# Existence of Three Solutions for a Nonlinear Fractional Boundary

Authors: Chuanzhi Bai
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is concerned with the existence of three solutions to a nonlinear fractional boundary value problem as follows:(d/dt)((1/2)0Dtα-1(0CDtαu(t))-(1/2)tDTα-1(tCDTαu(t)))+λa(t)f(u(t))=0,a.e.t∈[0,T],u(0)=u(T)=0, where α∈(1/2,1], and λ is a positive real parameter. The approach is based on a critical-points theorem established by G. Bonanno.


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## Body

## 1. Introduction

Differential equations with fractional order have recently proved to be strong tools in the modeling of many physical phenomena in various fields of physical, chemical, biology, engineering, and economics. There has been significant development in fractional differential equations, one can see the monographs [1–5] and the papers [6–20] and the references therein.Critical-point theory, which proved to be very useful in determining the existence of solution for integer-order differential equation with some boundary conditions, for example, one can refer to [21–25]. But till now, there are few results on the solution to fractional boundary value problem which were established by the critical-point theory, since it is often very difficult to establish a suitable space and variational functional for fractional boundary value problem. Recently, Jiao and Zhou [26] investigated the following fractional boundary value problem:(1.1)ddt(120Dt-β(u′(t))+12tDT-β(u′(t)))+∇F(t,u(t))=0,a.e.t∈[0,T],u(0)=u(T)=0
by using the critical point theory, where 0Dt-β and tDT-β are the left and right Riemann-Liouville fractional integrals of order 0≤β<1, respectively, F:[0,T]×RN→R is a given function and ∇F(t,x) is the gradient of F at x.In this paper, by using the critical-points theorem established by Bonanno in [27], a new approach is provided to investigate the existence of three solutions to the following fractional boundary value problems:
(1.2)ddt(120Dtα-1(D0Ctau(t))-12tDTα-1(DtCTau(t)))+λa(t)f(u(t))=0,a.e.t∈[0,T],u(0)=u(T)=0,
where α∈(1/2,1], 0Dtα-1 and tDTα-1 are the left and right Riemann-Liouville fractional integrals of order 1-α respectively, 0cDtα and tcDTα are the left and right Caputo fractional derivatives of order α respectively, λ is a positive real parameter, f:R→R is a continuous function, and a:R→R is a nonnegative continuous function with a(t)≢0.


## 2. Preliminaries

In this section, we first introduce some necessary definitions and properties of the fractional calculus which are used in this paper.Definition 2.1 (see [5]).
Letf be a function defined on [a,b]. The left and right Riemann-Liouville fractional integrals of order α for function f denoted by aDt-αf(t) and tDb-αf(t), respectively, are defined by
(2.1)aDt-αf(t)=1Γ(α)∫at‍(t-s)α-1f(s)ds,t∈[a,b],α>0,tDb-αf(t)=1Γ(α)∫tb‍(s-t)α-1f(s)ds,t∈[a,b],α>0,
provided the right-hand sides are pointwise defined on [a,b], where Γ(α) is the gamma function.Definition 2.2 (see [5]).
Letγ≥0 and n∈N.
(i) Ifγ∈(n-1,n) and f∈ACn([a,b],RN), then the left and right Caputo fractional derivatives of order γ for function f denoted by aCDtγf(t) and tCDbγf(t), respectively, exist almost everywhere on [a,b], aCDtγf(t) and tCDbγf(t) are represented by
(2.2)aCDtγf(t)=1Γ(n-γ)∫at‍(t-s)n-γ-1f(n)(s)ds,t∈[a,b],tCDbγf(t)=(-1)nΓ(n-γ)∫tb‍(s-t)n-γ-1f(n)(s)ds,t∈[a,b],
respectively.
(ii) Ifγ=n-1 and f∈ACn-1([a,b],RN), then aCDtn-1f(t) and tCDbn-1f(t) are represented by
(2.3)aCDtn-1f(t)=f(n-1)(t),tCDbn-1f(t)=(-1)(n-1)f(n-1)(t),t∈[a,b].
With these definitions, we have the rule for fractional integration by parts, and the composition of the Riemann-Liouville fractional integration operator with the Caputo fractional differentiation operator, which were proved in [2, 5].Property 1 (see [2, 5]).
We have the following property of fractional integration:(2.4)∫ab‍[aDt-γf(t)]g(t)dt=∫ab‍[tDb-γg(t)]f(t)dt,γ>0
provided that f∈Lp([a,b],RN), g∈Lq([a,b],RN), and p≥1, q≥1, 1/p+1/q≤1+γ or p≠1, q≠1, 1/p+1/q=1+γ.Property 2 (see [5]).
Letn∈N and n-1<γ≤n. If f∈ACn([a,b],RN) or f∈Cn([a,b],RN), then
(2.5)aDt-γ(DaCtγf(t))=f(t)-∑j=0n-1‍f(j)(a)j!(t-a)j,tDb-γ(DtCbγf(t))=f(t)-∑j=0n-1‍(-1)jf(j)(b)j!(b-t)j,
for t∈[a,b]. In particular, if 0<γ≤1 and f∈AC([a,b],RN) or f∈C1([a,b],RN), then
(2.6)aDt-γ(DaCtγf(t))=f(t)-f(a),Dtb-γ(DtCbγf(t))=f(t)-f(b).Remark 2.3.
In view of Property1 and Definition 2.2, it is obvious that u∈AC([0,T]) is a solution of BVP (1.2) if and only if  u is a solution of the following problem:
(2.7)ddt(12D0t-β(u′(t))+12DtT-β(u′(t)))+λa(t)f(u(t))=0,a.e.t∈[0,T],u(0)=u(T)=0,
where β=2(1-α)∈[0,1).In order to establish a variational structure for BVP (1.2), it is necessary to construct appropriate function spaces.Denote byC0∞[0,T] the set of all functions g∈C∞[0,T] with g(0)=g(T)=0.Definition 2.4 (see [26]).
Let0<α≤1. The fractional derivative space E0α is defined by the closure of C0∞[0,T] with respect to the norm
(2.8)∥u∥α=(∫0T‍|D0Ctαu(t)|2dt+∫0T‍|u(t)|2dt)1/2,∀u∈E0α.Remark 2.5.
It is obvious that the fractional derivative spaceE0α is the space of functions u∈L2[0,T]having an α-order Caputo fractional derivative 0CDtαu∈L2[0,T] and u(0)=u(T)=0.Proposition 2.6 (see [26]).
Let0<α≤1.  The fractional derivative space E0α is reflexive and separable Banach space.Lemma 2.7 (see [26]).
Let1/2<α≤1.  For all u∈E0α,  one has the following:(i)
(2.9)∥u∥L2≤TαΓ(α+1)∥D0Ctαu∥L2.(ii)
(2.10)∥u∥∞≤Tα-1/2Γ(α)(2(α-1)+1)1/2∥D0Ctαu∥L2.By (2.9), we can consider E0α with respect to the norm
(2.11)∥u∥α=(∫0T‍|D0Ctαu(t)|2dt)1/2=∥D0Ctαu∥L2,∀u∈E0α
in the following analysis.Lemma 2.8 (see [26]).
Let1/2<α≤1,  then for all any u∈E0α,  one has
(2.12)|cos(πα)|∥u∥α2≤-∫0T‍D0Ctαu(t)·tCDTαu(t)dt≤1|cos(πα)|∥u∥α2.Our main tool is the critical-points theorem [27] which is recalled below.Theorem 2.92.9(see [27]).
LetX be a separable and reflexive real Banach space; Φ:X→R be a nonnegative continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on X*; Ψ:X→R be a continuously Gateaux differentiable function whose Gateaux derivative is compact. Assume that there exists x0∈X such that Φ(x0)=Ψ(x0)=0, and that(i)
lim∥x∥→+∞(Φ(x)-λΨ(x))=+∞, for all  λ∈[0,+∞].  Further, assume that there are r>0, x1∈X such that(ii)
r<Φ(x1);(iii)
supx∈Φ-1(]-∞,r[)¯wΨ(x)<(r/(r+Φ(x1)))Ψ(x1). Then, for each
(2.13)λ∈Λ1=rsupx∈Φ-1(]-∞,r[)¯wΨ(x)]Φ(x1)Ψ(x1)-supx∈Φ-1(]-∞,r[)¯wΨ(x),rsupx∈Φ-1(]-∞,r[)¯wΨ(x)[rsupx∈Φ-1(]-∞,r[)¯wΨ(x),
the equation
(2.14)Φ′(x)-λΨ′(x)=0
has at least three solutions in X and, moreover, for each h>1, there exists an open interval
(2.15)Λ2⊂[0,hr(r(Ψ(x1)/Φ(x1)))-supx∈Φ-1(]-∞,r[)¯wΨ(x)]
and a positive real number σ such that, for each λ∈Λ2, (2.14) has at least three solutions in X whose norms are less than σ.


## 3. Main Result

For givenu∈E0α, we define functionals Φ,Ψ:Eα→R as follows:
(3.1)Φ(u)∶=-12∫0T‍D0Ctαu(t)·DtCTαu(t)dt,Ψ(u)∶=∫0T‍a(t)F(u(t))dt,
where F(u)=∫0u‍f(s)ds. Clearly, Φ and Ψ are Gateaux differentiable functional whose Gateaux derivative at the point u∈E0α are given by
(3.2)Φ′(u)v=-12∫0T‍(D0Ctαu(t)·DtCTαv(t)+DtCTαu(t)·D0Ctαv(t))dt,Ψ′(u)v=∫0T‍a(t)f(u(t))v(t)dt=-∫0T‍∫0t‍a(s)f(u(s))ds·v′(t)dt,
for every v∈E0α. By Definition 2.2 and Property 2, we have
(3.3)Φ′(u)v=∫0T‍(12D0tα-1(D0Ctαu(t))-12DtTα-1(DtCTαu(t)))·v′(t)dt.
Hence, Iλ=Φ-λΨ∈C1(E0α,R). If u*∈E0α is a critical point of Iλ, then
(3.4)0=Iλ′(u*)v=∫0T(∫0t‍12D0tα-1(D0Ctαu*(t))-12DtTα-1(DtCTαu*(t))+λ∫0t‍a(s)f(u*(s))ds)·v′(t)dt,
for v∈E0α. We can choose v∈E0α such that
(3.5)v(t)=sin2kπtTorv(t)=1-cos2kπtT,k=1,2,….
The theory of Fourier series and (3.4) imply that
(3.6)12D0tα-1(D0Ctαu*(t))-12DtTα-1(DtCTαu*(t))+λ∫0t‍a(s)f(u*(s))ds=C
a.e. on [0,T] for some C∈R. By (3.6), it is easy to know that u*∈E0α is a solution of BVP (1.2).By Lemma2.7, if α>1/2, we have for each u∈E0α that
(3.7)∥u∥∞≤Ω(∫0T‍|D0Ctαu(t)|2dt)1/2=Ω∥u∥α,
where
(3.8)Ω=Tα-1/2Γ(α)2(α-1)+1.Given two constantsc≥0 and d≠0, with c≠(2A(α)/|cos(πα)|)Ω·d, where Ω as in (3.8).For convenience, set(3.9)A(α)∶=8Γ2(2-α)Γ(4-2α)T1-2α((1+33-2α)24α-5-22α-3-1).Theorem 3.1.
Letf:ℝ→ℝ be a continuous function,  a:R→R be a nonnegative continuous function with a(t)≢0, and 1/2<α≤1. Put F(x)=∫0x‍f(s)ds for every x∈ℝ, and assume that there exist four positive constants c,d,μ, and  p,  with c<(2A(α)/|cos(πα)|)Ω·d and p<2, such that(H1)
F(x)≤μ(1+|x|p),  for  all x∈ℝ;(H2)
F(x)≥0 for all x∈[0,Γ(2-α)d], and
(3.10)F(x)<|cos(πα)|c2(|cos(πα)|c2+2Ω2A(α)d2)∫0T‍a(t)dt×[F(Γ(2-α)d)∫T/43T/4‍a(t)dt+T4Γ(2-α)d∫0Γ(2-α)d‍b(s)F(s)ds],∀x∈[-c,c],where b(s)=a((T/4Γ(2-α)d)s)+a(T-(T/4Γ(2-α)d)s). Then, for each
(3.11)λ∈Λ1=c2|cos(πα)|2Ω2∫0T‍a(t)dt·max|x|≤cF(x)]A(α)d2ℜa+ℜ∫0Γ(2-α)d‍b(x)F(x)dx-∫0T‍a(t)dt·max|x|≤cF(x),c2|cos(πα)|2Ω2∫0T‍a(t)dt·max|x|≤cF(x)[c2|cos(πα)|2Ω2∫0T‍a(t)dt·max|x|≤cF(x,,,,,,,,,,,,,,,),where ℜa and  ℜ denote F(Γ(2-α)d)∫T/43T/4a(t)dt‍  and  T/(4Γ(2-α)d) respectively,the problem (1.2) admits at least three solutions in E0α and, moreover, for each h>1, there exists an open interval
(3.12)Λ2⊂[0,hA(α)d2ℜa+ℜ∫02Γ(2-α)‍b(x)F(x)dx-(2Ω2A(α)d2/c2|cos(πα)|)∫0T‍a(t)dt·max|x|≤cF(x)]
such that, for each λ∈Λ2, the problem (1.2) admits at least three solutions in E0α whose norms are less that σ.Proof.
LetΦ,Ψ be the functionals defined in the above. By the Lemma 5.1 in [26], Φ is continuous and convex, hence it is weakly sequentially lower semicontinuous. Moreover, Φ is coercive, continuously Gateaux differentiable functional whose Gateaux derivative admits a continuous inverse on E0α. The functional Ψ is well defined, continuously Gateaux differentiable and with compact derivative. It is well known that the critical point of the functional Φ-λΨ in E0α is exactly the solution of BVP (1.2).From (H1) and (2.12), we get
(3.13)lim∥u∥α→+∞(Φ(u)-λΨ(u))=+∞,
for all  λ∈ [0,+∞[. Put
(3.14)u1(t)={4Γ(2-α)dTt,t∈[0,T4[,T4,Γ(2-α)d,t∈[T4,3T4],4Γ(2-α)dT(T-t),t∈3T4]T4,T].
It is easy to check that u1(0)=u1(T)=0 and u1∈L2[0,T]. The direct calculation shows
(3.15)0CDtαu1(t)={4dTt1-α,t∈[0,T4[,T4,4dT(t1-α-(t-T4)1-α),t∈[T4,3T4],4dT(t1-α-(t-T4)1-α-(t-3T4)1-α),t∈3T4]3T4,T],∥u1∥α2=∫0T‍(D0Ctαu1(t))2dt=∫0T/4‍+∫T/43T/4‍+∫3T/4T‍(D0Ctαu1(t))2dt=16d2T2[∫0T‍t2(1-α)dt+∫T/4T‍(t-T4)2(1-α)dt+∫3T/4T‍(t-3T4)2(1-α)dt-2∫T/4T‍t1-α(t-T4)1-αdt-2∫3T/4T‍t1-α(t-3T4)1-αdt+2∫3T/4T‍(t-T4)1-α(t-3T4)1-αdt]=16d2T2[(1+(34)3-2α+(14)3-2α)T3-2α3-2α-2∫T/4T‍t1-α(t-T4)1-αdt-2∫3T/4T‍t1-α(t-3T4)1-αdt+2∫3T/4T‍(t-T4)1-α(t-3T4)1-αdt]<∞.
That is, 0CDtαu1∈L2[0,T]. Thus, u1∈E0α. Moreover, the direct calculation shows
(3.16)tCDTαu1(t)={4dT((T-t)1-α-(3T4-t)1-α-(T4-t)1-α),t∈[0,T4[T4,4dT((T-t)1-α-(3T4-t)1-α),t∈[T4,3T4],4dT(T-t)1-α,t∈3T4]3T4,T],Φ(u1)=-12∫0T‍D0Ctαu1(t)·DtCTαu1(t)dt=-8d2T2[∫0T/4‍t1-α((T-t)1-α-(3T4-t)1-α-(T4-t)1-α)dt+∫T/43T/4‍(t1-α-(t-T4)1-α)((T-t)1-α-(3T4-t)1-α)dt+∫3T/4T‍(t1-α-(t-T4)1-α-(t-3T4)1-α)(T-t)1-αdt]=-8d2T2[∫0T‍t1-α(T-t)1-αdt-∫0T/4‍t1-α(T4-t)1-αdt+∫T/43T/4‍(t-T4)1-α(3T4-t)1-αdt-∫3T/4T‍(t-3T4)1-α(T-t)1-αdt-∫03T/4‍t1-α(3T4-t)1-α-∫T/4T‍(t-T4)1-α(T-t)1-αdt]=8Γ2(2-α)Γ(4-2α)T1-2αd2((1+33-2α)24α-5-22α-3-1)=A(α)d2,Ψ(u1)=∫0T‍a(t)F(u1(t))dt=∫0T/4‍a(t)F(4Γ(2-α)dTt)dt+∫T/43T/4‍a(t)F(Γ(2-α)d)dt+∫3T/4T‍a(t)F(4Γ(2-α)dT(T-t))dt=F(Γ(2-α)d)∫T/43T/4‍a(t)dt+T4Γ(2-α)d∫0Γ(2-α)d‍b(x)F(x)dx.Letr=(|cos(πα)|/2Ω2)c2. Since c<(2A(α)/|cos(πα)|)Ω·d, we obtain r<Φ(u1).By (2.12) and (3.7), one has Φ(u)≤r⇒∥u∥∞≤c. Thus,
(3.17)supu∈Φ-1(]-∞,r[)¯wΨ(u)=supu∈Φ-1(]-∞,r])Ψ(u)≤max|x|≤cF(x)∫0T‍a(t)dt.Moreover, we have(3.18)rr+Φ(u1)Ψ(u1)=(|cos(πα)|/2Ω2)c2(|cos(πα)|/2Ω2)c2+A(α)d2×[F(Γ(2-α)d)∫T/43T/4‍a(t)dt+T4Γ(2-α)d∫0Γ(2-α)d‍b(x)F(x)dx]=|cos(πα)|c2|cos(πα)|c2+2Ω2A(α)d2×[F(Γ(2-α)d)∫T/43T/4‍a(t)dt+T4Γ(2-α)d∫0Γ(2-α)d‍b(x)F(x)dx].Hence, from (H2) one has(3.19)supu∈Φ-1(]-∞,r[)¯wΨ(u)<rr+Φ(u1)Ψ(u1).Now, taking into account that(3.20)Φ(u1)Ψ(u1)-supu∈Φ-1(]-∞,r[)¯wΨ(u)≤A(α)d2ℜa+ℜ∫0Γ(2-α)d‍b(x)F(x)dx-∫0T‍a(t)dt·max|x|≤cF(x),rsupu∈Φ-1(]-∞,r[)¯wΨ(u)≥c2|cos(πα)|2Ω2∫0T‍a(t)dt·max|x|≤cF(x),hrr(Ψ(u1)/Φ(u1))-supu∈Φ-1(]-∞,r[)¯wΨ(u)≤hA(α)d2ℜa+ℜ∫02Γ(2-α)‍b(x)F(x)dx-(2Ω2A(α)d2/c2|cos(πα)|)∫0T‍a(t)dt·max|x|≤cF(x)=m.
Thus, by Theorem 2.9 it follows that, for each λ∈Λ1, BVP (1.2) admits at least three solutions, and there exists an open interval Λ2⊂[0,m] and a real positive number σ such that, for each λ∈Λ2, BVP (1.2) admits at least three solutions in E0α whose norms are less than σ.Finally, we give an example to show the effectiveness of the results obtained here.Letα=0.8, T=1, a(t)≡1,  and f(u)=e-uu8(9-u)+u. Then BVP (1.2) reduces to the following boundary value problem:
(3.21)ddt(12D0t-0.2(D0Ct0.8u(t))-12Dt1-0.2(DtC10.8u(t)))+λ(e-uu8(9-u)+u)=0,a.e.t∈[0,1],u(0)=u(1)=0.Example 3.2.
Owing to Theorem3.1, for each λ∈]0.291,0.318[, BVP (3.21)  admits at least three solutions. In fact, put c=1 and d=2, it is easy to calculate that Ω=1.1089, A(0.8)=1.3313, and
(3.22)2A(0.8)|cos(0.8π)|Ω·d=4.0235>1=c.
Since(3.23)F(x)=∫0x‍f(s)ds=e-xx9+23x3/2,
we have that condition (H1) holds. Moreover, F(x)≥0 for each x∈[0,2Γ(1.2)], and
(3.24)|cos(0.8π)||cos(0.8π)|+2Ω2A(0.8)·22[12F(2Γ(1.2))+14Γ(1.2)∫02Γ(1.2)‍F(s)ds]>1.064>1.0345=e-1+23≥F(x),|x|≤1,
which implies that condition (H2) holds. Thus, by Theorem 3.1, for each λ∈]0.291,0.318[, the problem (3.21) admits at least three nontrivial solutions in E00.8. Moreover, for each h>1, there exists an open interval Λ⊂]0,3.4674h[ and a real positive number σ such that, for each λ∈Λ, the problem (3.21) admits at least three solutions in E00.8 whose norms are less than σ.



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