# Bifurcation of Positive Solutions for a Class of Boundary Value Problems of Fractional Differential Inclusions

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

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## Abstract

Using Krein-Rutman theorem, topological degree theory, and bifurcation techniques, this paper investigates the existence of positive solutions for a class of boundary value problems of fractional differential inclusions.


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## Body

## 1. Introduction

Fractional differential equations have been of great interest recently. Engineers and scientists have developed new models that involve fractional differential equations. These models have been applied successfully, for example, in mechanics (theory of viscoelasticity and viscoplasticity), (bio)chemistry (modelling of polymers and proteins), electrical engineering (transmission of ultrasound waves), medicine (modelling of human tissue under mechanical loads), and so forth. For details, see [1–7] and references therein. For example, in [5], Qiu and Bai considered the existence of positive solutions to BVP of the nonlinear fractional differential equation
(1)D0+αCu(t)+f(t,u(t))=0,0<t<1,u(0)=u′(1)=u′′(0)=0,
where 2<α≤3, f:(0,1]×[0,+∞)→[0,+∞), and CD0+α is the Caputo’s fractional derivatives. They obtained the existence of at least one positive solution by using Krasnoselskii’s fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone.In [8], Tian and Liu investigated the following singular fractional boundary value problem (BVP, for short) of the form
(2)D0+αCu(t)+λf(t,u(t))=0,0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0,
where 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, under some suitable assumptions, 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**.In this paper, we consider the following boundary value problem of fractional differential inclusions of the form(3)D0+αCu(t)∈-F(t,u(t)),0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0,
where n-1<α≤n, n≥4, D0+αC is the Caputo’s fractional derivatives, and F:J×ℝ+→2ℝ+.As mentioned in [9], the field of differential inclusions is a versatile and general area of mathematics that provides a framework for modelling physical processes that feature discontinuities. Examples of such phenomena include mechanical systems with Coulomb friction modeled as a force proportional to the sign of a velocity and systems whose control laws have discontinuities [10]. In addition, differential inclusions are a useful format for treating differential equations where the right-hand side may be inaccurately known [11]. Differential inclusions are also employed in the dynamic modelling of economic processes and game theory [12], control theory, optimization, partial differential equations, and the study of general evolution processes [13]. The types of the aforementioned applications naturally motivate a deeper theoretical analysis of the subject.Also there are some papers concerned with initial or boundary value problems of fractional differential inclusions (see, for instance, [9, 14–20] and references therein). The method used in these references is fixed point theorem. However, to the best of our knowledge, there is no paper studying such problems using bifurcation ideas. As we know, the bifurcation technique is widely used in solving boundary value problems (see, for instance, [21–24] and references therein). The purpose of present paper is to fill this gap. By using Krein-Rutman theorem, topological degree theory, and bifurcation techniques, the existence of positive solutions of BVP (3) is investigated.The paper is organized as follows. Section2 contains some preliminaries. In Section 3, by using bifurcation techniques, Krein-Rutman theorem, and topological degree theory, bifurcation results from infinity and trivial solution are established. Finally, in Section 4, the main results of the present paper are given and proved.


## 2. Preliminaries

For convenience, we present some necessary definitions and results from fractional calculus theory (see [6]).Definition 1.
The fractional (arbitrary) order integral of the functionh∈L1([a,b]) of order α∈ℝ+ is defined by
(4)Iaαh(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 2.
For a functionh given on the interval [a,b], the αth Caputo fractional-order derivative of h is defined by
(5)(Da+αCh)(t)=1Γ(n-α)∫at‍(t-s)n-α-1h(n)(s)ds.
Here n is the smallest integer greater than or equal α.Lemma 3.
Letα>0, then the differential equation
(6)CD0+αu(t)=0
has solutions u(t)=c0+c1t+c2t2+⋯+cn-1tn-1, for some ci∈ℝ, i=0,1,2,…,n-1, where n is the smallest integer greater than or equal to α.Lemma 4.
Assume thatu∈C(0,1)∩L1[0,1] with a derivative of order n that belongs to C(0,1)∩L1[0,1]. Then
(7)I0+αCD0+αu(t)=u(t)+c0+c1t+c2t2+⋯+cn-1tn-1.
for some ci∈ℝ, i=0,1,2,…,n-1, where n is the smallest integer greater than or equal α.Lemma 5.
The relation(8)I0+αI0+βφ=I0+α+βφ
is valid in the following case:
(9)Reβ>0,Re(α+β)>0,φ∈L1[a,b].For more detailed results of fractional calculus, we refer the reader to [6]. In addition, we need the following preliminaries on multivalued operators.Let(X,∥·∥) be a Banach space. Then a multivalued map Θ:X→2X is convex (closed) valued if Θ(x) is convex (closed) for all x∈X. Θ is bounded if Θ(B)=⋃x∈B‍Θ(x) is bounded in X for any bounded set B of X.Θ
:
D
→
2
X is said to be lower semicontinuous, l.s.c. for short, if Θ-1(V) is open in D whenever V⊂X is open.LetΘ:D→2X be a multivalued map and θ:D→X a single-valued function; if for allx∈D, θ(x)∈Θ(x), then θ is called a selection function of Θ. If in addition θ is continuous, then θ is called a continuous selection.The following lemmas are crucial in the proof of our main result.Lemma 6 (25, Lemma 2.1, page 14).
LetD≠∅ be a subset of a Banach space X, and Θ:D→2X a l.s.c. with closed convex values. Then, given (w0,x0)∈graph(Θ), Θ has a continuous selection θ such that θ(w0)=x0.For more details on multivalued maps, see the books of Deimling [25].Finally in this section, we list the following results on topological degree of completely operators.Lemma 7 (Schmitt and Thompson [26]).
LetV be a real reflexive Banach space. Let G:ℝ×V to V be completely continuous such that G(λ,0)=0,forallλ∈ℝ. Let a,b∈ℝ(a<b) be such that u=0 is an isolated solution of the equation
(10)u-G(λ,u)=0,u∈V,
for λ=a and λ=b, where (a,0), (b,0) are not bifurcation points of (10). Furthermore, assume that
(11)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
(12)𝒯={(λ,u):(λ,u)isasolutionof(2.1)withu≠0}¯∪([a,b]×0).
Then there exists a connected component𝒞 of 𝒯 containing [a,b]×0 in ℝ×V, and either(i)
𝒞is unbounded in ℝ×V or(ii)
𝒞∩[(ℝ∖[a,b])×0]≠∅.Lemma 8 (Schmitt [27]).
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 (10) is, a priori, bounded in V for λ=a and λ=b; that is, there exists an R>0 such that
(13)G(a,u)≠u≠G(b,u)
for all u with ∥u∥≥R. Furthermore, assume that
(14)deg(I-G(a,·),BR(0),0)≠deg(I-G(b,·),BR(0),0), for  sufficiently large R>0. Then there exists a closed connected set 𝒞 of solutions of (10) 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 9 (Guo [28]).
LetΩ be a bounded open set of infinite-dimensional real Banach space E, and let A:Ω-→E be completely continuous. Suppose that(i)
infx∈∂Ω∥Ax∥>0;(ii)
Ax=μx, x∈∂Ω⇒μ∉(0,1].
Then(15)deg(I-A,Ω,θ)=0.


## 3. Bifurcation Results

### 3.1. Assumptions and Conversion of BVP (3)

Suppose that the following two assumptions hold throughout the paper.(H1) LetF:J×ℝ+→2ℝ+ be a nonempty, closed and convex multivalued map such that F is l.s.c., where J=[0,1].(H2) There exist functionsa0,a0,b∞,b∞∈C(J,ℝ+) with a0(t),a0(t),b∞(t),b∞(t)≢0 in any subinterval of [0,1] such that
(16)F(t,u)⊂[a0(t)u-ξ1(t,u),a0(t)u+ξ2(t,u)]∩[b∞(t)u-ζ1(t,u),b∞(t)u+ζ2(t,u)],
for all(t,u)∈J×ℝ+, where ξi,ζi∈C(J×ℝ+) with ξi(t,u)=o(u) as u→0 uniformly with respect to t∈[0,1], (i=1,2), and ζi(t,u)=o(u) as u→+∞ uniformly with respect to t∈[0,1], (i=1,2).The basic space used in this paper isC[0,1]. Obviously, C[0,1] is a Banach space with norm ∥u∥=maxt∈J|u(t)| (for allu∈C[0,1]). Let
(17)Q:={u∈C[J,ℝ+]:u(t)≥t2u(s),∀t,s∈J}.
It is easy to see that Q is a cone of E. Moreover, from (17), we have for all u∈Q,
(18)u(t)≥t2∥u∥,∀t∈J.We first consider the following linear boundary problem of fractional differential equation:(19)D0+αCu(t)+g(t)=0,0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0,
where g∈C[0,1].Lemma 10 (Tian and Liu [8]).
Giveng∈C[0,1], the unique solution of (19) is
(20)u(t)=∫01‍G(t,s)g(s)ds,
where
(21)G(t,s)=1Γ(α){(α-1)(α-2)2t2(1-s)α-3-(t-s)α-1,s≤t;(α-1)(α-2)2t2(1-s)α-3,t≤s.Lemma 11 (Tian and Liu [8]).
The functionG(t,s) defined by (21) has the following properties:
(22)(i)G(t,s)>0,∀t,s∈[0,1];(ii)G(t,s)≤H(s)≤(1-s)α-32Γ(α-2),
where
(23)H(s)=1Γ(α){(α-1)(α-2)2s2(1-s)α-3-(1-s)α-1,s≤t,(α-1)(α-2)2s2(1-s)α-3,t≤s;(iii)G(t,s)≥t2G(τ,s),∀t,s,τ∈[0,1].For the sake of using bifurcation technique to investigate BVP (3), we study the following fractional boundary value problem with parameters:
(24)D0+αCu(t)∈-λF(t,u(t)),0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0.A function(λ,u) is said to be a solution of BVP (24) if (λ,u) satisfies (24). In addition, if λ>0, u(t)>0 for t∈(0,1), then (λ,u) is said to be a positive solution of BVP (24). Obviously, if λ>0, u∈Q∖{θ} is a solution of BVP (24), then by (18) we know that (λ,u) is a positive solution of BVP (24), where θ denotes the zero element of Banach space E.Fora∈C(J,ℝ+) with a(t)≢0 in any subinterval of J, define the linear operator La:C(J)→C(J) by
(25)Lau(t)=∫01‍G(t,s)a(s)u(s)ds,
where G(t,s) is defined by (21).From Lemmas10, 11, and the well-known Krein-Rutman Theorem, one can obtain the following lemma.Lemma 12.
The operator defined by (25) 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, we have ϕ(t)=λ1(a)Laϕ(t).Notice that the operatorLa can be regarded as La:L2[0,1]→L2[0,1]. This together with Lemma 12 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
(26)φ*(t)=λ1(a)La*φ*(t),∀t∈J.Note that condition (H1) implies thatF(t,u) is lower semicontinuous. Then, from Lemma 6, there exists a continuous function f:J×ℝ+→ℝ+ such that f(t,u)∈F(t,u) for all (t,u)∈J×ℝ+. Therefore, to solve BVP (24), we consider the problem
(27)D0+αCu(t)+λf(t,u(t))=0,0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0.Define(28)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. From Lemma 10, the solution of
(29)D0+αCu(t)+λf-(t,u(t))=0,0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0
is equivalent to the fixed point of operator
(30)Aλu(t)=λ∫01‍G(t,s)f-(s,u(s))ds,∀u∈C[0,1].LetΣ⊂ℝ+×C[0,1] be the closure of the set of positive solutions of BVP (27). From Lemma 11 and the definitions of f- and the cone Q, it is easy to see Σ⊂Q and Aλ:C[0,1]→Q. Moreover, we have the following conclusion.Lemma 13.
Forλ>0, (λ,u) is a positive solution of BVP (27) if and only if (λ,u) is a nontrivial solution of BVP (29); that is, u is a nontrivial fixed point of operator Aλ in Q. Therefore, the closure of the set of nontrivial solutions (λ,u) of BVP (29) in ℝ+×Q is exactly Σ.


### 3.2. Bifurcation from Infinity and Trivial Solution

Lemma 14.
Let[c,d]⊂ℝ+ be a compact interval with [λ1(b∞),λ1(b∞)]∩[c,d]=∅. Then there exists R1>0 such that
(31)u≠Aλu,∀λ∈[c,d],∀u∈C[0,1]with∥u∥≥R1.Proof.
Suppose, on the contrary, that there exist{(μn,un)}⊂[c,d]×C[0,1] with ∥un∥→∞(n→+∞) such that un=Aμnun. Without loss of generality, assume μn→μ∈[c,d]. Notice that un∈Q. By Lemma 13, (17), and (18), we have un(t)>0 in (0,1]. Set vn=un/∥un∥. Then vn=Aμnun/∥un∥. From the continuity of f-(t,u), it is easy to see that {vn} is relatively compact in C[0,1]. Taking a subsequence and relabeling if necessary, suppose vn→v in C[0,1]. Then ∥v∥=1 and v∈Q.
On the other hand, from (H2) we know(32)f(t,u)∈[b∞(t)u-ζ1(t,u),b∞(t)u+ζ2(t,u)],∀(t,u)∈J×ℝ+.
Therefore, by virtue of (30), we know
(33)vn(t)≤μn∫01‍G(t,s)(b∞(s)vn(s)+ζ2(s,un(s))∥un∥)ds,(34)vn(t)≥μn∫01‍G(t,s)(b∞(s)vn(s)-ζ1(s,un(s))∥un∥)ds.
Let ψ* and ψ* be the positive eigenfunctions of Lb∞*, Lb∞* corresponding to λ1(b∞) and λ1(b∞), respectively. Then from (33), it follows that
(35)〈vn,ψ*〉≤μn〈Lb∞vn,ψ*〉+μn∫01‍ψ*(t)∫01‍G(t,s)ζ2(s,un(s))∥un∥dsdt.
Letting n→+∞ and using condition (H2), we have
(36)〈v,ψ*〉≤μ〈Lb∞v,ψ*〉=μ〈v,Lb∞*ψ*〉=μ〈v,ψ*λ1(b∞)〉,
which implies μ≥λ1(b∞). Similarly, one can deduce from (34) that μ≤λ1(b∞).
To sum up,λ1(b∞)≤μ≤λ1(b∞), which contradicts with μ∈[c,d]. The conclusion of this lemma follows.Lemma 15.
Forμ∈(0,λ1(b∞)), there exists R1>0 such that
(37)deg(I-Aμ,BR,0)=1,∀R≥R1.Proof.
Notice that[0,μ]∩[λ1(b∞),λ1(b∞)]=∅. From Lemma 14, there exists R1>0 such that
(38)u≠Aλu,∀λ∈[0,μ],∀u∈C[0,1]with∥u∥≥R1,
which means
(39)u≠τAμu,∀τ∈[0,1],∀u∈C[0,1]with∥u∥≥R1.
Therefore, by the homotopy invariance of topological degree, we have(40)deg(I-Aμ,BR,0)=deg(I,BR,0)=1,∀R≥R1.Lemma 16.
Forλ>λ1(b∞), there exists R2>0 such that
(41)deg(I-Aλ,BR,0)=0,∀R≥R2.Proof.
We first prove that forλ>λ1(b∞), there exists R2>0 such that
(42)Aλu≠μu,∀μ∈(0,1],∀u∈C[0,1]with∥u∥≥R2.
Suppose, on the contrary, that there exist{(μn,un)}⊂(0,1]×C[0,1] with ∥un∥→∞(n→+∞) such that Aλun=μnun.
By Lemma13, un(t)>0 in (0,1]. Set vn=un/∥un∥; that is, μnvn=Aλun/∥un∥. Without loss of generality, assume μn→μ-∈[0,1]. First we show μ-≠0. From (32) and the continuity of f-(t,u), it is easy to see that Aλun/∥un∥ is relatively compact in C[0,1]. Suppose (Aλun/∥un∥)→y. Notice that vn∈Q and ∥vn∥=1. Therefore, vn(t)≥t2 for t∈(0,1]. Consequently,
(43)μnvn(t)=Aλun(t)∥un∥≥λ∫01‍G(t,s)(b∞(s)vn(s)-ζ1(s,un(s))∥un∥)ds≥λt2maxτ∈J∫01‍s2G(τ,s)b∞(s)ds-λ∫01‍G(t,s)ζ1(s,un(s))∥un∥ds.
From (H2) and Lemma 11, it is easy to see maxτ∈J∫01‍s2G(τ,s)b∞(s)ds>0. If μ=0, letting n→+∞ in the above inequality, we can obtain a contradiction. So μ∈(0,1] and it is reasonable to suppose vn→v (relabeling if necessary) in C[0,1]. By virtue of (32), we know
(44)〈vn,ψ*〉≥〈μnvn,ψ*〉=1∥un∥〈Aλun,ψ*〉≥λ〈Lb∞vn,ψ*〉-λ∫01‍ψ*(t)∫01‍G(t,s)ζ1(s,un(s))∥un∥dsdt.
Lettingn→+∞ and using condition (H2), we obtain that
(45)〈v,ψ*〉≥λ〈Lb∞v,ψ*〉=λ〈v,Lb∞*ψ*〉=λ〈v,ψ*λ1(b∞)〉,
which implies λ≤λ1(b∞). This is a contradiction. Therefore, (42) holds. By Lemma 9, for each λ>λ1(b∞), there exists R2>0 such that
(46)deg(I-Aλ,BR,0)=0,∀R≥R2.
The conclusion of this lemma follows.Theorem 17.
[
λ
1
(
b
∞
)
,
λ
1
(
b
∞
)
] is a bifurcation interval of positive solutions from infinity for BVP (27), 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 (27) which meets [λ1(b∞),λ1(b∞)]×∞ and is unbounded in λ direction.Proof.
From Lemma13, we need only to prove that the conclusion holds for (29).
For fixedn∈ℕ with λ1(b∞)-1/n>0, by Lemmas 15, 16, and their proof, there exists R>0 such that all of the conditions of Lemma 8 are satisfied with G(λ,u)=Aλu, a=λ1(b∞)-1/n, and b=λ1(b∞)+1/n. So, there exists a closed connected set 𝒞n of solutions of (29), which is unbounded in [λ1(b∞)-1/n,λ1(b∞)+1/n]×C[0,1]. From Lemma 14, the case (ii) of Lemma 8 cannot occur. Thus, 𝒞n bifurcates from infinity in [λ1(b∞)-1/n,λ1(b∞)+1/n]×C[0,1] and is unbounded in λ direction. In addition, for any closed interval [c,d]⊂[λ1(b∞)-1/n,λ1(b∞)+1/n]∖[λ1(b∞),λ1(b∞)], by Lemma 14, the set {u∈C[0,1]:(λ,u)∈Σ,λ∈[c,d]} is bounded in C[0,1]. Therefore, 𝒞n must be bifurcated from infinity in [λ1(b∞),λ1(b∞)]×C[0,1], which implies that 𝒞n can be regarded as 𝒞∞. Consequently, 𝒞∞ is unbounded in λ direction.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(a0),λ1(a0)]∩[c,d]=∅. Then there exists δ1>0 such that
(47)u≠Aλu,∀λ∈[c,d],∀u∈C[0,1]with0<∥u∥≤δ1.Lemma 19.
Forμ∈(0,λ1(a0)), there exists δ1>0 such that
(48)deg(I-Aμ,Bδ,0)=1,∀δ∈(0,δ1].Lemma 20.
Forλ>λ1(a0), there exists δ2>0 such that
(49)deg(I-Aλ,Bδ,0)=0,∀δ∈(0,δ2].Finally, using Lemmas18–20, Lemma 7, and the similar method used in the proof of Theorem 17, the following conclusion can be proved.Theorem 21.
[
λ
1
(
a
0
)
,
λ
1
(
a
0
)
] is a bifurcation interval of positive solutions from the trivial solution for BVP (27); that is, there exists an unbounded component 𝒞0 of positive solutions of BVP (27), 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)].


## 3.1. Assumptions and Conversion of BVP (3)

Suppose that the following two assumptions hold throughout the paper.(H1) LetF:J×ℝ+→2ℝ+ be a nonempty, closed and convex multivalued map such that F is l.s.c., where J=[0,1].(H2) There exist functionsa0,a0,b∞,b∞∈C(J,ℝ+) with a0(t),a0(t),b∞(t),b∞(t)≢0 in any subinterval of [0,1] such that
(16)F(t,u)⊂[a0(t)u-ξ1(t,u),a0(t)u+ξ2(t,u)]∩[b∞(t)u-ζ1(t,u),b∞(t)u+ζ2(t,u)],
for all(t,u)∈J×ℝ+, where ξi,ζi∈C(J×ℝ+) with ξi(t,u)=o(u) as u→0 uniformly with respect to t∈[0,1], (i=1,2), and ζi(t,u)=o(u) as u→+∞ uniformly with respect to t∈[0,1], (i=1,2).The basic space used in this paper isC[0,1]. Obviously, C[0,1] is a Banach space with norm ∥u∥=maxt∈J|u(t)| (for allu∈C[0,1]). Let
(17)Q:={u∈C[J,ℝ+]:u(t)≥t2u(s),∀t,s∈J}.
It is easy to see that Q is a cone of E. Moreover, from (17), we have for all u∈Q,
(18)u(t)≥t2∥u∥,∀t∈J.We first consider the following linear boundary problem of fractional differential equation:(19)D0+αCu(t)+g(t)=0,0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0,
where g∈C[0,1].Lemma 10 (Tian and Liu [8]).
Giveng∈C[0,1], the unique solution of (19) is
(20)u(t)=∫01‍G(t,s)g(s)ds,
where
(21)G(t,s)=1Γ(α){(α-1)(α-2)2t2(1-s)α-3-(t-s)α-1,s≤t;(α-1)(α-2)2t2(1-s)α-3,t≤s.Lemma 11 (Tian and Liu [8]).
The functionG(t,s) defined by (21) has the following properties:
(22)(i)G(t,s)>0,∀t,s∈[0,1];(ii)G(t,s)≤H(s)≤(1-s)α-32Γ(α-2),
where
(23)H(s)=1Γ(α){(α-1)(α-2)2s2(1-s)α-3-(1-s)α-1,s≤t,(α-1)(α-2)2s2(1-s)α-3,t≤s;(iii)G(t,s)≥t2G(τ,s),∀t,s,τ∈[0,1].For the sake of using bifurcation technique to investigate BVP (3), we study the following fractional boundary value problem with parameters:
(24)D0+αCu(t)∈-λF(t,u(t)),0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0.A function(λ,u) is said to be a solution of BVP (24) if (λ,u) satisfies (24). In addition, if λ>0, u(t)>0 for t∈(0,1), then (λ,u) is said to be a positive solution of BVP (24). Obviously, if λ>0, u∈Q∖{θ} is a solution of BVP (24), then by (18) we know that (λ,u) is a positive solution of BVP (24), where θ denotes the zero element of Banach space E.Fora∈C(J,ℝ+) with a(t)≢0 in any subinterval of J, define the linear operator La:C(J)→C(J) by
(25)Lau(t)=∫01‍G(t,s)a(s)u(s)ds,
where G(t,s) is defined by (21).From Lemmas10, 11, and the well-known Krein-Rutman Theorem, one can obtain the following lemma.Lemma 12.
The operator defined by (25) 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, we have ϕ(t)=λ1(a)Laϕ(t).Notice that the operatorLa can be regarded as La:L2[0,1]→L2[0,1]. This together with Lemma 12 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
(26)φ*(t)=λ1(a)La*φ*(t),∀t∈J.Note that condition (H1) implies thatF(t,u) is lower semicontinuous. Then, from Lemma 6, there exists a continuous function f:J×ℝ+→ℝ+ such that f(t,u)∈F(t,u) for all (t,u)∈J×ℝ+. Therefore, to solve BVP (24), we consider the problem
(27)D0+αCu(t)+λf(t,u(t))=0,0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0.Define(28)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. From Lemma 10, the solution of
(29)D0+αCu(t)+λf-(t,u(t))=0,0<t<1,u(j)(0)=0,0≤j≤n-1,j≠2,u′′(1)=0
is equivalent to the fixed point of operator
(30)Aλu(t)=λ∫01‍G(t,s)f-(s,u(s))ds,∀u∈C[0,1].LetΣ⊂ℝ+×C[0,1] be the closure of the set of positive solutions of BVP (27). From Lemma 11 and the definitions of f- and the cone Q, it is easy to see Σ⊂Q and Aλ:C[0,1]→Q. Moreover, we have the following conclusion.Lemma 13.
Forλ>0, (λ,u) is a positive solution of BVP (27) if and only if (λ,u) is a nontrivial solution of BVP (29); that is, u is a nontrivial fixed point of operator Aλ in Q. Therefore, the closure of the set of nontrivial solutions (λ,u) of BVP (29) in ℝ+×Q is exactly Σ.


## 3.2. Bifurcation from Infinity and Trivial Solution

Lemma 14.
Let[c,d]⊂ℝ+ be a compact interval with [λ1(b∞),λ1(b∞)]∩[c,d]=∅. Then there exists R1>0 such that
(31)u≠Aλu,∀λ∈[c,d],∀u∈C[0,1]with∥u∥≥R1.Proof.
Suppose, on the contrary, that there exist{(μn,un)}⊂[c,d]×C[0,1] with ∥un∥→∞(n→+∞) such that un=Aμnun. Without loss of generality, assume μn→μ∈[c,d]. Notice that un∈Q. By Lemma 13, (17), and (18), we have un(t)>0 in (0,1]. Set vn=un/∥un∥. Then vn=Aμnun/∥un∥. From the continuity of f-(t,u), it is easy to see that {vn} is relatively compact in C[0,1]. Taking a subsequence and relabeling if necessary, suppose vn→v in C[0,1]. Then ∥v∥=1 and v∈Q.
On the other hand, from (H2) we know(32)f(t,u)∈[b∞(t)u-ζ1(t,u),b∞(t)u+ζ2(t,u)],∀(t,u)∈J×ℝ+.
Therefore, by virtue of (30), we know
(33)vn(t)≤μn∫01‍G(t,s)(b∞(s)vn(s)+ζ2(s,un(s))∥un∥)ds,(34)vn(t)≥μn∫01‍G(t,s)(b∞(s)vn(s)-ζ1(s,un(s))∥un∥)ds.
Let ψ* and ψ* be the positive eigenfunctions of Lb∞*, Lb∞* corresponding to λ1(b∞) and λ1(b∞), respectively. Then from (33), it follows that
(35)〈vn,ψ*〉≤μn〈Lb∞vn,ψ*〉+μn∫01‍ψ*(t)∫01‍G(t,s)ζ2(s,un(s))∥un∥dsdt.
Letting n→+∞ and using condition (H2), we have
(36)〈v,ψ*〉≤μ〈Lb∞v,ψ*〉=μ〈v,Lb∞*ψ*〉=μ〈v,ψ*λ1(b∞)〉,
which implies μ≥λ1(b∞). Similarly, one can deduce from (34) that μ≤λ1(b∞).
To sum up,λ1(b∞)≤μ≤λ1(b∞), which contradicts with μ∈[c,d]. The conclusion of this lemma follows.Lemma 15.
Forμ∈(0,λ1(b∞)), there exists R1>0 such that
(37)deg(I-Aμ,BR,0)=1,∀R≥R1.Proof.
Notice that[0,μ]∩[λ1(b∞),λ1(b∞)]=∅. From Lemma 14, there exists R1>0 such that
(38)u≠Aλu,∀λ∈[0,μ],∀u∈C[0,1]with∥u∥≥R1,
which means
(39)u≠τAμu,∀τ∈[0,1],∀u∈C[0,1]with∥u∥≥R1.
Therefore, by the homotopy invariance of topological degree, we have(40)deg(I-Aμ,BR,0)=deg(I,BR,0)=1,∀R≥R1.Lemma 16.
Forλ>λ1(b∞), there exists R2>0 such that
(41)deg(I-Aλ,BR,0)=0,∀R≥R2.Proof.
We first prove that forλ>λ1(b∞), there exists R2>0 such that
(42)Aλu≠μu,∀μ∈(0,1],∀u∈C[0,1]with∥u∥≥R2.
Suppose, on the contrary, that there exist{(μn,un)}⊂(0,1]×C[0,1] with ∥un∥→∞(n→+∞) such that Aλun=μnun.
By Lemma13, un(t)>0 in (0,1]. Set vn=un/∥un∥; that is, μnvn=Aλun/∥un∥. Without loss of generality, assume μn→μ-∈[0,1]. First we show μ-≠0. From (32) and the continuity of f-(t,u), it is easy to see that Aλun/∥un∥ is relatively compact in C[0,1]. Suppose (Aλun/∥un∥)→y. Notice that vn∈Q and ∥vn∥=1. Therefore, vn(t)≥t2 for t∈(0,1]. Consequently,
(43)μnvn(t)=Aλun(t)∥un∥≥λ∫01‍G(t,s)(b∞(s)vn(s)-ζ1(s,un(s))∥un∥)ds≥λt2maxτ∈J∫01‍s2G(τ,s)b∞(s)ds-λ∫01‍G(t,s)ζ1(s,un(s))∥un∥ds.
From (H2) and Lemma 11, it is easy to see maxτ∈J∫01‍s2G(τ,s)b∞(s)ds>0. If μ=0, letting n→+∞ in the above inequality, we can obtain a contradiction. So μ∈(0,1] and it is reasonable to suppose vn→v (relabeling if necessary) in C[0,1]. By virtue of (32), we know
(44)〈vn,ψ*〉≥〈μnvn,ψ*〉=1∥un∥〈Aλun,ψ*〉≥λ〈Lb∞vn,ψ*〉-λ∫01‍ψ*(t)∫01‍G(t,s)ζ1(s,un(s))∥un∥dsdt.
Lettingn→+∞ and using condition (H2), we obtain that
(45)〈v,ψ*〉≥λ〈Lb∞v,ψ*〉=λ〈v,Lb∞*ψ*〉=λ〈v,ψ*λ1(b∞)〉,
which implies λ≤λ1(b∞). This is a contradiction. Therefore, (42) holds. By Lemma 9, for each λ>λ1(b∞), there exists R2>0 such that
(46)deg(I-Aλ,BR,0)=0,∀R≥R2.
The conclusion of this lemma follows.Theorem 17.
[
λ
1
(
b
∞
)
,
λ
1
(
b
∞
)
] is a bifurcation interval of positive solutions from infinity for BVP (27), 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 (27) which meets [λ1(b∞),λ1(b∞)]×∞ and is unbounded in λ direction.Proof.
From Lemma13, we need only to prove that the conclusion holds for (29).
For fixedn∈ℕ with λ1(b∞)-1/n>0, by Lemmas 15, 16, and their proof, there exists R>0 such that all of the conditions of Lemma 8 are satisfied with G(λ,u)=Aλu, a=λ1(b∞)-1/n, and b=λ1(b∞)+1/n. So, there exists a closed connected set 𝒞n of solutions of (29), which is unbounded in [λ1(b∞)-1/n,λ1(b∞)+1/n]×C[0,1]. From Lemma 14, the case (ii) of Lemma 8 cannot occur. Thus, 𝒞n bifurcates from infinity in [λ1(b∞)-1/n,λ1(b∞)+1/n]×C[0,1] and is unbounded in λ direction. In addition, for any closed interval [c,d]⊂[λ1(b∞)-1/n,λ1(b∞)+1/n]∖[λ1(b∞),λ1(b∞)], by Lemma 14, the set {u∈C[0,1]:(λ,u)∈Σ,λ∈[c,d]} is bounded in C[0,1]. Therefore, 𝒞n must be bifurcated from infinity in [λ1(b∞),λ1(b∞)]×C[0,1], which implies that 𝒞n can be regarded as 𝒞∞. Consequently, 𝒞∞ is unbounded in λ direction.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(a0),λ1(a0)]∩[c,d]=∅. Then there exists δ1>0 such that
(47)u≠Aλu,∀λ∈[c,d],∀u∈C[0,1]with0<∥u∥≤δ1.Lemma 19.
Forμ∈(0,λ1(a0)), there exists δ1>0 such that
(48)deg(I-Aμ,Bδ,0)=1,∀δ∈(0,δ1].Lemma 20.
Forλ>λ1(a0), there exists δ2>0 such that
(49)deg(I-Aλ,Bδ,0)=0,∀δ∈(0,δ2].Finally, using Lemmas18–20, Lemma 7, and the similar method used in the proof of Theorem 17, the following conclusion can be proved.Theorem 21.
[
λ
1
(
a
0
)
,
λ
1
(
a
0
)
] is a bifurcation interval of positive solutions from the trivial solution for BVP (27); that is, there exists an unbounded component 𝒞0 of positive solutions of BVP (27), 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)].


## 4. Main Results

The main results of this paper are the following two conclusions.Theorem 22.
Suppose that (H1) and (H2) hold. In addition, suppose either(i)
λ1(b∞)<1<λ1(a0) or(ii)
λ1(a0)<1<λ1(b∞).
Then BVP (3) has at least one positive solution.Proof.
We need only to prove that there is a component ofΣ that crosses the hyperplane {1}×C(J), where Σ⊂ℝ+×C[0,1] is the closure of the set of positive solutions of BVP (27). Notice that (0,0) is the only solution of (27) with λ=0. By Lemmas 14 and 18, for any component 𝒞 of  Σ, we have 𝒞∩({0}×C(J))=∅.
Case (i). Consider λ1(b∞)<1<λ1(a0).
From Theorem17, there exists an unbounded component 𝒞∞ of solutions of (27), which meets [λ1(b∞),λ1(b∞)]×∞ and is unbounded in λ direction.
If𝒞∞∩(ℝ+×{0})=∅, by 𝒞∞∩({0}×C(J))=∅ and Theorem 17, we know that 𝒞∞ must cross the hyperplane {1}×C(J).
If𝒞∞∩(ℝ+×{0})≠∅, by Theorem 21, 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∞)<1<λ1(a0) guarantees that 𝒞∞ crosses the hyperplane {1}×C(J).
Case (ii). Consider λ1(a0)<1<λ1(b∞).
From Theorem21, there exists an unbounded component 𝒞0 of positive solutions of BVP (27), 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)].
We show that𝒞0 must cross the hyperplane {1}×C(J). Suppose, on the contrary, 𝒞0∩{1}×C(J)=∅. From λ1(a0)<1, we know 𝒞0⊂[0,1]×C(J). Notice that 𝒞0 is unbounded. Then 𝒞0 must joint [0,1]×{∞}. By Theorem 17, it is a contradiction with λ1(b∞)>1. Thus the result follows.Theorem 23.
Suppose that (H1), (H2), and the following assumption holds.(H3)
There existR>0 and h∈L[0,1] such that for t∈J,
(50)supt2R≤u≤RF(t,u)≤h(t),maxt∈J∫01‍G(t,s)h(s)ds<R.
In addition, suppose(51)λ1(a0)<1,λ1(b∞)<1.
Then BVP (3) has at least two positive solutions.Proof.
From Theorems17 and 21, there exist two unbounded components 𝒞0 and 𝒞∞ of solutions of (27), which meet [λ1(a0),λ1(a0)]×{0} and [λ1(b∞),λ1(b∞)]×∞, respectively. It is sufficient to show that 𝒞0 and 𝒞∞ are disjoint in [0,1]×C(J) and both cross the hyperplane {1}×C(J).
For this sake, from assumption (H3), there existsε>0 such that
(52)(1+ε)maxt∈J∫01‍G(t,s)h(s)ds<R.
Now we show Σ∩([0,1+ε]×∂BR)=∅, where BR={u∈C(J):∥u∥<R}. Suppose that, on the contrary, (λ,u) is a solution of (27) such that 0≤λ≤1+ε and ∥u∥=R. Then by Lemma 13, we know u∈Q. Therefore, u(t)∈[t2R,R] for t∈J. From (H3), (30), and Lemma 13, it follows that
(53)R=∥u∥=maxt∈Jλ∫01‍G(t,s)f-(s,u(s))ds≤(1+ε)maxt∈J∫01‍G(t,s)h(s)ds<R,
which is a contradiction. Thus, Σ∩([0,1+ε]×∂BR)=∅, which implies
(54)𝒞0∩([0,1+ε]×∂BR)=∅,𝒞∞∩([0,1+ε]×∂BR)=∅.
Immediately,𝒞0 and 𝒞∞ are disjoint in [0,1]×C(J).
Notice that𝒞0 and 𝒞∞ are both unbounded. Moreover, 𝒞0∩({0}×C(J))=∅, 𝒞∞∩({0}×C(J))=∅, and 𝒞∞ is unbounded in λ direction. So 𝒞0 and 𝒞∞ both cross the hyperplane {1}×C(J). This means that there exist (1,u1)∈𝒞0 and (1,u2)∈𝒞∞ with ∥u1∥<R and ∥u2∥>R.
Consequently, BVP (3) has at least two positive solutions.


## 5. An Example

Letρ be the unique characteristic value of L1 corresponding to positive eigenfunctions with a(t)≡1 in (25). From Lemma 12 it follows that ρ exists.Example 24.
Consider the following boundary value problem of fractional differential inclusions(55)D0+3.5Cu(t)∈-F(t,u(t)),0<t<1,u(j)(0)=0,0≤j≤3,j≠2,u′′(1)=0,
where
(56)F(t,u)=[ρ4u-ξ(t,u),ρ2u+ξ2(t,u)]∩[2ρu-ζ1(t,u),3ρu+ξ(t,u)],ξ(t,u)={ρ4t2u3,t∈J,u∈[0,1],ρ4t2u,t∈J,u∈[1,+∞),ζ1(t,u)={2ρu,t∈J,u∈[0,1],2ρu,t∈J,u∈[1,+∞),ξ2(t,u)=5ρ2u+ξ(t,u).
Then BVP (55) has at least one positive solution.Proof.
BVP (55) can be regarded as the form (3). From (56), one can see that (H1) and (H2) are satisfied with a0(t)=ρ/4, b∞(t)=2ρ, a0(t)=ρ/2, b∞(t)=3ρ, and ξ1(t,u)=ζ2(t,u)=ξ(t,u).
By the definition ofρ, it is easy to see λ1(b∞)=1/2<1<2=λ1(a0).
Therefore, by Theorem22, BVP (55) has at least one positive solution.



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*Source: 942831-2013-03-28.xml*