Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 294694, 21 pages
doi:10.1155/2012/294694
Research Article
Uniqueness and Asymptotic Behavior of
Positive Solutions for a Fractional-Order Integral
Boundary Value Problem
Min Jia, Xin Liu, and Xuemai Gu
Communication Research Center, Harbin Institute of Technology, Harbin 150080, China
Correspondence should be addressed to Min Jia, jiamin@hit.edu.cn
Received 18 July 2012; Accepted 7 August 2012
Academic Editor: Xinguang Zhang
Copyright q 2012 Min Jia et al. 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 a model arising from porous media, electromagnetic, and signal processing of wireless
communication system −Dt α xt ft, xt, x t, x t, . . . , xn−2 t, 0 < t < 1, x0 x 0 · · · 1
xn−2 0 0, xn−2 1 0 xn−2 sdAs, where n − 1 < α ≤ n, n ∈ N and n ≥ 2, Dt α is the
1
standard Riemann-Liouville derivative, 0 xsdAs is linear functionals given by RiemannStieltjes integrals, A is a function of bounded variation, and dA can be a changing-sign measure.
The existence, uniqueness, and asymptotic behavior of positive solutions to the singular nonlocal
integral boundary value problem for fractional differential equation are obtained. Our analysis
relies on Schauder’s fixed-point theorem and upper and lower solution method.
1. Introduction
Recently, fractional-order models have proved to be more accurate than integer order
models, that is, there are more degrees of freedom in the fractional-order models.Indeed,
we can find numerous applications in viscoelasticity, electrochemistry control, porous media,
electromagnetic, and signal processing of wireless communication system. Especially, in
application of the digital signal processing, the fractional digital signal processing can
greatly improve the high frequency components of signal, enhance an intermediate frequency
component of signal, and reserve nonlinear low frequency signal. According to this analysis,
fractional differential equation applied to the edge information extraction will get higher
signal to noise ratio than that of the traditional method based on one- or-two order differential
equation.
2
Abstract and Applied Analysis
Many applications on digital signal processing were found recently, for example,
Hartley et al. 1 derived a closed-loop control electromagnetic and signal processing system
of equations with the Chua resistor
b
4xt − x3 t ,
Dt α xt aDt α−1 yt − xt −
7
dyt
xt − yt.
dt
1.1
Caputo 2 presented a fractional model system to describe the relationship between the
electric field E and the current density D in the study of a electrochemical polarization
medium,
γDt ν Dt αDt σEt δDt ν Et,
1.2
where γ, α, σ, δ are constants and ν is a real number. Anastasio 3 believed that signal
processing of communication system of vestibule visual reflex effect is fractional, which can
be described by the following model
τ1 r t rt τ1 τ2 Dt α1 vt τ1 Dt α vt,
1.3
where rt is the vestibular reflex nerve discharge rate, vt is the head of the rotational
angular velocity, α is fraction.
Motivated by systems 1.1–1.3 and their application background in electromagnetic
and signal processing of wireless communication system, in this paper, we consider the
existence, uniqueness, and asymptotic behavior of positive solutions for the higher nonlocal
fractional differential equation
−Dt α xt f t, xt, x t, x t, . . . , xn−2 t ,
x0 x 0 · · · xn−2 0 0,
xn−2 1 1
0 < t < 1,
xn−2 sdAs,
1.4
0
n − 1 < α ≤ n, n ∈ N and n ≥ 2, Dt α is the standard Riemann-Liouville derivative,
where
1
xsdAs is linear functionals given by Riemann-Stieltjes integrals, A is a function of
0
bounded variation, and dA can be a changing-sign measure, and f : 0, 1 × 0, ∞n−1 →
0, ∞ is continuous, f may be singular at xi 0 and t 0, 1.
The nonlocal integral-boundary value problems represent a class of interesting and
important problems arising in physical, biological, and chemical processes and have attracted
the attention of Khan 4, Gallardo 5, Karakostas and Tsamatos 6, Ahmad et al. 7, Feng
et al. 8, Corduneanu 9, and Agarwal and O’Regan 10.
Abstract and Applied Analysis
3
When α is an integer, a lot of work has been done dealing with nonlocal 3-point
boundary value problems, see 11, 12. In 12, Eloe and Ahmad studied the following nthorder nonlocal differential equation
un t atfu 0,
0 < t < 1,
u0 u 0 · · · un−2 0 0,
u1 ηuξ,
1.5
where 0 < ξ < 1, 0 < ηξn−1 < 1. By applying the fixed point theorem in cones, the authors
prove the existence of at least one positive solution when a and f is continuous and f is either
sublinear or superlinear. Recently, Hao et al. 13 studied the existence of positive solutions
for the BVP 1.5 with integer order n, and nonlinear term is replaced by atft, xt, where
a can be singular at t 0, 1, f can be singular at x 0 and no singularity at t 0, 1.
If α is fractional, Yuan 14 studied the n − 1, 1-type conjugate boundary value
problem
Dt α ut ft, ut 0,
0 < t < 1, n − 1 < α ≤ n, n ≥ 3,
u0 u 0 · · · un−2 0 0,
u1 0,
1.6
where f is continuous and semipositone, Dt α is the standard Riemann-Liouville derivative.
By giving properties of Green’s function and using the Guo-Krasnosel’skii fixed-point
theorem on cones, the existence of multiple positive solutions were obtained. More recently,
Zhang 15 considered the following BVP whose nonlinear term and boundary condition
contain derivatives of unknown functions
Dt α xt qtf x, x , . . . , xn−2 0,
0 < t < 1, n − 1 < α ≤ n,
x0 x 0 · · · xn−2 0 xn−2 1 0,
1.7
where Dt α is the standard Riemann-Liouville fractional derivative of order α, q may be
singular at t 0 and f may be singular at x 0, x 0, . . . , xn−2 0, by using fixed-point
theorem of the mixed monotone operator, the unique existence result of positive solution to
problem 1.7 was established. Other some recent results, see 16–20
Motivated by the results mentioned above, in this paper, we study the existence,
uniqueness, and asymptotic behavior of positive solutions for the BVP 1.4 where the
nonlinear terms and boundary conditions all involve derivatives of unknown functions and
with Riemann-Stieltjes integral boundary condition, moreover f may be singular at xi 0 and
t 0, 1. Our main tool relies on Schauder’s fixed-point theorem and upper and lower solution
method.
2. Preliminaries and Lemmas
In this section, we present here the necessary definitions from fractional calculus theory.
These definitions can be found in the recent literatures.
4
Abstract and Applied Analysis
Definition 2.1 see 21, 22. The Riemann-Liouville fractional integral of order α > 0 of a function x : 0, ∞ → R is given by the following:
I α xt 1
Γα
t
2.1
t − sα−1 xsds
0
provided that the right-hand side is pointwise defined on 0, ∞.
Definition 2.2 see 21, 22. The Riemann-Liouville fractional derivative of order α > 0 of a
function x : 0, ∞ → R is given by the following:
n t
d
1
Dt xt t − sn−α−1 xsds,
Γn − α dt
0
α
2.2
where n α 1, α denotes the integer part of number α, provided that the right-hand side
is pointwise defined on 0, ∞.
Proposition 2.3 see 21, 22. (1) If x ∈ L1 0, 1, ν > σ > 0, then
I ν I σ xt I νσ xt,
Dt σ I ν xt I ν−σ xt,
Dt σ I σ xt xt.
2.3
(2) If α > 0, σ > 0, then
Dt α tσ−1 Γσ σ−α−1
t
.
Γσ − α
2.4
Proposition 2.4 see 21, 22. Let α > 0, and fx is integrable, then
I α Dt α fx fx c1 xα−1 c2 xα−2 · · · cn xα−n ,
2.5
where ci ∈ R i 1, 2, . . . , n, n is the smallest integer greater than or equal to α.
Our discussion is based on the assumption −1 < α − n ≤ 0 in this paper.
Let xt I n−2 yt, yt ∈ C0, 1, by standard discuss, we easily reduce order the BVP
1.4 to the following equivalent BVP
−Dt α−n2 yt f t, I n−2 yt, I n−3 yt, . . . , I 1 yt, yt ,
y0 0,
y1 2.6
1
ysdAs.
0
Abstract and Applied Analysis
5
In fact, let xt I n−2 yt, yt ∈ C0, 1, then
Dt α xt dn n−α
dn
I xt n I n−α I n−2 yt
n
dt
dt
dn 2n−α−2
I
yt Dt α−n2 yt,
dtn
x t Dt 1 I n−2 yt I n−3 yt,
2.7
x t Dt 2 I n−2 yt I n−4 yt,
..
.
xn−3 t Dt n−3 I n−2 yt I 1 yt,
xn−2 t Dt n−2 I n−2 yt yt.
By 2.7, we have xn−2 0 y0 0 and
y1 1
2.8
ysdAs.
0
Moreover −Dt α−n2 yt ft, I n−2 yt, I n−3 yt, . . . , I 1 yt, yt. Thus, 1.4 is transformed
into 2.6.
On the other hand, if y ∈ C0, 1, 0, ∞ is a solution for problem 2.6. Then, from
Proposition 2.3 and 2.7, one has
−Dt α xt −Dt α−n2 yt f t, I n−2 yt, I n−3 yt, . . . , I 1 yt, yt
f t, xt, x t, x t, . . . , xn−2 t ,
2.9
0 < t < 1.
Notice for any α > 0,
I α yt 1
Γα
t
t − sα−1 ysds,
2.10
0
which implies that I α y0 0. So x0 0, and from 2.7, for i 1, 2, . . . , n − 2, we have
xi 0 0, and
x
n−2
1 1
xn−2 sdAs.
2.11
0
Consequently, the BVP 2.6 is transformed into the BVP 1.4.
Applying Propositions 2.3 and 2.4, by standard discuss, we have the following Lemma.
6
Abstract and Applied Analysis
Lemma 2.5. Given y ∈ L1 0, 1, then the problem
Dt α−n2 yt ht 0,
y0 0,
0 < t < 1,
y1 0,
2.12
has the unique solution
yt 1
2.13
Gt, shsds,
0
where Gt, s is the Green function of the BVP 2.12 and is given by the following:
⎧ α−n1
t
1 − sα−n1 − t − sα−n1
⎪
⎪
,
⎪
⎨
Γα − n 2
Gt, s ⎪
⎪ tα−n1 1 − sα−n1
⎪
⎩
,
Γα − n 2
0 ≤ s ≤ t ≤ 1,
2.14
0 ≤ t ≤ s ≤ 1.
By Proposition 2.4, the unique solution of the problem
Dt α−n2 yt 0,
y0 0,
0 < t < 1,
y1 1,
2.15
is tα−n1 . Let
C
1
tα−n1 dAt,
2.16
0
and define
GA s 1
Gt, sdAt.
2.17
0
Then the Green function for the nonlocal BVP 2.6 is the detail see 23 or 16
Kt, s tα−n1
GA s Gt, s.
1−C
Throughout paper, we always assume the following holds.
2.18
Abstract and Applied Analysis
7
H0A is a function of bounded variation such that GA s ≥ 0 for s ∈ 0, 1 and 0 ≤ C <
1, where C is defined by 2.16.
Lemma 2.6. Suppose H0 holds, then the Green function defined by 2.18 satisfies
1 Kt, s > 0, for all t, s ∈ 0, 1.
2
tα−n1
GA s ≤ Kt, s ≤ Hstα−n1 ,
1−C
2.19
where
Hs 1 − sα−n1 GA s
.
Γα − n 2 1 − C
2.20
Proof. 1 is obvious. For 2, by 2.14 and 2.18, we have
Ht, s tα−n1
tα−n1 1 − sα−n1 tα−n1
GA s Gt, s ≤
GA s
1−C
Γα − n 2
1−C
2.21
Hstα−n1 ,
Ht, s tα−n1
tα−n1
GA s Gt, s ≥
GA s.
1−C
1−C
Definition 2.7. A continuous function ψt is called a lower solution of the BVP 2.6, if it
satisfies
−Dt α−n2 ψtt ≤ f t, I n−2 ψt, I n−3 ψt, . . . , I 1 ψt, ψt ,
ψ0 ≥ 0,
ψ1 ≥
2.22
1
ψsdAs.
0
Definition 2.8. A continuous function φt is called a upper solution of the BVP 2.6, if it
satisfies
−Dt α−n2 φtt ≥ f t, I n−2 φt, I n−3 φt, . . . , I 1 φt, φt ,
φ0 ≤ 0,
φ1 ≤
2.23
1
φsdAs.
0
It follows from Lemma 2.5, we have the following maximum principle.
8
Abstract and Applied Analysis
Lemma 2.9 maximum principle. If y ∈ C0, 1, R satisfies
y0 0,
y1 1
ysdAs,
2.24
0
and Dt α−n2 yt ≤ 0 for any t ∈ 0, 1. Then
yt ≥ 0,
t ∈ 0, 1.
2.25
3. Main Results
Let E C0, 1, and
P y ∈ E : there exist positive numbers 0 < ly < 1, Ly > 1
such that ly tα−n1 ≤ yt ≤ Ly tα−n1 , t ∈ 0, 1 }
3.1
Clearly, tα−n1 ∈ P , so P is nonempty. For any y ∈ P , define an operator T by the following:
T y t 1
Kt, sf s, I n−2 ys, I n−3 ys, . . . , I 1 ys, ys ds.
3.2
0
Let
κi t Γα − n 2 α−i−1
t
,
Γα − i
i 0, 1, 2, . . . , n − 2,
3.3
then
κ0 t I n−2 sα−n1 , κ1 t I n−3 sα−n1 , . . . , κn−3 t I 1 sα−n1 , κn−2 t tα−n1 .
3.4
The conditions imposed on f are the following.
H1 f ∈ C0, 1 × 0, ∞n−1 , 0, ∞, and ft, x0 , x1 , x2 , . . . , xn−2 is decreasing in
xi > 0 for i 0, 1, 2, . . . , n − 2;
≡ 0, t ∈ 0, 1, and
H2 for any λi > 0, ft, λ0 tn−2 , λ1 tn−3 , . . . , λn−3 t, λn−2 /
1
0<
0
Hsfs, λ0 κ0 s, λ1 κ1 s, λ2 κ2 s, . . . , λn−2 κn−2 sds < ∞.
3.5
Abstract and Applied Analysis
9
Lemma 3.1. Suppose H0–H2 hold, then T is well defined and T P ⊂ P .
Proof. For any y ∈ P , by the definition of P , there exists two positive numbers 0 < ly < 1, Ly >
1 such that ly tα−n1 ≤ yt ≤ Ly tα−n1 for any t ∈ 0, 1. It follows from 2.19 and H1-H2
that
T y t 1
Kt, sf s, I n−2 ys, I n−3 ys, . . . , I 1 ys, ys ds
0
≤
1
Hsf s, I n−2 ly sα−n1 , . . . , I 1 ly sα−n1 , ly sα−n1 ds
3.6
0
1
Hsf s, ly κ0 s, ly κ1 s, . . . , ly κn−2 s ds < ∞.
0
Now take C maxt∈0,1 yt, then for any s ∈ 0, 1,
GA sf s,
C
C
C
sn−2 ,
sn−3 , . . . , s, C ≡
/ 0,
1!
n − 2!
n − 3!
3.7
by H2. Thus by the continuity of ft, x0 , x1 , . . . , xn−2 , we have
1
GA sf s,
0
C
C
C
sn−2 ,
sn−3 , . . . , s, C ds > 0.
1!
n − 2!
n − 3!
3.8
This yields
1
GA sf s, I n−2 C, I n−3 C, . . . , I 1 C, C ds
0
1
GA sf s,
0
C
C
C
sn−2 ,
sn−3 , . . . , s, C ds > 0.
1!
n − 2!
n − 3!
3.9
By 2.19 and 3.6–3.9, we have
T y t 1
Kt, sf s, I n−2 ys, I n−3 ys, . . . , I 1 ys, ys ds
0
tα−n1
≥
1−C
1
GA sf s, I
n−2
C, I
n−3
3.10
C, . . . , I C, C ds ≥
1
0
ly tα−n1 ,
where
ly
1
min 1,
1−C
1
0
GA sf s, I
n−2
C, I
n−3
1
C, . . . , I C, C ds .
3.11
10
Abstract and Applied Analysis
On the other hand, it follows from 2.19 that
T y t 1
Kt, sf s, I n−2 ys, I n−3 ys, . . . , I 1 ys, ys ds
0
≤ tα−n1
1
Hsf s, ly κ0 s, ly κ1 s, . . . , ly κn−2 s ds,
3.12
0
≤
Ly tα−n1 ,
where
Ly
1
max 1, Hsf s, ly κ0 s, ly κ1 s, . . . , ly κn−2 s ds .
3.13
0
It follows from 3.6–3.12 that T is well defined and T P ⊂ P .
Take
1
L max 1, Hsfs, κ0 s, κ1 s, . . . , κn−2 sds ,
0
1
l min 1,
1−C
1
3.14
GA sfs, κ0 s, κ1 s, . . . , κn−2 sds .
0
By H2 and Weierstrass distinguishing method, we know
at 1
Kt, sfs, κ0 s, κ1 s, . . . , κn−2 sds
3.15
0
is a continuous function on 0, 1, that is, at ∈ E.
Theorem 3.2 existence. Suppose H0–H2 hold, and if
1
0
L
L
L
L
GA sf s, κ0 s, κ1 s, . . . , κn−2 s, κn−1 s ds ≥ 1 − C.
l
l
l
l
3.16
Then the BVP 1.4 has at least a positive solution wt.
Proof. By 2.18 and 3.2, we have
−Dt α−n2 T y t f t, I n−2 yt, I n−3 yt, . . . , I 1 yt, yt ,
T y 0 0,
T y 1 1
0
T y sdAs.
3.17
Abstract and Applied Analysis
11
It follows from 2.19 that
ltα−n1 ≤ at ≤ Ltα−n1 ,
3.18
which implies that at ∈ P . Let
bt at
,
l
3.19
then we also have
tα−n1 ≤ bt ≤
L α−n1
t
.
l
3.20
Thus according to the fact that the operator T is nonincreasing relative to y, Lemma 3.1
and 3.20, we have
T bt ≤ T tα−n1 at ≤
at
bt.
l
3.21
and T bt ∈ P . By 2.19, 3.12, and 3.20, one has
1 α−n1
T bt ≥
t
1−C
≥
1 α−n1
t
1−C
≥t
α−n1
1
GA sf s, I n−2 bs, I n−3 bs, . . . , I 1 bs, bs ds
0
1
0
L
L
L
GA sf s, κ0 s, κ1 s, . . . , κn−2 s ds
l
l
l
3.22
.
Thus, by 3.20, 3.21 and the operator T is nonincreasing relative to y, we have
Dt α−n2 bt f t, I n−2 bt, I n−3 bt, . . . , I 1 bt, bt
1 − f s, I n−2 tα−n1 , . . . , I 1 tα−n1 , tα−n1 f t, I n−2 bt, . . . , I 1 bt, bt
l
1 ≤ − f s, I n−2 tα−n1 , . . . , I 1 tα−n1 , tα−n1 f s, I n−2 tα−n1 , . . . , I 1 tα−n1 , tα−n1
l
≤ 0,
Dt
α−n2
T bt f t, I
n−2
T bt, I
n−3
T bt, . . . , I T bt, T bt
1
−f t, I n−2 bt, I n−3 bt, . . . , I 1 bt, bt
f t, I n−2 T bt, I n−3 T bt, . . . , I 1 T bt, T bt
≥ 0,
t ∈ 0, 1.
3.23
12
Abstract and Applied Analysis
However 3.17 implies that bt, T bt satisfy boundary conditions of BVP 2.6. Let
ψt T bt,
φt 1 α−n1 1
Tt
at bt,
l
l
3.24
then 3.21, 3.23 imply that ψt, φt are lower and upper solution of BVP 2.6, respectively, and ψt, φt ∈ P .
Define the function F and the operator A in E by the following:
⎧ ⎪
f t, I n−2 ψt, I n−3 ψt, . . . , I 1 ψt, ψt , y < ψt,
⎪
⎪
⎨ F t, y f t, I n−2 yt, I n−3 yt, . . . , I 1 yt, yt , ψt ≤ y ≤ φt,
⎪
⎪
⎪
⎩f t, I n−2 φt, I n−3 φt, . . . , I 1 φt, φt, y > φt,
By t 1
Kt, sF s, ys ds,
∀y ∈ E.
3.25
3.26
0
Clearly, F : 0, 1 × 0, ∞ → 0, ∞ is continuous by 3.25. Consider the following boundary value problem
−Dt α−n2 yt F t, y , 0 < t < 1,
1
ysdAs.
y0 0,
y1 3.27
0
Obviously, a fixed point of the operator B is a solution of the BVP 3.27.
For all y ∈ E, it follows from Lemma 2.6 and 3.22 that
By t ≤
1
0
≤
1
HsF s, ys ds ≤
1
Hsf s, I n−2 ψs, I n−3 ψs, . . . , I 1 ψs, ψs ds
0
Hsf s, I n−2 sα−n1 , . . . , I 1 sα−n1 , sα−n1 ds
3.28
0
1
Hsfs, κ0 s, κ1 s, . . . , κn−2 sds < ∞.
0
So B is bounded. It is easy to see B : E → E is continuous from the continuity of F and K.
Let Ω ⊂ E be bounded, that is, there exists a positive constant N > 0 such that ||y|| ≤ N,
for all y ∈ Ω. Let L max0≤t≤1,0≤y≤N |Ft, y| 1, and since Kt, s is uniformly continuous on
0, 1 × 0, 1, then for any ε > 0 and s ∈ 0, 1, there exists δ > 0 such that
|Kt1 , s − Kt2 , s| <
ε
,
L
3.29
Abstract and Applied Analysis
13
for |t1 − t2 | < δ. Then
Byt1 − Byt2 ≤
1
Kt1 , s − Kt2 , sF s, ys ds < ε.
3.30
0
This implies that BΩ is equicontinuous.
By the means of the Arzela-Ascoli theorem, we have B : E → E is completely continuous. Thus, by using Schauder’s fixed-point theorem, B has at least a fixed-point w such that
w Bw.
Now we prove
ψt ≤ wt ≤ φt,
t ∈ 0, 1.
3.31
Let zt φt − wt, t ∈ 0, 1. By φt is upper solution of BVP 2.6 and w is fixed
point of B, we know
w0 0,
w1 1
wsdAs.
3.32
0
From the definition of F and 3.22, we obtain
f t, I n−2 φt, I n−3 φt, . . . , I 1 φt, φt ≤ F t, yt
≤ f t, I n−2 ψt, I n−3 ψt, . . . , I 1 ψt, ψt
≤ f t, I n−2 tα−n1 , . . . , I 1 tα−n1 , tα−n1 ,
3.33
t ∈ 0, 1.
Thus 3.16 and 3.33 imply
Dt α−n2 zt Dt α−n2 φt − Dt α−n2 wt
1 − f t, I n−2 tα−n1 , . . . , I 1 tα−n1 , tα−n1 Ft, wt
l
1 ≤ − f t, I n−2 tα−n1 , . . . , I 1 tα−n1 , tα−n1
l
f t, I n−2 tα−n1 , . . . , I 1 tα−n1 , tα−n1 ≤ 0, ∀t ∈ 0, 1.
3.34
By 3.32, 3.34, and Lemma 2.9, we know zt ≥ 0 which implies wt ≤ φt on 0, 1. By
the same way, it is easy to prove wt ≥ ψt on 0, 1. So we obtain
ψt ≤ wt ≤ φt,
t ∈ 0, 1.
3.35
14
Abstract and Applied Analysis
Consequently, Ft, wt ft, I n−2 wt, I n−3 wt, . . . , I 1 wt, wt, t ∈ 0, 1. Then wt is a
positive solution of the BVP 2.6, which implies that xt I n−2 wt is a positive solution of
the BVP 1.4.
Theorem 3.3 asymptotic behavior. Suppose the conditions of Theorem 3.2 are satisfied. Then
there exist two constants B1 , B2 such that the positive solution xt of the BVP 1.4 satisfies
B1 tα−1 ≤ xt ≤ B2 tα−1 ,
3.36
xt
0,
t → 0 tα−1
lim
where
B1 Γα − n 2
,
Γα
B2 LB1 .
3.37
Proof. By 3.22 and 3.35, we know
3.38
wt ≥ ψt ≥ tα−n1 .
On the other hand, it follows from 3.38 and 2.19 that
wt 1
Kt, sf s, I n−2 ws, I n−3 ws, . . . , I 1 ws, ws ds
0
≤ tα−n1
1
Hsf s, I n−2 sα−n1 , . . . , I 1 sα−n1 , sα−n1 ds
0
tα−n1
1
3.39
Hsfs, κ0 s, κ1 s, . . . , κn−2 s, κn−1 sds
0
≤ Ltα−n1 .
Then
3.40
tα−n1 ≤ wt ≤ Ltα−n1 .
Since
I
n−2 α−n1
t
1
Γn − 2
t
0
t − sn−3 sα−n1 ds Γα − n 2 α−1
t ,
Γα
3.41
so it follows from 3.40 that
Γα − n 2 α−1
Γα − n 2 α−1
t
t ,
≤ I n−2 wt xt ≤ L
Γα
Γα
3.42
Abstract and Applied Analysis
15
That is,
B1 tα−1 ≤ xt ≤ B2 tα−1 .
3.43
In the end, from n − 1 < α ≤ n and L’Hospital Rule, we have
xt
xn−2 t
lim
t → 0 tα−1
t → 0 α − 1α − 2 · · · α − n 2tα−n1
Lim
xn−1 t
lim
0.
t → 0 α − 1α − 2 · · · α − n 1tα−n
3.44
Theorem 3.4 uniqueness. Suppose the conditions of Theorem 3.2 are satisfied and α n. Then the
positive solution xt of the BVP 1.4 is unique.
Proof. We only need the BVP 2.6 has unique solution. Suppose that w1 , w2 are C0, 1 positive solutions to the BVP 2.6. We may assume, without loss of generality, that there exists
t∗ ∈ 0, 1 such that w2 t∗ − w1 t∗ max{w2 t − w1 t} > 0. Let
a inf{t1 | 0 ≤ t1 < t∗ , w2 t ≥ w1 t, t ∈ t1 , t∗ };
b inf{t2 | t∗ < t2 ≤ 1, w2 t ≥ w1 t, t ∈ t∗ , t2 };
zt w2 t − w1 t,
3.45
t ∈ 0, 1.
Evidently,
t∗ ∈ a, b,
w2 t ≥ w1 t,
f t, I n−2 w2 t, I n−3 w2 t, . . . , I 1 w2 t, w2 t
≤ f t, I n−2 w1 t, I n−3 w1 t, . . . , I 1 w1 t, w2 t ,
t ∈ a, b.
3.46
Thus
Dt α−n2 zt z t ≥ 0,
t ∈ a, b.
3.47
By the boundary conditions of the BVP 2.6, it is easy to check that there exist the
following two possible cases:
1 za 0,
zb 0;
2 za 0,
zb > 0;
3.48
Case 1. From z t ≥ 0 and za 0, zb 0, we get that zt ≤ 0, t ∈ a, b, which implies a
contradiction with w2 t∗ > w1 t∗ .
16
Abstract and Applied Analysis
Case 2. In this case we have b 1 and z t∗ 0. Since z t is increasing on a, b, we also
have z t is increasing on t ∈ a, t∗ , that is, z t ≤ z t∗ 0 for any t ∈ a, t∗ . Thus zt is
nonincreasing on t ∈ a, t∗ , which implies that zt∗ ≤ za 0, this is a contradiction with
w2 t∗ > w1 t∗ .
Therefore the BVP 2.6 has unique solution, and the BVP 1.4 also has unique
solution.
Remark 3.5. We only get the uniqueness of positive solution for the BVP 1.4 as α is integer,
but it remains unknown as to whether an analogous unique result holds for the case n − 1 <
α < n. We believe the result should also hold and try to study this problem, but it is a pity
that we failed to find the effective method to solve this problem.
Remark 3.6. The BVP 1.4 allows the nonlinearity ft, x0 , x2 , . . . , xn−2 has singularity at t 0, 1 and xi 0, i 0, 1, . . . , n−2. If ft, x0 , x2 , . . . , xn−2 is continuous at xi 0, i 0, 1, . . . , n−2,
that is
f ∈ C0, 1 × 0, ∞n−1 , 0, ∞, and ft, x0 , x1 , x2 , . . . , xn−2 is decreasing in xi > 0 for
i 0, 1, 2, . . . , n − 2;
1
0<
Hsfs, 0, 0, . . . , 0ds < ∞.
3.49
0
then the BVP 1.4 has at least a positive solution xt, and there exists a constant B3 > 0 such
that
0 ≤ xt ≤ B3 tα−1 .
3.50
If α n, the positive solution of the BVP 1.4 also is unique.
Proof. In fact, in Theorem 3.2 the set P is replaced by the following:
P1 {x ∈ E : xt ≥ 0, t ∈ 0, 1}
3.51
and 3.21 can be replaced by the following:
0 ≤ φt T 0,
0 ≤ ψt T φ t ≤ T 0 φt.
Clearly φt, ψt ∈ P1 , and
Dt α−n2 φt f t, I n−2 φt, I n−3 φt, . . . , I 1 φt, φt
−ft, 0, 0, . . . , 0 f t, I n−2 T 0, I n−3 T 0, . . . , I 1 T 0, T 0 ≤ 0,
3.52
Abstract and Applied Analysis
Dt α−n2 ψt f t, I n−2 ψt, I n−3 ψt, . . . , I 1 ψt, ψt
17
−f t, I n−2 φt, I n−3 φt, . . . , I1 φt, φt f t, I n−2 ψt, I n−3 ψt, . . . , I 1 ψt, ψt
≥ 0,
t ∈ 0, 1.
3.53
Thus the rest of proof is similar to those of Theorem 3.2.
On the other hand, by Lemma 2.6 and G2, the positive solution of the BVP 2.6
satisfies
0 ≤ wt ≤ φt T 0 ≤
1
1
Kt, sfs, 0, 0, . . . , 0ds
0
3.54
Hsfs, 0, 0, . . . , 0dstα−n1 ,
0
and then
3.55
0 ≤ I n−2 wt xt ≤ B3 tα−1 ,
where
Γα − n 2
B3 Γα
1
Hsfs, 0, 0, . . . , 0ds.
3.56
0
Moreover if α n, the positive solution of the BVP 1.4 also is unique.
Example 3.7. Consider the existence of positive solutions for the nonlinear fractional
differential equation
−1/3
−Dt 7/2 xt 10t−1/4 x
x−1/8 ,
x0 x 0 0,
x 1 1
0 < t < 1,
3.57
x sdAs,
0
where
⎧
⎪
⎪
0,
⎪
⎪
⎪
⎪
⎪
⎪
⎨3
At ,
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩1,
1
,
t ∈ 0,
2
1 3
t∈
,
,
2 4
3
,1 .
t∈
4
3.58
18
Abstract and Applied Analysis
Then the BVP 3.58 is equivalent to the following 4-point BVP with coefficients of both signs
−1/3
−Dt 7/2 xt 10t−1/4 x
x−1/8 ,
x0 x 0 0,
x 1 0 < t < 1,
1
3
3 1
x
− x
,
2
2
2
4
3.59
Conclusion: The BVP 3.59 has at least a positive solution xt such that
2 5/2
t ≤ xt ≤ 14.886t5/2 .
5
3.60
Proof. Clearly,
0≤C
1
t
5/2
dAt 1 −
0
3/4
1/2
3 5/2
dt 2
1
dt
5/2
≈ 0.0217 < 1,
3/4
⎧
t3/2 1 − s3/2
⎪
⎪
⎪
,
s
G
t,
⎨ 1
Γ5/2
Gt, s ⎪
t3/2 1 − s3/2 − t − s3/2
⎪
⎪
⎩G2 t, s ,
Γ5/2
0 ≤ t ≤ s ≤ 1,
3.61
0 ≤ s ≤ t ≤ 1.
Thus,
⎧
1
3
⎪
⎪
G
,
s
−
⎪2 2 2
⎪
⎪
⎪
⎪
⎪
⎪
1
⎨3
G
,
s
−
GA s 2 1 2
⎪
⎪
⎪
⎪
3
1
⎪
⎪
⎪
G
,
s
−
1
⎪
⎪
2
⎩2
Hs 1
3
G2
,s ,
2
4
1
3
G2
,s ,
2
4
1
3
G1
,s ,
2
4
1
,
2
1
3
≤s< ,
2
4
3
≤ s ≤ 1,
4
0≤s<
⎧
√
√ ⎪
2
−
3 1 − s3/2 23/4 − s3/2 − 61/2 − s3/2
⎪
3/4
3/2
⎪
⎪
⎪
,
√
⎪
⎪
⎪
3 π
⎪
⎪
⎪
⎪
⎨ 3/2√2 − 3/4√3 1 − s3/2 23/4 − s3/2
√
⎪
⎪
3 π
⎪
⎪
⎪
⎪
√
√
⎪
⎪
3/2 2 − 3/4 3 1 − s3/2
⎪
⎪
⎪
⎪
,
√
⎩
3 π
1 − s3/2 GA s
.
Γ5/2
0.9783
Clearly, H0 and H1 hold.
,
0≤s<
1
,
2
3
1
≤s< ,
2
4
3
≤ s ≤ 1,
4
3.62
Abstract and Applied Analysis
19
≡ 0,
t−3/8 λ−1/3
t−1/4 /
On the other hand, for any λi > 0, i 0, 1, ft, λ0 t, λ1 λ−1/8
0
1
t ∈ 0, 1, and
κ0 s 1
0<
0
10
2 5/2
s ,
5
κ1 s s3/2 ,
Hsfs, λ0 κ0 s, λ1 κ1 sds
1
0
1 − s3/2 GA s
Γ5/2
0.9783
2
λ0
5
3.63
−1/8
−9/16
s
λ−1/3
s−3/4
1
ds < ∞.
Thus H2 is satisfied.
Now we compute L, l. Since
1
GA sfs, κ0 s, κ1 sds 10
0
1
0
10
1/2
0
−1/8
2
−9/16
−3/4
ds
GA s
s
s
5
⎤
√
√ 3/2 2 − 3/4 3 1 − s3/2 23/4 − s3/2 − 61/2 − s3/2
⎥
⎢
√
⎦
⎣
3 π
⎡
−1/8
2
−9/16
−3/4
ds
×
s
s
5
3/4 3/2√2 − 3/4√3 1 − s3/2 23/4 − s3/2 −1/8
2
−9/16
−3/4
10
s
s
√
5
3 π
1/2
√
√ 1
3/2 2 − 3/4 3 1 − s3/2 2 −1/8
−9/16
−3/4
ds 2.600,
10
s
s
√
5
3 π
3/4
−1/8
1
1
2
−9/16
−3/4
ds
Hsfs, κ0 s, κ1 sds 10 Hs
s
s
5
0
0
−1/8
1
2
1 − s3/2 GA s
−9/16
−3/4
10
ds 37.215.
s
s
Γ5/2
0.9783
5
0
3.64
Thus
1
L max 1, Hsfs, κ0 s, κ1 sds max{1, 37.215} 37.215,
0
1
l min 1,
1−C
1
0
3.65
GA sfs, κ0 s, κ1 sds
min{1, 2.658} 2.658.
20
Abstract and Applied Analysis
So
1
0
1
L
L
2 −1/8 −9/16
−1/3 −3/4
ds
GA sf s, κ0 s, κ1 s ds 10 GA s 14 ×
s
14
s
l
l
5
0
3.66
4.540 ≥ 1 − C 0.9783,
which implies that 3.16 holds. Then the BVP 1.4 has at least a positive solution xt, and
there exist two constants
B1 Γα − n 2 2
,
Γα
5
B2 LB1 14.886,
3.67
such that
2 5/2
t ≤ xt ≤ 14.886t5/2 .
5
3.68
Acknowledgment
This work is supported by the National Natural Science Foundation of China 61201143
and the Fundamental Research Funds for the Central Universities Grant no. HIT. NSRIF.
2010091, the National Science Foundation for Postdoctoral Scientists of China Grant no.
2012M510956, and the Postdoctoral Funds of Heilongjiang Province Grant no. LBHZ11128.
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