Mathematical description of Volterra integral equation

Miller, R. K.

English

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On t h e  Linearization of Volterra  In t eg ra l  Equations 
R. K. Miller* 
Center f o r  Dynamical Systems 
Brown University 
Providence, Rhode Is land  
May, 1967 
(THRU) 
/ 
(CODE) 
* 
This Research w a s  supported in  p a r t  .,y the  A i r  Force Off,ze of 
Sc ien t i f i c  Research, Office o f  Aerospace Research, U. S. A i r  Force 
under MOSR Grant No., 693-67, i n  pa r t  by the  National Aeronautics 
and Space Administration, Huntsville, under Contract No. NAS8-11264 
and i n  p a r t  by the  National Aeronautics and Space Administration 
under Grant No. NGP, 40-002-015. 
https://ntrs.nasa.gov/search.jsp?R=19670027633 2020-03-12T12:19:17+00:00Z
On the  Linearization of Volterra In t eg ra l  Equations -- - 
I. Introduction. 
Given a nonlinear d i f f e ren t i a l  equation 
. (1) X' = CX + ~ ( I X I ) ,  (' = d/dt)  
it i s  well  known t h a t  t he  asymptotic s t a b i l i t y  of t he  l i n e a r  system 
yf = Cy implies the  l o c a l  asymptotic s t a b i l i t y  of t h e  t r i v i a l  solu- 
t i o n  of (1). All known proofs of t h i s  f a c t  depend on the  f a c t  t h a t  
solut ions of t he  l i n e a r  system decay exponentially o r  the  equivalent 
fact  t h a t  there  e x i s t s  a quadratic Lyapunov function f o r  t h e  l i n e a r  
system. 
Consider a system of n equations of t he  form 
t 
x ( t )  = f ( t )  + J a( t - s )g(x(s ) )ds ,  t 2 0 
0 
( 2  1 
where x,f and g a re  n-vectors, a ( t )  i s  an n x n matrix and 
g(0 )  = 0. If f i s  ' f s m a l l f f  t h i s  system i s  of ten replaced by t h e  
more eas i ly  analyzed l i n e a r  system 
where J 
Levin and Nohel have proved by example t h a t  solut ions of equations 
of t he  form (3) need not decay exponentially, c . f .  [l, p.333, l i n e  
(2.11)]. 
t h a t  solutions of t h e  l i n e a r  system ( 3 )  approximate those of  (2) i n  
i s  the  Jacobian matrix g' (0) = (dgi(0)/aj) .  However 
Therefore it has not been possible  up t o  now t o  show 
an appropriate. sense, except i n  the  case where solut ions of (3) 
decay exponentially . 
2 
Nohel [g,lO] has pointed out t h i s  gap i n  the  theory of Volterra 
i n t e g r a l  equations. 
i za t ion  can be developed when solut ions of (3) do not behave l i k e  solu- 
He  has asked whether or not a theory of l i nea r -  
t i ons  of an ordinary d i f f e r e n t i a l  equation. The purpose of t h i s  paper i s  
t o  provide i n  sect ion I1 below a theory of l i nea r i za t ion  f o r  equations 
(2) under very general assumptions on 
our analysis i s  Theorem 1 of [2] which we use i n  place of t h e  usual  
a( t )J .  The e s sen t i a l  t o o l  i n  
estimates from ordinary d i f f e r e n t i a l  equations. 
. __. 
The advantage of our method i s  t h a t  one can replace the  
loca l ,  nonlinear problem (2) by the  l i n e a r  equatlon ( 3 )  and the  
l i n e a r  equation f o r  i t s  resolvent. These l i n e a r  equations may be 
s tudied using known methods such as transform techniques. In  
Sections 111, I V  and V below we give some examples which i l l u s t r a t e  
t h i s .  
I n  t h e  sequel we s h a l l  need the  following notat ions and 
conventions. Let R" denote real  n-space with a norm 1x1. Let 
ID1 denote the  corresponding matrix norm. Let BC[O,w) be the  
space of bounded continuous functions on 0 5 t < 03 with norm 
Similarly BC(R) w i l l  be the  space of bounded continuous functions 
on -03 < t < m with norm 
-00 < t < m). 
11. General S t a b i l i t y  Conditions. 
Concerning equation (2) we assume: 
3 
1 
( A l )  a E LL(O,t)  f o r  each t > 0, 
(A2) f ( t )  BC[o>w)J 
(A4) the  Jacobian matrix J i s  nonsingular. 
Since we assume J i s  nonsingular, it i s  no loss  of gen- 
e r a l i t y  t o  assume J i s  the  n x n i d e n t i t y  matrix I. We need 
only replace a ( t )  by a ( t ) J  and g(x) by J-'g(x). Thus equa- 
t i o n  (3) may be rewri t ten i n  the form 
It i s  wel l  known t h a t  the unique solut ion of equation ( 3 ' )  
has the form 
t 
y ( t )  = f(t)  - J b ( t - s ) f ( s )ds ,  (t Z 0 )  
0 
(4) 
where the matrix b i s  the  resolvent kernel  determined by the 
matrix equation 
t 
b ( t )  = - a ( t )  + J b ( t - s ) a ( s )ds .  
0 
(5  1 
We assume t h a t  
(A>) the  matrix b determined by (3) e x i s t s  f o r  a l l  
1 t > 0 and Ib ( t ) l  E L ( 0 , ~ ) .  
Theorem 1. I f  assumptions (Al->) a re  s a t i s f i e d  then there  e x i s t s  - - --
E > 0 and cl > 0 such tha t  when the solut ion y ( t )  of ( 3 ' )  - ---- 0 
4 
1 .Proof. Since b E L ( 0 , ~ )  it follows t h a t  equation (2) i s  equiva- 
l e n t  t o  the  system 
where y i s  defined by l i n e  (4) and 
Pick cl > 0 such tha t  i f  1x1 5 el, then 
00 
21G(x)I J I b ( s ) l d s S  1., 
0 
03 
and 1 Ib(s) ldslg ' (x)  - 11 < 1. Pick eo = c1/2. Let Tx(t) be 
the function defined by the r ight  hand s ide  of equation (6). Let 
0 
Our estimates on E and cl imply t h a t  T i s  a contradiction 
map on S(0,c1). This proves Theorem 1. 
0 
Corollary 1. 
and > 0 such t h a t  when I I f I I  5 e2 t he  solut ion x ( t )  of 
(2) e x i s t s  f o r  a l l  t 2 o and s a t i s f i e s  llxllo 5 cl. 
If ( A l - 3 )  are  sat isf ied,  then there  e x i s t  cl > 0 - - --- 
--- - - 0 -
-- -
Proof. Pick E ~ .  such t h a t  
5 -  
where E i s  the  constant given i n  Theorem 1 
above implies llyllo 5 co. Thus Corollary 1 
above. 
1. Then equation (4) 
follows from Theorem 1 
Theorem 2. Let (Al-3) hold and l e t  eo and cl be given by 
Theorem 1 above. If llyll, 5 c0 
- --- - ---
and y ( t )  + O  as  t +Q), then - -- -
x ( t )  + O  as t +a. -
Proof. Let r be the posi t ive l i m i t  s e t  of the  solut ion x ( t ) ,  
t h a t  i s  r i s  the smallest  s e t  such t h a t  x ( t )  tends t o  r as 
t +m. Since x ( t )  i s  bounded it i s  e a s i l y  shown t h a t  i s  
nonempty, compact and connected. 
Since x ( t )  solves equation (6), y ( t )  + 0 and b E: L 1 (0,a) 
it follows from Theorem 1 of [2] t h a t  i s  the  union of solu- 
t i ons  of 
(7.1) 
t 
z ( t )  = - b(t-s)G(z(s))ds ,  
--03 
Let Tz( t )  be the function defined by the r i g h t  hand s ide 
of l i n e  (7.1) when z E: BC(--a),m) and 1 1 ~ 1 1 ~  P el. The estimates 
i n  the  proof of Theorem 1 above imply t h a t  T i s  a contract ion 
map. Thus z ( t )  0 i s  the  unique so lu t ion  of (7.1-2). This 
means t h a t  I' = EO). Thus x ( t )  4 0  and the  proof of Theorem 2 
i s  complete. 
Using Corollary 1 and Theorem 2 we obtain the  following 
r e s u l t .  
6 
be given by €2 -_.__- Corollary 2. Let ( A l - 3 )  hold and l e t  c1 and 
Corollary 1 above. If l l f l l o  S e2 y ( t )  -+ 0 as  t + CQ, then 
--- --
- -
x ( t )  4 0 .  
111. Applications: Integrable Kernels. 
The 
Section I1 
s h a l l  need 
Theorem 3 
purpose of t h i s  section i s  t o  apply the  theory i n  
with the  addi t ional  assumption t h a t  a E: L (0,m). We 
t h e  following result. 
1 
1 (Paley and Wiener). Let a E L ( 0 , ~ ) .  Then the  s o h -  - --- 
t i o n  b of equation (3) i s  Ll(0,m) i f  and only i f  the  determinant ----- - - 
03 
det  (I-! exp ( - s t ) a ( t )d t )  # 0, 
0 
(8 )  
i n  the  r i g h t  ha l f  plane Res L 0. ----- 
This theorem i s  proved by a t r i v i a l  modification of t he  
proof of Paley and Wiener of Theorem X V I I I  i n  [ 3 ,  p. 601. Paley 
and Wiener use Theorem 3 t o  study the  asymptotic behavior of s o h -  
t i ons  of equation (3' ) i n  case f ( t )  + o as t + 03. Their re- 
s u l t  has the  following nonlinear generalization. 
Theorem 4 .  Suppose (Al -4 )  hold (8) i s  s a t i s f i e d  f o r  Res h 0 
-9 - -
is given by Corollary 1 above. I f  IIfllo 6 c2 and - -  -and 
f( t)  3 0 as t 3 03, then x ( t )  3 0. 
- €2 --- 
- -
Proof. 
(4).  Since f( t)  -3 0 as t -+ w and b E: L (O,m), the  Lebesgue 
Dominated Convergence Theorem implies t h a t  y ( t )  + O .  An 
The solut ion of the  l inear ized  equation ( 3 1 )  i s  given by 
1 
7 
appl icat ion of Corollary 2 completes the  proof of Theorem 4. 
Levin [4] has obtained another nonlinear general izat ion of 
t h e  Paley-Wiener r e su l t .  His r e s u l t  i s  ne i ther  stronger nor weaker 
than Theorem 4 above. Levin s tudies  a sca l a r  equation ( n d )  
while we allow n > 1. O u r  hypothesis on a ( t )  i s  weaker than 
Levin's and our hypothesis on g(x) stronger.  Theorem 3 i s  a 
l o c a l  r e s u l t  while Levin's r e s u l t  i s  global.  
The condition f ( t )  4 0  i s  e s s e n t i a l  t o  t he  proof of 
Theorem 4 above. If f has a d i f f e ren t  type of asymptotic be- 
havior, it may' s t i l l  be possible t o  analyze t h e  l o c a l  behavior of 
solut ions of equation (2). f ( t )  For example i n  Theorem 5 below, 
i s  constant but  not necessarily zero. 
IV .  Applications: In tegrodi f fe ren t ia l  Equations. 
The purpose of this section i s  t o  apply the  theory of 
Section I1 t o  the  study of the l o c a l  behavior of i n t eg rod i f f e ren t i a l  
equations of the  form 
where k i s  l o c a l l y  integrable  and. m i s  a constant. We allow 
m = 0. 
one s e t s  f ( t )  F x and 
This system can be wri t ten i n  the  form of equation (2) i f  
0 
t 
a( t )  = m + k(s)ds .  
0 
We wish t o  invest igate  t h e  asymptotic behavior of solut ions 
of equation (9 )  when xo i s  small. We remark t h a t  the de f in i t i ons  
of s t a b i l i t y  and asymptotic s t a b i l i t y  of the  t r i v i a l  solut ion 
x = 0 of ( 9 )  are  the  same as f o r  ordinary d i f f e r e n t i a l  equations. 
Theorem 5.  Let f and a be as defined above. If (A3-4) -> hold 
a E L (0,co) and (8) --- i s  t r u e  fo r  Res 2: 0, -- then f o r  xo s u f f i -  
- - -- - -
1 -
c i e n t l y  small 
(i) 
(ii) 
the  t r i v i a l  solution of ( 9 )  i s  s tab le  and 
each solut ion of ( 9 )  tends t o  a constant as 
- - - -
t -+ 03. - --- -
Proof. It follows from the proof of Corollary 1 above t h a t  f o r  
each E, 0 < E < E ~ ,  there  ex i s t s  6 > 0 such t h a t  llxllo 5 E 
when IxoI 5 6. 
To prove p a r t  (ii) note t h a t  if lxol S c2 then [ x ( t ) l  5 el 
f o r  all t B 0. Moreover 
I Since b E L (o,~), 
e x i s t s .  By Theorem 1 of [ 2 ]  the  pos i t ive  l i m i t  s e t  of x ( t )  i s  
the  union of solut ions of 
00 t 
(10.1) Z ( t )  = (I-! b(s)dS)Xo - J b(t-S)G(z(S))dS, 
0 -co 
Let S(0,c1) be the closed sphere i n  BC(R) with center  
a t  the or ig in  and radius  E Let So be the  subset of S(0,c1) 1' 
9 
consis t ing of constant functions. The estimates on E i n  t he  
proof of Theorem 1 imply t h a t  the  r i g h t  s ide of (10.1) defines a 
1 
contract ion map on S(0,e1) and on So. Therefore t h e  unique 
so lu t ion  of (10.1-2) i s  a constant function 
pos i t ive  l i m i t  s e t  of x ( t )  i s  the  s ing le  point  z 0’ x ( t )  + zo 
as t + w, and Theorem 5 i s  proved. 
z ( t )  z 0’ ‘Thus the  
For xo s m a l l ,  the  l i m i t  zo i s  obtained by solving the  
equation 
00 00 
z = (I-J b(s)ds)xo - (J b(s)ds)G(zo). 
0 
0 
0 
L e t  t he  solut ion be zo = F(xo). Then F(0) = 0 and F maps a 
neighborhood of xo = 0 diffeomorphical lymto a neighborhood of 
z = 0. 
c a l l y  s tab le .  
This means t h a t  the  t r i v i a l  so lu t ion  cannot be asymptoti- 
0 
V. Applications: A Reactor Problem. 
The dynamic behavior of a continuous medium nuclear reac tor  
can be described, under cer ta in  simplifying assumptions, by the  
following in tegrodi f fe ren t ia l  equations f o r  theunknown u and T: 
(11.1) 
with the  i n i t i a l  conditions 
These equations have been extensively s tudied by Levin and 
Nohel, i n  t h e  l i n e a r  case g(u) = u c.f. [1,5] and i n  t h e  nonlinear 
10 
case zf. [6j. IE t he  reactor  problem g(u) = exp(u) - 1. 
We wish t o  study the  asymptotic behavior of solut ions of 
(11) using the  theory of Section 11. Our analysis  depends heavi ly  
on the  papers [l,?, and 61 both f o r  motivation and techniques. 
Since Levin and Nohel have t rea ted  the  uniqueness problem f o r  (11- 
12) we do not consider it further.  
Let * denote the  L2 Fourier transform. If f,a, and 71 
a r e  L2(R), then an elementary appl icat ion of transform theory 
shows t h a t  u ( t )  s a t i s f i e s  the equation 
t 
u ' ( t )  = -1 ml(t-s)g(u(s))ds - m2(t),u(0) = uo 
0 
(13) 
where f o r  j = 1,2. 
Using a Taubiswn theorem Levin and Nohel [l] study the  l i n e a r  equa- 
t i o n  
t 
vr (t) = -J ml(t-s)v(s)ds - m,(t) ,v(O) = vo. 
0 
(14) 
They prove 
Theorem 6 (Levin and Nohel). Suppose f,a - m d  7 sa t i s fy :  
11 
Then the  solut ion v ( t )  of (14) ex i s t s  f o r  a l l  t B 0 and v ( t )  = -- --- -
Corollary 3 .  
exists a pos i t ive  constant 
t h a t  f o r  a l l  t 2 0 
If  the  hypotheses of Theorem 6 are  s a t i s f i e d  then there  --- ---
K1 (independent of v and f )  such - 0 -  - 
---
Proof. Let v,(t) be the  solution of (14) when v = 1 and m 2 ( t )  E 0 
and l e t  v2( t )  be the  solution when v = 0. Then the  general  solut ion 
i s  vl(t)vo + v2( t ) .  By Theorem 6 v, (t) i s  bounded. 
0 
0 
Let V be t h e  
5.32 of [l] we see t h a t  
I 
Laplace transform of v2. Using l i n e s  5.28 and 
f o r  --CQ < y < 00 
w 0 J  
J exp(-(iy)1/21x-s,l )a(x)f(s)dxds 
-03 -co 
where H(y) i s  i n  LL(-mYco) and H depends only on a and 7. 
Lemmas 5.1-5.6 of [l] show t h a t  
of [8, p. 2661. Therefore 
V s a t i s f i e s  t he  hypotheses of Theorem 2 
12 
This proves Corollary 3. 
Using Theorem 2 and 6 we prove 
Theorem 7. Let f,a - and rj s a t i s f y  (A6-8). - Let g s a t i s f y  (A3) 
with gr (0) = 1. Then there  exis t  6 > 0 (depending only on qya 
and g) such t h a t  when IuoI 5 6 and ]If11 5 6, then the solution 
u(t)  of (13) ex i s t s  fo r  a l l  t 2 0 and u ( t )  0 as  t -+a. 
-
-
- - --
- -- - -
Proof. Let b ( t )  be the  resolvent kernel f o r  equation (14), t h a t  
i s  b ( t )  solves (14) i n  the special  case v = 0 and f = q. By 
Theorem 6 we see t h a t  
0 
v ( t )  and b ( t )  = u(t -3/2) as t + w. Thus 
I b i s  i n  L ( 0 , ~ ) .  We know from Corollary 3 t h a t  Iv( t ) l  i s  s m a l l  
when 
the  proof of Theorem 7. 
IuoI and IIfll are  s m a l l .  An application of Theorem 2 completes 
Corollary 4. Let the hypotheses of Theorem 7 hold. If 6 i s  given -- - - - -- - 
by - Theorem 7 - and IIfll, IuoI 5 6, - then u ( t ;  E L'(0,w). 
Proof. Fix uo and f with Iu0I and I I f l l  5 6. Let v ( t )  be the 
solut ion of (14). There ex is t s  K > 0 such t h a t  fo r  a l l  t 2 0 
I__ 
Ib( t ) l  5 K(t+l)-3/2,v(t)l 5 K ( t + l )  -3/2 . 
Since u ( t )  + 0, there  ex i s t s  T > 0 such t h a t  i f  t B T then 
Let K1 be a bound on G(u(t))l f o r  0 6 t < 03. For a l l  t B 0, 
T 
u(t+T) = v(t+T) - J b(t+T-s)g(u(s))ds 
0 
t 
- b(t-s)G(u(T+s))ds, 
0 
T 
Iu(t+T)I 5 K ( t + T + l )  -3/2 + KK, J ( t + T + l - ~ ) - ~ / ~ d s )  
0 
.J- 
t 
+ 1 K(t+1-~)-~/~1u(T+s)l/(4K)ds, 
0 
S K ( t + T + l )  -3/2 + 2KK1((t+l) - ( t+T+l )  
t 
+ ( t + l - ~ ) - ~ / ~ I  u(T+s) I /bds 
The comparison theorem of Nohel [7, Theorem 2.11 implies 
that for t h 0, lu(t+T)I 5 U ( t ) ,  where U solves 
t 
U ( t )  = H l ( t )  + H 2 ( t )  + 1 H3(t-s)U(s)ds. 
0 
Since fo r  any t > 0, 
t 
1 H2(s)ds = 4m1(=1 - 6 1  - 1 + 6 1 )  5 4KK1(&1 - l), 
0 
1 1 it follows t h a t  5 E: L ( 0 , ~ ) ) .  Clearly H1 and H E L (0,co) and 3 
Q) 
H3(s)ds 5 1/2. Thus the r i g h t  hand s ide of equation (13) determines 
0 
1 4  
a contraction map on L l ( 0 , ~ ) .  Since U ( t )  dominates I u(t+T) 1, 
u( t )  E. L ( 0 , ~ ) .  1 This completes the  proof of Corollary 3 .  
I n  order t o  study the asymptotic behavior of T(x,t) we 
need the following addi t ional  assmption: 
(Ag) f, 7 E C (R) and 7 i s  loca l ly  Holder continuous. 
Theorem 8. Suppose g s a t i s f i e s  (A3) - and g ' ( 0 )  = 1. - Let f,a 
and 17 s a t i s f y  (~6-9) .  Then for  u and IIfll su f f i c i en t ly  s m a l l  
0 -  ---
problem (11-12) - -  has a unique solution u( t ) ,  T(x, t) .  Moreover, 
1 and u( t )  4 0  as t + c o  with - u c L ( 0 , ~ ) .  - -
Proof. For uo and llfll suf f ic ien t ly  small Theorem 7 and Corollary 
3 imply the existence of  a solution of equation (13) such t h a t  u ( t )  
1 u E L ( 0 , ~ )  and u ( t )  -+ 0. Given t h i s  u ( t )  define T(x,t) on 
where G(x,t) = ( b ~ i - L ) - ~ / ~ e x p ( - x ~ / ( 4 t ) ) .  Using the  same proof as i n  
[7, p.2641 we ver i fy  t h a t  the p a i r  u(t),T(x, t) i s  a solution of (11) 
and (12). Moreover, f o r  any t > 0 
- W  -0 )  0 
1 Since g ( u ( t ) )  is L ( 0 , ~ )  it follows by dominated convergence t h a t  
* 
15 
t t -1/2 
s-1/21g(u(t-s))Ids = f ( t-s)  Ig(u(s ) ) [  ds + 0. 
0 0 
Therefore T(x,t) 4 0  as t --)a uniformly f o r  -Q) < x < Q). This 
proves Theorem 8. 
Theorem 8 i s  neither stronger nor weaker than the r e s u l t s  
i n  [ 6 ] .  The advantage of Theorem 8 i s  t h a t  we avoid a hypothesis on 
the  interconnection of f,a and 7, c . f .  16, l i n e  1.161. The main 
disadvantage of Theorem 8 i s  tha t  the r e s u l t  i s  l o c a l  while the  re -  
sults of [6] a re  global. 
References 
[l] J. J. Levin and J. A. Nohel, On a system of in tegrodi f fe r -  
e n t i a l  equations occurring i n  reac tor  dynamics, Journal  of 
Math. Mech, - 9 (1960), pp. 347-368. 
[2] 
I 
R. K. Miller, Asymptotic behavior of solut ions of nonlinear 
Volterra equations, Bull. h e r .  Math. SOC., 72 (1966), 
PP. 153-156. 
[ 3 ]  R. E. A. C. Paley and N. Wiener, Fourier Transforms -- i n  t he  
Complex Domain, h e r .  Math. S O ~ .  1934. 
[4] J. J. Levin, The qua l i t a t ive  behavior of a nonlinear 
Volterra equation, Proc. Amer. Math. Soc., - 16 (1965), pp. 
711-718. 
[ 5 ]  J. J. Levin and J. A. Nohel, On a system of in tegrodi f fe r -  
e n t i a l  equations occurring i n  reac tor  dynamics 11, Archive 
Rat. Mech. Anal. - 11 (1962), pp. 210-243. 
[ G I  , A system of nonlinear integro- 
d i f f e r e n t i a l  equations, Mich. Math. J. - 13 (1966), pp. 257- 
270. 
[7] J. A. Nohel, Some problems i n  nonlinear Volterra i n t e g r a l  
equations, Bull. Amer. Math. SOC. - 68 (1962), pp. 323-329. 
[8] G. Doetsch, Theorie und Anwendung der  Laplace - Transforma- 
t ion,  Berlin (1937). 
C 91 J. A. Nohel, Problems i n  qua l i t a t ive  behavior of solut ions 
of nonlinear Volterra equations, Nonlinear In t eg ra l  Equations 
edi ted by P. M. !inselme, University of Wisconsin Press, 
1964. 
C .lo 1 , Qual i ta t ive  behavior of solut ions of nonlinear 
Volterra equations, S t ab i l i t y  Problems of Solutions of 
D i f f e ren t i a l  Equations, Oderisi  Gubbio, 1966. 


On the linearization of Volterra integral equations

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