Positive Solutions Using Bifurcation Techniques for Boundary Value Problems of Fractional Differential Equations

Abstract

This paper is concerned with the existence of positive solutions for a class of boundary value problems of fractional differential equations with parameter. The main tools used here are bifurcation techniques and topological degree theory. Finally, an example is worked out to demonstrate the main result.

1. Introduction

During the last few decades, fractional calculus and fractional differential equations have been studied extensively since fractional-order models are found to be more adequate than integer-order models in some real-world problems. In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth involves derivatives of fractional order. For details and examples, see [1–7] and references therein. Recently, there have been a few papers which deal with the boundary value problem for fractional differential equation. For example, in [8], Tian and Liu investigated the following singular fractional boundary value problem (BVP, for short) of the form ^𝐶𝐷^𝛼₀⁺𝑢(𝑡)((𝑡𝑢+𝜆𝑓𝑡,𝑢))=0,0<𝑡<1,_(𝑗)(0)𝑢=0,0≤𝑗≤𝑛−1,𝑗≠2,_′′(1)=0, (1) where ^𝐶𝐷^𝛼₀⁺ is Caputo’s fractional derivatives, 𝑛−1<𝛼≤𝑛, 𝑛≥4, and 𝑓∶(0,1)×(0,+∞)→[0,+∞) is continuous; that is, 𝑓(𝑡,𝑢) may be singular at 𝑡=0,1 and 𝑢=0. By constructing a special cone, they obtained that there exist positive numbers 𝜆^∗ and 𝜆^∗∗ with 𝜆^∗<𝜆^∗∗ such that the above system has at least two positive solutions for 𝜆∈(0,𝜆^∗) and no solution for 𝜆>𝜆^∗∗ under some suitable assumptions such as the following.(A1) There exists an interval [𝑎,𝑏]⊂(0,1) such that lim_𝑢→+∞𝑓(𝑠,𝑢)/𝑢=+∞ uniformly with respect to 𝑠∈[𝑎,𝑏].

In [9], Bai and Lü consider the following nonlinear fractional differential equation Dirichlet-type boundary value problem: 𝐷^𝛼₀⁺𝑢(𝑡)((𝑡(),𝑢(0)(1)+𝑓𝑡,𝑢))=0,𝑡∈0,1=𝑢=0, (2) where 1<𝛼≤2 is a real number and 𝐷^𝛼₀⁺ is the standard Riemann-Liouville differentiation. The corresponding Green function is derived. By means of some fixed point theorems on cone, the existence and multiplicity of positive solutions for BVP (2) were investigated.

In [10], Jiang and Yuan further investigated BVP (2). Comparing with [9], they deduced some new properties of the Green function, which extended the results of integer-order Dirichlet boundary value problems. Based on these new properties and Krasnoselskii fixed point theorem, the existence and multiplicity of positive solutions for BVP (2) were considered.

In this paper, by using bifurcation techniques, we consider the following boundary value problem of fractional differential equation: 𝐷^𝛼₀⁺𝑢(𝑡)((𝑡(),𝑢(0)(1)+𝜂𝑓𝑡,𝑢))=0,𝑡∈0,1=𝑢=0, (3) where 1<𝛼≤2, 𝐷^𝛼₀⁺ is the standard Riemann-Liouville differentiation, 𝑟>0 is a given constant, and 𝑓∶[0,1]×ℝ⁺→ℝ⁺ is a given continuous function satisfying some assumptions that will be specified later.

It is remarkable that the method used in references mentioned above was fixed point theorems and the same kind of conditions was used such that the nonlinearity 𝑓(𝑡,𝑢) satisfies superlinear or sublinear condition at 0 and ∞, which is similar to (A1). To the best of our knowledge, there is no paper studying such fractional differential equations using bifurcation ideas. As we know, the bifurcation technique is widely used in solving BVP of integer-order differential equations (see, e.g., [11–13] and references therein). In [14], by virtue of bifurcation ideas, the authors studied a kind of BVP of differential inclusions. The purpose of present paper is to fill this gap. By using bifurcation techniques and topological degree theory, the existence of positive solutions of BVP (3) is investigated. The main features of present paper are as follows. First, the nonlinearity 𝑓(𝑡,𝑢) is asymptotically linear at 0 and ∞, not super-linear or sub-linear (see the condition (H1) in Section 2 and example in Section 4). Next, the main method used here is bifurcation techniques and topological degree, not fixed point theorem on cone, which is different from the references.

The paper is organized as described below. At the end of this section, for completeness, we list some results on bifurcation theory from interval and topological degree of completely continuous operators. Section 2 contains background materials and preliminaries. In Section 3, by using bifurcation techniques and topological degree theory, bifurcation results from infinity and trivial solution are established. Then the main results of present paper are given and proved. Finally in Section 4, an example is worked out to demonstrate the main results.

Lemma 1 (Schmitt and Thompson [15]). Let 𝑉 be a real reflexive Banach space. Let 𝐺∶ℝ×𝑉 to 𝑉 be completely continuous such that 𝐺(𝜆,0)=0, for all 𝜆∈ℝ. Let 𝑎,𝑏∈ℝ (𝑎<𝑏) be such that 𝑢=0 is an isolated solution of the equation 𝑢−𝐺(𝜆,𝑢)=0,𝑢∈𝑉, (4) for 𝜆=𝑎 and 𝜆=𝑏, where (𝑎,0), (𝑏,0) are not bifurcation points of (4). Furthermore, assume that deg𝐼−𝐺(𝑎,⋅),𝐵_𝑟(0),0≠deg𝐼−𝐺(𝑏,⋅),𝐵_𝑟(0),0, (5) where 𝐵_𝑟(0) is an isolating neighborhood of the trivial solution. Let 𝒯=)∶()(4)∪]).(𝜆,𝑢𝜆,𝑢𝑖𝑠𝑎𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑜𝑓𝑤𝑖𝑡ℎ𝑢≠0([𝑎,𝑏×0 (6)
Then there exists a connected component 𝒞 of 𝒯 containing [𝑎,𝑏]×0 in ℝ×𝑉, and either(i)𝒞 is unbounded in ℝ×𝑉 or(ii)𝒞∩[(ℝ⧵[𝑎,𝑏])×0]≠∅.

Lemma 2 (Schmitt [16]). Let 𝑉 be a real reflexive Banach space. Let 𝐺∶ℝ×𝑉 to 𝑉 be completely continuous, and let 𝑎,𝑏∈ℝ (𝑎<𝑏) be such that the solution of (4) is, a priori, bounded in 𝑉 for 𝜆=𝑎 and 𝜆=𝑏; that is, there exists an 𝑅>0 such that 𝐺(𝑎,𝑢)≠𝑢≠𝐺(𝑏,𝑢) (7) for all 𝑢 with ‖𝑢‖≥𝑅. Furthermore, assume that deg𝐼−𝐺(𝑎,⋅),𝐵_𝑅(0),0≠deg𝐼−𝐺(𝑏,⋅),𝐵_𝑅(0),0, (8) for 𝑅>0 sufficiently large. Then there exists a closed connected set 𝒞 of solutions of (4) that is unbounded in [𝑎,𝑏]×𝑉, and either(i)𝒞 is unbounded in 𝜆 direction or(ii)there exists an interval [𝑐,𝑑] such that (𝑎,𝑏)∩(𝑐,𝑑)=∅ and 𝒞 bifurcates from infinity in [𝑐,𝑑]×𝑉.

Lemma 3 (Guo [17]). Let Ω be a bounded open set of real Banach space 𝐸, and let 𝐴∶Ω→𝐸 be completely continuous. If there exists 𝑦₀∈𝐸, 𝑦₀≠𝜃 such that 𝑥∈𝜕Ω,𝜏≥0⟹𝑥−𝐴𝑥≠𝜏𝑦₀, (9) then deg(𝐼−𝐴,Ω,𝜃)=0. (10)

2. Background Materials and Preliminaries

For convenience, we present some necessary definitions and results on fractional calculus theory (see [6]).

Definition 4. The fractional (arbitrary) order integral of the function ℎ∈𝐿¹([𝑎,𝑏]) of order 𝛼∈ℝ⁺ is defined by 𝐼^𝛼_𝑎+ℎ(𝑡)=^𝑡_𝑎(𝑡−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠, (11) where Γ is the gamma function. When 𝑎=0, we write 𝐼^𝛼ℎ(𝑡)=[ℎ∗𝜑_𝛼](𝑡), where 𝜑_𝛼(𝑡)=𝑡^𝛼−1/Γ(𝛼) for 𝑡>0, and 𝜑_𝛼(𝑡)=0 for 𝑡≤0, and 𝜑_𝛼→𝛿(𝑡) as 𝛼→0, where 𝛿 is the delta function.

Definition 5. For a function ℎ given on the interval [𝑎,𝑏], the 𝛼th Riemann-Liouville fractional-order derivative of ℎ is defined by 𝐷^𝛼_𝑎+ℎ𝑡)=1(Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡_𝑎(𝑡−𝑠)^{𝑛−𝛼−1}ℎ(𝑠)𝑑𝑠, (12) where 𝑛=[𝛼]+1.

Lemma 6. Let 𝛼>0; then, the differential equation 𝐷^𝛼₀⁺𝑢(𝑡)=0. (13) has solutions 𝑢(𝑡)=𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑛𝑡^𝛼−𝑛, for some 𝑐_𝑖∈ℝ, 𝑖=0,1,2,…,𝑛, where 𝑛 is the smallest integer greater than or equal to 𝛼.

Notice that 𝐷^𝛼₀⁺𝐼^𝛼ℎ(𝑡)=ℎ(𝑡) for all ℎ∈𝐶(0,1)∩𝐿(0,1). From Lemma 6, we deduce the following result.

Lemma 7. Assume that 𝑢∈𝐶(0,1)∩𝐿¹[0,1] with a derivative of order 𝑛 that belongs to 𝐶(0,1)∩𝐿¹[0,1]. Then 𝐼^𝛼₀⁺𝐷^𝛼₀⁺𝑢(𝑡)=𝑢(𝑡)+𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑛𝑡^𝛼−𝑛, (14) for some 𝑐_𝑖∈ℝ, 𝑖=0,1,2,…,𝑛, where 𝑛 is the smallest integer greater than or equal to 𝛼.

For more detailed results of fractional calculus, we refer the reader to [6].

Now let us list the following assumption satisfied throughout the paper.(H1) There exist two positive numbers 𝑟,𝑅 with 𝑟<𝑅 and functions 𝑎₀,𝑎⁰,𝑏_∞,𝑎⁰∈𝐶(𝐽,ℝ⁺) with 𝑎₀(𝑡),𝑎⁰(𝑡),𝑏_∞(𝑡),𝑎⁰(𝑡)≢0 in any subinterval of [0,1] such that 𝑓(𝑡,𝑢)⊂𝑎₀(𝑡)𝑢−𝜉₁(𝑡,𝑢),𝑎⁰(𝑡)𝑢+𝜉₂(),∀()[𝑡,𝑢𝑡,𝑢∈𝐽×0,𝑟],𝑓(𝑡,𝑢)⊂𝑏_∞(𝑡)𝑢−𝜁₁(𝑡,𝑢),𝑏^∞(𝑡)𝑢+𝜁₂(),∀()𝑡,𝑢𝑡,𝑢∈𝐽×𝑅,+∞, (15)where 𝐽=[0,1], 𝜉_𝑖,𝜁_𝑖∈𝐶(𝐽×ℝ⁺) with 𝜉_𝑖(𝑡,𝑡^𝛼−2𝑢)=𝑜(𝑡^𝛼−2𝑢) as 𝑢→0 uniformly with respect to 𝑡∈[0,1] (𝑖=1,2), and 𝜁_𝑖(𝑡,𝑡^𝛼−2𝑢)=𝑜(𝑡^𝛼−2𝑢) as 𝑢→+∞ uniformly with respect to 𝑡∈[0,1] (𝑖=1,2).

To solve BVP (3), we first consider the following linear boundary problem of fractional differential equation: 𝐷^𝛼₀⁺𝑢(𝑡)(𝑡)(),𝑢(0)(1)+𝑔=0,𝑡∈0,1=𝑢=0, (16) where 𝑔∈𝐶[0,1]. We cite the following two lemmas from references.

Lemma 8 (see [9]). Given 𝑔∈𝐶[0,1], then 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑔(𝑠)d𝑠 (17) is a solution of (16), where 𝐺()=1𝑡,𝑠Γ(𝛼)[𝑡(1−𝑠)]^𝛼−1−(𝑡−𝑠)^𝛼−1[𝑡(,0≤𝑠≤𝑡≤1,1−𝑠)]^𝛼−1,0≤𝑡≤𝑠≤1. (18)

Lemma 9 (see [10]). The function 𝐺(𝑡,𝑠) defined by (18) has the following properties.(i)𝐺(𝑡,𝑠)>0, ∀𝑡,𝑠∈(0,1).(ii)The function 𝐺^∗(𝑡,𝑠)=∶𝑡^2−𝛼𝐺(𝑡,𝑠) has the following properties: 𝛼−1Γ(𝛼)𝑡(1−𝑡)𝑠(1−𝑠)^𝛼−1≤𝐺^∗()≤1𝑡,𝑠Γ(𝛼)𝑠(1−𝑠)^𝛼−1,∀𝑡,𝑠∈[0,1]. (19)

The basic space used in this paper is 𝐸=∶{𝑢∈𝐶[0,1]∶𝑢(0)=𝑢(1)=0}. (20) Obviously, 𝐸 is a Banach space with norm ‖𝑢‖=max_𝑡∈𝐽|𝑢(𝑡)| (for all 𝑢∈𝐸).

Let 𝑄∶={𝑣∈𝐸∶𝑣(𝑡)≥(𝛼−1)𝑡(1−𝑡)𝑣(𝑠)≥0,∀𝑠,𝑡∈(0,1)}. (21) It is easy to see that 𝑄 is a cone of 𝐸. Moreover, from (21), we have, for all 𝑣∈𝑄, 𝑣(𝑡)≥(𝛼−1)𝑡(1−𝑡)‖𝑣‖,∀𝑡∈[0,1]. (22)

For the sake of using bifurcation technique to investigate BVP (3), we study the following fractional boundary value problem with parameter 𝜆: 𝐷^𝛼₀⁺𝑦(𝑡)(𝑡(),𝑦(0)(1)+𝜆𝑓𝑡,𝑦)=0,𝑡∈0,1=𝑦=0. (23)

A function (𝜆,𝑢) is said to be a solution of BVP (23) if (𝜆,𝑢) satisfies (23). In addition, if 𝜆>0, 𝑢(𝑡)>0 for 𝑡∈(0,1), then (𝜆,𝑢) is said to be a positive solution of BVP (23). Obviously, if 𝜆>0, 𝑢∈𝑄⧵{𝜃} is a solution of BVP (23), then by (22), we know that (𝜆,𝑢) is a positive solution of BVP (23), where 𝜃 denotes the zero element of Banach space 𝐸.

Define 𝑓()=𝑓(),()𝑡,𝑢𝑡,𝑢𝑡,𝑢∈𝐽×ℝ⁺,𝑓(𝑡,0),(𝑡,𝑢)∈𝐽×(−∞,0). (24) Then 𝑓(𝑡,𝑢)≥0 on 𝐽×ℝ. Let 𝐴_𝜆𝑣(𝑡)=∶𝜆¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛼−2𝑣(𝑠)𝑑𝑠,∀𝑣∈𝑄. (25) By assumption (H1) and using a similar process of the proof of Lemma  4.1 in [10], we know that 𝐴_𝜆∶𝐶[0,1]→𝑄 is completely continuous.

From Lemma 8, if 𝑣∈𝐶[0,1] is the the fixed point of operator 𝐴_𝜆, then 𝑦(𝑡)=𝑡^𝛼−2𝑣(𝑡) is the solution of 𝐷^𝛼₀⁺𝑦(𝑡)+𝜆𝑓(𝑡(),𝑦(0)(1)𝑡,𝑦)=0,𝑡∈0,1=𝑦=0. (26)

Let Σ=∶{(𝜆,𝑣)∈ℝ⁺×𝐶[0,1]∶𝑣=𝐴_𝜆𝑣,𝑣≠𝜃}, (27) where 𝜃 is the zero element of 𝐶[0,1]. From Lemma 9 and the definitions of 𝑓 and the cone 𝑄, it is easy to see that Σ⊂𝑄. Moreover, we have the following conclusion.

Lemma 10. For 𝜆>0, if 𝑣 is a nontrivial fixed point of operator 𝐴_𝜆, then (𝜆,𝑡^𝛼−2𝑣(𝑡)) is a positive solution of BVP (23). Furthermore, (𝜆,𝑦) is a positive solution of BVP (23) if and only if (𝜆,𝑦) is a nontrivial solution of BVP (26).

For 𝑎∈𝐶(𝐽,ℝ⁺) with 𝑎(𝑡)≢0 in any subinterval of 𝐽, define the linear operator 𝐿_𝑎∶𝐶(𝐽)→𝐶(𝐽) by 𝐿_𝑎𝑢(𝑡)=¹₀𝐺^∗(𝑡,𝑠)𝑎(𝑠)𝑠^𝛼−2𝑢(𝑠)𝑑𝑠, (28) where 𝐺^∗(𝑡,𝑠) is defined by Lemma 9.

From Lemmas 8 and 9 and the well-known Krein-Rutman Theorem, one can obtain the following lemma.

Lemma 11. The operator 𝐿_𝑎 defined by (28) is completely continuous and has a unique characteristic value 𝜆₁(𝑎), which is positive, real, and simple and the corresponding eigenfunction 𝜙(𝑡) is of one sign in (0,1); that is, 𝜙(𝑡)=𝜆₁(𝑎)𝐿_𝑎𝜙(𝑡) for all 𝑡∈𝐽.

Notice that the operator 𝐿_𝑎 can be regarded as 𝐿_𝑎∶𝐿²[0,1]→𝐿²[0,1]. This together with Lemma 11 guarantees that 𝜆₁(𝑎) is also the characteristic value of 𝐿^∗_𝑎, where 𝐿^∗_𝑎 is the conjugate operator of 𝐿_𝑎. Let 𝜑^∗ denote the nonnegative eigenfunction of 𝐿^∗_𝑎 corresponding to 𝜆₁(𝑎). Then we have 𝜑^∗(𝑡)=𝜆₁(𝑎)𝐿^∗_𝑎𝜑^∗(𝑡),∀𝑡∈𝐽. (29)

3. Main Results

The main results of present paper are the following two theorems.

Theorem 12. Suppose that either(i)𝜂∈[𝜆₁(𝑎₀),𝜆₁(𝑏^∞)] or(ii)𝜂∈[𝜆₁(𝑏_∞),𝜆₁(𝑎⁰)].
Then BVP (3) has at least one positive solution.

Theorem 13. Suppose the following.(H2) There exist 𝑅>0 and ℎ∈𝐿[0,1] such that 𝑓()(𝑡)[][],Γ(𝛼)𝑡,𝑢≤ℎ𝑢,𝑓𝑜𝑟𝑡∈0,1,𝑢∈0,𝑅𝜂<∫¹₀[𝑠(1−𝑠)]^𝛼−1ℎ(𝑠).𝑑𝑠 (30)
In addition, if 𝜂>max𝜆₁𝑎₀,𝜆₁𝑏_∞, (31)
then BVP (3) has at least two positive solutions.

To prove Theorems 12 and 13, we first prove the following lemmas.

Lemma 14. Let [𝑐,𝑑]⊂ℝ⁺ be a compact interval with [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]∩[𝑐,𝑑]=∅. Then there exists 𝛿₁∈(0,𝑟) such that 𝑣≠𝐴_𝜆𝑣,∀𝜆∈[𝑐,𝑑],∀𝑣∈𝐸𝑤𝑖𝑡ℎ0<‖𝑣‖≤𝛿₁. (32)

Proof. If this is false, then there exist {(𝜇_𝑛,𝑣_𝑛)}⊂[𝑐,𝑑]×𝐶[0,1] with ‖𝑣_𝑛‖→0+(𝑛→+∞) such that 𝑣_𝑛=𝐴_𝜇𝑛𝑣_𝑛. Without loss of generality, assume 𝜇_𝑛→𝜇∈[𝑐,𝑑] and ‖𝑣_𝑛‖<𝑟 for all 𝑛. Notice that 𝑣_𝑛∈𝑄. By Lemma 10 and (21), we have 𝑣_𝑛(𝑡)>0 in (0,1). Set 𝑤_𝑛=𝑣_𝑛/‖𝑣_𝑛‖. Then 𝑤_𝑛=𝐴_𝜇𝑛𝑣_𝑛/‖𝑣_𝑛‖. From the definition of 𝑓(𝑡,𝑢), it is easy to see that {𝑤_𝑛} is relatively compact in 𝐶[0,1]. Taking a subsequence and relabeling if necessary, suppose 𝑤_𝑛→𝑤 in 𝐶[0,1]. Then ‖𝑤‖=1 and 𝑤∈𝑄.
On the other hand, from (H1), we know 𝑓(𝑡,𝑢)∈𝑎₀(𝑡)𝑢−𝜉₁(𝑡,𝑢),𝑎⁰(𝑡)𝑢+𝜉₂(),∀()[𝑡,𝑢𝑡,𝑢∈𝐽×0,𝑟]. (33) Therefore, by virtue of (25), we know 𝑤_𝑛(𝑡)≤𝜇_𝑛¹₀𝐺^∗(𝑡,𝑠)𝑎⁰(𝑠)𝑠^𝛼−2𝑤_𝑛(𝑠)+𝜉₂𝑠,𝑠^𝛼−2𝑣_𝑛(𝑠)𝑣_𝑛𝑑𝑠,𝑤_𝑛(𝑡)≥𝜇_𝑛¹₀𝐺^∗(𝑡,𝑠)𝑎₀(𝑠)×𝑠^𝛼−2𝑤_𝑛(𝑠)−𝜉₁𝑠,𝑠^𝛼−2𝑣_𝑛(𝑠)𝑣_𝑛𝑑𝑠. (34)(35) Let 𝜓^∗ and 𝜓_∗ be the positive eigenfunctions of 𝐿^∗_𝑎0,𝐿^∗_𝑎0 corresponding to 𝜆₁(𝑎⁰) and 𝜆₁(𝑎₀), respectively. Then from (34), it follows that 𝑤_𝑛,𝜓^∗≤𝜇_𝑛𝐿_𝑎0𝑤_𝑛,𝜓^∗+𝜇_𝑛¹₀𝜓^∗(𝑡)×¹₀𝐺^∗(𝑡,𝑠)𝑎₀(𝑠)𝜉₂𝑠,𝑠^𝛼−2𝑣_𝑛(𝑠)𝑣_𝑛𝑑𝑠𝑑𝑡. (36) Letting 𝑛→+∞ and using condition (H1), we have 𝑣,𝜓^∗≤𝜇𝐿_𝑎0𝑣,𝜓^∗=𝜇𝑣,𝐿^∗_𝑎0𝜓^∗𝜓=𝜇𝑣,^∗𝜆₁𝑎⁰, (37) which implies 𝜇≥𝜆₁(𝑎⁰). Similarly, one can deduce from (35) that 𝜇≤𝜆₁(𝑎₀).
Consequently, 𝜆₁(𝑎⁰)≤𝜇≤𝜆₁(𝑎₀), which contradicts 𝜇∈[𝑐,𝑑]. Therefore, there exists 𝛿₁∈(0,𝑟) such that 𝑣≠𝐴_𝜆𝑣,∀𝜆∈[𝑐,𝑑],∀𝑣∈𝐸with0<‖𝑣‖≤𝛿₁. (38)

Lemma 15. For 𝜇∈(0,𝜆₁(𝑎⁰)), there exists 𝛿₁∈(0,𝑟) such that deg𝐼−𝐴_𝜇,𝐵_𝛿,0=1,∀𝛿∈0,𝛿₁. (39)

Proof. Notice that [0,𝜇]∩[𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]=∅. From Lemma 14, there exists 𝛿₁∈(0,𝑟) such that 𝑣≠𝐴_𝜆𝑣,∀𝜆∈0,𝜇,∀𝑣∈𝐶[0,1]with0<‖𝑣‖≤𝛿₁, (40) which means 𝑣≠𝜏𝐴_𝜇𝑣,∀𝜏∈[0,1],∀𝑣∈𝐶[0,1]with0<‖𝑣‖≤𝛿₁. (41)
Therefore, by the homotopy invariance of topological degree, we have deg𝐼−𝐴_𝜇,𝐵_𝛿,0=deg𝐼,𝐵_𝛿,0=1,∀𝛿∈0,𝛿₁. (42)

Lemma 16. For 𝜆>𝜆₁(𝑎₀), there exists 𝛿₂∈(0,𝑟) such that deg𝐼−𝐴_𝜆,𝐵_𝛿,0=0,∀𝛿∈0,𝛿₂. (43)

Proof. First we prove that for 𝜆>𝜆₁(𝑎₀), there exists 𝛿₂∈(0,𝑟) such that 𝑣−𝐴_𝜆𝑣≠𝜏𝜑₀,∀𝜏≥0,∀𝑣∈𝐶[0,1]with0<‖𝑣‖≤𝛿₂, (44) where 𝜑₀ is the positive eigenfunctions of 𝐿_𝑎0 corresponding to 𝜆₁(𝑎₀).
If this is false, then there exist 𝑣_𝑛∈𝐶[0,1] with ‖𝑣_𝑛‖→0 (𝑛→+∞) and 𝜏_𝑛≥0 such that 𝑣_𝑛−𝐴_𝜆𝑣_𝑛=𝜏_𝑛𝜑₀. (45)
By Lemma 10, we have 𝑣_𝑛(𝑡)>0 in (0,1). From 𝜆>𝜆₁(𝑎₀), there exists 𝜎>0 satisfying 𝜆>(1−𝜎)𝜆₁(𝑎₀). Then condition (H1) guarantees that there exists 𝛿∈(0,𝑟) such that |𝜉₁(𝑡,𝑡^𝛼−2𝑢)|<𝜎𝑡^𝛼−2𝑢 for 𝑢∈(0,𝛿). Noticing ‖𝑣_𝑛‖→0(𝑛→+∞), there exists 𝑁>0 such that ‖𝑣_𝑛‖<𝛿 for 𝑛>𝑁. Consequently, by virtue of (25) and (45), for 𝑛>𝑁, we know 𝑣_𝑛(𝑡)=𝐴_𝜆𝑣_𝑛(𝑡)+𝜏_𝑛𝜑₀≥𝜆¹₀𝐺^∗(𝑡,𝑠)𝑎₀(𝑠)×𝑠^𝛼−2𝑣_𝑛(𝑠)−𝜉₁𝑠,𝑠^𝛼−2𝑣_𝑛(𝑠)()𝑑𝑠≥𝜆1−𝜎¹₀𝐺^∗(𝑡,𝑠)𝑎₀(𝑠)𝑠^𝛼−2𝑣_𝑛(𝑠)()𝐿𝑑𝑠=𝜆1−𝜎_𝑎0𝑣_𝑛(𝑡). (46) Let 𝜓_∗ be the positive eigenfunction of 𝐿^∗_𝑎0 corresponding to 𝜆₁(𝑎₀). Then 𝑣_𝑛,𝜓_∗≥𝜆(1−𝜎)𝐿_𝑎0𝑣_𝑛,𝜓_∗=𝜆(1−𝜎)𝑣_𝑛,𝐿^∗_𝑎0𝜓_∗=𝜆(1−𝜎)𝑣_𝑛,𝜓_∗𝜆₁𝑎₀. (47)
This together with ⟨𝑣_𝑛,𝜓_∗⟩>0 guarantees that 𝜆(1−𝜎)≤𝜆₁𝑎₀, (48) which is a contradiction. Therefore, (44) holds. By Lemma 3, for each 𝜆>𝜆₁(𝑎₀), there exists 𝛿₂>0 such that deg𝐼−𝐴_𝜆,𝐵_𝛿,0=0,∀𝛿∈0,𝛿₂. (49)

Theorem 17. [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)] is a bifurcation interval of positive solutions from the trivial solution for BVP (23); that is, there exists an unbounded component 𝒞₀ of positive solutions of BVP (23), which meets [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0}. Moreover, there exists no bifurcation interval of positive solutions from the trivial solution which is disjointed with [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)].

Proof. By virtue of (27) and Lemma 10, we need only to prove that there exists an unbounded component 𝒞₀ of Σ, which meets [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0}, and there exists no bifurcation interval of Σ from the trivial solution which is disjointed with [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)].
For fixed 𝑛∈ℕ with 𝜆₁(𝑎⁰)−(1/𝑛)>0, by Lemmas 15 and 16 and their proof, there exists 𝑟>0 such that all of the conditions of Lemma 1 are satisfied with 𝐺(𝜆,𝑢)=𝐴_𝜆𝑢, 𝑎=𝜆₁(𝑎⁰)−(1/𝑛), and 𝑏=𝜆₁(𝑎₀)+(1/𝑛). This together with Lemma 10 guarantees that there exists a closed connected set 𝒞_𝑛 of Σ containing [𝜆₁(𝑎⁰)−(1/𝑛),𝜆₁(𝑎₀)+(1/𝑛)]×0 in ℝ⁺×𝐶[0,1]. From Lemma 14, the case (ii) of Lemma 1 cannot occur. Thus, 𝒞_𝑛 bifurcates from [𝜆₁(𝑎⁰)−(1/𝑛),𝜆₁(𝑎₀)+1/𝑛]×0 and is unbounded in ℝ⁺×𝐶[0,1]. Moreover, for any closed interval [𝑐,𝑑]⊂[𝜆₁(𝑎⁰)−1/𝑛,𝜆₁(𝑎₀)+1/𝑛]⧵[𝜆₁(𝑎⁰),𝜆₁(𝑎₀)], by Lemma 14, there exists 𝛿₁>0 such that the set {𝑣∈𝐶[0,1]∶(𝜆,𝑣)∈Σ,0<‖𝑣‖≤𝛿₁,𝜆∈[𝑐,𝑑]}=∅. Therefore, 𝒞_𝑛 must be bifurcated from [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0}, which implies that 𝒞_𝑛 can be regarded as 𝒞₀. In addition, using Lemma 14 again, there exists no bifurcation interval of positive solutions from the trivial solution which is disjointed with [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)].

By a process similar to the above, one can obtain the following conclusions.

Lemma 18. Let [𝑐,𝑑]⊂ℝ⁺ be a compact interval with [𝜆₁(𝑏^∞),𝜆₁(𝑏_∞)]∩[𝑐,𝑑]=∅. Then there exists 𝑅₁>𝑅 such that 𝑢≠𝐴_𝜆𝑢,∀𝜆∈[𝑐,𝑑],∀𝑢∈𝐶[0,1]𝑤𝑖𝑡ℎ‖𝑢‖≥𝑅₁. (50)

Lemma 19. For 𝜇∈(0,𝜆₁(𝑏^∞)), there exists 𝑅₁>𝑅 such that deg𝐼−𝐴_𝜇,𝐵_𝑅,0=1,∀𝑅≥𝑅₁. (51)

Lemma 20. For 𝜆>𝜆₁(𝑏_∞), there exists 𝑅₂>𝑅 such that deg𝐼−𝐴_𝜆,𝐵_𝑅,0=0,∀𝑅≥𝑅₂. (52)

Theorem 21. [𝜆₁(𝑏^∞),𝜆₁(𝑏_∞)] is a bifurcation interval of positive solutions from infinity for BVP (23), and there exists no bifurcation interval of positive solutions from infinity which is disjoint with [𝜆₁(𝑏^∞),𝜆₁(𝑏_∞)]. More precisely, there exists an unbounded component 𝒞^∞ of solutions of BVP (23) which meets [𝜆₁(𝑏^∞),𝜆₁(𝑏_∞)]×∞ and is unbounded in 𝜆 direction.

Now we are in position to prove Theorems 12 and 13.

Proof of Theorem 12. Obviously the solution of the form (𝜂,𝑦)(𝑦≠𝜃) of (23) is a positive solution of BVP (3). So by Lemma 10, it is sufficient to show that there is a component 𝒞 of Σ that crosses the hyperplane {𝜂}×𝐶[0,1], where Σ⊂ℝ⁺×𝐶[0,1] is defined by (27).
Case i  (𝜆₁(𝑎₀)<𝜂<𝜆₁(𝑏^∞)). By Theorem 17, there exists an unbounded component 𝒞₀ of positive solutions of BVP (23), which bifurcates from [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{𝜃}. Therefore, there exists (𝜇_𝑛,𝑣_𝑛)∈𝒞₀ such that 𝜇_𝑛+𝑣_𝑛⟶+∞as𝑛⟶+∞. (53)
If there exists some 𝑛∈ℕ such that 𝜇_𝑛≥𝜂, the conclusion follows. Suppose, on the contrary, 𝜇_𝑛<𝜂 for all 𝑛∈ℕ. Since (0,𝜃) is the only solution of (23) with 𝜆=0, by Lemmas 14 and 18, we know 𝒞₀∩({0}×𝐶[0,1])=∅. Therefore, 𝜇_𝑛∈(0,𝜂) for all 𝑛∈ℕ. Taking a subsequence and relabeling if necessary, suppose 𝜇_𝑛→𝜇^∗ as 𝑛→+∞. Then 𝜇^∗∈[0,𝜂]. This together with (53) guarantees that ‖𝑣_𝑛‖→+∞.
Choose [𝑐,𝑑]=[0,𝜆₁(𝑏^∞)−(1/𝑚)] for 𝑚∈ℕ. From Lemma 18, it follows that 𝜇^∗>𝜆₁(𝑏^∞)−(1/𝑚) for each 𝑚∈ℕ, which means 𝜇^∗≥𝜆₁(𝑏^∞)>𝜂. This is a contradiction.
Case ii  (𝜆₁(𝑏_∞)<𝜂<𝜆₁(𝑎⁰)). From Theorem 21, there exists an unbounded component 𝒞^∞ of solutions of (23) which bifurcates from [𝜆₁(𝑏^∞),𝜆₁(𝑏_∞)]×∞ and is unbounded in 𝜆 direction.
If 𝒞^∞∩(ℝ⁺×{0})=∅, using the fact that 𝒞^∞∩({0}×𝐶[0,1])=∅ and 𝒞^∞ is unbounded in 𝜆 direction, we know that 𝒞^∞ must cross the hyperplane {𝜂}×𝐶[0,1].
If 𝒞^∞∩(ℝ⁺×{0})≠∅, by 𝒞^∞∩({0}×𝐶[0,1])=∅ and Theorem 17, we know 𝒞^∞∩(ℝ⁺×{0})∈[𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0}. Therefore, 𝒞^∞ joins [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0} to [𝜆₁(𝑏^∞),𝜆₁(𝑏_∞)]×∞. This together with 𝜆₁(𝑏_∞)<𝜂<𝜆₁(𝑎⁰) guarantees that 𝒞^∞ crosses the hyperplane {𝜂}×𝐶[0,1].

Proof of Theorem 13. First we show that there exists 𝜀>0 such that Σ∩0,𝜂+𝜀×𝜕𝐵_𝑅=∅, (54) where 𝐵_𝑅={𝑣∈𝐶[0,1]∶‖𝑣‖<𝑅}, Σ⊂ℝ⁺×𝐶[0,1] is defined by (27).
In fact, from assumption (H2), it follows that there exists 𝜀>0 such that 𝜂+𝜀Γ(𝛼)¹₀[𝑠(1−𝑠)]^𝛼−1ℎ(𝑠)𝑑𝑠<1. (55) If there is a solution (𝜆,𝑣) of 𝑣=𝐴_𝜆𝑣 such that 0≤𝜆≤𝜂+𝜀 and ‖𝑣‖=𝑅, then 0≤𝑣(𝑡)≤‖𝑣‖=𝑅for𝑡∈[0,1]. (56) By virtue of (25) and Lemma 9, we have 𝑅=‖𝑣‖=max_𝑡∈𝐽𝜆¹₀𝐺^∗(𝑡,𝑠)𝑓𝑠,𝑠^𝛼−2𝑣(𝑠)≤𝑑𝑠𝜂+𝜀𝑅max_𝑡∈𝐽¹₀𝐺^∗(𝑡,𝑠)𝑠^𝛼−2ℎ(𝑠)≤𝑅𝑑𝑠𝜂+𝜀Γ(𝛼)¹₀[𝑠(1−𝑠)]^𝛼−1ℎ(𝑠)𝑑𝑠<𝑅, (57) which is a contradiction. Thus, Σ∩([0,𝜂+𝜀]×𝜕𝐵_𝑅)=∅.
Next, from Theorem 17, there exist unbounded components 𝒞₀ of solutions of (23), which meet [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0}. By (54), we know 𝒞₀∩([0,𝜂+𝜀]×𝜕𝐵_𝑅)=∅. This together with the fact that 𝒞₀ is unbounded, 𝜆₁(𝑎₀)<𝜂, and 𝒞₀∩({0}×𝐶[0,1])=∅ guarantees that 𝒞₀ crosses the hyperplane {𝜂}×𝐶[0,1]. Then BVP (23) has a positive solution 𝑣₁ with (𝜂,𝑣₁)∈𝒞₀ and ‖𝑣₁‖<𝑅. By Lemma 10, 𝑡^𝛼−2𝑣₁(𝑡) is a positive solution of BVP (3).
Similarly, by Theorem 21 and (54), BVP (3) has a positive solution 𝑡^𝛼−2𝑣₂(𝑡) with (𝜂,𝑣₂)∈𝒞^∞ and ‖𝑣₂‖>𝑅. The conclusion follows.

Immediately, from the proof of Theorem 13, we have the following corollary.

Corollary 22. Suppose that assumption (H2) holds. In addition, suppose that one of the following two conditions holds:(i)𝜆₁(𝑎₀)<𝜂;(ii)𝜆₁(𝑏_∞)<𝜂.
Then BVP (3) has at least one positive solution.

Remark 23. Corollary 22 is different from Theorem 12 though their results are similar.

4. An Example

Let 𝜌 be the unique characteristic value of 𝐿₁ corresponding to positive eigenfunctions with 𝑎(𝑡)≡𝑡 in (28). From Lemma 11, it follows that 𝜌 exists. Now we are ready to give the following example.

Example 1. Consider the following boundary value problem of fractional differential inclusions: 𝐷₀^1.5+𝑢(𝑡)((𝑡(),𝑢(0)(1)+𝜂𝑓𝑡,𝑢))=0,𝑡∈0,1=𝑢=0, (58) where 𝑓()ℎ(𝑢)1𝑡,𝑢=𝜌𝑡𝑢+sin𝑢(,ℎ(𝑢)=3+𝑡sin𝑡𝑢)21,𝑢∈0,2;11+𝑢,𝑢∈2;[).,34,𝑢∈3,+∞ (59)
Then BVP (58) has at least one positive solution as 𝜂∈[1/3,2/5].

Proof. BVP (58) can be regarded as the form (3). Let 𝑓(𝑡,𝑢)=0 for 𝑢=0; then, 𝑓(𝑡,𝑢) is continuous.
From (59), choose 𝑟=1/2,𝑅=3, 𝑎₀(𝑡)=𝜌𝑡/2, 𝑎⁰(𝑡)=(5/2)𝜌𝑡, 𝑏_∞(𝑡)=3𝜌𝑡, 𝑏^∞(𝑡)=5𝜌𝑡, 𝜉₁(𝑡,𝑢)=−2𝑡𝑢sin(𝑡𝑢), 𝜉₂(𝑡,𝑢)=(2/5)𝑡𝑢sin(𝑡𝑢), 𝜁₁(𝑡,𝑢)=−(1/3)𝑢sin(1/𝑢), 𝜁₂(𝑡,𝑢)=(1/5)𝑢sin(1/𝑢).
It is easy to see 𝜉_𝑖(𝑡,𝑡^𝛼−2𝑢)=𝑜(𝑡^𝛼−2𝑢) as 𝑢→0 and 𝜁_𝑖(𝑡,𝑡^𝛼−2𝑢)=𝑜(𝑡^𝛼−2𝑢) as 𝑢→+∞ both uniformly with respect to 𝑡∈[0,1], (𝑖=1,2).
Therefore, (H1) is satisfied.
By the definition of 𝜌, it is easy to see 𝜆₁(𝑎⁰)=2/5, 𝜆₁(𝑏_∞)=1/3.
As a result, by Theorem 12, BVP (58) has at least one positive solution as 𝜂∈[1/3,2/5].

Acknowledgments

The author wishes to thank the anonymous referees for their valuable suggestions. Research is supported by the NNSF of China (11171192), the Graduate Educational Innovation Foundation of Shandong Province (SDYY1005), and the Natural Science Foundation of Shandong Province (ZR2013AM005).