A N N A L E S SO C IE T A T IS M A TH EM A T IC A E PO LO N A E Series I : C O M M EN TA TIO N ES M A TH EM A TIC A E X I X (1977) RO CZN IK I P O L SK IE G O TO W A RZY STW A M A TEM A TYCZNEGO
Séria I : P E A C E M A TEM A TY CZN E X I X (1977)
Ta d e u sz Ja n k o w sk i (G dan sk)
On periodic solutions of the equation x'(t) = F (t, x(t), x'(t)) In [1] T. Burton considered a system of non-linear differential equations
( 1 ) œ'(t) = f(t,œ (t)),
where
æ'(t) == (x[{t), ...,œ'r(t)), f{ ) = (Л( ) , . . . , / r( )).
He proved (by the successive approximations method) under certain quite simple assumptions that all solutions of ( 1 ) are either T-periodic.
In this paper we considered a system of non-linear differential equations
(2) æ'{t) = F (t,x (t),æ '{t)),
where
я' (t) = {æ[ ( t ) ,..., œ'r(t)), F ( ) = ( F 1{ ) , . . . , F r{ )).
B y ''substitution y (t)= æ '(t), equation ( 2 ) with initial condition æ(*o) = #o is equivalent to the following one
(3) - where
t
y { t ) = F \ t , jy {x )d r + oc0, y(«)),
*0
t t t
Vi*) = 2 /r(*)), fy (T )d r = ( f y 1(T)dT,..., J y r(r)dr).
<0 ^0 *0
If the function y (t,t0,x 0) is a solution of (3), then the function t
®0) = j У i?) Xo)dr + x0, 1о
is a solution of ( 2 ) with initial condition x(t0, t0, æ0) = æ0.
In this paper we proved (by similar assumptions as in [ 1 ]) that all
solutions of (2) (or (3)) are either T-periodic. The paper contains a
generalization of results of [ 1 ].
250
T. Jankowski1. Periodic solutions — ordinary iterations. We introduce A
ssu m p t io nH . 1° The vector function F { t , x x, x 2) is defined and continuous for te(
—oo,
o o ),x {
=(xxi, . . . , x ri), х{сВг
,г = 1 , 2 ,
2 ° for any (t, x x, x z)€( — о о, o o ) x B r x Br we have (i) F {t, X1, X2)eBr,
(ii) F { —t , x 1, —x 2) = —F ( t , x 1, x 2),
3° there exists Te(
0 , o o )such that for any
(t , xx, x 2)e{
— o o , o o )x Br x Br F{t + T, хг, x 2) = F{t, x x, x 2).
!Now we can formulate the theorem on the periodic solutions of ( 2 ) (or (3)). We have
T
h e o r e m1. I f assumption H is fulfilled and if
1 ° for any (t0, x 0)
e ( — o o , o o )x Br there exists a unique solution ÿ { t, t0, x 0) of equation (3),
2° for any (t0, x 0)e(
— o o , o o )x Br the sequence defined by the relations Уо(Мо»®о) = 0> 0 = ( O , . . . , O ) ,
t
Уп+i it? ®o)
=
jJ
Уni1' ? to
jx o) dr
-j- x 0,yn(t, t0, x0) j ,
n = 0, 1,l0
is convergent uniformly for t€[t0, ^0 + T] to y(t, t0, x0), then {do) for any (t, tQ, x 0)e( — o о, со) x ( — oo, oo) x Br we have
У ni t? to, x o) — У nit, t0, X0) , 71 = 0, 1, ..., yn(t + T, t0,
x 0)= yn{t, t0, x 0), n = 0 , 1 , ...,
(b) the solution y(t, t0, x0) of (3) is continuous and for any (t, t0, x 0) e( —
o o , o o )x ( —
o o , o o )x Br ive have
y{ t, t0,
x0) = y (t, t0, xf), y{t + T, t0, x Q) = y(t, t0, x 0),
(c) for any {t, t0, a?0)e( —
o o , o o )x ( —
o o , o o )x B r we have x{ t, t0, X
q) = x{t, t0, x0) ,
x{t + T , t0, xf) = x { t , t0, x0) ,
where x(t, t0, x 0) the solution of (2) with the initial condition x(t0, t0, x0) = x 0, i.e.,
t
x{t, t0, x 0) = J y{r, t0, x 0)dr + x Q.
h
Periodic solutions o f an equation 251
Proof. Adapting induction we can easy prove part (a). Now if w->oo, then yn(t, t0, a>0)-+y(t, tQ, x0). The continuity of y(t, t0, xQ) follows from the uniform convergence of the sequence {yn(t, t0, ж0)} and continuity of all functions yn(t,t0, x0). Further, if n^-oo, then from (a) we have part (b).
The proof of part (c) is obvious.
E e m ark 1 . If the function F { t ,x x,<ict) is independent of x z, then from Theorem 1 we get the result contained in theorem of [1].
E e m ark 2. We can obtain of the uniformly convergence of the sequence {yn(t, t0, xf)} by the analogous assumptions as in paper [ 2 ].
E e m ark 3. If assumptions of Theorem 1 (except 2 °) are satisfied, and if
(i) the sequence {yn(t,t0, x 0)}, te[t0, t0-\-T] is convergent to the function y(t, tQ, x0),
(ii) there exist a Lebesgue-integrable function G: [t0, t 0 + T]->[ 0 , oo) such that
\F{t, хг, a? 2 ) I < G{t), t€[t0,t 0 + T], æ1, x 2eBr,
then ÿ {t,t0, x 0) — y(t, t0, x0), and the assertion of Theorem 1 is true with measurability instead of continuity of y (t,t0, x0).
2. Periodic solutions — approximate iterations. The yn(t,t0,x 0) is approximate solution of (3). If we want to profit by the approximate solution of (3), then we observe that we cannot appoint exactly the se
quence {yn{t, t0, £P0)}. Hence in place of the sequence {yn{t, t0, æ0)} usually another sequence {yn{t, t0, a?0)}, and consequently the relation given by part (a) of Theorem 1 is useless.
The sequence of the approximate iterations {yn{t, t0, a?0)} may be defined
t
Voit, h ,
®o) =
Q, Уп+i it,t0, x0) = F n{t,
f уп{т,<0, ®0)йт + h
~ \ ~ ^ 0 , ÿ n ( t , t o , ® o )) , n = 0 , 1 , • . • ,
where {F n} is a sequence convergent to the function F . Eow we can formulate the following
T
heobem2 . I f assumption H is fulfilled and if
1° F n: ( — oo, oo) x B r X B r-+Br, n = 0, 1, ..., and F n are continuous functions and satisfying assumption H,
2° for any (t0, x 0 )e( — oo, oo) x B r the sequence {yn(t,t0,x 0)} defined ty (4) is convergent uniformly for te[t0, t0-\-T] to the unique solution y(t,t0,x o) of (3), then the assertion of Theorem 1 is true with yn(t, t0, x0) instead of y j t , t0, x0).
6 Prace M a te m a tyczne 19 z. 2
252 T. Jankowski
R em ark 4. If there exists t0e( — oo, oo) such that for any (t , x , y)e
£ [|0, | 0 + 1 ]
х й г х5 г,
F n(t, X, y) — F (t, x, y), n = 0 ,1 ,..., then yn(t, t0, x0) = yn(<, ®ob w = 0 , 1 , ...
References
