Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 404917, 16 pages
doi:10.1155/2011/404917
Research Article
Nonlocal Boundary Value Problem for Impulsive
Differential Equations of Fractional Order
Liu Yang1, 2 and Haibo Chen1
1
2
Department of Mathematics, Central South University, Changsha, Hunan 410075, China
Department 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
Copyright q 2011 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, see 1–3.
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, see 4–11 and the
references therein. As pointed out in 12, the nonlocal boundary condition can be more useful than the standard condition to describe some physical phenomena. There are three noteworthy papers dealing with the nonlocal boundary value problem of fractional differential
equations. Benchohra et al. 12 investigated the following nonlocal boundary value problem
c
Dα ut ft, ut 0,
u0 gu,
0 < t < T, 1 < α ≤ 2,
uT uT ,
where c Dα denotes the Caputo’s fractional derivative.
1.1
2
Advances in Difference Equations
Zhong and Lin 13 studied the following nonlocal and multiple-point boundary value
problem
c
Dα ut ft, ut 0,
u0 u0 gu,
0 < t < 1, 1 < α ≤ 2,
u 1 u1 m−2
bi u ξi .
1.2
i1
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
Sivasundaram 15, 16 studied the following impulsive hybrid boundary value problems for
fractional differential equations, respectively,
Dq ut ft, ut 0, 1 < q ≤ 2, t ∈ J1 0, 1 \ t1 , t2 , . . . , tp ,
Δutk Ik u t−k , Δu tk Jk u t−k , tk ∈ 0, 1, k 1, 2, . . . , p,
c
u0 u 0 0,
1.3
u1 u 1 0,
Dq ut ft, ut 0, 1 < q ≤ 2, t ∈ J1 0, 1 \ t1 , t2 , . . . , tp ,
Δutk Ik u t−k , Δu tk Jk u t−k , tk ∈ 0, 1, k 1, 2, . . . , p,
1
1
q1 usds,
q2 usds.
αu0 βu 0 αu1 βu 1 c
0
1.4
0
Motivated by the facts mentioned above, in this paper, we consider the following
problem:
Dq ut f t, ut, u t , 1 < q ≤ 2, t ∈ J1 0, 1 \ t1 , t2 , . . . , tp ,
Δutk Ik u t−k , Δu tk Jk u t−k , tk ∈ 0, 1, k 1, 2, . . . , p,
c
αu0 βu 0 g1 u,
1.5
αu1 βu 1 g2 u,
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 → 0 utk h, ut−k limh → 0− utk h, k 1, 2, . . . , p, 0 t0 < t1 < t2 < · · · < tp < tp1 1, α > 0, β ≥ 0, and
g1 , g2 : PCJ, Ê → Ê are two continuous functions. We will define PCJ, Ê in Section 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.
Advances in Difference Equations
3
In addition, the nonlinear term ft, ut, u t involves u t. 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 improved see Remark 3.6
The organization of this paper is as follows. In Section 2, we will give some lemmas
which are essential to prove our main results. In Section 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, ∞ → Ê is given by
α
I0
yt 1
Γα
t
t − sα−1 ysds,
2.1
0
where the right side is pointwise defined on 0, ∞.
Definition 2.2 see 1–3. The Caputo fractional derivative of order α > 0 of a function y :
0, ∞ → Ê is given by
c
Dα ut 1
Γn − α
t
t − sn−α−1 yn sds,
2.2
0
where n α1, α denotes the integer part of the number α, and the right side is pointwise
defined on 0, ∞.
Lemma 2.3 see 1–3. Let α > 0, then the fractional differential equation
solutions
ut c0 c1 t c2 t2 · · · cn−1 tn−1 ,
c
Dq ut 0 has
2.3
where ci ∈ Ê, i 0, 1, . . . , n − 1, n q 1.
Lemma 2.4 see 1–3. Let α > 0, then one has
α c α
I0
D ut ut c0 c1 t c2 t2 · · · cn−1 tn−1 ,
2.4
where ci ∈ Ê, i 0, 1, . . . , n − 1, n q 1.
Second, we define
PCJ, Ê {x : J → Ê; x ∈ Ctk , tk1 , Ê, k 0, 1, . . . , p 1 and xtk , xt−k exist
with xt−k xtk , k 1, . . . , p}.
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Advances in Difference Equations
PC1 J, Ê {x ∈ PCJ, Ê; x t ∈ Ctk , tk1 , Ê, k 0, 1, . . . , p 1, x tk ,
x t−k exist, and x is left continuous at tk , k 1, . . . , p}. Let C PC1 J, Ê; it
is a Banach space with the norm x supt∈J {xtPC , x tPC }, where xPC supt∈J |xt|.
Like Definition 2.1 in 16, 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 of 1.5 if it satisfies 1.5.
To deal with problem 1.5, we first consider the associated linear problem and give its
solution.
Lemma 2.6. Assume that
Ji Xt ⎧
⎨t0 , t1 ,
i 0,
⎩
ti , ti1 ,
i 1, 2, . . . , p,
⎧
⎨0,
t ∈ J0 ,
⎩
1,
t ∈ J0 .
2.5
For any σ ∈ C0, 1, the solution of the problem
Dq ut σt, 1 < q ≤ 2, t ∈ J1 0, 1 \ t1 , t2 , . . . , tp ,
Δutk Ik u t−k , Δu tk Jk u t−k , tk ∈ 0, 1, k 1, 2, . . . , p,
c
αu0 βu 0 g1 u,
αu1 βu 1 g2 u
is given by
ut t
t − sq−1 σs
ds
Γ q
ti
1
β
β 1 1 − sq−2 σs
1 − sq−1 σs
−t
ds ds
α
α tp
Γ q
Γ q−1
tp
0<tk <1
tk
tk−1
− tk − sq−1 σs
ds Ik u tk
Γ q
tk
β
− tk − sq−2 σs
1 − tk
ds Jk u tk
α
Γ q−1
tk−1
0<tk <1
2.6
Advances in Difference Equations
tk tk − sq−1 σs
− Xt
ds Ik u tk
Γ q
tk−1
0<tk <t
t
k
− tk − sq−2 σs
Xt
ds Jk u tk
t − tk Γ q−1
tk−1
0<tk <t
1
αg1 u Xt αt − β g2 u − g1 u ,
2
α
5
for t ∈ Ji , i 0, 1, . . . , p.
2.7
Proof. By Lemmas 2.3 and 2.4, the solution of 2.6 can be written as
q
ut I0 σt − b0 − b1 t t
0
t − sq−1
σsds − b0 − b1 t,
Γ q
t ∈ 0, t1 ,
2.8
where b0 , b1 ∈ Ê. Taking into account that c Dq I0 ut ut, I0 I0 ut I0 ut for p, q > 0,
we obtain
q
u t t
0
q
p
pq
t − sq−2 σs
ds − b1 .
Γ q−1
2.9
Using αu0 βu 0 g1 u, we get
ut t
0
β
1
t − sq−1
− t g1 u,
σsds b1
α
α
Γ q
t ∈ 0, t1 .
2.10
If t ∈ t1 , t2 , then we have
ut t
t1
t − sq−1 σs
ds − c0 − c1 t − t1 ,
Γ q
2.11
where c0 , c1 ∈ Ê. In view of the impulse conditions Δut1 ut1 −ut−1 I1 ut−1 , Δu t1 u t1 − u t−1 J1 ut−1 , we have
t1
β
1
t − sq−1 σs
t1 − sq−1 σs
− t g1 u I1 u t−1
ds ds b1
ut α
α
Γ
q
Γ
q
t1
0
t
1
− t1 − sq−2 σs
, t ∈ t1 , t2 .
ds J1 u t1
t − t1 Γ q−1
0
t
2.12
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Advances in Difference Equations
Repeating the process in this way, the solution ut for t ∈ tk , tk1 can be written as
ut t
β
1
t − sq−1 σs
− t g1 u
ds b1
α
α
Γ
q
tk
tk tk − sq−1 σs
− ds Ik u tk
Γ q
tk−1
0<tk <t
t
k
− tk − sq−2 σs
,
ds Jk u tk
t − tk Γ q−1
tk−1
0<tk <t
2.13
t ∈ tk , tk1 .
Applying the boundary condition αu1 βu 1 g2 u, we find that
b1 1
tp
1 − sq−1 σs
ds
Γ q
β
α
1
tp
0<tk <1
1 − sq−2 σs
ds
Γ q−1
t
k
tk−1
− tk − sq−1 σs
ds Ik u tk
Γ q
2.14
tk
β
− tk − sq−2 σs
1 − tk
ds Jk u tk
α
Γ q−1
tk−1
0<tk <1
1
g1 u − g2 u .
α
Substituting the value of b1 into 2.10 and 2.13, we obtain 2.7.
Now, we introduce the fixed point theorem which was established by O’Regan in 22.
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
φz < z for z > 0, such that F2 x − F2 y ≤ 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.
Advances in Difference Equations
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3. Main Results
In order to apply Lemma 2.7 to prove our main results, we first give F, F1 , F2 as follows. Let
Ωr {u ∈ C : u ≤ r}, r > 0,
F1 ut t
t − sq−1 fs, xs, x s
ds
Γ q
ti
1
β
β 1 1 − sq−2 fs, xs, x s
1 − sq−1 fs, xs, x s
−t
ds ds
α
α tp
Γ q
Γ q−1
tp
tk tk − sq−1 fs, xs, x s
− dsIk u tk
Γ q
tk−1
0<tk <1
tk
β
− tk − sq−2 fs, xs, x s
1 − tk
dsJk u tk
α
Γ q−1
tk−1
0<tk <1
tk tk − sq−1 fs, xs, x s
− Xt
ds Ik u tk
Γ q
tk−1
0<tk <t
t
k
− tk − sq−2 fs, xs, x s
,
ds Jk u tk
Xt
t − tk Γ q−1
tk−1
0<tk <t
for t ∈ Ji ,
F2 ut 1
αg1 u Xt αt − β g2 u − g1 u ,
2
α
i 0, 1, . . . , p,
for t ∈ Ji , i 0, 1, . . . , p,
F F1 F2 .
3.1
Clearly, for any t ∈ Ji , i 0, 1, . . . , p,
F1 u t F2 u t t
t − sq−2 fs, xs, x s
ds
Γ q−1
ti
1
β 1 1 − sq−2 fs, xs, x s
1 − sq−1 fs, xs, x s
−
ds ds
α tp
Γ q
Γ q−1
tp
tk tk − sq−1 fs, xs, x s
− ds Ik u tk
Γ q
tk−1
0<tk <1
tk
β
− tk −sq−2 fs, xs, x s
1 − tk
ds Jk u tk
α
Γ q−1
tk−1
0<tk <1
tk tk − sq−2 fs, xs, x s
− ,
Xt
ds Jk u tk
Γ q−1
tk−1
0<tk <t
1
Xt g2 u − g1 u .
α
Now, we make the following hypotheses.
3.2
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Advances in Difference Equations
A1 f : 0, 1 × Ê × Ê → Ê is continuous. There exists a nonnegative function pt ∈
C0, 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 that α β/α2 l1 l2 L < 1.
Moreover, g1 0 0, g2 0 0, and
g1 u − g1 v ≤ l1 u − v,
A3 Ik , Jk : Ê →
g2 u − g2 v ≤ l2 u − v,
∀u, v ∈ C.
3.3
Ê are continuous. There exists a positive constant M such that
|Ik u| ≤ M,
|Jk u| ≤ M,
k 1, 2, . . . , p.
3.4
Let
2
β
β
1 Mp 1 Mp 2pM,
α
α
β
1 Mp pM,
H2 Mp α
1
1 − sq−1 ps
K1 ds
Γ q
0
1
β
β 1 1 − sq−2 ps
1 − sq−1 ps
1
ds ds
α
α tp
Γ q
Γ q−1
tp
H1 tk
tk tk − sq−1 ps
β
tk − sq−2 ps
1
ds ds
α
Γ q
Γ q−1
tk−1
0<tk <1 tk−1
0<tk <1
tk tk − sq−2 ps
tk tk − sq−1 ps
ds ds,
Γ q
Γ q−1
0<tk <1 tk−1
0<tk <1 tk−1
1
β 1 1 − sq−2 ps
1 − sq−1 ps
P
ds ds
K2 α tp
Γ q
Γ q
Γ q−1
tp
tk
tk tk − sq−1 ps
β
tk − sq−2 ps
1
ds
ds
α
Γ q
Γ q−1
tk−1
0<tk <1 tk−1
0<tk <1
tk tk − sq−2 ps
ds,
Γ q−1
0<tk <1 tk−1
where P maxs∈0,1 ps.
Now, we state our main results.
3.5
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Theorem 3.1. Assume that A1 , A2 , and A3 are satisfied; moreover, supr∈0,∞ r/H Kψr > 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 Mr maxs∈0,1 {|fs, xs, x s|, x ∈ Ωr }. In fact, by A1 , Mr can be replaced by
P ψr. For any u ∈ Ωr , we have
t − sq−1 fs, xs, x s
ds
|F1 ut| ≤
Γ q
ti
1
1 − sq−1 fs, xs, x s
β
1
ds
α
Γ q
tp
β 1 1 − sq−2 fs, xs, x s
ds
α tp
Γ q−1
tk tk − sq−1 fs, xs, x s
− ds Ik u tk
Γ q
tk−1
0<tk <1
β
1 − tk
α
0<tk <1
t
k
− tk − sq−2 fs, xs, x s
ds Jk u tk
×
Γ q−1
tk−1
tk tk − sq−1 fs, xs, x s
− Xt
ds Ik u tk
Γ q
tk−1
0<tk <t
t
k
− tk − sq−2 fs, xs, x s
Xt
ds Jk u tk
t − tk Γ q−1
tk−1
0<tk <t
1
1 − sq−1
ds
≤ Mr
Γ q
0
1
1
β
β
1 − sq−1
1 − sq−2
1 Mr
ds Mr
ds
α
α
Γ q
tp
tp Γ q − 1
tk
tk − sq−1
Mr
ds M
Γ q
tk−1
0<tk <1
tk
β
tk − sq−2 Mr
1
ds M
α
Γ q−1
tk−1
0<tk <1
tk tk − sq−1 Mr
ds M
Γ q
tk−1
0<tk <1
tk tk − sq−2 Mr
ds M
Γ q−1
tk−1
0<tk <1
t
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β
β
1
1
1
1
1 Mr ≤ Mr Mr p Mr M
α
α
Γ q1
Γ q1
Γ q
Γ q1
β
1
1
Mr p
α
Γ q M
1
1
p Mr M p Mr M , for t ∈ Ji , i 0, 1, . . . , p,
Γ q1
Γ q
t
t − sq−2 fs, xs, x s
F1 u t ≤
ds
Γ q−1
ti
1
1 − sq−1 fs, xs, x s
β 1 1 − sq−2 fs, xs, x s
ds ds
α tp
Γ q
Γ q−1
tp
t
k
tk−1
0<tk <1
− tk − sq−1 fs, xs, x s
ds Ik u tk
Γ q
tk
β
− tk − sq−2 fs, xs, x s
1 − tk
ds Jk u tk α
Γ q−1
tk−1
0<tk <1
Xt
t
k
0<tk <t
1
≤ Mr Γ q
tk−1
1
tp
− tk − sq−2 fs, xs, x s
ds Jk u tk
Γ q−1
β
1 − sq−1 Mr
ds α
Γ q
0<tk <1
t
k
tk−1
1
tp
1 − sq−2 Mr
ds
Γ q−1
tk − sq−1 Mr
ds M
Γ q
tk
β
tk − sq−2 Mr
1 − tk
ds M
α
Γ q−1
tk−1
0<tk <1
tk tk − sq−2 Mr
ds M
Γ q−1
tk−1
0<tk <1
β
1
1
1
1
Mr p Mr M
≤ Mr Mr α
Γ q
Γ q1
Γ q
Γ q1
β
1
1
Mr p
α
Γ q M
1
p Mr M , for t ∈ Ji , i 0, 1, . . . , p.
Γ q
3.6
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These imply that F1 ut ≤ B, where B is a positive constant, that is, F1 is uniformly
bounded. In addition, for any u ∈ Ωr , for all τ1 , τ2 ∈ Ji , τ1 < τ2 , we can obtain
|F1 uτ1 − F1 uτ2 |
τ2
τ2 − sq−1 fs, xs, x s
ds
≤
Γ q
τ1
τ1 τ2 − sq−1 − τ1 − sq−1 fs, xs, x s
ds
Γ q
0
1
1 − sq−1 fs, xs, x s
ds
τ2 − τ1 Γ q
tp
β 1 1 − sq−2 fs, xs, x s
ds
α tp
Γ q−1
tk tk − sq−1 fs, xs, x s
− ds Ik u tk
Γ q
tk−1
0<tk <1
tk
β
− tk − sq−2 fs, xs, x s
1 − tk
ds Jk u tk α
Γ q−1
tk−1
0<tk <1
t
k
− tk − sq−2 fs, xs, x s
ds Jk u tk τ2 − τ1 Γ q−1
tk−1
0<tk <τ1
1
τ2 − τ1 q
≤ Mr Mr −τ2 − τ1 q τ2 q − τ1 q Γ q1
Γ q1
1
β 1 1 − sq−2 Mr
1 − sq−1 Mr
τ2 − τ1 ds ds
α tp Γ q − 1
Γ q
tp
tk tk − sq−1 Mr
ds M
Γ q
tk−1
0<tk <1
tk
β
tk − sq−2 Mr
1 − tk
ds M
α
Γ q−1
tk−1
0<tk <1
t
k
tk − sq−2 Mr
τ2 − τ1 ds M ,
Γ q−1
tk−1
0<tk <τ1
F1 u τ1 − F1 u τ2 τ2
τ2 − sq−2 fs, xs, x s
≤
ds
Γ q−1
τ1
τ1 τ1 − sq−2 − τ2 − sq−2 fs, xs, x s
ds
Γ q−1
ti
≤
τ2 − τ1 q−1
τ2 − τ1 q−1 τ1 − ti q−1 − τ2 − ti q−1
Mr .
Mr Γ q
Γ q
3.7
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Taking into account the uniform continuity of the function tq , tq−1 on 0, 1, we get that F1 is
equicontinuity on Ωr . By the Lemma 5.4.1 in 23, we have F1 Ωr as relatively compact. Due
to the continuity of f, Ik , Jk , it is clear that F1 is continuous. Hence, we complete the proof of
Step 1.
Step 2. FU is bounded.
From supr∈0,∞ r/H Kψr > 1/1 − L, it follows that there exists a positive
constant r0 , such that
r0
1
.
>
H Kψr0 1 − L
3.8
Now, we verify the validity of all the conditions in Lemma 2.6 with respect to the operator
F1 , F2 , and F. Let Ωr0 U. From A2 , we have
|F2 ut| ≤
≤
1 α1 − t β g1 u − g1 0 αt − βg2 u − g2 0
2
α
1
α β l1 r0 l2 r0 ,
α2
F2 u t 1 l2 r0 l1 r0 ,
α
for t ∈ Ji ,
3.9
for t ∈ Ji , i 0, 1, . . . , p.
Combining with the property that F1 U is bounded Step 1, we have F bounded on U.
Hence, we can assume that FU ≤ G, G > 0 is a constant.
Step 3. F2 is a nonlinear contraction.
Let Y Ωr1 , r1 max{G, r0 }, E C. By A2 , we obtain |F2 ut − F2 vt| ≤
1/α2 |α1 − t βg1 u − g1 v| |αt − βg2 u − g2 v| ≤ α β/α2 l1 l2 u − v,
and |F2 u t − F2 v t| ≤ 1/αl1 l2 u − v ≤ Lu − v, for t ∈ Ji . Since L < 1, we have
F2 u − F2 v ≤ φu − v, that is, F2 is a nonlinear contraction φz Lz.
Step 4. C2 in Lemma 2.7 does not occur.
To this end, we perform the argument by contradiction. Suppose that C2 holds, then
there exist λ ∈ 0, 1, u ∈ ∂Ωr0 , such that u λFu. Hence, we can obtain u r0 and
|u| ≤
t
t − sq−1 psψr0 ds
Γ q
ti
1
β 1 1 − sq−2 psψr0 β
1 − sq−1 psψr0 1
ds ds
α
α tp
Γ q
Γ q−1
tp
tk tk − sq−1 psψr0 ds M
Γ q
tk−1
0<tk <1
tk
β
tk − sq−2 psψr0 1 − tk
ds M
α
Γ q−1
tk−1
0<tk <1
Advances in Difference Equations
tk tk − sq−1 psψr0 Xt
ds M
Γ q
tk−1
0<tk <t
13
t
k
tk − sq−2 psψr0 ds M
Xt
t − tk Γ q−1
tk−1
0<tk <t
1
α β l1 r0 l2 r0 2
α
2
β
β
1
1 Mp 1 Mp 2pM
≤ 2 α β l1 l2 r0 α
α
α
ψr0 1
1 − sq−1 ps
ds
Γ q
0
β
1
α
1
tp
1
β
1 − sq−1 ps
ds α
Γ q
tp
1 − sq−2 ps
ds
Γ q−1
tk tk − sq−1 ps
ds
Γ q
0<tk <1 tk−1
tk
β
tk − sq−2 ps
1
ds
α
Γ q−1
tk−1
0<tk <1
tk tk − sq−1 ps
tk tk − sq−2 ps
ds ds
Γ q
Γ q−1
0<tk <1 tk−1
0<tk <1 tk−1
≤ Lr0 H1 K1 ψr0 ,
u ≤
t
t − sq−2 psψr0 ds
Γ q−1
ti
1
β 1 1 − sq−2 psψr0 1 − sq−1 psψr0 ds ds
α tp
Γ q
Γ q−1
tp
t
0<tk <1
k
tk−1
tk − sq−1 psψr0 ds M
Γ q
tk
β
tk − sq−2 psψr0 1 − tk
ds M
α
Γ q−1
tk−1
0<tk <1
Xt
0<tk <t
≤
t
k
tk−1
tk − sq−2 psψr0 ds M
Γ q−1
β
1
α
β
1
Mp
pM
l
Mp
r
l
1
2 0
α
α2
1
l1 r0 l2 r0 α
14
Advances in Difference Equations
P
ψr0 Γ q
1
tp
β
1 − sq−1 ps
ds α
Γ q
1
tp
1 − sq−2 ps
ds
Γ q−1
tk tk − sq−1 ps
ds
Γ q
0<tk <1 tk−1
tk
β
tk − sq−2 ps
1
ds
×
α
Γ q−1
tk−1
0<tk <1
tk tk − sq−2 ps
ds
Γ q−1
0<tk <1 tk−1
≤ Lr0 H2 K2 ψr0 .
3.10
Therefore, r0 ≤ Lr0 H Kψr0 . However, it contradicts with 3.8.
Hence, by using Steps 1–4, Lemmas 2.6 and 2.7, F has at least one fixed point u ∈ Ωr0 ,
which is the solution of problem 1.5.
Next, we will give some corollaries.
Corollary 3.2. Assume that A1 , A2 , and A3 are satisfied; moreover, lim supr∈0,∞ r/H Kψr ∞, where H max{H1 , H2 }, K max{K1 , K2 }; then the problem 1.5 has at least
one solution.
Assume that,
A1 sublinear growth, f : 0, 1 × Ê × Ê → Ê is continuous. There exists a nonnegative
function pt ∈ C0, 1 with pt > 0 on a subinterval of 0, 1. Also there exists a
constant γ ∈ 0, 1, such that |ft, u, v| ≤ pt|u|γ for any t, u, v ∈ 0, 1 × Ê × Ê.
Corollary 3.3. Assume that A1 , A2 , and A3 are satisfied, then the problem 1.5 has at least
one solution.
Assume that
B1 f : 0, 1 × Ê → Ê is continuous. There exists a nonnegative function pt ∈ C0, 1
with pt > 0 on a subinterval of 0, 1. Also there exists a constant γ ∈ 0, 1 such
that |ft, u| ≤ pt|u|γ for any t, u ∈ 0, 1 × Ê,
B2 there exist two positive constants l1 , l2 such that α β/α2 l1 l2 L < 1.
Moreover, q1 0 0, q2 0 0, and
q1 u − q1 v ≤ l1 u − v,
B3 Ik , Jk : Ê →
q2 u − q2 v ≤ l2 u − v,
∀u, v ∈ C,
3.11
Ê are continuous. There exists a positive constant M, such that
|Ik u| ≤ M,
|Jk u| ≤ M,
k 1, 2, . . . , p.
3.12
Advances in Difference Equations
15
Corollary 3.4. Assume that B1 , B2 , and B3 are satisfied, then the problem 1.4 has at least one
solution.
Assume that
B1 f : 0, 1 × Ê → Ê is continuous. There exists a nonnegative function pt ∈ C0, 1
with pt > 0 on a subinterval of 0, 1. |ft, u| ≤ pt for any t, u ∈ 0, 1 × Ê.
Corollary 3.5. Assume that B1 , B2 , and B3 are satisfied, then the problem 1.4 has 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 ≤ L1 u − v, Ik u − Ik V ≤ L2 u − v, and Jk u − Jk V ≤ L2 u − v.
Moreover, we only need α β/α2 l1 l2 L < 1.
Example 3.7. Consider the following problem:
1
0 < t < 1, t ∈ J1 0, 1 \
D
,
2
1
u2
u2
1
,
Δu
,
Δu
2
2
1 u2
2 u2
1
1
|us|
|us|
ds,
u1 u 1 ds,
u0 u 0 8
8
|us|
|us|
0
0
c
3/2
ut θu sin u t ,
2
2 3.13
where θ > 0. Here, α β 1, p 1, q 3/2. Let ps ≡ θ P and ψu u2 , then we can
see that A1 holds. Choosing l1 l2 1/8, L 1/2, we can easily obtain√that A2 holds. Let
√
H 8, K 4 26 2/3 πθ. Hence,
M 1, then we have that A3 also holds.
Moreover,√
√
we get supr∈0,∞ r/H Kψr 1/2 8θ4 26 2/3 π > 1/1 − L 2 for any given
√
√
Theorem 3.1, the above problem 3.13 has at
0 < θ < 3 π/1284 26 2. Therefore, By √
√
least one solution for 0 < θ < 3 π/1284 26 2.
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