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Volume 2011, Article ID 784161, 17 pagesdoi:10.1155/2011/784161 Research Article Nonlocal Impulsive Cauchy Problems for Evolution Equations 1 Department of Mathematics, Shanghai Jiao Ton

Volume 2011, Article ID 784161, 17 pages

doi:10.1155/2011/784161

Research Article

Nonlocal Impulsive Cauchy Problems for

Evolution Equations

1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

2 Department of Mathematics, Changshu Institute of Technology, Suzhou, Jiangsu 215500, China

Correspondence should be addressed to Jin Liang,jinliang@sjtu.edu.cn

Received 17 October 2010; Accepted 19 November 2010

Academic Editor: Toka Diagana

Copyrightq 2011 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 6 and 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 12 and 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:

dt ut  Ft, ut  Aut  ft, ut, 0 ≤ t ≤ K, t / t i ,

u0  gu  u0,

Δut i   I i ut i , i  1, 2, , p, 0 < t1 < t2< · · · < t p < K,

1.1

where−A : DA ⊆ X → X is the infinitesimal generator of an analytic semigroup {Tt; t ≥

0} and X is a real Banach space endowed with the norm  · ,

Δut i   ut

i



− ut−

i



,



u

t

i



 lim

t → t

i

ut, ut−

i



 lim

t → t−

i

ut



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, K with the norm

u  max{ut; t ∈ 0, K}, 2.1

and by L10, K, X the space of X-valued Bochner integrable functions on 0, K with the

normf L10K ftdt Let

PC0, K, X : {u : 0, K → X; ut is continuous at t / t i , left continuous at t  t i ,

and the right limit u

t

i



exists for i  1, 2, , p.

2.2

It is easy to check that PC0, K, X is a Banach space with the norm

uPC sup

In this paper, for r > 0, let B r : {x ∈ X; x ≤ r} and

W r : {u ∈ PC0, K, X; ut ∈ B r , ∀t ∈ 0, K}. 2.4 Throughout this paper, we assume the following

H1 The operator −A : DA ⊆ X → X is the infinitesimal generator of a compact

analytic semigroup{Tt : t ≥ 0} on Banach space X and 0 ∈ ρA the resolvent set

of A.

In the remainder of this work, M : sup0≤t≤KTt < ∞.

Under the above conditions, it is possible to define the fractional power A α : DA α ⊂

X → X, 0 < α < 1, of A as closed linear operators And it is known that the following

properties hold

Theorem 2.1 see 13, Pages 69–75 Let 0 < α < 1 and assume that (H1) holds Then,

1 DA α  is a Banach space with the norm x α  A α x for x ∈ DA α ,

2 Tt : X → DA α  for t > 0,

3 A α Ttx  TtA α x for x ∈ DA α  and t ≥ 0,

4 for every t > 0, A α Tt is bounded on X and there exists C α > 0 such that

A α Tt ≤ C t α α , 0 < t ≤ K, 2.5

5 A −α is a bounded linear operator in X with DA α   ImA −α ,

6 if 0 < α < β ≤ 1, then DA β  → DA α .

We denote by X α that the Banach space DA α endowed the graph norm from now on

Definition 2.2 A function u ∈ PC0, K, X is said to be a mild solution of 1.1 on 0, K if the function s → ATt − sFs, us is integrable on 0, t for all t ∈ 0, K and the following integral equation is satisfied:

ut  Ttu0 F0, u0 − gu − Ft, ut  t

0ATt − sFs, usds

 t 0

Tt − sfs, usds  

0<t i <t Tt − t i I i ut i , 0 ≤ t ≤ K.

2.6

To discuss the compactness of subsets of PC0, K, X, we let t0 0, t p1  K,

J0 t0 , t1, J1 t1 , t2, , Jpt p , t p1

For D ⊆ PC0, K, X, we denote by D| J ithe set

D| J i u ∈ Ct i , t i1 , X; ut i   vt

i



, ut  vt, t ∈ J i , v ∈ D, 2.8

i  0, 1, 2, , p Then it is easy to see that the following result holds.

Lemma 2.3 A set D ⊆ PC0, K, X is precompact in PC0, K, X if and only if the set D| J i is precompact in Ct i , t i1 , X for every i  0, 1, 2, , p.

Next, we recall that the Hausdorff measure of noncompactness α· on each bounded subsetΩ of Banach space Y is defined by

αΩ  inf{ε > 0; Ω has a finite ε-net in Y}. 2.9

Some basic properties of α· are given in the following Lemma.

Lemma 2.4 see 14 Let Y be a real Banach space and let B, C ⊆ Y be bounded Then,

1 B is precompact if and only if αB  0;

2 αB  αB  αconvB, where B and convB mean the closure and convex hull of B,

respectively;

3 αB ≤ αC when B ⊆ C;

4 αB  C ≤ αB  αC, where B  C  {x  y; x ∈ B, y ∈ C};

5 αB ∪ C ≤ max{αB, αC};

6 αλB  |λ|αB for any λ ∈ R;

7 let Z be a Banach space and Q : DQ ⊆ Y → Z Lipschitz continuous with constant k.

Then αQB ≤ kαB for all B ⊆ DQ being bounded.

We note that a continuous map Q : W ⊆ Y → Y is an α-contraction if there exists a positive constant k < 1 such that αQC ≤ kαC for all bounded closed C ⊆ W.

Lemma 2.5 see Darbo-Sadovskii’s fixed point theorem in 14 If W ⊆ Y is bounded closed

and convex, and Q : W → W is an α-contraction, then the map Q has at least one fixed point in W.

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 item g and the impulsive functions I iare only assumed to be continuous

in PC0, K, X and X, respectively

In practical applications, the values of ut for t near zero often do not affect gu For

example, it is the case when

gu 

q

j1

c j u

s j

, 0 < s1< s2< · · · < s q < K. 3.1

So, to prove our main results, we introduce the following assumptions

H2 g : PC0, K, X → X is a continuous function, and there is a δ ∈ 0, t1 such that

gu  gv for any u, v ∈ PC0, K, X with us  vs, s ∈ δ, K Moreover,

there exist L1, L

1> 0 such that gu ≤ L1uPC L

1for any u ∈ PC0, K, X.

H3 There exists a β ∈ 0, 1 such that F : 0, K × X → X β is a continuous function,

and F·, u·  F·, v· for any u, v ∈ PC0, K, X with us  vs, s ∈ δ, K Moreover, there exist L2, L3> 0 such that

β Ft, x1 − Aβ Ft, x2 2x1− x2 3.2 for any 0≤ t ≤ K, x1 , x2∈ X, and

for any 0≤ t ≤ K, x ∈ X.

H4 The function ft, · : X → X is continuous a.e t ∈ 0, K; the function f·, x :

0, K → X is strongly measurable for all x ∈ X Moreover, for each l ∈ N, there exists a function ρ l ∈ L10, K, R such that ft, x ≤ ρ l t for a.e t ∈ 0, K and all x ∈ B l, and

γ : lim inf

l → ∞

1

l

K

0

ρ l sds < ∞. 3.4

H5 I i : X → X is continuous for every i  1, 2, , p, and there exist positive numbers

L4, L

4such thatI i x ≤ L4x  L

4for any x ∈ X and i  1, 2, , p.

We note that, byTheorem 2.1, there exist M0 > 0 and C1−β> 0 such that M0 A −β and

1−βTt C1−β

For simplicity, in the following we set L  max{L1 , L2, L3, L4} and will substitute L1, L2, L3, L4

by L below.

Theorem 3.1 Let (H1)–(H5) hold Then the nonlocal impulsive Cauchy problem 1.1 has at least

one mild solution on 0, K, provided

L0  ML  M0L  γ  pL M0 L  LC1−βK

β

To prove the theorem, we need some lemmas Next, for n ∈ N, we denote by Q n the

maps Q n: PC0, K, X → PC0, K, X defined by

Q n ut  Tt



u0 F0, u0 − T

 1

n



gu



− Ft, ut  t

0

ATt − sFs, usds

 t

0

Tt − sfs, usds  

0<t <t Tt − t i T

 1

n



I i ut i , 0 ≤ t ≤ K.

3.7

In addition, we introduce the decomposition Q n  Q n1  Q n2  Q n3  Q n4, where

Q n1 ut  Tt



u0− T

 1

n



gu



,

Q n2 ut  

0<t i <t Tt − t i T

 1

n



I i ut i ,

Q n3 ut  TtF0, u0 − Ft, ut  t

0

ATt − sFs, usds,

Q n4 ut  t

0

Tt − sfs, usds

3.8

for u ∈ PC0, K, X and t ∈ 0, K.

Lemma 3.2 Assume that all the conditions in Theorem 3.1 are satisfied Then for any n ≥ 1, the map

Q n defined by3.7 has at least one fixed point u n ∈ PC0, K, X.

Proof To prove the existence of a fixed point for Q n, we will use Darbu-Sadovskii’s fixed point theorem

Firstly, we prove that the map Q n3 is a contraction on PC0, K, X For this purpose,

let u1 , u2∈ PC0, K, X Then for each t ∈ 0, K and by condition H3, we have

Q n3 u1t − Qn3 u2t

≤ MF0, u10 − F0, u20  Ft, u1t − Ft, u2t

 t

0

ATt − sFs, u1s − Fs, u2sds

≤ M −β A β F0, u10 − A−β A β F0, u20 −β A β Ft, u1t − A−β A β Ft, u2t

 t

0

1−βTt − sA β Fs, u1s − Aβ Fs, u2s

≤ MM0 Lu1− u2  M0 Lu1t − u2t  t

0

C1−β

t − s1−βLu1s − u2sds.

3.9 Thus,

Q n3 u1− Q n3 u2PC≤



M  1M0 L  LC1−βK β

β



u1 − u2, 3.10

which implies that Q n3is a contraction by condition3.6

Secondly, we prove that Q n4 , Q n1 , Q n2 are completely continuous operators Let

{u m}∞

m1be a sequence in PC0, K, X with

lim

in PC0, K, X By the continuity of f with respect to the second argument, we deduce that

for each s ∈ 0, K, fs, u m s converges to fs, us in X, and we have

Q n4 u m − Q n4 uPC≤ M K

0

m s − fs, us ds,

Q n1 u m − Q n1 uPC≤ M m  − gu

Q n2 u m − Q n2 uPC≤ M

p

i1

I i u m t i  − I i ut i .

3.12

Then by the continuity of f, g, I i, and using the dominated convergence theorem, we get

lim

m → ∞ Q n4 u m  Q n4 u, lim

m → ∞ Q n1 u m  Q n1 u, lim

m → ∞ Q n2 u m  Q n2 u 3.13

in PC0, K, X, which implies that Qn4 , Q n1 , Q n2are continuous on PC0, K, X

Next, for the compactness of Q n4we refer to the proof of4, Theorem 3.1

For Q n1 and any bounded subset W of PC0, K, X, we have

Q n1 ut  Ttu0− T

 1

n



Ttgu, t ∈ 0, K, u ∈ W, 3.14

which implies that Q n1 Wt is relatively compact in X for every t ∈ 0, K by the

compactness of T 1/n On the other hand, for 0 ≤ s ≤ t ≤ K, we have

Q n1 ut − Q n1 us ≤



u0− T

 1

n



gu 3.15 Since{T1/ngu; u ∈ W} is relatively compact in X, we conclude that

Q n1 ut − Q n1 us −→ 0 uniformly as t −→ s and u ∈ W, 3.16

which implies that Q n1 W is equicontinuous on 0, K Therefore, Q n1is a compact operator

Now, we prove the compactness of Q n2 For this purpose, let

J0 0, t1, J1  t1 , t2, , Jpt p , K

Note that

Q n2 ut 

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

Tt − t1T

 1

n



I1ut1, t ∈ J1,

· · ·

p

i1 Tt − t i T

 1

n



I i ut i , t ∈ J p

3.18

Thus according toLemma 2.3, we only need to prove that

{Q n2 u; u ∈ W}| J1 



T· − t1T

 1

n



I1ut1; · ∈ J1, u ∈ W



3.19

is precompact in Ct1 , t2, X, as the remaining cases for t ∈ Ji , i  2, 3, , p, can be dealt with in the same way; here W is any bounded subset in PC0, K, X And, we recall that

v  Q n2 u| J

1, u ∈ W, which means that

vt1  Qn2 ut

1



 T

 1

n



I1ut1,

vt  Q n2 ut  Tt − t1T

 1

n



I1ut1, t ∈ J1.

3.20

Thus, by the compactness of T 1/n, we know that {Q n2 u; u ∈ W}| J

1t is relatively compact in X for every t ∈ J1

Next, for t1≤ s ≤ t ≤ t2, we have

1T

 1

n



I1ut1 − Ts − t1T

 1

n



I1ut1

 1Tt − s − T0T

 1

n



I1ut1

≤ M

 1

n



I1ut1

3.21

Thus, the set{Q n2 u; u ∈ W}| J

1 ⊆ Ct1 , t2, X is equicontinuous due to the compactness of

{T1/nI1ut1; u ∈ W} and the strong continuity of operator T· By the Arzela-Ascoli

theorem, we conclude that{Q n2 u; u ∈ W}| J

1 is precompact in Ct1 , t2, X The same idea can be used to prove that{Q n2 u; u ∈ W}| J i is precompact for each i  2, 3, , p Therefore, {Q n2 u; u ∈ W} is precompact in PC0, K, X, that is, the operator Q n2 : PC0, K, X → PC0, K, X is compact

Thus, for any bounded subset W ⊆ PC0, K, X, we have byLemma 2.4,

αQ n W ≤ αQ n1 W  αQ n3 W  αQ n4 W  αQ n2 W ≤ L0αW. 3.22

Hence, the map Q n is an α-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 that Q n W r ⊆ W r Suppose this is not true; then for each positive integer r, there are u r ∈ W r and t r ∈ 0, K such that Q n u r t r  > r Then

r < Q n u r t r

 Tt r



u0− T

 1

n



gu r   F0, u r0



− Ft r , u r t r  t

r

0

ATt r − sFs, u r sds

 t

r

0

Tt r − sfs, u r sds  

0<t i <t r Tt r − t i T

 1

n



I i u r t i

≤ Mu0  Lr  L

1 M0 Lr  1  M0 Lr  1  t

0

C1−β

t − s1−βLr  1ds

 M t

0

ρ r sds  MpLr  L

4



≤ Mu0  Lr  L

1



 1  MM0 Lr  1  LC1−ββ K β r  1

 M K

0

ρ r sds  MpLr  L

4



.

3.23

Dividing on both sides by r and taking the lower limit as r → ∞, we obtain that

L0  ML  M0L  γ  pL M0 L  LC1−ββ K β ≥ 1. 3.24

This is a contradiction with inequality 3.6 Therefore, there exists r > 0 such that the mapping Q n maps W r into itself By Darbu-Sadovskii’s fixed point theorem, the operator

Q n has at least one fixed point in W 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 PC h, K, X for all h ∈ 0, δ, where

D : {u n ; u n ∈ PC0, K, X coming from Lemma 3.2, n ≥ 1}, 3.25

and δ is the constant in (H2).

Proof The proof will be given in several steps In the following h is a number in 0, δ.

Step 1 D| h,t1is precompact in Ch, t1, X.

For u ∈ PC0, K, X, define Q F1: PC0, K, X → PC0, K, X by

Q F1 ut  TtF0, u0, t ∈ 0, K. 3.26

For u ∈ Ch, t1, X, let ut  ut, t ∈ h, t1, ut  uh, t ∈ 0, h, and we define

Q F2 : Ch, t1, X → Ch, t1, X by

Q F2 ut  −Ft, ut  t

0

ATt − sFs, usds, t ∈ h, t1. 3.27

By conditionH3, Q F2 is well defined and for u ∈ D, we have

Q n3 ut  Q F1 ut Q F2 u| h,t1t, t ∈ h, t1. 3.28

On the other hand, for u n ∈ D, n ≥ 1, we have Q n2 u n t  0, t ∈ h, t1 So,

u n t  Q n1 u n t  Q F1 u n t Q F2 u n|h,t1t  Q n4 u n t, t ∈ h, t1. 3.29 Now, for{Q n1 u n ; n≥ 1}, we have

Q n1 u n t  Ttu0 − TtT

 1

n



gu n , t ∈ h, t1. 3.30

By the compactness of T t, t > 0, we get that {Q n1 u n t; n ≥ 1} is relatively compact in X for every t ∈ h, t1 and {Q n1 u n ; n≥ 1}|h,t1is equicontinuous onh, t1, which implies that {Q n1 u n ; n≥ 1}|h,t1is precompact in Ch, t1, X.

By the same reasoning,{Q F1 u n ; n≥ 1}|h,t1is precompact in Ch, t1, X.

For Q F2 , we claim that Q F2 : Ch, t1, X → Ch, t1, X is Lipschitz continuous with constant M0 L  LC1−βK β /β In fact, H3 implies that for every u, v ∈ Ch, t1, X and

t ∈ h, t1,

Q F2 ut − Q F2 vt

≤ Ft, ut − Ft, vt  t

0

ATt − sFs, us − Fs, vsds

≤ M0 Lut − vt  t

0

C1−β

t − s1−βL ds max

0≤t≤t 1

ut − vt

≤ M0 Lut − vt  LC1−ββ K βmax

h≤t≤t ut − vt,

3.31

when the nonlocal item g and the impulsive functions I iare only assumed... strongly measurable for all x ∈ X Moreover, for each l ∈ N, there exists a function ρ l ∈ L10, K, R such that ft, x ≤ ρ l t for a.e t ∈ 0,... L4

by L below.

Theorem 3.1 Let (H1)–(H5) hold Then the nonlocal impulsive Cauchy problem 1.1 has at least

one mild solution on 0, K,