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

Authors: Yansheng Liu
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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
(1)CD0+αu(t)+λf(t,u(t))=0,0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0,
where CD0+α 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 limu→+∞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:
(2)D0+αu(t)+f(t,u(t))=0,t∈(0,1),u(0)=u(1)=0,
where 1<α≤2 is a real number and D0+α 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:(3)D0+αu(t)+ηf(t,u(t))=0,t∈(0,1),u(0)=u(1)=0,
where 1<α≤2, D0+α 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 nonlinearityf(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. Section2 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]).
LetV 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
(4)u-G(λ,u)=0,u∈V,
for λ=a and λ=b, where (a,0), (b,0) are not bifurcation points of (4). Furthermore, assume that
(5)deg(I-G(a,·),Br(0),0)≠deg(I-G(b,·),Br(0),0),
where Br(0) is an isolating neighborhood of the trivial solution. Let
(6)𝒯={(λ,u):(λ,u)isasolutionof(4)withu≠0}¯∪([a,b]×0).
Then there exists a connected component𝒞 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]).
LetV 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
(7)G(a,u)≠u≠G(b,u)
for all u with ∥u∥≥R. Furthermore, assume that
(8)deg(I-G(a,·),BR(0),0)≠deg(I-G(b,·),BR(0),0),
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]).
LetΩ 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
(9)x∈∂Ω,τ≥0⟹x-Ax≠τy0,
then
(10)deg(I-A,Ω,θ)=0.


## 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 functionh∈L1([a,b]) of order α∈ℝ+ is defined by
(11)Ia+αh(t)=∫at‍(t-s)α-1Γ(α)h(s)ds,
where Γ is the gamma function. When a=0, we write Iαh(t)=[h*φα](t), where φα(t)=tα-1/Γ(α) for t>0, and φα(t)=0 for t≤0, and φα→δ(t) as α→0, where δ is the delta function.Definition 5.
For a functionh given on the interval [a,b], the αth Riemann-Liouville fractional-order derivative of h is defined by
(12)(Da+αh)(t)=1Γ(n-α)(ddt)n∫at‍(t-s)n-α-1h(s)ds,
where n=[α]+1.Lemma 6.
Letα>0; then, the differential equation
(13)D0+αu(t)=0.
has solutions u(t)=c1tα-1+c2tα-2+⋯+cntα-n, for some ci∈ℝ, i=0,1,2,…,n, where n is the smallest integer greater than or equal to α.Notice thatD0+αIαh(t)=h(t) for all h∈C(0,1)∩L(0,1). From Lemma 6, we deduce the following result.Lemma 7.
Assume thatu∈C(0,1)∩L1[0,1] with a derivative of order n that belongs to C(0,1)∩L1[0,1]. Then
(14)I0+αD0+αu(t)=u(t)+c1tα-1+c2tα-2+⋯+cntα-n,
for some ci∈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 numbersr-,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
(15)f(t,u)⊂[a0(t)(u-ξ1(t,u)),a0(t)(u+ξ2(t,u))],ppppppppppppppppppp∀(t,u)∈J×[0,r-],f(t,u)⊂[b∞(t)(u-ζ1(t,u)),b∞(t)(u+ζ2(t,u))],pppppppppppppppp∀(t,u)∈J×[R-,+∞),where J=[0,1], ξi,ζi∈C(J×ℝ+) with ξi(t,tα-2u)=o(tα-2u) as u→0 uniformly with respect to t∈[0,1] (i=1,2), and ζi(t,tα-2u)=o(tα-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:
(16)D0+αu(t)+g(t)=0,t∈(0,1),u(0)=u(1)=0,
where g∈C[0,1]. We cite the following two lemmas from references.Lemma 8 (see [9]).
Giveng∈C[0,1], then
(17)u(t)=∫01‍G(t,s)g(s)ds
is a solution of (16), where
(18)G(t,s)=1Γ(α){[t(1-s)]α-1-(t-s)α-1,0≤s≤t≤1,[t(1-s)]α-1,0≤t≤s≤1.Lemma 9 (see [10]).
The functionG(t,s) defined by (18) has the following properties.(i)
G(t,s)>0, ∀t,s∈(0,1).(ii)
The functionG*(t,s)=∶t2-αG(t,s) has the following properties:
(19)α-1Γ(α)t(1-t)s(1-s)α-1≤G*(t,s)≤1Γ(α)s(1-s)α-1,ppppp∀t,s∈[0,1].The basic space used in this paper is(20)E=:{u∈C[0,1]:u(0)=u(1)=0}.
Obviously, E is a Banach space with norm ∥u∥=maxt∈J|u(t)| (for all u∈E).Let(21)Q:={v∈E:v(t)≥(α-1)t(1-t)v(s)≥0,∀s,t∈(0,1)}.
It is easy to see that Q is a cone of E. Moreover, from (21), we have, for all v∈Q,
(22)v(t)≥(α-1)t(1-t)∥v∥,∀t∈[0,1].For the sake of using bifurcation technique to investigate BVP (3), we study the following fractional boundary value problem with parameter λ:
(23)D0+αy(t)+λf(t,y(t))=0,t∈(0,1),y(0)=y(1)=0.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(24)f-(t,u)={f(t,u),(t,u)∈J×ℝ+,f(t,0),(t,u)∈J×(-∞,0).
Then f-(t,u)≥0 on J×ℝ. Let
(25)Aλv(t)=:λ∫01‍G*(t,s)f-(s,sα-2v(s))ds,∀v∈Q.
By assumption (H1) and using a similar process of the proof of Lemma  4.1 in [10], we know that Aλ:C[0,1]→Q is completely continuous.From Lemma8, if v∈C[0,1] is the the fixed point of operator Aλ, then y(t)=tα-2v(t) is the solution of
(26)D0+αy(t)+λf-(t,y(t))=0,t∈(0,1),y(0)=y(1)=0.Let(27)Σ=∶{(λ,v)∈ℝ+×C[0,1]:v=Aλv,v≠θ}¯,
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.
Forλ>0, if v is a nontrivial fixed point of operator Aλ, then (λ,tα-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).Fora∈C(J,ℝ+) with a(t)≢0 in any subinterval of J, define the linear operator La:C(J)→C(J) by
(28)Lau(t)=∫01‍G*(t,s)a(s)sα-2u(s)ds,
where G*(t,s) is defined by Lemma 9.From Lemmas8 and 9 and the well-known Krein-Rutman Theorem, one can obtain the following lemma.Lemma 11.
The operatorLa 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)Laϕ(t) for all t∈J.Notice that the operatorLa can be regarded as La:L2[0,1]→L2[0,1]. This together with Lemma 11 guarantees that λ1(a) is also the characteristic value of La*, where La* is the conjugate operator of La. Let φ* denote the nonnegative eigenfunction of La* corresponding to λ1(a). Then we have
(29)φ*(t)=λ1(a)La*φ*(t),∀t∈J.


## 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 existR>0 and h∈L[0,1] such that
(30)f(t,u)≤h(t)u,fort∈[0,1],u∈[0,R],η<Γ(α)∫01‍[s(1-s)]α-1h(s)ds.
In addition, if(31)η>max{λ1(a0),λ1(b∞)},
then BVP (3) has at least two positive solutions.To prove Theorems12 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
(32)v≠Aλv,∀λ∈[c,d],∀v∈Ewith0<∥v∥≤δ1.Proof.
If this is false, then there exist{(μn,vn)}⊂[c,d]×C[0,1] with ∥vn∥→0+(n→+∞) such that vn=Aμnvn. Without loss of generality, assume μn→μ∈[c,d] and ∥vn∥<r- for all n. Notice that vn∈Q. By Lemma 10 and (21), we have vn(t)>0 in (0,1). Set wn=vn/∥vn∥. Then wn=Aμnvn/∥vn∥. From the definition of f-(t,u), it is easy to see that {wn} is relatively compact in C[0,1]. Taking a subsequence and relabeling if necessary, suppose wn→w in C[0,1]. Then ∥w∥=1 and w∈Q.
On the other hand, from (H1), we know(33)f(t,u)∈[a0(t)(u-ξ1(t,u)),a0(t)(u+ξ2(t,u))],∀(t,u)∈J×[0,r-].
Therefore, by virtue of (25), we know
(34)wn(t)≤μn∫01‍G*(t,s)a0(s)(ξ2(s,sα-2vn(s))∥vn∥sα-2wn(s)ppppppppppppppppppp+ξ2(s,sα-2vn(s))∥vn∥)ds,(35)wn(t)≥μn∫01‍G*(t,s)a0(s)pppppp×(sα-2wn(s)-ξ1(s,sα-2vn(s))∥vn∥)ds.
Let ψ* and ψ* be the positive eigenfunctions of La0*,La0* corresponding to λ1(a0) and λ1(a0), respectively. Then from (34), it follows that
(36)〈wn,ψ*〉≤μn〈La0wn,ψ*〉+μn∫01‍ψ*(t)×∫01‍G*(t,s)a0(s)ξ2(s,sα-2vn(s))∥vn∥dsdt.
Letting n→+∞ and using condition (H1), we have
(37)〈v,ψ*〉≤μ〈La0v,ψ*〉=μ〈v,La0*ψ*〉=μ〈v,ψ*λ1(a0)〉,
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
(38)v≠Aλv,∀λ∈[c,d],∀v∈Ewith0<∥v∥≤δ1.Lemma 15.
Forμ∈(0,λ1(a0)), there exists δ1∈(0,r-) such that
(39)deg(I-Aμ,Bδ,0)=1,∀δ∈(0,δ1].Proof.
Notice that[0,μ]∩[λ1(a0),λ1(a0)]=∅. From Lemma 14, there exists δ1∈(0,r-) such that
(40)v≠Aλv,∀λ∈[0,μ],∀v∈C[0,1]with0<∥v∥≤δ1,
which means
(41)v≠τAμv,∀τ∈[0,1],∀v∈C[0,1]with0<∥v∥≤δ1.
Therefore, by the homotopy invariance of topological degree, we have(42)deg(I-Aμ,Bδ,0)=deg(I,Bδ,0)=1,∀δ∈(0,δ1].Lemma 16.
Forλ>λ1(a0), there exists δ2∈(0,r-) such that
(43)deg(I-Aλ,Bδ,0)=0,∀δ∈(0,δ2].Proof.
First we prove that forλ>λ1(a0), there exists δ2∈(0,r-) such that
(44)v-Aλv≠τφ0,∀τ≥0,∀v∈C[0,1]with0<∥v∥≤δ2,
where φ0 is the positive eigenfunctions of La0 corresponding to λ1(a0).
If this is false, then there existvn∈C[0,1] with ∥vn∥→0 (n→+∞) and τn≥0 such that
(45)vn-Aλvn=τnφ0.
By Lemma10, we have vn(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,tα-2u)|<σtα-2u for u∈(0,δ). Noticing ∥vn∥→0(n→+∞), there exists N>0 such that ∥vn∥<δ for n>N. Consequently, by virtue of (25) and (45), for n>N, we know
(46)vn(t)=Aλvn(t)+τnφ0≥λ∫01‍G*(t,s)a0(s)×(sα-2vn(s)-ξ1(s,sα-2vn(s)))ds≥λ(1-σ)∫01G*(t,s)a0(s)sα-2vn(s)ds=λ(1-σ)La0vn(t).
Let ψ* be the positive eigenfunction of La0* corresponding to λ1(a0). Then
(47)〈vn,ψ*〉≥λ(1-σ)〈La0vn,ψ*〉=λ(1-σ)〈vn,La0*ψ*〉=λ(1-σ)〈vn,ψ*λ1(a0)〉.
This together with〈vn,ψ*〉>0 guarantees that
(48)λ(1-σ)≤λ1(a0),
which is a contradiction. Therefore, (44) holds. By Lemma 3, for each λ>λ1(a0), there exists δ2>0 such that
(49)deg(I-Aλ,Bδ,0)=0,∀δ∈(0,δ2].Theorem 17.
[
λ
1
(
a
0
)
,
λ
1
(
a
0
)
] 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 fixedn∈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)=Aλu, a=λ1(a0)-(1/n), and b=λ1(a0)+(1/n). This together with Lemma 10 guarantees that there exists a closed connected set 𝒞n 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, 𝒞n 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, 𝒞n must be bifurcated from [λ1(a0),λ1(a0)]×{0}, which implies that 𝒞n 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
(50)u≠Aλu,∀λ∈[c,d],∀u∈C[0,1]with∥u∥≥R1.Lemma 19.
Forμ∈(0,λ1(b∞)), there exists R1>R- such that
(51)deg(I-Aμ,BR,0)=1,∀R≥R1.Lemma 20.
Forλ>λ1(b∞), there exists R2>R- such that
(52)deg(I-Aλ,BR,0)=0,∀R≥R2.Theorem 21.
[
λ
1
(
b
∞
)
,
λ
1
(
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 Theorems12 and 13.Proof of Theorem12.
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 (μn,vn)∈𝒞0 such that
(53)μn+∥vn∥⟶+∞asn⟶+∞.
If there exists somen∈ℕ such that μn≥η, the conclusion follows. Suppose, on the contrary, μn<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, μn∈(0,η) for all n∈ℕ. Taking a subsequence and relabeling if necessary, suppose μn→μ* as n→+∞. Then μ*∈[0,η]. This together with (53) guarantees that ∥vn∥→+∞.
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.
If𝒞∞∩(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].
If𝒞∞∩(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 Theorem13.
First we show that there existsε>0 such that
(54)Σ∩([0,η+ε]×∂BR)=∅,
where BR={v∈C[0,1]:∥v∥<R}, Σ⊂ℝ+×C[0,1] is defined by (27).
In fact, from assumption (H2), it follows that there existsε>0 such that
(55)η+εΓ(α)∫01‍[s(1-s)]α-1h(s)ds<1.
If there is a solution (λ,v) of v=Aλv such that 0≤λ≤η+ε and ∥v∥=R, then
(56)0≤v(t)≤∥v∥=Rfort∈[0,1].
By virtue of (25) and Lemma 9, we have
(57)R=∥v∥=maxt∈Jλ∫01‍G*(t,s)f-(s,sα-2v(s))ds≤(η+ε)Rmaxt∈J∫01‍G*(t,s)sα-2h(s)ds≤(η+ε)RΓ(α)∫01‍[s(1-s)]α-1h(s)ds<R,
which is a contradiction. Thus, Σ∩([0,η+ε]×∂BR)=∅.
Next, from Theorem17, there exist unbounded components 𝒞0 of solutions of (23), which meet [λ1(a0),λ1(a0)]×{0}. By (54), we know 𝒞0∩([0,η+ε]×∂BR)=∅. 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, tα-2v1(t) is a positive solution of BVP (3).
Similarly, by Theorem21 and (54), BVP (3) has a positive solution tα-2v2(t) with (η,v2)∈𝒞∞ and ∥v2∥>R. The conclusion follows.Immediately, from the proof of Theorem13, 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.
Corollary22 is different from Theorem 12 though their results are similar.


## 4. An Example

Letρ 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:(58)D0+1.5u(t)+ηf(t,u(t))=0,t∈(0,1),u(0)=u(1)=0,
where
(59)f(t,u)=ρtu[h(u)+sin1u+tsin(tu)],h(u)={32,u∈(0,12];1+u,u∈[12,3);4,u∈[3,+∞).
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 seeξi(t,tα-2u)=o(tα-2u) as u→0 and ζi(t,tα-2u)=o(tα-2u) as u→+∞ both uniformly with respect to t∈[0,1], (i=1,2).
Therefore, (H1) is satisfied.
By the definition ofρ, it is easy to see λ1(a0)=2/5, λ1(b∞)=1/3.
As a result, by Theorem12, BVP (58) has at least one positive solution as η∈[1/3,2/5].



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*Source: 162418-2013-11-14.xml*