Semigroups and differential equations with infinite delay

Nonlinear Analysis, Theory, Printed in Great Britain

Methods

SEMIGROUPS

& Applications.

Vol. 5, No. 7, pp. 737-756,

0362-546X.81/070737-20 @ 1981 Pergamon

1981.

AND DIFFERENTIAL EQUATIONS DELAY

$02.00/O Ltd.

bZS8

WITH INFINITE

T. HAV_%RNEANU Universitatea “Al. 1 Cuza”, Seminarvi Matematic, Iasi-RX,

Romania

(Received 24 March 1980)

Key wor& and phrases: Differential neutral systems, infinite delay, Laplace transform, (C,) semigroup, resolvent of (C,) semigroup, Banach \pace.

1. INTRODUCTION CORDUNEANU [l-5] and Luca [13-143 h ave considered integro-differential delays of the form:

a(t) = (Ax)(t), a(t) = (W(t)

systems with infinite

t E R,,

(S,)

+ f(t),

t E R,,

k(t) = (Ax)(t) + f(t; x),

(9

t E R,,

(S,)

where A is the operator defined by the formula (Ax)(t) = y j=O

’ B(t -

A,x(t - hj) +

r) x(r)

dr,

teR+.

s0

(1.1)

In (1.1) (Aj}j’_“ois a given sequence of constant matrices of order n, {tj)j+=coo is an increasing sequence of real numbers (bounded or not), B(t) is a given matrix-function of order n. In [l-5], and [13-141, the authors have studied the existence, uniqueness and behaviour of the solution of the systems (S&S,). The obtained results are mainly based on a variation of constants formula, and on some theorems of harmonic and functional analysis. In papers [8,9, 151, linear autonomous functional differential equations of the form dx x = Lx*,

(E-1)

are studied, where x:[(T-r,0.+A]-+W”, x,: [-r,O]

+ V,

Odrd+co, (T < t < Cr+ A,

A>O, x,(e) = x(t + e),

eE[--,Ol,

and L is a bounded linear operator on some function space 92, into the n-dimensional complex vector space W. On this function space the solution operator T(t), t 2 0, of equation (E.l) is a strongly continuous semigroup of bounded linear operators. In these papers some results are obtained on the spectrum and the resolvent set of the inlinitesima1 generator of T(t). 737

738

T. HAI~KNEANU

In [8,10]

linear autonomous

neutral

systems, of the form

03.2) are considered

where

D and L are linear bounded

d z

x(t) -

unbounded because the problem: t

B&t

- hj)

j=l

A may be a linear

= Ax(t) + 2 1

x(t) = 4(t), where

on

C([-r;O],R").

C = In our paper. L is generally Datko [6] has considered

operators

A,#

unbounded

- hj),

operator

t 3 0,

j=l

on 6.

(1.2)

t E( - 03,0].

A is the infinitesimal generator of a (C,) {Aj}yz {Bj>yz are linear operators from L@'), semigroup of operators from L(X), and hj, j = 1, m are given positive numbers. We can assume that ‘fj < hj+ 1,j = 1, (m - 1). Using some results of semigroups of operators theory and some hypotheses on the operators {Aj>j”_ 1 and (Bj}j”_ r, Datko [6] has proved the existence and the uniqueness of the solution for problem (1.2). Other problems are considered in [7]. In the present paper we have solved the same problems for the case when the set of delays {h,t is infinite. 1,

1

2.

PRELIMINARIES

We shall use the following notations: %?= the complex plane; R = the real axis; R, = {tE R; t 3 O}; _'& = a Banach space with the norm I.1 ; function such that 3 ,tli=, x(t) E%}, C = C[R_,%] = {x1x:(-co;O]+& is a continuous and the norm in C is defined as (141/ = sup {J4(4(?

tE(-co;O]),

&EC.

If Y is a Banach space then L[I:Y] = [Y]is the Banach space of linear bounded operators from Y into itself with the norm 11.I). Let x: [O; + co) -+_E and S: [0, + co) + [%I be two functions measurable and strongly measurable respectively. It is known that if there exists real constants M, a, with M 3 0,such that Ix(t)1 d M ear, then x and S have Laplace

IlS(t)I/ < M e”‘,

transforms X(S) = Z(x)(s)

and S(s) = Z(S)(s),

and these transforms are defined for s E %?with Re s > CI.The inverse of Laplace transform will be denoted by _9?‘(?)(t). For the next section we shall be in need of the following known results: THEOREM

function

2.1. (See [ll].) Let {T(t); t > 0} c [X] respectively. If there exists

and x:

lim>[x(r) - x(t,)] f-f” t - t,

R, -+X be a (CJ-semigroup

= x’(t,) E-E,

and a

Semigroups

and differential

equations

139

with infinite delay

then the equalities [T(r)x(t) lim 1 f-f0 t - 4l lim ~ l PW(r) *+fo t - r0

-

- W,)x(r)]

T(t,)x(t,)]

= CT(t)x(t)l:=~“~

= AW,)x(r,)

+ T(t,)x’(t,)

are true. LEMMA2.1. (See [6].) If f: R, then the inequality

-+ [..?G] is a strongly

measurable

function

and

1)f(t))) < M ecu,

t E R,,

< [M”t”-l/(n

Ij[f”]*(t)II holds. Here [f”]*

-

l)!] eat,

Vn EN

f* . . . *f and “*” means the convolution ntimes satisfying the condition:

denotes

Let A be an operator

A: 9(A) c _% -+.% is the infinitesimal

generator

operation

of a (C,) semigroup

of two function.

of operators

from [XI. (2.1)

Therefore, these exists M, 3 1, and a1 E R, s.t. 11 T(t)11 < M, e”“. Let {Aj},!=~ and {Bj}j+=“o,be two sequences of constant operators from [X] such that: Range Bi E 9(A), Y

&$E[%],

VjEN

(2.2)

)IAjll < + 00,

(2.3)

llBjll

(2.4)

j=l ~ j= 1

<

+

co,

+c” I\AB,(I < + 00. j=l

Let {hj},?=y be a strictly increasing in the next sections.

sequence

3. EXISTENCE, UNIQUENESS, AND A LINEAR AUTONOMOUS

In the Banach

space E we consider

of real positive

with the initial

[

Our results are contained

THE

SOLUTIONS SYSTEM

OF

the system:

x(t) - 1 B,x(t - hj) = Ax(t) + j=l

numbers.

SOME PROPERTIES OF NEUTRAL DIFFERENTIAL

+cO

;

0.5)

1

‘c” A,x(t

- hj),

t 2 0,

(3.la)

j= 1

condition: x(t) = 4(t),

t E R,

I$ E C.

(3.lb)

One can easily establish that the problem (3.laH3.lb) can be written in an integral form. Indeed, using the hypotheses (2.3H2.5), and Theorem 2.1, for each solution x(t) of the problem (3.la)-(3.lb) we may write:

740

T. HA~KRNEANU

$

T(t -

a)x(o)

-

[

U)BjX(U - Il.)

+y T(t j=l

= - AT(t - u)x(u) + y

BjX(U - hj) + T(t - a)

$ x(a)

-

AT(t - a)

j= I

1

+f T(t - U)Bj

$ x(a

- hj) = - AT(t

- a)x(o)

j=l

1 +j=l ‘c” T(t -a)ABjx(a -hj) -tT(t -a)Ax(a) +j= ”7yt -a)Ajx(o -hj) 1

+ +f qt

&

- U)ABjX(U - hj) + T(t -

x(u) -

j=l

+f BjX(cJ - hj)

= - AT(t - a)x(a)

j= 1

1

=

T(t

-

a)

‘c” (Aj

+

ABj)X(G

-

hi)

)

[ j=l

that is ;

ir(t - 0)X(G) -

y

[

j=l

Integrating

the two members

x(t, 0,4)

j=

I(t -

G)BjX(G-

]

of (3.2) between

= ‘c” B,X(C - hj; 0,4)

+ r(t)

4(O) -

I

X(6 094)

=

$(t),

-

G)

+c” Bj4(4rj)

y [j=,

(x‘ij +

ABj)X(G

-

hj)

1.

tT(r 1 s -CT) +

t 3 0, t E (-

(3.3a) =J,

01, 4 Ec,

(3.3b)

which justifies the above mentioned statement. In the following, we shall study problem (3.3b) and we shall consider that any solution of this problem is a solution for the problem (3.lb). Now we can formulate the following result: 3.1. For every 4 E C, the system 2: R + --) S is a continuous map. THEOREM

(3*2)

0

j= 1

+ ABj)x(o - h,,O,@) do,

[ j=l

= T(t

0 and t, and using (3.lb) we obtain

I

1

y(Aj

$)

(3.3aH3.3b)

has a unique

solution

(3.3a)(3.lab

Z( .). Moreover,

: qt, 0, qb) =j= +c” Bjf$(L -hj) +T(t) 4(O) - +fBj@I( -hj) I +~~T(t-~)[~~(Ai+R~j)~(~-hi)]d.i =[O,$].

Proof: Let t E [O; hi] be arbitrarily becomes

1

fixed. Then, taking

[

into account

(3.3b), the relation

(3.3a)

j= 1

Therefore Z(t, 0,4) is determined for t E [0, h,]. Because T(t) and 4(t) are continuous on their domains, it follows that 3(t, 0, 4) is continuous for t E [O; h,]. Substituting $(t), in the right member of (3.3a) with Z(t,O, 4), we determine the solution 2 on [hi, 2/r,], and so on. Definition

3.1. Let x(t, 0,4) be the solution

of the problem

(3.3aH3.3b).

For t 2 to, we define the

741

Semigroups and differential equations with infinite delay

application

: (3.4)

x,(4, to) = {x0 + 0, t,, 41, (7E (- cfz; Ol>v and the map r: R, -+ [C], by the formula:

(3.5)

r(r) cp = x,(&O) = {x(t + e,o, 6)> c E ( - KJ; 01). PROPOSITION 3.1. The map z, defined

Proof: For proof, it is sufftcient

by the relation

(3.5), is a semigroup

on C.

to point out that the relation

r(t - qJx,“(4~0)= r(t) 6 i.e.

is true. Let t 2 t, 2 0. Then, we obtain

vf$ E c

= z(t)c#.& Vt 3 t, 2 0,

z(t - t,)r(t,)fJ

from (3.3a), with x,,(4,0)

instead

of 4, and from Theorem

3.1,

[ j=l t--to -y (Aj +ABj)X(cJ -t,-a)‘c” -hj, 0,X,oc~, 0)) -hj, 0,~) 1+s T(t 1do j=l‘jx(t,

‘(t - ‘09‘3 ‘,,(6, O)) -

+f B~x(t - ‘0 -

hj>

‘3

x,,(4Y

‘1)

0

from which, taking

into account

=

T(t

-

tJ

x(t,,

‘2

4)

j=l

(3.3a), we obtain

x(t - to,0, x&b, 0)) - +f B,+ - to - hj, 0, x,(+,0)) = W - to) j=l

or

x(t - $3 0, x,(+, 0))- +f B,x(t - to - $70, x,($,0)) j=l

= T(t) 4(O) - +f Bj4( - hj) j= 1

1

+~~T(t-n,[:Ti(Aj+~~j)x(u-hj,O,d)]du +~~T(~-~)[~~(Aj+‘~j)x(a-to-‘j,O,X~,!~,O)))d~~ (3.6) One can easily establish

that we have:

x(a - t, - hi, 0, X,(&J, 0)) = X*,(4>O)(a - ‘0 - hj) = X(t, + (a - ‘0 - hj), 024) = x(a - hi,O,r$),a~[O,tO].

(3.7)

142

T. HAVAKNEANU

From (3.5) and (3.7) we obtain

+c” B,x(t -

x(t - to,0, xt&$, 0)) -

From the uniqueness

[

of the solution

of the problem

into account

j=l

(3.3aH3.3b),

x(t - t,, 0, x&A 0)) = x(t, 0, $)> But, taking

1

to - hj, 0>xtoG#h 0))= Vt) $43 - y Bjd4- hj)

j=l

we have

t 3 r,.

(3.4) and (3.5), this means that

r(r - to) x,,(A 0) = r(t) 4,

i.e.,

r(t - to) r(t,) 4 = r(t) 6

Remurk 3.1. The hypotheses (2.2H2.5) ensure both the existence of the change of variables. The hypotheses and the continuity convergence of the used series.

of used integrals, and the validity of x(t, 0,4) ensure the uniform

THEOREM3.2. The map z: [O; + co) -+ [C] is a (CJ-semigroup

on C.

ProoJ: From (a)

the Definition

t(O) 6 = #,

3.1 we have

v6 E C.

Now, we have to prove that (b)

lim T(t) qh = qb, Vcj E C.

t-to+

Using formula

=

(3.5) we obtain

sup (x0 +

oc[mr,o]

D,

0,4) - WI < sup 1-a+ at[-1;0]

g,o,4) - W)) +

SUP I(Jw) E-f; OJ

d@)l.

(3.9) Because $(a) and x(t, 0, ) are continuous

at 0, we have

lim /If(t) f$ - $11 = 0, 1-*0+ Because r(t) is a (Co)-semigroup,

there exist two constants /r(t))) < M, e@‘,

Let c1be the infinitesimal THEOREM

3.3 The domain

generator

M,

3

1, and a2 E R, such that

t 3 0.

of z(t).

of E is the set of those 4 E C which satisfy the conditions:

(i) (ii)

there exists @‘, and 4’ E C; 4(O) E g(A);

(iii)

4’(O) = +c” Bjf$‘(-h,) j=l

V$ E C.

+ y j=l

Aj$(-hi)

f @(O).

(3.10)

743

Semigroups and differential equations with infinite delay

Proqf (i). Because

for cr < 0 we have a(#&> = lim $+(r@) fj)ro) - (P(G)] = lim $&(h t a) - Qjfc~)]= i@(o). h-+0+

h-+0+

(iij(iii). From (3.la) and (3.lb) it follows that 4(O) E 9(A), and

or #‘(O) - 5 B, qb’(- hj) = A&O) f j= 1

*c” Aj Cp(- hj), j= 1

which implies (iii). 4. A REPRESENTATION

FORMULA

FOR THE (3.3B)

SOLUTION

OF THE

PROBLEM

(3.3A)-

In this section we shah give a representation formula for the solution of the problem (3.3a)(3.3b). Because r(t) is a (C~)-semigroup, it follows that, for every $ E C, the solution x(t, 0, 4) of the problem (3.3#--(3.3b) has Laplace transform Z(‘(s, tp), By applying formally the Laplace transfo~ to both parts of equation (3.3a), taking into account the properties of Laplace tr~sform, and the fact that Z(T)(s) = (sl - A)-’ = &f(s, A), Re s > E~, (4.1) we obtain

(4.21 It is clear that the hypotheses (2.lH2.5) and the properties of Laplace transform ensure the existence of the two members of the equality (4.2). Let us consider the equation I _

‘Eme-%

j=l Denoting

Bj

-

(sl - A)- ’ y (Aj + ABj) e-““j S(s) = (sl - A)- l. 1 j= 1

B(s) = I -

y

.j=1

Bj eeshj,

Res > aI,

(4.3)

(4.4)

T. HAV~RNEANU

144

d(s) = (sl - A)-’

+f (Aj + ABl)e-““I,

Res > z1

(4.5)

j=l

and taking into account the hypotheses clj > max {a,,~,} such that

(2.2)(2.5)

it follows that there exists a positive

j/Z- B(s)/ d eehlRes

number

Re s > CL, j=l

from which it follows that lim

(I - B(s)) = 0,

(4.6)

sZ(S) = 0.

(4.7)

Res-r+cc

and lim Res++

The relation

(4.6) implies

the existence

z

of a number

111- B(s))) < 1, which implies

the existence

LX~> CI~,such that Res > a,,

of B- l(s) and the representation +EJ = I + c (I - B(s))k,

B-‘(s)

Res > Q.

(4.8)

k=l

Then for Re s > x4, we have I/B-

%I(

d 1+

cc” pk= &,

(4.9)

k=t

where )/I - B(s)]/ < p < 1,

for

Res > x4.

From (4.7) and (4.9) we obtain lim

d(s) B-‘(s)

(4.10)

= 0,

Res-+a

which implies the existence

of a number

u5 > CL~,such that

I)&(s) B-‘(s)/ Therefore

[I - d(s) B-‘(s)]-’

< 1,

Res > CX~.

will exist for Re s > z5, and moreover,

[I - d(s) B-‘(s)]

- ’ = I + y

[d(s)B-

it has the representation (4.11)

‘(s)]~.

k=l Using

the previous

notations S(S)

=

and results we can write equation

B-‘(s)[Z

- d(s)B-l(s)]-‘(sl

- A)-‘,

(4.3) in

the

form:

Res > a5.

(4.12)

Semigroups

and differential

equations

with infinite

745

delay

Substituting B(s) from (4.4) in (4.8) we obtain: (4.13) In the formula (4.13) we shall make the Cauchy product, then we shall group the terms with the same exponent of “e”. Finally, we shall order the exponents and we shall obtain 0 < h, = or < o2 < . . . < 0” < . . . , where fX ok = C nr’hi

(a formal series),

i=l r(“) E N u (O}, and only a finite number of ny’ are positive. Therefore, on + + 00, for n + + co. Then, the relation (4.13) can be written as Hke-“*“.

B-l(s) = I + ‘f

(4.14)

k=l Now,

we shall prove that the relation (4.14) is valid. Indeed, using our hypotheses, it follows that there exists a number Q > c(~,such that

1,:

Bjeeshj/l < (y

~~Sj~~)e~hlRes< C < 1,

(4.15)

Res 2 Us.

j=l

Then, from (4.15) we obtain JIB-l(s)\\ < 1 + y

Ck < + co,

Re s 2 Q.

k=l

The series ,zr ( x

eehjRes)

and

IPill eWhjRes>*(which is the Cauchy product of order k of the series +c” llBill j=

u

=

y(y k=l

are uniformly

(IB~JI ewhjRes)

and absolutely

convergent

1

for

j=l

Re s > CL~, and consequently we can write u

=

+xm

D~~-““R’S

k=l

=

y( +c” IIBj[l eehjRes)k. k=l

j=l

Then, using the formula (4.14), we obtain: (jB-l(s)(j < 1 + +c” (IH,[[ e-“‘kRes< 1 + y

Dke-OkReS < + co, for Re s > Q,

k=l

k=l

from which we have +CC

)]B-‘(s))] < 1 + 1 k=l

)]H,)J e-ON6 < + 00, for Re s > us,

(4.16)

T. HAVKRNEANU

746

Using (4.5) and (4.14) we now obtain d(s)B-l(~)

= (d - A)-*

(’

I(

y (Aj + ABJe-“hJ

j=l

+c” 8,ewmk”,

= (~1 - A)-’

H, eoks

I + y k=l

(4.17)

Re s B CZ~,

k=l

where fi, are constructed

to H,. Thus, we obtain

analogously

from which we derive the inequality

Let 1 be the characteristic and (4.17) we obtain

function

Z-‘[II-‘(s)(sZ

of R +. Then using a well-known

result (see [12]), from (4.14)

= T(t) -+- +c” x(t - o,)H,T(t

- A)-‘](t)

- ok),

(4.18)

k=l

LP[d(s)B-‘(s)](t)

=

y

x(t

-

Ok)

T(t

-

(4.19)

Ok)Icik.

k=l

Taking

into account

the previous

11Y491[B-1(s)(sZ - A)-‘](t)l( = M, e”6’[ 1 + y

reasoning,

from (4.18) and (4.19) we obtain

d M, eu6’ + M, eu6’,F

l/Hkll e-mka6] = M, east,

l(H,(I e-“kn6

t 3 0,

(4.20)

k=l

Ilk-‘[~(s)B-‘(s)](t)ll

< M, ea6’ y

llfikll e-n6wr = M, east,

t 3 0.

(4.21)

k=l

Next, we shall need the following LEMMA

4.1. Under

hypotheses

result:

(2.1H2.5)

the following

inequalities

are valid:

l)_P- ‘[(XI(s) B- ‘(s)y(sz - A)- ‘]@)I[ = 11 i’ 04”- l[(SQs) B- ‘(S,)/](D) -0

x

T(t - o)da

Proof: Taking,

II

< (Mj4M,/j!)tjeu6’,

in Lemma

2.1, f(t)

j = 1,2,. . .

= Y-‘[d(s)B-‘(s)](t),

(4.22)

M = M4,cx = ccc, and taking

into

747

Semigroups and differential equations with infinite delay

consideration

t J

the fact 11 T(t)(( < M, e**”< M, ea6’,we obtain

.=Y-‘[d(s) B- l(s)i](~) T(t - cr)do

< M,Mi (jh

ea6’ ea6r-z6agi- * da

.

0

MIM$i t’ east.

=

From (4.21) we obtain

12-fy (d(s)

a)‘](t)/l zgMl(E

B-l(s)~(sZ -

tj)=Ml

Tea6’

ea6’(eM4’- 1) G M, ea7’,

*

j=l

(4.23) where LX,= u6 + M, Using once again Lemma 2.1, with f(t) = ~-‘~~-‘(s)~(s)~~) iI ,,_T’

[

y

(~-l(~~~~s~~]~t~~ G z

j=l

J

and n = j, we get

Mij+ tjea6’ c Ft,

t 2 0.

(4.24)

1

Because of the equality B-l(s~~(s)~-l(s~~(s~

- A)_’ = (P(s)

&(s))i B-f(s)-@1 - A)-%,

(4.25)

we can write

I’ [ z-1

B--‘(s)

II

‘c” (~(s)B-l(s)~(sz-

A)_’] (t) i/=

p-‘[]g(B-‘wm~)i

j=l

. B- ‘(s)(sZ - A)-l

1II @)I

On the basis of (4.20), and the properties of Laplace transform, we obtain j/8-1 B- l(s) ‘c” (d(s)Bj=l /’ [

‘(s))i(sl - A)-’

]

(t)/( d M, ea7’,

t 2 0.

(4.26)

Using now the relations (4.1 I), (4.20), and (4.26), from (4.12) we obtain ll~-‘~~(s)]~~)~~ G M, e”*,

t 2 0.

(4.27)

De~~itfo~ 4.1. The map S(t) = _Y!-“(S(s))(t), for t 2 0, S(t) = 0 for t < 0 will be named the fun~fff~e~t~f trff~sfor~~~tfon for the system (3.3a)-(3.3b). PROPOSITION 4.1. The fundamental transformation S(t)x,

=

T(t)x,

-I-

‘c”

x(t

-

hj)

B,S(t

satisfies the identity

-hj)x,+SdT(t-~)[~~Xb-hji

j=l

. (Aj

+

AB,.)

S(cr

-

hj)

o

da.

(4.28)

P Proof. From Definition 4.1 and inequality (4.27) it follows that 64- ‘[S(s)](t) = S(t) exists, and

T. HAL~RNEANU

748

has Laplace transform. Applying to both members of the relation (4.3) the inverse Laplace transform, we obtain the equality (4.28). Let us suppose that we know T(t). Proposition 4.1 permits us to calculate S(t), step by step, on the intervals [nh,, (n + I)/+], Vn EN. PROFQSITION 4.2. The fundamental tion of the points

transformation

wk, k EN. Moreover,

is a strongly continuous map with the excep +a the map S(t) - 1 B,S(t - hj) is a strongly continuous j=

1

map on R,. Proof: The proof of the first affirmation is based on the formula (4.28). Because T(t) is a strong continuous map and because S(t) = 0, for t < 0, it follows that S(t) = T(t), t E (0, h,), which implies that S(t) is a strongly continuous map on (0, h,). Inductively, one can easily establish that S(t) is a strongly continuous map on every open interval (oj, oj+ l),j E N. The second affirmation of this equality

(because

the last term of (4.28) is a strongl;%ntinuous

PROFWSITION4.3. X(s, 4) defined by the formula X(s,$) =

S(s) f#J(O) - +f B.4(-h.) j=l

’

+S(s)[Jz sIhj Bj eeshj

The proof of this proposition (4.2) may be written as

.e-Fb$(c)da

- Bj#(-hj)

(4.2) can be written

map). as

‘c” J~“,je--‘“iio)do] ‘1 fjzl A. -SIX Je

+ S( )

s ees”4(o) dcr].

(4.29)

results easily if we observe

that, using (4.3) and (4.12), the relation

. 11

THEOREM 4.1. If &,E C is an absolutely (3.3aH3.3b) may be written as x(t, o, ~$1= [s(t) -

‘f BjS(t - hj) from the two members

results from (4.28), if we subtract

+f sft - hj) ~,]b(o)

continuous

then the solution

+ +c” Lo s(t - 0 - hj). j=l

j=l

function,

[Aj$(4

+

of the problem

Bj4’Wl

do.

u-h,

(4.30) Proof: Because e -shj

s emSo4(c)

da = $( -hj)

- e-“hjr$(0) + e-Shj

’ s -hj

e-‘“4(o)

da,

Semigroups

and differential

equations

with infinite

149

delay

and because the function = f$(t - hJ,

f-t)

t

E

co,hj],

f(t)

= 0,

t > hj

verifies the equality

s

1

0

’ S(t - 0 - hj)f,(o) do (s) = S(s) eBShj [S - hj if we take into account (4.29), and the fact that there exists 9

e-“”

4(c)

do,

-hj

44 (44) = =5- yqs,@Xt), we obtain x(t, 0,4) = 9-‘(Z(s,

++f5

@)(t) =

S(t) -

1

+f S(t - hj)Bj &O) j=l

0

j=

1

s(t

-

o

-

hj)

[Aj#(a)

+

da.

~~qt~‘(a)]

- hj

LEMMA2. If 4 E C, then the function Ii/(t) =

(4.31)

O S(t - hj - a)B,r$(a)da s -hi

is differentiable ae. on [0, + co).

t 1s

Proof: Taking into account (4.1 l), (4.12), and (4.18), one can easily establish that S(t)Bj may be written as

S(t)B, =

T(t) + y [

x(t - ok) H,T(t

Q(a)T(t - a)Bjda,

- 03. Bj +

(4.32)

0

k=l

where Q: [O; + co) + X is a strongly continuous map, with possible exception of the points “lj,j = 1,2,. . . So, if x0 E X and t E R - {oj>,!=~, then from (4.32) it follows that S(t)Bjxo is differentiable in t, and moreover, the equality $[S(t)Bjxo]

= S(t)AB,x,

+ Q(t)Bjxo

(4.33)

holds. Then from (4.3 1) we obtain

s 0

a - hj)Bj$(a)] da =

s 0

-I-

S(t - a - hj)ABj&a) da

-

kj

Q(t - a - hj)Bj+(a)da.

(4.34)

-hi

COROLLARY 4.1. Because S(t) and Q(t) have Laplace transforms, then Y(t) and (d/dt)[S(t) B,x,] have Laplace transforms. Definition

4.2. We define the derivative of S(t)B, as

T. HAV~RNEANU

750

$[S(t)Bj]

= S(t)ABj + Q(t)Bj,

for

where Q(t) is given by the formula

+oz

r

Q(t) = 2-l

c

B-‘(s)

(d(s)B-‘(s))~

k=l

t > 0.

J

(t),

(4.36)

From the Corollary 4.1 we see that (d/dt)[S(t)fj] has Laplace transform. Because S(t)B, is a continuous map on the open intervals (oj, wj+ 1),~ = 1,2,. . . , and because by virtue of the Corollary 4.1 (d/dt[S(t)Bj] has Laplace transform, we can write

s +CC

(‘1 =

yCas(t)BjJ

1 01

e-“‘$[S(f)Bj]

ewst $[S(t)Bj]

=

dt

0

dt + y

s 0

{Ok

k=2

e-“’ &S(t)Bj]

dt

WE-,

=e -‘O* lim S(t)Bj - Bj + +c” eeswr lim s(t)B, k=2 t+wr, t+wi lim S(t)B, + s9[S(t)Bj](s), 1-C& 1

(4.37)

for Re s > a7.

From (4.32) we obtain e --swk lim S(t)Bj - lim S(t)B, *-ro& t-oc Then (4.37) may be written 8

aS(t)Bj)

= -e-““kHkBj

(4.38)

I

(s) = &‘[S(t)B,](s)

- Bj -

y

edsok HkBj,

k=l

equivalently, s6P[S(t)Bj](s)

1

$S(t)Bj)

= 2

(s) + Bj + ‘f

[

’ [s’(s)f(s)](t) =

‘$S(t s 0 dt

Applying

= T-

(4.39)

emsok H,B,

k=l

f :R+ + X, where f is a continuous

Thus, if f(s) is the Laplace transform of the function on the open intervals (oj, oj+ 1), j E N, then y-

1

as

[ or

+c’ emswk-l k=2

function

’ [sU[S(t)](s)Bjf(s)](t)

- a)Bj]f(a)

dc f Bjf (t) + +c”x(t - ok)HkBjf(t

- ok).

the formula

(4.40) to the function fict) = Ht -

hjl[l

-

Xtt

-

hJ],

4

"2

we obtain Ye

‘[sS(s)Bj

(4.40)

k=l

eeShj j:,j

ees”c$(a) d,](t)

= @j

-[S(t d”,

- 0 - hj)Bj] . @(CT) da + Bj#(t - hj)

(4.41)

Semigroups and differential equations with infinite delay

x [l - x(t - hj)] + ‘c” x(t - oJ[l

- x(t - hj - w,)]. H,Bj4(t

751

(4.42)

- hj - WJ.

k=l

Definition

4.3. For j = 1,2,. . . we define the map Rj:[O, + 00) + [C, X]

as Ri(t)4 = _P-‘[

THEOREM 4.2. The solution

x(t, 0, 4) = S(t) #J(O)-

sS(s)Bi eeShj j”,;, e-‘“&a)

of the problem

+f Bj&41j)

(3.3aH3.3b)

+ +f

j=l

j=l

1

can be written

O [S

s(t -

CJ -

(4.43)

do](t).

in the form

hj) . Aj&a) da + Rj(t)c$ ,

V4 E C

1

-hj

(444) ProoJ: Taking

into account

(4.43) and observing

that for j = 1,2,. . . we have

~-‘[s(s)Aj eeshj{Ihj e-s”c#do)do](r) = sfhj S(t and applying

the inverse

Laplace

transform

to both members

(4.45)

hj - o)Aj$(a) do,

of (4.29), we obtain

(4.44).

Remark 4.1. If Aj = 8 E [%I, j B (m + l), then the results contained in the Theorems and 4.1-4.2 are the same as those obtained by Datko [6] (Theorems l-5).

3.1-3.3

We remark here that there exists a difference between the results on the system (3.la)--(3.lb), and the results on the systems from [2, 131, which contain a Volterra integral part.

Remark 4.2. If Aj, B. and A are constant of the solution

matrices

and_‘& = R”, we obtain

a representation

formula

oft h e problem: i

x(t) -

+f B&t

- hj)

j=l

= Ax(t) + +f A,x(t - hj),

1

t B 0,

j=O

where T(t) = eAt. 5. SOME

EXTENSIONS

OF

THE

RESULTS

Let A, Aj, B,, j = 1, co be operators satisfying part of (2.2). Denoting by Cj(t) the operators CJt) = T(t)AB,

IN

THE

the conditions

SECTIONS

3 AND

4

(2.1), (2.3), (2.4), and the first

t 2 0, jEN,

we suppose that: Cj(r) E [XI, r E R+,

Cj: R, + [ST], Cj is a strong

continuous

map on R,,

Vj E IV; (5.1)

T. HAV~~RNEANU

752

There is fi E R, such that

(5.2) j=l

We consider the problem (3.3a)-(3.3b) in the above-mentioned hypotheses. It is obvious that all results which were established in Sections 3 and 4 remain true. Using a reasoning similar to the one in the previous sections, we shall give a representation formula of the solution of the problem (3.3aH3.3b). Applying the Laplace transform to the two members of the equation (3.3a), we obtain 1

_

+c” e-shj

- (~1 -

B,

= (sl -

+‘f

j=l

A)-‘[,g AjeeshjJy,ij

Bj + S(Cj)(s)

+c” Bj@( 41~)

e --So@(a) da + 4(O) -

j=l

1s

1

eeShj

e _ ““@(a)

do.

A)-’

Aj

(5.3)

- I.,

Let us consider _

1

_Y(Cj)(s) eeshj X(s, 4)

y j=l

0

j= 1

1

+c” Aj eCshj -

A)-’

>

j= 1

the equation y

Bje-shj

(~1 -

j=l

y

II

9(Cj)(s) eeShj

emShj+ y

j=l

j=l

S(s) =

51.

(5.4)

With the notation +m

d(s) = (sl - A)-’

+CC

j=l

and taking

into account

(4.4), equation

(5.4) becomes = $I.

Taking into account the above-mentioned hypotheses (4.11) are true for Re s > y where y is large enough. Then, formula (4.12) becomes sts) = B-‘(S)

I + +f

s(s) = f B-

(5.6)

we can conclude

bW-

k=l

(5.7) can be written

(5.5)

j=l

[B(s) - d(s)]s(s)

The relations

C Z(Cj) ewShj,

1 A, eFshj +

that the formula

1

‘Hk ;. 1

(4.6)-

(5.7)

as l(S) + i

r

B-‘(S)(d(S)B-

‘b))iT

forRes

(5.8)

J-1

large enough.

Because 9-l [

AB-l(s) S

1

(t) = x(t)e’l + +f x(t - wk)Hke’-“” k=l

(5.9)

Semigroups

and differential

equations

153

with infinite delay

we have

k=l

where a6 > 1, or (5.10) where N, is a real positive constant. Similarly to (4.26) we can prove that

‘j=l

_!Z’-’ L ‘f B-‘(s)(d(s)B-‘(s)y

1

(t) < IV, eat,

for a 6 large enough, from which it follows that there exists _.%-‘[S(s)](t) = S(t), and moreover, I\S(t>IId NJePf,

(5.11)

t > 0.

The relation (4.29) becomes x(s, 4) = S(s){s(sI - A)-

‘[:F: Aj e-+

+ 4(o) - ”

Bj$(-hj)

j:kj e-‘“M

da

1s

1 [

0

+ s +f B, + U(Cj)(s) ewshj

(5.12)

-hi

j=l

j=l

and the Proposition 4.1 becomes

r

PROPOSITION 5.1. S(t) = piP [S(s)](t) satisfies the equation.

SW,

= e’ x0 + ‘? x(t - hj)BjS(t - hj)xo j=l

(5.13)

x(,~Cfi-Irj)drjS(t-rs)xoda PROPOSITION 5.2. The fundamental transformation exception of points {cII,},‘=“, .

S(t)x, is differentiable on R, with the possible

Proof. Because +a: S(t)x, = _S?-‘[S(s)](t)x,

= efxo +

C x(t - cok)Hkxoecrmwk)+ k=l

r’

Q(a)x, e’-“da,

x0 ES,

J 0

(5.14)

T. HAVKRNEANU

754

where Q(t) = 2-l

B-‘(s) ‘c” (&(s)B-r(s))i C

t),

t 2 0,

j=l

it follows that Q(t) is a strongly continuous map, with possible exception of points {II+},‘=“,, and then it follows that S(t)x, is differentiable at every point t E R - (oJ~}~?=~. We have = s(r)x, + QW,.

&@)+,I

(5.15)

COROLLARY 5.1. The maps 0 and

S(t - hj - a)~#+) do,

$(t) =

4 EC,

J -hi have Laplace transforms. Applying the same reasoning as that for obtaining formula (4.39), we find &)x0

= Y -$ S(t)x, [

1

(5.16) k=l

for Re s large enough. PROPOSI~ON5.3. Let f: [0, ; + co) --*E be a continuous function on the intervals (ol, oj+ r), j E N and suppose that f has the Laplace transform f(s). Then the equality p-

lCmml(t) = y- ‘fsTw~s~S(~)l(t) =

J~~,S~t

4f WI

(5.17)

do +f x(t - wJ&f(t - WJ+ xW(t) k=l

holds. The proof is immediate. We shall write (5.12) as ?i(s, 4) =

+

sS(s)[(sI - A)-(gAje-~h~~~,~je-~u~(,)d.ll

1 [j=I + d(s)

d(s)

5

(Bj + LY(Cj)b)).

s-hi 1 0

emshJ

e-‘O

&r)da

.

(5.18)

Denoting by fr(s) = (sl - a)-1[~I,4je-sh~

S_oce-s~~(~)do],

j=l 1 f&i) =j=l y(Bj + -hi

f&s) = (sl - A)_’ 4(O) - ‘c” Bjr$(_hj) ,

(5.19) (5.20)

[

9(Cj)(s)) eeshJ

’

s

e-”

$(a) du,

(5.21)

Semigroups

and differential

equations

with infinite delay

IS5

we obtain j-r(t) = Z-‘[.&(s)](t)

=

fi(t) = ~-‘C~~(s)l(t)

=

1

Aj~(a - h,)(l - ~(0 - hj)) do,

1,

Cjct - CJ- h,)+(o)da

(5.22) (5.23)

+ y Bj4(t - h,)(l - x(t - hj)).

(5.24)

j=l

Taking into account (5.17) and (5.22)-(5.24), from (5.18) we get

46 034) = y- lCswf~(~)l(o + L?- ‘[sS(s)f,(s)](t) =

+ 55 'bwf,(s)l(t)

s’d

S@- &fl(4 + f,(4 +

f&41

da

0dt

+

kFl x(t -

qyqYi(t - qJ + f,
Ql

(5.25)

+ fi(4 + f,(t) + f,CS~ t 2 0, x(t, 0, 4) = 4(t),

t d 0.

Therefore (5.25) is a representation of the solution of the problem (3.3+(3.3b) in the hypotheses of this section. Next, we shall give two examples. Let f be an analytic function in a neighborhood V of the spectrum of A. Let r be a closed curve which contains no point of the spectrum of A. We suppose that f is analytic on a neighbourhood of I-, and the point at cc is not inside of r. We define f(A)

= (27ri)-’

f(o)@1 - A)-’ do.

(5.26)

sr We consider the following functional differential equations :

$x(t)]= Ax(t) + y Fxbr - n), n=1

x(t)

+m c

-

n=l

We shall calculate the fundamental (4.3) becomes

(f(W 7

x(t .

transformations

.

-

n)

1 =

Ax(t).

(5.27) (5.28)

for (5.27) and (5.28). For (5.27) formula

(5.29) or, if Re s is large enough, we obtain s(,)

=

(sl

_

A)-

1 +

(5.30)

756

T. HAVARNEANU

Therefore the formula (4.28) becomes S(t)x,

= T(t)x,

+

+c”y ye-s” j(sl-

St?-’ [

G

j=l

~1

>

'

A)-U+l)

1

(t)~,

(5.31)

- A)-‘.

(5.32)

We consider now (5.28). The formula (4.3) becomes

On the basis of identity

5

[

I + (SZ - A)-IA

(f(A))”

=

s

e-Sn(.yz _ A)-’

L!I$E,

(5.33)

1

formula (5.32) can be written as 1

[

_

s

(

+f

ewns

~)(sz-A)-‘]S(s)=(sz-A)-’

(5.34)

?I=1

and so we have: s(s) = (~1 -

A)-l

+ y

_ A)-(j+l)

,yj

(5.35)

j= 1

for Re s large enough. From (5.35) it follows S(t)x, = T(t)x,

c

‘c”

+ dp- l j=l d y [ =1

epsntY!3$K)j(~, _ A)- (j+ l’ 1 (t)x,.

1

(5.36)

These examples are similar to those of Datko [6]. REFERENCES 1. CORDUNEANU C., Some differential equations with delay, Proc. EQUADIFF3 (Czechoslouak Con@wzce on Dij‘erential Equations. Bmo, 1972), pp. 105-l 14. 2. CORDUNEANUC. & LUCA N., The stability of some feedback systems with delay, J. muth. Analysis Applic. 51, 377-393 (1975). 3. CORDUNEANUC., Stability problems for some classes of feedback systems with delay, in Equations d&‘j. et &ct. nonlin&ires, pp. 398-405, Herman, Paris (1973). 4. CORDUNEANU C., Asymptotic behaviour for some systems with infinite delay, in Proceedings, KNO VVII, Berlin, (1975). 5. CORDUNEANU C., Functional equations with infinite delay, Boll. Un mat. Ital. (4) 11, f.3, pp. 173-181 (1975). 6. DATKO R., Linear autonomous neutral differential equations in a Banach space, J. di&I Eqns 25, 2, 258-274 1977. 7. HALANAY A., Differential Equations: Stability, Oscillations, Time Lugs (translation), Academic Press, Inc., New York (1966). 8. HALE J. K., Theory of Functional Dqjerential Equations, Springer, New York (1977). 9. HALE J. K. & KATO J., Phase space for retarded equations with infinite delay, Funkrialaj Ekwacioj 21, 1141 (1978). 10. HENRY D., Linear autonomous neutral functional differential equations, J. dif; Eqns 15, 106-128 (1974). 11. HILLE E. & PHILLIPS R. S., Functional analysis and semi-groups, Am. math., Sot., Providence (1957). 12. KREIN S. G., Lineijnie differencialnye urauneniju c Bunahom prostranstoe, Izd. “Nauka”, Glavnaja Redakcija Fiziko-Matematiceskoj Literatury, Moskwa, 1967. 13. LUCA N., Integrodifferential systems with infinitely many delays, Ann&i Mat. pura uppl. (IV), CXVI, 177-188. 14. LUCA N., The stability of the solution of a class of integrodifferential systems with infinite delays, J. muth. Analysis Applic. 62, 323-329 1979. 15. NAITO T., On linear autonomous retarded equations with an abstract phase space for infinite delay, J. d$/ Eqns 33, 7&91 (1979).