Almost periodic behavior of solutions of nonlinear Volterra syste

Miller, R. K.

English

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R^ K. biiller^
CPnt-,Er for ilynazni cal Systems
Divi si.on of App1iF•3. Mathematics
RrOt•T:"1 L1rii yFi^i^
Providence Rhode Tsla.nd
'phis re,s3rch was supported in pa-rt by +he i^latiora' Aerenau+ics
wnd Sp^.cs A^im..i.nzs ^ration^ under gra yit no ^;GL 40- 002-01^ ar_d in
p4.rt by the Ai.r Force OFfi .e of Scientific Researc^i "^ --- "Ter ^rarlt
no. AF-A]~'GS^ 693 -c'7g ^	 ^ r^ ^
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(TH U)
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(CATEGORY)
ABSTRACT
Th-is paper studies forced almost periodic oscillations
in a nonlinear systern of two 	 1terra integral equations. It is a
a
	 soquel paper to an curlier paper nn the same topic. Earlier re-
sults are improved In two way	 First it Is shown that the
oscillatory solution is Lya.punov stable under small perturbations
in the coefficients of the equation. Secondly it is shown that
whenever the coefficients are qua,siperiodic and analytic, the almost
periodic oscillation is in the surie class.
A.hi,S. Subject Classification; Primary 451, and 4530
Key Words: Almost periodic ) Volterra integral equations, stability,
quasiperiodic.
I t,Uj0S'1'1,;'ffODIC BEHAVIOR OY SOLU'TION'S OP A
NONLINEAR VOLTERBA SYSTV-1. II
R. K. Miller
I. Introduction.
In this paper we continue the study of forced oscilla-
tions in a nonlinear system of Volterra integral equation.,, of the
form
t
( 1 . 1 )	 xl ( t ) - fl ( t ) - f al(t-s)gl(S,xl(s),x2(s))ds
0
t
O a2(t-s)g2(sgx1(s),x2(,;))ds,
t
x2(t) -!-- f2 ( t ) - f a2(t-s)(;l(s,xl(s)1x2(s))ds
0
t
f a,, (t-s)g2(s1xi(s)1x2(J))ds.
0
In an earlier paper [11 sufficient conditions were given so that
the solutions x1 (t) and x2 (t) tend to certain almost periodic
limiting functions P l (t) and P2 (t) as t -+ co. In this paper
we shall im prove the previous results in two ways. Firstly, it
will be shown that this oscillatory behavior is stable under small
perturbations in the functions f i and g^, That is the solution
of the perturbed problem is oscillatory and is near the solution
r
`l
_-
2of the unperturbed problem.
conditions in order that the
are analytic in t.
We shall follow th,
results froca that paper. In
vector form
Seconds:,,) we give rather weak sufficient'
limiting functions Pl (t) and P,(t)
notation in [1] and shall freely use
Particular we rewrite (1.1) in the
t
(^)	 x(t) = f(t) _ j A(t- s)G(s,x(s))ds
0
where A(c) is the appropriate matrix and x,f and G are two
dimensional column victors. The vector norm will be
x^
I X l = max ( I x I, l xl	 n I), i f x	 xC	 2
The symbol Q will always denote the s pecial matrix
(1.2)	 Q _ 
2 -112	 1	 1
1	 -1 )
Note that Q = Q 1 and that	 N/2. For any N > 0 the re-
solvent of NQA(t)Q will be denoted by RN (t). Using RN (t), the
variation of constants formula and the change of variables
x = Qy + f(t) one car rewrite system (E) in the equivalent form
r!
.^
t
0
where 
Gr1
(t,s•) :.: ^tG (t,^2y+ f (t) ) rr- l.
Sys^,cra (1.1) arise:, in a nr^t;ural_ wny from thA initial -
boundary valve p,^obl^.m
(1.3)
	
ut ` uxx
u(O,x) - F(x)	 (U < x < ^r),
.r
and
u Y (t,G) = gl(t,u(t,0),u(t^^r) ), ux (t,'rr) _
-^2(t,u(t^0)^u(t,^)),
for fyll t > 0. In particular, we nave in mind boundary con-
ditions motivated by C. C. T,in's theor;^ of superflutdity (see [^]
°" [^])^
(l.. li )	 gl(t^xl) = Bl (xl- cl sin klt) 3 ^ ^(t^x2 ) -
= B2 (x2-c2 sin k2t)3.
As an application of the res^llts prove: here and in [1) we shall
prove the follo;ling r :salt.
,m^, _ _
	 -	 --	
-	 -_ -
^^. .
T^ieorem 1. Consider the ^robl«r^ (1.3 -^+) wlth ^'0 E C2 [O^ir]. Z
^;ivF^n any B > 0 thf:re exi ats E > 0 such that 3 f (A i -33^ < E
^__. _._.
for i^ 1 2 anti. if F F C? [U T^ ] Frith
w ^ ^w
^^^ 0 mr,.•1	 Ux
then the boitndax•y functions	 u(t^0) and	 u(t,^r) tend asyrcpt^^tice.11;^
as t -a o^	 to a]rtost periodic	 1 ilciit3.nE fu ticti.ons P^ (t)	 And	 ' 2 (t) .y	 ^...^-
Z^ne functions	 u(t,^),	 u(t,^r)^
	 Pl (t) nd	 F2 (t) a ll very coa-
ti nuously with	 ^'^ B^ ^ B2 ^ C 1	 and C2 .	 2-lvreover,	 each	 Pi (t)	 has
the form 	 Pi (t) = F'i (klt^ k2 t)	 where Pi (8 1' 82) is real an^.l.yti_c
in (© 1,82 )	 and is	 2^►'-periodic	 iri e ach of its two variab le s.
P'
5I7. t irturb^ation Results.
Assume; that Lhe coefficient functions f,C^ arod A ii ►
(E) s:^,tisfy file fol.t^w in^ hyhoth;:ses:
(A1) a^ (t) ^ 1 -^- ? Fn=l ^'x 1J (- r^2t) ' a^(t) - 1 + 2 ^1'ri-1^ ].)nPxl,(-n^t)
and
``^ fit)
	
a^ (t) ^
^(t) :^
^2 (t)
	
al(t)
(A2) f (^.) is continuous and bounded on 0 ^ t < ^.
:; I
	
(A3) G(t,x) is continuous in (t a x) for all t ? 0, ^x^ < ^^
,^" I	 G i5 locally Lipschitz continuous in x A'ld G(t,0) ^ 0.
,,
(ASE) The function G (t, x l , x2 ) has the . ^ec ial form
e(t^xl)
G (t, xl , x2 ) _
g(t,x2)
where g(t,y) is an odd nondecreasing function of y which is
bounded in t E (-^, «^) uniformly for y on any comeact :.upset
.^/+
G'die fol.].owing hypotheses are rf:lated to (11^•)
(A^) Tt^erf exi sti T osi.tive numbers T1 and K such thfit if
y^ ^ K thra ^ y-urn (t r y) ^ < K uniformly for. all t E (-^, ^^) .
Here y ^ R^ i r a two di.mens ional coJ_tunn vector and G N i ::
the fur:eLion def'iiied below (^^fJ).
(AFB ) Theme ex7st posi.t;ive tnunber^ N, K O ar^d Kl sucJl that if
Y ^ ^ K, then ^ y-C N (t, 5') i ^ I:0 < Kl uniformly for a.11
The proof of Ler•^ ma 3 in [ 1] is actL^.ally a E^,roof of the
following stronger result.
.E.
^^
LerunK 1. 3uppo^r G s atisfies (A3) an^1 (A^^) and b ?
sup (^ f (^) ^ : t ? 0} . Then for any M > V ^ b and ''or any e ir.
the j.nt-erval 0 < e < b there exists a nlunber rJ > 0 such that
(Aj) is true Wlth K = M + ^. Moreover, if L is a, constant such
that
^	 L ? sup (^G(t^y)^ . -^ < t < ^^ ^x^`• 5M }^
then ii depends only on the Hampers N,^^ and L.
Using Lemma. 1 we now rroveo
Lomma 2_. Suppose G satisfies (A3) and (A1+). Then G satisfies
(A^ 1 wh^^i •r^ N ctnd K .: K
	
t+.re the :z^.mb^i • s obtp i reed in '^c^ramn 1
.^.._.._	 ...._.	 1	 	 ____^
t^oove.
____.__
Proof. Let K =^ Kl and N be given by Lemrrla ] .
We must shag that there exirti,s a rnunr,er K O < K 1 such
^.	 that (A6) iFi t2•u^^. Fir a contradictiuri we supposE there is no
such K0. Tl,^:n for each pcsitivc integer n there exists, tiumUers
yn and to such thet
Yn - GN (t n ^Yn ) I ^ K - 1^n.
BV ^uss'Lly twtln^ a suLaequence we rosy assume that y n -^ y0^
GN (t rl , yn ) -i g0 and f(tn ) -^ f0 as n -+ ^. Note that ^y0 ^ s K.
Define xn ^: ^.Yn + f ( gin ) so the.t xn -► x0 = Qy0 + f0.
Write x0 end Y = NQuO ir, terms of their components
x0 _ col(xOl,xr2 ),	 r = NQgO = col(Yl,YL ) .
Def^_ne
U	 if z - 0
g'^ (z) _
	 ^ rl '	 i f z = ^ x01_ I	 .
Y2 ^	 i f z ^- ^ x02
r
8^xten3 g*(^)
extend ^" as
Z ^ 0 bncl .l^t
col (g^ (xl )^ ^;
N^ZEO .
linearly btt^^r^cn the I^oints
n constant or the reraaininG
F^ (-^) ^ -g" ( •r,) when z <
"(x,^)) Tor all xl and x2
0^ ^xp11	 and IY021,
Dart of thf^ hal f ling
0. Define G * (xl^x2 ) r
Nate that G*(x0) _
The function G * define3 ire this way sa^,^sfies (A3) and
(/1^+) and has th^^ same urger bound h a g the function G in
Lcrrunas 1 and 2. Since
	 ^ y0 1 ^ K, Lt ► en Lewna 1 imp] ies that
I YO - QG* (Q v^o+ro )r^-1 l < K.
Ori the other hf^nd it follows by tt:e construction of G* and the
choice of y0 and g0 that
IYO - QG^(Qy0+f0 ) N 1 ^ = Iy0 - QG* (x0 ) N 11
' y0
 - 60 I ? K .
This contradiction completes the proof. Q.E.D.
Theorem 2. Suite se (Al_4) arz true and b = sup (i r'(t) I : t ? 0} .
^,	 Consider the perturbation problem
t
`	 (^)	 x(t) = F(t) - I A(t-^) (^( J ,x(^)) T P(s,x(s)) }as,
0
^:^
"^
^;
9..r
wh
..^
P is continuous in (t,x).	 i •n^n for• Hny	 E > 0	 the
exiatg f^ n^^1b^r•	S > 0 such that ^ f
sup ( ^ ? (t , x )^ • t ? 0, ^Q(x-^'(t))^ ^ V•' b i ^ E;) < S
then the solution X(t) of (P'F) exists for atl t -^ 0 and
sat:ist'ics the inequality
Pruuf. Define II(t,x) 	 G(t,x) + P(t;,x) and let IN(t,y) =
QtI(t,Qy { F(t))N 1 . Given Kl = v^ b + ^ chocsc N an3 KU
using Lerruna 2. Choose fi < N(Kl -KO ) I Qj 1 so that
QP(t^^2Y - ^ ' (t ) )N l ^ ^ ^ Qj bT^-]. < Kl - K0.
If ^y^	 K1 then by the choice of o one has
^hN1(t,Y)^ s I GN (t ^Y)I + ^Q-°( t^Q,Y - F(t))N-ll
2'hus :i(t,Y) satisfies (A'^) 3nd (A5). Now apply Theorerl ^+ of
[1], Q.E.D.
10
r
Z`^icorcin ;,	 ;:os^th _coefficients 1'^A c_	 G	 f (^,) sa.t
(A 1 - 4 ). D'^ ^^ f^^ s s^tp (^ f(i,) ^ ^ t ? 0} . Su^ that g{v!•n
a_riy A > 0 thE^re e_x3st: a pcsitiv^, cunt{nuous,^i+nc:rc:.lsin^ flu^..tion
Cx(u) such that
Cs(t,U+x) _ ^(t,x)}/u ^ a (I L^I)	 (lUl ' A)
vniforml_y for all t ? 0 and all x such that ^Q(x-f(t))^ s
^^
f	 + 
1+ .^^	 Then ^;ive ri F^cry ^ > 0 there exists a positive riuniber
----- — -
b such ttitit whcrrever
..^	 _^_.
i) F (t) is any conti.nlzous function : ati sfyinE; !^ f-^'^^
suP (^f(^.) - F(t)o t ' 0} < S^
ii) P(t,x) is sny continuous i^unction satisfyinG
sup	 {IP(t,x)I ; t ? 0, IQ(x	
-
f ( t ))^
	
^-	 Vcilfll	 t	 1+} < 8,
and
iii) X(t) is the uriiaue solution of^^^
then	 ^ x (t)
	 - X (t) ^ s ^	 for al_l t ? 0.
Proof. Define y(t) = x(t) - X(t)^ ^(t) = f(±) - F(t) and
H(t^y) = G(t,y+X(t)) - G(t,X (t)) . Then one has
t
y(t) _ ^(t) _ f A(t_s){H(s,y(s)) - P(s,x(s))}ds
0
or a;,rrstolic^lly
11
1Pt Y =^ Qy^ ^.IJ 	NC^Ia end f{I^(,L^Y) _^ Q^{(t^aY)I1 1 so th^.t
(? • 1 )	 Y = (^ + ^^' (Y - xN (Y) + QI'(x)) Iv 1 ) - _-I,^Y•
If ^;N is the reso.Lvent of ^' that is
then any equation of the form Y = S - A NA Y ma.y be written in the
equivalent forri Y = S - ^ 1^5. Applying this to (2.]) and using
the rel_atien (2^2) one cari calculate
^d^AT1*(Y - --1V (Y) {^ QP(X)N 1 ^ ^
(^.3)
	 Y - ^ - 
- ^T1#^ + - ^V*(Y - ^1 (Y) + QP(X)N 1}
pr
t
( 2 .3')	 Y(t) _ ^( t ) - @ f ^(t- s)^(s)ds
0
t	 _
f HN(t-s){Y(s) _ ^^,J(s^Y(s)) + QP(s^X(s)N 1}ds.
0
^:
- --^---- -----
	
- •IEEE
Mj (t,x) _ (g(t,x j +Xj (^)) - g(t,X j (t))}^xj (j ^ 1,2).
12
Define SO = ((ti,x): t Z 0, IQ(x - F(t)))	 fIIFII + 1}
and let sl = ((t,x): t ^ o, IQ (x - f(t))I ^ f ilf il + 4}. If
li^ll = II f_FII < 1., and i f (t, x) E so then	 .
I Q(x - f ( t )) I f I Q(x - ^' (t)) I i I ^I II f -FII
^2iIFIl + 1 + ^•1 ^ ^(IIfII + 1) + 1 ^-
x/211 fll + ^ .
Therefore, SO L S, if Ilcpll < 1. By 'I'tieorem 2 tYiere exists a
number SO > 0 such that if I T' (t, x) I < ^0 or, SO then X (t )
exists for s.11 t ? 0 and (t,X(t)) e S0.
Write N(t,x) in the fot^n H(t,x) _= col (P41x1,M2rc)
where
Then Y - T-iN (t,Y) can be mitten in the form
M 1+M2
1	 ^]V
Y - FiN (t,Y) = A(t,Y)Y, A =
Ml-M2
.^:
M1-ML
^N
Ml+M2
1 _ .^_
If M(t,x) = col (MI (t,xl ), Mc (t,x2 )) and if IM(t,QYI < rT then
the norm of the matrix A is
13
E,^ _ 1 - ^ M(t,QY) ^ ^N = max (1-6'.l^I^', 1-I vi2^N} . ^'or any number
K > 0 if ^Y^ s K^2, then since (A^ < 1 one has ^AY) ^ K^ 2 . If
K^2
	
^ Y ^ s K, then either ^ Y1+Y2 ^ or ( Y 1-Y2 ^ ^ K^2. Therefor•e^
the i^ypotheses of the theorem imply that ^M(t^QY)I ? o:(K^2) > 0
for some function cx(u). This means that
Y-^( t , Y ) 1 1 — a (K/2 ) /N	 ^K/2 s ^ Y^ s K) .
Consequently for any given ^ > 0, if K = ^^ ^ then there exist
positive numbers N and KO such that if ^Y^ s ^^ ^ then
^Y - HN (t,Y)^ ^ KO < ^^ ^2. The number 8 in the conclusion of the
present theorem will be chosen so that 8 s min {50 1} acid such that
4^^f - F;^ + 2^ P(t^x) IN 1 s oS s e _ ^,^ KO for all (t a x) E Sl . For
this choice of 8 we shall show t:^at ^y(t) ^ _ ^ x(t) - X(t) ^
	 ^
for all t ? 0. Equivalently since Q = Q 1 and ^Q^ _ ^^
then we may show that ^ Y(t) ^ _ ^ Q(x(t) - X(t)) ^ s ^^^.
Let W = (z E C[0^^) o ^ z(t) ^ s ^^^ far all t ? 0}
and let TZ be the map defined by the right hand side of (2.3)^
thst is
t
Tz(t) = c^ (t) _ Q f ^(t- s)Q(s)ds +
0
t	 _
f RN (t s){Z(s) - ^1(s,Z(s)) - Qr(s,X(s))N 1 }ds.
0
..^ -	 •= •^
1^
t
IIy Lerntna 1 of [ 1] the matr. ix KN t L1 (0, «^) cnJ. f I ^ZN (t- s) i d s < 1.
0
for alt ^: 2 0. t'^ierafore, i f z f W,
t
0
N V ^ nlax P ^; x	 RN t-s ds ^+	 hN t-s KOd^
S1	 0	 0
s P V^ ^I^D^I + ^/^ tax I F(i.,h) I N 1 + y,0
S1
^^6s+ KO = e% ^.
Tflis shows that Tz E W if z E W. If the space C[0,^) is given
the topology of uniform convergence on bounded subsets of [0,^),
then it becomes a locally convex linear tc;^oloSical space with the
additional property that T: C[0,«^) -^ C[0,^) is a. coronletely con-
tinuous ma,n. Since W is a. closed bounded convex subset of
C[0,^) and T tnana W into itself then the Scha^^ r fixed point
theorem applies. This means that (2 0 30 h:^.s at least ene sol^:tion
Y(t) such that i i(t) I
	 S^V G for all t ? G, But the function
H(t,Y) is locally ^,ipschitz continuous ir_ Y so that the solution
of (2.3^) is unique that is Y(t) = Q(x(t) - hit)). Q.E.D.
15
III. Quasi-Periodic Fut,etiuris.
L!^t kl,k ,^ , ...,ktn be positive constants r:hich are
linearly ind^:r^^endcnt over the integers. Let k denote the vector
k = ( kl , k2^ ..., k^) with m z 1.
Definition. A function ^ (t) wi.11 be called quasi^^eriodic with
fundarnentaL f'r'equencies Y, if and only if' there exists e. function
_____
m (p) = m ( gl, ©2 , . ^ ., ©rn ) continuous in 8 and reriodic ir. each
variable 8. of period 27r such that
J
cp(t) = 4(kt) = ^D(klt^k2t^...,kmt)^ 	 -^ < L < ^.
E_;ch quasiperiodic is easily seen to be almost periodic.
If m = 1 su tha.t k = kl then the quasiperiodic function is
actually periodic.
According to the results, in [1] if x Qy + f(t) then
for ary N > 0 the function y(t) solves (E N ) . Conditions are
given which guarantee that y(t) tends asymptotically to an almost
periodic function p(t) where
t
(3^ 1 )	 P(t) = f ^(t-s)^P(s) - GN(s^p(s)))ds^ -^o < t < ^^
_^
T'ne unction p is ±he unique solution of (j.l) if N is
sufficiently large.
16
The uim iii ^iiis section i^ to give sufficient con^iitioiis
in order that p(t) = P(^:t) is q^zr^.si^:eriodie rind P( p ) is analytic
^.n p . Assiune:
(A"j) G(t^x) = Y(^a^x) and f(t) = cp(kt) are quasiperio^.iic in t
with fun^ir^mente.l- frequencies k. P^oreover^ Y(O^x) and y(0) are
real analytic ft^r^^:ti.ons of
	
(6,x) and p respectively in regions
(3.2)	 U(^G ) = ((O^x): ^7ni 6 j ^, ^lm x i ^ < S^j, .^o < Re x l ^ Re Oj < o0
for 1 ^ j ^ m and i = 1^2}
and
(3.3)	 D(8^) -- (0: ^ Im Oj ^ < b0^ _^ < Re 8 j < ^ for 1 s j s m}.
Unc?er t^iis assumption it foll.o^rrs t^lat the function
_,
YN (kt ,Y) = QY'(kt^Q„Y + ^(kt))N
is also quasiperodic and anal^►tic in U(8^). If the solution of
(3 ,1) wa.s quas ipErodic ^ say p (t) = P (kt) . then (3.1) co^_ald be
rewritten as
t
?(Kt) = i t^.(t-s){P(ks) _ YN(ks^P(ks)i}ds
_^
Gf R11 (s) (P(kt-ks) - rTJ("t-ks^P(kt-ks))}dso
1'j
Since k = (^:1^ k2^ ..., kn	 is a vcc ^^^r of .^ ire ..^_z • ly indc;^ ^,n^i^_:^t
fregtiericj es ar11 F^(E^) is continuovu in 0^ then tYri..s is equivalent
to the equation
(^.^^)	 P(0) ^ f0 I2N (s) (F(0-ks) - r^1(4—k:;,F(0-k ;} } }ds.
Converscly^ if P(G) is any continuous solution of (3.^) such that
P(0) is 2;^_periodic in each variable 0^, then p(t) ^ p(kt)
will solve (3.1). Th^refore^ our proble.^n is reduced to finding an
analytic an3 periodic solution of (j.l^).
For any S > 0 the symbols D(8) or U(8) will denote
regions defined in the manner of (3.2) and (3.3). Using this no-
tation we now prove the f'ollowin^;
Ti:20rem 4. 3unpose (!'►1..,) a.Zd (A6 -7) are true. Ther. tY^ere exists
a 5 > 0 such that (3,4) has a solution P(8) which is real
analytic in © E D(S) ar.d 2;r_per;oaic in each variable 0^. 	 '
Prc^i. Let N, KO and K1 be the number:: given by (Ao). For any
8 in the interval 0 < 5 < 80 let ,}(S) denote the set of a
functions 7(B) real analytic in 8 E D(s) and 21r-periodic in each
variable 8.. If .^(8) is given the tocolog,^ of uniform con-
,
vergence on compact subsets of D(8), ther_ this family beccx^es s
lccally con ^•;.x linear topologica.'.. apace over the real numbers.
Define
..
18
S u ('L ^ .^(&): ^L(0)^ ^ K1 for al.l 0 c D(5)}
where K1 is the constant in (A6). T'ien S is a closed convex
nonempty and com^^act subset of ^(fi). Since (AE) is true for
G (t, x)	 r(kt^:c) and since k =- ( kl , k2 , , . ^, ^cm) is a vector•
 with
linearly independent components, tht:n
I Y - Y(e ^Y) I ^ KO < K^ if ^ y^ s Kl ,	 (g,Y) E ^(8)
and (8,y) is real. By continuity there exists a number fi with
0 < 5 < SO such that
Y - r( e , y ) I ^ Kl	 if ^ yl s Kl 	srid (O,Y) E U(s)
where (6,y) is now allowed to be complex. This is tre 3pproprizte S.
For any Z E S define
TZ(6) = Of i^(s){Z(8-ks) - YN (0-1:s,Z(6-ks))}ds, © E D(8).
By Lemma 1 of [1) the matrix RN (t) E Ll (0,^) with f ^B^(t)^dt s 1.
0
This meats that TZ(6) is well defined, T7. E t(8) and
^TZ(e)^ s 0 ^^(s)^Klds s K.l.
i
- ---
	
- — —
	
-
--	
--	
_	
;..
f'
^9
In pai• t.icular Z' map:; S into S cont.'.nuously.. By the Schnuder	 ^ .
fixed point theorotn T has a fixed point. ^.E.I).
i
r	 r
•^^
J. Outline of the Proof of 'i'heorert 1.
7'}^e re^ults jn sCCtion 2 of [1] show that (1.3)	 is
equivalent to (t^.)	 with	 xl (t)	 n	 u('.:,0) end	 x^(t) ^ u(t;,'►r),	 with
°°	 2	 °^f^ (t) = F'0/2 + ^; F'n exp(-r, t), f^(t) = F^2+ ^; F' n (_1) nexp (-n?t)
n=1	 k=1
erid with
7r
Fn	 (2^7r) f F'(r.) cos nxdx.
0
It is easy to prove that
the uni`'orm norcu over 0
C 2 [O,^r]. The res7xlts ^.n
x2
 (t) vary continuously
as f anti g vary.
fl(t) and f2 (t) vary continuously in
t < ^o as F varies in the norm of
section 2 above show that x l (t) and
(again in the uniform norm over 0 s t < ^o)
The results in section 6 of [1] are sufficient to see
that xl (t) and x2 (t) are asymptotic to almost periodic functions
pl (t) and p2 (t) such that p i t) = col (p^(t),p2(t)) solves
(3.1). If k i and k2
 are linearly indeaPnd^nt, then Theorem 4
above implies that p(t,) is anal;^ic and quasiperiodic with
fundamental fz^equencies k l and k2 . Finally, note that since
p(t) - x(t) ^ -^ 0 as t -+ ^ wtiera p(t) is almost periodic and
x(t) varies continuously with f and g, then p(t) varies con-
tenuously (in the uniform norm over -^o < t < ^) :rith f and g.
r21
kl ana k.^ are linearly dependent over the inteEers^ there theL
:^me conclusion follo^.,rs but ^ritiYi F ^t) a periodic f'unctior..
,^
^-	 A
w
22
RCf'erences
1. k. K. t•iiller.^ Almost periodic behsvior of solutions of a non-
linear Volterra syst;e^n^ Quarterly oi' Applied riathema.tics^ to
appear.
2. C. Co Lin s F-^ydro^.fynarnics of liquid. helitun II I Phys. Rev.
Letters 2 0.959), 245_216.
3. N. Levinson A nonlineRr Volterra equation arising in the
theory of sujaer^ltzidity^ J. t.ath. Ans.1. AFpl. ]_(1,0-60), 1_11.
^l
.'^"


Almost periodic behavior of solutions of a nonlinear Volterra system, 2

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