doi:10.1155/2011/784161
Research Article
Nonlocal Impulsive Cauchy Problems for
Evolution Equations
Jin Liang
1and Zhenbin Fan
1, 21Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
2Department of Mathematics, Changshu Institute of Technology, Suzhou, Jiangsu 215500, China
Correspondence should be addressed to Jin Liang,[email protected]
Received 17 October 2010; Accepted 19 November 2010
Academic Editor: Toka Diagana
Copyrightq2011 J. Liang and Z. Fan. 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.
Of concern is the existence of solutions to nonlocal impulsive Cauchy problems for evolution equations. Combining the techniques of operator semigroups, approximate solutions, noncompact measures and the fixed point theory, new existence theorems are obtained, which generalize and improve some previous results since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required. An application to partial differential equations is also presented.
1. Introduction
Impulsive equations arise from many different real processes and phenomena which appeared in physics, chemical technology, population dynamics, biotechnology, medicine, and economics. They have in recent years been an object of investigations with increasing interest. For more information on this subject, see for instance, the paperscf., e.g.,1–6and references therein.
On the other hand, Cauchy problems with nonlocal conditions are appropriate models for describing a lot of natural phenomena, which cannot be described using classical Cauchy problems. That is why in recent years they have been studied by many researcherscf., e.g.,
4,7–12and references therein.
d
dtut Ft, ut Aut ft, ut, 0≤t≤K, t /ti,
u0 gu u0,
Δuti Iiuti, i1,2, . . . , p, 0< t1< t2<· · ·< tp< K,
1.1
where−A:DA⊆X → Xis the infinitesimal generator of an analytic semigroup{Tt; t≥ 0}andXis a real Banach space endowed with the norm · ,
Δuti uti
−ut− i
,
ut i
lim
t→t i
ut, ut− i
lim
t→t−
i
ut
, 1.2
F,f,g,Iiare given continuous functions to be specified later.
By going a new way, that is, by combining operator semigroups, the techniques of approximate solutions, noncompact measures, and the fixed point theory, we obtain new existence results for problem1.1, which generalize and improve some previous theorems since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required in the present paper.
The organization of this work is as follows. InSection 2, we recall some definitions, and facts about fractional powers of operators, mild solutions and Hausdorff measure of noncompactness. In Section 3, we give the existence results for problem 1.1 when the nonlocal item and impulsive functions are only assumed to be continuous. InSection 4, we give an example to illustrate our abstract results.
2. Preliminaries
Let X, · be a real Banach space. We denote by C0, K, X the space of X-valued continuous functions on0, Kwith the norm
umax{ut;t∈0, K}, 2.1
and byL10, K, Xthe space ofX-valued Bochner integrable functions on0, Kwith the
normfL1
K
0 ftdt. Let
PC0, K, X:{u:0, K → X; utis continuous att /ti, left continuous attti,
and the right limitutiexists fori1,2, . . . , p.
2.2
It is easy to check that PC0, K, Xis a Banach space with the norm
uPC sup
t∈0,K
In this paper, forr >0, letBr :{x∈X;x ≤r}and
Wr :{u∈PC0, K, X;ut∈Br, ∀t∈0, K}. 2.4
Throughout this paper, we assume the following.
H1The operator −A : DA ⊆ X → X is the infinitesimal generator of a compact analytic semigroup{Tt:t≥0}on Banach spaceXand 0∈ρA the resolvent set ofA.
In the remainder of this work,M:sup0≤t≤KTt<∞.
Under the above conditions, it is possible to define the fractional powerAα :DAα⊂
X → X, 0 < α < 1, of A as closed linear operators. And it is known that the following
properties hold.
Theorem 2.1see13, Pages 69–75. Let0< α <1and assume that (H1) holds. Then,
1DAαis a Banach space with the normxαAαxforx∈DAα,
2Tt:X → DAαfort >0,
3AαTtxTtAαxforx∈DAαandt≥0,
4for everyt >0,AαTtis bounded onXand there existsCα>0such that
AαTt ≤ Cα
tα , 0< t≤K, 2.5
5A−αis a bounded linear operator inXwithDAα ImA−α,
6if0< α < β≤1, thenDAβ→DAα.
We denote byXαthat the Banach spaceDAαendowed the graph norm from now on.
Definition 2.2. A functionu ∈PC0, K, Xis said to be a mild solution of1.1on0, Kif the functions → ATt−sFs, usis integrable on0, tfor allt∈0, Kand the following integral equation is satisfied:
ut Ttu0F0, u0−gu −Ft, ut
t
0
ATt−sFs, usds
t
0
Tt−sfs, usds
0<ti<t
Tt−tiIiuti, 0≤t≤K.
2.6
To discuss the compactness of subsets of PC0, K, X, we lett00,tp1K,
J0 t0, t1, J1 t1, t2, . . . , Jp
tp, tp1 . 2.7
ForD⊆PC0, K, X, we denote byD|Jithe set
D|Ji u∈Cti, ti1, X;uti vti
, ut vt, t∈Ji, v∈D, 2.8
Lemma 2.3. A setD ⊆PC0, K, Xis precompact in PC0, K, Xif and only if the setD|Ji is precompact inCti, ti1, Xfor everyi0,1,2, . . . , p.
Next, we recall that the Hausdorffmeasure of noncompactnessα·on each bounded subsetΩof Banach spaceYis defined by
αΩ inf{ε >0; Ωhas a finiteε-net inY}. 2.9
Some basic properties ofα·are given in the following Lemma.
Lemma 2.4see14. LetY be a real Banach space and letB, C⊆Ybe bounded. Then,
1Bis precompact if and only ifαB 0;
2αB αB αconvB, whereBand convBmean the closure and convex hull ofB, respectively;
3αB≤αCwhenB⊆C;
4αBC≤αB αC, whereBC{xy;x∈B, y∈C};
5αB∪C≤max{αB, αC};
6αλB |λ|αBfor anyλ∈R;
7letZbe a Banach space andQ:DQ ⊆Y → ZLipschitz continuous with constantk. ThenαQB≤kαBfor allB⊆DQbeing bounded.
We note that a continuous mapQ :W ⊆ Y → Y is anα-contraction if there exists a positive constantk <1 such thatαQC≤kαCfor all bounded closedC⊆W.
Lemma 2.5see Darbo-Sadovskii’s fixed point theorem in14. IfW ⊆ Y is bounded closed and convex, andQ:W → Wis anα-contraction, then the mapQhas at least one fixed point inW.
3. Main Results
In this section, by using the techniques of approximate solutions and fixed points, we establish a result on the existence of mild solutions for the nonlocal impulsive problem1.1
when the nonlocal itemg and the impulsive functionsIiare only assumed to be continuous
in PC0, K, XandX, respectively.
In practical applications, the values ofutfortnear zero often do not affectgu. For example, it is the case when
gu
q
j1
cjusj, 0< s1< s2<· · ·< sq< K. 3.1
So, to prove our main results, we introduce the following assumptions.
H2g : PC0, K, X → Xis a continuous function, and there is aδ ∈0, t1such that
H3There exists aβ ∈ 0,1such thatF : 0, K×X → Xβ is a continuous function,
andF·, u· F·, v·for anyu, v ∈PC0, K, Xwithus vs,s ∈δ, K. Moreover, there existL2, L3>0 such that
AβFt, x1−AβFt, x2≤L
2x1−x2 3.2
for any 0≤t≤K,x1, x2∈X, and
AβFt, x≤L3x1 3.3
for any 0≤t≤K,x∈X.
H4The functionft,· : X → X is continuous a.e.t ∈ 0, K; the functionf·, x :
0, K → X is strongly measurable for allx ∈ X. Moreover, for eachl ∈ N, there exists a functionρl ∈L10, K,Rsuch thatft, x ≤ ρltfor a.e.t∈0, Kand
allx∈Bl, and
γ:lim inf
l→ ∞ 1
l
K
0
ρlsds <∞. 3.4
H5Ii :X → Xis continuous for everyi1,2, . . . , p, and there exist positive numbers
L4, L4such thatIix ≤L4xL4for anyx∈Xandi1,2, . . . , p.
We note that, byTheorem 2.1, there existM0>0 andC1−β>0 such thatM0A−βand
A1−βTt≤ C1−β
t1−β, 0< t≤K. 3.5
For simplicity, in the following we setLmax{L1, L2, L3, L4}and will substituteL1, L2, L3, L4
byLbelow.
Theorem 3.1. Let (H1)–(H5) hold. Then the nonlocal impulsive Cauchy problem1.1has at least one mild solution on0, K, provided
L0 M
LM0LγpL
M0L
LC1−βKβ
β <1. 3.6
To prove the theorem, we need some lemmas. Next, forn∈ N, we denote byQn the
mapsQn: PC0, K, X → PC0, K, Xdefined by
Qnut Tt
u0F0, u0−T
1
n
gu
−Ft, ut
t
0
ATt−sFs, usds
t
0
Tt−sfs, usds
0<ti<t
Tt−tiT
1
n
Iiuti, 0≤t≤K.
In addition, we introduce the decompositionQnQn1Qn2Qn3Qn4, where
Qn1ut Tt
u0−T
1
n
gu
,
Qn2ut
0<ti<t
Tt−tiT
1
n
Iiuti,
Qn3ut TtF0, u0−Ft, ut
t
0
ATt−sFs, usds,
Qn4ut
t
0
Tt−sfs, usds
3.8
foru∈PC0, K, Xandt∈0, K.
Lemma 3.2. Assume that all the conditions inTheorem 3.1are satisfied. Then for anyn≥1, the map Qndefined by3.7has at least one fixed pointun∈PC0, K, X.
Proof. To prove the existence of a fixed point forQn, we will use Darbu-Sadovskii’s fixed point
theorem.
Firstly, we prove that the mapQn3is a contraction on PC0, K, X. For this purpose,
letu1, u2∈PC0, K, X. Then for eacht∈0, Kand by conditionH3, we have
Qn3u1t−Qn3u2t
≤MF0, u10−F0, u20Ft, u1t−Ft, u2t
t
0
ATt−sFs, u1s−Fs, u2sds
≤MA−βAβF0, u10−A−βAβF0, u20A−βAβFt, u1t−A−βAβFt, u2t
t
0
A1−βTt−sAβFs, u1s−AβFs, u2sds
≤MM0Lu1−u2M0Lu1t−u2t
t
0
C1−β
t−s1−βLu1s−u2sds.
3.9
Thus,
Qn3u1−Qn3u2PC≤
M1M0L
LC1−βKβ
β
u1−u2, 3.10
Secondly, we prove that Qn4, Qn1, Qn2 are completely continuous operators. Let
{um}m∞1be a sequence in PC0, K, Xwith
lim
m→ ∞umu 3.11
in PC0, K, X. By the continuity offwith respect to the second argument, we deduce that for eachs∈0, K,fs, umsconverges tofs, usinX, and we have
Qn4um−Qn4uPC≤M
K
0
fs, ums−fs, usds,
Qn1um−Qn1uPC≤Mgum−gu,
Qn2um−Qn2uPC≤M
p
i1
Iiumti−Iiuti.
3.12
Then by the continuity off,g,Ii, and using the dominated convergence theorem, we get
lim
m→ ∞Qn4umQn4u, mlim→ ∞Qn1umQn1u, mlim→ ∞Qn2umQn2u 3.13
in PC0, K, X, which implies thatQn4, Qn1, Qn2are continuous on PC0, K, X.
Next, for the compactness ofQn4we refer to the proof of4, Theorem 3.1.
ForQn1and any bounded subsetWof PC0, K, X, we have
Qn1ut Ttu0−T
1
n
Ttgu, t∈0, K, u∈W, 3.14
which implies that Qn1Wt is relatively compact in X for every t ∈ 0, K by the
compactness ofT1/n. On the other hand, for 0≤s≤t≤K, we have
Qn1ut−Qn1us ≤
Tt−Ts
u0−T
1
n
gu. 3.15
Since{T1/ngu; u∈W}is relatively compact inX, we conclude that
Qn1ut−Qn1us −→0 uniformly ast−→sandu∈W, 3.16
which implies thatQn1Wis equicontinuous on0, K. Therefore,Qn1is a compact operator.
Now, we prove the compactness ofQn2. For this purpose, let
Note that
Thus according toLemma 2.3, we only need to prove that
Thus, for any bounded subsetW ⊆PC0, K, X, we have byLemma 2.4,
αQnW≤αQn1W αQn3W αQn4W αQn2W≤L0αW. 3.22
Hence, the mapQnis anα-contraction in PC0, K, X.
Now, in order to apply Lemma 2.5, it remains to prove that there exists a constant
r > 0 such thatQnWr ⊆ Wr. Suppose this is not true; then for each positive integerr, there
This is a contradiction with inequality 3.6. Therefore, there exists r > 0 such that the mapping Qn maps Wr into itself. By Darbu-Sadovskii’s fixed point theorem, the operator
Qnhas at least one fixed point inWr. This completes the proof.
Lemma 3.3. Assume that all the conditions in Theorem 3.1 are satisfied. Then the set D|h,K is precompact in PCh, K, Xfor allh∈0, δ, where
D:{un;un∈PC0, K, Xcoming from Lemma3.2, n≥1}, 3.25
andδis the constant in (H2).
Step 1. D|h,t1is precompact inCh, t1, X.
Foru∈PC0, K, X, defineQF1: PC0, K, X → PC0, K, Xby
QF1ut TtF0, u0, t∈0, K. 3.26
Foru∈ Ch, t1, X, letut ut,t ∈ h, t1,ut uh,t ∈ 0, h, and we define
QF2:Ch, t1, X → Ch, t1, Xby
QF2ut −Ft, ut
t
0
ATt−sFs, usds, t∈h, t1. 3.27
By conditionH3,QF2is well defined and foru∈D, we have
Qn3ut QF1ut
QF2u|h,t1
t, t∈h, t1. 3.28
On the other hand, forun∈D,n≥1, we haveQn2unt 0,t∈h, t1. So,
unt Qn1unt QF1unt
QF2un|h,t1
t Qn4unt, t∈h, t1. 3.29
Now, for{Qn1un;n≥1}, we have
Qn1unt Ttu0−TtT
1
n
gun, t∈h, t1. 3.30
By the compactness ofTt,t >0, we get that{Qn1unt;n≥ 1}is relatively compact inX
for everyt ∈h, t1and{Qn1un;n≥ 1}|h,t1is equicontinuous onh, t1, which implies that
{Qn1un;n≥1}|h,t1is precompact inCh, t1, X.
By the same reasoning,{QF1un;n≥1}|h,t1is precompact inCh, t1, X.
ForQF2, we claim thatQF2 :Ch, t1, X → Ch, t1, Xis Lipschitz continuous with
constantM0L LC1−βKβ/β. In fact, H3implies that for every u, v ∈ Ch, t1, Xand
t∈h, t1,
QF2ut−QF2vt
≤ Ft, ut−Ft, vt
t
0
ATt−sFs, us−Fs, vsds
≤M0Lut−vt
t
0
C1−β
t−s1−βLdsmax0≤t≤t1ut−vt
≤M0Lut−vt
LC1−βKβ
β hmax≤t≤t1
ut−vt,
According to the proof ofStep 1, we know that
Lemma 2.3, it remains to prove that
Next, fort1≤s≤t≤t2, we have Thus, byLemma 2.4, we obtain
αD|h,t2
By Lemma 3.3, without loss of generality, we may suppose that un → u ∈
asn → ∞. Thus,
Lettingn → ∞in both sides, we obtain
u∗t Ttu
which implies that u∗ is a mild solution of the nonlocal impulsive problem 1.1. This completes the proof.
Remark 3.4. From Lemma 3.3and the above proof, it is easy to see that we can also prove
Theorem 3.1by showing thatD|0,his precompact in PC0, h, X.
The following results are immediate consequences ofTheorem 3.5.
Theorem 3.6. Assume (H1), (H2), (H4), and (H5) hold. IfF≡0, then the nonlocal impulsive problem
1.1has at least one mild solution on0, K, providedMLγpL<1.
Theorem 3.7. Assume (H1), (H4), and (H5) hold. Ifg ≡0, F ≡0, then the impulsive problem1.1
has at least one mild solution on0, K, providedMγpL<1.
Remark 3.8. Theorems 3.5-3.6 are new even for many special cases discussed before, since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required.
4. Application
In this section, to illustrate our abstract result, we consider the following differential system:
∂ ∂t
wt, x
π
0
λt, x, ywt, ydy
∂2
∂x2wt, x vt, wt, x,
0≤t≤1, 0≤x≤π, t /ti,
wt,0 wt, π 0, 0≤t≤1,
wt i
−wt− i
Iiwti, i1, . . . , p, 0< t1<· · ·< tp<1,
w0, x
q
j1
cjwsj, xw0x, 0< s1<· · ·< sq <1, 0≤x≤π,
4.1
wherew0 ∈ L20, π,ti, sj, cj are given real numbers fori 1, . . . , p,j 1, . . . , q, andλ : 0,1×0, π×0, π → Randv:0,1×R → Rare functions to be specified below.
To treat the above system, we takeX L20, πwith the norm · and we consider
the operatorA:DA⊆X → Xdefined by
Az−z 4.2
with domain
DA z∈X; z, zarea absolutely continuous, z∈X, z0 zπ 0. 4.3
The operator−Ais the infinitesimal generator of an analytic compact semigroupTtt≥0on
X. Moreover,Ahas a discrete spectrum, the eigenvalues aren2,n∈N, with the corresponding
normalized eigenvectorsenx
2/πsinnx, and the following properties are satisfied.
aIfz∈DA, thenAz∞n1n2z, e
nen.
bFor eachz∈X,Ttz∞n1exp−n2tz, enen. Moreover,Tt ≤1 for allt≥0.
cFor eachz∈X,A−1/2z∞
n11/nz, enen. In particular,A−1/21. dA1/2 is given by A1/2z ∞
n1nz, enen with the domain DA1/2 {z ∈
Assume the following.
From the definition ofFand assumption1, it follows that
F·, u1· F·, u2· withu1t u2t, t∈δ,1, foru1, u2∈PC0,1, X,
Thus, system4.1can be transformed into the abstract problem1.1, and conditionsH2,
If3.6holdsit holds when the related constants are small, then according toTheorem 3.1, the problem4.1has at least one mild solution in PC0,1, X.
Acknowledgments
The authors would like to thank the referees for helpful comments and suggestions. J. Liang acknowledges support from the NSF of China10771202and the Specialized Research Fund for the Doctoral Program of Higher Education of China2007035805. Z. Fan acknowledges support from the NSF of China11001034and the Research Fund for Shanghai Postdoctoral Scientific Program10R21413700.
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