0<𝛿₂<𝛾 such that ^𝛿2₀𝑠(1−𝑠)^𝛽−1Γ𝛽𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2<𝜀𝑑𝑠6. (54)
Since ((𝑡₂(1−𝑠)^𝛽−1−(𝑡₂−𝑠)^𝛽−1𝑡₂^2−𝛽)/Γ(𝛽))𝑓(𝑠,(𝑥(𝑠)+𝑒/𝑛)𝑠^𝛽−2) is bounded for 𝑠∈[𝑡₁,𝑡₂], there exists 𝛿₃>0, such that ^𝑡2_𝑡1𝑡₂(1−𝑠)^𝛽−1−𝑡₂−𝑠^𝛽−1𝑡₂^2−𝛽Γ𝛽𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2<𝜀𝑑𝑠6,𝑡₂−𝑡₁<𝛿₃. (55)
For |𝑡₂−𝑡₁|<𝛿₄∶=𝛾𝜀/(6∫¹₀(𝑠(1−𝑠)^𝛽−1/Γ(𝛽))𝑘(𝑠)𝑞[(𝛽−1)𝑠(1−𝑠)^𝛽−1𝑟,𝑅]𝑑𝑠), one can see that ¹_𝑡2𝑡₂−𝑡₁(1−𝑠)^𝛽−1Γ𝛽𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2<𝜀𝑑𝑠6. (56) Similarly, for |𝑡₂−𝑡₁|<𝛿₅∶=𝛿₂𝜀/(6∫¹₀(𝑠(1−𝑠)^𝛽−1/Γ(𝛽))𝑘(𝑠)𝑞[(𝛽−1)𝑠(1−𝑠)^𝛽−1𝑟,𝑅]𝑑𝑠), we have ^𝑡1_𝛿2𝑡₂−𝑡₁(1−𝑠)^𝛽−1Γ𝛽𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2<𝜀𝑑𝑠6. (57)
Since ^𝑡1_𝛿2𝑡₂^2−𝛽𝑡₂−𝑠^𝛽−1−𝑡₁^2−𝛽𝑡₁−𝑠^𝛽−1Γ𝛽𝑥(𝑠)+𝑒×𝑓𝑠,𝑛𝑠^𝛽−2≤1𝑑𝑠𝛿₂1−𝑡₁^{𝛽−1𝑡1𝛿}₂𝑡₂^2−𝛽𝑡₂−𝑠^𝛽−1−𝑡₁^2−𝛽𝑡₁−𝑠^𝛽−1Γ𝛽×𝑠(1−𝑠)^𝛽−1𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2𝑑𝑠, (58)𝑓(𝑡,(𝑥(𝑡)+𝑒/𝑛)𝑡^𝛽−2) is bounded on 𝑡∈[𝛿₂,𝑡₁], and (𝑡₂^2−𝛽(𝑡₂−𝑠)^𝛽−1−𝑡₁^2−𝛽(𝑡₁−𝑠)^𝛽−1)/Γ(𝛽)→0 as 𝑡₁→𝑡₂, there exists 𝛿₆>0 such that ^𝑡1_𝛿2𝑡₂^2−𝛽𝑡₂−𝑠^𝛽−1−𝑡₁^2−𝛽𝑡₁−𝑠^𝛽−1Γ𝛽𝑥(𝑠)+𝑒×𝑓𝑠,𝑛𝑠^𝛽−2<𝜀𝑑𝑠6,𝑡₂−𝑡₁<𝛿₆. (59)
We choose 𝛿^′=min{𝛿_𝑖∶𝑖=2,3,4,5,6}. For |𝑡₂−𝑡₁|<𝛿^′, we have 𝑥𝑡₂−𝑥𝑡₁=¹₀𝐺^∗𝑡₂𝑓𝑥(𝑠)+𝑒,𝑠𝑠,𝑛𝑠^𝛽−2−𝑑𝑠₁₀𝐺^∗𝑡₁𝑓𝑥(𝑠)+𝑒,𝑠𝑠,𝑛𝑠^𝛽−2≤𝑑𝑠_𝛿20𝐺^∗𝑡₂𝑓𝑥(𝑠)+𝑒,𝑠𝑠,𝑛𝑠^𝛽−2+𝑑𝑠_𝛿20𝐺^∗𝑡₁𝑓𝑥(𝑠)+𝑒,𝑠𝑠,𝑛𝑠^𝛽−2+𝑑𝑠_𝑡2𝛿2𝐺^∗𝑡₁𝑓𝑥(𝑠)+𝑒,𝑠𝑠,𝑛𝑠^𝛽−2+𝑑𝑠_1𝑡2𝐺^∗𝑡₂𝑓𝑥(𝑠)+𝑒,𝑠𝑠,𝑛𝑠^𝛽−2𝑑𝑠−^𝑡1_𝛿2𝐺^∗𝑡₁𝑓𝑥(𝑠)+𝑒,𝑠𝑠,𝑛𝑠^𝛽−2𝑑𝑠−¹_𝑡1𝐺^∗𝑡₁𝑓𝑥(𝑠)+𝑒,𝑠𝑠,𝑛𝑠^𝛽−2𝑑𝑠≤2^𝛿2₀𝑠(1−𝑠)^𝛽−1Γ𝛽𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2+𝑑𝑠_𝑡2𝛿2𝑡₂(1−𝑠)^𝛽−1−𝑡₂−𝑠^𝛽−1𝑡₂^2−𝛽Γ𝛽𝑥(𝑠)+𝑒×𝑓𝑠,𝑛𝑠^𝛽−2+𝑑𝑠_𝑡1𝑡2𝑡₁(1−𝑠)^𝛽−1Γ𝛽𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2+𝑑𝑠_𝑡1𝑡2𝑡₂−𝑡₁(1−𝑠)^𝛽−1Γ𝛽𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2+𝑑𝑠_𝑡1𝛿2𝑡₂−𝑡₁(1−𝑠)^𝛽−1Γ𝛽𝑓𝑥(𝑠)+𝑒𝑠,𝑛𝑠^𝛽−2+𝑑𝑠_𝑡1𝛿2𝑡₂^2−𝛽𝑡₂−𝑠^𝛽−1−𝑡₁^2−𝛽𝑡₁−𝑠^𝛽−1Γ𝛽𝑥(𝑠)+𝑒×𝑓𝑠,𝑛𝑠^𝛽−2𝑑𝑠<𝜀. (60) This together with (49) and (50) implies that 𝐷 is equicontinuous for 𝑡∈[0,1].
Now, we show that 𝐷(𝑡) is relatively compact. By Lemma 7, we have 𝛼(𝐷(𝑡))≤𝛼¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑥_𝑛(𝑠)+𝑒𝑛𝑠^𝛽−2𝑑𝑠∶𝑛≥𝑛₀≤2¹₀𝐺^∗()𝑓𝑥𝑡,𝑠×𝛼𝑠,_𝑛(𝑠)+𝑒𝑛𝑠^𝛽−2∶𝑛≥𝑛₀𝑑𝑠≤2𝐿¹₀𝐺^∗(𝑡,𝑠)𝛼(𝐷(𝑠))𝑑𝑠, (61) which together with Lemma 5 and (H4) implies that 𝛼_𝑐(𝐷)1≤2𝐿Γ𝛽¹₀𝑠(1−𝑠)^𝛽−1𝑑𝑠𝛼_𝑐(𝐷). (62) Thus, 𝛼_𝑐(𝐷)=0. It follows from Lemma 8 that there is a convergent subsequence of {𝑥_𝑛}. Without loss of generality, we assume that {𝑥_𝑛} itself converges to some 𝑥∈𝑄. Then, the dominated convergence theorem and (16) imply that 𝑥(𝑡)=¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛽−2𝑥(𝑠)𝑑𝑠. (63) Thus, the singular BVP (1) has a positive solution 𝑡^𝛽−2𝑥(𝑡). Similarly, {𝑦_𝑛} has a convergent subsequence, which converges to 𝑦(𝑡). Then, 𝑡^𝛽−2𝑦(𝑡) is also a positive solution to BVP (1).
Finally, we show 𝑥≠𝑦. We only need to prove that the operator 𝐴^∗ has no fixed point in 𝜕𝑄_𝑅.
In fact, if it is not true, then we assume that 𝑧 is a fixed point of the operator 𝐴^∗ in 𝜕𝑄_𝑅. Then 𝑧(𝑡)=¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛽−2𝑧(𝑠)𝑑𝑠, (64) with (𝛽−1)𝑡(1−𝑡)^𝛽−1𝑅≤‖𝑧(𝑡)‖≤𝑅+1.
By (H1) and (H2), one can see that 𝑅=max_𝑡∈𝐽‖𝑧(𝑡≤)‖₁₀𝑠(1−𝑠)^𝛽−1𝑘(𝑠)𝑞𝛽−1𝑠(1−𝑠)^𝛽−1𝑅,𝑅𝑑𝑠≤¹₀𝑠(1−𝑠)^𝛽−1𝑘(𝑠)𝑞𝛽−1𝑠(1−𝑠)^𝛽−1𝑅,𝑅+1𝑑𝑠<𝑅, (65) which is a contradiction. The proof of this theorem is completed.

Remark 14. It is noted that the singularity of 𝑓(𝑡,𝑥) at 𝑡=0,1 is overcome by (H2), while the singularity at 𝑥=𝜃 is handled by solving the approximate problem and using the sequential-based technique.

4. Example

Consider the following BVP: −𝐷₀^3/2₊𝑥_𝑛(𝑡)=12𝜋𝑡()×11−𝑡∑^𝑚_𝑖=1𝑥_𝑖(𝑡)𝑡^1/2+√𝑡𝑥_𝑛+1(𝑡)+𝑡𝑥²_𝑛+2(𝑡),(),𝑥𝑡∈0,1_𝑛(0)=𝑥_𝑛(1)=0,𝑛=1,2,…,𝑚, (66) where 𝑥_𝑚+𝑛=𝑥_𝑛. Then, BVP (66) has at least two positive solutions.

Proof. We consider the problem (66) in 𝑚-dimensional Euclidean space 𝑅^𝑚={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚)∶𝑥_𝑛∈𝑅,𝑛=1,2,…,𝑚} with the norm ‖𝑥‖=∑^𝑚_𝑛=1|𝑥_𝑛|. Then, BVP (66) has the form of (1) with 𝑥(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡),…𝑥_𝑚(𝑡),𝑓_𝑛()=1𝑡,𝑥2𝜋𝑡()11−𝑡∑^𝑚_𝑖=1𝑥_𝑖𝑡^1/2+√𝑡𝑥_𝑛+1+𝑡𝑥²_𝑛+2. (67)
It is easy to see that 𝑓(𝑡,𝑥) is singular at 𝑡=0,1 and 𝑥=𝜃. Set 𝑘(𝑡)=1/(2𝜋𝑡(1−𝑡)) and 𝑞(𝑥)=(1/|∑^𝑚_𝑖=1𝑥_𝑖|)+𝑥_𝑛+1+𝑥²_𝑛+2. One can easily see that (H1) holds.
We choose 𝑅=1 and 𝜑^∗=𝜙^∗=(1,1,…,1). Considering that ¹₀𝑠()𝜋1−𝑠𝑑𝑠=8, (68) we can easily check that (H2)–(H6) hold. From Theorem 13, BVP (66) has at least two positive solutions 𝑥=(𝑥₁,𝑥₂,…𝑥_𝑚) and (𝑦₁,𝑦₂,…𝑦_𝑚) which satisfy 0<‖𝑥‖<1<‖𝑦‖<+∞.

Acknowledgments

The authors wish to thank referees and Dr. Haitao Li for their valuable suggestions. The project was supported by the National Natural Science Foundation of China (11171192), Graduate Educational Innovation Foundation of Shandong Province (SDYY1005), and the Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2010SF025).