Volume 2008, Article ID 437453,15pages doi:10.1155/2008/437453
Research Article
Positive Solutions for Boundary Value
Problems of
N
-Dimension Nonlinear Fractional
Differential System
Aijun Yang and Weigao Ge
Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China
Correspondence should be addressed to Aijun Yang,[email protected]
Received 27 October 2008; Accepted 18 December 2008
Recommended by Zhitao Zhang
We study the boundary value problem for a kindN-dimension nonlinear fractional differential system with the nonlinear terms involved in the fractional derivative explicitly. The fractional differential operator here is the standard Riemann-Liouville differentiation. By means of fixed point theorems, the existence and multiplicity results of positive solutions are received. Furthermore, two examples given here illustrate that the results are almost sharp.
Copyrightq2008 A. Yang and W. Ge. 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.
1. Introduction
We are interested in the followingN-dimension nonlinear fractional differential system:
Dα1
0x1t f1
t, x2t, Dμ01x2t
0,
.. .
DαN−1
0 xN−1t fN−1
t, xNt, DμN0−1xNt
0,
DαN0xNt fN
t, x1t, DμN0x1t
0,
0< t <1, 1.1
that is subject to the boundary conditions
x10 x20 · · ·xN0 0,
x11 x21 · · ·xN1 0,
whereD0αiis the standard Riemann-Liouville fractional derivative of orderαi,fi ∈C0,1×
R×R,R, 1< αi <2,μi>0,i1,2, . . . , N, andαi−μi−1>1,i1,2, . . . , N,μ0μN.
Recently, fractional differential equationsin short FDEs have been studied exten-sively. The motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, engineering, and so on. For an extensive collection of such results, we refer the readers to the monographs by Samko et al.1, Podlubny2, Miller and Ross3, and Kilbas et al.4.
Some basic theory for the initial value problems of FDE involving Riemann-Liouville differential operator has been discussed by Lakshmikantham 5–7, El-Sayed et al. 8, 9, Diethelm and Ford10, and Bai11, and so on. Also, there are some papers which deal with the existence and multiplicity of solutions for nonlinear FDE boundary value problems
in short BVPs by using techniques of topological degree theory. For example, Su 12 considered the BVP of the coupled system
Dαut ft, vt, Dμvt, 0< t <1,
Dβvt gt, ut, Dνut, 0< t <1,
u0 u1 v0 v1 0.
1.3
By using the Schauder fixed point theorem, one existence result was given. In13, Bai and L ¨u obtained positive solutions of the two-point BVP of FDE
Dα
0ut f
t, ut, 0< t <1, 1< α≤2,
u0 u1 0 1.4
by means of Krasnosel’skii fixed point theorem and Leggett-Williams fixed point theorem.
D0αis the standard Riemann-Liouville fractional derivative.
Zhang discussed the existence of solutions of the nonlinear FDE
cDα
0ut ft, ut, 0< t <1, 1< α≤2 1.5
with the boundary conditions
u0 ν /0, u1 ρ /0, 1.6
u0 u0 0, u1 u1 0, 1.7
in14,15, respectively. Since conditions 1.6 and1.7are nonzero boundary values, the Riemann-Liouville fractional derivativeD0αis not suitable. Therefore, the author investigated
the BVPs1.5-1.6and1.5–1.7by involving in the Caputo fractional derivativecDα
From above works, we can see a fact, although the BVPs of nonlinear FDE have been studied by some authors, to the best of our knowledge, higher-dimension fractional equation systems are seldom considered. Su in12studied the two-dimension system, however, the Schauder fixed point theorem cannot ensure the solutions to be positive. Since only positive solutions are useful for many applications, we investigate the existence and multiplicity of positive solutions for BVP 1.1-1.2in this paper. In addition, two examples are given to demonstrate our results.
2. Preliminaries
For the convenience of the reader, we first recall some definitions and fundamental facts of fractional calculus theory, which can be found in the recent literatures1–4.
Definition 2.1. The fractional integral of orderτ >0 of a functionf:0,∞ → Ris given by
I0τfx 1
Γτ
x
0
ft
x−t1−τdt, x >0, 2.1
provided that the integral exists, whereΓτis the Euler gamma function defined by
Γz
∞
0
tz−1e−tdt, z >0, 2.2
for which, the reduction formula
Γz1 zΓz, Γ1 1, Γ
1 2
√π, 2.3
the Dirichlet formula
1
0
tz−11−tω−1dt ΓzΓω
Γzω,
z, ω /∈Z−0
2.4
hold.
Definition 2.2. The fractional derivative of orderτ >0 of a continuous functionf:0,∞ → R
can be written as
D0τfx 1
Γn−τ
d dx
nx
0
ft
x−tτ1−ndt, n τ 1, 2.5
whereτdenotes the integer part ofτ, provided that the right side is pointwise defined on
Remark 2.3. The following properties are useful for our discussion:
I0τI0σft I0τσft, Dτ0I0τft ft, τ >0, σ >0, f ∈L0,1,
I0τD0τft ft c1tτ−1c2tτ−2· · ·cntτ−n, ci∈R, i1,2, . . . , n,
I0τ:C0,1−→C0,1, D0τf∈C0,1∩L0,1, τ >0, f ∈C0,1.
2.6
In the following, we present the useful lemmas which are fundamental in the proof of our main results.
Lemma 2.4see16. LetCbe a convex subset of a normed linear spaceEandUbe an open subset ofCwithp∗∈U. Then every compact continuous mapN:U → Chas at least one of the following two properties:
A1Nhas a fixed point;
A2there is anx∈∂Uwithx 1−λp∗λNx,for some0< λ <1.
Definition 2.5. The mapαis said to be a nonnegative continuous concave functional on a cone
Pof a real Banach spaceEprovided thatα:P → 0,∞is continuous and
αtx 1−ty≥tαx 1−tαy, 2.7
for allx, y∈P,andt∈0,1.
Let α and β be nonnegative continuous convex functionals on the cone P, ψ be a nonnegative continuous concave functional onP. Then for positive real numbersr > aand
L, one defines the following convex sets:
Pα, r;β, L {x∈P:αx< r, βx< L},
Pα, r;β, L {x∈P:αx≤r, βx≤L},
Pα, r;β, L;ψ, a {x∈P:αx< r, βx< L, ψx> a},
Pα, r;β, L;ψ, a {x∈P:αx≤r, βx≤L, ψx≥a}.
2.8
The assumptions below about the nonnegative continuous convex functionalsα,βwill be used as follows:
B1there existsM >0 such thatx ≤Mmax{αx, βx},for allx∈P;
B2Pα, r;β, L/∅,for allr >0, L >0.
Lemma 2.6 see 17. Let P be a cone in a real Banach space E,r2 ≥ d > b > r1 > 0, and
L2 ≥ L1 > 0. Assume thatαandβare nonnegative continuous convex functionals satisfying (B1)
y ∈ Pα, r1;β, L1and T : Pα, r2;β, L2 → Pα, r2;β, L2,is a completely continuous operator.
Suppose
C1{y∈Pα, d;β, L2;ψ, b:ψy> b}/ ∅,ψTy> b, fory∈Pα, d;β, L2;ψ, b;
C2αTy< r1,βTy< L1, for ally∈Pα, r1;β, L1;
C3ψTy> b,for ally∈Pα, d;β, L2;ψ, bwithαTy> d.
ThenT has at least three fixed pointsy1, y2, y3∈Pα, r2;β, L2with
y1∈P
α, r1;β, L1
,
y2∈
y∈Pα, r2;β, L2;ψ, b
:ψy> b,
y3∈P
α, r2;β, L2
\Pα, r2;β, L2;ψ, b∪Pα, r1;β, L1
.
2.9
3. Related lemmas
LetXX1×X2× · · · ×XNwith the norm
xmax xi Xi :i1,2, . . . , N
, forxx1, x2, . . . , xN
∈X, 3.1
whereXi{xi ∈C0,1:D0μi−1xi∈C0,1},i1,2, . . . , Nwith
xi Xi xi ∞ Dμi−1xi ∞, 3.2
where · ∞is the standard sup norm of the spaceC0,1. Throughout, we denoteμ0 μN
andxN1 x1. ThenXis a Banach spacesee12. Define the coneP ⊂Xby
P xx1, x2, . . . , xN
∈X:xit≥0, xi0 0, t∈0,1, i1,2, . . . , N
. 3.3
Lemma 3.1. Ifx∈P, thenxi∞≤1/Γ1μi−1Dμi−1x
i∞,i1,2, . . . , N.
Proof. Forx x1, x2, . . . , xN∈P, we have
xit I0μi−1D0μi−1xit
≤ 1
Γμi−1 t
0
Dμi−1x
is
t−s1−μi−1ds
≤ 1
Γ1μi−1
Dμi−1x
i ∞, i1,2, . . . , N.
3.4
That is,xi∞≤1/Γ1μi−1Dμi−1x
It is well known that the solution for the system BVP1.1-1.2is equivalent to the fixed point of the following integral system:
T1x2t
1
0
G1t, sf1
s, x2s, Dμ01x2s
ds,
.. .
TN−1xNt
1
0
GN−1t, sfN−1s, xNs, DμN0−1xNs
ds,
TNx1t
1
0
GNt, sfN
s, x1s, DμN0x1s
ds,
0< t <1, 3.5
forx∈X, where
Git, s 1
Γαi
⎧ ⎨ ⎩
t1−sαi−1−t−sαi−1, 0≤s≤t≤1,
t1−sαi−1, 0≤t≤s≤1. 3.6
DenoteTx: T1x2, . . . , TN−1xN, TNx1, we can see
Tixi1t tαi−1I0αifi
1, xi11, Dμixi11
−I0αifi
t, xi1t, Dμixi1t
, 3.7
i1,2, . . . , N. For the Green functionsGit, s,i1,2, . . . , N, we can obtain
iGit, s≥0,fort, s ∈0,1,γisGis, s≤Git, s≤Gis, s,fort, s∈θ,1−θ×
0,1,θ∈0,1/2, where
γis
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
1−θ1−sαi−1−1−θ−sαi−1
s1−sαi−1 , 0< s≤ri,
θαi−1
sαi−1, ri≤s <1,
3.8
here,ri∈θ,1−θis the unique solution of the equation
1−θ1−sαi−1−1−θ−sαi−1θ1−sαi−1; 3.9
iimaxt∈0,101Git, sds αi−1
αi−1/ααi
i Γαi1 :ρi1and mint∈θ,1−θ
1
0Git, sds
θ1−θαi−1/Γαi1 :ρi2.
Proof. We divide the proof into three steps.
Step 1. T :P → P. In fact, for anyx ∈P, sincefit, xi1t, Dμi0xi1t ≥0 fort ∈0,1and
Git, s ≥ 0,fort, s ∈ 0,1,Tixi1t ≥ 0,fort ∈ 0,1. Moreover,G0, s 0 implies that
Tixi10 0.
Step 2. T is continuous onP, which is valid due to the continuity of the functionf.
Step 3. We will show thatT is relatively compact. For any given bounded setU ⊂ P, there existsM >0 such thatx ≤M,for allx∈U. We takeκimax{|fit, u, v|:t∈0,1, |u| ≤
M, |v| ≤M}.Forx∈U, lett1, t2∈0,1be such thatt1< t2, we have
Tixi1
t1
−Tixi1
t2tαi1−1−tαi2−1
I0αifi
1, xi11, D0μixi11
−I0αifi
t1, xi1
t1
, D0μixi1
t1
−I0αifi
t2, xi1
t2
, D0μixi1
t2
≤tαi1−1−tαi2−1 1
Γαi
1
0
1−sαi−1fi
s, xi1s, D0μixi1s
ds
1
Γαi
t2
0
t2−s
αi−1f i
s, xi1s, D0μixi1s
ds
− 1
Γαi
t1
0
t1−s
αi−1f
i
s, xi1s, D0μixi1s
ds
≤ κi
Γαi1
tαi1−1−tαi2−1
κi
Γαi
t2
t1
t2−s
αi−1dst1
0
t2−sαi−1−t 1−s
αi−1ds
κi
Γαi1
tαi2−1−tαi1−1tαi2 −tαi1−→0, ast2−t1−→0.
3.10 Notice that
Dμi−1
0 Tixi1t I
αi
0fi
1, xi11, Dμixi11
·Dμi−1
0 tαi−1−I
αi−μi−1
0 fi
t, xi1t, Dμi−1xi1t
,
3.11 one gets
Dμi−1
0 Tixi1
t1
−Dμi−1
0 Tixi1
t2
I0αifi
1, xi11, Dμixi11
Dμi−1
0 t
αi−1 1 −D
μi−1
0 t
αi−1 2
−Iαi−μi−1
0 fi
t1, xi1
t1
, Dμixi1
t1
−Iαi−μi−1
0 fi
t2, xi1
t2
, Dμixi1
t2
≤ κi
αiΓ
αi−μi−1
tαi−μi−1−1
1 −t
αi−μi−1−1
2
κi
Γαi−μi−1
t2
t1
t2−s
αi−μi−1−1ds
t1
0
t2−s
αi−μi−1−1−t
1−s
αi−μi−1−1ds
κi
αiΓ
αi−μi−1
tαi−μi−1−1
2 −t
αi−μi−1−1
1
κi
Γαi−μi−11
tαi−μi−1
2 −t
αi−μi−1
1
−→0, ast2−t1−→0,
3.12
wherei1,2, . . . , N, we can see thatTUis an equicontinuous set. Now, we proof thatT is uniformly bounded. For anyx∈U,
Tixi1ttαi−1Iαi
0fi
1, xi11, D0μixi11
−Iαi0fi
t, xi1t, D0μixi1t
≤ 1
Γαi
1
0
1−sαi−1fi
s, xi1s, D0μixi1s
ds
1
Γαi
t
0
t−sαi−1fi
s, xi1s, D0μixi1s
ds
≤ 2κi
Γαi1
<∞,
Dμi−1
0 Tixi1tI0αifi
1, xi11, Dμixi11
Dμi−1
0 tαi−1−I0αi−μi−1fi
t, xi1t, Dμixi1t
≤ κi
αiΓ
αi
Γαi
Γαi−μi−1
κi
Γαi−μi−1 t
0
t−sαi−μi−1−1ds
≤ κi
2αi−μi−1
αiΓ
αi−μi−11 <∞,
3.13
wherei 1,2, . . . , N. That is,TUis uniformly bounded. Thus, T is relatively compact. By means of the Arzela-Ascoli theorem,T :P → Pis completely continuous.
4. The existence of one positive solution
Theorem 4.1. If there existai, bi, ci∈C0,1,R,i1,2, . . . , Nsatisfying
bi
∞ ci ∞<min
ααii Γαi1
αi−1
αi−1,
αiΓ
αi−μi−11
2αi−μi−1
, 4.1
such that
fit, x, y≤ait bitxcity. 4.2
Proof. Lemma 3.2indicates thatT :P → Pis completely continuous.
Fori1,2, . . . , N, let
Qi >max
αi−1
αi−1 a
i ∞
ααii Γαi1
−αi−1
αi−1 b
i ∞ ci ∞
,
2αi−μi−1 ai ∞
αiΓ
αi−μi−11−2αi−μi−1 bi ∞ ci ∞
,
QmaxQi:i1,2, . . . , N
.
4.3
DefineΩ {x x1, x2, . . . , xN∈P :xiXi < Qi, i1,2, . . . , N}, thenx< Q. For∀x∈∂Ω,
xiXi Qi. Thus,xi∞≤QiandD0μi−1xi∞≤Qi:
Tixi1t1 0
Git, sfi
s, xi1s, D0μixi1s
ds
≤ 1
0
Git, s
ais bisxi1s cisDμi0xi1s
ds
≤ ai ∞ bi ∞ ci ∞
Qi
αi−1
αi−1
ααii Γαi1
< Qi,
Dμi−1
0 Tixi1tI0αifi
1, xi11, Dμixi11
Dμi−1
0 tαi−1−I0αi−μi−1fi
t, xi1t, Dμixi1t
≤ 1
Γαi
1
0
1−sαi−1fi
s, xi1s, D0μixi1s
ds· Γ
αi
Γαi−μi−1t
αi−μi−1−1
1
Γαi−μi−1
t
0
t−sαi−μi−1−1f
i
s, xi1s, D0μixi1s
ds
≤ ai ∞ bi ∞ ci ∞
Qi
αiΓ
αi−μi−1
ai
∞ bi ∞ ci ∞
Qi
Γαi−μi−11
ai ∞ bi ∞ ci ∞
Qi
2αi−μi−1
αiΓ
αi−μi−11 < Qi
4.4
indicate thatTixi1Xi < Qi, and thenTx max{Tixi1Xi : i 1,2, . . . , N} < Q. Take
p∗ 0 inLemma 2.4, for anyx∈∂Ω,xλTx0< λ <1does not hold. Hence, the operator
Thas at least a fixed point, then the BVP1.1-1.2has at least one positive solution.
Example 4.2. Consider the problem
D50/3x1t f1
t, x2t, D01/4x2t
0, 0< t <1,
D30/2x2t f2
t, x1t, D01/3x1t
0, 0< t <1,
x10 x11 x20 x21 0,
where
f1t, u, v
10 9 Γ 2 3 −1 9Γ 1 3 1 2
√
π
Γ1/4
t1
9Γ
1 3
1 12
√
π
5Γ1/4 t2 1 9Γ 1 3 √
tu1
9Γ
1 3
t3/4v,
f2t, u, v
3 4
√
π−1
2
1Γ2/3
Γ1/3
t 1
4
25Γ2/3 2Γ1/3
t21
2t
1/3u 1 4t
2/3v,
α1
5
3, α2 3
2, μ1 1
4, μ2 1 3.
4.6
Choose
a1t
10 9 Γ 2 3 1 9Γ 1 3 1 12
√
π
5Γ1/4
t2, b
1t 1 9Γ 1 3 √
t, c1t
1 9Γ 1 3
t3/4,
a2t 3
4 √
π1
4
25Γ2/3 2Γ1/3
t2, b
2t 1 2t
1/3, c 2t 1
4t 2/3.
4.7
It is easy to check that4.1holds. Thus, byTheorem 4.1, the BVP4.5has at least one positive solution. In fact,xt t3/21−t, t1/21−tis such a solution.
5. The existence of triple positive solutions
Let the nonnegative continuous convex functionals α, β and the nonnegative continuous concave functionalψbe defined on the coneP by
αx max xi ∞:i1,2, . . . , N
,
βx max Dμi−1
0 xi ∞:i1,2, . . . , N
,
ψx min min
θ≤t≤1−θ
xit:i1,2, . . . , N
.
5.1
Obviously,αandβsatisfyB1andB2,ψx≤αx,for allx∈P. For simplicity, we denote
ρi3:
1−θ
θ
γisGis, sds, ρi4:
2αi−μi−1
αiΓ
αi−μi−11,
σ:max
1
Γ1μi
:i1,2, . . . , N
.
Theorem 5.1. Assume that there exist constantsσL≥b/θ > b > σl >0such thatbΓμi1≤θL,
fori1,2, . . . , N. Suppose
H1fit, u, v≤min{σL/ρi1, L/ρi4},t, u, v∈0,1×0, σL×−L, L;
H2fit, u, v> b/ρi2,t, u, v∈0,1×b, b/θ×−L, L;
H3fit, u, v<min{σl/ρi1, l/ρi4},t, u, v∈0,1×0, σl×−l, l;
H4fit, u, v> b/ρi3,t, u, v∈θ,1−θ×b, σL×−L, L.
Then the BVP1.1-1.2has at least three positive solutionsx x1, x2, . . . , xN,y y1, y2, . . . ,
yN,andz z1, z2, . . . , zNsuch that
0≤xit≤σl, 0≤yit≤σL, σl≤zit≤σL, t∈0,1,
Dμi−1
0 xi ∞≤l, D
μi−1
0 yi ∞≤L, −l≤D
μi−1
0 zit≤L, t∈0,1,
yit> b, zit≤b, t∈θ,1−θ, fori1,2, . . . , N.
5.3
Proof. Lemma 3.2has showed thatT :P → Pis completely continuous. Now, we will verify that all the conditions ofLemma 2.6are satisfied. The proof is based on the following steps.
Step 1. We will show thatH1impliesT :Pα, σL;β, L → Pα, σL;β, L.
In fact, forx∈Pα, σL;β, L,αx≤σL,βx≤L, and thenxi∞≤σL,D0μi−1xi∞≤L,
i1,2, . . . , N. In view ofH1, we have
Tixi1 ∞max 0≤t≤1
1
0
Git, sfi
s, xi1s, D0μixi1s
ds
≤ max
t,u,v∈0,1×0,σL×−L,Lfit, u, v·max0≤t≤1 1
0
Git, sds
≤ σL
ρi1 ·
ρi1σL,
Dμi−1T
ixi1 ∞max 0≤t≤1I
αi
0fi
1, xi11, Dμixi11
·Dμi−1
0 tαi−1−I0αi−μi−1fi
t, xi1t, Dμixi1t
≤ max
t,u,v∈0,1×0,σL×−L,Lft, u, v
·max 0≤t≤1
1
Γαi
1
0
1−sαi−1ds Γ
αi
Γαi−μi−1t
αi−μi−1−1
1
Γαi−μi−1 t
0
t−sαi−μi−1−1ds
≤ L
ρi4 ·
ρi4 L.
5.4
Step 2. To check the conditionC1inLemma 2.6, we choosex∗t b/θtμN,b/θtμ1, . . . ,
b/θtμN−1,t∈0,1. It is easy to see that
αx∗max
max
t∈0,1
θbtμi:i1,2, . . . , N
b
θ,
βx∗max
max
t∈0,1
θbDμi0tμi:i1,2, . . . , N
max
b
θΓ
1μi
:i1,2, . . . , N
≤L,
ψx∗min
min
t∈θ,1−θ
bθtμi:i1,2, . . . , N
min
b
θθ
μi :i1,2, . . . , N
> b.
5.5
Consequently,{x∈Pα, b/θ;β, L;ψ, b:ψx> b}/∅. For anyx∈Pα, b/θ;β, L;ψ, b, from
H2, one gets
min
t∈θ,1−θ
Tixi1t min
t∈θ,1−θ
1
0
Git, sfi
s, xi1s, D0μixs
ds
≥ min
t,u,v∈0,1×b,b/θ×−L,Lfit, u, v·t∈minθ,1−θ
1
0
Git, sds
> b
ρi2 ·
ρi2b,
5.6
then we can obtainψTx> b.
Step 3. It is similar toStep 1that we can proveT :Pα, σl;β, l → Pα, σl;β, lby condition
H3, that is,C2inLemma 2.6holds.
Step 4. We verify thatC3inLemma 2.6is satisfied. Forx∈Pα, σL;β, L;ψ, bwithαTx> b/θ, we have
min
t∈θ,1−θTixi1t≥
1
0
γisGis, sfi
s, xi1s, D0μixi1s
ds
≥ 1−θ
θ
γisGis, sds· min
t,u,v∈θ,1−θ×b,σL×−L,Lfit, u, v
> ρi3·
b
ρi3
b.
5.7
Thus,ψTx> b,C3inLemma 2.6is satisfied.
Therefore, the operatorThas three pointsx, y, z∈Pα, σL;β, Lwith
x∈Pα, σl;β, l, y∈Pα, σL;β, L;ψ, b,
z∈Pα, σL;β, L\Pα, σL;β, L;ψ, b∪Pα, σl;β, l.
Then the BVP1.1-1.2has three positive solutionsx, y, z∈Pα, σL;β, Lsuch that
0≤xit≤σl, 0≤yit≤σL, σl≤zit≤σL, t∈0,1,
Dμi−1
0 xi ∞≤l, D0μi−1yi ∞≤L, −l≤Dμi0−1zit≤L, t∈0,1,
yit> b, zit≤b, t∈θ,1−θ, fori1,2, . . . , N.
5.9
Example 5.2. Consider the problem
D30/2x1t f1
t, x2t, D01/2x2t
0, 0< t <1,
D70/4x2t f2
t, x1t, D01/4x1t
0, 0< t <1,
x10 x11 x20 x21 0,
5.10
where
f1t, u, v
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 2 t u2 102 |v|
106, u∈
0,56
25 , 1 2 t 213749 24890 u 2 |v|
106 3136 62500− 3136 625 · 213749 24890 , u∈
56 25,3
, 1 2 t 1070313 31250 |v|
106, u∈3,∞,
f2t, u, v
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 5 t u2 103 v2
1010, u∈
0,56
25 , 1 5 t 14740614 2489000 u 2 v2
1010 3136 625000− 3136 625 · 14740614 2489000, u∈
56 25,3
, 1 5 t 2359 100 v2
1010, u∈3,∞.
5.11
Here, we haveα1 3/2,α27/4,μ11/2,μ21/4. By choosingθ1/4 and the definition ofσandρij,i1,2,j1,2,3,4, one gets
σmax
1
Γ11/2, 1
Γ11/4
Γ 1
11/2 1
1/2√π ≈1.12,
ρ11
3/2−13/2−1
3/23/2Γ3/21 8
ρ12
1/41−1/43/2−1
Γ3/21
1
2√3π ≈0.16,
ρ13
3/4
1/4
γ1sG1s, sds
2 √
π
3/4
1/4 1 2
√
1−sds √2 π
1−√ 3/6
1/4
3
4−sds≈0.12,
ρ14
3−1/4
3/2Γ3/2−1/41 88
15Γ1/4 ≈1.61,
ρ21≈0.18, ρ22≈0.12, ρ23≈0.06, ρ24≈1.53.
5.12
Takingl2,b3,andL1000, we have
f1t, u, v≤min
σL
ρ11
, L
ρ14
≈621.11, fort, u, v∈0,1×0,1120×−1000,1000,
f1t, u, v> b
ρ13 ≈
16.67, fort, u, v∈
1 4,
3 4
×3,1120×−1000,1000,
f1t, u, v<min
σl
ρ11
, l
ρ14
≈1.24, fort, u, v∈0,1×
0,56
25
×−3,3,
f1t, u, v>
b
ρ12 ≈
18.75, fort, u, v∈0,1×3,12×−1000,1000,
5.13
that is,f1 satisfies the conditionsH1–H4ofTheorem 5.1. Similarly, we can show thatf2 satisfiesH1–H4. Thus, byTheorem 5.1, the BVP5.10has at least three positive solutions
x x1, x2,y y1, y2,andz z1, z2such that
0≤xit≤2.24, 0≤yit≤1120, 2.24≤zit≤1120, t∈0,1, i1,2,
D1/4
0 x1 ∞≤2, D01/2x2 ∞≤2, D01/4y1 ∞≤1000, D01/2y2 ∞≤1000, −2≤D10/4z1t≤1000, −2≤D01/2z2t≤1000, t∈0,1,
yit>3, zit≤3, t∈
1 4,
3 4
, i1,2.
5.14
Remark 5.3. The particular caseN 2 has been studied by 12 for the existence of one solution, our paper generalizes12 for the obtaining of one and three positive solutions. ForN1, we develop13–15by the nonlinear termsfiinvolved in theμi-order
Acknowledgments
This work is supported by National Natural Science Foundation of ChinaNNSF 10671012 and the Specialized Research Fund for the Doctoral Program of Higher EducationSRFDP of China20050007011.
References
1 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, NY, USA, 1993.
2 I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
3 K. S. Miller and B. Ross,An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
4 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,Theory and Applications of Fractional Differential Equa-tions, vol. 204 ofNorth-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
5 V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.
6 V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3337–3343, 2008.
7 V. Lakshmikantham and A. S. Vatsala, “General uniqueness and monotone iterative technique for fractional differential equations,”Applied Mathematics Letters, vol. 21, no. 8, pp. 828–834, 2008. 8 A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “On the fractional-order logistic
equation,”Applied Mathematics Letters, vol. 20, no. 7, pp. 817–823, 2007.
9 A. M. A. El-Sayed and E. M. El-Maghrabi, “Stability of a monotonic solution of a non-autonomous multidimensional delay differential equation of arbitrary fractional order,” Electronic Journal of Qualitative Theory of Differential Equations, no. 16, pp. 1–9, 2008.
10 K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,”Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.
11 C. Bai, “Positive solutions for nonlinear fractional differential equations with coefficient that changes sign,”Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 4, pp. 677–685, 2006.
12 X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,”
Applied Mathematics Letters, vol. 22, no. 1, pp. 64–69, 2009.
13 Z. Bai and H. L ¨u, “Positive solutions for boundary value problem of nonlinear fractional differential equation,”Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005. 14 S. Zhang, “Existence of solution for a boundary value problem of fractional order,”Acta Mathematica
Scientia, vol. 26, no. 2, pp. 220–228, 2006.
15 S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,”Electronic Journal of Differential Equations, vol. 2006, no. 36, pp. 1–12, 2006.
16 J. Mawhin,Topological Degree Methods in Nonlinear Boundary Value Problems, vol. 40 ofCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1979.
