AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 942831 10.1155/2013/942831 942831 Research Article Bifurcation of Positive Solutions for a Class of Boundary Value Problems of Fractional Differential Inclusions 0000-0002-4911-9016 Liu Yansheng Yu Huimin Ahmad Bashir 1 Department of Mathematics Shandong Normal University Jinan 250014 China sdnu.edu.cn 2013 28 3 2013 2013 11 01 2013 27 02 2013 04 03 2013 2013 Copyright © 2013 Yansheng Liu and Huimin Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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  and references therein. For example, in , 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<α3, 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 , 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)=0,0jn-1,j2,u(1)=0, where n-1<αn, n4, 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)=0,0jn-1,j2,u(1)=0, where n-1<αn, n4, D0+αC is the Caputo’s fractional derivatives, and F:J×+2+.

As mentioned in , 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 . In addition, differential inclusions are a useful format for treating differential equations where the right-hand side may be inaccurately known . Differential inclusions are also employed in the dynamic modelling of economic processes and game theory , control theory, optimization, partial differential equations, and the study of general evolution processes . 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, 1420] 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,  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. Section 2 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 ).

Definition 1.

The fractional (arbitrary) order integral of the function hL1([a,b]) of order α+ 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 t0 and φαδ(t) as α0, where δ is the delta function.

Definition 2.

For a function h 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.

Let α>0, then the differential equation (6)  CD0+αu(t)=0 has solutions u(t)=c0+c1t+c2t2++cn-1tn-1, for some ci, i=0,1,2,,n-1, where n is the smallest integer greater than or equal to α.

Lemma 4.

Assume that uC(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, 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,φL1[a,b].

For more detailed results of fractional calculus, we refer the reader to . In addition, we need the following preliminaries on multivalued operators.

Let (X,·) be a Banach space. Then a multivalued map Θ:  X2X is convex (closed) valued if Θ(x) is convex (closed) for all xX. Θ is bounded if Θ(B)=xBΘ(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 VX is open.

Let Θ:  D2X be a multivalued map and θ:  DX a single-valued function; if for allxD, 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 (<xref ref-type="bibr" rid="B7">25</xref>, Lemma 2.1, page 14).

Let D be a subset of a Banach space X, and Θ:  D2X 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 .

Finally in this section, we list the following results on topological degree of completely operators.

Lemma 7 (Schmitt and Thompson [<xref ref-type="bibr" rid="B25">26</xref>]).

Let V be a real reflexive Banach space. Let G:  ×V to V be completely continuous such that G(λ,0)=0,forallλ. Let a,b(a<b) be such that u=0 is an isolated solution of the equation (10)u-G(λ,u)=0,uV, 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)is  a  solution  of  (2.1)with  u0}¯([a,b]×0).

Then there exists a connected component 𝒞 of 𝒯 containing [a,b]×0 in ×V, and either

𝒞  is unbounded in ×V or

𝒞[([a,b])×0].

Lemma 8 (Schmitt [<xref ref-type="bibr" rid="B24">27</xref>]).

Let V be a real reflexive Banach space. Let G:×V to V be completely continuous, and let a,b(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)uG(b,u) for all u with uR. 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

𝒞 is unbounded in λ direction or

there exists an interval [c,d] such that (a,b)(c,d)= and 𝒞 bifurcates from infinity in [c,d]×V.

Lemma 9 (Guo [<xref ref-type="bibr" rid="B10">28</xref>]).

Let Ω be a bounded open set of infinite-dimensional real Banach space E, and let A:Ω-E be completely continuous. Suppose that

infxΩAx>0;

Ax=μx, xΩμ(0,1].

Then (15)deg(I-A,Ω,θ)=0.

3. Bifurcation Results 3.1. Assumptions and Conversion of BVP (<xref ref-type="disp-formula" rid="EEq1.1">3</xref>)

Suppose that the following two assumptions hold throughout the paper.

(H1) Let F:J×+2+ be a nonempty, closed and convex multivalued map such that F is l.s.c., where J=[0,1].

(H2) There exist functions a0,a0,b,bC(J,+) 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×+, where ξi,ζiC(J×+) with ξi(t,u)=o(u) as u0 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 is C[0,1]. Obviously, C[0,1] is a Banach space with norm u=maxtJ|u(t)| (for alluC[0,1]). Let (17)Q:={uC[J,+]:u(t)t2u(s),t,sJ}. It is easy to see that Q is a cone of E. Moreover, from (17), we have for all uQ, (18)u(t)t2u,tJ.

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)=0,0jn-1,j2,u(1)=0, where gC[0,1].

Lemma 10 (Tian and Liu [<xref ref-type="bibr" rid="B26">8</xref>]).

Given gC[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,st;(α-1)(α-2)2t2(1-s)α-3,ts.

Lemma 11 (Tian and Liu [<xref ref-type="bibr" rid="B26">8</xref>]).

The function G(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,st,(α-1)(α-2)2s2(1-s)α-3,ts;(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)=0,0jn-1,j2,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, uQ{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.

For aC(J,+) 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 Lemmas 10, 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 operator La 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),tJ.

Note that condition (H1) implies that F(t,u) is lower semicontinuous. Then, from Lemma 6, there exists a continuous function f:J×++ such that f(t,u)F(t,u) for all (t,u)J×+. Therefore, to solve BVP (24), we consider the problem (27)D0+αCu(t)+λf(t,u(t))=0,0<t<1,u(j)(0)=0,0jn-1,j2,u(1)=0.

Define (28)f-(t,u)={f(t,u),(t,u)J×+,f(t,0),(t,u)J×(-,0). Then f-(t,u)0 on J×. From Lemma 10, the solution of (29)D0+αCu(t)+λf-(t,u(t))=0,0<t<1,u(j)(0)=0,0jn-1,j2,u(1)=0 is equivalent to the fixed point of operator (30)Aλu(t)=λ01G(t,s)f-(s,u(s))ds,uC[0,1].

Let Σ+×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 ΣQ and Aλ:C[0,1]Q. Moreover, we have the following conclusion.

Lemma 13.

For λ>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 +×Q is exactly Σ.

3.2. Bifurcation from Infinity and Trivial Solution Lemma 14.

Let [c,d]+ be a compact interval with [λ1(b),λ1(b)][c,d]=. Then there exists R1>0 such that (31)uAλu,λ[c,d],uC[0,1]withuR1.

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 μnμ[c,d]. Notice that unQ. 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 vnv in C[0,1]. Then v=1 and vQ.

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×+. Therefore, by virtue of (30), we know (33)vn(t)μn01G(t,s)(b(s)vn(s)+ζ2(s,un(s))un)ds,(34)vn(t)μn01G(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*μnLbvn,ψ*+μn01ψ*(t)01G(t,s)ζ2(s,un(s))undsdt. Letting n+ and using condition (H2), we have (36)v,ψ*μLbv,ψ*=μv,Lb*ψ*=μv,ψ*λ1(b), which implies μλ1(b). Similarly, one can deduce from (34) that μλ1(b).

To sum up, λ1(b)μλ1(b), which contradicts with μ[c,d]. The conclusion of this lemma follows.

Lemma 15.

For μ(0,λ1(b)), there exists R1>0 such that (37)deg(I-Aμ,BR,0)=1,RR1.

Proof.

Notice that [0,μ][λ1(b),λ1(b)]=. From Lemma 14, there exists R1>0 such that (38)uAλu,λ[0,μ],uC[0,1]withuR1, which means (39)uτAμu,τ[0,1],uC[0,1]withuR1.

Therefore, by the homotopy invariance of topological degree, we have (40)deg(I-Aμ,BR,0)=deg(I,BR,0)=1,RR1.

Lemma 16.

For λ>λ1(b), there exists R2>0 such that (41)deg(I-Aλ,BR,0)=0,RR2.

Proof.

We first prove that for λ>λ1(b), there exists R2>0 such that (42)Aλuμu,μ(0,1],uC[0,1]withuR2.

Suppose, on the contrary, that there exist {(μn,un)}(0,1]×C[0,1] with un(n+) such that Aλun=μnun.

By Lemma 13, un(t)>0 in (0,1]. Set vn=un/un; that is, μnvn=Aλun/un. Without loss of generality, assume μnμ-[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 vnQ and vn=1. Therefore, vn(t)t2 for t(0,1]. Consequently, (43)μnvn(t)=Aλun(t)unλ01G(t,s)(b(s)vn(s)-ζ1(s,un(s))un)dsλt2maxτJ01s2G(τ,s)b(s)ds-λ01G(t,s)ζ1(s,un(s))unds. From (H2) and Lemma 11, it is easy to see maxτJ01s2G(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 vnv (relabeling if necessary) in C[0,1]. By virtue of (32), we know (44)vn,ψ*μnvn,ψ*=1unAλun,ψ*λLbvn,ψ*-λ01ψ*(t)01G(t,s)ζ1(s,un(s))undsdt.

Letting n+ and using condition (H2), we obtain that (45)v,ψ*λLbv,ψ*=λv,Lb*ψ*=λv,ψ*λ1(b), which implies λλ1(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,RR2.

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 Lemma 13, we need only to prove that the conclusion holds for (29).

For fixed 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 {uC[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]+ be a compact interval with [λ1(a0),λ1(a0)][c,d]=. Then there exists δ1>0 such that (47)uAλu,λ[c,d],uC[0,1]with0<uδ1.

Lemma 19.

For μ(0,λ1(a0)), there exists δ1>0 such that (48)deg(I-Aμ,Bδ,0)=1,δ(0,δ1].

Lemma 20.

For λ>λ1(a0), there exists δ2>0 such that (49)deg(I-Aλ,Bδ,0)=0,δ(0,δ2].

Finally, using Lemmas 1820, 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

λ1(b)<1<λ1(a0) or

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 of  Σ that crosses the hyperplane {1}×C(J), where Σ+×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 Theorem 17, there exists an unbounded component 𝒞 of solutions of (27), which meets [λ1(b),λ1(b)]× and is unbounded in λ direction.

If 𝒞(+×{0})=, by 𝒞({0}×C(J))= and Theorem 17, we know that 𝒞 must cross the hyperplane {1}×C(J).

If 𝒞(+×{0}), by Theorem 21, we know 𝒞(+×{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 Theorem 21, 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 that 𝒞0 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.

There exist R>0 and hL[0,1] such that for tJ, (50)supt2RuRF(t,u)h(t),maxtJ01G(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 Theorems 17 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 exists ε>0 such that (52)(1+ε)maxtJ01G(t,s)h(s)ds<R. Now we show Σ([0,1+ε]×BR)=, where BR={uC(J):u<R}. Suppose that, on the contrary, (λ,u) is a solution of (27) such that 0λ1+ε and u=R. Then by Lemma 13, we know uQ. Therefore, u(t)[t2R,R] for tJ. From (H3), (30), and Lemma 13, it follows that (53)R=u=maxtJλ01G(t,s)f-(s,u(s))ds(1+ε)maxtJ01G(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 that 𝒞0 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

Let ρ 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)=0,0j3,j2,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,tJ,u[0,1],ρ4t2u,tJ,u[1,+),ζ1(t,u)={2ρu,tJ,u[0,1],2ρu,tJ,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 of ρ, it is easy to see λ1(b)=1/2<1<2=λ1(a0).

Therefore, by Theorem 22, BVP (55) has at least one positive solution.

Acknowledgments

The research is supported by NNSF of China (11171192) and the Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2010SF025).

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