# Multiple Solutions for a Class of Fractional Boundary Value Problems

Authors: Ge Bin
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the multiplicity of solutions for the following fractional boundary value problem:(d/dt)((1/2)0Dt-β(u'(t))+(1/2)0DT-β(u'(t)))+λ∇F(t,u(t))=0,a.e.t∈[0,T],u(0)=u(T)=0, where 0Dt-β and 0DT-β are the left and right Riemann-Liouville fractional integrals of order 0≤β<1, respectively, λ>0 is a real number, F:[0,T]×ℝN→ℝ is a given function, and ∇F(t,x) is the gradient of F at x. The approach used in this paper is the variational method. More precisely, the Weierstrass theorem and mountain pass theorem are used to prove the existence of at least two nontrivial solutions.


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## Body

## 1. Introduction

In this paper, we consider the fractional boundary value problem of the following form:(P)ddt(12D0t-β(u′(t))+12D0T-β(u′(t)))+λ∇F(t,u(t))=0,a.a.t∈[0,T],u(0)=u(T)=0,
where D0t-β and D0T-β are the left and right Riemann-Liouville fractional integrals of order 0≤β<1, respectively, λ>0 is a real number, F:[0,T]×ℝN→ℝ is a given function, and ∇F(t,x) is the gradient of F at x.In particular, ifλ=1, the problem (P1) reduces to the standard second-order boundary value problem of the following form:
(P1)ddt(12D0t-β(u′(t))+12D0T-β(u′(t)))+∇F(t,u(t))=0,a.a.t∈[0,T],u(0)=u(T)=0.Fractional calculus and fractional differential equations can find many applications in various fields of physical science such as viscoelasticity, diffusion, control, relaxation processes, and modeling phenomena in engineering, see [1–12]. Recently, many results were obtained dealing with the existence and multiplicity of solutions of nonlinear fractional differential equations by use of techniques of nonlinear analysis, such as fixed-point theory (including Leray-Schauder nonlinear alternative) (see [13, 14]), topological degree theory (including coincidence degree theory) (see [15, 16]), and comparison method (including upper and lower solutions methods and monotone iterative method) (see [17, 18]). However, it seems that the popular methods mentioned above are not appropriate for discussing (P) and (P1), as the equivalent integral equation is not easy to be obtained.In the past, there were investigations of the eigenvalue problems for fractional differential equations. For more detailed information on this topic, we refer to Zhang et al. [19–21], Wang et al. [22, 23], and Jiang et al. [24].Recently, there are many papers dealing with the existence of solutions for problem(P1). In [25], Jiao and Zhou obtained the existence of solutions for (P1) by mountain pass theorem under the Ambrosetti-Rabinowitz condition. Chen and Tang [26] studied the existence and multiplicity of solutions for the system (P1) when the nonlinearity F(t,·) is superquadratic, asymptotically quadratic, and subquadratic, respectively.But so far, few papers discuss the two solutions of the system(P) via critical point theory. The aim of the present paper is to study the existence of at least two solutions for the system (P) as the parameter λ>λ0 for some constant λ0.The paper is organized as follows. We first introduce some basic preliminary results and a well-known lemma in Section2, including the fractional derivative space Eα, where α∈(1/2,1]. In Section 3, we give the main result and its proof. In Section 4, we give the summary of this paper.


## 2. Preliminary

In this section, we recall some related preliminaries and display the variational setting which has been established for our problem.Definition 2.1 (see [8]).
Letf(t) be a function defined on [a,b] and τ>0. The left and right Riemann-Liouville fractional integrals of order τ for function f(t) denoted by Dat-τf(t) and Dtb-τf(t), respectively, are defined by
(2.1)Dat-τf(t)=1Γ(τ)∫at‍(t-s)τ-1f(s)ds,t∈[a,b],Dtb-τf(t)=1Γ(τ)∫tb‍(t-s)τ-1f(s)ds,t∈[a,b],
provided the right-hand sides are pointwise defined on [a,b], where Γ is the gamma function.Definition 2.2 (see [8]).
Letf(t) be a function defined on [a,b]. The left and right Riemann-Liouville fractional derivatives of order τ for function f(t) denoted by Dabτf(t) and Dtbτf(t), respectively, are defined by
(2.2)aDtτf(t)=dndtnDatτ-nf(t)=1Γ(n-τ)dndtn(∫at‍(t-s)n-τ-1f(s)ds),tDbτf(t)=(-1)ndndtnDtbτ-nf(t)=1Γ(n-τ)dndtn(∫tb‍(t-s)n-τ-1f(s)ds),
where t∈[a,b], n-1≤τ<n, and n∈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 belong-ing to the space of absolutely continuous functions, which we denote byAC([a,b],ℝN). ACk([a,b],ℝN)(k=1,2,…) is the space of functions f such that f∈Ck([a,b],ℝN). In particular, AC([a,b],ℝN)=AC1([a,b],ℝN).Definition 2.3 (see [8]).
Letτ≥0 and n∈ℕ. If τ∈[n-1,n) and f(t)∈ACn([a,b],ℝN), then the left and right Caputo fractional derivatives of order τ for function f(t) denoted by acDtτf(t) and tcDbτf(t), respectively, which exist a.e. on [a,b]. acDtτf(t) and tcDbτf(t) are represented by
(2.3)Dactτf(t)=Datτ-nf(n)(t)=1Γ(n-τ)(∫at‍(t-s)n-τ-1f(n)(s)ds),Dtcbτf(t)=(-1)Dtnbτ-nf(n)(t)=1Γ(n-τ)(∫tb‍(t-s)n-τ-1f(n)(s)ds),
respectively, where t∈[a,b].Definition 2.4 (see [25]).
Define0<α≤1 and 1<p<∞. The fractional derivative space E0α,p is defined by the closure of C0∞([0,T],ℝN) with respect to the norm
(2.4)∥u∥α,p=(∫0T‍|u(t)|pdt+∫0T‍|D0ctαu(t)|pdt)1/p,∀u∈E0α,p,
where C0∞([0,T],ℝN) denotes the set of all functions u∈C∞([0,T],ℝN) with u(0)=u(T)=0. It is obvious that the fractional derivative space E0α,p is the space of functions u∈Lp([0,T],ℝN) having an α-order Caputo fractional derivative D0ctαu∈Lp([0,T],ℝN) and u(0)=u(T)=0.Proposition 2.5 (see [25]).
Let0<α≤1 and 1<p<∞. The fractional derivative space E0α,p is a reflexive and separable space.Proposition 2.6 (see [25]).
Let0<α≤1 and 1<p<∞. For all u∈E0α,p, one has
(2.5)∥u∥Lp≤TαΓ(α+1)∥D0ctαu∥Lp.
Moreover, if α>1/p and 1/p+1/q=1, then
(2.6)∥u∥∞≤T(α-1)/pΓ(α)((α-1)q+1)1/q∥D0ctαu∥Lp.
According to (2.6), one can consider E0α,p with respect to the norm
(2.7)∥u∥α,p=∥D0ctαu∥Lp=(∫0T‍|D0ctαu|pdt)1/p.Proposition 2.7 (see [25]).
Define0<α≤1 and 1<p<∞. Assume that α>1/p, and the sequence uk converges weakly to u∈E0α,p, that is, uk⇀u. Then uk→u in C([0,T],ℝN), that is, ∥uk-u∥∞→0, as k→∞.Making use of Definition2.3, for any u∈AC([0,T],ℝN), problem (P) is equivalent to the following problem:
(P2)ddt(12D0tα-1(D0ctαu(t))-12DtTα-1(DtcTαu(t)))+λ∇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 problem(P2) in the Hilbert space Eα=E0α,2 with the corresponding norm ∥u∥α=∥u∥α,2. It follows from [25, Theorem 4.1] that the functional φ given by
(2.8)φ(u)=∫0T‍[-12(D0ctαu(t),DtcTαu(t))]dt-λ∫0T‍F(t,u(t))dt
is continuously differentiable on Eα. Moreover, for u,v∈Eα, we have
(2.9)〈φ′(u),v〉=-∫0T‍12[(D0ctαu(t),DtcTαv(t))+(DtcTαu(t),D0ctαv(t))]dt-λ∫0T‍(∇F(t,u(t)),v(t)]dt.Definition 2.8 (see [25]).
A functionu∈AC([0,T],ℝN) is called a solution of (P2) if
(i)Dα(u(t)) is derivative for a.e. t∈[0,T],
(ii)u satisfies (P2), where Dα(u(t)):=(1/2)D0tα-1(0cDtαu(t))-(1/2)0tDTα-1(tcDTαu(t)).Proposition 2.9 (see [25]).
If1/2<α≤1, then for any u∈Eα, one has
(2.10)|cos(πα)|∥u∥α2≤-∫0T‍(D0ctαu(t),tcDTαu(t))dt≤1|cos(πα)|∥u∥α2.Proposition 2.10 (see [25]).
Let1/2<α≤1 be satisfied. If u∈Eα, then the functional J:Eα→ℝ denoted by
(2.11)J(u)=-12∫0T‍(D0ctαu(t),DtcTαu(t))dt
is convex and continuous on Eα.In order to prove the existence of two solutions for problem(P), firstly, we recall some well-known results. Their proofs can be found in many books. Please refer to the references and its references therein.Lemma 2.11 (see [27]).
IfX is a Banach space, φ∈C1(X,ℝ), e∈X, and r>0, such that ∥e∥>r and
(2.12)b:=inf∥u∥=rφ(u)>φ(0)≥φ(e),
and if φ satisfies the PS condition, with
(2.13)c:=infγ∈Γmaxt∈[0,1]φ(γ(t)),Γ:={γ∈C([0,1],X):γ(0)=0,γ(1)=e},
then c is a critical value of φ.


## 3. The Main Result and Proof of the Theorem

In this part, we will prove that for(P) there also exist two solutions for the general case.Our hypotheses on nonsmooth potentialF(t,x) are as follows.H
(
F
)
1: F:[0,T]×ℝN→ℝ is a function such that F(t,0)=0 a.e. on [0,T] and satisfies the following facts:(i)
for allx∈ℝN, t↦F(t,x) is measurable,(ii)
for a.a.t∈[0,T], x↦F(t,x) is continuously differentiable,(iii)
there existc∈C([0,T],ℝ) and 0<α0<2, such that
(3.1)|∇F(t,x)|,|F(t,x)|≤c(t)(1+|x|α0),for  a.a.t∈[0,T],allx∈ℝN,(iv)
there existγ>2 and μ∈L∞([0,T]), such that
(3.2)lim  sup|x|→0(∇F(t,x),x)|x|γ<μ(t),uniformly  for  a.a.t∈[0,T],(v)
there existξ0∈ℝN, t0∈(0,T), and r0>0, such that
(3.3)F(t,ξ0)>δ0>0,a.e.t∈Br0(t0),
where Br0(t0):={t∈[0,T]:|t-t0|≤r0}⊂[0,T].Remark 3.1.
It is easy to verify thatF:[0,T]×ℝN→ℝ satisfies the whole assumption in [25, Theorem 4.1]. So, u∈Eα is a solution of the corresponding Euler equation φ′(u)=0, then u is a solution of problem (P2) which, of course, corresponds to the solution of problem (P). For great details, please see [25, Theorem 4.2].Theorem 3.2.
Suppose thatH(F)1 holds. Then there exists λ0>0 such that for each λ>λ0, the problem (P2) has at least two nontrivial solutions, which correspond to the two solutions of problem (P).Proof.
The proof is divided into four steps as follows.
Step
1. We will show that φ is coercive in this step.
Firstly, byH(F)1(iii), (2.6), and (2.10), we have
(3.4)φ(u)=∫0T[-12(D0ctαu(t),DtcTαu(t))]‍dt-λ∫0T‍F(t,u(t))dt≥|cos(πα)|2∥u∥α2-c0λT-c0λ∫0T‍|u(t)|α0dt≥|cos(πα)|2∥u∥α2-c0λT-c0λT(T(α-1)/22α-1Γ(α))α0∥u∥αα0→∞,as∥u∥α→∞,
where c0=maxt∈[0,T]|c(t)|.
Step
2. We will show that the φ is weakly lower semicontinuous.
Letun⇀u weakly in Eα, and by Proposition 2.7, we obtain the following results:
(3.5)Eα↪C([0,T],ℝN),un(t)→u(t)fora.e.t∈[0,T],F(t,un(t))→F(t,u(t))fora.e.t∈[0,T].
By Fatou's lemma,(3.6)lim  supn→∞∫0T‍F(t,un(t))dt≤∫0T‍F(t,u(t))dt.
On the other hand, by Proposition 2.10, we have limn→∞J(un)=J(u), that is,
(3.7)limn→∞∫0T‍[-12(D0ctαun(t),tcDTαun(t))]dt=∫0T‍[-12(D0ctαu(t),tcDTαu(t))]dt.
Thus,(3.8)lim  infn→∞φ(un)=lim  infn→∞∫0T‍[-12(D0ctαun(t),DtcTαun(t))]dt-lim  supn→∞λ∫0T‍F(t,un(t))dt≥∫0T‍[-12(D0ctαu(t),DtcTαu(t))]dt-λ∫0T‍F(t,u(t))dt=φ(u).
Hence, by the Weierstrass theorem, we deduce that there exists a global minimizeru0∈Eα such that
(3.9)φ(u0)=minu∈Eαφ(u).
Step
3. We will show that there exists λ0>0 such that for each λ>λ0, φ(u0)<0.
By the condition byH(F)1 (v), there exists ξ0∈ℝN such that F(t,ξ0)>δ0>0, a.e. t∈Br0(t0). It is clear that
(3.10)0<M1:=max|x|≤|ξ0|{c0+c0|x|α0}<+∞.
Now we denote(3.11)κ0=M1δ0+M1,λ0=maxκ∈[κ1,κ2]|ξ0|2T3-2α2Γ2(1-α)r03(1-κ)2(1-α)2(3-2α)(δ0κ-M1+κM1)|cos(πα)|,
where κ0<κ1<κ2<1, and δ0 is given in the condition H(F)1 (v). A simple calculation shows that the function κ↦δ0κ-M1+κM1 is positive whenever κ>κ0 and δ0κ0-M1+κ0M1=0. Thus, λ0 is well defined and λ0>0.
We will show that for eachλ>λ0, the problem (P2) has two nontrivial solutions. In order to do this, for t∈[t1,t2], let us define
(3.12)ηκ(t)={0ift∈[0,T]/Br0(t0),ξ0ift∈Bκr0(t0),ξ0r0(1-κ)(r0-|t-t0|)ift∈Br0(t0)/Bκr0(t0).
Then|ηκ′(t)|≤|ξ0|/r0(1-κ) and
(3.13)|D0ctαηκ(t)|≤1Γ(1-α)∫0t‍(t-s)-α|ηκ′(s)|ds≤1Γ(1-α)|ξ0|r0(1-κ)t1-α1-α.
Hence,
(3.14)∥ηι∥α2=∫0T‍|D0ctαηκ(t)|2dt≤|ξ0|2T3-2αΓ2(1-α)r02(1-κ)2(1-α)2(3-2α).
By H(F)1 (iii) and (v), we have
(3.15)∫0T‍F(t,ηκ(t))dt=∫Bκr0(t0)‍F(t,ηκ(t))dt+∫Br0(t0)∖Bκr0(t0)‍F(t,ηκ(t))dt≥2κr0δ0-M1(2r0-2κr0)=2r0(δ0κ-M1+κM1).
Forκ∈[κ1,κ2], by Proposition 2.9, we have
(3.16)φ(ηκ)=∫0T‍[-12(D0ctαηκ(t),DtcTαηκ(t))]dt-λ∫0TF(t,ηκ(t))dt≤1|cos(πα)|∥ηκ∥α2-2λr0(δ0κ-M1+κM1)≤1|cos(πα)||ξ0|2T3-2αΓ2(1-α)r02(1-κ)2(1-α)2(3-2α)-2λr0(δ0κ-M1+κM1),
so that φ(ηt)<0 whenever λ>λ0.
Step
4. We will check the PS condition in the following.
Suppose that{un}n≥1⊆Eα such that φ(un)→c and ∥φ′(un)∥α→0.
Sinceφ is coercive and {un}n≥1 is bounded in Eα and passed to a subsequence, which still denote {un}n≥1, we may assume that there exists u∈Eα, such that un⇀u weakly in Eα; thus, we have
(3.17)〈φ′(un)-φ′(u),un-u〉=〈φ′(un),un-u〉-〈φ′(u),un-u〉≤∥φ′(un)∥∥un∥α-〈φ′(u),un-u〉→0,
as n→∞. Moreover, according to Proposition 2.7, we have ∥un-u∥∞→0, as n→∞. Observing that
(3.18)〈φ'(un)-φ'(u),un-u〉=-∫0T‍(D0ctα(un(t)-u(t)),DtcTα(un(t)-u(t)))dt-∫0T‍(∇F(t,un(t))-F(t,u(t)),un(t)-u(t))dt≥|cos(πα)|∥un-u∥α2-∫0T‍|∇F(t,un(t))-F(t,u(t))|dt∥un-u∥∞,
combining this with (3.17), it is easy to verify that ∥un-u∥α→0, as n→∞, and hence that un→u in Eα. Thus, φ satisfies the PS condition.Step
5. We will show that there exists another nontrivial weak solution of problem (P2).From the mean value theorem andH(F)1 (v), we have
(3.19)F(t,x)-F(t,0)=∫01‍(∇F(t,θx),x)dθ=∫01‍(∇F(t,θx),θx)θdθ≤∫01‍μ(t)θ|θx|γdθ=μ(t)|x|γ∫01‍θγ-1dθ=μ(t)|x|γγ,
for all |x|<β, a.e. t∈[0,T].It follows from the conditionsH(F)1 (iii) and all |x|≥β and a.e. t∈[0,T] that
(3.20)|F(t,x)|≤c0|t|+c0|x|α0≤c0|xβ|+c0|x|α0≤(c0βα0+c0)|x|α0-γ|x|γ≤(c0βγ+c0βγ-α0)|x|γ,
and this together with (3.19) yields that for all x∈ℝN and a.e. t∈[0,T],
(3.21)|F(t,x)|≤(μ(t)γ+c0βγ+c0βγ-α0)|x|γ(x)≤c1|x|γ,
for some positive constant c1.For allλ>λ0, ∥u∥α<1, and |u|∞<1, we have
(3.22)φ(u)=∫0T‍[-12(D0ctαu(t),tcDTαu(t))]dt-λ∫0T‍F(t,u(t))dt≥|cos(πα)|∥u∥α2-λc1∫0T‍|u(t)|γdt≥|cos(πα)|∥u∥α2-λc1[T(α-1)/2Γα(α)2α-1]γ∣∥u∥αγ.So, forρ>0 small enough, there exists a ν>0 such that
(3.23)φ(u)>ν,for∥u∥α=ρ,
and ∥u0∥α>ρ. So by the mountain pass theorem (cf. Lemma 2.11), we can get u1∈Eα which satisfies
(3.24)φ(u1)=c>0,φ′(u1)=0.Therefore,u1 is another nontrivial critical point of φ.Remark 3.3.
We can find a potential function satisfying the hypothesis of our Theorem3.2. For great details, please see Section 4(B) in Summary.So far, the results involved potential functions exhibiting sublinear. The next theorem concerns problems where the potential function is superlinear.Our hypotheses on nonsmooth potentialF(t,x) are as follows.H
(
F
)
2: F:[0,T]×ℝN→ℝ is a function such that F(t,0)=0 a.e. on [0,T] and satisfies the following facts:(i)
for allx∈ℝN, t↦F(t,x) is measurable,(ii)
for a.a.t∈[0,T], x↦F(t,x) is continuously differentiable,(iii)
there existc¯∈C([0,T],ℝ) and α0>2, such that
(3.25)|∇F(t,x)|,|F(t,x)|≤c¯(t)(1+(t)|x|α0),for a.a.t∈[0,T],allx∈ℝN,(iv)
there existγ>2 and μ∈L∞([0,T]), such that
(3.26)lim  sup|x|→0(∇F(t,x),x)|x|γ<μ(t),uniformly for a.a.t∈[0,T],(v)
there existξ0∈ℝN, t0∈(0,T), and r0>0, such that F(t,ξ0)>δ0>0, a.e. t∈Br0(t0),where Br0(t0):={t∈[0,T]:|t-t0|≤r0}⊂[0,T],(vi)
for a.a.t∈[0,T] and all x∈ℝN, we have
(3.27)F(t,x)≤ν(t)withν∈Lβ([0,T],ℝ),1≤β<2.Theorem 3.4.
Suppose thatH(F)2 holds. Then there exists a λ0>0 such that for each λ>λ0, the problem (P2) has at least two nontrivial solutions, which correspond to the two solutions of problem (P).Proof.
The steps are similar to those of Theorem3.2. In fact, we only need to modify Step 1 and Step 4 as follows: 1′ shows that φ is coercive under the condition H(F)2 (vi); 4′ shows that there exists a second nontrivial solution under the conditions H(F)2 (iii) and H(F)2 (iv). Then from Steps 1′, 2, 3, and 4′ above, problem (P) has at least two nontrivial solutions.
Step
1'. By H(F)2 (vi), for all u∈Eα, ∥u∥α>1, we have
(3.28)φ(u)=∫0T‍[-12(D0ctαu(t),DtcTαu(t))]dt-λ∫0T‍F(t,u(t))dt≥|cos(πα)|∥u∥α2-λc¯0∫0T‍ν(t)dt→∞,as∥u∥α→∞,
where c¯0=maxt∈[0,T]|c¯(t)|.
Step
4'. Because of hypothesis H(F)2 (iii), we have
(3.29)F(t,x)≤c¯0+c¯0|x|α0≤c¯0|xβ|α0+c¯0|x|α0=c¯0|1β|α0|x|α0+c¯0|x|α0=c2|x|α0,
for a.e. t∈[0,T] and all |x|≥β with c2>0.
Combining (3.19) and (3.29), it follows that
(3.30)|F(t,x)|≤μ(t)γ|x|γ+c2|x|α0,
for a.e. t∈[0,T] and all x∈ℝN.
Thus, for allλ>λ0, ∥u∥α<1 and |u|∞<1, we have
(3.31)φ(u)=∫0T‍[-12(D0ctαu(t),DtcTαu(t))]dt-λ∫0T‍F(t,u(t))dt≥|cos(πα)|∥u∥α2-λ∫0T‍μ(t)γ|u(t)|γdt-λc2∫0T‍|u(t)|α0dt≥|cos(πα)|∥u∥α2-λc3[T(α-1)/2Γ(α)2α-1]γ|∥u∥αγ-λc4[T(α-1)/2Γ(α)2α-1]α0|∥u∥αα0,
where c3 and c4 are positive constants.
So, forρ>0 small enough, there exists a ν>0 such that
(3.32)φ(u)>ν,for∥u∥α=ρ,
and ∥u0∥α>ρ.
Arguing as in proof of Step 4 of Theorem3.2, we conclude that φ satisfies the PS condition. So by the mountain pass theorem (cf. Lemma 2.11), we can get that u1∈Eα satisfies
(3.33)φ(u1)=c>0,φ′(u1)=0.
Therefore,u1 is another nontrivial critical point of φ.Remark 3.5.
We will give some examples, which satisfy the hypothesis of our Theorem3.4. For great details, please see Section 4(C) in Summary.


## 4. Summary

(A) Ifβ=0, then α=1-β/2=1. Therefore, by Theorems 3.2 and 3.4, we actually obtain the existence of two weak solutions of the following eigenvalue problem:
(P3)u′′(t)+λ∇F(t,u(t))=0,a.a.t∈[0,T],u(0)=u(T)=0,
where λ>0 is a real number, F:[0,T]×ℝN→ℝ is a given function, and ∇F(t,x) is the gradient of F at x. Although many excellent results have been worked out on the existence of solutions for second-order system (P3) (e.g., [28, 29]), it seems that no similar results were obtained in the literature for fractional system (P).(B) We give an example in the following to illustrate our viewpoint in Remark3.1. We consider
(4.1)F(t,x)={a(t)sin(π2|x|γ),0≤|x|<1,a(t),|x|≥1,
where γ>2, a∈C([0,T],ℝ), a(t)>0 for all t∈[0,T].Obviously, hypothesesH(F)1 (i), (ii), and (v) are satisfied. Moreover,
(4.2)|F(t,x)|≤{a(t)π2|x|γ≤a(t)π2|x|γ-α0|x|α0≤a(t)π2|x|α0≤a(t)πγ2|x|α0,0≤|x|<1,a(t)≤a(t)πγ2,|x|≥1,|∇F(t,x)|={|a(t)πγ2cos(π2|x|γ)|x|γ-2x|≤a(t)πγ2|x|γ-1≤a(t)πγ2,0≤|x|<1,0,|x|≥1.Therefore,(4.3)|∇F(t,x)|,|F(t,x)|≤πγa(t)2+πγa(t)2|x|α0.
So, condition H(F)1(iii) holds.On the other hand,(4.4)lim  sup|x|→0〈∇F(t,x),x〉|x|γ=lim  sup|x|→0γ(π/2)a(t)cos((π/2)|x|γ)|x|γ|x|γ=π2γa(t),
uniformly for a.a. t∈[0,T], so condition H(F)1 (iv) holds.(C) We can find the following potential functions satisfying the conditions stated in Theorem3.4:
(4.5)F1(t,x)={-a(t)γ|x|γ,|x|<1,a(t)πcos(π2|x|2)-a(t)γ,|x|≥1,F2(t,x)={a(t)sin(π2|x|γ),|x|<1,a(t)[1|x|+12|x|2-12],|x|≥1,F3(t,x)={a(t)γ|x|γ,|x|<1,-2a(t)|x|+a(t)(2+1γ),|x|≥1,
where a∈C([0,T],ℝ), a(t)>0 for all t∈[0,T].It is clear thatFi(x,0)=0 (i=1,2,3) for a.e. t∈[0,T], and hypotheses H(F)2 (i) and H(F)2 (ii) are satisfied. A direct verification shows that conditions H(F)2 (v) and H(F)2 (vi) are satisfied. Note that
(4.6)|∇F1(t,x)|={|-a(t)|x|γ-2x|≤a(t)|x|γ-1≤2a(t),0≤|x|<1,|-a(t)sin(π2|x|2)x|≤a(t)|x|≤2a(t)|x|γ,|x|≥1,|∇F2(t,x)|={|γa(t)cos(π2|x|γ)π2|x|γ-2x|≤γa(t)π2|x|γ-1≤a(t)πγ2,0≤|x|<1,|a(t)(x-x|x|3)|≤a(t)(|x|+1|x|2)≤a(t)+a(t)|x|γ,|x|≥1,|∇F3(t,x)|={|a(t)|x|γ-2x|≤|a(t)|x|2(5/2)|≤a(t),0≤|x|<1,|a(t)|x|-5/2x|≤a(t)|x|≤a(t)|x|γ,|x|≥1,|F1(t,x)|≤a(t)|x|γ+a(t)(1+1)=2a(t)+2a(t)|x|γ,|F2(t,x)|≤a(t)|sin(π2|x|γ)|+a(t)[1+12|x|γ-12]≤3a(t)+a(t)|x|γ2,|F3(t,x)|≤a(t)1γ|x|γ+2a(t)|x|+a(t)(2+1γ)≤5a(t)+a(t)|x|γ.
So,
(4.7)|∇F1(t,x)|,|F1(t,x)|≤2a(t)(1+|x|γ),|∇F2(t,x)|,|F2(t,x)|≤a(t)πγ2(1+|x|γ),|∇F3(t,x)|,|F3(t,x)|≤5a(t)(1+|x|γ),lim  sup|x|→0(∇F1(t,x),x)|x|γ=lim|x|→0-a(t)|x|γ|x|γ=-a(t),lim  sup|x|→0(∇F2(t,x),x)|x|γ=lim|x|→0a(t)cos((π/2)|x|γ)(π/2)γ|x|γ|x|γ=lim|x|→0a(t)cos(π2|x|γ)π2γ=γπa(t)2,lim  sup|x|→0(∇F3(t,x),x)|x|γ=lim|x|→0a(t)|x|γ|x|γ=a(t),
which shows that assumptions H(F)2 (iii) and H(F)2 (iv) are fulfilled.



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*Source: 468980-2012-10-24.xml*