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Volume 2010, Article ID 293410, 15 pagesdoi:10.1155/2010/293410 Research Article Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces He Yang De

Volume 2010, Article ID 293410, 15 pages

doi:10.1155/2010/293410

Research Article

Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces

He Yang

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Correspondence should be addressed to He Yang,yanghe256@163.com

Received 29 December 2009; Revised 20 July 2010; Accepted 3 September 2010

Academic Editor: Alberto Cabada

Copyrightq 2010 He Yang 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

By constructing a mixed monotone iterative technique under a new concept of upper and lower

solutions, some existence theorems of mild ω-periodic L-quasi solutions for abstract impulsive

evolution equations are obtained in ordered Banach spaces These results partially generalize and extend the relevant results in ordinary differential equations and partial differential equations

1 Introduction and Main Result

Impulsive differential equations are a basic tool for studying evolution processes of real life phenomena that are subjected to sudden changes at certain instants In view of multiple applications of the impulsive differential equations, it is necessary to develop the methods for their solvability Unfortunately, a comparatively small class of impulsive differential equations can be solved analytically Therefore, it is necessary to establish approximation methods for finding solutions The monotone iterative technique of Lakshmikantham et

al see 1 3 is such a method which can be applied in practice easily This technique combines the idea of method of upper and lower solutions with appropriate monotone conditions Recent results by means of monotone iterative method are obtained in4 7 and the references therein In this paper, by using a mixed monotone iterative technique in the

presence of coupled lower and upper L-quasisolutions, we consider the existence of mild

ω-periodicL-quasisolutions for the periodic boundary value problem PBVP of impulsive

evolution equations

ut  Aut  ft, ut, ut, a.e t ∈ J, Δu| t t k  I k ut k , ut k , k  1, 2, , p,

u 0  uω

1.1

in an ordered Banach space X, where A : DA ⊂ X → X is a closed linear operator and

−A generates a C0-semigroup Tt t ≥ 0 in X; f : J × X × X → X only satisfies weak Carath´eodory condition, J  0, ω, ω > 0 is a constant; 0  t0 < t1 < t2 < · · · < t p < t p1 ω;

I k : X × X → X is an impulsive function, k  1, 2, , p; Δu| t t k denotes the jump of ut

at t  t k, that is, Δu| t t k  ut

k  − ut−

k , where ut

k  and ut−

k represent the right and left

limits of ut at t  t k , respectively Let PC J, X : {u : J → X | ut is continuous at t / t k and left continuous at t  t k , and ut

k  exists, k  1, 2, , p} Evidently, PCJ, X is a Banach

space with the normuPC supt ∈J ut Let J J \ {t1, t2, , t p }, J J \ {0, t1, t2, , t p}

Denote by X1the Banach space generated by DA with the norm  · 1   ·   A ·  An abstract function u ∈ PCJ, X ∩ C1J, X  ∩ CJ, X1 is called a solution of the PBVP1.1 if

u t satisfies all the equalities of 1.1

Let X be an ordered Banach space with the norm ·  and the partial order “≤”, whose

positive cone K : {u ∈ X | u ≥ 0} is normal with a normal constant N Let L ≥ 0 If functions

v0, w0∈ PCJ, X ∩ C1J, X  ∩ CJ, X1 satisfy

v0t  Av0t ≤ ft, v0t, w0t  Lv0t − w0t, t ∈ J,

Δv0|t t k ≤ I k v0t k , w0t k , k  1, 2, , p,

v00 ≤ v0ω,

1.2

w0t  Aw0t ≥ ft, w0t, v0t  Lw0t − v0t, t ∈ J,

Δw0|t t k ≥ I k w0t k , v0t k , k  1, 2, , p,

w00 ≥ w0ω,

1.3

we call v0, w0 coupled lower and upper L-quasisolutions of the PBVP1.1 Only choosing

“” in 1.2 and 1.3, we call v0, w0 coupled ω-periodic L-quasisolution pair of the

PBVP1.1 Furthermore, if u0 : v0 w0, we call u0an ω-periodic solution of the PBVP1.1

Definition 1.1 Abstract functions u, v ∈ PCJ, X are called a coupled mild ω-periodic

L-quasisolution pair of the PBVP1.1 if ut and vt satisfy the following integral equations:

u tTtB1u, v

t

0

T t−sG1u, vsds

0<t k <t

T t − t k I k ut k , vt k , t ∈ J,

v t  TtB1v, u

t

0

T t−sG1v, usds

0<t k <t

T t−t k I k vt k , ut k , t ∈ J,

1.4

where B1x, y  I − Tω−1ω

0 T ω − sG1x, ysds p

k1T ω − t k I k xt k , yt k and

G1x, ys  fs, xs, ys  Lxs − ys for any x, y ∈ PCJ, X, I is an identity operator.

Ifu : u  v, then u is called a mild ω-periodic solution of the PBVP1.1

Without impulse, the PBVP1.1 has been studied by many authors, see 8 11 and the references therein In particular, Shen and Li11 considered the existence of coupled mild

ω-periodic quasisolution pair for the following periodic boundary value problemPBVP in

X:

ut  Aut  ft, ut, ut, t ∈ J,

where f : J × X × X → X is continuous Under one of the following situations:

i Tt t ≥ 0 is a compact semigroup,

ii K is regular in X and Tt is continuous in operator norm for t > 0,

they built a mixed monotone iterative method for the PBVP1.5, and they proved that, if the PBVP1.5 has coupled lower and upper quasisolutions i.e., L ≡ 0 and without impulse

in1.2 and 1.3 v0 and w0 with v0 ≤ w0, nonlinear term f satisfies one of the following

conditions:

F1 f : J × X × X → X is mixed monotone,

F2 There exists a constant M1> 0 such that

f t, u2, w  − ft, u1, w  ≥ −M1u2− u1, ∀t ∈ J, v0t ≤ u1≤ u2≤ w0t, v0t ≤ w ≤ w0t,

1.6

and ft, u, v is nonincreasing on v.

Then the PBVP1.5 has minimal and maximal coupled mild ω-periodic quasisolutions between v0and w0, which can be obtained by monotone iterative sequences from v0and w0 But conditionsi and ii are difficult to satisfy in applications except some special situations

In this paper, by constructing a mixed monotone iterative technique under a new

concept of upper and lower solutions, we will discuss the existence of mild ω-periodic

L-quasi solutions for the impulsive evolution Equation1.1 in an ordered Banach space X In

our results, we will delete conditionsi and ii for the operator semigroup Tt t ≥ 0, and

improve conditionsF1 and F2 for nonlinearity f In addition, we only require that the nonlinear term f : J × X × X → X satisfies weak Carath´eodory condition:

1 for each u, v ∈ X, f·, u, v is strongly measurable.

2 for a.e.t ∈ J, ft, ·, · is subcontinuous, namely, there exists e ⊂ J with mes e  0 such

that

f t, u n , v nweak

for any t ∈ J \ e, and u n → u, v n → v n → ∞.

Our main result is as follows:

Theorem 1.2 Let X be an ordered and weakly sequentially complete Banach space, whose positive

cone K is normal, A : D A ⊂ X → X be a closed linear operator and −A generate a positive C0 -semigroup T t t ≥ 0 in X If the PBVP1.1 has coupled lower and upper L-quasisolutions v0and

w0with v0≤ w0, nonlinear term f and impulsive functions I k ’s satisfy the following conditions

H1 There exist constants M > 0 and L ≥ 0 such that

f t, u2, v2 − ft, u1, v1 ≥ −Mu2− u1  Lv2− v1 1.8

for any t ∈ J, and v0t ≤ u1≤ u2≤ w0t, v0t ≤ v2 ≤ v1≤ w0t.

H2 Impulsive function I k ·, · is continuous, and for any u i , v i ∈ X i  1, 2, it satisfies

I k u1, v1 ≤ I k u2, v2, k  1, 2, , p 1.9

for any t ∈ J, and v0t ≤ u1≤ u2≤ w0t, v0t ≤ v2 ≤ v1≤ w0t.

then the PBVP1.1 has minimal and maximal coupled mild ω-periodic L-quasisolutions

between v0 and w0, which can be obtained by monotone iterative sequences starting from v0 and

w0.

Evidently, condition H1 contains conditions F1 and F2 Hence, even without impulse in PBVP1.1,Theorem 1.2still extends the results in10,11

The proof of Theorem 1.2 will be shown in the next section In Section 2, we also

discuss the existence of mild ω-periodic solutions for the PBVP1.1 between coupled lower

and upper L-quasisolutions see Theorem 2.3 In Section 3, the results obtained will be applied to a class of partial differential equations of parabolic type

2 Proof of the Main Results

Let X be a Banach space, A : DA ⊂ X → X be a closed linear operator, and −A generate a

C0-semigroup Tt t ≥ 0 in X Then there exist constants C > 0 and δ ∈ R such that

Definition 2.1 A C0-semigroup Tt t ≥ 0 is said to be exponentially stable in X if there exist constants C ≥ 1 and δ > 0 such that

Let I0  t0, T  Denote by CI0, X  the Banach space of all continuous X-value functions on interval I0 with the normu C  maxt ∈I0ut It is well-known 12, Chapter

4, Theorem 2.9 that for any x0∈ DA and h ∈ C1I0, X, the initial value problemIVP of linear evolution equation

ut  Aut  ht, t ∈ I0,

u t0  x0

2.3

has a unique classical solution u ∈ C1I0, X  ∩ CI0, X1 expressed by

u t  Tt − t0x0

t

t0

If x0∈ X and h ∈ CI0, X , the function u given by 2.4 belongs to CI0, X We call it a mild solution of the IVP2.3

To proveTheorem 1.2, for any h ∈ PCJ, X, we consider the periodic boundary value

problemPBVP of linear impulsive evolution equation in X

ut  Aut  ht, t ∈ J, t / t k ,

Δu| t t k  y k , k  1, 2, , p,

u 0  uω,

2.5

where y k ∈ X, k  1, 2, , p.

Lemma 2.2 Let Tt t ≥ 0 be an exponentially stable C0-semigroup in X Then for any h ∈

PC J, X and y k ∈ X, k  1, 2, , p, the linear PBVP2.5 has a unique mild solution u ∈ PCJ, X

given by

u t  TtBh 

t

0

T t − shsds  

0<t k <t

T t − t k y k , t ∈ J, 2.6

where B h  I − Tω−1ω

0 T ω − shsds p

k1T ω − t k y k .

Proof For any h ∈ PCJ, X, we first show that the initial value problem IVP of linear

impulsive evolution equation

ut  Aut  ht, t ∈ J, t / t k ,

Δu| t t k  y k , k  1, 2, , p,

u 0  x0

2.7

has a unique mild solution u ∈ PCJ, X given by

u t  Ttx0

t

0

T t − shsds  

0<t k <t

T t − t k y k , t ∈ J, 2.8

where x0∈ X and y k ∈ X, k  1, 2, , p.

Let J k  t k , t k1, k  0, 1, 2, , p Let y0 0 If u ∈ PCJ, X is a mild solution of the

linear IVP2.7, then the restriction of u on J ksatisfies the initial value problemIVP of linear evolution equation without impulse

ut  Aut  ht, t k < t ≤ t k1,

u

tk

Hence, ont k , t k1, ut can be expressed by

u t  Tt − t ku t k   y k





t

t k

Iterating successively in the above equality with ut j  for j  k, k − 1, , 1, 0, we see that u

satisfies2.8

Inversely, we can verify directly that the function u ∈ PCJ, X defined by 2.8 is

a solution of the linear IVP2.7 Hence the linear IVP2.7 has a unique mild solution u ∈

PCJ, X given by 2.8

Next, we show that the linear PBVP2.5 has a unique mild solution u ∈ PCJ, X given

by2.6

If a function u ∈ PCJ, X defined by 2.8 is a solution of the linear PBVP2.5, then

x0 uω, namely,

I − Tωx0 

ω

0

T ω − shsds 

p



k1

Since Tt t ≥ 0 is exponentially stable, we define an equivalent norm in X by

|x|  sup

Thenx ≤ |x| ≤ Cx and |Tt| < e −δt t ≥ 0, and especially, |Tω| < e −δω < 1 It follows

that I − Tω has a bounded inverse operator I − Tω−1, which is a positive operator when

T tt ≥ 0 is a positive semigroup Hence we choose x0  I − Tω−1ω

0 T ω − shsds 

p

k1T ω − t k y k   Bh Then x0 is the unique initial value of the IVP2.7 in X, which satisfies u0  x0 uω Combining this fact with 2.8, it follows that 2.6 is satisfied

Inversely, we can verify directly that the function u ∈ PCJ, X defined by 2.6 is a solution of the linear PBVP2.5 Therefore, the conclusion ofLemma 2.2holds

Evidently, PCJ, X is also an ordered Banach space with the partial order “≤” reduced

by positive function cone KPC: {u ∈ PCJ, X | ut ≥ 0, t ∈ J} KPCis also normal with the

same normal constant N For v, w ∈ PCJ, X with v ≤ w, we use v, w to denote the order

interval{u ∈ PCJ, X | v ≤ u ≤ w} in PCJ, X, and vt, wt to denote the order interval {u ∈ X | vt ≤ u ≤ wt} in X FromLemma 2.2, if Tt t ≥ 0 is a positive C0-semigroup,

h ≥ 0 and y k ≥ 0, k  1, 2, , p, then the mild solution u ∈ PCJ, X of the linear PBVP2.5

satisfies u≥ 0

Proof of Theorem 1.2 We first show that f t, h1t, h2t ∈ L1J, X for any t ∈ J and

h1t, h2t ∈ v0t, w0t Since v0t ≤ h1t ≤ w0t, v0t ≤ h2t ≤ w0t for any t ∈ J,

from the assumptionH1, we have

f t, h1t, h2t  M  Lh1t − Lh2t

≤ ft, w0t, v0t  Lw0t − v0t  Mw0t

≤ w

0t  A  MIw0t  h0t,

f t, h1t, h2t  M  Lh1t − Lh2t

≥ ft, v0t, w0t  Lv0t − w0t  Mv0t

≥ v

0t  A  MIv0t  g0t.

2.13

Namely, g0t ≤ ft, h1t, h2t  M  Lh1t − Lh2t ≤ h0t, t ∈ J From the normality of cone K in X, we have

f t, h1t, h2t  M  Lh1t − Lh2t ≤ N h0− g0

PC g0

PC M∗. 2.14

Combining this fact with the fact that ft, h1t, h2t is strongly measurable, it follows that

f t, h1t, h2t ∈ L1J, X Therefore, for any h1t, h2t ∈ v0t, w0t, t ∈ J, we consider

the periodic boundary value problemPBVP of impulsive evolution equation in X

ut  A  MIut  Gh1, h2t, a.e t ∈ J, Δu| t t k  I k h1t k , h2t k , k  1, 2, , p,

u 0  uω,

2.15

where Gh1, h2t  ft, h1t, h2t  M  Lh1t − Lh2t Let M > 0 be large enough such that M > δ otherwise, replacing M by M  δ, the assumption H1 still holds Then

−AMI generates an exponentially stable C0-semigroup St  e −Mt T t t ≥ 0 Obviously,

S t t ≥ 0 is a positive C0-semigroup andSt ≤ Ce −M−δt for t≥ 0 FromLemma 2.2, the PBVP2.15 has a unique mild solution u ∈ PCJ, X given by

u t  StBh1, h2 

t

0

S t − sGh1, h2sds  

0<t k <t

S t − t k I k h1t k , h2t k , t ∈ J,

B h1, h2  I − Sω−1 ω

0

S ω − sGh1, h2sds 

p



k1

S ω − t k I k h1t k , h2t k

.

2.16

Let D  v0, w0 We define an operator Q : D × D → PCJ, X by

Q h1, h2t  StBh1, h2 

t

0

S t − sGh1, h2sds

0<t k <t

S t − t k I k h1t k , h2t k , t ∈ J. 2.17

Then the coupled mild ω-periodic L-quasisolution of the PBVP1.1 is equivalent to the

coupled fixed point of operator Q.

Next, we will prove that the operator Q has coupled fixed points on D For this purpose, we first show that Q : D × D → PCJ, X is a mixed monotone operator and

v0 ≤ Qv0, w0, Qw0, v0 ≤ w0 In fact, for any t ∈ J, v0t ≤ u1t ≤ u2t ≤ w0t, v0t ≤

v2t ≤ v1t ≤ w0t, from assumptions H1 and H2, we have

G u1, v1t ≤ Gu2, v2t,

I k u1t k , v1t k  ≤ I k u2t k , v2t k , k  1, 2, , p. 2.18 Since Stt ≥ 0 is a positive C0-semigroup, it follows thatI − Sω−1  ∞

n0S nω is

a positive operator Then Bu1, v1 ≤ Bu2, v2 Hence from 2.17 we see that Qu1, v1 ≤

Q u2, v2, which implies that Q is a mixed monotone operator Since

ϕ t  v

0t  A  MIv0t ≤ Gv0, w0t, t ∈ J, 2.19 fromLemma 2.2and1.2, we have

v0t  Stv00 

t

0

S t − sϕsds  

0<t k <t

S t − t k Δv0|t t k

≤ Stv00 

t

0

S t − sGv0, w0sds  

0<t k <t

S t − t k I k v0t k , w0t k

2.20

for t ∈ J Especially, we have

v0ω ≤ Sωv00 

ω

0

S ω − sGv0, w0sds 

p



k1

S ω − t k I k v0t k , w0t k . 2.21

Combining this inequality with v00 ≤ v0ω, it follows that

v00 ≤ I − Sω−1 ω

0

S ω − sGv0, w0sds 

p



k1

S ω − t k I k v0t k , w0t k

 Bv0, w0.

2.22

On the other hand, from2.17, we have

Q v0, w0t  StBv0, w0 

t

0

S t − sGv0, w0sds

0<t k <t

S t − t k I k v0t k , w0t k , t ∈ J. 2.23

Therefore, Qv0, w0t − v0t ≥ StBv0, w0 − v00 ≥ 0 for all t ∈ J It implies that

v0≤ Qv0, w0 Similarly, we can prove that Qw0, v0 ≤ w0

Now, we define sequences{v n } and {w n} by the iterative scheme

v n  Qv n−1, w n−1, w n  Qw n−1, v n−1, n  1, 2, 2.24

Then from the mixed monotonicity of operator Q, we have

v0≤ v1≤ v2≤ · · · ≤ v n ≤ · · · ≤ w n ≤ · · · ≤ w2≤ w1≤ w0. 2.25

Therefore, for any t ∈ J, {v n t} and {w n t} are monotone order-bounded sequences in X Noticing that X is a weakly sequentially complete Banach space, then {v n t} and {w n t} are relatively compact in X Combining this fact with the monotonicity of2.25 and the

normal-ity of cone K in X, it follows that {v n t} and {w n t} are uniformly convergent in X Let

v∗t  lim

n→ ∞v n t, w∗t  lim

Then v∗, w∗: J → X are strongly measurable, and v0t ≤ v∗t ≤ w∗t ≤ w0t for any t ∈ J Hence, v∗, w∗∈ L1J, X.

At last, we show that v∗ and w∗are coupled mild ω-periodic L-quasisolutions of the

PBVP1.1 For any φ ∈ X∗, from subcontinuity of f and continuity of I k ’s, there exists e ⊂ J with mes e 0 such that

φ Gv n , w n t −→ φGv∗, w∗t, n −→ ∞, t ∈ J \ e,

I k v n t k , w n t k  −→ I k v∗t k , w∗t k , n −→ ∞, k  1, 2, , p. 2.27

Hence, for any t ∈ J and s ∈ 0, t \ e, denote by S∗t − s the adjoint operator of St − s, then

S∗t − s ∈ X∗, and

φ St − sGv n , w n s  S∗t − sφGv n , w n s

−→ S∗t − sφGv∗, w∗s  φSt − sGv∗, w∗s, n −→ ∞,

φ

0<t k <t

S t − t k I k v n t k , w n t k

−→ φ



0<t <t

S t − t k I k v∗t k , w∗t k , n −→ ∞.

2.28

On the other hand, we have

φ St − sGv n , w n s ≤ φ · St − s · Gv n , w n s ≤ φ CM∗ M∗∗. 2.29 From Lebesgue’s dominated convergence theorem, we have

φ Bv n , w n   φ



I − Sω−1ω

0

S ω − sGv n , w n sds



p



k1

S ω − t k I k v n t k , w n t k

−→ φ



I − Sω−1

ω

0

S ω − sGv∗, w∗sds

p

k1

S ω − t k I k v∗t k , w∗t k

 φBv∗, w∗, n −→ ∞.

2.30

Hence, from2.17, we have

φ v n1t  φQv n , w n t  φStBv n , w n   φ

t

0

S t − sGv n , w n sds

 φ

0<t k <t

S t − t k I k v n t k , w n t k

−→ φStBv∗, w∗  φ

t

0

S t − sGv∗, w∗sds

 φ

0<t k <t

S t − t k I k v∗t k , w∗t k

 φ

S tBv∗, w∗ 

t

0

S t − sGv∗, w∗sds  

0<t k <t

S t − t k I k v∗t k , w∗t k

 φQv∗, w∗t, n −→ ∞.

2.31

On the other hand, it follows from 2.26 that limn→ ∞v n1t  v∗t, t ∈ J Hence

φ v n1t → φv∗t n → ∞ By the uniqueness of limits, we can deduce that

φ Qv∗, w∗t  φv∗t, t ∈ J, φ ∈ X∗. 2.32

2.28

On the other hand, we have

φ St...

2.31

On the other hand, it follows from 2.26 that limn→ ∞v n1t  v∗t, t ∈ J Hence

φ... CM∗ M∗∗. 2.29 From Lebesgue’s dominated convergence theorem, we have

φ Bv n , w n   φ