FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY IN BANACH SPACES

. In this paper, a definition of the fundamental operator for the linear autonomous functional differential equation with infinite delay in a Banach space is given, and some sufficient and necessary conditions of the fundamental operator being exponentially stable in abstract phase spaces which satisfy some suitable hypotheses are obtained. Moreover, we discuss the relation between the exponential asymptotic stability of the zero solution of nonlinear functional differential equation with infinite delay in a Banach space and the exponential stability of the solution semigroup of the corresponding linear equation, and find thatthe exponential stability problem of the zero solution for the nonlinear equation can be discussed only in the exponentially fading memory phase space.


INTRODUCTION
We consider the functional differential equation with delay -f(t,x,), ( where x,(0) x(t + 0),-o < 0 0, and x takes values in a Banach space E. Because many phenomena in nature which vary in time can be written in the form of (1.1), the study of (1.1) has been a significant and interesting subject.Since the 70s, the theory of the functional differential equation with delay has been developed swiftly, a lot of important results have been obtained (see ]).However, as stated in [1], most of the papers dealing with this subject require that E be a finite-dimensional space.Therefore, the case when E is an infinite-dimensional space must be researched further.
The main motivation for this paper was a desire to take a step in this direction.We investigated the exponential stability problem for linear or nonlinear functional differential equations with infinite delay whenE is an infinite-dimensional space.A definition of the fundamental operator for the linear autonomous functional differential equation with infinite delay in a Banach space is given in Section 2, which si the generalization of the fundamental matrix (see [1,3]).In Section 3, we overcame successfully the difficulty caused by infiniteness of dimension of the space, and obtained some sufficient and necessary conditions for the fundamental operator to be exponentially stable in an abstract phase space which satisfy some suitable hypotheses.Some examples of the phase spaces are given in Section 4. In Section 5, we discussed the relation between the exponential asymptotic stability of the zero solution of a nonlinear functional differential equation with infinite delay in a Banach space and the exponential stability of the solution semigroup of the corresponding linear equation, and found that the exponential stability problem for the nonlinear equations can be discussed only in the exponentially fading memory phase spaces.

2.
THE DEFINITION AND PROPERTIES or THE FUNDAMENTAL OPERATOR Let E be a Banach space, denote by II" a norm in E, and by R, R/, R_ the real, the non-negative real, the non-positive real numbers, respectively.Let B be a linear vector space of functions mapping (_o%0] into E, and assume that B is a Banach space with the norm II" IL and satisfies the following hypotheses: <H2) x<0)II K x IL for all x in B and some constant K.
It is easy to verify that for each IEB, the solution of (2.1)x(t)-x(t,) exists uniquely for lER/.Moreover, by (HI), the solution operators T(t), O, defined by (T(t)C)(O)-x,(O)-x,(O,) for 9IEB, is strongly continuous semigroup of bounded linear operators on B. If L -'0, we denote the solution semigroup by S(t).
By the arguments similar to the Theorem 4.4 in [8], we have LEMMA 2.1.There exists a real to, such that exp(:k -)b lEB for b tEE and Re.> tot, and it is an analytical function of k.
LEMMA 2.2.There exists a constant oJ o such that (M L (exp( o)))-exists forRe.> oJ, where L(exp( .))is a linear operator from E to E defined by L(exp( .))bL(exp(-)b) for b E. PROOF.If (HI) (H2) (H3) hold, by (H3) we have that there exists O.oat.o, such that exp( ,)b II (b CE)is uniformly bounded for Re > t.Oo.So, by the boundedness of L and the Banach- Steinhaus theorem, we get that L(exp(.)) is uniformly bounded for Re.> t.o0.If follows that there is a sufficiently large co z co such that (L/" -L(exp(..)))exists for Re.> o.If (H1) (H2) (H4) hold, by (H4) we have for Re.Oo and b E, where o3 > co.According to the same reason as above, we obtain the conclusion.
Then i) There exists real Ix > e.o, such that A-X()= i exp(-Lt)X(t)dt for ii) X(t) is a continuous function of in [0,oo).
iii) For any e > 0, there exists a C(e) such that IIx(/) II C(e)exp(vt + 0t for 0.

THE EXPONENTIAL STABILITY OF THE FUNDAMENTAL OPERATOR
The fundamental operator X(t) is said to exponentially stable if there exist positive numbers G and r, such that x(t)I1": G exp(-rt) for 0. (H6) The operator-value function L(_,.01(')) mapping ER/ into B(E,E), defined by L (Z-,,01('))b L (Z-,.0(')b) for b E E, is left continuous, where B(E,E) stands for the space of all bounded linear operators form E to E.
Set Y,()b X,(o)b _,.01(-)b forb E, then by (HI), Y,()b B. Thus, by (H5),L(X,()b)is well defined for > 0 and b E. Thereby, by virtue of i) and the definition of the fundamental operator of (2.1), for any b E, taking the Laplace inverse transforms of the two sides of A-(X)b [b / L we have X(t)b-b+ f L(X(s+O)b)ds-b+ I L(X,)bds fort>0, -0 where 0 ER_.It implies X(t) is the solution of (3.1), so ii) holds.REMARK, ii) of Lemma 3.1 implies that the fundamental operator is, the generalization of the fundamental matrix defined by ( 40) and (41) in [1].THEOREM 3.2.Let B satisfy (1-11)-(1-13) (H5), (1-16) hold, L(exp(X -)) be analytic for ReX > 0 and continuous for ReX O, A-I(X) exist for ReX > 0 and there exist positive numbers W, Q, N and M such that Ig(t)l W, In(t)lQ(g(t), M(t)is the functions in (H3), tR+), Then the fundamental operator of (2.1) is exponentially stable if and only if there is a positive constant such that: i) A-I(X) can be extended analytically to the half plane ReX > -e, ii) Lira sup a-'(v + io,)II-0 fo v,,v= (-, ), I--.**,,t iii) For any f E E" (the dual space of E), b E E and v > -e, there is a constant J such that In order to prove this theorem, we shall use the following well known result.LEMMA 3.3 [5, P. 409] Suppose that f(R)]f(x)]dx < oo and sup{f(x)} < o% and that the Fourier R transform ](y)is real-valued and non-negative.Then f(R)f(y)dy 2nsup{f(x)}.

R
The proof of eorem 3.2.Necessity..2) Suppose X(t) is exponentially stable, that is, there exist positive constants G and r such that 11X(t) 11< G exp(-rt) for a 0. Then forRe X > -r, the integral on the right side of (3.2) converges absolutely and defines an analytic function of X in the half plane ReX > -r.It is just the analytic extension of A-I(X) to the half plane ReX >-r, that is, i) holds.
For any-r <v < v < v < o% Hence, which implies that ii) holds.
SUFFICIENCY.By i) and ii), we get that f *'r ex)A-( From the aumptions of this eorem, we deduce that there is a constant D such that D -)II + Il for g 0. ( where T 0 is a constant.Since -(im) is continuous and bounded on [-T T (im)-L(exim "))- (i) i integrable on [T ) and (-,-r,] by means of L(exNim -))I1 and (3.4), making e of he arguments similar o those in the proof of ii) of the necessity of this eorem, and noting ha (integrating by s) im + 0, we obtain Lim IIx(t)II-o.Hence, there is a constant Do>0 such that IIx<t)IIDo.us, by virtue of Lemma 3.1, 3), IK(t)I W, IM(t)I Q, and IIXo IIN, we have <t)I1-1 L(<'))I1 W IlL IIDo +M IlL IIN D, Therefore, thanks to Plancherel formula [10] and iii), we obtain J( p))-' lll bll .(3.6) Taking to 0, then there is t > to with t-to s such that lf(X(fi)12 ex-2pt(e p))-I111 b In fact, if not, that is, for any [to, t0 + 1], If(S(t)b)12> exp(-2pt)J(rt(e-p)) -l f [I-ll b If-', thus, to+l i exp(2pt)l f(X(t)b)l 2dt I exp(2pt)l fCXCt)b )l 2dt J (t(e p))-' f I111 b , which is in contradiction with (3.6)., we can choose a strictly increasing number sequence {t=} with 0 and t..
This ends the proof of this Theorem.
REMARK 1.If E is a Hilbert space, then the iii) in Theorem 3.2 can be changed into the following iii') There exists a constant J such that A-'(v + ico)II do, for v > -e.
PROOF.Sufficiency is obvious.Noting that -'( + ico) I1'-= ep(-0: +ico)t)X(t)dt, exp(-(v +ico)s)X(s)ds where (-, -) stands for the inner product of E, by the arguments similar to those in the proof of the necessity of Theorem 3.2, we can prove the necessity.
RE1WhRK 2. It is clear that if we substitute (H3) with (H4) in Lemma 3.1, the conclusions of this lemma are also true, and if we substitute (H4) and .0K-)bIL N IIb (v is a constant, b E E, > 0) for (H3), [K(t)1 W, [g(t)l< Q and IIx0 IIN in Theorem 3.2, the conclusion of this theorem is true also.
THEOREM 3.4.Let B satisfy (H1)-(H3),/ a linear vector space (/ D B), L a linear continuous operator from/ to E, (HS') and (H6) hold.Let L(exp(X .))be analytic for ReX > 0 and continuous for ReXz 0, A-(X)exist for ReXz0 and there exists g such that (=xp0,-)) IIg.Suppose X(t)is the fundamental operator of equation (q(0) EB) and L<X,)II C sup X(t> ll,"here C i a constant."I'hn the cocluion of Lemma 3.1 and Theorem 3.2 t,0 hold. The proof is similar to the proof of Lemma 3.1 and Theorem 3.2, so, we omit it.According to the arguments similar to those in the proof [8, Proposition 6.4], we can obtain the following THEOREM 3.5 Let (H1)-(H3) or (H1)(H2)(H4) hold for B, there exist positive constant C and r such that L(exp(X .))canbe extended analytically to the half plane ReX>-r, and (,xp0, .))IIc fo any ReX >-r.Then the fundamental operator of (2.1) is exponentially stable if there exists a positive number r E (0,r) such that A-(X) exists in the half plane ReX -ft.

i-I
Clearly, B is a Banach space.From (4.1) it follows that L is a linear bounded operator, L and B satisfy (H1)-(H4), (H5')and 0-16).Furthermore, the corresponding fundamental operator satisfies I y((t)" i-x A'X(t -t i)+ i 'B(s)X(t -s)ds, t>O, IX(0 +) 1, X(t) O, < O Also by (4.1), we get that there is a constant C such that IlL(X,)IIc sup{llx(/)II, to}.It is easy to verify that the hypotheses of Theorem 3.5 are satisfied.Hence, if there exists a positive number rl E (0,r) such that A-I(:L) exist for Re), > -rl, thenX(t) is exponentially stable.EXAMPLE 3. Let B {q(0); q}(0) is bounded uniformly continuous function from R_ to E}, with the norm q}(')l" sup (0)II,Z is the operator of Example 2 as r 0. OR_ It is easy to prove that L and B satisfy the hypotheses of Theorem 3.4, therefore, X(t) is exponentially stable if and only if there exists a positive number e such that i), ii) and iii) of Theorem 3.2 hold.
If there exist positive numbers C and r such that x'# I1,< c xp<-at)II I1 for any q} B and > 0, then the phase space B is called to be an exponentially fading memory space, where II,-iqll w II w B and ap(0) (0) for any 0 E (-oo,-t]}. THEOREM 5.1.LetB be a phase space and the solutionx 0 of (5.1) is exponentially asymptotically stable.Then B is an exponentially fading memory space.
So, for 1 U(t 1 ex-(5-CNM)t).is fact implies that the conclusion of the theorem holds.
According to Bellmen inequality, we obtain thus, by CNM < ag, the solution semigroup of (5.3) is exponentially stable and the proof is complete.ACOWLEME.We are Nateful to our advir Profeor F. L. Huang for his generous guidance.