ON OSCILLATIONS OF PERTURBED SECOND ORDER DIFFERENTIAL EQUATION

Vol.3 NO.2 1983

~ ~ ~

~

~

II

Acta Mathematica Scientia

ON OSCILLATIONS OF PERTURBED SECOND ORDER DIFFERENTIAL EQUATION-H. El-Owaidy and A.A.S. Zagroutv Math, Dept. FacuJty of Science, King Abdul-Aziz University Jeddah , SAUDI ARABIA. A perturbed second order linear differential equation is considered. A criteria concerning the oscillatory behaviour of solutions of the perturbed equation based on the perturbing function and the solutions of the associated unperturbed equation is given.

In this paper we' shall consider the linear differential equations (P(t)x' (t»' + q(t)x(t) = 0 (1),' = d jdt (P(t) y' (t»' + q(t) y(t) = 1(t) (2) where p,q and.,1 are real-valued continuous functions on [a,oo) where a is any real number, PCt»O for r~o, I(t) =1=0 on [a,oo). The problem of determining oscillation c ri te r ia for second order linear differential equations has received a great deal of attention in the last twenty years-see for example LlI, [2J, [3J, [4J, [5J. Before proceeding, we shall require some definitions and lemmas: Deflnltlont' 1): A solu tion of (1) ( or (2) ) is said to be nonoscillatory on [a, 00 ) if it has 0111y a finite number of zeros on [al,oo) for some a l a and is osci l latory if it has an infinite number of zeros on [a 1 ,00). Equation ( 1 ) (or ( 2 » is oscillatory if it has at least 'one oscillatory solution on [a,oo) and is nonoscillatory if all solu tipns are no nosci llator y It is well known from ( 4 ) that if Xl and X 2 are solution basis for equation ( 1), then the general solution of ( 2 ) is gi yen by:

>

• permanent Address. Math. Dept. Faculty of Science, Al-Azhar University, Nasr-City, Cairo, EgJ pt • •• Received

122

where

ACTA MATHEMATIC A SCIENTIA

y(t)=C t xt(f)+c 2 x

C1

and

C2

2(f)+yp,

Vol.3

(3)

arbitrary constants and 2

YP =

t

Ex i (!)f/(S)W iSfl_ds,

i-I

a

W(X t,X 2)(S)

where w(x t ,X 2) (t) is the quasi-Wronskian of the solutions x 1 and and w;(f) is the determinant obtained from w(x 1 ,x 2 ) by replacing the i t h column with vector (O,l)T, i = 1,2 i ,e

"2

No.2

123

On Oscilations of Perturbed Second Order Differential Equation

Assume x(t) is an oscillatory solution of (1) and y(t) is a n onosci Ila tor y solution of ( 2 ). Then there exists a sequences ,{tt}, tt~oo as i~oo such that(x p y'-ypx')(t i) =0 for all i , I Proof The agreement is similar" to the proof of Sturm separation theorem and therefore is omitted. I Theorem 3 Let all solutions of (2)' be nonoscillatory and of the same sign. If x(t) solution of (1) such that lim inf P(t) >0, then Theorem 2

t-+oo

the particular solution YP .of (2)

has the

property that

I yp(t) 1= 00.

lim t-+

oo

Proof Let y t (t) = x(t) + yp(t) denote any nonoscillatory solution of (2) . Since all solutions of (2) are of the same sign on [a ,00), f 0 rea ch { the r e ex is t sap 0 in t tiS U ch t hat y i (t) ~ 0 0 n [t i ,00 ) . Wi thou t loss of generali ty . we can assume tha t y i (t) >0 on [t i ,00) for all s, Then it is clear that yp(t»-c x(t) on [ti' 00), and henceIYp(t)I>lx(t)l,for large t. since lim inf Ix(t)l>o as t ~ 00, it follows that Iyp(t) I~oo as t~oo and the result follows. I

Theorem 4

If S:(l/P(t))dh

oo

and solutions of (1) are bounded,

then (2) , has at most one nonoscillatory solution. It follows from the variation of parameters (constants) formula and hypotheses, that all solutions of (2) are bounded. Assume that Yl and Y2 are any solutions of (2), and hence their difference Yt - Y2 = x(t) is a solution of (1). Thus from equation ( 2 ) we obtain.

Proof

Y2(Py~)/_Yl(Py~)/=(YI-Y2)j(t)

(5)

The left hand side of (5) can be written (Yl(PY2)-Y2(PY1»)' and hence equation ( 5 ) takes the form (Yl(PY~)-Y2(PY~»)'=x(t) j(t)

(6)

from hypotheses on x(t) and f(t), it follows that the left hand side of (6) has a limit as t~oo. To prove that equation (2) has at most one nono sci ll a tory solution: On the contrary assume that there exists two" distinct nonosci l Iator y solutions Yl an d Y2 of equation (2). Let x , = YI-Y2 and x 2 (f,) be a normalized solution basis for equa t iorr/ 1-) i . e W(X 1 , X 2 ) ( t ) = 1 =(XIPX~-X2PX~). Let Y3=Y'2+X2 andz(t) =tan-1'(j"3!Yl)then: ". z ' (t) = '( YIP y~' - P 3 P Y~ )/ P( Y i + Y ~ ) ~ ( 7·) Since Y 1 (t) is nonoscilla tory solu ti on of. ( 2), by assumption, there exists t 1 > 0 such that yi>O for f>t 1 , which means that z(t) and, consequently, z(t) are well defined for all t>t 1. The numerator in the right hand side of ( 7 ) can be rewri tten as

124

ACTA MA THEMA TICA SCIENTIA

(y IPY~ - Y3Pyi)

= ex 1 PX~ -x 2PX~) + (x 1 -

X

2

)PY;

-Y2P(X 1 - X ; ) /

Assume x(t) = x 1 -x 2 ,noting that x 1 and X 2 are normalized solution baris of (1) and using theorem 2 ,it follows that (YIPY~-Y3PY~) (t) has the limi t 1 as t~oo. If we choose t 2 >t 1 so that

(YIPY~-Y3PyD>~

for t>t 2

then equation ( 7 ) takes the form Z'(t»1/3P(Y~+Y~»O,

t>t 2

>

Thus z(t) is strictly monotone for t t 2. In addi tion, since Y 1 and Y 3 are bounded, there exists a posi ti ve number M such that and hence

yi + y;
z(t»z(t 1 ) +

1 -M

3p

Jt

t 1

ds,

1

/3pM ( 8)

Since the integral on the right of (8) diverges and hence z(t) becomes unbounded as t-+ oo • This implies that both Yl(t) and Y3(t) must oscillate which contradicts that Y 1 is nonoscillatory solution' of ( 2 ). I

References ( 1)

J. Barrett: Oscillation Theory of Ordinary Differential Equations, Advanced in ~aths. 31(1969),415-509.

G.J •Butler: A Note on Nonoscillatory Solutions of Sublinear Hill's Equation. J. Math , Anal.. Appl , 63(1978),43-49. ( 3) H'El-Owa idy, On Oscillations of Second Order Differential Equations. Kyung pook Math. J. korea 21(1981) 117-121. ( 4) R. Grimmer and W. Patula: Nonoscillatory Solutions of Forced Second Order Linear Equation. J. Math.Anal. Appl , 56(1976),452-459. ( 5) S.Pankin: Oscillation Theorems for Second Order Norr-homogeneous linear Differential Equation. J. Math, Anal. 53(1976) 550-553. ( 2)