Summary on autonomous linear functional equation

Hale, J. K.

English

NASA Technical Reports Server

I 
1 '  * .r 
I 
A Class of Linear Functional Equations 
s GPO PRICE 
CFSTI PRICE(S) $ 
J. K. Hale* 
Hard copy (HC) /o+ / 
Microfiche (MF) /sa 
ff 653 July 65 
(THRUI 
I N66 28498 
(ACCESSION NUMPERI I 
6 
- IPA&Sl 
i 
> 
P- 73 7 ' 2 s  / z 
(NASA CR O R  TMX OR AD NUMBER1 
. 
*This research was supported in part by National Aeronautics and Space 
Administration under Contract No. 
United States Army Research Office, Durham, under Contract No. DA-31-124- 
ARO-D-270 and in part by the Office of Aerospace Research, United States 
Air Force, under AFOSR Grant No. 693-66. 
NGR 40-002-015, in part by the 
https://ntrs.nasa.gov/search.jsp?R=19660019208 2020-03-16T19:31:37+00:00Z
A Class of Linear Functional Equations 
by 
J.K.  Hale 
This r e F r t  is a s-amm-y of some unpublished r e s u l t s  of 
K. Meyer, C. Perel l6  and the  author concerning a c l a s s  of autonomous 
l i nea r  functional equations which includes as  spec ia l  cases autonomous 
l i nea r  funct ional  d i f f e r e n t i a l  equations of retarded and neu t r a l  type 
as wel l  as  funct ional  difference equations. 
Let Rn be a r e a l  o r  complex n-dimensional l i nea r  vector 
space of column vectors with norm I I and l e t  Cr([-ryO],Rn) be 
t h e  Banach space of continuous functions mapping [ - r , O ]  i n t o  Rn wi th  
t he  norm llcpll fo r  cp i n  C defined by llcpll = max (lcp(e>l, 8 i n  
[ - r ,o]) .  If g, f are  continuous l i nea r  mappings of c i n t o  R ~ ,  
then there  ex i s t  n X n matrices p,q whose elements a re  of bounded 
var ia t ion  on [ - r , O ]  such t h a t  
r r r 
I 
fo r  a l l  cp i n  Cr. We s h a l l  suppose t h a t  t he  measure p i s  nonatomic 
a t  0 and more spec i f ica l ly  t h a t  there i s  a continuous nondecreasing 
2 
function 6( s) ,  0 I s 5 r, such t h a t  6(0) = 0 and 
f o r  a l l  cp i n  Cr.  
For any CP i n  Cr, def ine u(Cp) = (p(0)-g((p). For any con- 
tinuous function h mapping IO,.) i n t o  Rn and any fixed element 
rp i n  Cr, consider t h e  func t iona l  i n t e g r a l  equation 
x ( t )  = rp(t) , -r 5 t 5 0 9 
( 3 )  
t t 
Cy and i s  defined by x (0)  = t where, for  each fixed t L 0, x i s  i n  
= x(t+e) , -r 5 e I 0. 
continuous function sa t i s fy ing  the  above r e l a t ion .  
t 
BY a solut ion of ( 3 ) ,  we w i l l  always mean a 
For g G 0, equation (3) i s  equivalent t o  t h e  func t iona l  
d i f f e r e n t i a l  equation of retarded type 
with t h e  i n i t i a l  condition a t  t = 0 given by Cp. If f 0 and h 0, 
equation (3) i s  a func t iona l  difference equation of retarded type and, 
3 
i n  par t icu lar ,  includes difference equations. For both f and g not 
i den t i ca l ly  zero, equation (3) corresponds t o  a retarded equation of 
neu t r a l  type. I n  fac t ,  formal d i f fe ren t ia t ion  of t h e  equation yields  
where k i s  defined a s  5 ( e )  = %(t+6),  -r 6 8 S 0. A l s o ,  i f  one 
begins w i t h  ( 5 )  and defines a solution with i n i t i a l  function Cp at  0 
t o  be a continuous function sat isfying ( 5 )  almost everywhere, then an 
. in tegrat ion yields  ( 3 )  w i t h  
t t 
r(p) = Cp(0)-g((p). 
This l a t t e r  remark i s  precisely t h e  reason f o r  considering 
the  equation ( 3 )  ra ther  than ( 5 ) .  If one attempts t o  discuss  equation 
( 5 )  di rec t ly ,  then t h e  f i r s t  problem t h a t  i s  encountered i s  a precise  
de f in i t i on  of a solut ion and a precise de f in i t i on  of the  topology t o  
be induced on the  space i n  which the solut ion w i l l  l i e .  Such a topology 
w i l l  necessar i ly  include the  f i r s t  der iva t ive  of  
whereas, i f  we consider equation ( 3 ) ,  t h e  simpler space 
employed. 
x i n  some way; 
Cr can be 
If h i n  (3)  i s  iden t i ca l ly  zero, we w i l l  say equation ( 3 )  
i s  homogeneous and, otherwise, it is nonhomogeneous. 
THEOREM 1. For any given cp i n  Cr, t he re  i s  a unique function x(Q) 
defined and continuous on [-r,-) such t h a t  x(q) s a t i s f i e s  ( 3 )  on 
h 
[O,m). Furthermore, t he re  i s  a constant I3 > 0 such t h a t  
T h i s  theorem i s  proved by using t h e  nonatomic property of 
p a t  0 together with t h e  contraction mapping pr inc ip le  t o  f i r s t  
show t h a t  ( 3 )  has a solut ion on a small i n t e r v a l  t o  t h e  r i g h t  of 
t = 0. 
then allows one t o  obtain t h e  estimate (6) f o r  
An appl icat ion of a r e s u l t  on t h e  continuation of t h e  solut ion 
t 2 0. 
I f  h i s  iden t i ca l ly  zero and x = x(cp) i s  a solut ion of 
t h e  homogeneous equation 
x = c p  
0 
(7) 
then it follows from the  uniqueness of t h e  solut ion t h a t  
continuous l i nea r  mapping of 
and xt(cp) s a t i s f i e s  t h e  semigroup property. I f  we define t h e  l i nea r  
operator T( t) by 
xt(cp) i s  a 
i n t o  Cr f o r  each fixed t B 0, ‘r 
5 
then we can prove 
TEIEOREX 2. The family of l i nea r  operators [ T ( t ) ,  t 2 0) mapping Cr 
i n t o  Cr i s  a strongly continuous semigroup on [O,m) with T(0) = I. 
If, i n  addition the  function 6( s) i n  (2)  s a t i s f i e s  lims q s y s  = 0, 
then t h e  inf in i tes imal  generator A of T ( t )  i s  given by 
and t h e  domain of A, g ( A )  
a continuous f i r s t  der iva t ive  and i ( 0 )  = g(@)  + f(rp). 
consis ts  of a l l  functions Cp i n  Cr with 
It i s  in t e re s t ing  t o  note t h a t  i f  Cp i s  i n  g (A) ,  then T ( t ) p  
i s  ac tua l ly  a continuously d i f fe ren t iab le  solut ion of the  funct ional  
d i f f e r e n t i a l  equation ( 5 )  w i t h  h = 0. 
It i s  easy t o  show t h a t  the spectrum of A, a(A), cons is t s  of 
only point spectrum and t h a t  h i s  i n  a(A) i f  and only i f  h s a t i s f i e s  
t he  cha rac t e r i s t i c  equation 
hB 0 A8 0 (9) de t  ah) = 0 , A(A)  = AI - I Ae dp(6) - I e dV(6) . 
-r -r 
Also, because A i s  a closed operator and a root h of (8) has f i n i t e  
mult ipl ic i ty ,  one can show t h a t  t he  resolvent operator (A-AI) has 
0 
-1 
r 
b 
a pole of f i n i t e  order a t  lo and, thus, t h e  generalized eigenspace of 
has f i n i t e  dimension. If R(A) and -@(A) denote, respectively, 
IO 
t he  n u l l  space and range of an operator A and the  generalized eigen- 
space of  
space 
Q 'Ef g ( A - h .  I)k each of which i s  invariant under both A and T ( t ) ,  
t Z 0. When Cr i s  decomposed i n  t h i s  way, we s h a l l  say Cr i s  
decomposed by Xo as C = P B, Q and wri te  any element cp i n  
as  cp = cp + rpQ , cpp i n  P, cpQ i n  Q. 
then there  i s  a matric B such t h a t  AO = OB and, thus, @ ( e )  = O(0)e , 
k i s  given by %(A-h.,I) , then it can be shown t h a t  t h e  
xO 
d ef k i s  decomposed as  a d i r ec t  sum of the  subspaces P = %(A-XoI)  , 'r 
0 
'r r 
P If O i s  a bas i s  f o r  P, 
Be 
, - r 6 8 5 0 ,  -r 5 8 5 0. Also, one eas i ly  shows t h a t  T( t )QP(B)  = O(0)e B( t+e)  
which implies t h a t  t h e  solutions of (3) on t h e  generalized eigenspace 
of a solution of (9) can be defined on 
t h e  semigroup T ( t )  
ordinary d i f f e r e n t i a l  equation. 
t h e  same r o l e  as t h e  Jordan canonical form i n  ordinary d i f f e r e n t i a l  
equations. 
(-w,m) and t h a t  t h e  act ion of 
on t h i s  subspace i s  e s sen t i a l ly  the  same as an 
The decomposition outlined here plays 
I n  t h e  applications,  it i s  necessary t o  have an exp l i c i t  
representation for t he  project ion operator E associated with the  
above decomposition. This can be obtained from t h e  formula 
A 0  
I E cp = - ./ (A-XI)-+pdh. 
hO *= c 
where c i s  a c i r c l e  i n  t h e  complex plane which contains no point i n  
7 
a ( A )  except )Lo. As i n  the  case of retarded funct ional  d i f f e r e n t i a l  
equations ( see  [l] o r  [ 2 ] )  t h e  projection operator EA can a l so  be 
obtained i n  the following way. Rnl be the  n-dimensional r e a l  or  
0 
Let 
complex space of row vectors and define t h e  operator 
B ( A * )  C C([O,r],R n ) given by a l l  $ i n  C([O,r],Rn) which a re  con- 
A* w i t h  
0 0 
-r -r 
tzcuzusl;. differentizble w i t h  +(n) = +(-qaU(e) - J + q a q ( e )  a d  
f o r  $ i n  g ( A * ) ,  
For Jr i n  g ( A * ) ,  cp i n  g ( A ) ,  if follows t h a t  ( $ , A s p )  = (AV,cp). To  
obtain the  project ion operator 
0 = ((pl, ...cp ) be a bas is  f o r  t h e  generalized eigenspace P = %(A-XoI) k 
of Xo and l e t  Y = col(Jrl, ...,$ ) be a bas i s  f o r  
EA cp = 0(Y,cp) f o r  a l l  cp i n  C([-r,O],Rn). 
EA, one proceeds as  follows: i f  we l e t  
0 
P 
%(A*-XoI)k, then 
P 
0 
Another important re la t ion  i n  ordinary d i f f e r e n t i a l  equations 
i s  t h e  var ia t ion  of constants formula. By using t h e  f a c t  t he  solut ion 
. 
8 
x*(t,h) of (3) w i t h  cp = 0 i s  a continuous l i n e a r  mapping of 
C ( [ O , t ] , R n )  i n t o  Rn and t h e  Reisz representation theorem, it 
follows t h a t  
where W ( t , t )  = 0, W ( t , s )  i s  of bounded va r i a t ion  i n  s f o r  s i n  
[ O , t ]  and W(t,s) i s  continuous from t h e  r i g h t  i n  s f o r  s i n  
( 0 , t ) .  One can a l so  show t h a t  W ( t , s )  i s  continuous from t h e  r i g h t  
a t  s = 0, W( t, s )  = W ( t - s , O )  and - W ( t , O )  i s  t h e  matrix solut ion 
of (3) with Cp = 0 and f equal t o  t h e  i d e n t i t y  matrix. Because 
equation (3) i s  l inear ,  it follows t h a t  t h e  solut ion x = x(cp) s a t i s f i e s  
where we have defined V ( t )  = W ( t , O ) .  Using t h e  f a c t  t h a t  V ( t )  = 0 
f o r  -r 6 t 5 0, we a l so  obtain 
which can be wr i t ten  more compactly as 
9 
Equation (10) i s  called the  var ia t ion o f  constants formula f o r  (3). 
If xo i s  a solution of (8) and C i s  decomposed by A. 
-
as P %3 Q, then it can a l so  be shown t h a t  
I f  
i s  equivalent t o  (4)  and the  var ia t ion of constants formula (ID) can 
be wr i t ten  as 
g i s  iden t i ca l ly  zero i n  ( 3 ) ,  then we have seen t h a t  equation ( 3 )  
x -  t -  
t 
T(t)cp + 1 T(t-s)Kof(s)ds 
0 
where T ( t ) K o  i s  t h e  so .ution of (4)  with i n i t i a l  value a t  0 given 
by Ko(e) = 0 f o r  -r I 8 < 0, K (0) = I, t h e  iden t i ty  matrix. T h i s  
i s  t h e  standard manner of wri t ing t h e  var ia t ion  of constants formula 
fo r  (4) as given i n  [ 3 ]  and 1.41. 
i.e., f ,g  not i den t i ca l ly  zero, t h i s  formula a l so  coincides with t h e  
one given f o r  some spec ia l  cases i n  [ 3 ] .  
0 
For t h e  equation of neu t r a l  type; 
To apply these  r e s u l t s  i n  spec ia l  applications, it i s  necessary 
Q t o  obtain precise  estimates of T(t)cpQ and Vt. We are  i n  t h e  process 
of obtaining these estimates by using the  general  representat ion theorems 
10 
fo r  a semigroup i n  terms of t h e  inverse Laplace transform of t h e  
resolvent of t h e  in f in i t e s ima l  generator. 
prove tha t  
An eas ie r  way would be t o  
a ( T ( t ) )  = e + ( 0 )  , 
a f ac t  which seems t o  be t r u e  f o r  t h e  pa r t i cu la r  A given above. 
BIBLIOGRAPHY 
[ 13. Hale, J., Linear func t iona l  d i f f e r e n t i a l  equations w i t h  constant 
coef f ic ien ts ,  Contributions t o  D i f f e r e n t i a l  Equations 2(1963), 
291-3 19. 
[2]. Shimanov, S.N., On t h e  theory of l i n e a r  d i f f e r e n t i a l  equations 
w i t h  re ta rda t ions .  Di f fe ren tz ia ln ie  Uravneniya 1( 1965), 102-116. 
[ 31. Bellman, R. and K. Cooke, Differential-Difference Equations, 
Academic Press, 1963. 
Halanay, A., D i f f e r e n t i a l  Equations, Academic Press, 1965. [ 41. 


A Class of Linear Functional Equations

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server