R O O Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P R A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )
Z. K
amontand W. P
aw elski(Gdansk)
On a certain case of asymptotic stability of the integral у = О of the differential equation d y l d x = g ( y / x )
In [2] snfficent conditions of asymptotic stability of the integral у — у о of the differential equation
dy
have been given. They were stated in Theorem 3 of that paper and are quoted here in the following theorem.
T
heorem1. Let us assume that :
1° function f (oc, y) is defined and continuous in the plane set D
={{x,y): X e ( a ,
+oo), y e
( i / o - a ,y0
+ a ) }{where a > 0; in particular it may be a = + oo);
2° f(%, y) < 0 in D 1 and f { x , y) > 0 in D 2, where
■2>x = {(я, У): + °° ) , У е (Уо, 2/o+a)}>
J>2 = {( x , y): xe (a , + o o ), т/e (y0- a 5 y 0)};
3° Йеге the limits : . •
lim f{x, y) = <55 (5 < 0 /or ÿe (y0, y 0 + a),
x->+oo
гЛг/
lim f(x, у) = у, у > 0 for each ÿ e (y0 — a , y 0) .
X — * + c o
V / y
Then y = y 0 is a solution of (1) which is asymptotic uniformly stable with respect to the initial conditions given on every fixed segment
К = {{x, y): X = x0, ye <y0-jf f , Уо + Р>], where 0 < ft < a and xQ > a.
In the case when <5 = 0 or у = 0 or else ô = y = 0 a stable solution
У = y0 may be or may not be asymptotic stable. Corresponding examples
can be found in 2 and 3 of [2].
We shall consider here the case when both limits in 3 are equal zero and we shall give a sufficient condition, which, for homogeneous equations will decide of asymptotic stability of the stable integral y = 0. This stability need not be, as we shall show, uniform with respect to the initial conditions.
T
h e o r e m2. I f
1° function g(u) is continuous in ( — u0, u0), where u 0 > 0 (in partic
ular u 0 = + oo) ;
2° g(u) < 0 for U€ (0, u0), g(u) > 0 for uz ( — u0, 0);
3° through every point of the set JB 1 u U 2, where
exists, then the integral y = 0 of (2) is asymptotic stable uniformly with respect to the initial conditions given on every fixed segment
where x 0 < 0, 0 < y 0 < u 0 x0.
Proof. That the integral y = 0 of (2) is asymptotic stable follows in a simple manner from Theorem 6 of [1].
Indeed, putting in that Theorem a — 0 we see that integral y = 0 of (2) is asymptotic stable.
From condition 3° of our theorem we get that this stability is uniform with respect to initial conditions given on every fixed interval of the type К г.
Re ma r k 1. Theorem 1 must not be applied for investigation of asym
ptotic stability of equation (2) since then we have for every ÿ > 0 E x = {(a>, y): ccz (0, +oo), 0 < y < u 0 x},
E
2= {(x, у): же (0, +oo), — u0x < у < 0}
passes exactly one integral of the equation
( 2 )
4° for every b Ф 0
K x = {(x,y): x = x0, y e ( ~ y 0 , y 0>}
and for every у < 0
V/V
We shall now show that in this case asymptotic stability of the in
tegral y = 0 of (2) does not have to be uniform (with respect to the initial conditions given simultaneously on all segments K x for x 0 > 0, 0 < y 0
< u 0 x0).
The integral y = 0 of (2) is asymptotic stable uniformly with respect to the initial conditions given simultaneously on all segments of the form K x if
1. this integral is asymptotic stable,
2. for every e > 0 there exists a constant A > 0 depending only of s such that for x > xQ-\- A holds the inequality \(p(x) \ < e for all integrals
<p {x) originating from any segment of the form K x = {{x, y): x = a>0, y e ( - y 0, y 0>}
for every x 0 > 0 and fixed y0e(0, u 0 x0).
T heorem 3. Assume that g(u) is a function satisfying conditions of Theorem 2, and also, let g(u) be strictly decreasing in ( — u 01 u0). Then asym
ptotic stability of the integral y = 0 of (2) is not uniform with respect to the initial conditions.
Proof. We shall first show that for fixed y 0 and any 0 < e < y 0 there is no constant A > 0 depending only on e such that x > x 0 -j- A would imply |p(x)\ < s for all integrals cp(x) of (2) originating from any segment of the form
K l = {(x, y): X = X 0 , ye ( ~ y 0, y 0}}, where xQ > 0, u 0 xQ > y 0.
To show this we shall construct a sequence of segments
<#<>) АоУ, ( x x, xx -}- Af) , ... , (xn, xn Ч~АПУ, ... ,
where A 0 > 0, A x > 0, ..., A n > 0, ... and a sequence of integrals
<pQ{x), <рг{х), ...,<pn{æ), ...
of (2), such that the following conditions hold:
(3) lim J.n = +oo,
П— >oo
(4) Pn(^n) У 01 tpni^n "b -^-n) =
where y 0 and e are fixed and such that 0 < e < y 0. We fix next a point
#0 > 0 such that x 0 > x, where y 0 = uQx, and <pQ{x) will denote an integral of (2) satisfying the initial conditions
(5) Poi^o) — Уо
and a constant J.0 > 0 is chosen so that
(6) y 0 (x 0 + A 0) = e.
Such a constant exists, since 0 < s < y 0 and <p 0 (x) decreases asympto
tically to zero when x -» + oo. Continuing our procedure we denote by xx a coordinate of the point in which the straight lines у = ---x s
x 0 -\-A 0 and у — y 0 cross, and by y x(x) the integral of (2) satisfying the initial condition
(7) — У о*
The constant A x > 0 is chosen so that
(8) y x{xx + A x) = e.
Since
е < У о = t
л/
jlX q -^A q
then
(9) xx > #0 + A 0.
We shall now prove that
(10) A X> A 0.
To show this, in view of the strict monotonicity of y x{x) and of the equality
<Pi(®iAAx) — £, it is enough to prove that (11) <px{æx + A 0) > s.
It is easy to verify that the integral y = y x{x) of (2) can be represented as y = — y 1 0 (cx), where
c
( 12 )
Thus
х 0 -\-А 0 s Уо ’
(13) y x(x) = — у A --- x , N Vo lœo + A 0 e \ x.
Assembling (5), (6), (7), (12), (13) together we obtain (14) <Pi{ccx + A 0)
1 1
= —(ро(схг + сА0) --- ^(e®!) - [y0(®o + ^o) - VoW] + «
G G
= A 0 <Po(cxx + excA0) — A 0 y' 0 (x 0 + e 2 A 0) + e
— A q \<P q (X q -\- A q -f- c 6 xA q) —cpo{xQ-j- 02Ao)] + s
— A q [<P q ( x )~]—( p 0 { x )'\ -f- e,
where 0 < 0X < 1, 0 < 02 < 1, x = ж0 + А 0 + свхА 0, x — ж0+ 62 A 0.
We see that x > x and since <p 0 (x) and g(u) are strictly decreasing we get
Й И - Й И = 9 <Po(x)
X
<Po(x)
X
> 0 ,
and from this last inequality combined with (14) follows (11), because, according to our assumptions A 0 > 0.
In general, after defining the (n — l)-th interval (xn_1, xn__ 1 -}-An_1) and the (n — l)-th integral <pn-i(x) so that
9 V -l( ^ r a -l) == Уо and
tPn—i i ^ n —i A A n _y) e
I
hold, we define next interval <xn, xn + A ny and next integral <pn{x) in the following fashion:
Let xn designates a coordinate of the point in which the straight у - --- x e
'X'n—i A n_ j and у = y 0 meet.
Let, further, <pn(x) designate an integral of (2) satisfying the initial condition
(15) <РпЫ = Уо-
The constant A n > 0 is chosen so that the equality
(16) cpn{xnA A^) = s
is satisfied.
For every n such a constant A n exists, since 0 < s < y 0 and <pn{x) asymptotically decreases to zero when x -> + oo.
Since s < у о = ---— xn, then e Я'п— 1 -^-n — 1
(-17) æn - \A A n_i,
(18) A n > A n_l .
The proof of this last inequality is similar to that of (10), so we shall omit it here. We shall now show that condition (3) holds.
In fact, from the mean value theorem and from (15) and (16) it follows that
Уп i^n ~b A n) 9^n(*Li) ■^-пУп(^п)
where (19)
and further
( 20 )
£ ïl X П " h A n 5
A m = У о
<Pn ( £n )
f i - y 0
(17), (18), (19) yield together a sequence of inequalities
> Хп__г + А п_ 1 > жп_2 -f- A n_2 + ^ln_ 1 > . . . > X 0 + ^d0 + -di + + . . . -\-Ап_х > X 0 + m A q , from which we have lim = + oo, and
n->oo
lim
Пг^СО
rfJ J n )
£» - 0.
since e<<Pni£n)<yо thus
îsow, from the above and (20) we obtain (3) since g(u) is continuous in ( - u 0, u0).
Condition (4) is clearly satisfied for every n = 0 , 1 , 2 , . . . To end the proof of Theorem 3 it suffices to notice that for fixed y Q and 0 < s < y 0, for every number A > 0, we may find, by (3) and (4) such A n > A, a point xn and an integral <pn(x) of (2) originating from the end of the interval
K n = {(ж, у): x = xn, у * ( - у о, у o>}, that the following holds
*Pn {'fin ~b A^) £.
From this it obviously follows
Vni'fin "4" A) £,
and so stability of the solution у = 0 is not uniform with respect to the initial conditions.
Re ma r k 2. From the way the theorem 3 was proved it is evident that it may be slightly generalized if we assume that the integral у = 0 of (2) is asymptotic stable in place of Condition 4 of Theorem 2.
E
x a m p l e. Theorem 3 implies that the solution у — 0 of the equation dy _ _ y _
dx x
is asymptotic stable in the set
D = {(x, y): Xe (a, +oo), y e ( - o o , +oo)},
where a > 0, but this stability is not uniform with respect to the initial
conditions.
R e f e r e n c e s
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