Approximate controllability of second-order evolution differential inclusions in Hilbert spaces

0.21 MB·English

by  N. I. Mahmudov

Approximate controllability of second-order
evolution differential inclusions in Hilbert spaces
5
1
N.I. Mahmudov
0
2 Department of Mathematics, Eastern Mediterranean University, Gazimagusa,

n T.R. North Cyprus, Mersin 10, Turkey. email: [email protected]
a
V. Vijayakumar
J
1 Department of Mathematics, Info Institute of Engineering, Kovilpalayam,
3 Coimbatore - 641 107, Tamil Nadu, India. email: [email protected]

R. Murugesu
]
S Department of Mathematics, SRMV College of Arts and Science,
D
Coimbatore - 641 020, Tamil Nadu, India. email: [email protected]
.
h
t
a
m
[
Abstract

1
In this paper, we consider a class of second-order evolution differential inclusions in
v
0 Hilbert spaces. This paper deals with the approximate controllability for a class of second-
8 order control systems. First, we establish a set of sufficient conditions for the approximate
0 controllability for a class of second-order evolution differential inclusions in Hilbert spaces.
0
We use Bohnenblust-Karlin’s fixed point theorem to prove our main results. Further, we
0
extend the result to study the approximate controllability concept with nonlocal conditions
.
2 and extend the resultto study the approximatecontrollabilityfor impulsive controlsystems
0
with nonlocal conditions. An example is also given to illustrate our main results.
5
1
Keywords: Approximatecontrollabiliy, Secondorderdifferentialinclusions,Cosinefunction
:
v
of operators, Impulsive systems, Evolution equations, Nonlocal conditions.
i
X 2010 Mathematics Subject Classification: 26A33, 34B10, 34K09, 47H10.
r
a
1 Introduction
Controllability is one of the elementary concepts in mathematical control theory. This is a
qualitative property of dynamical control systems and it is of particular importance in control
theory. Roughly speaking, controllability generally means, that it is possible to steer dynamical
controlsystemfromanarbitraryinitialstatetoanarbitraryfinalstateusingthesetofadmissible
controls. Most of the criteria, which can be met in the literature, are formulated for finite
dimensional systems. It should be pointed out that many unsolved problems still exist as far as
controllability of infinite dimensional systems are concerned.
In the case of infinite dimensional systems two basic concepts of controllability can be dis-
tinguished which are exact and approximate controllability. This is strongly related to the fact
1Corresponding author: N.I. Mahmudov
1
that in infinite dimensional spaces there exist linear subspaces, which are not closed. Exact con-
trollability enables to steer the system to arbitrary final state while approximate controllability
means that system can be steered to arbitrary small neighborhood of final state. In other words
approximate controllability gives the possibility of steering the system to states which form the
dense subspace in the state space. In recent years, controllability problems for various types of
nonlineardynamicalsystemsininfinitedimensionalspacesbyusingdifferentkindsofapproaches
have been considered in many publications, see [1, 4, 5, 13, 19, 24, 32–37, 41–43, 51, 52, 54, 55]
and the references therein.
Recently, in [37], Mahmudov et al. studied the approximate controllability of second-order
neutralstochasticevolutionequationsusingsemi-groupmethods,togetherwiththeBanachfixed
point theorem. In [43] Sakthivel et al. studied the approximate controllability of second-order
systems with state-dependent delay by using Schauder fixed point theorem. In [21], Henr´ıquez
studied the existence of solutions of non-autonomous second order functional differential equa-
tions with infinite delay by using Leray-Schauder’s Alternative fixed point theorem. In [54],
Yan studied the approximate controllability of fractional neutral integro-differential inclusions
with state-dependent delay in Hilbert spaces by using Dhage’s fixed point theorem. In [52],
Vijayakumar et al. discussed the approximate controllability for a class of fractional neutral
integro-differential inclusions with state-dependent delay by using Dhage’s fixed point theorem.
In[41]Sakthivel etal. studiedtheapproximate controllability of fractional nonlinear differential
inclusions with initial and nonlocal conditions by using Bohnenblust-Karlin’s fixed point the-
orem. Very recently, in [4] Arthi et al. established the sufficient conditiond for controllability
of second-order impulsive evolution systems with infinite delay by using Leray-Schauder’s fixed
point theorem and in [5], proved existence and controllability results for second-order impul-
sive stochastic evolution systems with state-dependent delay by using Leray-Schauder’s fixed
point theorem. In [19], Guendouzi investigated the approximate controllability for a class of
fractional neutral stochastic functional integro-differential inclusions Bohnenblust-Karlin’s fixed
point theorem.
Inspired by the above works, in this paper, we establish sufficient conditions for the approxi-
mate controllability for a class of second-order evolution differential inclusions in Hilbert spaces
of the form
x′′(t) A(t)x(t)+F(t,x(t))+Bu(t), t I = [0,b], (1)
∈ ∈
x(0) =x X, x′(0) = y X. (2)
0 0
∈ ∈
In this equation, A(t) : D(A(t)) X X is a closed linear operator on a Hilbert space X and
⊆ →
the control function u() L2(I,U), a Hilbert space of admissible control functions. Further, B
· ∈
is a bounded linear operator from U to X, and F : I X 2X is a nonempty, bounded,
× → \{∅}
closed and convex multivalued map.
Converting a second-order system into a first-order system and studying its controllability
maynotyielddesiredresultsduetothebehaviorofthesemigroupgeneratedbythelinearpartof
the converted first order system. So, in many cases, it is advantageous to treat the second-order
abstract differential system directly rather than to convert that into first-order system. To the
best of our knowledge, the study of the approximate controllability for a class of second-order
evolution differential systems in Hilbert spaces treated in this paper, is an untreated topic in
the literature, to fill the gap, we study this interesting paper.
This paper is organized as follows. In Section 3, we establish a set of sufficient conditions
for the approximate controllability for a class of second-order evolution differential inclusions
in Hilbert spaces. In Section 4, we establish a set of sufficient conditions for the approximate
controllabilityforaclassofsecond-orderevolutiondifferentialinclusionswithnonlocalconditions
2
in Hilbert spaces. In Section 5, we establish a set of sufficient conditions for the approximate
controllability foraclass ofsecond-orderimpulsiveevolution differentialinclusionswithnonlocal
conditions in Hilbert spaces. An example is presented in Section 6 to illustrate the theory of
the obtained results.
2 Preliminaries
In this section, we mention a few results, notations and lemmas needed to establish our
main results. We introduce certain notations which will be used throughout the article without
any further mention. Let (X, ) and (Y, ) be Hilbert spaces, and (Y,X) be the
X Y
k · k k · k L
Banach space of bounded linear operators from Y into X equipped with its natural topology;
in particular, we use the notation (X) when Y = X. By ρ(A), we denote the resolvent set of
L
a linear operator A. Throughout this paper, B (x,X) will denote the closed ball with center at
r
x and radius r > 0 in a Hilbert space X. We denote by , the Banach space C(J,X) endowed
C
with supnorm given by x sup x(t) , for x .
C t∈I
k k ≡ k k ∈ C
Inrecenttimestherehasbeenanincreasinginterestinstudyingtheabstractnon-autonomous
second order initial value problem
x′′(t) =A(t)x(t)+f(t), 0 s,t b, (3)
≤ ≤
x(s)=x , x′(s)= y , (4)
0 0
whereA(t) :D(A(t)) X X,t I = [0,b]isaclosed denselydefinedoperatorandf :I X
⊆ → ∈ →
is an appropriate function. Equations of this type have been considered in many papers. The
reader is referred to [7, 16, 30, 38, 39] and the references mentioned in these works. In the most
of works, the existence of solutions to the problem (3) (4) is related to the existence of an
−
evolution operator S(t,s) for the homogeneous equation
x′′(t) =A(t)x(t), 0 s,t b, (5)
≤ ≤
Let as assume that the domain of A(t) is a subspace D dense in X and independent of t, and
for each x D the function t A(t)x is continuous.
∈ 7→
Following Kozak [26], in this work we will use the following concept of evolution operator.
Definition 1. A family S of bounded linear operators S(t,s) : I I (X) is called an
× → L
evolution operator for (5) if the following conditions are satisfied:
(Z1) For each x X, the mapping [0,b] [0,b] (t,s) S(t,s)x X is of class C1 and
∈ × ∋ → ∈
(i) for each t [0,b], S(t,t) =0,
∈
(ii) for all t,s [0,b], and for each x X,
∈ ∈
∂ ∂
S(t,s)x = x, S(t,s)x = x.
∂t t=s ∂t t=s −
(cid:12) (cid:12)
(cid:12) (cid:12)
(Z2) For all t,s [0,b], if x D(A), the(cid:12)n S(t,s)x D(A), the m(cid:12)apping [0,b] [0,b] (t,s)
∈ ∈ ∈ × ∋ →
S(t,s)x X is of class C2 and
∈
(i) ∂2 S(t,x)x = A(t)S(t,s)x,
∂t2
(ii) ∂2 S(t,x)x = S(t,s)A(s)x,
∂s2
3
(iii) ∂ ∂ S(t,x)x = 0.
∂s∂t
t=s
(cid:12)
(Z3) For all t,s ∈ [0,b](cid:12)(cid:12), if x ∈ D(A), then ∂∂sS(t,s)x ∈D(A), then ∂∂t22∂∂sS(t,s)x, ∂∂s22∂∂tS(t,s)x
and
(i) ∂2 ∂ S(t,s)x = A(t) ∂ S(t,s)x,
∂t2∂s ∂s
(ii) ∂2 ∂ S(t,s)x = ∂ S(t,s)A(s)x,
∂s2∂t ∂t
and the mapping [0,b] [0,b] (t,s) A(t) ∂ S(t,s)x is continuous.
× ∋ → ∂s
Throughout this work we assume that there exists an evolution operator S(t,s) associated
∂S(t,s)
to the operator A(t). To abbreviate the text, we introduce the operator C(t,s) = .
− ∂s
In addition, we set N and N for positive constants such that sup S(t,s) N and
0≤t,s≤bk k ≤
sup C(t,s) N. Furthermore, we denote by N a positive constant such that
0≤t,s≤bk k ≤ 1
e
S(t+h,s) S(t,s) N h,
e 1
k − k ≤ | |
for all s,t,t+h [0,b]. Assuming that f : I X is an integrable function, the mild solution
∈ →
x : [0,b] X of the problem (3) (4) is given by
→ −
t
x(t) = C(t,s)x +S(t,s)y + S(t,τ)h(τ)dτ.
0 0
Z0
In the literature several techniques have been discussed to establish the existence of the
evolution operator S(, ). In particular, a very studied situation is that A(t) is the perturbation
· ·
of an operator A that generates a cosine operator function. For this reason, below we briefly
review some essential properties of the theory of cosine functions. Let A : D(A) X X be
⊆ →
the infinitesimal generator of a strongly continuous cosine family of bounded linear operators
(C0(t))t∈R on Hilbert space X. We denote by (S0(t))t∈R the sine function associated with
(C0(t))t∈R which is defined by
t
S (t)x = C(s)xds, x X, t R.
0
∈ ∈
Z0
We refer the reader to [17, 47, 48] for the necessary concepts about cosine functions. Next we
only mention a few results and notations about this matter needed to establish our results. It
is immediate that
t
C (t)x x =A S(s)xds,
0
−
Z0
for all X. The notation [D(A)] stands for the domain of the operator A endowed with the graph
norm x = x + Ax , x D(A). Moreover, in this paper the notation E stands for the
A
k k k k k k ∈
space formed by the vectors x X for which the function C()x is a class C1 on R. It was
∈ ·
proved by Kisyn´ski [25] that the space E endowed with the norm
x = x + sup AS(t,0)x , x E,
E
k k k k k k ∈
0≤t≤1
is a Hilbert space. The operator valued function
C (t) S (t)
G(t) = 0 0
AS (t) C (t)
0 0
(cid:20) (cid:21)
4
is a strongly continuous group of linear operators on the space E X generated by the operator
×
0 I
= defined on D(A) E. It follows from this that AS (t) : E X is a bounded
A A 0 × 0 →
(cid:20) (cid:21)
linear operator such that AS (t)x 0 as t 0, for each x E. Furthermore, if x :[0, ) X
0
→ → t ∈ ∞ →
is a locally integrable function, then z(t) = S(t,s)x(s)ds defines an E-valued continuous
0
function.
R
The existence of solutions for the second order abstract Cauchy problem
x′′(t)=Ax(t)+h(t), 0 t b, (6)
≤ ≤
x(0) =x , x′(0) = y , (7)
0 0
whereh :[0,b] X is anintegrable function, has beendiscussedin[47]. Similarly, theexistence
→
of solutions of the semilinear second order Cauchy problem it has been treated in [48]. We only
mention here that the function x() given by
·
t
x(t) =C (t s)x +S (t s)y + S (t τ)h(τ)dτ, 0 t b, (8)
0 0 0 0 0
− − − ≤ ≤
Zs
is called the mild solution of (6) (7) and that when x E, x() is continuously differentiable
0
− ∈ ·
and
t
x′(t)= AS (t s)x +C (t s)y + C (t τ)h(τ)dτ, 0 t b.
0 0 0 0 0
− − − ≤ ≤
Zs
In addition, if x D(A), y E and f is a continuously differentiable function, then the
0 0
∈ ∈
function x() is a solution of the initial value problem (6) (7).
· −
Assumenow thatA(t) = A+B(t)whereB() :R (E,X) is amapsuch thatthefunction
· → L
t B(t)xiscontinuouslydifferentiableinX foreachx E. IthasbeenestablishedbySerizawa
→ ∈
[44] that for each (x ,y ) D(A)e E the noenautonomous abstract Cauchy problem
0 0
∈ ×
e
x′′(t) =(A+B(t))x(t), t R, (9)
∈
x(0) =x , x′(0) = y , (10)
0 0
e
has a unique solution x() such that the function t x(t) is continuously differentiable in E. It
· 7→
isclear thatthesameargumentallows ustoconcludethatequation (9)withtheinitial condition
(7) has a unique solution x(,s) such that the function t x(t,s) is continuously differentiable
· 7→
in E. It follows from (8) that
t
x(t,s) = C (t s)x +S (t s)y + S(t τ)B(τ)x(τ,s)dτ.
0 0 0 0
− − −
Zs
In particular, for x = 0 we have e
0
t
x(t,s) = S (t s)y + S(t τ)B(τ)x(τ,s)dτ.
0 0
− −
Zs
Consequently, e
t
x(t,s) S (t s) y + S (t s) B(τ) x(s,τ) dτ.
1 0 L(X,E) 0 0 L(X,E) L(X,E) 1
k k ≤ k − k k k k − k k k k k
Zs
e
5
and, applying the Gronwall-Bellman lemma we infer that
x(t,s) M y , s,t I.
1 0
k k ≤ k k ∈
We define the operator S(t,s)y = x(t,s). It follows from the previous estimate that S(t,s) is
0 f
a bounded linear map on E. Since E is dense in X, we can extend S(t,s) to X. We keep the
notation S(t,s) for this extension. It is well known that, except in the case dim(X) < , the
∞
cosine function C (t) cannot be compact for all t R. By contrast, for the cosine functions
0
∈
that arise in specific applications, the sine function S (t) is very often a compact operator for
0
all t R.
∈
Theorem 2. [21, Theorem 1.2]. Under the preceding conditions, S(, ) is an evolution operator
· ·
for (9) (10). Moreover, if S (t) is compact for all t R, then S(t,s) is also compact for all
0
− ∈
s t.
≤
We also introduce some basic definitions and results of multivalued maps. For more details
on multivalued maps, see the books of Deimling [14] and Hu and Papageorgious [45].
A multivalued map G : X 2X is convex (closed) valued if G(x) is convex (closed)
→ \{∅}
for all x X. G is bounded on bounded sets if G(C) = G(x) is bounded in X for any
∈ x∈C
bounded set C of X, i.e., sup sup y : y G(x) < .
x∈C S
{k k ∈ } ∞
n o
Definition 3. G is called upper semicontinuous (u.s.c. for short) on X if for each x X, the
0
∈
set G(x ) is a nonempty closed subset of X, and if for each open set C of X containing G(x ),
0 0
there exists an open neighborhood V of x such that G(V) C.
0
⊆
Definition 4. G is called completely continuous if G(C) is relatively compact for every bounded
subset C of X.
If the multivalued map G is completely continuous with nonempty values, then G is u.s.c.,
if and only if G has a closed graph, i.e., x x , y y , y Gx imply y Gx . G has a
n ∗ n ∗ n n ∗ ∗
→ → ∈ ∈
fixed point if there is a x X such that x G(x).
∈ ∈
Definition 5. A function x is said to be a mild solution of system (1)-(2) if x(0) = x ,
0
∈ C
x′(0) = y and there exists f L1(I,X) such that f(t) F(t,x(t)) on t I and the integral
0
∈ ∈ ∈
equation
t t
x(t) = C(t,0)x +S(t,0)y + S(t,s)f(s)ds+ S(t,s)Bu(s)ds, t I.
0 0
∈
Z0 Z0
is satisfied.
In order to address the problem, it is convenient at this point to introduce two relevant
operators and basic assumptions on these operators:
b
Υb = S(b,s)BB∗S∗(b,s)ds :X X,
0 →
Z0
R(a,Υb) = (aI +Υb)−1 : X X.
0 0 →
It is straightforward that the operator Υb is a linear bounded operator.
0
To investigate the approximate controllability of system (1)-(2), we impose the following
condition:
6
H0 aR(a,Υb) 0 as a 0+ in the strong operator topology.
0 → →
In view of [32], Hypothesis H0 holds if and only if the linear system
x′′(t) =Ax(t)+(Bu)(t), t [0,b], (11)
∈
x(0) =x x′(0) = y , (12)
0 0
is approximately controllable on [0,b].
Some of our results are proved using the next well-known results.
Lemma 6. [28, Lasota and Opial] Let I be a compact real interval, BCC(X) be the set of
all nonempty, bounded, closed and convex subset of X and F be a multivalued map satisfying
F : I X BCC(X) is measurable to t for each fixed x X, u.s.c. to x for each t I, and
× → ∈ ∈
for each x the set
∈ C
S = f L1(I,X) :f(t) F(t,x(t)), t I
F,x
{ ∈ ∈ ∈ }
is nonempty. Let F be a linear continuous from L1(I,X) to , then the operator
C
F S : BCC( ), x (F S )(x) = F(S ),
F F F,x
◦ C → C → ◦
is a closed graph operator in .
C ×C
Lemma 7. [9, Bohnenblust and Karlin]. Let be a nonempty subset of X, which is bounded,
D
closed, and convex. Suppose G : 2X is u.s.c. with closed, convex values, and such
D → \{∅}
that G( ) and G( ) is compact. Then G has a fixed point.
D ⊆ D D
3 Approximate controllability results
In this section, first we establish a set of sufficient conditions for the approximate controlla-
bility for a class of second order evolution differential inclusions of the form (1)-(2) in Hilbert
spaces by using Bohnenblust-Karlin’s fixed point theorem. In order to establish the result, we
need the following hypotheses:
H1 S0(t), t > 0 is compact.
H2 For each positive number r and x with x C r, there exists Lf,r() L1(I,R+) such
∈ C k k ≤ · ∈
that
sup f :f(t) F(t,x(t)) L (t),
f,r
{k k ∈ } ≤
for a.e. t I.
∈
H3 The function s Lf,r(s) L1([0,t],R+) and there exists a γ > 0 such that
→ ∈
t
L (s)ds
lim 0 f,r =γ < + .
r→∞ r ∞
R
It will be shown that the system (1)-(2) is approximately controllable, if for all a > 0, there
exists a continuous function x() such that
·
t t
x(t)=C(t,0)x +S(t,0)y + S(t,s)f(s)ds+ S(t,s)Bu(s,x)ds, f S , (13)
0 0 F,x
∈
Z0 Z0
u(t,x) = B∗S(b,t)R(a,Υb)p(x()), (14)
0 ·
7
where
t
p(x()) = x C(b,0)x S(b,0)y S(b,s)f(s)ds.
b 0 0
· − − −
Z0
Theorem 8. Suppose that the hypotheses H0-H3 are satisfied. Assume also
1
Nγ 1+ N2M2b < 1, (15)
α B
h i
where M = B , Then system (1)-e(2) has a seolution on I.
B
k k
Proof. The main aim in this section is to find conditions for solvability of system (1)-(2) for
a > 0. We show that, using the control u(x,t), the operator Γ : 2C, defined by
C →
t
Γ(x)= ϕ :ϕ(t) = C(t,0)x +S(t,0)y + S(t,s)[f(s)+Bu(s,x)]ds, f S ,
0 0 F,x
∈ C ∈
n Z0 o
has a fixed point x, which is a mild solution of system (1)-(2).
We now show that Γ satisfies all the conditions of Lemma 7. For the sake of convenience,
we subdivide the proof into five steps.
Step 1. Γ is convex for each x .
∈C
In fact, if ϕ , ϕ belong to Γ(x), then there exist f , f S such that for each t I, we
1 2 1 2 F,x
∈ ∈
have
t t
ϕ (t)=C(t,0)x +S(t,0)y + S(t,s)f (s)ds+ S(t,s)BB∗S∗(b,t)R(a,Υb) x C(b,0)x
i 0 0 i 0 b− 0
Z0 Z0 h
b
S(b,0)y S(b,η)f (η)dη (s)ds, i= 1,2.
0 i
− −
Z0 i
Let λ [0,1]. Then for each t I, we get
∈ ∈
t
λϕ (t)+(1 λ)ϕ (t) =C(t,0)x +S(t,0)y + S(t,s)[λf (s)+(1 λ)f (s)]ds
1 2 0 0 1 2
− −
Z0
t
+ S(t,s)BB∗S∗(b,t)R(a,Υb) x C(b,0)x S(b,0)y
0 b− 0 − 0
Z0 "
b
S(b,s)[λf (s)+(1 λ)f (s)]ds (s)ds.
1 2
− −
Z0 #
It is easy to see that S is convex since F has convex values. So, λf +(1 λ)f S . Thus,
F,x 1 2 F,x
− ∈
λϕ +(1 λ)ϕ Γ(x).
1 2
− ∈
Step 2. For each positive number r > 0, let B = x : x r . Obviously, B is a
r C r
{ ∈ C k k ≤ }
bounded, closed and convex set of . We claim that there exists a positive number r such that
C
Γ(B ) B .
r r
⊆
If this is not true, then for each positive number r, there exists a function xr B , but
r
∈
Γ(xr) does not belong to B , i.e.,
r
Γ(xr) sup ϕr :ϕr (Γxr) > r
C C
k k ≡ k k ∈
n o
8
and
t t
ϕr(t) =C(t,0)x +S(t,0)y + S(t,s)fr(s)ds+ S(t,s)Bur(s,x)ds,
0 0
Z0 Z0
for some fr SF,xr. Using H1-H3, we have
∈
r < Γ(xr)(t)
k k
t t
C(t,0)x + S(t,0)y + S(t,s)fr(s) ds+ S(t,s)Bur(s,x) ds
0 0
≤k k k k k k k k
Z0 Z0
t
N x +N y +N L (s)ds
0 0 f,r
≤ k k k k
h Z0 i
1 e e b
+ N2M2b x +N x +N y +N L (s)ds .
α B k bk k 0k k 0k f,r
" Z0 #
e e e
Dividing both sides of the above inequality by r and taking the limit as r , using H3, we
→ ∞
get
1
Nγ 1+ N2M2b 1.
α B ≥
h i
This contradicts with the condition e(15). Hencee, for some r > 0, Γ(Br) Br.
⊆
Step 3. Γ sends bounded sets into equicontinuous sets of . For each x B , ϕ Γ(x), there
r
C ∈ ∈
exists a f S such that
F,x
∈
t t
ϕ(t) =C(t,0)x +S(t,0)y + S(t,s)f(s)ds+ S(t,s)Bu(s,x)ds.
0 0
Z0 Z0
Let 0< ε < 0 and 0< t < t b, then
1 2
≤
ϕ(t ) ϕ(t ) = C(t ,0) C(t ,0) x + S(t ,0) S(t ,0) η
1 2 1 2 0 1 2
| − | | − || | | − || |
t1−ε t1
+ [S(t ,s) S(t ,s)]f(s)ds + [S(t ,s) S(t ,s)]f(s)ds
1 2 1 2
− −
(cid:12)Z0 (cid:12) (cid:12)Zt−ε (cid:12)
(cid:12) t2 t1−ε (cid:12) (cid:12) (cid:12)
+(cid:12) S(t ,s)f(s)ds + [S((cid:12)t ,(cid:12)η) S(t ,η)]Bu(η,x)dη (cid:12)
2 1 2
−
(cid:12)Zt1 (cid:12) (cid:12)Z0 (cid:12)
(cid:12) t1 (cid:12) (cid:12) t2 (cid:12)
+(cid:12) [S(t ,η) S(t(cid:12),η)(cid:12)]Bu(η,x)dη + S(t ,η)Bu(η,x)(cid:12)dη
1 2 2
−
(cid:12)Zt−ε (cid:12) (cid:12)Zt1 (cid:12)
(cid:12) t1−ε (cid:12) (cid:12) (cid:12)
+(cid:12) [S(t ,η) S(t ,η)]Bu(η,x)d(cid:12)η (cid:12) (cid:12)
1 2
−
(cid:12)Z0 (cid:12)
C((cid:12)t ,0) C(t ,0) x + S(t ,0) S(t(cid:12),0) η
≤| (cid:12)1 − 2 || 0| | 1 − 2(cid:12) || |
t1−ε
+ S(t ,s) S(t ,s)L (s)ds
1 2 f,r
| − |
Z0
t1 t2
+ S(t ,s) S(t ,s)L (s)ds+N L (s)ds
1 2 f,r f,r
| − |
Zt−ε Zt1
t1−ε
e
+M S(t ,η) S(t ,η) u(η,x) dη
B 1 2
| − |k k
Z0
t1 t2
+M S(t ,η) S(t ,η) u(η,x) dη +NM u(η,x) dη.
B 1 2 B
| − k k k k
Zt−ε Zt1
e
9
The right-hand side of the above inequality tends to zero independently of x B as (t
r 1
∈ −
t ) 0 and ε sufficiently small, since the compactness of the evolution operator S(t,s) implies
2
→
thecontinuity intheuniformoperatortopology. ThusΓ(xr)sendsB intoequicontinuousfamily
r
of functions.
Step 4. The set Π(t) = ϕ(t) : ϕ Γ(B ) is relatively compact in X.
r
∈
Let t (0,b] be fixed and ε a real number satisfying 0 < ε < t. For x B , we define
r
∈ (cid:8) (cid:9) ∈
t−ε t−ε
ϕ (t) = C(t,0)x +S(t,0)y + S(t,s)f(s)ds+ S(t,η)Bu(η,x)dη.
ε 0 0
Z0 Z0
Since S (t) is a compact operator, the set Π (t) = ϕ (t) : ϕ Γ(B ) is relatively compact in
0 ε ε ε r
{ ∈ }
X for each ε, 0 < ε < t. Moreover, for each 0 < ε < t, we have
t t
ϕ(t) ϕ (t) N L (s)ds+NM u(η,x) dη.
ε f,r B
| − | ≤ k k
Zt−ε Zt−ε
Hence there exist relatively compacet sets arbitrarily cloese to the set Π(t) = ϕ(t) : ϕ Γ(B ) ,
r
{ ∈ }
and the set Π(t) is relatively compact in X for all t [0,b]. Since it is compact at t = 0, hence
∈
Π(t) is relatively compact in X for all t [0,b].
∈
Step 5. Γ heas a closed graph.
Let x x as n , ϕ Γ(x ), and ϕ ϕ as n . We will show thatϕ Γ(x ).
n ∗ n n n ∗ ∗ ∗
→ → ∞ ∈ → → ∞ ∈
Since ϕ Γ(x ), there exists a f S such that
n ∈ n n ∈ F,xn
t t
ϕ (t) =C(t,0)x +S(t,0)y + S(t,s)f (s)ds+ S(t,s)BB∗S∗(b,t)R(a,Υb) x
n 0 0 n 0 b
Z0 Z0 h
b
S(b,0)x S(b,η)f (η)dη (s)ds.
0 n
− −
Z0 i
We must prove that there exists a f S such that
∗ ∈ F,x∗
t t
ϕ (t)=C(t,0)x +S(t,0)y + S(t,s)f (s)ds+ S(t,s)BB∗S∗(b,t) y S(b,0)x
∗ 0 0 ∗ 0 0
−
Z0 Z0 h
b
S(b,η)f (η)ds (s)ds.
∗
−
Z0 i
Set
u (t)= B∗S∗(b,t)[x C(b,0)x S(b,0)y ](t).
x b 0 0
− −
Then
u (t) u (t), for t I, as n .
xn → x∗ ∈ → ∞
Clearly, we have
t
ϕ C(t,0)x +S(t,0)y S(t,s)BB∗S∗(b,t)R(a,Υb) x S(b,0)x
n− 0 0− 0 b− 0
(cid:13)(cid:16) Z0 h
(cid:13) b t
(cid:13) S(b,η)f (η)dη (s)ds ϕ C(t,0)x +S(t,0)y S(t,s)BB∗S∗(b,t) y
n ∗ 0 0 0
− − − −
Z0 i (cid:17) (cid:16) Z0 h
b
C(b,0)x S(b,0)y S(b,η)f (η)ds (s)ds 0 as n .
0 0 ∗
− − −Z0 i (cid:17)(cid:13)C → → ∞
(cid:13)
(cid:13)
10