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• ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACF. MATEMATYCZNE XXI (1979)

Ka z im ie r z Zim a (Rzeszôw)

A certain theorem on the oscillation of solutions of equation ( p ( t ) x j + q( t ) x = 0

1. Introduction. We shall consider the equation

(1) (p(t)x')’ + q(t)x = 0 , t e [ 0 , oo),

where the functions p and q are continuous in the interval [0 , oo) and, moreover, p(t) > 0 for re [0 , oo).

De f in it io n. The non-trivial solution x of equation (1) will be called oscillating if the set N = { te [0 , oo): x(t) = 0} is infinite.

Let H denote a family of functions h : (0, oo)->(0, oo) of class C 1 such that:

1° h’(u) < 0 for u e (0 , oo), 2° lim h'(u) = — oo,

ц-> 0

3° lim h(u) = + oo.

n~+o

For h s H define f h: [0 , oo ) x [ 0 , oo)->( — oo, + oo) by the formula:

/h(L«) 0 for и = 0 , t ^ 0 ,

(Ь'(м))-1 • [A(t) sin2(h(u))+В (tj] for и Ф 0, t ^ 0, where A(t) = q(t)-(p{t)) \ B(t) = (p(t)) \ t e[0, oo).

The purpose of this paper is to prove the following theorem:

Th e o r e m 1. The solutions o f equation (1) are oscillating in the interval [0, oo) if and only if for any h e H the trivial solution o f the equation

(2) u’ = f h(t, u)

is asymptotically stable for + o o .

2. Auxiliary remarks. Applying the so-called Priifer Transformation ([1], p. 332): q = [х2+(рх')2]*, (p — Arc tan (x/px') to equation (1), we obtain the system

(3) (p' = A(t) sin2(p+B(t), q' = —A(t)g sin (p cos (p.

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392 K. Z im a

We have obviously x(t ) = g(t)-sinç>(t) for te [ 0 , oo) and we get g(t) > 0 in the interval [0, oo) for each non-trivial solution x of equation (1). It is easy to see that each solution q> of the equation

(4) <p' = A(t) sin2 <p + B(f), fe [0 ,o o ),

is defined in the whole interval [0 , oo) and if q> is an unbounded solution of (4), then lim (p (t) = + oo [2].

*-► ao

Since the function A(t) sin2 (p + B(t) satisfies the Lipschitz condition with respect to q> and is periodic in cp, either each solution of (4) is unbounded or each solution of (4) is bounded [2].

Finally, the solution x = q • sin q> of equation (1) oscillates in the inter­

val [0 , oo) if and only if the function (p is an unbounded solution of equation (4) (cf. [2]).

3. The proof of Theorem 1. The proof of Theorem 1 is based on the above-given properties of solutions of equation (4). Indeed, let h e H and assume that и is a solution of equation (2). Then the function (p = h(u) is a solution of equation (4). Conversely, if q> is a solution of equation (4), then и = h ~ l ((p) is a solution of (2). Finally, we conclude for (p = h(u) that the solutions of equation (1) are oscillating. Indeed, taking into consid- ded solution of equation (4), i.e.. lim (p(t) = +oo. But this equality — by

I-+ со

virtue of 1°—3° — is equivalent to:

lim h ~ l ((p(t)) = lim u(t) = 0 . Thus the proof of Theorem 1 is complete.

t

Example 1. If lim inf [B{t)-\ A(ty] e x p ( - f H(s)|ds) = a > 0, then

f - a o 0

the solutions of equation (1) are oscillating. Indeed, taking into consid­

eration equation (2) corresponding to h(u) = u~2, we see that V{t,u) = t

U2 • exp ( — J \A (5)| ds) is the Lapunov function for this equation : 0

dV

~dt (2) df dV dV

dt ^ du •Ait, и)

= и2 • ( \А (f)|) • exp ( J |^4 (s)| ds) 0

t

— и4 • exp ( — J \A (s)| ds) • [B (t) -I- A (t) sin2 (и ~ 2)]

0

^ — и4 [B (t) — \A (£)|] exp ( — J И (s)| ds) ^ - ^ a u 4 0

for t ^ T, T a suitably chosen constant.

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Theorem on the oscillation of solutions 39 3

Example 2. If A(t) > 0, A'(t) ^ 0 for t ^ 0 and lim inf [ £ (t)/A(r)] > 0, t-* 00

then the solutions of equation (1) are oscillating. In this case for h(u) = u~2, V( t , u ) = (Л(0)-1 •и2 is the Lapunov function for equation (2).

References

[1] Ph. H artm an , Ordinary differential equations, John Wiley and Sons, New Y or-L ondon- Sydney 1964.

[2] K. Z im a, O oscylacyjnosci rozwiqzan rôwnania rôzniczkowego drugiego rzçdu (in Polish), Rocznik Naukowo-Dydaktyczny Wyzszej Szkoly Pedagogicznej w Rzeszowie, Matematyka Z. 4/32 (1977).

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