A Continuation Result for a Bidimensional System of Differential Equations

Revista INTEGRACIÓN

Universidad Industrial de Santander Escuela de Matemáticas

Vol. 13, No 2,p. 49-54, julio-diciembre de 1995

A Continuation Result for a

Bidimensional System of Differential

Equations

In this note we present 'sorne sufficient conditions for the global existence of all solutions of the bidirnensional system x' =Q (y) - IJ(y) f (x), y'

=

-a(t) 9(3:), which contains the classical Lieuard's equation.

x' = o(y) -{3(y)f(x),

y' = -a(t)g(x),

where the dots indicate differentiation with respect to t and o, {3,

f

and 9 are continuous real-valued functions and a is a positive continuously differentiable function on [O,+oJ, and define

g(x) =

fox

g(s)ds, A(y) =

foY

o(r)dr.

If in (1) o(y) = y, {3(y)

==

1 and a(t)

==

1 this system reduces to well-known generalized Lienard's equation:

Let C(lR) and C I(lR) denote the family of continuous functions and continuous increasing functions on lR, and let:

CS(lR) = {h E C(lR) :xh(x)

>

Ofor x

#

O},

CC(lR) = {h E: (CI(R)nCS(lR)}, CP(lR) = {hEC(lR):h(x»Ofora11x}.

Some attempts [1,2,4,5,1l,12} have been made to find.~ufficient condi~ions on

¡

and 9 for solutions of (2) and its non-autonomous form:

to be continued in the future urider the condition 9 EC S(I R). In [5} and [9} we gave some conditions so that a11solutions of (1) are continuable to the future, considering that

¡,

9 EC S(IR). In this paper we obtain sufficient conditions for the global existence of solutions of (1) without make use of above condition. The problem of continuability of solutions is of particular importance in the qualitative theory. So, in various earlier papera (see, for example [6-1O,13}) we studied various qualitative properties of solutions of (1), taking in account the results of [5} and [9}.

On the other hand, the goal of this work is to illustrate how theresults obtained in [3}for the equation (2), can be generalized to the system (1). In this paper, we assume the uniqueness of the solutions of (1) and z(t; ta, zo) denote the unique solution of (1) with z(to; to,zo) =zo where z(-) =(x(-), y(-)).

Throughout this paper we also assume the following conditions hold:

a) a E CC(lR),

b) {3E C P(lR),

e)

¡,

9 E C(lR),

d) a E C P([O,+00

1)

n

C1([O,+00 )) where C1denote the family of functions

3. Bf(x}g(x} ~ - [G(x}

+

'Y] lor &11x.

4. A (±oo) =±oo.

a'(t)

5. a(t} > -1lor al1t,

Proof.

that:

for sorne T ~ tO. Choose To < T sufficiently close to T so that sorne solution

i (t) = (x(t ),

y

(t}) of (1) exist on [To, T] by an application oí local existence theorern.

Define L

=

{z~ =Az(To}

+

(1 - A)i(To} :O ~ A ~ 1}and let:

Since zo =i(To}, by continuous dependence we see that O

<

A. ~ 1. We clairn that z(t; To, z~.) does not exist up to t = T. If it did, then by continuous dependence, there would exist a neighborhood of z~. such that a11 solutions passing through that neighborhood exist at t =T,contradicting the definition of A•. This establishes the clairn. Thus we have:

for sorne To < T. ~ T.

We sha11 proof that the set 6. = {z(T.; TO,Z~.) : O ~ A < A.} is unbounded. By continuous dependence and (4), there exist sequence {tn} and {An} such that tn -- T.- YAn -- A. as n -- +00 and

for some p E 6.. On the other hand by local existence at (T.,p), the solution

:(t; T., p) exists on T. - f ~ t ~ T. for some f > O. Therefore by continuous

dependence:

contradicting (5). Hence 6. is unbounded.

Thus we can choose a sequence {z(T.;TQ, ZAn)} of solutions such that:

A(y) V (t, x, y) = a(t)

+

G(x)

+'Y.

I a'(t) A(y) V(l)(t, x, y) = - -(-) -(-) - {3(y)f (x )g(x), a t a t V(1)(t,x,y) ~ V(t,x,y).

I3y the continuity of V and the compactness of L,there exists WQ > O such

ror

all n, whiclr ilJa contradiction to (9). Hence, there exists a constant k > O euch that:

Iv(T.¡To,zAn)1 ~ k for 8011 n.

From this and by the continuity ofQ we have that:

Integrating the first equation of (1) on to ~ t

<

T and taking in account (10)

we obtain:

xo

+

a( -M)(t - to) -

[t

(3(y(r))! (x (r)) dr

lto

~ x (t)

~ XQ

+

a(M)(t - to) -

[t

(3(y(r))! (x (r)) dr.

lto

Supposc that x(t) --+ +00 as t -- T-, then exists é > O such tliat z(t} > O

on [T- 6,T). Makingt -- T- in the right hand side of the abov~ inequality

wc havc:

1:

¡3(y (T))! (x (T)) dr

l

T

-f¡

hT

=

{3(Y(T))!(x(r))dr+ {3(y(T))!(X(T))dr ~. T-f¡ ~ -00

[T

(3(y(r))!(x(r))dr >

o,

lT-f¡

contradicting (11). Thus x(t)

f-+

+00 ast -- T-. By similar way we obtain

that x(t)

f-+

-00 as t -- T-. This is a contradiction with (3). Hence, 8011

solutions of(1) exist in the future .•

The fo11owingexample does not satisfy the conditions of the Lemma

1

oí

[9J,

but do ours in the above theorem:

{X

+

2c, !(x) =x(x - d)(x

+

d) and g(x) = . -x, x - 2c, x <-c

Ixl ~

c c < x

Remark 1. lEQ(y) = y, {3(y)

==

1 and a(t)

==

1, the conditions oE Theorem

reduce to the obtained in [3,Th5.1].

,

Remark 2. Notice the advantages oEpresent prooEon the [3,T h.5.1], taking a unique Liapunov's Eunction (7).

[1] T. A. BURTON, A Continuation result 101' differential equations, Proc. Amer. Math. Soc., 67 (1977), pp. 272-276.

[2] J. R. GRAEF, On the generalized Liéna1'd equation with negative damping , .J. Differential Equations, 12 (1972), pp.34-62.

[3J T. HARA, T. YONEYAMA ANO .J.SUGIE, Continuation r'esults 101' differential e~lJ,tions by two Liapunov lunctions, A~m. Mat. pura appl. (IV), XXXIII (1983), pp.79-92.

[.(] - Necessary and Sufficíent Conditions lor' the Continuability 01 solutions 01 the system x'

=

y - F(x), y'

=

-g(;¡;), Appl. Anal., 19 (1985), pp.169-180. [5] J. E. N ÁPOLES. On the continuability 01 solutions 01 bidimensional system",

submited for publication.

[6] On the global stability 01 non autonomous systems, submited for publication. [7] J. E. N ÁPOLES ANO .J.A. REPILAOO, On the continuation and non oscillation

01 solutions 01 bidimensional systems x'

=

a(y) - l3(y) f(;¡;), y'

=

-a(t)g(:¡;),

Rey. Ciencias Matemáticas, XV, Nos 2-3 (1994),Uniy. of Hayana, to appear. [8] On the boundedness and the asymptotic stability in the whole 01

so-lutions 01 a bidimensional system 01 differ'ential equations" Rey. Ciencias Matemáticas,Uniy. of Hayana, to appear.

[9] Continuability, Oscíllability and Boundedness 01 bidimensional system ;1;'

=

a(y) - l3(y) f(x), y'

==

-a,(t)g(x), Rev. Ciencias Matemáticas, XV, Nos 2 3 (1994), Uniy. of Hayana, to appear.

[10] Sufficíent conditions 101' the oscillability 01 solutions 01 system x' =a(y) -l3(y) f(x), y' = -a(t)g(x), Rey. Ciencias Matemáticas, XV, Nos 2-3 (1994), Uniy. of Hayana, to appear.

[11] J. E. NÁPOLES ANO A. 1. RUIZ, On the behaviouT' olthe solutions olthe differ'-ential equation x" +g(x)x' +a(t)f(;¡;) =O,(1) , Rey. Ciencias Matemáticas, VI, No.1 (1985), Univ. of Habana, 1985, pp. 65-71 (spanish).

[12] - On the behavioUT' 01 the solutions of the differ'ential equation ;¡;"

+

g(x );¡;'

+

a(t) f(x)

=

O, (I1), Rey. Ciencias Matemáticas, VII, No.3 (1986), Uniy. of Hayana, pp. 35-39 (spanish).

[13] A. 1 RUIZ ANOJ. E. N ÁPOLES, Conver'gence in nonlinear' system with 10r'CÍ'nq ter'm, submited for publication.