Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 423796, 24 pages
doi:10.1155/2012/423796
Research Article
Existence Results for Fractional Differential
Inclusions with Multivalued Term Depending on
Lower-Order Derivative
Xiaoyou Liu1 and Zhenhai Liu2
1
School of Mathematical Science and Computing Technology, Central South University,
Changsha, Hunan 410075, China
2
School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China
Correspondence should be addressed to Xiaoyou Liu, liuxiaoyou2002@hotmail.com
Received 24 October 2012; Accepted 14 December 2012
Academic Editor: Svatoslav Staněk
Copyright q 2012 X. Liu and Z. Liu. 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.
This paper is concerned with a class of fractional differential inclusions whose multivalued term
depends on lower-order fractional derivative with fractional nonseparated boundary conditions.
The cases of convex-valued and non-convex-valued right-hand sides are considered. Some
existence results are obtained by using standard fixed point theorems. A possible generalization
for the inclusion problem with integral boundary conditions is also discussed. Examples are given
to illustrate the results.
1. Introduction
Recently, the subject of fractional differential equations has emerged as an important area
of investigation. Indeed, we can find numerous applications of fractional-order derivatives
in the mathematical modeling of physical and biological phenomena in various fields of
science and engineering 1–3. A variety of results on initial and boundary value problems
of fractional differential equations and inclusions can easily be found in the literature on
this topic. For some recent results, we can refer to, for instance, 4–20 equations 21–27
inclusions and the references therein.
Ahmad and Ntouyas 22 considered a boundary value problem of fractional
differential inclusions with fractional separated boundary conditions given by
Dq xt ∈ Ft, xt, t ∈ 0, 1, 1 < q ≤ 2,
α2 x1 β2 c Dp x1 γ2 ,
α1 x0 β1 c Dp x0 γ1 ,
c
0 < p < 1,
1.1
2
Abstract and Applied Analysis
where c Dq denotes the Caputo fractional derivative of order q, F : 0, 1 × R → 2R is a
0.
multivalued map, and αi , βi , γi i 1, 2 are real constants, with α1 /
In Cernea 24, the following multipoint boundary value problem for a fractionalorder differential inclusion was studied
Dα xt ∈ F t, xt, x t
x0 x 0 0,
a.e. t ∈ 0, 1, 2 < α ≤ 3,
x1 −
m
1.2
ai xξi λ,
i1
where Dα is the standard Riemann-Liouville fractional derivative, m ≥ 1, 0 < ξ1 < ξ2 < · · · <
α−1
< 1, λ > 0, ai > 0, i 1, 2, . . . , m, and F : 0, 1 × R × R → 2R is a multivalued
ξm < 1, m
i1 ai ξi
map.
In Khan et al. 11, the authors studied the existence and uniqueness results of
nonlinear fractional differential equation of the type
Dq xt f t, xt, c Dσ xt , t ∈ 0, T ,
T
T
gs, xds,
γxT δx T hs, xds,
αx0 − βx 0 c
0
1.3
0
where 0 < σ < 1, 1 < q < 2, α, δ > 0, β, γ ≥ 0 or α, δ ≥ 0, β, γ > 0 and c Dq , c Dσ are the Caputo
fractional derivatives. The results in 11, 22, 24 are obtained by using appropriate standard
fixed point theorems.
Motivated by the papers cited above, in this paper, we consider the existence results
for a new class of fractional differential inclusions of the form
c
Dα xt ∈ F t, xt, c Dβ xt ,
a.e. t ∈ 0, T ,
1.4
where c Dα denotes the Caputo fractional derivative of order α, F : 0, 1 × R × R → 2R is a
multivalued map, 1 < α ≤ 2, 0 < β ≤ 1, and T > 0. We study 1.4 subject to two families of
boundary conditions:
1 separated boundary conditions
a1 x0 b1 c Dγ x0 c1 ,
a2 xT b2 c Dγ xT c2 ,
1.5
a2 c Dγ x0 b2 c Dγ xT c2 ,
1.6
2 Nonseparated boundary conditions
a1 x0 b1 xT c1 ,
where ai , bi , ci , i 1, 2 are real constants and 0 < γ < 1.
Abstract and Applied Analysis
3
The results of this paper can easily to be generalized to the boundary value problems
of fractional differential inclusions 1.4 with the following integral boundary conditions:
c
γ
a1 x0 b1 D x0 c1
T
gs, xsds,
0
c
γ
a2 xT b2 D xT c2
a1 x0 b1 xT c1
T
hs, xsds,
0
gs, xsds,
0
c
γ
c
γ
1.7
T
a2 D x0 b2 D xT c2
1.8
T
hs, xsds,
0
where g, h : 0, T × R → R are given functions.
We remark that when the third variable of the multifunction F in 1.4 vanishes, the
problem 1.4, 1.5 reduces to the case considered in 22. When a1 b1 1, a2 b2 1,
and c1 c2 0, the problem 1.4, 1.6 reduces to an antiperiodic fractional boundary value
problem the case of F f a given continuous function was studied in 4, 15. Our results
generalize some results from the literature cited above and constitute a contribution to this
emerging field of research.
The rest of the paper is organized as follows: in Section 2 we present the notations
and definitions and give some preliminary results that we need in the sequel, Section 3 is
dedicated to the existence results of the fractional differential inclusion 1.4 with boundary
conditions 1.5 and 1.6, in Section 4 we indicate a possible generalization for the inclusion
problem 1.4 with integral boundary conditions 1.7 and 1.8, and two illustrative
examples are given in Section 5.
2. Preliminaries
In this section, we introduce notations, definitions, and preliminary facts that will be used in
the remainder of this paper.
Let X, · be a normed space. We use the notations P X {Y ⊆ X : Y /
∅}, Pcl X {Y ∈ P X : Y closed}, Pb X {Y ∈ P X : Y bounded}, Pcp X {Y ∈ P X : Y compact},
Pcp,c X {Y ∈ P X : Y compact, convex}, and so on.
Let A, B ∈ Pcl X; the Pompeiu-Hausdorff distance of A, B is defined as
hA, B max sup da, B, sup db, A .
a∈A
2.1
b∈B
A multivalued map F : X → P X is convex closed valued if Fx is convex closed
for all x ∈ X. F is said to be completely continuous if FB is relatively compact for every
B ∈ Pb X. F is called upper semicontinuous on X, if for every x0 ∈ X, the set Fx0 is a
nonempty closed subset of X, and for every open set O of X containing Fx0 , there exists an
open neighborhood U0 of x0 such that FU0 ⊆ O. Equivalently, F is upper semicontinuous if
the set {x ∈ X : Fx ⊆ O} is open for any open set O of X. F is called lower semicontinuous
4
Abstract and Applied Analysis
if the set {x ∈ X : Fx ∩ O /
∅} is open for each open set O in X. If a multivalued map F
is completely continuous with nonempty compact values, then F is upper semicontinuous if
and only if F has a closed graph, that is, if xn → x∗ and yn → y∗ , then yn ∈ Fxn implies
y∗ ∈ Fx∗ 28.
A multivalued map F : 0, T → Pcl X is said to be measurable if, for every x ∈ X,
the function t → dx, Ft inf{dx, y : y ∈ Ft} is a measurable function.
Definition 2.1. A multivalued map F : X → Pcl X is called
1 γ-Lipschitz if there exists γ > 0 such that
h Fx, F y ≤ γd x, y ,
for each x, y ∈ X,
2.2
2 a contraction if it is γ-Lipschitz with γ < 1.
Definition 2.2. A multivalued map F : 0, T × R × R → P R is said to be Carathéodory if
1 t → Ft, x, y is measurable for each x, y ∈ R;
2 x, y → Ft, x, y is upper semicontinuous for a.e. t ∈ 0, T .
Further, a Carathéodory function F is said to be L1 - Carathéodory if
3 for each l > 0, there exists ϕl ∈ L1 0, T , R such that
F t, x, y sup |v| : v ∈ F t, x, y ≤ ϕl t
2.3
for all |x| ≤ l, |y| ≤ l and a.e. t ∈ 0, T .
Lemma 2.3 see 29. Let X be a Banach space. Let G : 0, T × X → Pcp,c X be an L1 Carathéodory multivalued map and Γ a linear continuous map from L1 0, T , X to C0, T , X,
then the operator
Γ ◦ SG : C0, T , X −→ Pcp,c C0, T , X,
y −→ Γ ◦ SG y Γ SG,y
2.4
is a closed graph operator in C0, T , X × C0, T , X.
Here SG,y {v ∈ L1 0, T , X : vt ∈ Gt, yt for a.e. t ∈ 0, T }.
Definition 2.4 see 30. The Riemann-Liouville fractional integral of order q for a function f
is defined as
1
I q ft Γ q
provided the integral exists.
t
0
fs
t − s1−q
ds,
q > 0,
2.5
Abstract and Applied Analysis
5
Definition 2.5 see 30. For at least n-times differentiable function f, the Caputo derivative
of order q is defined as
c
1
D ft Γ n−q
q
t
n − 1 < q < n, n q 1,
t − sn−q−1 f n sds,
2.6
0
where q denotes the integer part of the real number q.
Lemma 2.6 see 20. Let α > 0; then the differential equation
c
Dα ht 0
2.7
has solutions ht c0 c1 t c2 t2 · · · cn−1 tn−1 and
I αc Dα ht ht c0 c1 t c2 t2 · · · cn−1 tn−1 ;
2.8
here ci ∈ R, i 0, 1, 2, . . . , n − 1, n α 1.
The following lemma obtained in 6 is useful in the rest of the paper.
Lemma 2.7 see 6. For a given y ∈ C0, T , R, the unique solution of the fractional separated
boundary value problem
Dα xt yt, t ∈ 0, T , 1 < α ≤ 2,
a2 xT b2 c Dγ xT c2 ,
a1 x0 b1 c Dγ x0 c1 ,
c
0 < γ < 1,
2.9
is given by
xt t
t − sα−1
ysds
Γα
0
T
T
t
c1
T − sα−1
T − sα−γ−1
−
a2
ysds b2
ysds v2 t ,
v1
Γα
a1
Γ α−γ
0
0
2.10
where
a2 T Γ 2 − γ b2 T 1−γ
,
v1 Γ 2−γ
v2 a1 c2 − a2 c1
.
a1 v1
2.11
We notice that the solution 2.10 of the problem 2.9 does not depend on the
parameter b1 , that is to say, the parameter b1 is of arbitrary nature for this problem. And
− b2 .
0 and a2 T γ Γ2 − γ /
by 2.10, we should assume that a1 /
6
Abstract and Applied Analysis
Lemma 2.8. For any y ∈ C0, T , R, the unique solution of the fractional nonseparated boundary
value problem
Dα xt yt, t ∈ 0, T , 1 < α ≤ 2,
a2 c Dγ x0 b2 c Dγ xT c2 ,
a1 x0 b1 xT c1 ,
c
0 < γ < 1,
2.12
is given by
xt T
tΓ 2 − γ
tΓ 2 − γ c2
t − sα−1
T − sα−γ−1
ysds −
ysds
Γα
T 1−γ
T 1−γ b2
Γ α−γ
0
0
T
T T − sα−γ−1
b1
T − sα−1
γ
ysds − T Γ 2 − γ
−
ysds
a1 b1
Γα
Γ α−γ
0
0
b1 c 2 T γ Γ 2 − γ
1
− c1 .
−
a1 b1
b2
t
2.13
Proof. For 1 < α ≤ 2, by Lemma 2.6, we know that the general solution of the equation
Dα xt yt can be written as
c
xt I yt − k1 − k2 t α
t
0
t − sα−1
ysds − k1 − k2 t,
Γα
2.14
where k1 , k2 ∈ R are arbitrary constants. Since c Dγ k 0 k is a constant, c Dγ t t1−γ /Γ2 − γ,
c γ α
D I yt I α−γ yt see 30, from 2.14, we have
c
k2 t1−γ
Dγ xt I α−γ yt − Γ 2−γ
t
0
k2 t1−γ
t − sα−γ−1
ysds − .
Γ α−γ
Γ 2−γ
2.15
Using the boundary conditions, we obtain
a1 −k1 b1
T
0
a2 × 0 b2
T
0
T − sα−1
ysds − k1 − k2 T
Γα
k2 T 1−γ
T − sα−γ−1
ysds − Γ α−γ
Γ 2−γ
c1 ,
2.16
c2 .
Abstract and Applied Analysis
7
Therefore, we have
b1 c 2 T γ Γ 2 − γ
− c1
b2
T
T T − sα−γ−1
b1
T − sα−1
γ
ysds − T Γ 2 − γ
ysds ,
a1 b1
Γα
Γ α−γ
0
0
T
Γ 2−γ
c2
T − sα−γ−1
k2 .
ysds −
b2
T 1−γ
Γ α−γ
0
1
k1 a1 b1
2.17
Substituting the values of k1 , k2 in 2.14, we obtain 2.13. This completes the proof.
From the proof of the above lemma, we notice that the solution 2.13 of the problem
2.12 does not depend on the parameter a2 , that is to say, the parameter a2 is of arbitrary
0 and b2 /
0.
nature for this problem. In this situation, we need to assume that a1 b1 /
Let us define what we mean by a solution of the problem 1.4, 1.5 and the problem
1.4, 1.6.
Definition 2.9. A function x ∈ AC1 0, T , R is a solution of the problem 1.4, 1.5 if it
satisfies the boundary conditions 1.5 and there exists a function f ∈ L1 0, T , R such that
ft ∈ Ft, xt, c Dβ xt a.e. on t ∈ 0, T and
t
t − sα−1
xt fsds
Γα
0
T
T
t
c1
T − sα−1
T − sα−γ−1
2.18
−
a2
fsds b2
fsds v2 t .
v1
Γα
a
Γ α−γ
1
0
0
Definition 2.10. A function x ∈ AC1 0, T , R is a solution of the problem 1.4, 1.6 if it
satisfies the boundary conditions 1.6 and there exists a function f ∈ L1 0, T , R such that
ft ∈ Ft, xt, c Dβ xt a.e. on t ∈ 0, T and
T
t
tΓ 2 − γ
tΓ 2 − γ c2
t − sα−1
T − sα−γ−1
xt fsds −
fsds Γα
T 1−γ
T 1−γ b2
Γ α−γ
0
0
T
T T − sα−γ−1
b1
T − sα−1
γ
fsds − T Γ 2 − γ
−
fsds
2.19
a1 b1
Γα
Γ α−γ
0
0
b1 c 2 T γ Γ 2 − γ
1
− c1 .
−
a1 b1
b2
Let C0, T , R be the space of all continuous functions defined on 0, T . Define the
space X {x : x and c Dβ x ∈ C0, T , R} 0 < β ≤ 1 endowed with the norm x maxt∈0,T |xt| maxt∈0,T |c Dβ xt|. We know that X, · is a Banach space see 14.
We end this section with two fixed point theorems, which will be used in the sequel.
8
Abstract and Applied Analysis
Theorem 2.11 nonlinear alternative of Leray-Schauder type 31. Let X be a Banach space, C
a closed convex subset of X, U an open subset of C with 0 ∈ U. Suppose that F : U → Pcp,c C is an
upper semicontinuous compact map. Then either 1 F has a fixed point in U, or 2 there is a x ∈ ∂U
and λ ∈ 0, 1 such that x ∈ λFx.
Theorem 2.12 Covitz and Nadler Jr. 32. Let X, d be a complete metric space. If F : X →
Pcl X is a contraction, then F has a fixed point.
3. Existence Results
In this section, we will give some existence results for the problems 1.4, 1.5 and 1.4, 1.6.
For each x ∈ X, define the set of selections of F by
SF,x v ∈ L1 0, T , R : vt ∈ F t, xt, c Dβ xt for a.e. t ∈ 0, T .
3.1
In view of Lemmas 2.7 and 2.8, we define operators N, M : X → P X as
Nx {h ∈ X : ht Sut, u ∈ SF,x },
3.2
Mx {h ∈ X : ht Kut, u ∈ SF,x }
3.3
with
t
t − sα−1
usds
Γα
0
T
T
t
c1
T − sα−1
T − sα−γ−1
−
a2
usds b2
usds v2 t ,
v1
Γα
a1
Γ α−γ
0
0
t
T
tΓ 2 − γ
tΓ 2 − γ c2
t − sα−1
T − sα−γ−1
usds −
usds Kut 1−γ
Γα
T
T 1−γ b2
Γ α−γ
0
0
T
T T − sα−γ−1
b1
T − sα−1
γ
usds − T Γ 2 − γ
−
usds
a1 b1
Γα
Γ α−γ
0
0
b1 c 2 T γ Γ 2 − γ
1
− c1 .
−
a1 b1
b2
Sut 3.4
It is clear that if x ∈ X is a fixed point of the operator N the operator M, then x is a solution
of the problem 1.4, 1.5 the problem 1.4, 1.6.
Now we are in a position to present our main results. The methods used to prove the
existence results are standard; however, their exposition in the framework of problems 1.4,
1.5 and 1.4, 1.6 is new.
Abstract and Applied Analysis
9
3.1. Convex Case
We consider first the case when F is convex valued.
H1: 1 F : 0, T × R × R → Pcp,c R is a Carathéodory multivalued map; 2 there exist
m ∈ L∞ 0, T , R and ϕ, ψ : 0, ∞ → 0, ∞ continuous, nondecreasing such that
F t, x, y sup |v| : v ∈ F t, x, y ≤ mt ϕ|x| ψ y
3.5
for x, y ∈ R and a.e. t ∈ 0, T .
Theorem 3.1. Assume that (H1) is satisfied and there exists L > 0 such that
L
> 1,
P ϕL ψL mL∞ Q
3.6
where
|c1 | |v2 |T 1−β
,
|a1 | Γ 2 − β
T α−β
T
Tα
T −β
|b2 |T α−γ
|a2 |T α
Q 1 .
Γα 1 |v1 |
Γα 1 Γ α − γ 1
Γ α−β1
Γ 2−β
P |v2 |T 3.7
Then the problem 1.4, 1.5 has at least one solution on 0, T .
Proof. Consider the operator N : X → P X defined by 3.2. From H1, we have for each
x ∈ X, the set SF,x is nonempty 29. For x ∈ X, let u ∈ SF,x and h Su, that is, h ∈ Nx, we
have
c
β
D ht t
0
kt1−β
t − sα−β−1
usds − ,
Γ α−β
Γ 2−β
3.8
where k is a constant given by
1
k
v1
T
a2
0
T − sα−1
usds b2
Γα
T
0
T − sα−γ−1
usds − v2 .
Γ α−γ
3.9
Hence we know that the operator N : X → P X is well defined.
We put Su S1 u S2 u where
S1 ut t
0
t − sα−1
usds,
Γα
S2 ut −ku t Here ku means that the constant k defined by 3.9 is related to u.
c1
.
a1
3.10
10
Abstract and Applied Analysis
We will show that N satisfies the requirements of the nonlinear alternative of LeraySchauder type. The proof will be given in five steps.
Step 1 Nx is convex valued. Since F is convex valued, we know that SF,x is convex and
therefore it is obvious that Nx is convex for each x ∈ X.
Step 2 N maps bounded sets into bounded sets in X. Let Br be a bounded subset of X such
that for any x ∈ Br , x ≤ r, r > 0. We prove that there exists a constant l > 0 such that for
each x ∈ Br , one has h ≤ l for each h ∈ Nx. Let x ∈ Br and h ∈ Nx, then there exists
u ∈ SF,x such that
ht Sut
for t ∈ 0, T .
3.11
By simple calculations, we have
t
Tα
t − sα−1
,
|us|ds ≤ ϕr ψr mL∞
Γα
Γα 1
0
ϕr ψr mL∞ T
|b2 |T α−γ
|a2 |T α
|c1 |
,
|v2 |T |S2 ut| ≤
Γα 1 Γ α − γ 1
|v1 |
|a1 |
|S1 ut| ≤
t t − sα−β−1
|ku |T 1−β
c β
|us|ds D ht ≤
Γ 2−β
0 Γ α−β
3.12
T α−β
|v2 |T 1−β
≤ ϕr ψr mL∞ Γ α−β1
Γ 2−β
ϕr ψr mL∞ T 1−β
|b2 |T α−γ
|a2 |T α
.
Γα 1 Γ α − γ 1
|v1 |Γ 2 − β
Hence we obtain
|c1 | |v2 |T 1−β ϕr ψr mL∞
|a1 | Γ 2 − β
Tα
T
T α−β
T 1−β
|b2 |T α−γ
|a2 |T α
×
Γα 1
Γα 1 Γ α − γ 1
|v1 | |v1 |Γ 2 − β
Γ α−β1
h ≤ |v2 |T ≤ P ϕr ψr mL∞ Q l
a constant.
3.13
Step 3 N maps bounded sets into equicontinuous sets in X. Let Br be a bounded set of X as
in Step 2. Let 0 ≤ t1 < t2 ≤ T and x ∈ Br . For each h ∈ Nx, then there is u ∈ SF,x such that
ht Sut. Since
Abstract and Applied Analysis
11
t1
t2 t − sα−1
t2 − sα−1 − t1 − sα−1
2
usds usds
|S1 ut2 − S1 ut1 | t1
Γα
Γα
0
ϕr ψr mL∞ ≤
t2 − t1 α tα2 − t2 − t1 α − tα1 Γα 1
ϕr ψr mL∞ tα2 − tα1
≤
,
Γα 1
c1
c1 ku t1 − |S2 ut2 − S2 ut1 | −ku t2 a1
a1
ϕr ψr mL∞
|b2 |T α−γ
|a2 |T α
≤
|v2 | t2 − t1 ,
Γα 1 Γ α − γ 1
|v1 |
1−β 1−β
t
k
t
k
u
u
c β
D ht2 − c Dβ ht1 I α−β ut2 − 2 − I α−β ut1 1 Γ 2−β
Γ 2−β ≤
ϕr ψr mL∞
|a2 |T α
|b2 |T α−γ
Γα 1 Γ α − γ 1
|v1 |Γ 2 − β
α−β α−β ϕr
ψr
t2 − t1
m
∞
L
|v2 |
1−β
1−β
t2 − t1
,
Γ 2−β
Γ α−β1
3.14
we obtain that since α > 1, α − β > 0 and 1 − β ≥ 0
c β
|ht2 − ht1 | −→ 0,
D ht2 − c Dβ ht1 −→ 0
as t2 −→ t1
3.15
and the limits are independent of x ∈ Br and h ∈ Nx.
Step 4 N has a closed graph. Let xn → x∗ , hn ∈ Nxn , and hn → h∗ ; we need to show
h∗ ∈ Nx∗ . Now hn ∈ Nxn implies that there exists un ∈ SF,xn such that hn t Sun t for
t ∈ 0, T . Let us consider the continuous linear operator Γ : L1 0, T , R → X given by
Γut t
0
t
t − sα−1
usds −
Γα
v1
T
a2
0
T − sα−1
usds b2
Γα
T
0
T − sα−γ−1
usds
Γ α−γ
3.16
and denote wt v2 t c1 /a1 . Then hn t − wt Γun t and
hn − h∗ max |hn t − wt − h∗ t − wt|
t∈0,T max c Dβ hn − wt − c Dβ h∗ − wt −→ 0 as n −→ ∞.
t∈0,T 3.17
12
Abstract and Applied Analysis
We apply Lemma 2.3 to find that Γ ◦ SF has closed graph and from the definition of Γ we get
hn − w ∈ Γ ◦ SF xn . Since xn → x∗ , hn − w → h∗ − w, it follows the existence of u∗ ∈ SF,x∗
such that h∗ − w Γu∗ . This means that h∗ ∈ Nx∗ .
Step 5 a priori bounds on solutions. Let x ∈ λNx for some λ ∈ 0, 1. Then there exists
u ∈ SF,x such that xt λSut for t ∈ 0, T . With the same arguments as in Step 2 of our
proof, for each t ∈ 0, T , we obtain
|xt| c Dβ xt ≤ P ϕx ψx mL∞ Q.
3.18
x ≤ P ϕx ψx mL∞ Q.
3.19
U {x ∈ X : x < L}.
3.20
Thus
Now we set
Clearly, U is an open subset of X and 0 ∈ U. As a consequence of Steps 1–4, together with
the Arzela-Ascoli theorem, we can conclude that N : U → Pcp,c X is upper semicontinuous
and completely continuous. From the choice of the U, there is no x ∈ ∂U such that x ∈
λNx for some λ ∈ 0, 1. Therefore, by the nonlinear alternative of Leary-Schauder type
Theorem 2.11, we deduce that N has a fixed point x ∈ U, which is a solution of the problem
1.4, 1.5. This completes the proof.
Theorem 3.2. Assume that (H1) is satisfied and there exists L1 > 0 such that
L1
> 1,
P1 ϕL1 ψL1 mL∞ Q1
3.21
b1 c2 T γ Γ2 − γ T γ Γ 2 − γ |c2 |
c1 T −β
−
P1 1 ,
a1 b1 b2
a1 b1 |b2 |
Γ 2−β
Γ 2−γ
1
|b1 |
α
Q1 T 1 Γα 1 Γ α − γ 1
|a1 b1 |
Γ 2−γ
1
α−β
T
.
Γ α−β1
Γ 2−β Γ α−γ 1
3.22
where
Then the problem 1.4, 1.6 has at least one solution on 0, T .
Proof. To obtain the result, the main aim is to study the properties of the operator M defined
in 3.3. The proof of them is similar to those of Theorem 3.1, so we omit the details. Here
Abstract and Applied Analysis
13
we just give some estimations, which are needed in the following theorems. Let x ∈ X and
h ∈ Mx; then there exists u ∈ SF,x such that
ht Kut,
for t ∈ 0, T .
3.23
We put Ku K1 u K2 u and
K1 ut t
0
t − sα−1
usds,
Γα
K2 ut −k2u t − k1u ;
3.24
here k1u and k2u are constants given by
k1u
b1 c2 T γ Γ 2 − γ
c1
−
a1 b1
a1 b1 b2
T
T T − sα−γ−1
b1
T − sα−1
γ
usds − T Γ 2 − γ
usds ,
a1 b1
Γα
Γ α−γ
0
0
T
Γ 2−γ
c2
T − sα−γ−1
u
k2 .
usds −
b2
T 1−γ
Γ α−γ
0
3.25
By simple calculations, we have
t
Tα
t − sα−1
,
|us|ds ≤ ϕx ψx mL∞
Γα
Γα 1
0
|K2 ut| ≤ T k2u k1u ,
α−γ
u
T
|
|c
2
γ
T k2 ≤ T Γ 2 − γ
ϕx ψx mL∞ ,
|b2 |
Γ α−γ 1
u
Γ 2 − γ Tα
Tα
k ≤ |b1 |
ϕx ψx mL∞ ϕx ψx mL∞
1
Γα 1
|a1 b1 |
Γ α−γ 1
b1 c2 T γ Γ2 − γ c1 −
,
a1 b1 b2
a1 b1 |K1 ut| ≤
t t − sα−β−1
c β
D
ht
|us|ds ≤
0 Γ α−β
u 1−β
k T
2
Γ 2−β
Γ 2 − γ T γ−β
T α−β
≤ ϕx ψx mL∞ Γ α−β1
Γ 2−β
T α−γ
|c2 |
×
.
ϕx ψx mL∞ |b2 |
Γ α−γ 1
3.26
14
Abstract and Applied Analysis
Hence we obtain
b1 c2 T γ Γ2 − γ T γ Γ 2 − γ |c2 |
T −β
c1 1 −
h ≤
a1 b1 b2
a1 b1 |b2 |
Γ 2−β
Γ 2−γ
1
|b1 |
α
ϕx ψx mL∞ T 1 Γα 1 Γ α − γ 1
|a1 b1 |
Γ 2−γ
1
α−β
ϕx ψx mL∞ T
.
Γ α−β1
Γ 2−β Γ α−γ 1
3.27
This is the end of the proof.
3.2. Nonconvex Case
Now we study the case when F is not necessarily convex valued.
A subset A of L1 0, T , R is decomposable if for all u, v ∈ A and J ⊆ 0, T Lebesgue
measurable, then uχJ vχ0,T −J ∈ A, where χ stands for the characteristic function.
H2: F : 0, T × R × R → Pcp R is a multivalued map such that 1 t, x, y → Ft, x, y
is Σ ⊗ BR ⊗ BR measurable; 2 x, y → Ft, x, y is lower semicontinuous for a.e.
t ∈ 0, T .
Theorem 3.3. Let (H1)2, (H2), and relation 3.6 hold; then the problem 1.4, 1.5 has at least
one solution on 0, T .
Proof. From H12, H2, and 33, Lemma 4.1, the map
F : X → P L1 0, T , R ,
x −→ Fx SF,x
3.28
is lower semicontinuous and has nonempty closed and decomposable values. Then from a
selection theorem due to Bressan and Colombo 34, there exists a continuous function f :
X → L1 0, T , R such that fx ∈ Fx for all x ∈ X. That is to say, we have fxt ∈
Ft, xt, c Dβ xt for a.e. t ∈ 0, T . Now consider the problem
c
Dα xt fxt,
t ∈ 0, T 3.29
with the boundary conditions 1.5. Note that if x ∈ X is a solution of the problem 3.29,
then x is a solution to the problem 1.4, 1.5.
Problem 3.29 is then reformulated as a fixed point problem for the operator N1 :
X → X defined by
N1 xt Sfx t.
3.30
It can easily be shown that N1 is continuous and completely continuous and satisfies all
conditions of the Leray-Schauder nonlinear alternative for single-valued maps 31. The
Abstract and Applied Analysis
15
remaining part of the proof is similar to that of Theorem 3.1, so we omit it. This completes
the proof.
Theorem 3.4. Let (H1)2, H2, and relation 3.21 hold, then the problem 1.4, 1.6 has at least
one solution on 0, T .
The proof of this theorem is similar to that of Theorem 3.3.
H3: F : 0, T × R × R → Pcp R is a multivalued map such that 1 F is integrably
bounded and the map t → Ft, x, y is measurable for all x, y ∈ R; 2 there exists
m ∈ L∞ 0, T , R such that for a.e. t ∈ 0, T and all x1 , x2 , y1 , y2 ∈ R,
h F t, x1 , y1 , F t, x2 , y2 ≤ mt |x1 − x2 | y1 − y2 .
3.31
Theorem 3.5. Let (H3) hold, if, in addition,
mL∞
Tα
T α−β
Γα 1 Γ α − β 1
T
T 1−β
|v1 | |v1 |Γ 2 − β
|a2 |T α
|b2 |T α−γ
Γα 1 Γ α − γ 1
< 1,
3.32
then the problem 1.4, 1.5 has at least one solution on 0, T .
Proof. From H3, we have that the multivalued map t → Ft, xt, c Dβ xt is measurable
28, Proposition 2.7.9 and closed valued for each x ∈ X. Hence it has measurable selection
28, Theorem 2.2.1 and the set SF,x is nonempty. Let N be defined in 3.2. We will show that,
under this situation, N satisfies the requirements of Theorem 2.12.
Step 1. For each x ∈ X, Nx ∈ Pcl X. Let hn ∈ Nx, n ≥ 1 such that hn → h in X. Then
h ∈ X and there exists un ∈ SF,x , n ≥ 1 such that
hn t Sun t,
t ∈ 0, T .
3.33
By H3, the sequence un is integrable bounded. Since F has compact values, we may pass to
a subsequence if necessary to get that un converges to u in L1 0, T , R. Thus u ∈ SF,x and for
each t ∈ 0, T hn t −→ ht Sut.
3.34
This means that h ∈ Nx and Nx is closed.
Step 2. There exists ρ < 1 such that
h Nx, N y ≤ ρx − y,
∀x, y ∈ X.
3.35
Let x, y ∈ X and h1 ∈ Ny; then there exists u1 ∈ SF,y such that
h1 t Su1 t,
t ∈ 0, T .
3.36
16
Abstract and Applied Analysis
From H32, we deduce
h F t, xt, c Dβ xt , F t, yt, c Dβ yt
≤ mt xt − yt c Dβ xt − c Dβ yt .
3.37
Hence, for a.e. t ∈ 0, T , there exists v ∈ Ft, xt, c Dβ xt such that
|u1 t − v| ≤ mt xt − yt c Dβ xt − c Dβ yt .
3.38
Consider the multivalued map V : 0, T → P R given by
V t u ∈ R : |u1 t − u| ≤ mt xt − yt c Dβ xt − c Dβ yt .
3.39
Since u1 t, αt mt|xt − yt| |c Dβ xt − c Dβ yt| are measurable, 35, Theorem III.41
implies that V is measurable. It follows from H3 that the map t → Ft, xt, c Dβ xt is
measurable. Hence by 3.38 and 28, Proposition 2.1.43, the multivalued map t → V t ∩
Ft, xt, c Dβ xt is measurable and nonempty closed valued. Therefore, we can find u2 t ∈
Ft, xt, c Dβ xt such that for a.e. t ∈ 0, T ,
|u1 t − u2 t| ≤ mt xt − yt c Dβ xt − c Dβ yt .
3.40
Let h2 t Su2 t, that is, h2 ∈ Nx. Since
t t − sα−1
u1 s − u2 sds
|S1 u1 t − S1 u2 t| 0 Γα
≤
t
0
≤
t − sα−1
ms xs − ys c Dβ xs − c Dβ ys ds
Γα
mL∞ T α x − y,
Γα 1
|S2 u1 t − S2 u2 t| |tku1 − ku2 |
a T T − sα−1
2
≤ T
u1 s − u2 sds
v1 0
Γα
T − sα−γ−1
u1 s − u2 sds
Γ α−γ
0
mL∞ T
|b2 |T α−γ
|a2 |T α
≤
x − y,
Γα 1 Γ α − γ 1
|v1 |
b2
v1
T
Abstract and Applied Analysis
1−β
1−β t
t
k
k
u
u
c β
β
− I α−β u2 t 2
D h1 t − c D h2 t I α−β u1 t − 1
Γ 2−β
Γ 2−β 17
m ∞ T α−β ≤ L
x − y Γ α−β1
mL∞ T 1−β
|b2 |T α−γ
|a2 |T α
x − y,
Γα 1 Γ α − γ 1
|v1 |Γ 2 − β
3.41
we obtain
h1 − h2 ≤ mL∞
Tα
T α−β
Γα 1 Γ α − β 1
T
T 1−β
|a2 |T α
|b2 |T α−γ
x − y.
Γα 1 Γ α − γ 1
|v1 | |v1 |Γ 2 − β
3.42
Denote
ρ mL∞
Tα
T α−β
Γα 1 Γ α − β 1
T
T 1−β
|v1 | |v1 |Γ 2 − β
|a2 |T α
|b2 |T α−γ
.
Γα 1 Γ α − γ 1
3.43
By using an analogous relation obtained by interchanging the roles of x and y, we get
h Nx, N y ≤ ρx − y.
3.44
Therefore, from condition 3.32, Theorem 2.12 implies that N has a fixed point, which is a
solution of the problem 1.4, 1.5. This completes the proof.
Theorem 3.6. Let (H3) hold, if, in addition,
T
1
mL∞
α
|b1 |
|a1 b1 |
mL∞ T α−β
Γ 2−γ
1
Γα 1 Γ α − γ 1
Γ 2−γ
1
< 1,
Γ α−β1
Γ 2−β Γ α−γ 1
then the problem 1.4, 1.6 has at least one solution on 0, T .
3.45
18
Abstract and Applied Analysis
Using the arguments employed in the proof of Theorem 3.5, we can prove this theorem
similarly. Hence the details are omitted here.
4. Integral Boundary Conditions
In this section, the existence results of the problems 1.4, 1.5 and 1.4, 1.6 obtained in
the previous section will be extended to the ones of the problems of fractional differential
inclusions 1.4 subject to the integral boundary conditions 1.7 and 1.8.
Lemma 4.1. For any y, ξ, χ ∈ C0, T , R, the unique solution of the fractional separated integral
boundary value problem,
Dα xt yt,
t ∈ 0, T , 1 < α ≤ 2,
T
c γ
ξsds,
a1 x0 b1 D x0 c1
c
a2 xT b2
c
D xT c2
γ
0
T
χsds,
4.1
0 < γ < 1,
0
is given by
xt t
t − sα−1
ysds
Γα
0
T
T
t
T − sα−1
T − sα−γ−1
−
a2
ysds b2
ysds
v1
Γα
Γ α−γ
0
0
c2 t
v1
T
0
c1 v1 − a2 t
χsds a1 v1
4.2
T
ξsds.
0
Lemma 4.2. For any y,ξ,χ ∈ C0, T , R, the unique solution of the fractional nonseparated integral
boundary value problem,
c
Dα xt yt,
t ∈ 0, T , 1 < α ≤ 2,
T
a1 x0 b1 xT c1
ξsds,
0
c
γ
a2 D x0 b2
c
D xT c2
γ
T
χsds,
0
0 < γ < 1,
4.3
Abstract and Applied Analysis
19
is given by
xt T
tΓ 2 − γ
tΓ 2 − γ c2 T
t − sα−1
T − sα−γ−1
ysds −
ysds
χsds
Γα
T 1−γ
T 1−γ b2
Γ α−γ
0
0
0
T
T T − sα−γ−1
b1
T − sα−1
γ
ysds − T Γ 2 − γ
−
ysds
a1 b1
Γα
Γ α−γ
0
0
T
b1 T γ Γ 2 − γ c2 T
c1
χsds ξsds.
−
b2 a1 b1 a1 b1 0
0
t
4.4
To obtain the existence results of the problems 1.4, 1.7 and 1.4, 1.8, in view of
Lemmas 4.1 and 4.2, we define two operators Π, Ω : X → P X as
Πx {h ∈ X : h Hv, v ∈ SF,x },
4.5
Ωx {h ∈ X : h Zv, v ∈ SF,x }
4.6
with
Hvt t
t − sα−1
vsds
Γα
0
T
T
t
T − sα−1
T − sα−γ−1
−
a2
vsds b2
vsds
v1
Γα
Γ α−γ
0
0
c1 T
c1 a2 t T
gs, xsds −
gs, xsds,
a1 0
a1 v1 0
0
T
t
tΓ 2 − γ
t − sα−1
T − sα−γ−1
vsds −
vsds
Zvt Γα
T 1−γ
Γ α−γ
0
0
tΓ 2 − γ c2 T
hs, xsds
T 1−γ b2
0
T
T T − sα−γ−1
b1
T − sα−1
γ
−
vsds − T Γ 2 − γ
vsds
a1 b1
Γα
Γ α−γ
0
0
T
b1 T γ Γ 2 − γ c2 T
c1
−
hs, xsds gs, xsds.
b2 a1 b1 a1 b1 0
0
c2 t
v1
T
hs, xsds 4.7
Observe that if x ∈ X is a fixed point of the operator Π the operator Ω, that is, x ∈ Πx
x ∈ Ωx, then x is a solution of the problem 1.4, 1.7 the problem 1.4, 1.8.
From the definitions of the operators N, Π see 3.2, 4.5, we know that the
difference between them is very apparent, that is, c1 , c2 in 3.2 were replaced by
T
T
c1 0 gs, xsds and c2 0 hs, xsds in 4.5. This fact is also true for the operators M,
Ω see 3.3, 4.6.
20
Abstract and Applied Analysis
In the following, we state some existence results for the problems 1.4, 1.7 and 1.4,
1.8. We omit the proofs as these are similar to the ones given in Section 3.
A1: The functions g, h : 0, T × R → R are continuous. There exist functions m2 , m3 ∈
L1 0, T , R and ϕ2 , ϕ3 : 0, ∞ → 0, ∞ continuous, nondecreasing such that
gt, x ≤ m2 tϕ2 |x|,
|ht, x| ≤ m3 tϕ3 |x|
4.8
for all x ∈ R and a.e. t ∈ 0, T .
Theorem 4.3. Assume that (H1) and (A1) hold. If there exists a constant I > 0 such that
I
> 1,
ϕI ψI mL∞ Q ϕ3 Im3 L1 R ϕ2 Im2 L1 W
4.9
here Q is defined by 3.7 and
|c2 |T
R
|v1 |
T −β
1 Γ 2−β
,
|c1 |
W
|a1 |
|a2 |T
1
|v1 |
T −β
1 Γ 2−β
.
4.10
Then the boundary value problem 1.4, 1.7 has at least one solution on 0, T .
Theorem 4.4. Assume that (H1) and (A1) hold. If there exists a constant I1 > 0 such that
I1
> 1,
ϕI1 ψI1 mL∞ Q1 ϕ3 I1 m3 L1 R1 ϕ2 I1 m2 L1 W1
4.11
here Q1 is defined by 3.22 and
|c2 |T γ Γ 2 − γ
T −β
|b1 |
R1 1
,
|b2 |
|a1 b1 | Γ 2 − β
W1 |c1 |
.
|a1 b1 |
4.12
Then the boundary value problem 1.4, 1.8 has at least one solution on 0, T .
Theorem 4.5. Assume that (H1)2, (H2), (A1) and condition 4.9 hold. Then the boundary value
problem 1.4, 1.7 has at least one solution on 0, T .
Theorem 4.6. Assume that (H1)2, (H2), (A1) and condition 4.11 hold. Then the boundary value
problem 1.4, 1.8 has at least one solution on 0, T .
A2: The functions g, h : 0, T × R → R are continuous and satisfy
gt, x − g t, y ≤ m2 tx − y,
ht, x − h t, y ≤ m3 tx − y
for all x, y ∈ R and a.e. t ∈ 0, T ; here m2 , m3 ∈ L1 0, T , R .
4.13
Abstract and Applied Analysis
21
Theorem 4.7. Assume that (H3) and (A2) hold. If, in addition,
mL∞ Q m3 L1 R m2 L1 W < 1,
4.14
here Q is defined by 3.7 and R, W are defined by 4.10, then the boundary value problem 1.4,
1.7 has at least one solution on 0, T .
Theorem 4.8. Assume that (H3) and (A2) hold. If, in addition,
mL∞ Q1 m3 L1 R1 m2 L1 W1 < 1,
4.15
here Q1 is defined by 3.22 and R1 , W1 are defined by 4.12, then the boundary value problem 1.4,
1.8 has at least one solution on 0, T .
5. Examples
In this section, we give two simple examples to show the applicability of our results.
Example 5.1. Consider the following fractional boundary value problem:
c
D3/2 xt ∈ F t, xt, c D3/4 xt ,
t ∈ 0, 1,
1 c 1/2
D x0 2.5,
2
1
1 c 1/2
2x1 D x1 − ,
3
3
x0 −
5.1
where α 3/2, β 3/4, γ 1/2, a1 1, b1 −1/2, c1 2.5, a2 2, b2 1/3, c2 −1/3, T 1,
and F : 0, 1 × R × R → P R is a multivalued map given by
F t, x, y u∈R:e
−|x|
y
|x|
3
sin t ≤ u ≤ 5 −
6t cos y .
1 x2
1 y
5.2
In the context of this problem, we have
F t, x, y sup |v| : v ∈ F t, x, y ≤ 7 6t3 ≤ 13,
for t ∈ 0, 1, x, y ∈ R.
5.3
It is clear that F is convex compact valued and is of Carathéodory type. Let mt ≡ 1 and
ϕ|x| ≡ 3, ψ|y| ≡ 10; we get for t ∈ 0, 1, x, y ∈ R
F t, x, y sup |v| : v ∈ F t, x, y ≤ mt ϕ|x| ψ y .
5.4
22
Abstract and Applied Analysis
As for the condition 3.6, since P ϕ|x| ψ|y|mL∞ Q P 13Q P , Q defined in 3.6
is a constant, we can choose L large enough so that
L
> 1.
P ϕL ψL mL∞ Q
5.5
Thus, by the conclusion of Theorem 3.1, the boundary value problem 5.1 has at least one
solution on 0, 1.
Example 5.2. Consider the following fractional differential inclusion with integral boundary
conditions:
c
D7/4 xt ∈ F t, xt, c D1/2 xt ,
1
3x0 x1 3
t ∈ 0, 1,
1
gs, xsds,
5.6
0
1 1
2 c D1/4 x0 3 c D1/4 x1 hs, xsds,
4 0
where α 7/4, β 1/2, γ 1/4, T 1, a1 3, b1 1/3, c1 1, a2 2, b2 3, c2 1/4,
1 y
1 l2 t ,
F t, x, y −l1 t −
0,
− 2, −
10
16 1 y
4 t2
sin x
gt, x 1
3 t2
5.7
ht, x x,
cos x,
and l1 , l2 ∈ L1 0, 1, R .
From the data given above, we have for t ∈ 0, 1, x, y ∈ R,
sup |u| : u ∈ F t, x, y ≤ 3 h F t, x1 , y1 , F t, x2 , y2 ≤
gt, x − g t, y ≤
1
3 t2
1
4 t2
1
4 t
x − y,
2
l1 t l2 t,
|x1 − x2 | 1 y1 − y2 ,
16
5.8
ht, x − h t, y ≤ x − y.
Then let m2 t 1/3 t2 , m3 t 1, and mt 1/16 1/4 t2 ; we have
h F t, x1 , y1 , F t, x2 , y2 ≤ mt |x1 − x2 | y1 − y2 ,
mL∞ Q1 m3 L1 R1 m2 L1 W1 ≤
1
3
1
× 3.1071 1 × 0.1707 ×
0.5924 < 1.
8
9 10
5.9
Abstract and Applied Analysis
23
Here Q1 is defined by 3.22 and R1 , W1 are defined by 4.12. Hence all the assumptions of
Theorem 4.8 are satisfied, and by the conclusion of it, the boundary value problem 5.6 has
at least one solution on 0, 1.
Acknowledgment
This project was supported by NNSF of China Grants nos. 11271087 and 61263006.
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