AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation64863510.1155/2012/648635648635Research ArticleExistence and Multiplicity of Solutions for Some Fractional Boundary Value Problem via Critical Point TheoryChenJing1,2TangX. H.1PereraKanishka1School of Mathematical Sciences and Computing TechnologyCentral South UniversityChangsha Hunan 410083Chinacsu.edu.cn2School of Mathematics and Computing SciencesHunan University of Science and TechnologyXiangtanHunan 411201Chinahnust.edu.cn201214122011201218102011271120112012Copyright © 2012 Jing Chen and X. H. Tang.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.

We study the existence and multiplicity of solutions for the following fractional boundary value problem: (d/dt)((1/2)0Dt-β(u(t))+(1/2)tDT-β(u(t)))+F(t,u(t))=0,  a.e.  t[0,T],  u(0)=u(T)=0, where F(t,·) are superquadratic, asymptotically quadratic, and subquadratic, respectively. Several examples are presented to illustrate our results.

1. Introduction and Main Results

Consider the fractional boundary value problem (BVP for short) of the following form: ddt(120Dt-β(u(t))+12tDT-β(u(t)))+F(t,u(t))=0,a.e.  t[0,T],u(0)=u(T)=0, where   0Dt-β and   tDT-β are the left and right Riemann-Liouville fractional integrals of order 0β<1, respectively, F:[0,T]×N satisfies the following assumptions.

F(t,x) is measurable in t for every xN and continuously differentiable in x for a.e. t[0,T], and there exist aC(+,+), bL1(0,T;+), such that

|F(t,x)|a(|x|)b(t),|F(t,x)|a(|x|)b(t), for all xN and a.e. t[0,T]. In particular, if β=0, BVP (1.1) reduces to the standard second-order boundary value problem of the following form: u(t)+F(t,u(t))=0,a.e.  t[0,T],u(0)=u(T)=0.

Differential equations with fractional order are generalization of ordinary differential equations to noninteger order. This generalization is not mere mathematical curiosities but rather has interesting applications in many areas of science and engineering such as in viscoelasticity, electrical circuits, and neuron modeling. The need for fractional order differential equations stems in part from the fact that many phenomena cannot be modeled by differential equations with integer derivatives. Such differential equations got the attention of many researchers and considerable work has been done in this regard, see the monographs of Kilbas et al. , Miller and Ross , Podlubny , Samko et al. , and the papers  and the references therein.

Recently, there are many papers dealing with the existence of solutions (or positive solutions) of nonlinear initial (or singular and nonsingular boundary) value problems of fractional differential equation by the use of techniques of nonlinear analysis (fixed-point theorems , Leray-Schauder theory [15, 16], lower and upper solution method, monotone iterative method , Adomian decomposition method , etc.), see  and the references therein.

Variational methods are very powerful techniques in nonlinear analysis and are extensively used in many disciplines of pure and applied mathematics including ordinary and partial differential equations, mathematical physics, gauge theory, and geometrical analysis. The existence and multiplicity of solutions for Hamilton systems, Schrödinger equations, and Dirac equations have been studied extensively via critical point theory, see .

In , Jiao and Zhou obtained the existence of solutions for BVP (1.1) by Mountain Pass theorem under the Ambrosetti-Rabinowitz condition (denoted by A.R. condition). Under the usual A.R. condition, it is easy to show that the energy functional associated with the system has the Mountain Pass geometry and satisfies the (PS) condition. However, the A.R. condition is so strong that many potential functions cannot satisfy it, then the problem becomes more delicate and complicated.

In this paper, in order to establish the existence and multiplicity of solutions for BVP (1.1) under distinct hypotheses on potential function by critical point theory, we introduce some functional space Eα, where α(1/2,1], and divide the problem into the following three cases.

1.1. The Superquadratic Case

For the superquadratic case, we make the following assumptions.

lim|x|0F(t,x)/|x|2=0, liminf|x|F(t,x)/|x|2L>π2/|cos(πα)|Γ2(2-α)T2α(3-2α) uniformly for some L>0 and a.e. t[0,T].

limsup|x|+F(t,x)/|x|rM<+ uniformly for some M>0 and a.e. t[0,T].

liminf|x|+((F(t,x),x)-2F(t,x))/|x|μQ>0 uniformly for some Q>0 and a.e. t[0,T],

where r>2 and μ>r-2. We state our first existence result as follows.

Theorem 1.1.

Assume that (A1)–(A3) hold and that F(t,x) satisfies the condition (A). Then BVP (1.1) has at least one solution on Eα.

1.2. The Asymptotically Quadratic Case

For the asymptotically quadratic case, we assume the following.

limsup|x|+F(t,x)/|x|2M<+ uniformly for some M>0 and a.e. t[0,T].

There exists τ(t)L1(0,T;+) such that (F(t,x),x)-2F(t,x)τ(t) for all xN and a.e. t[0,T].

lim|x|+[(F(t,x),x)-2F(t,x)]=+ for a.e. t[0,T].

Our second and third main results read as follows.

Theorem 1.2.

Assume that F(t,x) satisfies (A), (A1), (A2'), (A4), and (A5). Then BVP (1.1) has at least one solution on Eα.

Theorem 1.3.

Assume that F(t,x) satisfies (A), (A1), (A2'), and the following conditions:

there exists τ(t)L1(0,T;+) such that (F(t,x),x)-2F(t,x)τ(t) for all xN and a.e. t[0,T];

lim|x|+[(F(t,x),x)-2F(t,x)]=- for a.e. t[0,T].

Then BVP (1.1) has at least one solution on Eα.

1.3. The Subquadratic Case

For the subquadratic case, we give the following multiplicity result.

Theorem 1.4.

Assume that F(t,x) satisfies the following assumption:

F(t,x):=a(t)|x|γ, where a(t)L(0,T;+) and 1<γ<2 is a constant.

Then BVP (1.1) has infinitely many solutions on Eα.

2. Preliminaries

In this section, we recall some background materials in fractional differential equation and critical point theory. The properties of space Eα are also listed for the convenience of readers.

Definition 2.1 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let f(t) be a function defined on [a,b] and q>0. The left and right Riemann-Liouville fractional integrals of order q for function f(t) denoted by   aDt-qf(t) and   tDb-qf(t), respectively, are defined by   aDt-qf(t)=1Γ(q)at(t-s)q-1f(s)ds,  tDb-qf(t)=1Γ(q)tb(t-s)q-1f(s)ds, provided the right-hand sides are pointwise defined on [a,b], where Γ is the gamma function.

Definition 2.2 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let f(t) be a function defined on [a,b] and q>0. The left and right Riemann-Liouville fractional derivatives of order q for function f(t) denoted by   aDtqf(t) and   tDbqf(t), respectively, are defined by aDtqf(t)=dndtnaDtq-nf(t)=1Γ(n-q)dndtn(at(t-s)n-q-1f(s)ds),tDbqf(t)=(-1)ndndtntDbq-nf(t)=1Γ(n-q)(-1)ndndtn(tb(s-t)n-q-1f(s)ds), where t[a,b], n-1q<n and n.

The left and right Caputo fractional derivatives are defined via the above Riemann-Liouville fractional derivatives. In particular, they are defined for the function belonging to the space of absolutely continuous functions, which we denote by AC([a,b],N). ACk([a,b],N)(k=1,) is the space of functions f such that fCk-1([a,b],N) and f(k-1)AC([a,b],N). In particular, AC([a,b],N)=AC1([a,b],N).

Definition 2.3 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let q0 and n. If q[n-1,n) and f(t)ACn([a,b],N), then the left and right Caputo fractional derivative of order q for function f(t) denoted by   acDtqf(t) and   tcDbqf(t), respectively, exist almost everywhere on [a,b].   acDtqf(t) and   tcDbqf(t) are represented by acDtqf(t)=aDtq-nf(n)(t)=1Γ(n-q)(at(t-s)n-q-1f(n)(s)ds),tcDbqf(t)=(-1)ntDbq-nf(n)(t)=(-1)nΓ(n-q)(tb(s-t)n-q-1f(n)(s)ds), respectively, where t[a,b].

Property 2.4 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, that is,   aDt-q1(aDt-q2f(t))=aDt-q1-q2f(t),tDb-q1(tDb-q2f(t))=tDb-q1-q2f(t),q1,q2>0.

Definition 2.5 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

Define 0<α1 and 1<p<. The fractional derivative space E0α,  p is defined by the closure of C0([0,T],N) with respect to the norm uα,p=(0T|u(t)|pdt+0T  |0cDtαu(t)|pdt)1/p,uE0α,p, where C0([0,T],N) denotes the set of all functions uC([0,T],N) with u(0)=u(T)=0. It is obvious that the fractional derivative space E0α,p is the space of functions uLp(0,T;N) having an α-order Caputo fractional derivative   0cDtαuLp(0,T;N) and u(0)=u(T)=0.

Proposition 2.6 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

Let 0<α1 and 1<p<. The fractional derivative space E0α,p is a reflexive and separable space.

Proposition 2.7 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

Let 0<α1 and 1<p<. For all uE0α,p, one has uLpTαΓ(α+1)0cDtαuLp. Moreover, if α>1/p and 1/p+1/q=1, then uTα-1/pΓ(α)((α-1)q+1)1/q0cDtαuLp.

According to (2.8), we can consider E0α,p with respect to the norm uα,p=0cDtαuLp=(0T|0cDtαu(t)|pdt)1/p.

Proposition 2.8 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

Define 0<α1 and 1<p<. Assume that α>1/p and the sequence {uk} converges weakly to u in E0α,p, that is, uku. Then uku in C([0,T],N), that is, u-uk0, as k.

Making use of Property 2.4 and Definition 2.3, for any uAC([0,T],N), BVP (1.1) is equivalent to the following problem: ddt(120Dtα-1(0cDtαu(t))-12tDTα-1(tcDTαu(t)))+F(t,u(t))=0,a.e.  t[0,T],u(0)=u(T)=0, where α=1-β/2(1/2,1].

In the following, we will treat BVP (2.10) in the Hilbert space Eα=E0α,2 with the corresponding norm uα=uα,2. The variational structure of BVP (2.10) on the space Eα has been established.

Lemma 2.9 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

Let L:[0,T]×N×N×N be defined by L(t,x,y,z)=-12(y,z)-F(t,x), where F:[0,T]×N satisfies the assumption (A).

If 1/2<α1, then the functional defined by φ(u)=0TL(t,u(t),0cDtαu(t),tcDTαu(t))dt is continuously differentiable on Eα, and u,vEα, we have φ(u),v=0T(DxL(t,u(t),0cDtαu(t),tcDTαu(t)),v(t))dt+0T(DyL(t,u(t),0cDtαu(t),tcDTαu(t)),0cDtαv(t))dt+0T(DzL(t,u(t),0cDtαu(t),tcDTαu(t)),tcDTαv(t))dt.

Definition 2.10 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

A function uAC([0,T],N) is called a solution of BVP (2.10) if

Dα(u(t)) is derivative for almost every t[0,T],

u satisfies (2.10),

where Dα(u(t)):=(1/2)0Dtα-1(0cDtαu(t))-(1/2)tDTα-1(tcDTαu(t)).

Lemma 2.11 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

Let 1/2<α1 and φ be defined by (2.12). If assumption (A) is satisfied and uEα is a solution of corresponding Euler equation φ(u)=0, then u is a solution of BVP (2.10) which corresponding to the solution of BVP (1.1).

By Lemma 2.11, it means that the solutions for BVP (1.1) correspond to the critical points of the functional φ. We need the following estimate and known results for the sequel.

Proposition 2.12 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

If 1/2<α1, then for any uEα, one has |cos(πα)|uα2-0T(0cDtαu(t),tcDTαu(t))dt1|cos(πα)|uα2.

Lemma 2.13 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

Let X be a real Banach space, Φ:X is differentiable. One says that Φ satisfies the (PS) condition if any sequence {uk} in X such that {Φ(uk)} is bounded and Φ(uk)0 as k contains a convergent subsequence.

Lemma 2.14 (Mountain Pass theorem [<xref ref-type="bibr" rid="B24">24</xref>]).

Let X be a real Banach space and Φ:X is differentiable and satisfies the (PS) condition. Suppose that

Φ(0)=0,

there exist ρ>0 and σ>0 such that Φ(z)σ for all zX with z=ρ,

there exists z1 in X with z1ρ such that Φ(z1)<σ.

Then Φ possesses a critical value cσ. Moreover, c can be characterized as c=infgΩmaxzg([0,1])Φ(z), where Ω̅={gC([0,1],X):g(0)=0,  g(1)=z1}.

Lemma 2.15 (Clark theorem  [<xref ref-type="bibr" rid="B24">24</xref>]).

Let X be a real Banach space, ΦC1(X,) with Φ even, bounded below, and satisfying the (PS) condition. Suppose Φ(0)=0, there is a set KX such that K is homeomorphic to Sm-1, m, by an odd map, and supKΦ<0. Then Φ possesses at least m distinct pairs of critical points.

3. Proof of the Theorems

For uEα, where Eα:={uL2(0,T;RN):0cDtαuL2(0,T;RN)} is a reflexive Banach space with the norm defined by uα=0cDtαuL2,u:=maxt[0,T]|u(t)|.

It follows from Lemma 2.9 that the functional φ on Eα given by φ(u)=0T[-12(0cDtαu(t),tcDTαu(t))-F(t,u(t))]dt is continuously differentiable on Eα. Moreover, we have φ(u),v=-0T12[(0cDtαu(t),tcDTαv(t))+(tcDTαu(t),0cDtαv(t))]dt-0T(F(t,u(t)),v(t))dt.

Recall that a sequence {un}Eα is said to be a (C) sequence of φ if φ(un) is bounded and (1+unα)φ(un)α0 as n. The functional φ satisfies condition (C) if every (C) sequence of φ has a convergent subsequence. This condition is due to Cerami .

3.1. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>

We will first establish the following lemma and then give the proof of Theorem 1.1.

Lemma 3.1.

Assume (A), (A2), and (A3) hold, then the functional φ satisfies condition (C).

Proof of Lemma <xref ref-type="statement" rid="lem3.1">3.1</xref>.

Let {un}Eα be a (C) sequence of φ, that is, φ(un) is bounded and (1+unα)φ(un)α0 as n. Then there exists M0 such that |φ(un)|M0,(1+unα)φ(un)αM0, for all n.

By (A2), there exist positive constants B1 and M1 such that F(t,x)B1|x|r, for all |x|M1 and a.e. t[0,T].

It follows from (A) that |F(t,x)|maxs[0,M1]a(s)b(t), for all |x|M1 and a.e. t[0,T]. Therefore, we obtain F(t,x)B1|x|r+maxs[0,M1]a(s)b(t), for all xN and a.e. t[0,T].

Combining (2.14) and (3.8), we get |cos(πα)|2unα2φ(un)+0TF(t,un(t))dtM0+maxs[0,M1]a(s)0Tb(t)dt+B10T|un(t)|rdt.

On the other hand, by (A3), there exist η>0 and M2>0 such that (F(t,x),x)-2F(t,x)η|x|μ, for a.e. t[0,T] and |x|M2.

By (A), we have |(F(t,x),x)-2F(t,x)|(2+M2)maxs[0,M2]a(s)b(t), for all |x|M2 and a.e. t[0,T].

Therefore, we obtain (F(t,x),x)-2F(t,x)η|x|μ-ηM2μ-(2+M2)maxs[0,M2]a(s)b(t), for all xN and a.e. t[0,T].

It follows from (3.5) and (3.12) that 3M02φ(un)-φ(un),un=20T[-12(0cDtαun(t),tcDTαun(t))-F(t,un(t))]dt-0T[-(0cDtαun(t),tcDTαun(t))-(F(t,un(t)),un(t))]dt=0T[(F(t,un(t)),un(t))-2F(t,un(t))]dtη0T|un(t)|μdt-(2+M2)maxs[0,M2]a(s)0Tb(t)dt-ηM2μT, thus, 0T|un(t)|μdt is bounded.

If μ>r, then 0T|un(t)|rdtT(μ-r)/μ(0T|un(t)|μdt)r/μ, which, combining (3.9), implies that unα is bounded.

If μr, then 0T|un(t)|rdtunr-μ0T|un(t)|μdtC1r-μunαr-μ0T|un(t)|μdt, where C1:=Tα-1/2Γ(α)(2α-1)1/2, by (2.8).

Since μ>r-2, it follows from (3.9) that unα is bounded too. Thus unα is bounded in Eα.

By Proposition 2.8, the sequence {un} has a subsequence, also denoted by {un}, such that unuweakly  in  Eα,unustrongly  in  C([0,T],RN).

Then we obtain unu in Eα by use of the same argument of Theorem  5.2 in . The proof of Lemma 3.1 is completed.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>.

By (A1), there exist ϵ1(0,|cos(πα)|) and δ>0 such that F(t,x)(|cos(πα)|-ϵ1)Γ2(α+1)2T2α|x|2, for a.e. t[0,T] and xN with |x|δ.

Let ρ=Γ(α)(2(α-1)+1)1/2Tα-1/2δ,σ=ϵ1ρ22>0. Then it follows from (2.8) that uTα-1/2Γ(α)(2(α-1)+1)1/2uα=δ, for all uEα with uα=ρ.

Therefore, we have φ(u)=0T[-12(0cDtαu(t),tcDTαu(t))-F(t,u(t))]dt|cos(πα)|2uα2-(|cos(πα)|-ϵ1)Γ2(α+1)2T2α0T|u(t)|2dt|cos(πα)|2uα2-|cos(πα)|-ϵ12uα2=ϵ12uα2=σ, for all uEα with uα=ρ. This implies that (ii) in Lemma 2.14 is satisfied.

It is obvious from the definition of φ and (A1) that φ(0)=0, and therefore, it suffices to show that φ satisfies (iii) in Lemma 2.14.

By (A1), there exist ϵ2>0 and M3>0 such that F(t,x)>(π2|cos(πα)|Γ2(2-α)T2α(3-2α)+ϵ2)|x|2, for all |x|M3 and a.e. t[0,T].

It follows from (A) that |F(t,x)|maxs[0,M3]a(s)b(t), for all |x|M3 and a.e. t[0,T].

Therefore, we obtain F(t,x)(π2|cos(πα)|Γ2(2-α)T2α(3-2α)+ϵ2)(|x|2-M32)  -maxs[0,M3]a(s)b(t), for all xN and a.e. t[0,T].

Choosing u0=((T/π)sin(πt/T),0,,0)Eα, then u0L22=T32π2,u0α2T3-2αΓ2(2-α)(3-2α).

For ς>0 and noting that (3.24) and (3.25), we have φ(ςu0)=0T[-12(0cDtαςu0(t),tcDTαςu0(t))-F(t,ςu0(t))]dtς22|cos(πα)|u0α2-(ς2π2|cos(πα)|T2αΓ2(2-α)(3-2α)+ς2ϵ2)0T|u0(t)|2dt+C2ς22|cos(πα)|T3-2αΓ2(2-α)(3-2α)-ς2π2|cos(πα)|T2αΓ2(2-α)(3-2α)T32π2-ς2ϵ2T32π2+C2-, as ς, where C2 is a positive constant. Then there exists a sufficiently large ς0 such that φ(ς0u0)0. Hence (iii) holds.

Finally, noting that φ(0)=0 while for critical point u, φ(u)σ>0. Hence u is a nontrivial solution of BVP (1.1), and this completes the proof.

3.2. Proof of Theorem <xref ref-type="statement" rid="thm1.2">1.2</xref>

The following lemmata are needed in the proof of Theorem 1.2.

Lemma 3.2.

Assume (A5), then for any ε>0, there exists a subset Eε[0,T] with meas([0,T]Eε)<ε such that lim|x|[(F(t,x),x)-2F(t,x)]=+, uniformly for tEε.

Proof of Lemma <xref ref-type="statement" rid="lem3.2">3.2</xref>.

The proof is similar to that of Lemma  2 in  and is omitted.

Lemma 3.3.

Assume (A), (A2’), (A4), and (A5), then the functional φ satisfies condition (C).

Proof of Lemma <xref ref-type="statement" rid="lem3.3">3.3</xref>.

Suppose that {un}Eα is a (C) sequence of φ, that is, φ(un) is bounded and (1+unα)φ(un)α0 as n. Then we have liminfn[φ(un),un-2φ(un)]>-, which implies that limsupn0T[(F(t,un),un)-2F(t,un)]dt<+.

We only need to show that {un} is bounded in Eα. If {un} is unbounded, we may assume, without loss of generality, that unα as n. Put zn=un/unα, we then have znα=1. Going to a sequence if necessary, we assume that znz weakly in Eα, znz strongly in C([0,T],N) and L2(0,T;N).

By (A2), it follows that there exist constants B2>0 and M4>0 such that F(t,x)B2|x|2, for all |x|M4 and a.e. t[0,T].

By assumption (A), it follows that |F(t,x)|maxs[0,M4]a(s)b(t), for all |x|M4 and a.e. t[0,T]. Therefore, we obtain F(t,x)B2|x|2+maxs[0,M4]a(s)b(t) for all xN and a.e. t[0,T]. Therefore, we have φ(u)=0T[-12(0cDtαu(t),tcDTαu(t))-F(t,u(t))]dt|cos(πα)|2uα2-B20T|u|2dt-maxs[0,M4]a(s)0Tb(t)dt, from which, it follows that φ(un)unα2|cos(πα)|2-B2znL22-1unα2maxs[0,M4]a(s)0Tb(t)dt.

Passing to the limit in the last inequality, we get |cos(πα)|2-B2zL220, which yields z0. Therefore, there exists a subset E[0,T] with meas(E)>0 such that z(t)0 on E.

By virtue of Lemma 3.2, for ε=(1/2) meas(E)>0, we can choose a subset Eε[0,T] with meas([0,T]Eε)<ε such that lim|x|[(F(t,x),x)-2F(t,x)]=+, uniformly for tEε.

We assert that meas(EEε)>0. If not, meas(EEε)=0.

Since E=(EEε)(EEε), it follows that 0<meas(E)=meas(EEε)+meas(EEε)meas([0,T]Eε)<ε=12meas(E), which leads to a contradiction and establishes the assertion.

By (A4), we obtain thye following: 0T[(F(t,un),un)-2F(t,un)]dt=EEε[(F(t,un),un)-2F(t,un)]dt+[0,T](EEε)[(F(t,un),un)-2F(t,un)]dtEEε[(F(t,un),un)-2F(t,un)]dt-0T|τ(t)|dt.

By (3.36), (3.38), and Fatou's lemma, it follows that limn0T[(F(t,un),un)-2F(t,un)]dt=+, which contradicts (3.29). This contradiction shows that unα is bounded in Eα, and this completes the proof.

By virtue of Lemmas 3.2 and 3.3, the rest of the proof is similar to Theorem 1.1. Theorem 1.3 can be proved similarly.

3.3. Proof of Theorem <xref ref-type="statement" rid="thm1.4">1.4</xref>

The proof of Theorem 1.4 is divided into a sequence of lemma.

Lemma 3.4.

The functional φ is bounded below on Eα.

Proof of Lemma <xref ref-type="statement" rid="lem3.4">3.4</xref>.

By (2.8) and (2.14), for every uEα, we have φ(u)=-0T12(0cDtαu(t),tcDTαu(t))dt-0TF(t,u(t))dt=-0T12(0cDtαu(t),tcDTαu(t))dt-0Ta(t)|u(t)|γdt|cos(πα)|2uα2-a0uγT|cos(πα)|2uα2-a0TC1γuαγ, where a0=esssup{a(t):t[0,T]}. The proof of Lemma 3.4 is complete.

Lemma 3.5.

The functional φ satisfies the (PS) condition.

Proof of Lemma <xref ref-type="statement" rid="lem3.5">3.5</xref>.

Let {un} be a Palais-Smale sequence in Eα, that is, φ(un)  is  bounded  and    φ(un)0as    n+.

Suppose that {un} is unbounded in Eα, that is, unα+ as n+. Since φ(un),un-γφ(un)=(-1+γ2)0T(0cDtαun(t),tcDTαun(t))dt.

However, from (3.42), we have -γφ(un)(1-γ2)|cos(πα)|unα2-φ(un)unα, thus unα is a bounded sequence in Eα. Since Eα is a reflexive space, going, if necessary, to a subsequence, we can assume that unu in Eα, thus we have φ(un)-φ(u),un-u=φ(un),un-u-φ(u),un-uφ(un)αun-uα-φ(u),un-u0, as n. Moreover, according to (2.8) and Proposition 2.8, we have that {un} is bounded in C([0,T],N) and un-u0 as n.

Noting that φ(un)-φ(u),un-u=-0T(0cDtα(un(t)-u(t)),0cDTα(un(t)-u(t)))dt-0T(F(t,un(t))-F(t,u(t)),un(t)-u(t))dt|cos(πα)|un-uα2-|0T(F(t,un(t))-F(t,u(t)))dt|un-u. Combining (3.44) and (3.45), it is easy to verify that un-uα0 as n, and hence that unu in Eα. Thus, {un} admits a convergent subsequence. The proof of Lemma 3.5 is complete.

Lemma 3.6.

For any m, there exists a set KEα which is homeomorphic to Sm-1 by an odd map, and supkφ<0.

Proof of Lemma <xref ref-type="statement" rid="lem3.6">3.6</xref>.

For every m, define ui(t)=(siniπtT,0,,0),i=1,2,,m,Em=span{u1,,um},Km,β={uEm:uα=β}, where β is a positive number to be chosen later.

For any uEm, there exist λi, i=1,2,,m, such that u=i=1mλiui(t),uα2=0T|0cDtαu(t)|2dt=0T(0cDtαu(t),0cDtαu(t))dt=0T(λ10cDtαu1(t)++λm0cDtαum(t),λ10cDtαu1(t)++λm0cDtαum(t))dt=i=1mj=1maijλiλj=F(λ1,,λm), where aij=0T(0cDtαui(t),0cDtαuj(t))dt and F(λ1,,λm) is a real quadratic form.

Since F(λ1,,λm)=i=1mλiui(t)α20,(λ1,,λm)TRm,F(λ1,,λm)=0i=1mλiui(t)0λ1=λ2==λm=0. So, F(λ1,,λm) is a real positive definite quadratic form. Then there exist an invertible matrix Cm×m and μi, i=1,2,,m, such that (λ1,λ2,,λm)T=C(μ1,μ2,,μm)T,F(λ1,,λm)=i=1mμi2.

It is easy to prove that the odd mapping Ψ:Km,βSm-1 defined by Ψ(u)=β-1(μ1,,μm) is a homeomorphism between Km,β and Sm-1.

Since EmEα is a finite dimensional space, there exists ε(m)>0 such that meas{t[0,T]:a(t)|u(t)|γεuαγ}ε,uEm{0}.

Otherwise, for any positive integer n, there exists unEm{0} such that meas{t[0,T]:a(t)|un(t)|γ1nunαγ}<1n.

Set vn(t):=un(t)/unαEm{0}, then vnα=1 for all n and meas{t[0,T]:a(t)|vn(t)|γ1n}<1n.

Since dimEm<, it follows from the compactness of the unit sphere of Em that there exists a subsequence, denoted also by {vn}, such that {vn} converges to some v0 in Em. It is obvious that v0α=1.

By the equivalence of the norms on the finite dimensional space, we have vnv0 in L2(0,T;N), that is, 0T|vn-v0|2dt0as  n.

By (3.54) and Hölder inequality, we have 0Ta(t)|vn-v0|γdt(0Ta(t)2/(2-γ)dt)(2-γ)/2(0T|vn-v0|2dt)γ/2=a(2-γ)/2(0T|vn-v0|2dt)γ/20,as  n.

Thus, there exist ξ1,ξ2>0 such that meas{t[0,T]:a(t)|v0(t)|γξ1}ξ2.

In fact, if not, we have meas{t[0,T]:a(t)|v0(t)|γ1n}=0, for all positive integer n.

It implies that 00Ta(t)|v0|γ+2dt<Tnv02C12Tnv0α20, as n. Hence v0=0 which contradicts that v0α=1. Therefore, (3.56) holds.

Now let Ω0={t[0,T]:a(t)|v0(t)|γξ1},Ωn={t[0,T]:a(t)|vn(t)|γ<1n}, and Ωnc=[0,T]Ωn={t[0,T]:a(t)|vn(t)|γ1/n}.

By (3.53) and (3.56), we have meas(ΩnΩ0)=meas(Ω0(ΩncΩ0))meas(Ω0)-meas(ΩncΩ0)ξ2-1n, for all positive integer n. Let n be large enough such that ξ2-1n12ξ2,12γ-1ξ1-1n12γξ1, then we have 0Ta(t)|vn-v0|γdtΩnΩ0a(t)|vn-v0|γdt12γ-1ΩnΩ0a(t)|v0|γdt-ΩnΩ0a(t)|vn|γdt(12γ-1ξ1-1n)meas(ΩnΩ0)ξ12γξ22=ξ1ξ22γ+1>0, for all large n, which is a contradiction to (3.55). Therefore, (3.51) holds.

For any uKm,β, we have φ(u)=-0T12(0cDtαu(t),tcDTαu(t))dt-0TF(t,u(t))dt12|cos(πα)|uα2-0Ta(t)|u(t)|γdt12|cos(πα)|uα2-εuαγmeas(Ωu)12|cos(πα)|uα2-ε2uαγ, by (3.51), where Ωu:={t[0,T]:a(t)|u(t)|γεuαγ}.

Choosing β=(|cos(πα)|ε2)1/(2-γ), we conclude supKm,βφ<-ε2βγ/2<0 which completes the proof.

Now from the assertion of Lemma 2.15, we know that φ has at least m distinct pairs of critical points for every m, therefore, BVP (1.1) possesses infinitely many solutions on Eα. The proof of Theorem 1.4 is completed.

4. Examples

In this section, we give some examples to illustrate our results.

Example 4.1.

In BVP (1.1), let F(t,x)=ln(1+2|x|2)|x|2  .

These show that all conditions of Theorem 1.1 are satisfied, where r=2.5,μ=2.

By Theorem 1.1, BVP (1.1) has at least one solution uEα.

Example 4.2.

In BVP (1.1), let T=2π and F(t,x)=κf(x)(2+sint)arctan|x|2, where κ>0 and f(x) will be specified below.

Let f(x)=|x|2+ln(1+|x|2). Noting that 0ln(1+|x|2)|x|2, we see that (A) and (A2′) hold. It is also easy to see that (A1) holds for κ>(2π)1-2α|cos(πα)|Γ2(2-α)(3-2α). Furthermore, we have (f(x),x)-2f(x)=2|x|21+|x|2-2ln(1+|x|2)-, as |x|+. Therefore, we have (F(t,x),x)-2F(t,x)=κ2|x|21+|x|4f(x)(2+sint)+κ[(f(x),x)-2f(x)](2+sint)arctan|x|2-, uniformly for all t[0,2π] as |x|+. Thus (A4′) and (A5′) hold. By virtue of Theorem 1.3, we conclude that BVP (1.1) has at least one solution on Eα.

If f(x)=|x|2-ln(1+|x|2), then exactly the same conclusions as above hold true by Theorem 1.2.

Example 4.3.

In BVP (1.1), let F(t,x)=a(t)|x|3/2 where a(t)={T,t=02t,0<tT2-2(t-T),T2<t<TT,t=T.

By Theorem 1.4, BVP (1.1) has infinite solutions on Eα.

Acknowledgment

This paper is partially supported by the NNSF (no. 11171351) of China.

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