Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 641368, 11 pages
doi:10.1155/2009/641368
Research Article
Existence Results for Second-Order Impulsive
Neutral Functional Differential Equations with
Nonlocal Conditions
Meili Li and Chunhai Kou
Department of Applied Mathematics, Donghua University, Shanghai 201620, China
Correspondence should be addressed to Meili Li, stylml@dhu.edu.cn
Received 2 September 2009; Accepted 26 October 2009
Recommended by Guang Zhang
The existence of mild solutions for second-order impulsive semilinear neutral functional
differential equations with nonlocal conditions in Banach spaces is investigated. The results are
obtained by using fractional power of operators and Sadovskii’s fixed point theorem.
Copyright q 2009 M. Li and C. Kou. 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.
1. Introduction
The study of impulsive functional differential equations is linked to their utility in simulating
processes and phenomena subject to short-time perturbations during their evolution. The
perturbations are performed discretely and their duration is negligible in comparison with
the total duration of the processes. That is why the perturbations are considered to take
place “instantaneously” in the form of impulses. The theory of impulsive differential and
functional differential equations has been extensively developed; see the monographs of
Bainov and Simeonov 1, Lakshmikantham et al. 2, and Samoilenko and Perestyuk 3,
where numerous properties of their solutions are studied, and detailed bibliographies are
given.
This paper is devoted to extending existing results to second-order differential
equations. To be precise, in 4, the authors used Sadovsii’s fixed point theorem for a
condensing map to establish existence results for first-order impulsive semilinear neutral
functional differential inclusions with nonlocal conditions. Here, we obtain existence results
for second-order semilinear impulsive differential equations with nonlocal conditions of the
form
d tk ,
x t − Ft, xh1 t Axt Gt, xh2 t, t ∈ J 0, b, t /
dt
− Δx|ttk Ik x tk , k 1, . . . , m,
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Δx ttk I k x t−k ,
k 1, . . . , m,
x 0 η,
x0 gx x0 ,
1.1
where A is the infinitesimal generator of a strongly continuous cosine family Ct, t ∈ R,
of bounded linear operators in X. Also, 0 t0 < t1 < · · · < tm < tm1 b, Δx|ttk xtk − xt−k , Δx |ttk x tk − x t−k . Finally, F, G, g, Ik , I k k 1, . . . , m and h1 , h2 are
given functions to be specified later.
Other results on second order functional differential equations with and without
impulsive effect can be founded in the monographs 5–8.
This paper is organized as follows. In Section 2, we recall briefly some basic definitions
and lemmas. The existence theorem for 1.1 and its proof are arranged in Section 3.
Our approaches are based on Sadovskii’s fixed point theorem, and the theory of strongly
continuous cosine families.
2. Preliminaries
Definition 2.1 see 9. A one-parameter family Ct, t ∈ R, of bounded linear operators in
the Banach space X is called a strongly continuous cosine family if and only if
i Cs t Cs − t 2CsCt for all s, t ∈ R;
ii C0 I;
iii Ctx is strongly continuous in t on R for each fixed x ∈ X.
We define the associated sine family St, t ∈ R, by
Stx t
Csxds,
x ∈ X, t ∈ R.
2.1
0
We make the following assumption on A:
H1 A is the infinitesimal generator of a strongly continuous cosine family Ct, t ∈ R,
of bounded linear operators from X into itself.
The infinitesimal generator of a strongly continuous cosine family Ct, t ∈ R is the
operator A : X → X defined by
d2
Ax 2 Ctx ,
dt
t0
x ∈ DA,
2.2
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3
where
DA x ∈ X : Ctx is twice continuously differentiable in t .
2.3
We define
E x ∈ X : Ctx is once continuously differentiable in t .
2.4
Lemma 2.2 see 9. If Ct, t ∈ R, be a strongly continuous cosine family in X, then
i there exist constants K ≥ 1 and ω ≥ 0 so that Ct ≤ Keω|t| , for all t ∈ R, and
t2
ω|s|
St1 − St2 ≤ K e ds,
t1
∀t1 , t2 ∈ R;
2.5
ii if x ∈ E, then Stx ∈ DA and d/dtCtx AStx.
It is proved in 10 that for 0 ≤ α ≤ 1, the fractional powers −Aα exist as close linear
operator in X, D−Aα ⊂ D−Aβ , for 0 ≤ β ≤ α ≤ 1, and −Aα −Aβ −Aαβ for
0 ≤ α β ≤ 1.
We assume in addition the following assumption:
H2 for 0 ≤ α ≤ 1, −Aα maps onto X and is 1 − 1, so that D−Aα is a Banach space
when endowed with the form xα −Aα x, x ∈ D−Aα . We denote this
Banach space by Xα .
Denote J0 0, t1 , Jk tk , tk1 , k 1, 2, . . . , m. We define the following classes of
functions:
P CJ, Xα {x : J → Xα : xk ∈ CJk , Xα , k 0, 1, . . . , m and there exist
xtk , xt−k , k 1, . . . , m with xtk xt−k }:
P C1 J, Xα {x ∈ P CJ, Xα : xk ∈ CJk , Xα , k 0, 1, . . . , m and there exist
x tk , x t−k , k 1, . . . , m with x tk x t−k }, where xk and xk represent the restriction
of x and x to Jk , respectively, k 0, . . . , m, and xk Jk sups∈Jk xk sα .
Obviously, P CJ, Xα is a Banach space with the norm xP C max{xk Jk , k 0, . . . , m}, and P C1 J, Xα is also a Banach space with the norm xP C1 max{xP C , x P C }.
Definition 2.3. A function x· ∈ P C1 J, Xα is said to be a mild solution of 1.1 if
i x0 gx x0 , x 0 η;
ii Δx|ttk Ik xt−k , k 1, . . . , m;
iii Δx |ttk I k xt−k , k 1, . . . , m;
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iv the restriction of x· to the interval Jk k 0, . . . , m is continuous and the
following integral equation is verified:
xt Ct x0 − gx St η − F0, xh1 0 t
Ct − sFs, xh1 sds
0
St − sGs, xh2 sds 0
t
Ct − tk Ik x t−k
2.6
0<tk <t
St − tk I k x t−k ,
t ∈ J.
0<tk <t
For 1.1, we assume that the following hypotheses are satisfied: for some α ∈ 0, 1,
H3 there exists a constant β ∈ 0, 1 such that F : J × Xα → Xβ is a continuous function,
and −Aβ F : J × Xα → Xα satisfies the Lipschitz condition, that is, there exists a
constant L > 0 such that
−Aβ Ft1 , x1 − −Aβ Ft2 , x2 ≤ L|t1 − t2 | x1 − x2 α ,
α
2.7
for any 0 ≤ t1 , t2 ≤ b, x1 , x2 ∈ Xα . Moreover, there exists a constant L1 > 0 such that the
inequality
−Aβ Ft, x ≤ L1 xα 1
α
2.8
holds for any x ∈ Xα ;
H4 the function G : J × Xα → X satisfies the following conditions:
i for each t ∈ J, the function Gt, · : Xα → X is continuous, and for each x ∈ Xα ,
the function G·, x : J → X is strongly measurable,
ii for each positive number l ∈ N, there is a positive function wl ∈ L1 J such
that
sup Gt, x ≤ wl t
a.e. on J,
xα ≤l
1
lim inf
l→∞ l
b
wl sds γ < ∞,
2.9
0
where
xα sup xsα ;
0≤s≤b
H5 hi ∈ CJ, J, i 1, 2. g : P C1 J, Xα → Xα is continuous and satisfies that
2.10
Discrete Dynamics in Nature and Society
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i there exist positive constants L2 and L2 such that
gu ≤ L2 u 1 L
PC
2
α
∀u ∈ P C1 J, Xα ,
2.11
ii g is a completely continuous map;
H6 Ik , I k ∈ CXα , Xα , k 1, . . . , m are all bounded, that is, there exist constants
dk , dk , k 1, . . . , m, such that Ik xα ≤ dk , I k xα ≤ dk , for x ∈ Xα ;
H7 Ct, t ∈ J, is completely continuous.
3. Main Result
Theorem 3.1. Let x0 ∈ Xα . If the hypotheses H1 –H7 are satisfied, then 1.1 has a mild solution
provided that
L0 : 2M0 LMb < 1,
3.1
M L2 2M0 bL1 bγ < 1,
3.2
where
M sup{Ct : t ∈ J},
M sup C t : t ∈ J ,
M0 −A−β .
3.3
Proof. Consider the space B P C1 J, Xα with morm xP C1 max{xP C , x P C }. We
should now show that the operator P defined by
P xt Ct x0 − gx St η − F0, xh1 0 t
Ct − sFs, xh1 sds
0
t
0
St − sGs, xh2 sds Ct − tk Ik x t−k St − tk I k x t−k
0<tk <t
0<tk <t
3.4
has a fixed point. This fixed point is then a solution of 2.6.
For each positive number l, let Bl {x ∈ B : xtα ≤ l, t ∈ J}. Then for each l, Bl is
clearly a bounded close convex set in B. We claim that there exists a positive integer l such
that P Bl ⊆ Bl . If it is not true, then for each positive integer l, there is a function xl · ∈ Bl , but
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Discrete Dynamics in Nature and Society
∈ Bl , that is, P xl tα > l for some tl ∈ J, where tl denotes t is dependent on l.
P xl · /
However, on the other hand, we have
l < P xl tα Ct x0 − gxl St η − F0, xl h1 0
t
Ct − sFs, xl h1 sds 0
t
St − sGs, xl h2 sds
0
− − Ct − tk Ik xl tk St − tk I k xl tk <t
0<t <t
0<tk
α
k
≤ Ct x0 − gxl α St η − −A−β −Aβ F0, xl h1 0 α
t
−β
β
Ct − s−A −A Fs, xl h1 sds
0
3.5
α
t
St − sGs, xl h2 sds
0
α
− Ct − tk Ik xl t− −
t
xl tk I
St
k
k
k
α
0<tk <t
α
0<tk <t
≤ M x0 α L2 l L2 Mb ηα M0 L1 l 1
MM0 bL1 l 1 Mb
b
wl sds M
0
m
m
dk M b − tk dk .
k1
k1
Dividing on both sides by l and taking the lower limits as l → ∞, we get ML2 2M0 bL1 bγ ≥ 1. This is a contradiction with the formula 3.2. Hence for some positive
integer l, P Bl ⊆ Bl .
Next we will show that the operator P has a fixed point on Bl , which implies that 1.1
has a mild solution. For this purpose, we decompose P as P P1 P2 , where the operators
P1 , P2 are defined on Bl , respectively, by
P1 xt t
Ct − sFs, xh1 sds − StF0, xh1 0,
0
P2 xt Ct x0 − gx Stη t
St − sGs, xh2 sds
0
Ct − tk Ik x t−k St − tk I k x t−k ,
0<tk <t
0<tk <t
3.6
Discrete Dynamics in Nature and Society
7
for t ∈ J, and we will verify that P1 is a contraction while P2 is a completely continuous
operator.
To prove that P1 is a contraction, we take x1 , x2 ∈ Bl arbitrarily. Then for each t ∈ J and
by condition H3 , we have that
t
P1 x1 t − P1 x2 tα ≤ Ct − sFs, x1 h1 s − Fs, x2 h1 sds
0
α
StF0, x1 h1 0 − F0, x2 h1 0α
t
Ct − s−A−β −Aβ Fs, x1 h1 s − Fs, x2 h1 sds
0
α
St−A−β −Aβ F0, x1 h1 0 − F0, x2 h1 0 α
≤ 2M0 LMb sup x1 s − x2 sα
0≤s≤b
L0 sup x1 s − x2 sα .
0≤s≤b
3.7
Thus P1 x1 − P1 x2 α ≤ L0 x1 − x2 α . Therefore, by assumption 0 < L0 < 1 see 3.1,
we see that P1 is a contraction.
To prove that P2 is completely continuous, firstly we prove that P2 is continuous on Bl .
Let xn → x∗ , xn ∈ Bl , then by H4 i, we have Gs, xn h2 s → Gs, x∗ h2 s, n → ∞.
Since Gs, xn h2 s − Gs, x∗ h2 s ≤ 2wl s, by the dominated convergence theorem,
we have
P2 xn − P2 x∗ P C
t
supCt gx∗ − gxn St − sGs, xn h2 s − Gs, x∗ h2 sds
0
t∈J − − − −
Ct−tk Ik xn tk −Ik x∗ tk
St−tk I k xn tk −I k x∗ tk <t
0<t <t
0<tk
≤ Mgx∗ − gxn α m
k1
k
b
0
St − sGs, xn h2 s − Gs, x∗ h2 sα ds
MIk xn t−k − Ik x∗ t−k α
α
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m
Mb − tk I k xn t−k − I k x∗ t−k −→ 0
α
k1
as n −→ ∞,
P2 xn − P2 x∗ PC
t
supC t gx∗ − gxn Ct − sGs, xn h2 s − Gs, x∗ h2 sds
0
t∈J C t − tk Ik xn t−k − Ik x∗ t−k
0<tk <t
− −
Ct − tk I k xn tk − I k x∗ tk <t
0<tk
α
≤ M gx∗ − gxn α m
k1
m
b
0
Ct − sGs, xn h2 s − Gs, x∗ h2 sα ds
M Ik xn t−k − Ik x∗ t−k α
MI k xn t−k − I k x∗ t−k −→ 0 as n −→ ∞.
α
k1
3.8
Thus, P2 is continuous.
Next, we prove that {P2 x : x ∈ Bl } is a family of equicontinuous functions. Let τ1 , τ2 ∈
J, τ1 < τ2 . Then for each t ∈ J, we have
P2 xτ2 − P2 xτ1 α
≤ Cτ2 − Cτ1 x0 − gx α Sτ2 η − Sτ1 ηα
τ1
τ2
Sτ2 − s − Sτ1 − sGs, xh2 sds Sτ2 − sGs, xh2 sds
τ1
0
α
α
− − Cτ2 − tk Ik x tk Cτ2 − tk − Cτ1 − tk Ik x tk 0<t <τ
τ ≤t <τ
k
1
α
1
k
2
− − Sτ2 − tk I k x tk Sτ2 − tk − Sτ1 − tk I k x tk 0<t <τ
τ ≤t <τ
k
1
α
1
k
2
α
α
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≤ Cτ2 − Cτ1 x0 − gx α Sτ2 η − Sτ1 ηα
τ1
0
τ2
Sτ2 − s − Sτ1 − sGs, xh2 sα ds Sτ2 − sGs, xh2 sds
τ1
α
Cτ2 − tk − Cτ1 − tk dk Cτ2 − tk dk
τ1 ≤tk <τ2
0<tk <τ1
Sτ2 − tk − Sτ1 − tk dk Sτ2 − tk dk ,
τ1 ≤tk <τ2
0<tk <τ1
3.9
and similarly
P2 x τ2 − P2 x τ1 α
≤ C τ2 − C τ1 x0 − gx α S τ2 − S τ1 ηα
τ1
τ2
Cτ2 − s − Cτ1 − sGs, xh2 sds Cτ2 − sGs, xh2 sds
0
τ
α
1
α
− − C τ2 − tk Ik x tk C τ2 − tk − C τ1 − tk Ik x tk 0<t <τ
τ ≤t <τ
k
1
α
1
α
1
k
2
− − S τ2 − tk I k x tk S τ2 − tk − S τ1 − tk I k x tk 0<t <τ
τ ≤t <τ
k
1
k
2
α
α
≤ C τ2 − C τ1 x0 − gx α S τ2 − S τ1 ηα
τ1
τ2
Cτ2 − s − Cτ1 − sGs, xh2 sα ds Cτ2 − sGs, xh2 sds
0
τ1
C τ2 − tk − C τ1 − tk dk C τ2 − tk dk
0<tk <τ1
α
τ1 ≤tk <τ2
S τ2 − tk − S τ1 − tk dk S τ2 − tk dk .
0<tk <τ1
τ1 ≤tk <τ2
3.10
The right-hand sides are independent of x ∈ Bl and tend to zero as τ2 − τ1 → 0, since
Ct, St, C t, S t are uniformly continuous for t ∈ J and the compactness of Ct, St
for t > 0 implies the continuity in the uniform operator topology.
The compactness of St follows from that of Ct and Lemma 2.2.
This shows that P2 maps Bl into a family of equicontinuous functions.
It remains to prove that V t {P2 xt : x ∈ Bl } is relatively compact in X.
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Obviously, by condition H5 ii, V 0 is relatively compact in B. Let 0 < t ≤ b be fixed
and 0 < < t. For x ∈ Bl , we define
ccP2, xt Ct x0 − gx Stη t−
St − sGs, xh2 sds
0
Ct − tk Ik x t−k St − tk I k x t−k .
0<tk <t
3.11
0<tk <t
Since Ct, St are compact operators, the set V t {P2, xt : x ∈ Bl } is relatively
compact in B for every , 0 < < t. Moreover, for every x ∈ Bl , we have
t
t
St − sGs, xh2 sds ≤
P2 xt − P2, xtα St − swl sds,
t−
t−
α
t
t
P2 x t − P2, x t Ct
−
sGs,
xh
≤
sds
Ct − swl sds.
2
α
t−
t−
α
3.12
Therefore, there are relatively compact sets arbitrarily close to the set V t. Hence, the set V t
is relatively compact in B.
Thus, by Arzela-Ascoli theorem, P2 is a completely continuous operator. Those
arguments enable us to conclude that P P1 P2 is a condensing map on Bl , and by the fixed
point theorem of Sadovskii, there exists a fixed point x· for P on Bl . Therefore, the nonlocal
Cauchy problem with impulsive effect 1.1 has a mild solution. The proof is completed.
Acknowledgments
The work of the M. Li is supported by NNSF of China no. 10971139. The work of C. Kou is
supported by NNSF of China nos. 10701023 and 10971221. The authors would like to thank
the anonymous referee for his/her remarks about the evaluation of the original version of the
manuscript.
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