Hindawi Publishing Corporation
Journal of Inequalities and Applications
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,
Received 29 December 2009; Revised 20 July 2010; Accepted 3 September 2010
Academic Editor: Alberto Cabada
Copyright q 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 in 4–7 and
the references therein. In this paper, by using a mixed monotone iterative technique i n the
presence of coupled lower and upper L-quasisolutions, we consider the existence of mild ω-
t
k
,u
t
k
,k 1, 2, ,p,
u
0
u
ω
1.1
2 Journal of Inequalities and Applications
in an ordered Banach space X, where A : DA ⊂ X → X is a closed linear operator and
−A generates a C
0
-semigroup Ttt ≥ 0 in X; f : J × X × X → X only satisfies weak
Carath
´
eodory condition, J 0,ω, ω>0 is a constant; 0 t
0
<t
1
<t
limits of ut at t t
k
, respectively. Let PCJ, X : {u : J → X | ut is continuous at t
/
t
k
and left continuous at t t
k
,andut
k
exists, k 1, 2, ,p}. Evidently, PCJ, X is a Banach
space with the norm u
PC
sup
t∈J
ut.LetJ
J \{t
1
,t
2
, ,t
p
}, J
J \{0,t
1
,t
2
,X
1
satisfy
v
0
t
Av
0
t
≤ f
t, v
0
t
,w
0
t
L
v
0
,k 1, 2, ,p,
v
0
0
≤ v
0
ω
,
1.2
w
0
t
Aw
0
t
≥ f
t, w
0
w
0
t
k
,v
0
t
k
,k 1, 2, ,p,
w
0
0
≥ w
0
ω
,
1.3
we call v
0
,w
0
coupled lower and upper L-quasisolutions of the PBVP1.1. Only choosing
t
0
T
t−s
G
1
u, v
s
ds
0<t
k
<t
T
t − t
k
I
k
u
G
1
v, u
s
ds
0<t
k
<t
T
t−t
k
I
k
v
t
k
,u
t
k
Journal of Inequalities and Applications 3
Without impulse, the PBVP1.1 has been studied by many authors, see 8–11 and the
references therein. In particular, Shen and Li 11 considered the existence of coupled mild
ω-periodic quasisolution pair for the following periodic boundary value problem PBVP in
X:
u
t
Au
t
f
t, u
t
,u
t
,t∈ J,
u
0
u
t, u
2
,w
− f
t, u
1
,w
≥−M
1
u
2
− u
1
, ∀t ∈ J, v
0
t
≤ u
1
≤ u
2
≤ w
0
improve conditions F
1
and F
2
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, f t, ·, · is subcontinuous, namely, there exists e ⊂ J with mes e 0 such
that
f
t, u
n
,v
n
weak
−→ f
t, u, v
,
n −→ ∞
, 1.7
for any t ∈ J \ e,andu
n
− f
t, u
1
,v
1
≥−M
u
2
− u
1
L
v
2
− v
1
1.8
for any t ∈ J, and v
0
t ≤ u
1
≤ u
2
≤ w
u
2
,v
2
,k 1, 2, ,p 1.9
for any t ∈ J, and v
0
t ≤ u
1
≤ u
2
≤ w
0
t,v
0
t ≤ v
2
≤ v
1
≤ w
0
t.
then the PBVP1.1 has minimal and maximal coupled mild ω-periodic L-quasisolutions
between v
0
and w
0
, which can be obtained by monotone iterative sequences starting from v
0
Definition 2.1. A C
0
-semigroup Ttt ≥ 0 is said to be exponentially stable in X if there exist
constants C ≥ 1andδ>0 such that
T
t
≤ Ce
−δt
,t≥ 0.
2.2
Let I
0
t
0
,T. Denote by CI
0
,X the Banach space of all continuous X-value
functions on interval I
0
with the norm u
C
max
t∈I
0
ut. It is well-known 12, Chapter
4, Theorem 2.9 that for any x
2.3
Journal of Inequalities and Applications 5
has a unique classical solution u ∈ C
1
I
0
,X ∩ CI
0
,X
1
expressed by
u
t
T
t − t
0
x
0
t
t
0
T
t − s
,t∈ J, t
/
t
k
,
Δu|
tt
k
y
k
,k 1, 2, ,p,
u
0
u
ω
,
2.5
where y
k
∈ X, k 1, 2, ,p.
Lemma 2.2. Let Ttt ≥ 0 be an exponentially stable C
0
-semigroup in X. Then for any h ∈
PCJ, X and y
k
<t
T
t − t
k
y
k
,t∈ J,
2.6
where BhI − Tω
−1
ω
0
Tω − shsds
p
k1
Tω − t
k
y
k
.
Proof. For any h ∈ PCJ, X, we first show that the initial value problem IVP of linear
impulsive evolution equation
u
t
T
t
x
0
t
0
T
t − s
h
s
ds
0<t
k
<t
T
t − t
k
t
h
t
,t
k
<t≤ t
k1
,
u
t
k
u
t
k
y
k
.
2.9
Hence, on t
k
,t
s
ds. 2.10
Iterating successively in the above equality with ut
j
for j k,k − 1, ,1, 0, we see that u
satisfies 2.8.
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
by 2.6.
If a function u ∈ PCJ, X defined by 2.8 is a solution of the linear PBVP2.5, then
x
0
uω, namely,
I − T
ω
x
0
ω
0
T
x. 2.12
Then x≤|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
Ttt ≥ 0 is a positive semigroup. H ence we choose x
0
I − Tω
−1
ω
0
Tω − shsds
p
k1
Tω − t
k
y
k
Bh. Then x
0
is the unique initial value of the IVP2.7 in X, which
satisfies u0x
0
t ∈ v
0
t,w
0
t. Since v
0
t ≤ h
1
t ≤ w
0
t,v
0
t ≤ h
2
t ≤ w
0
t for any t ∈ J,
from the assumption H
1
, we have
f
t, h
1
t
,h
2
w
0
t
− v
0
t
Mw
0
t
≤ w
0
t
A MI
w
0
t
t
≥ f
t, v
0
t
,w
0
t
L
v
0
t
− w
0
t
Mv
0
t
1
t − Lh
2
t ≤ h
0
t,t∈ J. From the normality of
cone K in X, we have
f
t, h
1
t
,h
2
t
M L
h
1
t
t,h
2
t is strongly measurable, it follows that
ft, h
1
t,h
2
t ∈ L
1
J, X. Therefore, for any h
1
t,h
2
t ∈ v
0
t,w
0
t,t ∈ J, we consider
the periodic boundary value problemPBVP of impulsive evolution equation in X
u
t
A MI
u
,k 1, 2, ,p,
u
0
u
ω
,
2.15
where Gh
1
,h
2
tft, h
1
t,h
2
t M Lh
1
t − Lh
2
t.LetM>0 be large enough
such that M>δotherwise, replacing M by M δ, the assumption H
1
still holds. Then
−AMI generates an exponentially stable C
0
-semigroup Ste
G
h
1
,h
2
s
ds
0<t
k
<t
S
t − t
k
I
k
h
1
t
k
,h
1
,h
2
s
ds
p
k1
S
ω − t
k
I
k
h
1
t
k
,h
2
t
k
t
0
S
t − s
G
h
1
,h
2
s
ds
0<t
k
<t
S
t − t
k
I
k
. In fact, for any t ∈ J, v
0
t ≤ u
1
t ≤ u
2
t ≤ w
0
t,v
0
t ≤
v
2
t ≤ v
1
t ≤ w
0
t, from assumptions H
1
and H
2
, we have
G
u
1
,v
1
t
2
t
k
,v
2
t
k
,k 1, 2, ,p.
2.18
Since Stt ≥ 0 is a positive C
0
-semigroup, it follows that I − Sω
−1
∞
n0
Snω is
a positive operator. Then Bu
1
,v
1
≤ Bu
2
,v
2
v
0
,w
0
t
,t∈ J, 2.19
from Lemma 2.2 and 1.2, we have
v
0
t
S
t
v
0
0
t
0
S
t
0
S
t − s
G
v
0
,w
0
s
ds
0<t
k
<t
S
t − t
k
I
k
ω
0
S
ω − s
G
v
0
,w
0
s
ds
p
k1
S
ω − t
k
I
k
v
0
G
v
0
,w
0
s
ds
p
k1
S
ω − t
k
I
k
v
0
t
k
,w
0
v
0
,w
0
t
0
S
t − s
G
v
0
,w
0
s
ds
0<t
k
<t
S
0
− v
0
0 ≥ 0 for all t ∈ J. It implies that
v
0
≤ Qv
0
,w
0
. Similarly, we can prove that Qw
0
,v
0
≤ w
0
.
Now, we define sequences {v
n
} and {w
n
} by the iterative scheme
v
n
Q
v
n−1
,w
n−1
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 of 2.25 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
∗
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
G
v
n
,w
n
t
−→ φ
G
v
∗
,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
S
t − s
G
v
n
,w
n
s
S
φ
S
t − s
G
v
∗
,w
∗
s
,n−→ ∞ ,
φ
0<t
k
<t
S
t − t
k
I
k
t
k
,w
∗
t
k
,n−→ ∞ .
2.28
10 Journal of Inequalities and Applications
On the other hand, we have
φ
S
t − s
G
v
n
,w
n
CM
∗
M
∗∗
. 2.29
From Lebesgue’s dominated convergence theorem, we have
φ
B
v
n
,w
n
φ
I − Sω
−1
ω
0
S
ω − s
G
t
k
−→ φ
I − Sω
−1
ω
0
S
ω − s
G
v
∗
,w
∗
s
ds
p
∗
,n−→ ∞ .
2.30
Hence, from 2.17, we have
φ
v
n1
t
φ
Q
v
n
,w
n
t
φ
S
t
B
k
<t
S
t − t
k
I
k
v
n
t
k
,w
n
t
k
−→ φ
S
t
B
k
<t
S
t − t
k
I
k
v
∗
t
k
,w
∗
t
k
φ
S
t
B
t − t
k
I
k
v
∗
t
k
,w
∗
t
k
φ
Q
v
∗
,w
∗
t
t
,t∈ J, φ ∈ X
∗
. 2.32
Journal of Inequalities and Applications 11
By the arbitrariness of φ ∈ X
∗
, we have
v
∗
Q
v
∗
,w
∗
. 2.33
Similarly, we can prove that w
∗
Qw
∗
,v
∗
. Therefore, v
∗
,w
∗
is coupled mild ω-periodic
t, where M
0
I − Sω
−1
,
H
4
there exist positive constants τ
k
k 1, 2, ,p with
p
k1
τ
k
< 1 − ωNCM 2L −
RCM
0
1/CNCM
0
1 such that
I
k
u, v
− I
k
v, u
1
∗
there exist constants M>0 and 0 ≤ L<min{M, 1/ωNCCM
0
1} such that
f
t, u
2
,v
2
− f
t, u
1
,v
1
≥−M
u
2
− u
1
L
v
,w
0
.
12 Journal of Inequalities and Applications
Proof. From the proof of Theorem 1.2, when the conditions H
1
∗
and H
2
are satisfied, the
iterative sequences {v
n
} and {w
n
} defined by 2.24 satisfy 2.25. We show t hat there exists
auniqueu
∗
∈ PCJ, X such that u
∗
Qu
∗
,u
∗
. For any t ∈ J,fromH
3
, H
4
, 2.17, 2.24
and 2.25, we have
t
B
w
n−1
,v
n−1
− B
v
n−1
,w
n−1
t
0
S
t − s
G
w
n−1
,v
n−1
k
,v
n−1
t
k
− I
k
v
n−1
t
k
,w
n−1
t
k
≤ S
t
B
w
s
ds
0<t
k
<t
S
t − t
k
τ
k
w
n−1
t
k
− v
n−1
t
k
.
2.37
− B
v
n−1
,w
n−1
M 2L − R
t
0
S
t − s
w
n−1
s
− v
n−1
s
ds
B
w
n−1
,v
n−1
− B
v
n−1
,w
n−1
NC
M 2L − R
ω
w
n−1
− v
n−1
PC
NC
p
C 1
w
n−1
− v
n−1
PC
.
2.38
Therefore
w
n
− v
n
PC
≤
NC
M
0
C 1
ω
NC
M
0
C 1
ω
M 2L − R
p
k1
τ
k
n
w
0
− v
0
PC
−→ 0 2.40
as n →∞. Then there exists a unique u
∗
boundary value problem of parabolic type:
∂
∂t
u −∇
2
u f
1
x, t, u
f
2
x, t, u
, ∀x ∈ Ω, a.e.t∈ J,
Δu|
tt
k
I
k,1
u
x, t
k
I
k,2
the L
2
-norm ·
2
, K : {u ∈ Xux ≥ 0, a.e.x∈ Ω}. Then K is a generating normal cone in
X. Consider the operator A : DA ⊂ X → X defined by
D
A
u ∈ X |∇
2
u ∈ X, u
|
∂Ω
0
,Au −∇
2
u. 3.2
Then −A generates an analytic semigroup Ttt ≥ 0 in X. By the maximum principle of the
equations of parabolic type, it is easy to prove that Ttt ≥ 0 is a positive C
0
-semigroup in
X.Letλ
1
be the first eigenvalue of operator A and e
1
x, x ∈ Ω, t ∈ J
, I
k,1
e
1
x I
k,2
00,
x ∈ Ω.
iia The partial derivative of f
1
x, t, u on u is continuous on any bounded domain.
b The partial derivative of f
2
x, t, u on u has upper bound, and
sup∂/∂uf
2
x, t, u ≤ L.
iii For any u
1
,u
2
∈ 0,e
1
with u
1
≤ u
2
, we have
u
1
x, t
k
,x∈Ω,k1, 2, ,p. 3.3
14 Journal of Inequalities and Applications
Let f : J × X × X → X and I
k
: X × X → X be defined by ft, u, uf
1
·,t,u·
f
2
·,t,u· and by I
k
u, uI
k,1
u· I
k,2
u·. Then the problem 3.1 can be transformed
into the PBVP1.1. Assumption i implies that v
0
≡ 0andw
0
≡ e
1
are coupled lower and
upper L-quasisolutions of the PBVP1.1. From assumption iia, there exists a constant
x, t, u
1
∂
∂u
f
1
x, t, ξ
u
2
− u
1
≤ M
u
2
2
− f
1
x, t, u
1
≥−M
u
2
− u
1
. 3.6
Therefore, for any u
i
,v
i
∈ X with 0 ≤ u
1
≤ u
2
≤ e
1
, 0 ≤ v
2
≤ v
1
2
·
− f
1
·,t,u
1
·
− f
2
·,t,v
1
·
f
1
·,t,u
2
·
2
·
− u
1
·
sup
∂
∂u
f
2
·,t,ξ
v
2
·
− v
1
·
≥−M
u
coupled lower and upper quasisolutions of the PBVP1.1. Since condition H
1
contains
conditions F
1
and F
2
, even without impulse in PBVP1.1, the results in this paper still
extend the results in 10, 11.
Journal of Inequalities and Applications 15
Acknowledgments
The author is very grateful to the reviewers for their helpful comments and sugges-
tions. Research supported by NNSF of China 10871160, the NSF of Gansu Province
0710RJZA103, and Project of NWNUKJCXGC-3-47.
References
1 S. W. Du and V. Lakshmikantham, “Monotone iterative technique for differential equations in a
Banach space,” Journal of Mathematical Analysis and Applications, vol. 87, no. 2, pp. 454–459, 1982.
2 V. Lakshmikantham and A. S. Vatsala, “Quasisolutions and monotone method for systems of
nonlinear boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 79, no.
1, pp. 38–47, 1981.
3 G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear
Differential Equations, vol. 27 of Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics,
Pitman, Boston, Mass, USA, 1985.
4 H Y. Lan, “Monotone method for a system of nonlinear mixed type implicit impulsive integro-
differential equations in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 222,
no. 2, pp. 531–543, 2008.
5 B. Ahmad and S. Sivasundaram, “The monotone iterative technique for impulsive hybrid set valued
integro-differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 12, pp.
2260–2276, 2006.
6 Y. Li and Z. Liu, “Monotone iterative technique for addressing impulsive integro-differential
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