Advances in Difference Equations Volume 2011, Article ID 404917,16pages doi:10.1155/2011/404917
Research Article
Nonlocal Boundary Value Problem for Impulsive
Differential Equations of Fractional Order
Liu Yang
1, 2and Haibo Chen
11Department of Mathematics, Central South University, Changsha, Hunan 410075, China 2Department of Mathematics and Computational Science, Hengyang Normal University,
Hengyang, Hunan 421008, China
Correspondence should be addressed to Liu Yang,yangliu19731974@yahoo.com.cn
Received 18 September 2010; Accepted 4 January 2011
Academic Editor: Mouffak Benchohra
Copyrightq2011 L. Yang and H. Chen. 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 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, see4–11and 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.12investigated the following nonlocal boundary value problem
cDαut ft, ut 0, 0< t < T, 1< α≤2, u0 gu, uT uT,
1.1
Zhong and Lin13studied the following nonlocal and multiple-point boundary value problem
cDαut ft, ut 0, 0< t <1, 1< α≤2,
u0 u0gu, u1 u1
m−2
i 1
biuξi.
1.2
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 Sivasundaram15,16studied the following impulsive hybrid boundary value problems for fractional differential equations, respectively,
cDqut ft, ut 0, 1< q≤2, t∈J
1 0,1\
t1, t2, . . . , tp
,
Δutk Ikut−k, Δutk Jkut−k, tk∈0,1, k 1,2, . . . , p,
u0 u0 0, u1 u1 0,
1.3
cDqut ft, ut 0, 1< q≤2, t∈J
1 0,1\
t1, t2, . . . , tp
,
Δutk Ikut−k, Δutk Jkut−k, tk∈0,1, k 1,2, . . . , p,
αu0 βu0
1
0
q1usds, αu1 βu1
1
0
q2usds.
1.4
Motivated by the facts mentioned above, in this paper, we consider the following problem:
cDqut ft, ut, ut, 1< q≤2, t∈ J
1 0,1\
t1, t2, . . . , tp
,
Δutk Ikut−k, Δutk Jkut−k, tk∈0,1, k 1,2, . . . , p,
αu0 βu0 g1u, αu1 βu1 g2u,
1.5
where J 0,1, f : J × Ê × Ê → Ê is a continuous function, and Ik, Jk : Ê → Ê are continuous functions, Δutk utk−ut−k with utk limh→0utk h, ut−
k
limh→0−utkh, k 1,2, . . . , p, 0 t0 < t1 < t2 < · · · < tp < tp1 1, α > 0, β ≥ 0, and g1, g2: PCJ,Ê → Êare two continuous functions. We will define PCJ,ÊinSection 2.
In addition, the nonlinear term ft, ut, utinvolves 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 functiony:0,∞ → Êis given by
I0αyt 1
Γα
t
0
t−sα−1ysds, 2.1
where the right side is pointwise defined on0,∞.
Definition 2.2see1–3. The Caputo fractional derivative of orderα > 0 of a functiony :
0,∞ → Êis given by
cDαut 1
Γn−α
t
0
t−sn−α−1ynsds, 2.2
wheren α1, αdenotes the integer part of the numberα, and the right side is pointwise defined on0,∞.
Lemma 2.3 see 1–3. Let α > 0, then the fractional differential equation cDqut 0 has
solutions
ut c0c1tc2t2· · ·cn−1tn−1, 2.3
whereci∈Ê, i 0,1, . . . , n−1, n q 1.
Lemma 2.4see1–3. Letα >0, then one has
I0αcDαut ut c0c1tc2t2· · ·cn−1tn−1, 2.4
whereci∈Ê, i 0,1, . . . , n−1, n q 1.
Second, we define
PCJ,Ê {x:J → Ê; x∈Ctk, tk1,Ê, k 0,1, . . . , p1 andxt
k, xt−kexist
PC1J,Ê {x ∈ PCJ,Ê; x
t ∈ Ctk, tk
1,Ê, k 0,1, . . . , p 1, x t
k, xt−k exist, andxis left continuous attk, k 1, . . . , p}. LetC PC1J,Ê; it is a Banach space with the normx supt∈J{xtPC,xtPC}, wherexPC
supt∈J|xt|.
Like Definition 2.1 in16, we give the following definition.
Definition 2.5. A functionu ∈ C with its Caputo derivative of order qexisting on J1 is a solution of1.5if it satisfies1.5.
To deal with problem1.5, we first consider the associated linear problem and give its solution.
Lemma 2.6. Assume that
Ji
⎧ ⎨ ⎩
t0, t1, i 0,
ti, ti1, i 1,2, . . . , p,
Xt
⎧ ⎨ ⎩
0, t∈J0,
1, t ∈ J0.
2.5
For anyσ∈C0,1, the solution of the problem
cDqut σt, 1< q≤2, t∈ J
1 0,1\
t1, t2, . . . , tp
,
Δutk Ikut−k, Δutk Jkut−k, tk∈0,1, k 1,2, . . . , p,
αu0 βu0 g1u, αu1 βu1 g2u
2.6
is given by
ut
t
ti
t−sq−1σs
Γq ds
β α−t
1
tp
1−sq−1σs
Γq ds β α
1
tp
1−sq−2σs
Γq−1 ds
0<tk<1
tk
tk−1
tk−sq−1σs
Γq dsIk
ut−k
0<tk<1
β
α1−tk
tk
tk−1
tk−sq−2σs
Γq−1 dsJk
ut−k
Xt
0<tk<t
tk
tk−1
tk−sq−1σs
Γq dsIk
ut−k
Xt
0<tk<t
t−tk
tk
tk−1
tk−sq−2σs
Γq−1 dsJk
ut−k
1
α2
αg1u Xt
αt−βg2u−g1u
, fort∈Ji, i 0,1, . . . , p.
2.7
Proof. By Lemmas2.3and2.4, the solution of2.6can be written as
ut I0qσt−b0−b1t
t
0
t−sq−1
Γq σsds−b0−b1t, t∈0, t1, 2.8
whereb0, b1∈Ê. Taking into account that
cDqIq
0ut ut, I
q
0I
p
0ut I
pq
0 utforp, q >0, we obtain
ut
t
0
t−sq−2σs
Γq−1 ds−b1. 2.9
Usingαu0 βu0 g1u, we get
ut
t
0
t−sq−1
Γq σsdsb1
β
α−t
1
αg1u, t∈0, t1. 2.10
Ift∈t1, t2, then we have
ut
t
t1
t−sq−1σs
Γq ds−c0−c1t−t1, 2.11
wherec0, c1∈Ê. In view of the impulse conditionsΔut1 ut
1−ut−1 I1ut−1,Δut1
ut1−ut−1 J1ut−1, we have
ut
t
t1
t−sq−1σs
Γq ds
t1
0
t1−sq−1σs
Γq dsb1
β
α−t
1
αg1u I1
ut−1
t−t1
t1
0
t1−sq−2σs
Γq−1 dsJ1
ut−1
, t∈t1, t2.
Repeating the process in this way, the solutionutfort∈tk, tk1can be written as
ut
t
tk
t−sq−1σs
Γq dsb1
β α−t
1
αg1u
0<tk<t
tk
tk−1
tk−sq−1σs
Γq dsIk
ut−k
0<tk<t
t−tk
tk
tk−1
tk−sq−2σs
Γq−1 dsJk
ut−k
, t∈tk, tk1.
2.13
Applying the boundary conditionαu1 βu1 g2u, we find that
b1
1
tp
1−sq−1σs
Γq ds
β
α
1
tp
1−sq−2σs
Γq−1 ds
0<tk<1
tk
tk−1
tk−sq−1σs
Γq dsIk
ut−k
0<tk<1
β
α1−tk
tk
tk−1
tk−sq−2σs
Γq−1 dsJk
ut−k
1
α
g1u−g2u
.
2.14
Substituting the value ofb1into2.10and2.13, we obtain2.7.
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 2.7see13,22. Denote byUan open set in a closed, convex setY of a Banach spaceE. Assume that0∈ U. Also assume thatFUis bounded and thatF:U → Y is given byF F1F2, in whichF1 : U → Eis continuous and completely continuous and F2 : U → Eis a nonlinear contraction (i.e., there exists a nonnegative nondecreasing functionφ : 0,∞ → 0,∞satisfying
φz< zforz >0, such thatF2x−F2y ≤φx−yfor allx, y∈ U, then either
C1Fhas a fixed pointu∈ U, or
3. Main Results
A1f : 0,1×Ê×Ê → Ê is continuous. There exists a nonnegative functionpt ∈
Theorem 3.1. Assume that A1, A2, and A3 are satisfied; moreover, supr∈0,∞r/H
Kψr>1/1−L, whereH max{H1, H2}, K max{K1, K2}, then the problem1.5has at least one solution.
Proof. The proof will be given in several steps.
Step 1. The operatorF1 :Ωr → Cis completely continuous.
Taking into account the uniform continuity of the functiontq, tq−1on0,1, we get thatF
1is equicontinuity onΩr. By the Lemma 5.4.1 in23, we haveF1Ωras relatively compact. Due
to the continuity off, Ik, Jk, it is clear thatF1is continuous. Hence, we complete the proof of
Step 1.
Step 2. FUis bounded.
From supr∈0,∞r/H Kψr > 1/1−L, it follows that there exists a positive constantr0, such that
r0
HKψr0
> 1
1−L. 3.8
Now, we verify the validity of all the conditions inLemma 2.6with respect to the operator
F1, F2, andF. LetΩr0 U. FromA2, we have
|F2ut| ≤ 1
α2
α1−t βg1u−g10αt−βg2u−g20
≤ 1
α2
αβl1r0l2r0, fort∈Ji,
F2ut 1
αl2r0l1r0, fort∈Ji, i 0,1, . . . , p.
3.9
Combining with the property that F1U is boundedStep 1, we have F bounded on U. Hence, we can assume thatFU ≤G,G >0 is a constant.
Step 3. F2is a nonlinear contraction.
Let Y Ωr1, r1 max{G, r0}, E C. By A2, we obtain |F2ut−F2vt| ≤
1/α2|α1−t βg
1u−g1v||αt−βg2u−g2v|≤αβ/α2l1l2u−v, and|F2ut−F2vt| ≤1/αl1l2u−v ≤Lu−v, for t∈Ji. SinceL <1, we have
F2u−F2v ≤φu−v, that is,F2is a nonlinear contractionφz Lz.
Step 4. C2inLemma 2.7does not occur.
To this end, we perform the argument by contradiction. Suppose thatC2holds, then there existλ∈0,1, u∈∂Ωr0, such thatu λFu. Hence, we can obtainu r0and
|u| ≤
t
ti
t−sq−1psψr0
Γq ds
β α1
1
tp
1−sq−1psψr0
Γq ds β α
1
tp
1−sq−2psψr0
Γq−1 ds
0<tk<1
tk
tk−1
tk−sq−1psψr0
Γq dsM
0<tk<1
β
α1−tk
tk
tk−1
tk−sq−2psψr0
Γq−1 dsM
ψr0
P
Γq
1
tp
1−sq−1ps
Γq ds β α
1
tp
1−sq−2ps
Γq−1 ds
0<tk<1
tk
tk−1
tk−sq−1ps
Γq ds
×
0<tk<1
β α1
tk
tk−1
tk−sq−2ps
Γq−1 ds
0<tk<1
tk
tk−1
tk−sq−2ps
Γq−1 ds
≤Lr0H2K2ψr0.
3.10
Therefore,r0 ≤Lr0HKψr0. However, it contradicts with3.8.
Hence, by using Steps1–4, Lemmas2.6and2.7,Fhas at least one fixed pointu∈Ωr0,
which is the solution of problem1.5.
Next, we will give some corollaries.
Corollary 3.2. Assume that A1,A2, andA3 are satisfied; moreover,lim supr∈0,∞r/H
Kψr ∞, whereH max{H1, H2}, K max{K1, K2}; then the problem1.5has at least one solution.
Assume that,
A1 sublinear growth,f :0,1×Ê×Ê → Êis continuous. There exists a nonnegative functionpt ∈ C0,1withpt > 0 on a subinterval of0,1. Also there exists a constantγ∈0,1, such that|ft, u, v| ≤pt|u|γfor anyt, u, v∈0,1×
Ê×Ê.
Corollary 3.3. Assume thatA1,A2, andA3are satisfied, then the problem1.5has at least one solution.
Assume that
B1f :0,1×Ê → Êis continuous. There exists a nonnegative functionpt∈C0,1 withpt >0 on a subinterval of0,1. Also there exists a constantγ ∈ 0,1such that|ft, u| ≤pt|u|γfor anyt, u∈0,1×
Ê,
B2there exist two positive constants l1, l2 such that αβ/α2l1 l2 L < 1. Moreover,q10 0, q20 0, and
q1u−q1v≤l1u−v, q2u−q2v≤l2u−v, ∀u, v∈ C, 3.11
B3Ik, Jk:Ê → Êare continuous. There exists a positive constantM, such that
Corollary 3.4. Assume thatB1,B2, andB3are satisfied, then the problem1.4has at least one solution.
Assume that
B1f :0,1×Ê → Êis continuous. There exists a nonnegative functionpt∈C0,1 withpt>0 on a subinterval of0,1.|ft, u| ≤ptfor anyt, u∈0,1×Ê.
Corollary 3.5. Assume thatB1,B2, andB3are satisfied, then the problem1.4has at least one solution.
Remark 3.6. Compared with Theorem 3.2 in 16, Corollary 3.5 does not need conditions
ft, u−ft, v ≤L1u−v, Iku−IkV ≤L2u−v, andJku−JkV ≤L2u−v. Moreover, we only needαβ/α2l1l2 L <1.
Example 3.7. Consider the following problem:
cD3/2ut θu2sin2ut, 0< t <1, t∈ J
1 0,1\
1 2
,
Δu
1 2
u2
1u2, Δu
1 2
u2
2u2,
u0 u0
1
0
|us|
8|us|ds, u1 u
1 1 0
|us|
8|us|ds,
3.13
whereθ > 0. Here,α β 1, p 1, q 3/2. Letps≡θ P andψu u2, then we can see thatA1holds. Choosingl1 l2 1/8, L 1/2, we can easily obtain thatA2holds. Let
M 1, then we have thatA3also holds. Moreover,H 8, K 426
√
2/3√πθ. Hence, we get supr∈0,∞r/HKψr 1/2
8θ426√2/3√π >1/1−L 2 for any given 0 < θ <3√π/128426√2. Therefore, ByTheorem 3.1, the above problem3.13has at least one solution for 0< θ <3√π/128426√2.
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