Topological dynamics theory applied to nonlinear Volterra equation

Miller, R. K.

Sell, G. R.

English

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. -. , LI- - 
A Note on Volterra  Integral  Equations 
'* 
and Topological Dynamics 
by 
1 Richard K. Mi l le r  
and 
2 George R .  Sel l  
Division of Applied Mathematics 
Brown Universi ty  
GPO PRICE $ Providence, R .  1. 
1 
CSFTI PRICE(S) $ 
2School of Mathematics  
University of Minnesota 
Minneapolis, Minnesota 
and 
Hard copy (HC) ,,3/! a 
Microfiche (M F) - &s- 
ff 653 July 65 - ---versity of Southern California 
Los Angeles, California 
.I- -8- 
The  f i r s t  author  was  supported i n  p a r t  by the Ai r  F o r c e  Office of 
Scientific Research ,  Office of Aerospace  Resea rch ,  United S ta tes  
Ai r  F o r c e  under  AFOSR Grant No. AF-AFOSR-693-67 and by the 
National Aeronautics and Space Administration under Contract  No.  
NAS8-11264 and Grant No. NCR 40-002-015. The second author  
was  supported in par t  by the National Science Foundation under  
Grant  No. GP-7041X and the United States  A r m y  under  Contract  No. 
DA- 31 -124-ARO- D26 5. 
L 
https://ntrs.nasa.gov/search.jsp?R=19680011931 2020-03-12T08:35:16+00:00Z
. 
1. Introduction. In a recent  paper ,  G. R .  Sell  [ 5 ] ,  
[ 6 ] ,  has  developed methods which allow one to apply the theory  
of topological dynamics to a ve ry  genera l  c l a s s  of nonautono- 
mous ord inary  differential  equations. The  purpose of th i s  note 
is to i l lus t ra te  how the methods of Sell can be extended to 
nonlinear Vol te r ra  integral  equations of the f o r m  
A complete discussion of ou r  r e su l t s  along with the proofs of 
the theorems noted h e r e  wi l l  appear  in 131 and [4]. 
note we shal l  r e s t r i c t  ourself to a descr ipt ion of the semi-flow 
generated by (l), and we do this in  the case  where x , f , a  and 
g a r e  real-valued.  
In this 
Because of the generali ty of our  methods,  they can be 
applied to many problems.  Some of these applications a r e  
t rea ted  in [4]. We shall  i l lus t ra te  our  techniques by analyzing 
a problem of J .  Levin fl] in Section 5. 
2 .  Construction of the Semi-Flow.  A - flow is defir;ed 
to be a mapping TT : X x R d X ,  where  X is a topological 
space and R the r e a l s ,  that sa t i s f ies  (i) n ( x ,  0)  = x, 
(ii) n ( n ( x ,  t ) ,  s )  = n(x, t f s )  and (iii) TT is continuous. A 
(local)  flow w a s  'defined in [SI, and for  this note we need the 
concept of a ( local)  Senii-floLv, in which we r e s t r i c t  t to be 
I . . ,  ... 1. ,.._.... 
- 2 -  
nonnegative. 
that 'motions TT (x, t )  m a y  fail to exis t  fo r  all t ime  t . We 
A local  flow d i f fe rs  f r o m  a flow i n  the sense  
r e f e r  t h e ,  r e a d e r  to (51  and [7]  for  detai ls .  
F o r  Equation (11, the semi-flow i s  constructed as 
follows: Le t  V(t) = cp(f, g ,  a ; t)  denote the solution of (1). Under 
hypotheses on f , g  and a .which a r e  s ta ted  below, i t  is shown 
in [3 ] ,  that cp is uniquely determined and depends continuously 
on f,  g ,  a and t. Now define the function TTf by 
for  8 - > 0 and T in the interval of definition of 9. Define g, 
and a7 by 
fo r  T > 0,  0 < s < t < 00 and a l l  x. A topological space  X ,  
which i s  defined below, consis ts  of o rde red  t r ip les  ( f , g , a ) ,  and 
- - -  
the mapping n i s  defined by 
(3 )  
Our object is  to show that t h i  mapping n defines a 
semi-flow. Most of the defining conditions a r e  eas i ly  checked. 
The continuity of n is the only difficult i t em,  and th i s ,  of 
cour se ,  depends on the topology on X.  
3 .  Admissible and Compatible Topologies. Let  
= (Rt, R )  denote the space of real-valued continuous functions 
t defined on R . Assunic that has  the topology of uniform 
convergence on compact s e t s .  We sha l l  a s s u m e  that the t e r m  
f ( t ) , *  f r o m  (l), lies in  @. The  t e r m s  g(x,  t )  will  b e  a s sumed  
to be in  a l inear  topological space lt , 
4 
and the kerne ls  a ( t ,  s)  
belong to a l inear  topological space 
Definitions. The space d? is sa id  to be admiss ib le  i f  
the mapping 
( g ,  7)  ___) g7 
of b x Rt into ,& i s  continuous. 
admiss ib le  i f  the two mappings 
The space 4 is said to be 
+ 
of 4 . R  into & and RS into 4 a r e  continuous. W e  say 
that the pair  (b, 4) is compatible i f  
(1) 
( 2 )  F o r  every f e e, g e B, a B 4, Equation (1) 
Both & and 4 a r e  admiss ib le  and 
admi ts  a unique solution v(f ,  g , a ; t )  and fu r the rmore  y depends 
continuously on f ,  g , a  and t . 
Theorem 1. Let  - (&, 4) be compatible l i nea r  topological 
spaces .  Then the maDping given by ( 3 )  defines a semi-flon, 
on x = -
The question of admissible  and compatible spaces  is 
discussed a t  length i n  [ 3 ]  a n d  [-I]. 
examples  of compatible spaces.  These  are chosen because 
many applications fi t  into this format .  
We present  h e r e  jus t  two 
The space BP, 1 < D < a3 . The collection of a l l  
+ measurable  functions g ( x ,  t )  : R X R *I< such that for  each 
-4 -  
compact s e t  K C  R t he re  exis t  locally L functions m ( t )  
and k( t )  such that 
P 
t fo r  all x , y  in  K and t in R . We define a topology on 
& by saying 'th'at a generalized sequence Ig,) converges to 
g i f  fo r  eve ry  compact interval I C Rt and 'every compact  
s e t  c e(I, R) (where e(1. R )  denotes the Banach space of 
real-valued continuous functions defined on I) one has  
P 
uniformly for  x ( * )  i n  x. 
The space ,& . The collection of all continuous 
t functions g (x ,  t) : R X R *R such that for  each compact 
s e t  K C  R t he re  ex is t s  a continuous function k( t )  such that 
I g(x, t )  - g(yt  t )  I 2 k(t)  I x - Y I  
t 
for all x , y  in K and t i n  R . The topology on 
is the topology of unifortn convergence on compact  s e t s .  
The space ap, 1 < p < 00 . The collection of a l l  r e a l -  
valued measurable  functions a(t ,  s )  defined for  0 < s < t a3 
such that (i) for  each t ,  a( t ,  s )  is  locally L in s where  
p t q = 1 and ( i i )  for  e a c h  conipact interval  I C Rt and 
- - _  
9 
-1 -1 
every  t - > 0 one has  
n 
I a ( t  t h ,  s) - a ( t ,  s )  1' d s  4 0 
JI 
as  h 4 0 .  The  topology on a is defined by saying that 
a general ized sequence 
compact in te rva l  I C Rt one h a s  
P 
converges to  a i f  f o r  eve ry  i an )  
uniformly for  t i n  compact s e t s .  
The  space a,. The collection of all real-valued 
measu rab le  functions a ( t ,  s )  defined for  0 < s < t < 00 and 
such that (i) for  each t ,  a ( t ,  s)  i s  locally L1 i n  s and 
(ii) fo r  each compact interval 
i s  continuous in the weak*-topology on @ ( I , R ) * .  The  topology 
- -  
I C RS the mapping t *a( t ,* )  
f o r  
converges to a i f  fo r  every  a generalized sequence 
compact interval  I CR' and eve ry  x(-) 8 e(I, R )  one h a s  
urn, which i s  a weak topology, is  defined by saying that 
i a n 1  
n 
uniformly for  t i n  compact s e t s .  - 
Theorem 2. The spaces  ( hp, and ( & a,) 
a re  compatible. 
Other topologies a r e  given in [3 ]  and [4].  An interest ing 
fea ture  that a r i s e s  i n  our study is that a s  one weakens the 
topology on the kerne ls  
the topology on the t e rms  
compatibility of the D a i r  (8, a). 
a ( t ,  s ) ,  i t  is  necessa ry  to strengthen 
g(x, t )  in o rde r  to p r e s e r v e  the 
W e  r e f e r  the r eade r  to 
the main  papers  for  details .  
. . -  
- 0 -  
. . . 
4. - Compact Motions and Limiting Equations. Two of 
the bas ic  problems i n  applying topological dynamics is to 
determine conditions under- which a motion n(x, t) is compact ,  
that  is ,  n(x, t )  remains  i n  a compact se t  for  a l l  t - > 0 ,  and 
then to analyze the asymptotic behavior of n(x,t) i n  t e r m s  of 
the u) - l imit  se t .  In 141 sufficient (and somet imes  necessa ry )  
conditions a r e  given i n  o rde r  that  the motions g, and a, be  
compact in b, and 4. Because of lack  of space,  we will 
not formulate  these resul ts  h e r e .  A s  fo r  the -motion n ( f ,  g ,  a ; T ) ,  
one can prove the following resul t :  
Theorem 3 .  - Let  (&d)  be a compatible pa i r  and 
i n  & and a in 4 a r e  compact.  t -. -a s s u m e  that the motions 
If the solution cp(t) = cp(f,g,a;t) - of (1) l i e s  in  a compact  s e t  
fo r  a l l  t - > 0,  then the motion n(f,  g,  a ; t )  r ema ins  in  a compact 
s e t  for  a l l  t > 0 .  
gt - 
- 
The problem h e r e  is to show that the family ( T T f :  7 - > 0) 
l i e s  in a compact set  in  e . This  follows f r o m  the notion of 
compatibility and the fact  that TTf(8)  can be formulated a s  
(4) T,f(B) = cp(7 t 8) - a T ( e ,  s)gT(cp,(s), s m .  
Once one knows that a motion n(f,  g ,  a:,) is compact ,  then 
i t s  u- l imit  set  n i s  compact and nonempty. 
(F:K, g”, a:%) in fl i s  character ized a s  the l imit  of n ( f ,g , a ;Tn)  
fo r  some sequence 7 4 00. One can show that in this c a s e ,  
the t rans la tes  cp(t t T ) converge uniformly for  t in compact 
s e t s  to a function cp:k(t). Under appropr ia te  integrabili ty con- 
A typical point 
n 
n 
ditions on a ( t ,  s)  and g ( x ( s ) ,  s )  one can show that cpz:(t) sa t i s f ies  
- -  . 
where  
is  of the f o r m  
fT __+ f: in e .  In o ther  words ,  the function F* 
n 
This  general izes  a resu l t  of R .  K. Mil ler  [2]. 
5. An Application. The following problem, which is  
a generalization of a problem of J. Levin [l] follows eas i ly  
with ou r  techniques. 
Theorem 4. Consider the equation 
t 
n r  1 
x( t )  = f ( t )  - J b ( t - s )  t b(t ,  s)J g(x ( s ) )ds  
0 
where  (1) f ( t )  is continuous for  t > - -  0 and f ( t ) - +  f o  - as  
t d a, 
(2) a ( r )  @ ' [ O , m )  n L1(O,m) - and a ( r )  1 0 ,  
T > O ,  7 t 1  n 
uniformly f o r  r in compact s e t s ,  
(5)  g(x) i s  locally Lipschitzian and s t r ic t ly  increasing.  
Assume that the solution c,"(tY = rp(f, g ,  a ; t )  - of (6)  is  bounded 
be  t h e  solution of xO f o r  a l l  t - > 0 and let 
00 
where  A = 4 a ( r ) d r .  Then c p ( t ) 4 x 0  t + m .  
c , .  4 . .  
. .  
. - '  . 
- . -  
-8 - 
References - 
1. J. Levin. "The qualitative behavior of a nonlinear 
Volterra  equation. I t  Proc .  A m e r .  Math. SOC. 16 (1965). 
711-718. m 
2 .  R.  K. Miller.  "Asymptotic behavior of solutions of 
nonlinear Volterra  equations. ' I  Bull. Amer .  Math. SOC. 
72 (19661, 153-156. 
3 .  R.  K. Mil ler  and G .  R .  Sell.  "Existence,  uniqueness 
and continuity of solutions of integral  equations. I '  Ann. 
Mat Pura  Appl. ( to a p p e a r ) ,  
4. R.  K. iMiller and G. R .  Sell .  "Volterra  in tegra l  equations 
and topological dynamics. I '  ( to  appear )  
5 .  G. R .  , Sell. "Nonautonomous differential equations and 
topological dynamics I. The  basic theory.  ' I  T r a n s .  - Amer .  
Math. SOC. 127 (1967), 241-262. -
6. C. R .  Sell ,  "Nonautonomous differential  equations and 
topological dynamics 11. Limiting equations. I '  T r a n s .  Anier .  
Math. S O C .  127 (1967), 263-283. -
7 .  C. R .  Sell.  Lectures  o n  Topological Dynamics and Differential 
Equations. Van Nostrand, Pr inceton,  N. J . ,  (to appear )  


A note on Volterra integral equations and topological dynamics

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