Functional differential equations with state-dependent delay appear frequently in applications as models of equations and for this reason the study of this type of equations has received
Trang 1ELSEVIER
An International Journal Available online at www.sciencedirect.com computers &
with applicaUons
Computers and Mathematics with Applications 52 (2006) 411-420
www.elsevier.com/locate/camwa
E x i s t e n c e R e s u l t s for an I m p u l s i v e
A b s t r a c t Partial Differential E q u a t i o n
w i t h S t a t e - D e p e n d e n t D e l a y
E H E R N A N D E Z , M P I E R R I * AND G G O N C A L V E S t
ICMC, Universidade de S~o Paulo S£o Carlos, Cx 668, 13560-970, SP, Brazil
<lalohm><michelle><gabriel>©icmc sc usp br
(Received May 2005; revised and accepted March 2006)
A b s t r a c t - - I n this paper, we establish the existence of mild solutions for a class of impulsive abstract partial functional differential equation with state-dependent delay (~) 2006 Elsevier Ltd All rights reserved
K e y w o r d s - - A b s t r a c t functional differential equations, Impulsive differential equations, State de- pendent delay
1 I N T R O D U C T I O N
I n this p a p e r , we e s t a b l i s h t h e existence of m i l d solutions for an impulsive a b s t r a c t functional differential e q u a t i o n w i t h s t a t e - d e p e n d e n t d e l a y d e s c r i b e d b y
x'(t> = A x ( 0 + f ( t , xp(,,~,~), t e I = [0, a], (1.1)
where A is t h e infinitesimal g e n e r a t o r of a c o m p a c t C0-semigroup of b o u n d e d linear o p e r a t o r s
(T(t))~>o defined on a B a n a c h space X ; t h e functions x~ : ( - c ~ , 0 ] ~ X , xs(#) = x ( s + 0),
belongs to some a b s t r a c t phase space B d e s c r i b e d a x i o m a t i c a l l y ; 0 < t l < < tn < a are pre-fixed numbers; f : I × B -~ X , p : I × 5 -~ ( - c ~ , a], I i : B ~ X , i = 1 , , n , are a p p r o p r i a t e functions a n d t h e s y m b o l A ( ( t ) represents t h e j u m p of t h e function ~ a t t, which is defined b y
a ~ ( t ) = ~(t+) _ ~ ( t - )
T h e t h e o r y of impulsive differential e q u a t i o n s has b e c o m e an i m p o r t a n t a r e a of investigation in recent years s t i m u l a t e d b y t h e i r n u m e r o u s a p p l i c a t i o n s to p r o b l e m s arising in mechanics, electrical engineering, medicine, biology, ecology, etc Relative to o r d i n a r y impulsive differential equations,
*I wish to acknowledge the support of Fapep, Brazil, for this research
tI acknowledge the support of Capes, Brazil, for this research
The authors wish to thank to the referees for their comments, corrections, and suggestions
0898-1221/06/$ - see front matter (~) 2006 Elsevier Ltd All rights reserved Typeset by A j ~ - ~ X doi:10.1016/j.camwa.2006.03.022
Trang 2412 E HERN~.NDEZ et al
we cite among other works [1-5] First-order abstract partial differential equations with impulses are treated in [6-9]
The literature related to ordinary and partial functional differential equations with delay for which p(t, ¢) = t is very extensive and we refer the reader to [10,11] concerning this matter Functional differential equations with state-dependent delay appear frequently in applications
as models of equations and for this reason the study of this type of equations has received great attention in the last years, see, for instance, [12-20] and the references therein T h e literature related to partial functional differential equations with state-dependent delay is limited, to our knowledge, to the recent works [21,22] The study of impulsive partial functional differential equations with state-dependent delay is an untreated topic and it is the motivation of our paper
2 P R E L I M I N A R I E S
T h r o u g h o u t this paper, A : D ( A ) C X ~ X is the infinitesimal generator of a compact semigroup of linear operators (T(t))t>_o defined on a Banach spaces X a n d / ~ is a constant such
t h a t liT(t)II <- 2f/i for every t • I = [0, a] For related semigroup theory, we suggest [23]
To consider the impulsive condition (1.3), it is convenient to introduce some additional concepts and notations We say t h a t a function u : [c~, ~-] ~ X is a normalized piecewise continuous function
on [a, ~-] if u is piecewise continuous and left continuous on (a, ~-] We denote by 79C([a, T]; X) the space formed by the normalized piecewise continuous functions from [a, T] into X In particular,
we introduce the space PC formed by all functions u : [0, a] ~ X such t h a t u is continuous at
t ~ t i , u ( t T ) = u(t~) and u(t +) exists, for all i = 1 , ,n In this paper we always assume that
PC is endowed with the norm [[u][pc = supse[0,~ ] []u(s)[[ It is clear t h a t (PC, [[ IIpc) is a Banach space
To simplify the notations, we put to = 0, tn+l = a and for u • PC we denote by ~ • C([ti, ti+l]; X), i = 0, 1 , , n, the function given by
u(t), for t • (ti, t~+l],
~,(t) = u(t+), for t = t,
Moreover, for B C PC, we denote b y / ~ i , i = 0, 1 , ,n, the set B~ { ~ : u • B}
LEMMA 2.1 A set B C PC is relatively compact in PC if and only if, the set [~i is relatively compact in C([ti, ti+l]; X ) , for every i = 0, 1 , , n
In this work we will employ an axiomatic definition of the phase space B which is similar to
t h a t used in [24] Specifically, 13 will be a linear space of functions mapping ( - o o , 0] into X endowed with a seminorm [] []• and we will assume t h a t B satisfies the following axioms (A) If x : ( - c o , a + b] ~ X, b > 0, is such t h a t x[[o,~+b] • PC([a, a + b] : X ) and x~ • B , then for every t E [a, a + b] the following conditions hold:
(i) xt is in 13,
(ii) IIx(t)ll < gllx llB,
(iii) Ilxtlls <_ K ( t - a)sup{llx(s)] I : a < s < t} + M ( t - a)llx~lls, where g > 0 is
a constant; K, M : [0, c~) ~ [1, (x)), K is continuous, M is locally bounded and
H, K, M are independent of x(.)
(B) T h e space B is complete
EXAMPLE THE PHASE SPACES •Ch(X), "PC°(X) Let g : ( - ~ , 0] ~ [1, c~) be a continuous, nondecreasing function with g(0) = 1, which satisfies the conditions (g-l), (g-2) of [24] This means t h a t the function
g(t + e)
G(t) := sup
- ~ < e < - t g(O)
is locally bounded for t > 0 and t h a t l i m e ~ _ ~ g(0) = oc
Trang 3Existence Results 413
As usual, we said t h a t ~o : ( - c ~ , 0] * X is normalized piecewise continuous, if T is left continuous and the restriction of T to any interval I - r , 0] is piecewise continuous
Next, we modify slightly the definition of the spaces Cg, C ° in [24] We denote b y PCg(X) the
space formed b y the normalized piecewise continuous functions ~ such t h a t ~ / g is bounded on
( - o o , O] and b y T'C°(X) the subspace of ~Pgg(X) formed b y the functions ~o such t h a t
~ ( e ) -~ 0 g(e)
as 0 -~ - c ~ I t is easy to see t h a t 7)Cg(X) and PC°(X) endowed with the n o r m
II~llu := sup II~(e)ll
e<o g(O)
are phase spaces in the sense considered in this work Moreover, in these cases K ( s ) - 1 for
s > 0
EXAMPLE THE PHASE SPACE ~OCr X L2(g, X ) Let 1 < p < e~, 0 < r < ~ and g(.) be a Borel
nonnegative measurable function on ( - ~ , r) which satisfies the conditions (g-5)-(g-6) in the terminology of [24] Briefly, this means t h a t g(.) is locally integrable on ( - o % - r ) and t h a t there exists a nonnegative and locally b o u n d e d function G on ( - o o , 0] such t h a t g(~ + 0) <_ G(~)g(O)
for all ~ < 0 and 0 E ( - o o , - r ) \ N~, where N~ c_ ( - o % - r ) is a set with Lebesgue measure 0 Let /3 := ~oC, × LP(g; X ) , r > 0, p > 1, be the space formed of all classes of functions
T : ( - o % 0 ] + X such t h a t Tl[-r,0] E ~°C([-r, 0 ] , X ) , T(.) is Lebesgue-measurable on ( - ¢ ~ , - r ] and g I ~ I p is Lebesgue integrable on ( - c ~ , - r ] T h e s e m i n o r m in I1' Ilu is defined b y
lip
I]¢pI[B := sup Ilqo(O)ll + { / 9 ( e ) l l ~ ( e ) l l Pde}
\ a - o o / Proceeding as in the proof of [24, T h e o r e m 1.3.8], it follows t h a t B is a phase space which satisfies Axioms A and B Moreover, for r = 0 and p = 2, this space coincides with Co x L2(g, X ) , H = 1;
M ( t ) = G ( - t ) x/2 and g (t) = 1 + ( f ° t g(r) dT) 1/2, for t > 0
REMARK 1 In r e t a r d e d functional differential equations w i t h o u t impulses, the axioms of the
a b s t r a c t phase s p a c e / 3 include the continuity of the function t -~ xt, see [24,25] for details Due
to the impulsive effect, this p r o p e r t y is not satisfied in impulsive delay systems and, for this reason, has been eliminated in our a b s t r a c t description of/3
REMARK 2 Let qo E / 3 and t < 0 T h e notation ~t represents the function defined by ~t(0) =
~ ( t + 0) Consequently, if the function x(.) in Axiom A is such t h a t x0 = ~, t h e n xt = ~t We observe t h a t ~t is well defined for t < 0 since the d o m a i n of q~ is ( - o c , 0] We also note t h a t in general ~t ~/3; consider, for example, functions of the t y p e x ' ( t ) = ( t - #)-~X(~,0], # > 0, where
2¢(~,0 ] is the characteristic function of (/z, 0], # < - r and ap E (0, 1), in the space PCr x LV(g; X)
Additional terminologies and notations used in this p a p e r are s t a n d a r d in functional analysis
In particular, for B a n a c h spaces (Z, I1" [[z), (W, H" [[w), the n o t a t i o n £.(Z,W) stands for the
Banach space of b o u n d e d linear operators from Z into W and we a b b r e v i a t e t o £ ( Z ) whenever
Z = W Moreover, Br(x, Z) denotes the closed ball with center at x and radius r > 0 in Z
T h e p a p e r has four sections In Section 3 we establish the existence of mild solutions for system (1.1)-(1.3) Section 4 is reserved for examples
To conclude this section, we recall the following well-known result for convenience
THEOREM 2.1 (See [26, Theorem 6.5.4].) Let D be a dosed convex subset o f a Banach space Z and assume that 0 E D Let F : D + D be a completely continuous map Then, either the map F has a fixed point in D or {z E D : z = AF(z), 0 < A < 1} is unbounded
Trang 43 E X I S T E N C E R E S U L T S
In this section, we establish the existence of mild solutions for the impulsive abstract Canchy problem (1.1)-(1.3) To prove our results, we always assume t h a t p : I x B , ( - c % a ] is continuous and t h a t ~) and f satisfies the following conditions
H ~ Let TO(p-) = { p ( s , ¢ ) : ( s , ¢ ) • I x B, p ( s , ¢ ) < 0} The function t ~ ~)t is well defined from TO(p-) into B and there exists a continuous and bounded function J ~ : TO(p-) , such t h a t I]~tn~ _< J~(t)lI~llB for every t e TO(p-)
H1 The function f : I x / 3 , X satisfies the following conditions
(i) Let x : ( - o o , a] X be such t h a t x0 = ~ and xlI • :PC T h e function t -~
f ( t , xp(t,~,)) is measurable on [O,a] and the function t * f ( s , xt) is continuous on T~(p-) U [0, a] for every s • [0, a]
(ii) For each t • I, the function f ( t , .) : 13 X is continuous
(iii) There exists an integrable function rn : I ~ [0, co) and a continuous nondecreasing function W : [0, ~ ) ~ (0, cx~) such that
IIf(t,¢)ll _< m(t)W(llCH~), ( t , ¢ ) • I x B
REMARK 3 We point out here t h a t condition H~ is frequently satisfied by functions t h a t axe continuous and bounded In fact, assume t h a t the space of continuous and bounded functions
C b ( ( - ~ , 0], X) is continuously included in B Then, there exists L > 0 such t h a t
r sup0<_0 ll¢(8)ll ,,,,,
I l C t l l B ~ ]~]-~ il~,.m t < o , ¢ 5 0 , C e C b ( ( - - c ~ , O ] : X ) (3.1)
It is easy to see t h a t the space Cb((-cx~,O],X) is continuously included in :PCg(X) and :PC°(X) Moreover, if g(.) verifies (g-5)-(g-6) and g(.) is integrable on ( - c ~ , - r ] , then the space Cb(( oc, 0], X) is also continuously included in :PCr x LP(g; X ) For complementary details re- lated this matter, see Proposition 7.1.1 and Theorems 1.3.2 and 1.3.8 in [24]
REMARK 4 In delay differential equations without impulses, the function f is usually assumed
to be continuous This turns out to be a poor choice of a condition for an impulsive system since
in general, the function t ~ xt is discontinuous This fact is the justification for condition Hi-(i) Let x : ( - 0 % a] * X be a function such t h a t x, x' • :PC If x is a solution of (1.1)-(1.3), from the semigroup theory, we get
/:
x ( t ) = T ( t ) ~ , ( o ) + T ( t - s / f (s, xp( )) e s , t • [o,t~),
which implies t h a t
fo"
x(ti-) = T(t~)~(o) + T(t~ - ~)I (s, ~p(,,~.)) as
By using t h a t x ( t +) = x(t-~) + Ii(xt~), for t • (tl, t2) we find t h a t
x(t) = T ( t - t l ) x ( t +) + T ( t - s ) f (s, %(~,~.)) ds
= T ( t - t~)(x(t-~) + I~(xt~)) + T ( t - s ) f (s, %( )) ds
= T ( t - t l ) [T(tl)T(0) + ~ t , T ( t l - s ) f (s, xp( )) ds + Ii(xt~) 1
+ T ( t - s ) f (s, xp( )) ds
1
= T(t)~o(o) + T ( t - ~ ) f ( s , x~( )) as + T ( t - t~)Z~(x~)
Trang 5R e i t e r a t i n g these procedures, we can prove t h a t
L t
x(t) = T ( t ) ~ o ( 0 ) + T ( t - s ) f ( s , xp(,:,)) d s + E T ( t - t i ) I i ( x t ~ ) ,
0 < t i < t
t E I
This expression motivates the following definition
DEFINITION 3.1 A function x : ( - o o , a] -~ X is cMled a mild solution of the abstract Cauchy problem (1.1)-(1.3) if xo = ~, x:( ) E 13 for every s E I a n d
x(t) = T ( t ) ~ ( 0 ) + T ( t - s ) f (s, xp( )) ds + E T ( t - ti)Ii(xt,), t E I (3.2)
o<Q<t
T h e next l e m m a is proved using the phase spaces axioms
LEMMA 3.1 Let x : (-c~, a] ~ X be a function such that xo = ~ a n d xl[o,~l E PC Then
IIx~ll~ < (M~ + J~)II~IIB + K~ sup {11~(0)11; 0 E [0, ma~(0, s}]}, s ~ n ( O - ) U [0, a],
where Jo ~ = suPten(p_ ) J~'(t), M~ = s u p t e i M(t) and K~ = supte~ K(t)
REMARK 5 I n the sequel of this work, M~ = supte[o,~ ] M(t) and K~ = supte[o M K(t)-
Now, we can establish our first existence result
THEOREM 3.1 A s s u m e that there are constants Li, i = 1, 2 , , n, such that
/ f
then there exists a mild solution of (1.1)-(1.3)
PaOOF Let Y = {u E P C : u(0) = ~(0)} endowed with the uniform convergence topology On the space Y, we define t h e o p e r a t o r r : Y ~ Y b y
L t
rz(t) = T(t)~(O) + T(t - s)f(s,2p(s.~,))ds + ~ T(t - ti)Ii(Sa),
0<t~<t
t E I ,
where 2 : ( - c ~ , a] ~ X is such t h a t 2o = ~ and • = x on I F r o m A x i o m A, t h e strong continuity
of (T(t))t>_o and our a s s u m p t i o n s on ~ and f , we infer t h a t F x E 7)C
Next, we prove t h a t t h e r e exists r > 0 such t h a t P ( B r ( 0 , Y)) C Br(0, Y) If we assume this p r o p e r t y is false, t h e n for every r > 0 there exist x ~ E B~(0, Y) and t ~ E I such t h a t
r < IIr:(tr)ll Then, b y using L e m m a 3.1 we find t h a t
," < I I r : ( : ) l l
_ ~'/HII:IIB + Mfo m(s)W(ll~.(.,<~)~)ll :.B) ds + ~ (LdI~,,IIB + 111dO)ll)
i=I
fo:
+ IVI E (Li((Ma + Jg)ll~llB + K~r) + IIZi(0)ll),
i=i
Trang 6416 E HERNA.NDEZ et al
and hence,
I < IgIK= 1 f m(s) ds + L{ ,
which is contrary to our assumption
Let r > 0 be such t h a t F(Br(0, Y)) C B~(0, Y) Next, we will prove t h a t F is a condensing
m a p on Br(0, Y) Consider the decomposition F = F1 + F2 where
~0 t
F i x ( t ) = T(t)qo(0) + T ( t - s ) f (s, ~p(,,~,)) ds, t E I,
O<t~<t
STEP 1 T h e set FI(B~(O,Y))(t) : { F i x ( t ) : x E B~(0, Y)} is relatively compact in X for every
t E I The case t = 0 is obvious Let 0 < e < t < a If x E B~(O,Y), from L e m m a 3.1 it follows
t h a t
II~.(t:,)llB _< r* := (M~ + Jo~)ll:llB + K :
a n d so
O r T ( r - s)f(s, ~p(,,~)) ds _< r** : : lf/IW(r*) fo a re(s) ds,
Consequently, for x E B~(0, Y), we find t h a t
T E - [
F i x ( t ) : T(t)~(O) + T(e) T(t - e - s ) f (s, ~:(~,~s)) ds + T ( t - s ) f (s, ~p(~,~s)) ds
£
E {T(t)~o(0)} + T(e)B~ (0, X ) + C~,
where diam(C() _< 2/V/W(r*) f:_(re(s) ds, which proves t h a t r ~ ( B r ( 0 , Y))(t) is relatively compact
in X
STEP 2 T h e set of functions F ~ ( B r ( 0 , Y ) ) is equicontinuous on I Let 0 < t < a and e > 0 Since the semigroup (T(t))t>o is strongly continuous a n d F1 (Br(0, Y ) ) ( t ) relatively compact in X, there exists 0 < 6 < a - t such t h a t
] ] T ( h ) x - xll < e, x E FI(Br(O,Y))(t), 0 < h < 5 Under these conditions, for x E B~(0, Y) a n d 0 < h < 6, we get
f t+h
Ilhx(t+h)- rlz(t)ll < I](T(h)- I)r:(t)]l + T(t-s)f(s,.2p(s:s))ds
.It
< e + M W ( r * ) / re(s) ds,
Jt
where r* : (Ma + J0~)lI:lIB + Kar This proves t h a t F I ( B r ( 0 , Y ) ) is right equicontinuous at
t E (0, a) Similarly, we can prove the right equicontinuity at zero and the left equicontinuity at
t E (0, a] Thus, rl(Br(0, Y)) is equicontinuous on I
STEP 3 T h e m a p F I ( ' ) is continuous on B~(O,Y) Let (xn),eN be a sequence in B~(0, Y) and
x E B~(O,Y) such t h a t x n + x in PC From Axiom A, it is easy to see t h a t ( ~ ) , + 5s as
n + co uniformly for s E ( - c o , a] From this fact, condition H1 a n d the inequality
llf (s, :.(.,(:>.>) - f(., :.(.,:.))II -< llf (', 7%(.,(::).)) - : (., :.(.,(:>.>) If,,
+ II: (',:.(.,<::).>) - : (., :.(.,:.0 II.,
Trang 7we infer t h a t f(s,x-~p(~,(~-w),)) ~ f(~p(~,~,)) as n * c¢, for every s • I Now, a standard application of the Lebesgue dominated convergence theorem proves t h a t F i x = * F i x in Y Thus, F~(.) is continuous
STEP 4 T h e map F2(.) is a contraction on B~(0, Y)
T h e assertion follows directly from (3.3) and the estimate
n
][Fzx - Fzyllgc <_ Kal~/I E L~l[x - yllpc
i = 1
T h e previous steps prove t h a t F is a condensing operator from B~(0,Y) into Br(0, Y) Now, the existence of a mild solution is a consequence of [27, T h e o r e m 4.3.2] This completes the proof
THEOREM 3.2 Assume that p(t, ¢ ) <_ t for every (t, ¢ )
continuous, and that there are constants c~, i = 1, 2,
1 2 for every ¢ • 13 I f # = 1 - K a ~ / i ~ = l C i >
where
|
• I × B, the maps I~ are completely
, n , j = 1,2, such that [[I~(¢)[1 <
0 and
K ~ M re(s) ds <
W ( s ) '
C = 1 M~ + J~o + M H K a Ilqoll/3 + K~,~/ c
then there exists a mild solution of (1.1)-(1.3)
PROOF Let y : ( - c ¢ , a] ~ X the function defined by y(t) = ~(t) on ( - c ¢ , O] and y(t) = T(t)~(O)
on [0, a] On the space
~ c = ( ~ : ( - ~ , a ] -~ X ; ~ o = 0,~l[O,al e ~C}
endowed with the norm I1" live, we define the operator F : B79C ~ BT)C by
Fx(t) = f o T ( t - s ) f ( s , ~ p ( ~ , ~ ) ) d s + ~ T ( t - t i ) I i ( s t , ) , t e [0, a],
0 < t ~ < t
where ~ = y + x on ( - c ¢ , a] In order to use Theorem 2.1, we will establish a priori estimates for the solutions of the integral equation z = AFz, A E (0, 1) Assume t h a t x x, A • (0, 1), is a solution of z = )~Fz If a~(s) = sup0e[0,~l [IxX(0)ll, then from L e m m a 3.1 we have t h a t
IIx~(t)ll_< ~ m(s)W (M~+ J~')II~liB+K~ sup II~(0)ll ds
o~[o,~1
+ i f / ~ c~ ( M ~ + g ) l l ~ l l ~ + N o sup IIz~'(o)ll + ~ y - ~ c ~
< IVI ~otm(s)W ((M~ + J~ + -~/IHKa)l]~llt3 + Kaa;~(s)) ds
since p(s, ( = x ) ) _< s for each s • I If ~x(t) = (Mo + d~ + ~HKo)II~IIB + Koch(t), we obtain
t h a t
~ x ( t ) < _ ( M a + J ~ + ~ I / I H K ~ ) I I ~ I I B + K a 2 V I ~ c 2 + K ~ I V I m ( s ) W ( ~ ( s ) ) d s
i = l
+go~ ~ c~(t),
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a n d so,
~ ( t ) <_ ~ M~ + J~ + iQHK~ I1~11~ + Ka~I E c2i + m(s)W(~X(s)) ds
Denoting by t3~(t) the right-hand side of the last inequality, it follows that
Zi(t) < K ° M m ( t ) w ( z x ( t ) ) ,
#
and hence,
f ~(t) ds KaMfo~ / 2 ds x(o)=e W(s) <- ~ re(s) ds < w(~)'
which implies that the set of functions {/3~(.) : A • (0, 1)} is bounded in C(I : N) Thus, {x~(.) : A • (0, 1)} is bounded in B'PC
To prove that the map F is completely continuous, we introduce the decomposition Fx =
F i x + F2x where (Fix)0 = 0, i = 1, 2, and
~0 t
f i x ( t ) T ( t - s ) f (s,e.(s,~)) ds, t • z,
F2x(t) = E T ( t - ti)I,(~t,), t • I
O < t i < t
From the proof of Theorem 3.1, we deduce that I'1 is completely continuous Next, by using Lemma 2.1, we prove that F2 is also completely continuous The continuity of F2 can be proven using the phase space axioms From the definition of F2, for r > O, t • [t~,t~+l] N (0, a], i > 1, and u • B~ = B~(O, BPC), we find that
i
2 T ( t - tj)Ij(B~.(O;X)), t • (t~,t~+l),
j = l
r2u(t) • 2 T(t~+l - tj)Ij(B~.(O; X)), t = t~+~,
j=0 i-1
~2 T(t~ - tj)Xj(B~.(O;X)) + I~(B~.(O;X)), t = t~,
j = l
where r* := (Ma + gM)ll~lls + Kor, which proves that [r2(B~)]~(t) is relatively compact in
X, for every t • [t~,ti+l], since the maps Ij are completely continuous Moreover, using the compactness of the operators Ii and the strong continuity of (T(t))t>_o, we can prove that IF2 (B~)]i
is equicontinuous at t, for every t • [ti,ti+l] Now, from Lemma 2.1 we conclude that F2 is completely continuous
These remarks, in conjunction with Theorem 2.1, show that F has a fixed point x • B'PC
Clearly, the function u = x + y is a mild solution of (1.1)-(1.3) The proof is ended |
4 E X A M P L E
In this section, X = L2([0,~r]) and A : D(A) C X ~ X is the operator A f = f" with domain
D(A) := {f C X : f " E X, f(0) = f(Tr) = 0} It is well known that A is the infinitesimal generator
of a compact C0-semigroup of bounded linear operators (T(t))t>o on X Moreover, A has discrete spectrum, the eigenvalues are - n 2, n E N, with corresponding normalized eigenvectors
Trang 9Existence Results the set {zn : n E N} is an orthonormal basis of X and
T(t)x = ~ e-~2t(x,z~)z~,
n = l
for every x E X
Consider t h e differential
419
system,
= ~ ~u(t, ~) + fl(s - t ) u ( s -¢(Hu(t)ll),~)ds, (4.1)
o o
L
Au(tj, ~) = ~/j (tj - s)u(s, ~) ds, j = 1, , n, (4.4)
for (t,~) E [0, a] x [0,~], where ~ C B =PCo x L2(g,X) and 0 < tl < < t~ < a are pre-fixed
To study this system we impose the following conditions
(i) T h e functions/3 : R -~ IR, a : ~ -~ N + are continuous, b o u n d e d and
Lf = ( ~ ~2(S) ds) 1/2 <
(ii) The functions ~,j : R ~ ~ are continuous and
LJ := ( L (~J(-s))2 ds) 1/2 <
for every j = 1 , 2 , , n
By defining the functions p,f, Ij : B -* X by p ( t , ¢ ) = t - a(ll¢(0)ll),
f(¢)(~) = fl(s)~2(s, ~) ds
o ~
I j ( ¢ ) ( ~ ) = 7j(-s)¢(s,~)ds, j = 1 , 2 , , n ,
we can represent system (4.1)-(4.4) by the abstract impulsive C a u c h y problem (1.1)-(1.3) Moreover, the maps f, Ij, j = 1, ,n, are b o u n d e d linear operators, Ilflln(u,x) <- LI and II/jNL(B,x) < Lj for every j = 1 , ,n
PROPOSITION 4.1 Let p C B be such t h a t H ~ is valid and t -~ ~t is continuous on T~(p-) If
((; 1+ ag(T) d7 ) ) a L l + L~ <1, then there exists a mild solution of (4.1)-(4.3)
PROOF Let x : ( - o c , a] ~ X be such t h a t x0 = ~ and x]i 6 ~C A straightforward estimation permit to prove t h a t t * f(xt) is continuous on n ( p - ) x [0, a] and t h a t t ~ f(zp(t,x,)) is continuous on [0, a] Now, the existence of a mild solution for (4.1)-(4.3) is a consequence of
From R e m a r k 3, we obtain the next result
COROLLARY 4.1 Assume t h a t ~ E B is continuous and bounded on (-c~, 0] Then, there exists
a mild solution of (4.1)-(4.3)
Trang 10420 E HERN~NDEZ et al
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