Volume 2009, Article ID 641368,11pages 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,[email protected] 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.
Copyrightq2009 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 dt
xt−Ft, xh1t
Axt Gt, xh2t, t∈J 0, b, t /tk, Δx|ttk Ik
x t−k
, k1, . . . , m,
Δx
ttk Ik
x t−k
, k1, . . . , m, x0 gx x0, x0 η,
1.1
whereA is the infinitesimal generator of a strongly continuous cosine familyCt, t ∈ R, of bounded linear operators in X. Also, 0 t0 < t1 < · · · < tm < tm1 b, Δx|ttk xtk−xt−k, Δx|ttk xtk−xt−k.Finally, F, G, g, Ik, Ik 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 monographs5–8.
This paper is organized as follows. InSection 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 2.1see9. A one-parameter familyCt, t ∈R,of bounded linear operators in the Banach spaceXis called a strongly continuous cosine family if and only if
iCst Cs−t 2CsCtfor alls, t∈R;
iiC0 I;
iiiCtxis strongly continuous intonRfor each fixedx∈X.
We define the associated sine familySt, t∈R, by
Stx t
0
Csxds, x∈X, t∈R. 2.1
We make the following assumption onA:
H1Ais the infinitesimal generator of a strongly continuous cosine familyCt, t∈R, of bounded linear operators fromXinto itself.
The infinitesimal generator of a strongly continuous cosine familyCt, t ∈ Ris the operatorA:X → Xdefined by
Ax d2 dt2Ctx
t0
, x∈DA, 2.2
where
DA
x∈X:Ctxis twice continuously differentiable int . 2.3
We define
E
x∈X:Ctxis once continuously differentiable int . 2.4
Lemma 2.2see9. IfCt, t∈R,be a strongly continuous cosine family inX,then ithere exist constantsK≥1 andω≥0 so thatCt ≤Keω|t|,for allt∈R,and
St1−St2 ≤K
t2
t1
eω|s|ds
, ∀t1, t2∈R; 2.5
iiifx∈E,thenStx∈DAandd/dtCtxAStx.
It is proved in10that for 0≤α≤1,the fractional powers−Aαexist as close linear operator in X,D−Aα ⊂ D−Aβ,for 0 ≤ β ≤ α ≤ 1, and−Aα−Aβ −Aαβ for 0≤αβ≤1.
We assume in addition the following assumption:
H2for 0≤α≤1, −Aαmaps ontoXand is 1−1, so thatD−Aαis a Banach space when endowed with the form xα −Aαx, x ∈ D−Aα. We denote this Banach space byXα.
DenoteJ0 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, k1, . . . , mwithxtk xt−k}:
P C1J, Xα {x ∈ P CJ, Xα : xk ∈ CJk, Xα, k 0,1, . . . , m and there exist xtk, xt−k, k 1, . . . , mwith xtk xt−k},wherexk andxkrepresent the restriction ofxandxtoJk, respectively,k0, . . . , m,andxkJk sups∈J
kxksα.
Obviously, P CJ, Xα is a Banach space with the norm xP C max{xkJk, k 0, . . . , m},andP C1J, Xαis also a Banach space with the normxP C1 max{xP C,xP C}.
Definition 2.3. A functionx·∈P C1J, Xαis said to be a mild solution of1.1if ix0 gx x0, x0 η;
ii Δx|ttk Ikxt−k, k1, . . . , m;
iii Δx|ttk Ikxt−k, k1, . . . , m;
ivthe 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, xh10
t
0
Ct−sFs, xh1sds
t
0
St−sGs, xh2sds
0<tk<t
Ct−tkIk
x t−k
0<tk<t
St−tkIk
x t−k
, t∈J.
2.6
For1.1, we assume that the following hypotheses are satisfied: for someα∈0,1, H3there exists a constantβ∈0,1such thatF:J×Xα → Xβis a continuous function,
and−AβF : J×Xα → Xαsatisfies the Lipschitz condition, that is, there exists a constantL >0 such that
−AβFt1, x1−−AβFt2, x2
α≤L|t1−t2|x1−x2α, 2.7
for any 0 ≤ t1, t2 ≤ b, x1, x2 ∈ Xα.Moreover, there exists a constantL1 > 0 such that the inequality
−AβFt, x
α≤L1xα1 2.8
holds for anyx∈Xα;
H4the functionG:J×Xα → Xsatisfies the following conditions:
ifor eacht∈J,the functionGt,·:Xα → Xis continuous, and for eachx∈Xα, the functionG·, x:J → Xis strongly measurable,
iifor each positive numberl ∈ N, there is a positive functionwl ∈ L1Jsuch that
sup
xα≤lGt, x ≤wlt a.e.onJ, lim inf
l→ ∞
1 l
b
0
wlsdsγ <∞, 2.9
where
xα sup
0≤s≤bxsα; 2.10
H5hi∈CJ, J, i1,2. g:P C1J, Xα → Xαis continuous and satisfies that
ithere exist positive constantsL2andL2such that gu
α≤L2uP C1L2 ∀u∈P C1J, Xα, 2.11
iigis a completely continuous map;
H6Ik, Ik ∈ CXα, Xα, k 1, . . . , m are all bounded, that is, there exist constants dk, dk, k1, . . . , m,such thatIkxα≤dk, Ikxα≤dk,forx∈Xα;
H7Ct, t∈J,is completely continuous.
3. Main Result
Theorem 3.1. Letx0∈Xα.If the hypothesesH1–H7are satisfied, then1.1has a mild solution provided that
L0:2M0LMb <1, 3.1
M
L22M0bL1bγ
<1, 3.2
where
Msup{Ct:t∈J}, MsupCt:t∈J , M0−A−β. 3.3
Proof. Consider the space B P C1J, Xα with morm xP C1 max{xP C,xP C}. We should now show that the operatorPdefined by
P xt Ct
x0−gx St
η−F0, xh10
t
0
Ct−sFs, xh1sds
t
0
St−sGs, xh2sds
0<tk<t
Ct−tkIk
x t−k
0<tk<t
St−tkIk
x t−k
3.4
has a fixed point. This fixed point is then a solution of2.6.
For each positive numberl,letBl {x ∈B :xtα ≤l, t∈ J}.Then for eachl, Blis clearly a bounded close convex set in B. We claim that there exists a positive integerlsuch thatP Bl⊆Bl.If it is not true, then for each positive integerl,there is a functionxl·∈Bl,but
P xl·/∈Bl,that is,P xltα > lfor sometl∈J,wheretldenotestis dependent onl.
However, on the other hand, we have
l <P xltα Ct
x0−gxl St
η−F0, xlh10
t
0
Ct−sFs, xlh1sds t
0
St−sGs, xlh2sds
0<tk<t
Ct−tkIk
xl
t−k
0<tk<t
St−tkIk
xl
t−k
α
≤Ct
x0−gxl
αSt
η−−A−β−AβF0, xlh10
α
t
0
Ct−s−A−β−AβFs, xlh1sds
α
t
0
St−sGs, xlh2sds
α
0<tk<t
Ct−tkIk
xl t−k
α
0<tk<t
St−tkIk
xl t−k
α
≤M
x0αL2lL2
Mbη
αM0L1l1
MM0bL1l1 Mb b
0
wlsdsM m k1
dkM m k1
b−tkdk.
3.5
Dividing on both sides byland taking the lower limits asl → ∞,we getML2 2M0bL1 bγ ≥ 1.This is a contradiction with the formula3.2. Hence for some positive integerl, P Bl⊆Bl.
Next we will show that the operatorPhas a fixed point onBl,which implies that1.1 has a mild solution. For this purpose, we decomposeP asP P1P2,where the operators P1, P2are defined onBl,respectively, by
P1xt t
0
Ct−sFs, xh1sds−StF0, xh10,
P2xt Ct
x0−gx
Stη t
0
St−sGs, xh2sds
0<tk<t
Ct−tkIk
x t−k
0<tk<t
St−tkIk
x t−k
,
3.6
for t ∈ J, and we will verify that P1 is a contraction whileP2 is a completely continuous operator.
To prove thatP1is a contraction, we takex1, x2∈Blarbitrarily. Then for eacht∈Jand by conditionH3,we have that
P1x1t−P1x2tα≤
t
0
Ct−sFs, x1h1s−Fs, x2h1sds
α
StF0, x1h10−F0, x2h10α
t
0
Ct−s−A−β−AβFs, x1h1s−Fs, x2h1sds
α
St−A−β
−AβF0, x1h10−F0, x2h10
α
≤2M0LMbsup
0≤s≤bx1s−x2sα L0sup
0≤s≤bx1s−x2sα.
3.7
ThusP1x1−P1x2α ≤ L0x1−x2α.Therefore, by assumption 0< L0 <1see3.1, we see thatP1is a contraction.
To prove thatP2is completely continuous, firstly we prove thatP2is continuous onBl. Letxn → x∗, xn ∈Bl,then byH4i, we haveGs, xnh2s → Gs, x∗h2s, n → ∞.
SinceGs, xnh2s−Gs, x∗h2s ≤ 2wls,by the dominated convergence theorem, we have
P2xn−P2x∗P C
sup
t∈J
Ct
gx∗−gxn
t
0
St−sGs, xnh2s−Gs, x∗h2sds
0<tk<t
Ct−tk Ik
xn t−k
−Ik
x∗ t−k
0<tk<t
St−tk Ik
xn t−k
−Ik
x∗ t−k
α
≤Mgx∗−gxn
α
b
0
St−sGs, xnh2s−Gs, x∗h2sαds
m
k1
MIkxnt−k−Ik x∗
t−k
α
m
k1
Mb−tkIk
xn
t−k
−Ik
x∗ t−k
α−→0 as n−→ ∞, P2xn−P2x∗
P C
sup
t∈J
Ct
gx∗−gxn
t
0
Ct−sGs, xnh2s−Gs, x∗h2sds
0<tk<t
Ct−tk Ik
xn
t−k
−Ik
x∗ t−k
0<tk<t
Ct−tk Ik
xn t−k
−Ik x∗
t−k
α
≤Mgx∗−gxn
α
b
0
Ct−sGs, xnh2s−Gs, x∗h2sα ds
m
k1
MIk xn
t−k
−Ik x∗
t−k
α
m
k1
MIk
xn
t−k
−Ik
x∗ t−k
α−→0 asn−→ ∞.
3.8 Thus,P2is continuous.
Next, we prove that{P2x:x∈Bl}is a family of equicontinuous functions. Letτ1, τ2∈ J, τ1< τ2.Then for eacht∈J,we have
P2xτ2−P2xτ1α
≤Cτ2−Cτ1
x0−gx
αSτ2η−Sτ1η
α
τ1
0
Sτ2−s−Sτ1−sGs, xh2sds α
τ2
τ1
Sτ2−sGs, xh2sds
α
0<tk<τ1
Cτ2−tk−Cτ1−tkIk
x t−k
α
τ1≤tk<τ2
Cτ2−tkIk
x t−k
α
0<tk<τ1
Sτ2−tk−Sτ1−tkIk
x t−k
α
τ1≤tk<τ2
Sτ2−tkIk
x t−k
α
≤Cτ2−Cτ1
x0−gx
αSτ2η−Sτ1η
α
τ1
0
Sτ2−s−Sτ1−sGs, xh2sαds
τ2
τ1
Sτ2−sGs, xh2sds
α
0<tk<τ1
Cτ2−tk−Cτ1−tkdk
τ1≤tk<τ2
Cτ2−tkdk
0<tk<τ1
Sτ2−tk−Sτ1−tkdk
τ1≤tk<τ2
Sτ2−tkdk,
3.9
and similarly
P2xτ2−P2xτ1
α
≤Cτ2−Cτ1
x0−gx
αSτ2−Sτ1 η
α
τ1
0
Cτ2−s−Cτ1−sGs, xh2sds α
τ2
τ1
Cτ2−sGs, xh2sds
α
0<tk<τ1
Cτ2−tk−Cτ1−tk Ik
x t−k
α
τ1≤tk<τ2
Cτ2−tkIk
x t−k
α
0<tk<τ1
Sτ2−tk−Sτ1−tk Ik
x t−k
α
τ1≤tk<τ2
Sτ2−tkIk
x t−k
α
≤Cτ2−Cτ1
x0−gx
αSτ2−Sτ1 η
α
τ1
0
Cτ2−s−Cτ1−sGs, xh2sαds
τ2
τ1
Cτ2−sGs, xh2sds
α
0<tk<τ1
Cτ2−tk−Cτ1−tkdk
τ1≤tk<τ2
Cτ2−tkdk
0<tk<τ1
Sτ2−tk−Sτ1−tkdk
τ1≤tk<τ2
Sτ2−tkdk.
3.10
The right-hand sides are independent of x ∈ Bl and tend to zero as τ2 −τ1 → 0, since Ct, St, Ct, Stare uniformly continuous fort ∈ J and the compactness ofCt, St fort >0 implies the continuity in the uniform operator topology.
The compactness ofStfollows from that ofCtandLemma 2.2.
This shows thatP2mapsBlinto a family of equicontinuous functions.
It remains to prove thatVt {P2xt:x∈Bl}is relatively compact inX.
Obviously, by conditionH5ii,V0is relatively compact inB.Let 0< t≤bbe fixed and 0< < t.Forx∈Bl,we define
ccP2,xt Ct
x0−gx
Stη t−
0
St−sGs, xh2sds
0<tk<t
Ct−tkIk
x t−k
0<tk<t
St−tkIk
x t−k
.
3.11
Since Ct, St are compact operators, the set Vt {P2,xt : x ∈ Bl} is relatively compact inBfor every, 0< < t.Moreover, for everyx∈Bl,we have
P2xt−P2,xtα
t
t−St−sGs, xh2sds
α
≤ t
t−St−swlsds,
P2xt−P2,xt
α
t
t−Ct−sGs, xh2sds
α
≤ t
t−Ct−swlsds.
3.12
Therefore, there are relatively compact sets arbitrarily close to the setVt.Hence, the setVt is relatively compact inB.
Thus, by Arzela-Ascoli theorem, P2 is a completely continuous operator. Those arguments enable us to conclude thatP P1P2is a condensing map onBl,and by the fixed point theorem of Sadovskii, there exists a fixed pointx·forPonBl.Therefore, the nonlocal Cauchy problem with impulsive effect1.1has a mild solution. The proof is completed.
Acknowledgments
The work of the M. Li is supported by NNSF of Chinano. 10971139. The work of C. Kou is supported by NNSF of Chinanos. 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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