Volume 2011, Article ID 404917, 16 pagesdoi:10.1155/2011/404917 Research Article Nonlocal Boundary Value Problem for Impulsive Differential Equations of Fractional Order 1 Department of
Trang 1Volume 2011, Article ID 404917, 16 pages
doi:10.1155/2011/404917
Research Article
Nonlocal Boundary Value Problem for Impulsive Differential Equations of Fractional Order
1 Department of Mathematics, Central South University, Changsha, Hunan 410075, China
2 Department of Mathematics and Computational Science, Hengyang Normal University,
Hengyang, Hunan 421008, China
Received 18 September 2010; Accepted 4 January 2011
Academic Editor: Mouffak Benchohra
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
We study a nonlocal boundary value problem of impulsive fractional differential equations By means of a fixed point theorem due to O’Regan, we establish sufficient conditions for the existence
of at least one solution of the problem For the illustration of the main result, an example is given
1 Introduction
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics, mechanics, aerodynamics, chemistry, and engineering and biological sciences, involves derivatives of fractional order Fractional differential equations also provide an excellent tool for the description of memory and hereditary properties of many materials and processes In consequence, fractional differential equations have emerged as a significant development in recent years, see1 3
As one of the important topics in the research differential equations, the boundary value problem has attained a great deal of attention from many researchers, see4 11 and the references therein As pointed out in12, the nonlocal boundary condition can be more use-ful than the standard condition to describe some physical phenomena There are three note-worthy papers dealing with the nonlocal boundary value problem of fractional differential equations Benchohra et al.12 investigated the following nonlocal boundary value problem
c D α u t ft, ut 0, 0 < t < T, 1 < α ≤ 2,
where c D αdenotes the Caputo’s fractional derivative
Trang 2Zhong and Lin13 studied the following nonlocal and multiple-point boundary value problem
c D α u t ft, ut 0, 0 < t < 1, 1 < α ≤ 2,
u 0 u0 gu, u1 u1m−2
i 1
Ahmad and Sivasundaram 14 studied a class of four-point nonlocal boundary value problem of nonlinear integrodifferential equations of fractional order by applying some fixed point theorems
On the other hand, impulsive differential equations of fractional order play an important role in theory and applications, see the references 15–21 and references therein However, as pointed out in 15, 16, the theory of boundary value problems for nonlinear impulsive fractional differential equations is still in the initial stages Ahmad and Sivasundaram15,16 studied the following impulsive hybrid boundary value problems for fractional differential equations, respectively,
c D q u t ft, ut 0, 1 < q ≤ 2, t ∈ J1 0, 1 \t1, t2, , t p
,
Δut k I k
u
t−k
, Δut k J k
u
t−k
, t k ∈ 0, 1, k 1, 2, , p,
u 0 u0 0, u 1 u1 0,
1.3
c D q u t ft, ut 0, 1 < q ≤ 2, t ∈ J1 0, 1 \t1, t2, , t p
,
Δut k I k
u
t−k
, Δut k J k
u
t−k
, t k ∈ 0, 1, k 1, 2, , p,
αu 0 βu0
1 0
q1usds, αu 1 βu1
1 0
q2usds.
1.4
Motivated by the facts mentioned above, in this paper, we consider the following problem:
c D q u t ft, u t, ut, 1 < q ≤ 2, t ∈ J1 0, 1 \t1, t2, , t p
,
Δut k I k
u
t−k
, Δut k J k
u
t−k
, t k ∈ 0, 1, k 1, 2, , p,
αu 0 βu0 g1u, αu 1 βu1 g2u,
1.5
where J 0, 1, f : J × Ê × Ê → Ê is a continuous function, and I k , J k : Ê → Ê
are continuous functions, Δut k ut
k − ut−
k with ut
k limh→ 0 u t k h, ut−
k limh→ 0 −u t k h, k 1, 2, , p, 0 t0 < t1 < t2 < · · · < t p < t p1 1, α > 0, β ≥ 0, and
g1, g2: PCJ,Ê → Êare two continuous functions We will define PCJ,Ê inSection 2
To the best of our knowledge, this is the first time in the literatures that a nonlocal boundary value problem of impulsive differential equations of fractional order is considered
Trang 3In addition, the nonlinear term ft, ut, ut involves ut Evidently, problem 1.5 not only includes boundary value problems mentioned above but also extends them to a much wider case Our main tools are the fixed point theorem of O’Regan Some recent results in the literatures are generalized and significantly improvedseeRemark 3.6
The organization of this paper is as follows InSection 2, we will give some lemmas which are essential to prove our main results InSection 3, main results are given, and an example is presented to illustrate our main results
2 Preliminaries
At first, we present here the necessary definitions for fractional calculus theory These definitions and properties can be found in recent literature
Definition 2.1 see 1 3 The Riemann-Liouville fractional integral of order α > 0 of a function y : 0, ∞ → Êis given by
I0α y t 1
Γα
t 0
where the right side is pointwise defined on0, ∞.
Definition 2.2see 1 3 The Caputo fractional derivative of order α > 0 of a function y :
0, ∞ → Êis given by
c D α u t 1
Γn − α
t 0
where n α1, α denotes the integer part of the number α, and the right side is pointwise
defined on0, ∞.
Lemma 2.3 see 1 3 Let α > 0, then the fractional differential equation c D q u t 0 has
solutions
u t c0 c1t c2t2 · · · c n−1t n−1, 2.3
where c i∈Ê, i 0, 1, , n − 1, n q 1.
Lemma 2.4 see 1 3 Let α > 0, then one has
I0α c D α u t ut c0 c1t c2t2 · · · c n−1t n−1, 2.4
where c i∈Ê, i 0, 1, , n − 1, n q 1.
Second, we define
PCJ,Ê {x : J → Ê; x ∈ Ct k , t k1,Ê, k 0, 1, , p 1 and xt
k , xt−
k exist
with xt−
k xt k , k 1, , p}.
Trang 4PC1J,Ê {x ∈ PCJ,Ê; xt ∈ Ct k , t k1,Ê, k 0, 1, , p 1, xt
k,
xt−
k exist, and xis left continuous at t k , k 1, , p} Let C PC1J,Ê; it
is a Banach space with the normx sup t ∈J {xtPC, xtPC}, where xPC supt ∈J |xt|.
Like Definition 2.1 in16, we give the following definition
Definition 2.5 A function u ∈ C with its Caputo derivative of order q existing on J1 is a solution of1.5 if it satisfies 1.5
To deal with problem1.5, we first consider the associated linear problem and give its solution
Lemma 2.6 Assume that
J i
⎧
⎨
⎩
t0, t1, i 0,
t i , t i1, i 1, 2, , p, Xt
⎧
⎨
⎩
0, t ∈ J0,
1, t ∈ J0.
2.5
For any σ ∈ C0, 1, the solution of the problem
c D q u t σt, 1 < q ≤ 2, t ∈ J1 0, 1 \t1, t2, , t p
,
Δut k I k
u
t−k
, Δut k J k
u
t−k
, t k ∈ 0, 1, k 1, 2, , p,
αu 0 βu0 g1u, αu 1 βu1 g2u
2.6
is given by
u t
t
t i
t − s q−1σ s
Γq ds
β
α − t 1
t p
1 − s q−1σ s
Γq dsβ
α
1
t p
1 − s q−2σ s
Γq− 1 ds
0<t k <1
t k
t k−1
t k − s q−1σ s
Γq ds I k
u
t−k
0<t <1
β
α 1 − t k
t k
t k−1
t k − s q−2σ s
Γq− 1 ds J k
u
t−k
Trang 5Xt
0<t k <t
t k
t k−1
t k − s q−1σ s
Γq ds I k
u
t−k
Xt
0<t k <t
t − t k
t k
t k−1
t k − s q−2σ s
Γq− 1 ds J k
u
t−k
1
α2
αg1u Xtαt − βg2u − g1u, for t ∈ J i , i 0, 1, , p.
2.7
Proof By Lemmas2.3and2.4, the solution of2.6 can be written as
u t I q
0 σ t − b0− b1t
t 0
t − s q−1
Γq σsds − b0− b1t, t ∈ 0, t1, 2.8
where b0, b1∈Ê Taking into account that c D q I0qu t ut, I q
0I0pu t I p q
0 u t for p, q > 0,
we obtain
ut
t 0
t − s q−2σ s
Using αu0 βu0 g1u, we get
u t
t 0
t − s q−1
Γq σsds b1
β
α − t 1
α g1u, t ∈ 0, t1. 2.10
If t ∈ t1, t2, then we have
u t
t
t1
t − s q−1σ s
where c0, c1∈Ê In view of the impulse conditionsΔut1 ut
1−ut−
1 I1ut−
1, Δut1
ut
1 − ut−
1 J1ut−
1, we have
u t
t
t1
t − s q−1σ s
Γq ds
t1 0
t1− s q−1σ s
Γq ds b1
β
α − t
1
α g1u I1
u
t−1
t − t1 t1
0
t1− s q−2σ s
Γq− 1 ds J1
u
t−1
, t ∈ t1, t2.
2.12
Trang 6Repeating the process in this way, the solution ut for t ∈ t k , t k1 can be written as
u t
t
t k
t − s q−1σ s
Γq ds b1
β
α − t
1
α g1u
0<t k <t
t k
t k−1
t k − s q−1σ s
Γq ds I k
u
t−k
0<t k <t
t − t k
t k
t k−1
t k − s q−2σ s
Γq− 1 ds J k
u
t−k
, t ∈ t k , t k1.
2.13
Applying the boundary condition αu1 βu1 g2u, we find that
b1
1
t p
1 − s q−1σ s
Γq ds
β
α
1
t p
1 − s q−2σ s
Γq− 1 ds
0<t k <1
t k
t k−1
t k − s q−1σ s
Γq ds I k
u
t−k
0<t k <1
β
α 1 − t k
t k
t k−1
t k − s q−2σ s
Γq− 1 ds J k
u
t−k
1
α
g1u − g2u.
2.14
Substituting the value of b1into2.10 and 2.13, we obtain 2.7
Now, we introduce the fixed point theorem which was established by O’Regan in22 This theorem will be applied to prove our main results in the next section
Lemma 2.7 see 13,22 Denote by U an open set in a closed, convex set Y of a Banach space E.
Assume that 0 ∈ U Also assume that FU is bounded and that F : U → Y is given by F F1 F2,
in which F1 : U → E is continuous and completely continuous and F2 : U → E is a nonlinear
contraction (i.e., there exists a nonnegative nondecreasing function φ : 0, ∞ → 0, ∞ satisfying
φ z < z for z > 0, such that F2x − F2y ≤ φx − y for all x, y ∈ U, then either
C1 F has a fixed point u ∈ U, or
C2 there exists a point u ∈ ∂U and λ ∈ 0, 1 with u λFu, where U, ∂U represent the
closure and boundary of U, respectively.
Trang 73 Main Results
In order to applyLemma 2.7to prove our main results, we first give F, F1, F2as follows Let
Ωr {u ∈ C : u ≤ r}, r > 0,
F1u t
t
t i
t − s q−1f s, xs, xs
β
α − t 1
t p
1 − s q−1f s, xs, xs
α
1
t p
1 − s q−2f s, xs, xs
0<t k <1
t k
t k−1
t k − s q−1f s, xs, xs
u
t−k
0<t k <1
β
α 1 − t k
t k
t k−1
t k − s q−2f s, xs, xs
Γq− 1 ds J k
u
t−k
Xt
0<t k <t
t k
t k−1
t k − s q−1f s, xs, xs
u
t−k
Xt
0<t k <t
t − t k
t k
t k−1
t k − s q−2f s, xs, xs
Γq− 1 ds J k
u
t−k
,
for t ∈ J i , i 0, 1, , p,
F2u t 1
α2
αg1u Xtαt − βg2u − g1u, for t ∈ J i , i 0, 1, , p,
F F1 F2.
3.1
Clearly, for any t ∈ J i , i 0, 1, , p,
F1ut
t
t i
t − s q−2f s, xs, xs
t p
1 − s q−1f s, xs, xs
α
1
t p
1 − s q−2f s, xs, xs
0<t k <1
t k
t k−1
t k − s q−1f s, xs, xs
u
t−k
0<t k <1
β
α 1 − t k
t k
t k−1
t k −s q−2f s, xs, xs
Γq− 1 ds J k
u
t−k
Xt
0<t k <t
t k
t k−1
t k − s q−2f s, xs, xs
Γq− 1 ds J k
u
t−k
,
F2ut 1
α
Xtg2u − g1u.
3.2
Now, we make the following hypotheses
Trang 8A1 f : 0, 1 ×Ê×Ê → Ê is continuous There exists a nonnegative function pt ∈
C 0, 1 with pt > 0 on a subinterval of 0, 1 Also there exists a nondecreasing function ψ : 0, ∞ → 0, ∞ such that |ft, u, v| ≤ ptψ|u| for any t, u, v ∈
0, 1 ×Ê×Ê
A2 There exist two positive constants l1, l2 such thatα β/α2l1 l2 L < 1 Moreover, g10 0, g20 0, and
g1u − g1v ≤ l1u − v, g2u − g2v ≤ l2u − v, ∀u, v ∈ C. 3.3
A3 I k , J k:Ê → Êare continuous There exists a positive constant M such that
|I k u| ≤ M, |J k u| ≤ M, k 1, 2, , p. 3.4 Let
H1
β
α 1
Mp
β
α 1
2
H2 Mp
β
α 1
K1
1
0
1 − s q−1p s
Γq ds
β
α 1 1
t p
1 − s q−1p s
Γq ds β
α
1
t p
1 − s q−2p s
Γq− 1 ds
0<t k <1
t k
t k−1
t k − s q−1p s
0<t k <1
β
α 1
t k
t k−1
t k − s q−2p s
Γq− 1 ds
0<t k <1
t k
t k−1
t k − s q−1p s
0<t k <1
t k
t k−1
t k − s q−2p s
Γq− 1 ds,
K2 P
Γq 1
t p
1 − s q−1p s
Γq dsβ
α
1
t p
1 − s q−2p s
Γq− 1 ds
0<t k <1
t k
t k−1
t k − s q−1p s
0<t k <1
β
α 1
t k
t k−1
t k − s q−2p s
Γq− 1 ds
0<t k <1
t k
t k−1
t k − s q−2p s
Γq− 1 ds,
3.5
where P maxs ∈0,1 p s.
Now, we state our main results
Trang 9Theorem 3.1 Assume that A1, A2, and A3 are satisfied; moreover, sup r ∈0,∞ r/H
Kψ r > 1/1 − L, where H max{H1, H2}, K max{K1, K2}, then the problem 1.5 has at
least one solution.
Proof The proof will be given in several steps.
Step 1 The operator F1 :Ωr → C is completely continuous
Let M r maxs ∈0,1 {|fs, xs, xs|, x ∈ Ω r } In fact, by A1, M r can be replaced by
P ψ r For any u ∈ Ω r, we have
|F1u t| ≤
t
t i
t − s q−1f s, xs, xs
β
α 1 1
t p
1 − s q−1f s, xs, xs
β
α
1
t p
1 − s q−2f s, xs, xs
0<t k <1
t k
t k−1
t k − s q−1f s, xs, xs
Γq dsI k
u
t−k
0<t k <1
β
α 1 − t k
×
t k
t k−1
t k − s q−2f s, xs, xs
Γq− 1 dsJ k
u
t−k
Xt
0<t k <t
t k
t k−1
t k − s q−1f s, xs, xs
Γq dsI k
u
t−k
Xt
0<t k <t
t − t k
t k
t k−1
t k − s q−2f s, xs, xs
Γq− 1 dsJ k
u
t−k
≤ M r
1 0
1 − s q−1
Γq ds
β
α 1 M r
1
t p
1 − s q−1
Γq ds β
α M r
1
t p
1 − s q−2
Γq− 1 ds
0<t k <1
M r
t k
t k−1
t k − s q−1
Γq ds M
0<t k <1
β
α 1
t k
t k−1
t k − s q−2M r
Γq− 1 ds M
0<t k <1
t k
t k−1
t k − s q−1M r
Γq ds M
0<t <1
t k
t k−1
t k − s q−2M r
Γq− 1 ds M
Trang 10
≤ M r 1
Γq 1
β
α 1 M r 1
Γq 1
β
α M r
1
Γq p
M r 1
Γq 1 M
p
β
α 1
M r 1
Γq
M
p
M r 1
Γq 1 M
p
M r 1
Γq M
, for t ∈ J i , i 0, 1, , p,
F1ut ≤t
t i
t − s q−2f s, xs, xs
t p
1 − s q−1f s, xs, xs
α
1
t p
1 − s q−2f s, xs, xs
0<t k <1
t k
t k−1
t k − s q−1f s, xs, xs
Γq dsI k
u
t−k
0<t k <1
β
α 1 − t k
t k
t k−1
t k − s q−2f s, xs, xs
Γq− 1 dsJ k
u
t−k
Xt
0<t k <t
t k
t k−1
t k − s q−2f s, xs, xs
Γq− 1 dsJ k
u
t−k
≤ M r
1
Γq 1
t p
1 − s q−1M r
Γq ds β
α
1
t p
1 − s q−2M r
Γq− 1 ds
0<t k <1
t k
t k−1
t k − s q−1M r
Γq ds M
0<t k <1
β
α 1 − t k
t k
t k−1
t k − s q−2M r
Γq− 1 ds M
0<t k <1
t k
t k−1
t k − s q−2M r
Γq− 1 ds M
≤ M r
1
Γq M r
1
Γq 1
β
α M r
1
Γq p
M r
1
Γq 1 M
p
β
α 1
M r 1
Γq
M
p
M r 1
Γq M
, for t ∈ J i , i 0, 1, , p.
3.6
... represent theclosure and boundary of U, respectively.
Trang 73 Main Results
In...
2.12
Trang 6Repeating the process in this way, the solution ut for t ∈ t k...
t−k
Trang 5Xt
0<t k <t