Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 719192, 11 pages
doi:10.1155/2012/719192
Research Article
Nontrivial Solution of Fractional
Differential System Involving Riemann-Stieltjes
Integral Condition
Ge-Feng Yang
Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006, China
Correspondence should be addressed to Ge-Feng Yang, gefengyang 001@163.com
Received 14 September 2012; Accepted 25 October 2012
Academic Editor: Xinguang Zhang
Copyright q 2012 Ge-Feng Yang. 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 uniqueness of nontrivial solutions for a class of fractional differential
system involving the Riemann-Stieltjes integral condition, by using the Leray-Schauder nonlinear
alternative and the Banach contraction mapping principle, some sufficient conditions of the
existence and uniqueness of a nontrivial solution of a system are obtained.
1. Introduction
HIV is a retrovirus that targets the CD4 T lymphocytes, which are the most abundant white
blood cells of the immune system. To this day, there have already been over 16 million people
who died of AIDS. Although HIV infects other cells also, it wreaks the most havoc on the
CD4 T cells by causing their decline and destruction, thus decreasing the resistance of the
immune system 1–3. Mathematical models have been proven valuable in understanding
the dynamics of HIV infection 4–6. Perelson et al. 7, 8 developed a simple model for the
primary infection with HIV. In this model, four categories of cells were defined: uninfected
CD4 T cells, latently infected CD4 T cells, productively infected CD4 T cells, and virus
population. And the following two equations describe the evolution of the system:
dx
s − μx − βxy
dt
dy
βxy − υy,
dt
HIV-1
2
Abstract and Applied Analysis
where all parameters and variables are nonnegative. s is the assumed constant rate of the
production of CD4 T-cells, μ is their per capita death rate, βxy is the rate of infection of
CD4 T-cells by virus, and vy is the rate of disappearance of infected cells. Recently Arafa1
et al. introduced fractional order into a model of HIV-1 infection of CD4 T cells. The new
system is described by the following set of FODEs of order α1 , α2 , α3 > 0:
Dα1 T s − KV T − dT bI,
Dα2 I KV T − b δI,
HIV-2
Dα3 I NδI − cV.
T, I, and V denote the concentration of uninfected CD4 T cells, infected CD4 T cells,
and free HIV virus particles in the blood, respectively. δ represents death rate of infected
T cells and includes the possibility of death by the bursting of infected T cells, hence δd. The
parameter b is the rate at which infected cells return to uninfected class while c is the death
rate of virus and N is the average number of viral particles produced by an infected cell.
Motivated by HIV model, in this paper, we consider the existence of nontrivial solution
for fractional differential system
β
−Dtα xt λf t, xt, Dt xt, yt ,
β
Dt x0 0,
β
Dt x1 1
0
β
Dt xsdAs,
γ
−Dt yt gt, xt,
y0 0,
t ∈ 0, 1,
y1 1.1
1
ysdBs,
0
where λ is a parameter, 1 < γ < α ≤ 2, 1 < α − β < γ, 0 < β < 1, Dtα is the standard
1 β
Riemann-Liouville derivative. 0 Dt xsdAs denotes the Riemann-Stieltjes integral, and
A, B ∈ NBV 0, 1 are functions of bounded variation.
In the recent years, there has been a significant development in fractional order
differential equations involving fractional derivatives. For example, Ahmad and Nieto 9
considered a coupled system of nonlinear fractional differential equations with three-point
boundary conditions
p
Dtα ut f t, vt, Dt vt ,
u0 0,
β
q
Dt vt f t, ut, Dt ut ,
u1 γu η ,
v0 0,
t ∈ 0, 1,
v1 γv η ,
1.2
where α, β, p, q, γ, η satisfy certain conditions. Applying the Schauder fixed point theorem,
an existence result is proved provided that f, g : 0, 1 × R × R → R are given continuous
functions and satisfy some growth conditions. For a detailed description of recent work on
fractional differential equation, we refer the reader to some recent papers see 10–17.
The rest of the paper is organized as follows. Section 2 gives preliminaries and lemmas
about fractional calculus. In Section 3, we present the main results and the proof of the results.
In addition, an example is given to illustrate the application of the main results.
Abstract and Applied Analysis
3
2. Preliminaries and Lemmas
For the convenience of the reader, we present here some definitions from fractional calculus
which are to be used in the later sections.
Definition 2.1 see 18–20. The Riemann-Liouville fractional integral of order α > 0 of a
function x : 0, ∞ → R is given by
I α xt t
1
Γα
2.1
t − sα−1 xsds
0
provided that the right-hand side is pointwise defined on 0, ∞.
Definition 2.2 see 18–20. The Riemann-Liouville fractional derivative of order α > 0 of a
function x : 0, ∞ → R is given by
Dtα xt
n t
d
1
t − sn−α−1 xsds,
Γn − α dt
0
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, ∞.
Lemma 2.3 see 18–20. (1) If x ∈ L0, 1, ν > σ > 0, then
I ν I σ xt I νσ xt,
Dtσ I ν xt I ν−σ xt,
Dtσ I σ xt xt.
2.3
(2) If ν > 0, σ > 0, then
Dtν tσ−1 Γσ σ−ν−1
t
.
Γσ − ν
2.4
Lemma 2.4 see 18–20. Assume that x ∈ L1 0, 1 with a fractional derivative of order α > 0 that
belongs to L1 0, 1. Then
2.5
I α Dtα xt xt c1 tα−1 c2 tα−2 · · · cn tα−n ,
where ci ∈ R i 1, 2, . . . , n, n is the smallest integer greater than or equal to α.
Let xt I β vt, vt ∈ C0, 1; by standard discussion, we easily reduce the system
1.1 to the following modified problems:
α−β
−Dt
vt λf t, I β vt, vt, yt ,
v0 0,
v1 1
vsdAs,
γ
−Dt yt g t, I β vt ,
y0 0,
0
and the system 2.6 is equivalent to the system 1.1.
y1 t ∈ 0, 1,
2.6
1
ysdBs,
0
4
Abstract and Applied Analysis
Lemma 2.5 see 21. Let h ∈ L1 0, 1, if 1 < α − β, γ ≤ 2, then the unique solution of the linear
problems
α−β
−Dt
vt ht,
v0 0,
γ
−Dt yt ht,
y0 0,
t ∈ 0, 1,
v1 0,
2.7
t ∈ 0, 1,
y1 0,
is
vt 1
K1 t, shsds,
yt 1
0
K2 t, shsds,
2.8
0
respectively, where
0 ≤ t ≤ s ≤ 1,
t1 − sα−β−1 ,
1
K1 t, s α−β−1
α−β−1
Γ α − β t1 − s
− t − s
, 0 ≤ s ≤ t ≤ 1.
0 ≤ t ≤ s ≤ 1,
t1 − sγ−1 ,
1
K2 t, s γ−1
γ−1
Γ γ t1 − s − t − s , 0 ≤ s ≤ t ≤ 1,
2.9
are the Green functions of the boundary value problems 2.7.
By Lemma 2.4, the unique solution of the problem
α−β
Dt
vt 0,
v0 0,
0 < t < 1,
2.10
v1 1,
is tα−β−1 . Let
C
1
t
α−β−1
dAt,
0
B
1
tγ−1 dBt,
2.11
0
and define
GA s 1
0
K1 t, sdAt,
GB s 1
0
K2 t, sdBt.
2.12
Abstract and Applied Analysis
5
As in 14, if C /
0, B /
0, we can get that the Green function for the following nonlocal
system
α−β
−Dt
vt ht,
t ∈ 0, 1,
1
vsdAs,
v1 v0 0,
0
γ
−Dt yt
y0 0,
2.13
ht,
t ∈ 0, 1,
1
ysdBs,
y1 0
are given by, respectively,
Gt, s tα−β−1
GA s K1 t, s,
1−C
Ht, s tγ−1
GB s K2 t, s.
1−B
2.14
Clearly, Gt, s, Ht, s are continuous on 0, 1 × 0, 1; thus there exist positive
constants m, n such that
|Gt, s| ≤ m,
|Ht, s| ≤ n.
2.15
It is well known that v, y is a solution of the system 2.6 if and only if v, y ∈
C0, 1 × C0, 1 is a solution of the following nonlinear integral equation system:
vt λ
1
Gt, sf s, I β vs, vs, ys ds,
0
yt 1
β
2.16
Ht, sg s, I vs ds.
0
Obviously, the system 2.16 is equivalent to the following integral equation:
vt λ
1
Gt, sf
0
β
1
β
Hs, τg τ, I vτ dτ ds.
s, I vs, vs,
2.17
0
Lemma 2.6 see 22. Let X be a real Banach space and Ω a bounded open subset of X, where θ ∈ Ω;
T : Ω → X is a completely continuous operator. Then, either there exists x ∈ ∂Ω, λ > 1 such that
T x λx or there exists a fixed point x∗ ∈ Ω.
3. Main Results
The following definition introduces the Carathèodory conditions imposed on a map f.
6
Abstract and Applied Analysis
Definition 3.1. Let z z1 , z2 , . . . , zn . A map f : 0, 1 × Rn , t, z → ft, z is said to satisfy
the Carathéodory conditions if the following conditions hold:
i for each z ∈ Rn , the mapping t → ft, z is Lebesgue measurable;
ii for a.e. t ∈ 0, 1, the mapping z → ft, z is continuous on R.
Throughout the paper we always assume the following conditions hold.
H0 A, B are functions of bounded variation such that C, B /
1, where C, B are defined
by 2.11.
H1 f : 0, 1 × R3 → R and g : 0, 1 × R → R satisfy the Carathéodory condition.
Theorem 3.2. Suppose that H0-H1 hold. If ft, 0, 0, 0 /
≡ 0, and there exist nonnegative functions
pi i 1, 2, 3, q, r ∈ L1 0, 1 such that
ft, x1 , x2 , x3 ≤ p1 t|x1 | p2 t|x2 | p3 t|x3 | qt,
g t, y ≤ rty, a.e. t ∈ 0, 1.
a.e. t ∈ 0, 1,
3.1
In addition, there exists t0 ∈ 0, 1 such that pi0 t0 / 0 for some i0 ∈ {1, 2, 3}. Then there exists a
∗
∗
constant λ > 0, such that, for any 0 < λ ≤ λ , the system 1.1 has at least one nontrivial solution
x∗ , y∗ ∈ C0, 1 × C0, 1.
Proof. Let X C0, 1 be endowed with the ordering x ≤ y if xt ≤ yt for all t ∈ 0, 1, and
u maxt∈0,1 |ut| is defined as usual by maximum norm. Clearly, it follows that X, · is a Banach space. By 2.17 and 2.6, problem 1.1 has a solution x∗ , y∗ ∈ C0, 1 × C0, 1
β
if and only if v∗ Dt x∗ solves the following operator equation:
T vt λ
1
1
β
Gt, sf
β
Hs, τg τ, I vτ dτ ds,
s, I vs, vs,
0
3.2
0
in X. So we only need to seek a fixed point of T in X. By Ascoli-Arzela Theorem, it is obvious
that the operator T : X → X is a completely continuous operator.
1
Since ft, 0, 0, 0 /
≡ 0, and ft, 0, 0, 0 ≤ qt a.e. t ∈ 0, 1, we know 0 qtdt > 0. On the
other hand, by pi0 t0 / 0 for some i0 ∈ {1, 2, 3}, we have
1
p1 t p2 t p3 t dt > 0.
3.3
0
Let
1
R qsds
0
1
,
1
1/Γ β 1 n/Γ β
rsds
p
p
ds
p
s
s
s
1
2
3
0
0
3.4
where n is defined by 2.15. Define the set
Ω {v ∈ C0, 1 : ||v|| ≤ R}.
3.5
Abstract and Applied Analysis
7
Suppose v ∈ ∂Ω, μ > 1 such that T v μv. Then, for any v ∈ ∂Ω, and noticing that
1
1
||v||
β
t − sβ−1 vsds ≤ ,
I vτ Γ β 0
Γ β
3.6
we have
1
1 T v max |T vt| ≤ mλ f s, I β vs, vs, Hs, τg τ, I β vτ dτ ds
t∈0,1
0
0
1
1
β
β
≤ mλ
p1 tI vs p2 t|vs| p3 t Hs, τg τ, I vτ dτ qs ds
0
0
1
1
≤ mλ
p1 tI β vs p2 t|vs| np3 t rτI β vτdτ qs ds
0
0
p1 t||v||
np3 t||v|| 1
≤ mλ
rτdτ qs ds
p2 t||v|| Γ β
Γ β
0
0
1
1
1
n
≤ mλ
rsds
p1 s p2 s p3 s ds||v||
1 Γ β
Γ β 0
0
1
mλ
3.7
1
qsds.
0
Thus we take
1
λ 2
∗
m
1
n
1 Γ β
Γ β
1
1
rsds
0
−1
p1 s p2 s p3 s ds
3.8
.
0
Then for any 0 < λ ≤ λ∗ and v ∈ ∂Ω, by 3.7, one has
μ||v|| ≤ mλ
n
1
1 Γ β
Γ β
1
1
rsds
0
p1 s p2 s p3 s ds||v|| mλ
1
0
qsds
0
1
qsds
1
0
≤ ||v|| .
1
1
2
2 1/Γ β 1 n/Γ β
rsds
p
p
ds
p
s
s
s
1
2
3
0
0
3.9
Consequently,
μ≤
1 1
1.
2 2
3.10
8
Abstract and Applied Analysis
≡ 0, then
This contradicts μ > 1, by Lemma 2.6, T has a fixed point v∗ ∈ Ω. Since ft, 0, 0, 0 /
≡ 0, Thus let
v∗ /
x∗ I β v ∗ ,
y∗ 1
Ht, sgs, x∗ sds,
3.11
0
and then the system 1.1 has at least one nontrivial solution x∗ , y∗ ∈ C0, 1 × C0, 1 for
any 0 < λ ≤ λ∗ . This completes the proof of Theorem 3.2.
Theorem 3.3. Suppose that H0-H1 hold. If ft, 0, 0, 0 /
≡ 0, and there exist nonnegative functions
pi i 1, 2, 3, r ∈ L1 0, 1 such that
ft, x1 , x2 , x3 −f t, y1 , y2 , y3 ≤ p1 tx1 −y1 p2 tx2 −y2 p3 tx3 −y3 ,
g t, y1 − g t, y2 ≤ rty1 − y2 , a.e. t ∈ 0, 1.
a.e. t ∈ 0, 1,
3.12
In addition, there exists t0 ∈ 0, 1 such that pi0 t0 /
0 for some i0 ∈ {1, 2, 3}. Then there exists a
constant λ∗ > 0, such that, for any 0 < λ ≤ λ∗ , the system 1.1 has unique nontrivial solution
x∗ , y∗ ∈ C0, 1 × C0, 1.
Proof. Let T be given in Theorem 3.2; we will show that T is a contraction. In fact,
1
1 β
β
||T v1 − T v2 || ≤ mλ f s, I v1 s, v1 s, Hs, τg τ, I v1 τ dτ
0
0
−f
s, I v2 s, v2 s,
β
1
0
Hs, τg τ, I v2 τ dτ ds
β
1
≤ mλ
p1 tI β v1 s − v2 s p2 t|v1 s − v2 s|
0
np3 t
1
β
rτI v1 τ − v2 τdτ ds
0
p1 t||v1 − v2 ||
np3 t 1
≤ mλ
p2 t||v1 − v2 || rτdτ||v1 − v2 || ds
Γ β
Γ β
0
0
1
1
1
n
≤ mλ
rsds
p1 s p2 s p3 s ds||v1 − v2 ||.
1 Γ β
Γ β 0
0
1
3.13
If we choose
1
−1
1
1
1
n
m
rsds
.
λ p1 s p2 s p3 s ds
1 2
Γ β
Γ β 0
0
∗
3.14
Abstract and Applied Analysis
9
Then by 3.13 and 3.14, we have
||T v1 − T v2 || ≤
1
||v1 − v2 ||,
2
3.15
which implies that T is indeed a contraction. Finally, we use the Banach fixed point theorem
to deduce the existence of a unique solution to the system 1.1.
Corollary 3.4. Suppose that H0-H1 hold. If ft, 0, 0, 0 /
≡ 0,
lim sup
max
|x1 ||x2 ||x3 | → ∞ t∈0,1
ft, x1 , x2 , x3 < ∞.
|x1 | |x2 | |x3 |
3.16
In addition, there exists a nonnegative function r ∈ L1 0, 1 such that g satisfies
g t, y ≤ rty,
a.e. t ∈ 0, 1.
3.17
Then there exists a constant λ∗ > 0, such that, for any 0 < λ ≤ λ∗ , the system 1.1 has at least one
nontrivial solution x∗ , y∗ ∈ C0, 1 × C0, 1.
Proof. We prove f satisfies the conditions of Theorem 3.2. Let
W
ft, x1 , x2 , x3 ,
|x1 ||x2 ||x3 | → ∞ t∈0,1 |x1 | |x2 | |x3 |
lim sup
3.18
max
and choose ε > 0 such that W 1 − ε > 0. By 3.16, there exists N > 0 such that
ft, x1 , x2 , x3 ≤ W 1 − ε|x1 | |x2 | |x3 |,
|x1 | |x2 | |x3 | > N, t ∈ 0, 1.
3.19
Take M maxt∈0,1/Ω,|x1 ||x2 ||x3 |≤N , |ft, x1 , x2 , x3 |, and the measure of Ω is 0. Thus for any
t, x, x2 , x3 ∈ 0, 1 × R3 , we have
ft, x1 , x2 , x3 ≤ W 1 − ε|x1 | |x2 | |x3 | M,
a.e. t ∈ 0, 1.
3.20
From Theorem 3.2 we know the system 1.1 has at least one nontrivial solution.
Example 3.5. Consider the following fractional differential system:
−Dt3/2 x λ
Dt3/8 x0 0,
1
1
2
,
t2 sin t x − t1/2 cos2 Dt3/8 x √
sin
y
√
3
t
1−t
−Dt5/4 y t cos t lnt 1x,
1
3/8
Dt x1 Dt3/8 xtdAt,
y0 0,
0
3.21
y1 1
ytdBt,
0
10
Abstract and Applied Analysis
where
⎧
⎪
⎪
0,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨3
,
At ⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩1,
1
,
t ∈ 0,
2
1 3
t∈
,
,
2 4
3
t∈
,1 ,
4
⎧
⎪
⎪
0,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
Bt 1,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩2,
1
,
t ∈ 0,
4
1 3
,
,
t∈
4 4
3
t∈
,1 .
4
3.22
Then the system 3.21 is equivalent to the following 4-point BVP with coefficients of both
signs
−Dt3/2 x
λ
t sin t x − t
2
1/2
cos
2
Dt3/8 x
1
1
2 ,
√
sin y √
3
t
1−t
−Dt5/4 y t cos t lnt 1x,
3
1
3
1
Dt3/8 x1 Dt3/8
Dt3/8 x0 0,
− Dt3/8
,
2
2
2
4
3
1
y
.
y0 0,
y1 y
4
4
3.23
Clearly, H0 holds. Let
1
1
2,
ft, x1 , x2 , x3 t2 sin t x1 −t1/2 cos2 x2 √
sin x3 √
3
t
1−t
g t, y t cos tlnt1y.
3.24
Then H1 also is satisfied.
On the other hand, we have
ft, x1 , x2 , x3 ≤ t2 sin t |x1 | t1/2 |x2 | √1 |x3 | √ 1 2,
3
t
1−t
g t, y ≤ t cos t lnt 1y,
3.25
√
and ft, 0, 0, 0 1/ 1 − t 2 ≡
/ 0, which imply all conditions of Theorem 3.2 are satisfied, by
Theorem 3.2, there exists a constant λ∗ > 0, such that for any 0 < λ ≤ λ∗ , the system 3.21 has
at least one nontrivial solution x∗ , y∗ ∈ C0, 1 × C0, 1.
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Probability and Statistics
Volume 2014
The Scientific
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International Journal of
Differential Equations
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International Journal of
Advances in
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Mathematical Physics
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Journal of
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International
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Stochastic Analysis
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International Journal of
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Discrete Dynamics in
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Applied Mathematics
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Optimization
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