Nonlocal Impulsive Cauchy Problems for Evolution Equations

doi:10.1155/2011/784161

Research Article

Nonlocal Impulsive Cauchy Problems for Evolution Equations

Jin Liang

and Zhenbin Fan

1Department 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.

In4, the authors combined the two directions and studied firstly a class of nonlocal impulsive Cauchy problems for evolution equations by investigating the existence for mild in generalized sense solutions to the problems. In this paper, we study further the existence of solutions to the following nonlocal impulsive Cauchy problem for evolution equations:

d

dtut Ft, ut Aut ft, ut, 0≤t≤K, t /ti, u0 gu u0,

Δuti I_iuti, i1,2, . . . , p, 0< t₁< t₂<· · ·< t_p< 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 u t_i

−u t⁻_i

,

u t_i

lim

t→t_iut, u t⁻_i

lim

t→t⁻_iut

, 1.2

F,f,g,I_iare 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 normf_L1_K

0 ftdt. Let

PC0, K, X:{u:0, K → X; utis continuous att /ti, left continuous attti, and the right limitu

t_i

exists fori1,2, . . . , p .

2.2

It is easy to check that PC0, K, Xis a Banach space with the norm u_PC sup

t∈0,Kut. 2.3

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:sup_0≤t≤KTt<∞.

Under the above conditions, it is possible to define the fractional powerA^α :DA^α⊂ 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. Let 0< α <1 and assume that (H1) holds. Then, 1DA^αis a Banach space with the normx_αA^αxforx∈DA^α, 2Tt:X → DA^αfort >0,

3A^αTtxTtA^αxforx∈DA^αandt≥0,

4for everyt >0,A^αTtis bounded onXand there existsCα>0 such that A^αTt ≤ Cα

t^α , 0< t≤K, 2.5

5A^−αis a bounded linear operator inXwithDA^α ImA^−α, 6if 0< α < β≤1, thenDA^β→DA^α.

We denote byX_αthat the Banach spaceDA^αendowed 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 Tt

u₀F0, u0−gu −Ft, ut _t

0

ATt−sFs, usds

_t

0

Tt−sfs, usds

0<ti<t

Tt−t_iIiuti, 0≤t≤K.

2.6

To discuss the compactness of subsets of PC0, K, X, we lett00,t_p1K, J₀ t₀, t₁, J1 t₁, t₂, . . . , Jp

tp, t_p1 . 2.7

ForD⊆PC0, K, X, we denote byD|_Jithe set D|_Ji

u∈Cti, t_i1, X;uti v t_i

, ut vt, t∈Ji, v∈D

, 2.8

i0,1,2, . . . , p. Then it is easy to see that the following result holds.

Lemma 2.3. A setD ⊆PC0, K, Xis precompact in PC0, K, Xif and only if the setD|_Ji is precompact inCti, t_i1, 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

cju sj

, 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 gu gvfor anyu, v ∈ PC0, K, Xwithus vs,s ∈ δ, K. Moreover, there existL₁, L₁>0 such thatgu ≤L₁u_PCL₁for anyu∈PC0, K, X.

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

A^βFt, x1−A^βFt, x2≤L₂x1−x₂ 3.2

for any 0≤t≤K,x1, x2∈X, and

A^βFt, 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 functionρ_l ∈L¹0, K,Rsuch thatft, x ≤ ρ_ltfor a.e.t∈0, Kand allx∈B_l, 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 andC_1−β>0 such thatM₀A^−βand A^1−βTt≤ C_1−β

t^1−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

M0LLC_1−βK^β

β <1. 3.6

To prove the theorem, we need some lemmas. Next, forn∈ N, we denote byQn the mapsQ_n: 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−t_iT 1

n

I_iuti, 0≤t≤K.

3.7

In addition, we introduce the decompositionQnQn1Qn2Qn3Qn4, where

Qn1ut Tt

u0−T 1

n

gu

, Qn2ut

0<ti<t

Tt−t_iT 1

n

I_iuti,

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 Q_ndefined by3.7has at least one fixed pointu_n∈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 mapQ_n3is 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^−βA^βF0, u10−A^−βA^βF0, u20A^−βA^βFt, u1t−A^−βA^βFt, u2t

_t

0

A^1−βTt−s

A^βFs, u₁s−A^βFs, u₂sds

≤MM0Lu1−u2M0Lu1t−u2t _t

0

C_1−β

t−s^1−βLu1s−u2sds.

3.9

Thus,

Qn3u₁−Q_n3u_2PC≤

M1M0LLC_1−βK^β β

u1−u₂, 3.10

which implies thatQ_n3is a contraction by condition3.6.

Secondly, we prove that Qn4, Qn1, Qn2 are completely continuous operators. Let {um}^∞_m1be a sequence in PC0, K, Xwith

mlim→ ∞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−Qn4u_PC≤M _K

0

fs, ums−fs, usds,

Qn1u_m−Q_n1u_PC≤Mgum−gu, Qn2um−Qn2u_PC≤M

p i1

Iiumti−Iiuti.

3.12

Then by the continuity off,g,I_i, and using the dominated convergence theorem, we get

mlim→ ∞Qn4umQn4u, lim

m→ ∞Qn1umQn1u, lim

m→ ∞Qn2umQn2u 3.13

in PC0, K, X, which implies thatQ_n4, Q_n1, Q_n2are continuous on PC0, K, X.

Next, for the compactness ofQ_n4we 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 ofQ_n2. For this purpose, let J0 0, t1, J1 t1, t2, . . . , Jp

tp, K . 3.17

Note that

Qn2ut

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

0, t∈J0,

Tt−t1T 1

n

I1ut1, t∈J1,

· · · p

i1

Tt−t_iT 1

n

I_iuti, t∈J_p.

3.18

Thus according toLemma 2.3, we only need to prove that

{Qn2u;u∈W}|_J

1

T· −t₁T 1

n

I₁ut1; · ∈J₁, u∈W

3.19

is precompact inCt1, t2, X, as the remaining cases fort ∈ J_i,i 2,3, . . . , p, can be dealt with in the same way; hereW is any bounded subset in PC0, K, X. And, we recall that v Qn2u|_J

1,u∈W, which means that vt1 Qn2u

t₁ T

1 n

I₁ut1, vt Q_n2ut Tt−t₁T

1 n

I₁ut₁, t∈J₁.

3.20

Thus, by the compactness of T1/n, we know that {Qn2u;u ∈ W}|_J

1t is relatively compact inXfor everyt∈J₁.

Next, fort₁≤s≤t≤t₂, we have Tt−t₁T

1 n

I₁ut1−Ts−t₁T 1

n

I₁ut1

Ts−t1Tt−s−T0T 1

n

I1ut1

≤M

Tt−s−T0T 1

n

I₁ut1 .

3.21

Thus, the set{Qn2u;u∈ W}|_J

1 ⊆ Ct1, t₂, Xis equicontinuous due to the compactness of {T1/nI1ut1;u ∈ W} and the strong continuity of operatorT·. By the Arzela-Ascoli theorem, we conclude that{Qn2u;u ∈ W}|_J

1 is precompact inCt1, t₂, X. The same idea can be used to prove that{Qn2u;u ∈W}|_Ji is precompact for eachi 2,3, . . . , p. Therefore, {Qn2u;u ∈ W}is precompact in PC0, K, X, that is, the operatorQn2 : PC0, K, X → PC0, K, Xis compact.

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 thatQ_nW_r ⊆ W_r. Suppose this is not true; then for each positive integerr, there areur ∈Wrandt^r ∈0, Ksuch thatQnurt^r> r. Then

r <Qnu_rt^r

Tt^r

u₀−T 1

n

gur F0, ur0

−Ft^r, u_rt^r _t^r

0

ATt^r−sFs, ursds

_t^r

0

Tt^r−sfs, u_rsds

0<ti<t^r

Tt^r−t_iT 1

n

I_iu_rt_i

≤M

u₀LrL₁M₀Lr1 M₀Lr1 _t

0

C_1−β

t−s^1−βLr1ds M

_t

0

ρ_rsdsMp

LrL₄

≤M

u0LrL₁

1MM0Lr1 LC_1−βK^β

β r1

M _K

0

ρrsdsMp

LrL₄ .

3.23

Dividing on both sides byrand taking the lower limit asr → ∞, we obtain that

L₀ M

LM₀LγpL

M₀LLC_1−βK^β

β ≥1. 3.24

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 Q_nhas at least one fixed point inW_r. 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 Lemma 3.2, n≥1}, 3.25 andδis the constant in (H2).

Proof. The proof will be given in several steps. In the followinghis a number in0, δ.

Step 1. D|_h,t1is precompact inCh, t1, X.

Foru∈PC0, K, X, defineQ_F1: 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,Q_F2is 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{Qn1u_n;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 {Qn1u_n;n≥1}|_h,t1is precompact inCh, t1, X.

By the same reasoning,{QF1u_n;n≥1}|_h,t1is precompact inCh, t1, X.

ForQF2, we claim thatQF2 :Ch, t1, X → Ch, t1, Xis Lipschitz continuous with constantM₀L LC_1−βK^β/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

C_1−β

t−s^1−βLdsmax

0≤t≤t1

ut−vt

≤M₀Lut−vtLC_1−βK^β

β max

h≤t≤t1

ut−vt,

3.31

that is,

QF2u−Q_F2v_Ch,t1,X ≤

M₀LLC_1−βK^β β

u−v_Ch,t1,X. 3.32

Therefore,Q_F2 : Ch, t1, X → Ch, t1, Xis Lipschitz continuous with constantM₀L LC1−βK^β/β.

Clearly,{Qn4un;n≥1}is precompact in PC0, K, X, and so is{Qn4un;n≥1}|_h,t1in Ch, t1, X.

Thus, by3.29andLemma 2.4, we obtain

α

D|_h,t1

≤

M₀LLC_1−βK^β β

α

D|_h,t1

. 3.33

By 3.6,M0L LC1−βK^β/β < 1, which impliesαD|_h,t1 0. Consequently, D|_h,t1 is precompact inCh, t1, X.

Step 2. D|_h,t2is precompact in PCh, t2, X.

Foru∈PCh, t2, X, let

ut ut, t∈h, t2, ut uh, t∈0, h, 3.34

and defineQ_F2: PCh, t2, X → PCh, t2, Xby Q_F2 u

t −Ft, ut _t

0

ATt−sFs, usds, t∈h, t₂. 3.35

ByH3,Q_F2is well defined and foru∈D, we have Qn3ut QF1ut

Q_F2u|_h,t2

t, t∈h, t2. 3.36

So, forun∈D,n≥1, we have u_nt Qn1u_nt QF1u_nt

Q_F2 u_n|_h,t2

t Qn4u_nt Qn2u_nt, t∈h, t2, 3.37 where

Qn2u_nt

⎧⎪

⎨

⎪⎩

0, t∈h, t1,

Tt−t₁T 1

n

I₁unt1, t∈J₁. 3.38

According to the proof ofStep 1, we know that

{Qn1un;n≥1}|_h,t2, {QF1un;n≥1}|_h,t2, {Qn4un;n≥1}|_h,t2 3.39

are all precompact in PCh, t2, X and Q_F2 : PCh, t2, X → PCh, t2, X is Lipschitz continuous with constantM₀L LC_1−βK^β/β.

Next, we will show that{Qn2u_n;n ≥ 1}|_h,t2 is precompact in PCh, t2, X. Firstly, it is easy to see that{Qn2un;n ≥ 1}|_h,t1 is precompact inCh, t1, X. Thus according to Lemma 2.3, it remains to prove that

{Qn2un;n≥1}|_J1

T· −t1T 1

n

I1unt1; · ∈J₁, n≥1

3.40

is precompact inCt1, t2, X. And, we recall thatvn Qn2un|_J1,n≥1, which means that

vnt1 Qn2un t₁

T 1

n

I1unt1, v_nt Qn2u_nt Tt−t₁T

1 n

I₁unt1, t∈J₁.

3.41

ByStep 1,D|_h,t1 is precompact inCh, t1, X. Without loss of generality, we may suppose that

un|_h,t1−→w, asn−→ ∞inCh, t1, X. 3.42

Therefore,unt1 → wt1, asn → ∞inX. Thus, by the continuity ofI₁andTt, we get T

1 n

I₁unt1−I₁wt1

≤ T

1 n

I1unt1−T 1

n

I1wt1

T 1

n

I1wt1−I1wt1

≤MI1unt1−I₁wt1 T

1 n

I₁wt1−I₁wt1

−→0,

3.43

asn → ∞, which implies that{vnt1;n ≥ 1} is relatively compact inX. And, for t ∈ J₁, by the compactness ofTt,t > 0,{vnt;n ≥ 1}is also relatively compact inX. Therefore, {Qn2u_n;n≥1}|_J

1tis relatively compact inXfor everyt∈J₁.

Next, fort1≤s≤t≤t2, we have Tt−t₁T

1 n

I₁unt1−Ts−t₁T 1

n

I₁unt1

Ts−t₁Tt−s−T0T 1

n

I₁unt1

≤M

Tt−s−T0T 1

n

I₁unt1 .

3.44

Thus, the set{Qn2u_n;n≥1}|_J

1⊆Ct1, t₂, Xis equicontinuous onJ₁due to the compactness of{T1/nI1unt1; n≥1}and the strong continuity of operatorTt,t≥0. By the Arzela- Ascoli theorem, we conclude that{Qn2un;n≥1}|_J1is precompact inCt1, t2, X. Therefore, {Qn2u_n;n≥1}|_h,t2is precompact in PCh, t2, X.

Thus, byLemma 2.4, we obtain

α

D|_h,t2

≤

M0LLC_1−βK^β β

α

D|_h,t2

. 3.45

By 3.6,M0L LC_1−βK^β/β < 1, which impliesαD|_h,t2 0. Consequently, D|_h,t2 is precompact in PCh, t2, X.

Step 3. The same idea can be used to prove the compactness ofD|_h,ti in PCh, ti, X for i3, . . . , p, p1, wheret_p1K. This completes the proof.

Proof ofTheorem 3.1. Forun∈D,n≥1, let

u_nt

⎧⎨

⎩

u_nt, t∈δ, K,

u_nδ, t∈0, δ, 3.46

whereδcomes from the conditionH2. Then, by conditionH2,gun gun.

By Lemma 3.3, without loss of generality, we may suppose that un → u ∈ PC0, K, X, asn → ∞. Thus, by the continuity ofTtandg, we get

T 1

n

gun−gu

≤ T

1 n

gun−T 1

n

gu

T

1 n

gu−gu

≤Mgun−gu T

1 n

gu−gu

−→0,

3.47

asn → ∞. Thus,

{Qn1un;n≥1}

T·

u₀−T

1 n

gun

;n≥1

3.48 is precompact in PC0, K, X. Moreover, {Qn4un;n ≥ 1} and {Qn2un;n ≥ 1} are both precompact in PC0, K, X. And Q_n3 : PC0, K, X → PC0, K, X is Lipschitz continuous with constantM1M0L LC_1−βK^β/β. Note that

u_nt Q_nu_nt Q_n1u_nt Q_n3u_nt Q_n4u_nt Q_n2u_nt, t∈0, K. 3.49

Therefore, byLemma 2.4, we know that the setD is precompact in PC0, K, X. Without loss of generality, we may suppose thatun → u^∗in PC0, K, X. On the other hand, we also have

unt Tt

u₀F0, un0−T 1

n

gun

−Ft, unt

_t

0

ATt−sFs, unsds _t

0

Tt−sfs, unsds

0<ti<t

Tt−t_iT 1

n

I_iu_nt_i, 0≤t≤K.

3.50

Lettingn → ∞in both sides, we obtain u^∗t Tt

u₀F0, u^∗0−gu^∗ −Ft, u^∗t

_t

0

ATt−sFs, u^∗sds _t

0

Tt−sfs, u^∗sds

0<ti<t

Tt−t_iIiu^∗ti, 0≤t≤K,

3.51

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 3.5. Assume (H1), (H3)–(H5) hold. Ifg≡0, then the impulsive Cauchy problem1.1has at least one mild solution on0, K, provided

M

M₀LγpL

M₀LLC_1−βK^β

β <1. 3.52

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. Theorems3.5-3.6 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

λ t, x, y

w t, y

dy

∂²

∂x²wt, x vt, wt, x, 0≤t≤1, 0≤x≤π, t /ti, wt,0 wt, π 0, 0≤t≤1,

w t_i

−w t⁻_i

I_iwti, i1, . . . , p, 0< t₁<· · ·< t_p<1, w0, x

q j1

cjw sj, x

w₀x, 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 semigroupTt_t≥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, enen. 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.

dA^1/2 is given by A^1/2z _∞

n1nz, enen with the domain DA^1/2 {z ∈ X;_∞

n1nz, enen ∈X}.

Assume the following.

1The function λ : 0,1 ×0, π× 0, π → R is continuously differential with λt,0, y λt, π, y 0 fort ∈ 0,1,y ∈ 0, π, and there exists a real number δ∈0, s1such thatλt, x, y 0 fort∈0, δ,x, y∈0, π. Moreover,

Λ: sup

t∈0,1

_π

0

∂

∂x

λ

t, x, y² dxdy

_1/2

<∞. 4.4

2For eacht ∈ 0,1,vt,·is continuous, and for eachx ∈ R,v·, xis measurable and, there exists a functiona· ∈ L¹0,1,Rsuch that|vt, x| ≤ at|x|for a.e.

t∈0,1and allx∈R.

3Ii : X → X is a continuous function for eachi 1, . . . , p, and there exist positive numbersL4, L₄such thatIiz ≤L4zL₄for anyz∈Xandi1,2, . . . , p.

DefineF, f:0,1×X → Xandg: PC0,1, X → X, respectively, as follows. Forx∈0, π, Ft, zx

_π

0

λ t, x, y

z y

dy, ft, zx vt, zx, gux

q j1

cju sj

x.

4.5

From the definition ofFand assumption1, it follows that

F·, u1· F·, u2· withu₁t u₂t, t∈δ,1, foru₁, u₂∈PC0,1, X, Ft, z, en

_π

0

enx π

0

λ t, x, y

z y

dy

dx 1

n

π 0

∂

∂xλ t, x, y

z y

dy,

! 2

π cosnx

"

, A^1/2Ft, z1−A^1/2Ft, z2

∞ n1

nFt, z1−Ft, z2, enen

≤ _π

0

∂

∂x

λ

t, x, y² dydx

_1/2

× _π

0

z₁ y

−z2

y2dy 1/2

≤Λz₁−z₂.

4.6

Thus, system4.1can be transformed into the abstract problem1.1, and conditionsH2, H3,H4, andH5are satisfied with

L₁ q j1

##cj##, L₂L₃ Λ, ρlt lat, γ ₁

0

atdt. 4.7

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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