In this paper, we examine controllability problems of evolution inclusions with nonlocal conditions. Using the set-valued and single-valued Mönch fixe...

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Controllability of Impulsive Evolution Inclusions with Nonlocal Conditions1,2 M. GUO,3 X. XUE,4 AND R. LI5 Communicated by I. Galligani

Abstract. In this paper, we examine controllability problems of evolution inclusions with nonlocal conditions. Using the set-valued and single-valued Mo¨nch fixed-point theorem, we establish some sufficient conditions for the controllability under convex and nonconvex orientor fields respectively. Our approach is different from all previous approaches; we do not assume that the evolution system generates a compact semigroup; so, our method is applicable to a wide class of (impulsive) control systems and evolution inclusions in Banach spaces. Key Words. Controllability, evolution systems, nonlocal conditions, Mo¨nch fixed-point theorem.

1. Introduction Controllability for differential systems in Banach spaces has been studied by many authors. By using the Martelli fixed-point theorem, Benchohra and Ntouyas (Ref. 1) discussed a semilinear second-order differential system. In Ref. 2, by using the Schauder fixed-point theorem, Balachandran et al. proved controllability of nonlinear integrodifferential systems. In Ref. 3, Han and Park established sufficient conditions for the boundary controllability of differential equations with nonlocal conditions. 1

The authors wish to express their gratitude to Professor I. Galligani, who made many corrections, constructive criticisms, and suggestions. 2 This research was supported by Harbin Institute of Technology, Grant HIT. MD 2002.24. 3 Doctor in Mathematics, Lecture Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, China. 4 Professor, Department of Mathematics, Harbin Institute of Technology, Harbin, China. 5 Professor, Department of Mathematics, Harbin Institute of Technology, Harbin, China.

355 0022-3239=04=0200-0355=0 g 2004 Plenum Publishing Corporation

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In this paper, we will consider the following semilinear impulsive evolution systems with nonlocal conditions: x¢(t) – A(t)x(t)˛F (t, x(t)) + Bu(t), Dx(ti ) = li (x(ti )),

a:e: on T = [0, b],

i = 1, 2, . . ., s,

x(0) + M(x) = x0 ,

(1a) (1b) (1c)

where {A(t)}t ˛T is a family of linear operators which generates an evolution operator U: W = {(t, s): 0 # s# t# b} ﬁ L(X ); here, X is a Banach space, F : T · X ﬁ 2X is a multifunction (set-valued mapping), Dx(tk) = x(t+k ) – x(t–k), M: PC(T, X )ﬁ X, x0 ˛X, B is a bounded linear operator from a Banach space V to X, and the control function u() is given in L2(T, V ). Controllability of the system (1) without impulse was discussed by Li and Xue (Ref. 4) using the Kakutani-Fan fixed-point theorem and assuming that {A(t)}t ˛T is an infinitesimal generator of some compact operator semigroup. The nonlocal condition, which is a generalization of the classical initial condition, was motivated by physical problems. The pioneering work on nonlocal conditions is due to Byszewski (Ref. 5). In the past few years, several papers have been devoted to studying the existence of solutions for differential equations with nonlocal conditions. Among others, we refer to the papers by Balachandran and Chandrasekaran (Ref. 6), Balachandran and Ilamaran (Ref. 7), Ntouyas and Tsamatos (Refs. 8–9), Ntouyas (Ref. 10), and Benchohra et al. (Refs. 11–12). Since it is quite difficult to determine whether an operator semigroup is compact (see Ref. 13), unlike Refs. 1–2, 4, 8, 12, we do not assume that {A(t)}t ˛T generates a compact semigroup. This allows us to discuss some differential inclusions which contain a linear operator that generates a noncompact semigroup. The following are some simple examples. (i)

X = L2(–O, +O), the ordinary differential operator A1 = d=dt, with domain D(A1) = H1(–O, +O), generates a parallel-translation semigroup T1(t) defined by T1 (t)u(s) = u(t + s),

(ii)

for every u˛X :

The C0-semigroup T1(t) is not compact on X. X is an infinite-dimensional Banach space and A˜ is a bounded linear operator on X, the operator A2 = A˜ with D(A2) = X

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generates a uniformly continuous semigroup T2 (t) = exp(– A˜t), on X : T2(t) is noncompact. (iii) X = C(0, +O) and the differential operator A3 = d=dt, with domain D(A2) = C1(0, +O), generates a semigroup T3(t) defined by T3 (t)u(s) = u(t + s),

for each u˛X :

The C0-semigroup T3(t) is not compact on X. Unlike all previous papers, we will make use of the Mo¨nch fixed-point theorem to establish our results. The set-valued generalization of the theorem was obtained in Ref. 14, which includes the classical Martelli and KakutaniFan fixed-point theorems as special cases. Instead of the requirement of compactness for semigroups, we assume that the evolution system U(t, s) satisfies the following condition: a(U(t, s)D)# KU (t, s)a(D),

for every bounded subset D X ,

(2)

where a is the Kuratowski measure of noncompactness. Many evolution semigroups satisfy this condition. For instance, (a) the compact semigroup U(t, s) satisfies the hypothesis (2) with KU (t, s) ” 0, so our discussion includes the result in Ref. 4 as a special case; (b) the C0-semigroups T1(t) and T3(t) defined above satisfy the hypothesis (2) with KU(t, s) ” 1; (c) the uniformly continuous semigroup T2(t) defined above satisfies the hypothesis (2) when kA˜kL # a = const; here, KU (t) ” exp(a). Therefore, our discussion improves some results in Ref. 4. Since the method used in this paper is also available for (impulsive) evolution inclusions in Banach space, we can extend also the main results in Refs. 8 and 11–12.

2. Preliminaries and Basic Hypotheses Denote by R the one-dimensional real Euclidean space; let R+ = (0, +O), and R+ = [0, +O). Let X be a separable Banach space, let Xw denote the space X endowed with weak topology, and let conv(A) [resp. convw (A)] be the convex hull [resp. convex closed hull in Xw] of the subset A X. We will use the notations Pf (c) (X ) = {B ˝ X : B is nonempty, closed (convex) subset}, P(w)k(c) (X ) = {B ˝ X : B is nonempty, (w)-compact (convex) subset}:

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Let b ˛R+ :

T = [0, b],

A multifunction F : T ﬁ Pf(X ) is said to be measurable if, for every z˛X, the function tﬁ d(z, F (t)) = inf kz – xk x ˛F (t)

is measurable. This is equivalent to saying that GrF = {(t, x)˛T · X :x ˛F (t)}˛S · B(X), with S being the Lebesgue s -field of T and B(X) the Borel s-field of X (graph measurability), or saying that there exists a sequence fn: T ﬁ X, n $ 1, of measurable functions such that F (t) = { fn (t): n$ 1},

for all t˛T:

A graph measurable multifunction F : T · X ﬁPf (X ) has the property that, if x: T ﬁ X is measurable, then tﬁ F (t, x(t)) is measurable. So, by the Aumann selection theorem, we can find a measurable function g:Tﬁ X such that g(t) ˛F (t, x(t)),

a:e: on T:

Let Lp (T, X ),

p = 1, 2,

be the space of Lebesgue-Bochner integrable functions, which is a Banach space endowed with the norm ð 1=p kukLp = ku(t)kp dt : T

We denote by

S1E

the set of all integrable selectors of F (), i.e.,

SF1 = { f () ˛L1 (T, X ): f (t)˛F (t), a:e: on T}: It is easy to know that S1F is closed and is nonempty if and only if inf kxk˛L1 (T, R+ ):

x ˛F (t)

Using this set, we can define an integral for multifunctions, i.e., ð ð F (t)dt = f (t)dt: f ( )˛SF1 , T

T

where the vector-valued integral is in the sense of Bochner. We say that F ( ) is integrably bounded if and only if F () is measurable and jF ( )j˛L1 (T, R),

where jF (t)j = sup kxk: x ˛F (t)

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Finally the set S1F is decomposable in the sense that, if ( f1, f2, A)˛S1F · S1F · S, then f1 c A + f2 c Ac ˛SF1 : Let Y, Z be Banach spaces and let G:Y ﬁ Pf (Z). We say that G( ) is upper semicontinuous (u.s.c.) [resp. lower semicontinuous (l.s.c.)] if, for all C ˛Pf (Z ), we have that G – (C) = {y˛Y : G( y)˙ C „;} [resp: G+ (C) = {y˛Y : G( y)˝ C}] is closed in Y. The above definition of lower semicontinuity is equivalent to saying that, for all z˛Z, y ﬁ dZ (z, G(y)) = inf dZ (z, v) v˛G( y)

is upper semicontinuous as an R+-valued function. Also, if G(Y ) is compact, then the above definition of upper semicontinuity is equivalent to saying that G( ) has a closed graph in Y · Z. Moreover, for a sequence of nonempty sets {An} of some Banach space, if we define lim sup An = {z = lim znk : znk ˛Ank , k $ 1}, n ﬁ +O

n ﬁ +O

then G( ) is u.s.c. if and only if, for all vn ﬁ v, we have lim sup G(vn )˝ G(v): n ﬁ +O

Denote J0 = [0, t1 ], Jk = (tk , tk+1 ], L = {tk }, where 0 < tk < tk+1 # ts+1 = b < O,

k = 1, 2, . . ., s:

The following spaces will be used in the sequel: AC(T, X ) = {u: T ﬁ X : u is absolutely continuous on Jk , k = 0, 1, . . ., s}, PC(T, X ) = u: u˛C(TnL, X ), u(t–k ) = lim– u(t) = u(tk ), t ﬁ tk u(t+k ) = lim+ u(t) exists, k = 1, 2, . . ., s : t ﬁ tk

It is known that PC(I, X ) is a Banach space with norm kukPC = sup ku(t)k: t ˛T

For any x˛PC(T, X ), we denote Dx(tk ) = u(t+k ) – u(t–k ),

for k = 1, 2, . . . , s:

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Finally, we recall the definition of Kuratowski measure of noncompactness. Let D be a bounded subset of a Banach space X. Then, ( ) m [ a(D) = inf e > 0: D Mi and diam(Mi ) # e : i=1

A function x( )˛PC(T, X ) such that ðt x(t) = U(t, 0)x(0) + U(t, s)( f + Bu)(s)ds + li (x(ti )), Definition 2.1.

t˛T,

0

0

with f ˛S1F ( , x( )) and x(0) + M(x) = x0, is called a mild solution of the system (1). Definition 2.2. The system (1) is said to be nonlocally controllable on T if, for every x0, x1 ˛X, there exists a control u˛L2(T, V ) such that the mild solution x() of (1) satisfies x(b) + M(x) = x1. For the proof of the main results, we will need the following hypotheses: (H1)

F : T · X ﬁPkc(X ) is a multifunction such that: (i) for every x˛X, tﬁ F (t, x) is measurable; (ii) for a.e. t˛T, xﬁ F (t, x) is u.s.c. from X into Xw; (iii) sup |F (t, x)| # jn(t), a.e. on T, with jn() ˛L1(T, R+) and kxk # n ðb (3) lim inf (1=n) j n (s)ds = 0; n ﬁ +O

0

(iv) there is KF () ˛L1(T, R+ ) such that, for any bounded subset D X, a(F (t, D)) # KF (t)a(D), (H2)

t˛T:

(4)

Let L(X ) be the space of all linear operators on X with operator norm k kL. {A(t)}t ˛T is a family of linear densely defined operators that generate an evolution system U: W = {(t, s): 0# s# t # b} ﬁ L(X ) such that: (i)

there is a Caratheodory function KU: W ﬁ R+ [i.e., KU (, s) ˛ C (T, R+ ), for a.e. s˛T and KU (t, ) ˛L1(T, R+ ), for every t ˛T ] such that, for every (t, s)˛W and any bounded subset D X, a(U(t, s)D)# KU (t, s)a(D);

(5)

(ii)

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for any t1, t2 ˛T, with t2 # t1, ð t2 1=2 2 lim kU(t1 , s) – U(t2 , s)kL ds = 0:

(6)

t1 ﬁ t2

(H3)

M: PC(T, X ) ﬁ X is a compact operator such that lim

kykPC ﬁ +O

(H4)

0

kM( y)k=kykPC = 0:

(7)

Let V be a Banach space. B is a bounded linear operator from a Banach space V to X. W: L2(T, V ) ﬁX is a linear operator defined by ðb Wu = U(b, s)Bu(s)ds (8) 0

such that: the operator W has an invertible operator W –1, which takes values in L2(T, V )nkerW, and there exist positive constants LB and LW such that kBk# LB and kW –1k # LW; (ii) there are KW ˛L1(T, R+ ) and a constant KB $ 0 such that, for any bounded sets Q L2(T, V ) and H V, (i)

(H5)

a((W –1 Q)(t)) # KW (t)a(Q(t)),

(9a)

a(B(H)) # KB a V (H):

(9b)

Let li: X ﬁX, i = 1, 2, . . ., s, be a continuous operator such that: (i)

there are nondecreasing 1, 2, . . ., s, such that kli (x)k # Li (kxk)

and

functions

Li: R+ ﬁR+,

i=

lim inf Li (n)=n = 0, n ﬁ +O

i = 1, 2, . . . , s;

(10)

(ii) there exist constants Ki $ 0, i = 1, 2, . . ., s, such that a(li (D)) # Ki a(D),

(H6)

i = 1, 2, . . . , s: (11) Ðt Let k1(t, s) = KU (t, s) KF(s) and k2 (t) = 0 KU (t, s)KB (s)KW (s)ds: The following estimation holds true: ðb ðt 2 sup k1 (t, s)ds + 4 sup k2 (t) k1 (b, s)ds t ˛T 0 t ˛T 0 s (12) + 2 sup k2 (t) + 1 Ki < 1: t ˛T

i=1

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We need also the following lemma on the Kuratowski measure of noncompactness. Lemma 2.1. See Ref. 15. Let X be a Banach space and let C L1(T, X ) be a countable set with ku(t)k # h(t), for a.e. t ˛T and every u˛C, where h˛L1(T, R+). Then the function j (t) = a(C(t)) belongs to L1(T, R) and satisfies ð ð a u(t)dt: u˛C # 2 a(C(t))dt: T

T

Details of the Kuratowski measure of noncompactness can be found in Ref. 19. Our main tool is the following generalization of the Mo¨nch fixed-point theorem. Theorem 2.1. See Ref. 14. Let D be a closed convex subset of a Banach space X. Assume that N: D ﬁ Pkc(D) is upper semicontinuous and that, for some x0 ˛D, one has ¯ =C¯ with C M countable M D, M = conv({x0 } ¨ N(M)), M ¯ compact: M

(13)

Then, there exists x ˛D with x ˛N(x).

3. Main Results Theorem 3.1. If hypotheses (H1)–(H6) hold, then problem (1) is nonlocally controllable on T. Using hypothesis (H4) (i), for every y ˛PC(T, X ), define the

Proof. control uf (t) = W

–1

ðb s x1 – M( y) – U(b, 0)(x0 – M( y)) – U(b, s)f (s)ds – li ( y(ti )) (t), 0

i=1

where f ( )˛SF1 (, y()). We will show now that, when using this control, the multifunction G: PC(T, X )ﬁ 2PC(T, X ), defined by n G( y) = x˛PC(T, X ): x(t) = U(t, 0)(x0 – M( y)) + g ( f + Buf )(t) o 1 + li ( y(ti )), f ()˛SF(, y()) , 0 < ti # t

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where g ( f + Buf )˛C(T, X ) is defined by ðt g ( f + Buf )(t) = U(t, s)( f + Buf )(s)ds, 0

has a fixed point. This fixed point is then a solution of the system (1). Clearly, x1 – M ð yÞ˛(G( y))(b): Step 1. We claim that G() is nonempty closed convex valued. Let y()˛PC(T, X ) and let {sn} be a sequence of step functions such that ksn (t)k # ky(t)k:

sn (t)ﬁ y(t),

Then, by virtue of hypothesis (H1) (i), for every n$ 1, tﬁ F (t, sn(t)) admits a measurable selector fn(t), i.e., fn (t)˛F (t, sn (t)),

a:e: on T:

Let G(t) = conv

[

F (t, sn (t)):

n$1

Because of hypothesis (H1) (ii) and Theorem 7.4.2 of Ref. 16, we have that G(t) ˛Pwkc (X ), G( ) is clearly measurable [since for each n $ 1, tﬁ F (t, sn(t)) is measurable], and jG(t)j = sup kzk # j m (t), z ˛G(t)

a:e: t ˛T with kykC # m:

So, from Proposition 3.1 of Ref. 17, we have that SG1 ˛Pwkc (L1 (T, X ))

and

{ fn } ˝ SG1 ;

we may assume, by passing to a subsequence if necessary, that fn ﬁ f ,

weakly in L1 (T, X ):

Then, invoking Theorem 3.1 of Ref. 17, for a.e. t ˛T we have w w f (t) ˛conv lim sup { fn (t)} ˝ conv lim sup F (t, sn (t)) ˝ F (t, y(t)), n ﬁ +O

n ﬁ +O

where the last inclusion is a consequence of hypothesis (H1) (ii). Therefore, for y˛PC(T, X ), G( y)„ ;. Clearly, G( ) is convex valued and, because of Proposition 3.1 of Ref. 18, SF1 (, y()) ˛Pwkc (L1 (T, X ))

for every y˛PC(T, X );

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we deduce also that G() is closed valued; then, G: PC(T, X )ﬁ Pfc (PC(T, X )): Step 2. There exists a positive integer n0 $ 1 such that G(Bn0 )˝ Bn0 , where Bn0 = {y˛PC(T, X ): kykPC # n0 }: Suppose the contrary. Then, we can find yn ˛PC(T, X ), xn ˛G(yn) such that kyn kPC # n

and

kxn kPC > n:

Then, for every n$ 1 and some fn ˛SF (, yn()), we have that xn (t) = U(t, 0)(x0 – M( yn )) + g ( fn + Bufn )(t) +

li ( y(ti )):

0

So, we get that n < kxn kPC # LU (kx0 k + kM( yn )k) + kg ( fn )kC + kg (Bufn )kC s

(14)

+ kli ( y(ti ))k, i=1

where LU > 0 is such that kU(t, s)kL # LU : Note that kg ( fn )kC # sup

ðt

t ˛T

0

kg (Bufn )kC # sup t ˛T

kU(t, s)kL k fn (s)kds # LU

ðt 0

ðb

j n (s)ds,

(15)

0

kU(t, s)kL kBkkufn (s)kds# LB LU b1=2 kufn kL2 ,

(16)

s –1 kufn kL2 = W x1 – M( yn ) – U(b, 0)(x0 – M( yn )) – g ( fn )(b) – li ( yn (ti )) 2 i=1 L # LW kx1 k + LU kx0 k + (1 + LU )kM( yn )k ðb + LU 0

s j n (s)ds + Li (kyn (ti )k) : i=1

(17)

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Hence, by (14)–(17), we have n< (LU + LB LW L2U b1=2 )kx0 k + LB LW LU b1=2 kx1 k + (1 + LB LW b1=2 + LB LW LU b1=2 )LU kM( yn )k ðb s 2 1=2 + (LU + LB LW LU b ) j n (s)ds + (1 + LB LW LU b1=2 ) Li (ky(ti )k), i=1

0

which implies that ðb s 1< (1=n) C1 + C2 kM( yn )k + C3 j n (s)ds + C4 Li (n) , 0

(18)

i=1

where C1 = (LU + LW LB L2U b1=2 )kx0 k + LW LB LU b1=2 kx1 k, C2 = (1 + LW LB b1=2 + LW LB LU b1=2 )LU , C3 = LU + LW LB L2U b1=2 , C4 = 1 + LB LW LU b1=2 : Observing (3), (7), and (10), by passing to the limit as n ﬁ +O in (18), we get 1# 0, which is a contradiction. Thus, we deduce that there is n0 $ 1 such that G(Bn0 ) ˝ Bn0 . Step 3. G( ) is u.s.c. from Bn0 to Bn0 . To this end, we need only to show that G() has a closed graph. Let {ym } ˝ Bn0 , ym ﬁy,

in PC(T, X ),

xm ˛G( ym ), xm ﬁ x,

in PC(T, X ):

and let

Then, by definition, we have xm (t) = U(t, 0)(x0 – M( ym )) + g ( fm + Bufm )(t) +

li ( ym (ti )),

0

with fm ˛SF1 (, ym ( )) . Let [ F (t, ym (t)): G(t) = conv m$1

Because of hypothesis (H1) (ii) and Theorem 7.4.2 of Ref. 16, we have G(t) ˛Pwkc(X ), G( ) is clearly measurable [since for each m $ 1, t ﬁF (t, ym(t)) is measurable], and jG(t)j# j n0 (t),

a:e: on T:

So, from Proposition 3.1 of Ref. 17, we have that SG1 ˛Pwkc (L1 (T, X )),

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and since { fm} ˝ S1G, by passing to a subsequence, if necessary, we may assume that fm ﬁ f , weakly in L1 (T, X ): Then, invoking Theorem 3.1 of Ref. 17, we get that, for a.e. t˛T, f (t)˛convw lim sup { fm (t)} ˝ convw lim sup F (t, ym (t)) ˝ F (t, y(t)): m ﬁ +O

m ﬁ +O

Therefore, f ˛SF1 (, y()) : Also, we can verify easily that, for every t ˛T, g ( fm + Bufm )(t)ﬁ g ( f + Buf )(t), weakly in X , and by the continuity of li, li ( ym (ti )) ﬁ li ( y(ti )): 0

0

Hence, U(t, 0)(x0 – M( ym )) + g ( fm + Bufm )(t) + ﬁ U(t, 0)(x0 – M( y)) + g ( f + Buf )(t) +

li ( ym (ti ))

0

li ( y(ti )),

0

weakly in X, t˛T. Therefore, we get x(t) = U(t, 0)(x0 – M( y)) + g ( f + Buf )(t),

as m ﬁ O,

t ˛T,

with f ˛SF1 (, y()); i.e., G( ) has a closed graph and so in u.s.c. Step 4. Let ¯ =C¯, D Bn0 ˙ conv({0} ¨ G(D)) and D for some countable set C D. We claim that G satisfies the condition (13). At first, we show that D is equicontinuous on Jk, k = 0, 1, . . ., s. If we can show that G(D) is equicontinuous on every Jk, then from D conv({0} ¨ G(D)), it is easy to know that D is also equicontinuous on Jk. To this end, let x ˛G(D) and t¢, t† ˛Jk , t¢# t†; there are y ˛D and f˛S1F (, ()) such that kx(t†) – x(t)k = k[U(t†, 0) – U(t¢, 0)](x0 – M( y))k ð t† + U(t†, s)[ f (s) + Buf (s)]ds 0 ð t¢ U(t¢, s)[ f (s) + Buf (s)]ds – 0

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# k[U(t†, 0) – U(t¢, 0)](x0 – M( y))k ð t† + kU(t†, s)kL k f (s) + Buf (s)kds t¢

ð t¢ +

kU(t†, s) – U(t¢, s)( f (s) + Buf (s))kds

0

# k[U(t†, 0) – U(t¢, 0)]x0 k + k[U(t†, 0) – U(t¢, 0)]M( y)k ð t† ð t† + LU j n0 (s)ds + LU LB kuf (s)kds ð t¢ + 0

t¢

t¢

kU(t†, s) – U(t¢, s)kL (k f (s)k + kBuf (s)k)ds

# k[U(t†, 0) – U(t¢, 0)]x0 k + k[U(t†, 0) – U(t¢, 0)]M( y)k ð t† ð t† + LU j n0 (s)ds + LU LB kuf (s)kds ð t¢ +

t¢

kU(t†, s) –

0

t¢

U(t¢, s)k2L ds

1=2 (kj n0 kL2 + LB kuf (s)kL2 )

# k[U(t†, 0) – U(t¢, 0)]x0 k + k[U(t†, 0) – U(t¢, 0)]M( y)k ð t† ð t† + LU j n0 (s)ds + LU LB kuf (s)kds ð t¢ + 0

t¢

kU(t†, s) –

t¢

U(t¢, s)k2L ds

1=2 (kj n0 kL2 + LB C),

where C = LW kx1 k + LU kx0 k + (1 + LU )kM( y)k + LU

ðb 0

(19)

s

j n0 (s)ds + Li (n0 ) i=1

is a constant. With regard to (6), we know that the last item on the right-hand side of the inequality (19) tends to zero as t† ﬁ t¢. By the strongly continuity property of U(, s) and the absolute continuity of the Lebesgue integral, we can see that the first four items of the right-hand side of the inequality (19) also tend to zero as t† ﬁ t¢. Therefore, G(D) is equicontinuous and so D is equicontinuous on every Jk. From Step 2, we see that G is bounded on Bn0 . In order to apply the Mo¨nch fixed-point theorem, it remains to show that D(t) is relatively compact in X, for each t˛T. Since C D conv({0} ¨ G(D)), and since C is countable, we can find a countable set H = {hn} G(D) with C conv({0} ¨ H). Then, there exist wn ˛D with kwn k # jG(Bn0 )j

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and gn ˛SF1 ( , wn ( )) such that hn (t) = U(t, 0)(x0 – M(wn )) + g (gn + Bugn )(t) +

li (wn (ti )):

0

From D C¯ conv({0} ¨ H), we find that a(D(t)) # a(C¯(t)) # a(H(t)) #a U(t, 0)(x0 – M(wn )) + g (gn + Bugn )(t) +

li (wn (ti )): n$ 1

0

# a({U(t, 0)M(wn ): n $ 1}) + a({g gn (t): n $ 1}) + a({(g Bugn )(t): n $ 1}) +

a({li (wn (ti )): n $ 1}):

0

By Lemma 2.1, from (4) and (5) we have that a({g (gn )(t): gn ( )˛SF1 (, wn ()) , n$ 1}) ðt # 2 a({U(t, s)gn (s): gn ( )˛SF1 (, wn ()) , n$ 1})ds 0 ðt # 2 a({U(t, s)f (s): f () ˛SF1 (, y()) , y˛D})ds 0 ðt # 2 KU (t, s)a(F (s, D(s)))ds 0 ðt # 2 KU (t, s)KF (s)a(D(s))ds 0 ðt = 2 k1 (t, s)a(D(s))ds:

(20)

0

Moreover, from (9), (5), (11), (4), and Lemma 2.1, we obtain that 1 a V ({ugn (t): gn ( )˛SF(, wn ()) , n $ 1}) ðb n h = a V W –1 x1 – M(wn ) – U(b, 0)(x0 – M(wn )) – U(b, s)gn (s)ds 0

i o – li (wn (ti )(t) : gn ( )˛SF1 (, wn ( )) , n$ 1 i=1 ðb n # KW (t)a M(wn ) – U(b, 0)(M(wn )) – U(b, s)gn (s)ds s

s

– li (wn (ti )): i=1

gn ( )˛SF1 (, wn ()) , n$ 1

o

0

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# KW (t)a(M({wn : n$ 1})) + KW KU (b, 0)a(M({wn : n$ 1})) ðb + 2KW (t) a({U(b, s)gn (s): gn ( )˛SF1 (, wn ()) , n$ 1})ds 0

s

+ KW (t) a(li ({wn (ti ): n$ 1})) i=1

# KW (t)a(M({y: y˛D}) + KW (t)KU (b, 0)a(M({y: y ˛D})) ðb 1 + 2KW (t) KU (b, s)a({ f (s): f ( )˛SF(, y()) , y ˛D})ds 0 s

+ KW (t) a(li ({y(ti ): y ˛D})) i=1

# KW (t)a(M(D)) + KW (t)KU (b, 0)a(M(D)) ðb s + 2KW (t) KU (b, s)KF (s)a(D(s))ds + KW (t) Ki a(D(ti )): i=0

0

Together with Lemma 2.1 and (9), this implies that a({g (B(ugn (t))): gn () ˛SF1 (, wn ()) , n $ 1}) ðt # 2 a({U(t, s)B(ugn (s)): gn () ˛SF1 (, wn ( )), n$ 1})ds 0 ðt # 2 KU (t, s)KB (s)a({ugn (s): gn ( )˛SF1 (, wn ( )), n $ 1})ds 0 ðt # 2a(M(D)) KU (t, s)KB (s)KW (s)ds 0 ðt + 2KU (b, 0)a(M(D)) KU (t, s)KB (s)KW (s)ds 0 ð b ðt +4 KU (b, h)KF (h)a(D(h))dh KU (t, s)KB (s)KW (s)ds 0 0 s ðt + 2 Ki a(D(ti )) KU (t, s)KB (s)KW (s)ds i=0

0

= 2k2 (t)a(M(D)) + 2k2 (t)KU (b, 0)a(M(D)) ðb s + 4k2 (t) k1 (b, h)a(D(h))dh + 2k2 (t) Ki a(D(ti )): i=0

0

(21)

Also by (11), we get s

a({li (wn (ti )): n$ 1})# Ki a(D(ti )) # Ki a(D(ti )):

0

0

i=0

(22)

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From the equalities (20)–(22), we obtain that ðt a(D(t)) # KU (t, 0)a(M(D)) + 2 k1 (t, s)a(D(s))ds + 2k2 (t)a(M(D)) 0

ðb

+ 2KU (b, 0)k2 (t)a(M(D)) + 4k2 (t)

k1 (b, h)a(D(h))dh

0 s

s

+ 2k2 (t) Ki a(D(ti )) + Ki a(D(ti )): i=1

(23)

i=1

In view of the compactness of the operator M, from (5) we have a(U(t, 0)(x0 – M(wn )))# KU (t, 0)a(M(wn )) = 0: Now, according to the equality (23), we have that ðt ðb a(D(t)) # 2 k1 (t, s)a(D(s))ds + 4k2 (t) k1 (b, h)a(D(h))dh 0 s 0 + [2k2 (t) + 1] Ki a(D(ti )) : i=0

(24)

Since D is equicontinuous on every Jk, by Proposition 7.3 of Ref. 19, we find that a(D) = max a(D(JK )) = max max a(D(t)): 0 # k # s t ˛Jk

0#k#s

Then, the inequality (24) implies that ðb ðt a(D)# 2 sup k1 (t, s)ds + 4 sup k2 (t) k1 (b, s)ds t ˛T 0 t ˛T 0 s + 2 sup k2 (t) + 1 Ki a(D): i=1

t ˛T

¯ is ¯ =C According to the hypothesis (H6), a (D) = 0, so we have that D compact. By Theorem 2.1, we conclude that G( ) has a fixed point and thus the system (1) is nonlocally controllable on T. u Now, we consider the nonconvex version of the above result. Our hypothesis on the orientor field is the following: (H1¢) F :T · X ﬁ Pf (X ) is a multifunction such that (i) (ii)

(t, x)ﬁ F (t, x) is graph measurable and x ﬁÐF (t, x) is l.s.c.; b sup jF (t, x)j# j n (t), a:e: with lim inf (1=n) 0 j n (s)ds = 0; n ﬁ +O

kxk # n

1

(iii) there is KF () ˛L (T, R+ ) such that, for every t ˛T and every bounded subset D X, a (F (t, D)) # KF (t)a (D).

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Theorem 3.2. If hypotheses (H2)–(H4) and (H1¢) hold, then the system (1) is nonlocally controllable on T. Proof. Consider the multivalued Nemitsky operator N:PC(T, X )ﬁ 1 2L (T, X ) , defined by N(x) = S1F (, x( )). We will show that N( ) has nonempty closed decomposable values and is l.s.c. from PC(T, X ) to L1(T, X ). The nonemptyness, closedness, and decomposability of the values of N( ) are easy to check. To check the lower semicontinuity of N(), we need to show that, for every u ˛L1(T, X ), x ﬁ d(u, N(x)) is an upper semicontinuous R+-valued function. To this end, from Theorem 2.2 of Ref. 20, we have that d(u, N(x)) = inf ku – vk1 v˛N(x) ðb = inf ku(t) – v(t)kdt v˛N(x)

ðb

=

0

inf

0 v˛F(t, x(t)) ðb

=

ku(t) – vkdt

d(u(t), F (t, x, (t)))dt: 0

We will show that, for any l $ 0, the set Ul = {x ˛PC(T, X ): d(u, N(x)) $ l} is closed in PC(T, X ). For this purpose, let {xn} ˝ Ul and assume that xn ﬁ x in PC(T, X ). Then, for all t ˛T, xn (t) ﬁx(t),

in X :

By virtue of hypothesis (H1¢) (i), x ﬁ d(u(t), F(t, x)) is u.s.c. So, via the Fatou lemma, we have l # lim supn ﬁ +O d(u, N(xn )) ðb = lim sup d(u(t), F (t, xn (t)))dt n ﬁ +O

ðb

0

lim sup d(u(t), F (t, xn (t)))dt

#

0 n ﬁ +O ðb

d(u(t), F (t, x(t)))dt = d(u, N(x)):

# 0

Therefore, x ˛Ul and this proves the lower semicontinuity of N( ). This allows us to apply Theorem 3 of Ref. 21 and obtain a continuous map r: PC(T, X )ﬁ L1(T, X ) such that r(x)˛N(x),

for every x˛PC(T, X ):

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Consider a map p : PC(T, X )ﬁ PC(T, X ) defined by p (x)(t) = U(t, 0)(x0 – M(x)) + g (r(x) + Bur(x) ) +

li ( y(ti )):

0

As in the proof of Theorem 3.1, we can show that all the conditions of the Mo¨nch fixed-point theorem (Ref. 14, Theorem 1.1) are satisfied. Thus, applying the theorem, we get x = p (x). Then, the system (1) is nonlocally controllable on T. u Remark 3.1. Some conditions of Theorem 3.1 and 3.2 can be simplified in the following cases: (C1) If the multifunction F :X ﬁ 2X is completely continuous, i.e., if F maps a bounded set D X into a relatively compact subset of X, then the condition (12) can be rewritten as –1 s Ki < 2 sup k2 (t) + 1 : i=1

t ˛T

(C2) If the impulse effect does not occur in the system, then the relation (12) can be simplified as ðt ðb 2 sup k1 (t, s)ds + 4 sup k2 (t) k1 (b, s)ds < 1: t ˛T

0

t ˛T

0

(C3) If the evolution system generated by the family of operators {A(t)}t ˛T is a compact semigroup, then the condition (12) will become si=1 Ki < 1; in addition, if the system (1) is without any impulse, then the hypothesis (H6) can be removed. This is the case discussed in Ref. 4. (C4) If the evolution system is a uniformly continuous semigroup (see Ref. 13), then the hypothesis (H2) (ii) is satisfied automatically; in particular, a compact semigroup satisfies (H2).

4. Conclusions In this paper, controllability problems of first-order evolution inclusions with nonlocal conditions have been considered. Some sufficient conditions have been obtained. As pointed out in Section 1, these conditions are strictly weaker than most of the existing ones. In addition, the examples given in Section 1 show that our hypotheses are acceptable in practice and can be used in evolution systems containing noncompact semigroups. This paper does not concern with the controllability and existence of (impulse) delay inclusions in Banach space; such problems were studied in

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Refs. 22 and 12 respectively. The method used in this paper, with little modifications, will be also available for those problems; of course, the discussion will be based on a different function space. By the same idea, we can establish similar results for second-order evolution systems and integrodifferential inclusions. Although these equations are different in essence, the handling approach is analogous.

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