Some properties of oscillatory solutions o! certain differential equations of second order

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1967)

D. B^obkowski (Poznań)

Some properties of oscillatory solutions o! certain differential equations of second order

We shall consider the second-order ordinary differential equation

(0.1) teifix'I'+qitjX) = 0,

where x — x(t), p is a positive continuous function on the interval J = {t: a < t < oo}, and q is a real-valued continuous function on the strip В = {(t,x): taJ).

A solution x{t) of (0.1) is called oscillatory if for every t in / there are a, ft such that ft > a > t , x(a) — 0 and x(ft) Ф 0. A solution x(t) of (0.1) is called S-oscillatory if (i) it is oscillatory, (ii) the zeros of x have no accumulation point in «/, (iii) for every two points a, (3 such that a < a < (3 and x'(a) — x'(ft) = 0 there exists a у such that a < у < ft and x(y) = 0, i.e., the zeros of x and x ’ follow alternately.

Equation (0.1) is called oscillatory [S-oscillatory] if every non-zero solution of it is oscillatory [^-oscillatory, respectively]. It is well known that the equation

(0.2) x"-\-q(t)x = 0,

where q(t) > c > 0, is $-oscillatory; on the other hand, the solution x(t) = 3sint+sin3£ of the equation

15—18 sin21 x" H---x =0

1 +2cos 2t is oscillatory and is not $-oscillatory.

We shall show (in Lemma 2) that under certain assumptions any oscil­

latory solution of (0.1) is /^-oscillatory. Comparing (0.1) with another differential equations [P(t)x'Y +Q(t)x = 0 we get certain conditions sufficient in order that (0.1) be S-oscillatory. The results obtained here overlap some results of E. У. Petropavlovskaia [13].

In the second part of the paper we obtain a condition sufficient in OO

order that the series £ (tzn+z—hn) be convergent, where/0, f 2, ... denote

71 = 0

the successive zeros of an oscillatory solution x{t). This condition is a generalization of one given by E. Gagliardo [10].

The last part of the paper concerns the values of an $-oscillatory solution x(t) at the successive zeros of its derivative and the values of x'(t) at the successive zeros of x(t). Such investigation were initiated by G. Floquet [9] in 1883 who dealt with the solutions of (0.2) in the case where q is periodic. In 1919 W. F. Osgood [12] investigated this prob­

lem for nonperiodic q. Further papers have been published by M. Bier­

nacki [1], Z. Butlewski [2], [4], [5], [6], C. Taam [15], and others (cf. [14]

and [17]). The results obtained in § 3 are more general than those in [1], [2], [4], and are somewhat different from those in [6] and [15].

§ 1. Let — {p: реС{Ф), p(t) > 0 for t in J ) and let Jj be the set of all functions q in G(2) such that

(1.1) \q(t, Xj)— q(t, x 2)\ Ф m(t)\x1—x 2\

for some function m in C(S) depending on q. Here C{J) denotes the set of all real-valued continuous functions on J .

We begin by noticing that if p e^5 and q^21•> then for any initial conditions x(a) = c0, x'(a) = e1 there exists a unique solution of (0.1) satisfying these conditions and defined for all t in J (cf. L. Tonelli [16], B. Conti [7]).

Lemma 1. Suppose that pe^5, q eJ21} P , Q eO(«/), the inequalities (1.2) xq{t, x) > Q{t)xz, p { t ) ^ P ( t ) ,

(1.3) [P(<)~P{t)Y+[%q{t, x ) —Q{t)x*Y ф 0

are satisfied in 2) and u(t) is a solution of the differential equation

(1.4) [P(t)u'J Ą-Q(t)u = 0

such that u(a) = u((3) = 0, where a < a < /3. Furthermore, suppose that u{t) Ф 0 for some t in (a, ft). Then for every non-zero constant e the function x(t) = cu(t) is not a solution of the differential equation (0.1).

Proof. Suppose the contrary. Then [P{t)u'{t)Y = —Q(t)u{t) and [p{t)u'(t)]' ~ —c~1q(t, cu(t)), and the derivative of the function

V(t) = [P{t)-P(t)]u(t)u’(t)

would be nonnegative and not identically zero, because

V'(t) = [P(t)-p(t)][u'(t)Y+c~2{cu(t)q[t, cu{t)~]-Q(t)[cu(t)]*}.

Consequently, V{f) — V{a) would be positive, contradicting V{a) — F(/3)

= 0.

Lemma 2. Let x(t) be an oscillatory solution of (0.1) and let peg? and qe£2, where

&2 = {q: q e£ 1, q{t, 0) = 0, q(t, x) Ф 0 for t in and x Ф 0}.

Then x{t) is S-oscillatory.

Proof. Since q{t, 0) = 0 for t in */, the function 0 is a solution of (0.1); consequently, in virtue of the uniqueness of the solution, x(t) does not vanish identically on any interval. Similarly, since q(t,x) Ф0 for x Ф 0 ,x{t) does not vanish on any interval. An argument used by Z. Butlewski [3] shows that the zeros of x{t) must not have any accu­

mulation point in J . Indeed, if there existed a monotone sequence rx,

t2, ... of zeros of x(t) convergent to some r in Ф, then x(r) would be equal to 0,

x'(t) = lim n—>00

x( r) —x( Tn) T Tn

would also be equal to zero, and x(t) would be identically 0. Thus, the zeros of x(t) can be arranged into an increasing sequence tending to oo.

Now, if a, fi are two successive zeros of x'(t), then J q{t, x(t))dt = p(a)x' (a) — p(P)x' (/?) — 0

a

and q[t, x(t)) must vanish at a point у between a and since q(y, ос) Ф 0 if x Ф 0, x(y) must vanish as well. Thus x(t) is ^-oscillatory.

Theorem 1. Suppose t\at ре&, Pe£P, qe$2, QeCi^), the equation (1.4) is oscillatory and (1.2) holds. Then the equation (0.1) is S-oscillatory.

Proof. If (1.3) is not satisfied, then (0.1) is ^-oscillatory by Lemma 2 Now, if (1.3) does not hold, we argue as follows. Suppose that a, /5 are two successive zeros of a non-zero solution u(t) of the equation (1.4). We shall show that any solution x(t) of (0.1) has at least one zero in <a, /?>.

Indeed, if x(t) Ф 0 for t in <a, /? >, then the function S(t) = u{ t ) ^P { t ) u ' { t ) ~p{ t ) x ' { t ) ^^

is differentiable on <a, /?>. Taking into account (1.2) and (0.1), we infer that

, Г x'(t) u(t)-—x(t)u'(t) ~|2

s ’® - pH) — -- + [ P ( t ) - p ( m u ' ( t ) ] * +

L

J

Consequently, by (1.2) and (1.3), S'(t) is nonnegative and is not identi­

cally 0 because, by Lemma 1, x(t) is not identically equal to cu{t) for any c 0, and this contradicts S{a) = S(fi) = 0. We have thus shown that x{t) vanishes somewhere in <a,/3>; therefore x{t) is oscillatory and hence ^-oscillatory (by Lemma 2).

Corollary 1. Suppose that peg?, qe£2, limp(;£) < oo, and there exists t—> oo

a positive function у in C^(Y) such that

OO

J [?>(%"(«) +

^J

a

Then equation (0.1) is S-oscillatory.

Indeed, p(t) < К for some constant К and we may substitute P(t) = K , Q { t ) = KQiit), and apply the known criteria for the equation 3o"Jr Qi(t)x = 0 to be oscillatory (cf. [1 1] or [8]).

§ 2. Suppose now that x(t) is an ^-oscillatory solution of equation (0.1), реёР, and qe£>3, where

= {q: q e l 2, xq(t, x) > 0 for x Ф 0}.

Furthermore let t0, t2, t±, ... denote the successive zeros of x{t) and let L , , t$j ... denote the successive zeros of x'(t) (t2n < t2n+1 < t2n+2, for n = 0 , 1 , 2 , . . . ) . We shall use the following abbreviations:

(2 .1)

y(t) = p{t)x'{t), Y 2n = \y{t2n)\,

-^-2n+l~ l# ( W j-l) l i - Y 2 n — \X (t2n)\.

The following theorem is a generalization of a result of E. Gagliardo [10] who proved it for the linear equation x"-\-Q(t)x = 0.

Theorem 2. Let 0 < p 1 < p(t) < p 2 < oo, where p 1, p 2 ewe constants and peg?. Let x(t) be an S-oscillatory solution of (0.1) such that the integral

(2.2)

OO

J m(t)dt

a

is convergent, where m is any continuous function satisfying (1.1). Then the series

(2.3)

1

1

^2n + 2 I2 П is convergent.

P roof. Multiplying either side of (0.1) by y{t) and integrating from a to /9 we get

и у т г- ш ш г = - / « [ < > x(tnp(t)x'(t)dt.

a

Let a = ^t2n and p = <2»+i- Then

h n +1

(2.4) TL = 2 J q [ t , x ( t ) ] p ( t ) x ' ( t ) d t ( n = 0, 1, 2, . . . ) ;

Hn

similarly, setting a = <2n+1, p = ^t2n+2, we get

hn+2

(2.5) ^{Y 22n+2} = —.2 J g[«, ®(«)]p(<)®'(<)^ (w = 0 , 1 , 2 , . . . ) -

*271+1

Since ж(г) is ^-oscillatory, x { t ) x ' ( t ) > 0 for tin ( t 2n, t 2n+1} and x { t ) x ' ( t ) < 0 for t in <t2n+l, t 2n+2y. Integrating either side of (0.1) over <.t2n, t 2n+2> we get

*2тг+ 2

J q [ t , x ( t ) ] d t = Y 2nĄ - Y 2n+2 > P i { X 2n+ X 2n+2) if x ( t ) > 0

h n

and

*277+2

J*

—

—

(-T

, rf *®(^) ^ b.

hn

Hence, in virtue of X 2n — |гс'(<2п)| = Y 2nl p ( t 2n),

*277+2

(2.6) J" \ q [ t , x ( t ) ] \ d t > p 1( X 2n- \ - X 2n+2) . 4n

If a?(J) > 0 for ^t in ( t 2n, t 2n+2} , then

[ p { t ) x ’ ( t ) Y = —g[«, »(<)] <

and therefore p { t ) x ' ( t ) < ^{Y 2n} for ^t in ^{( t 2n,}^277+1/17 ’ {t) < p 2 ^ 2 n l P l fOr ^t in ‘(^тгj ^27i+i/’j and x (t) p2--T27H-2IPi for t in ■\4277+i7 ^271+2^* Integrat­

ing the respective sides of these inequalities over <<2n, <>, where t < <2n+ł, we get

ж(£) ^^/(Lti)H Х2п(<2»+1 ^ -3^271^271+1 ^2тг) •

Pi Pi

Similarly integrating over f t , t 2n+2>, where t > $ 2n+1, we get

*r(^) (^271+2) “f" -T^27l+2 (^277+2 ^ -T 2w+2 (^271+2 ^27l) •

Pi Pi

Consequently,

x(t) < — max(X2w, X 2n+2)(t2n+2—t2n) p 2 for n = 0 , 1 , 2 , . . Pi

Analogously, if x(if) < 0 for t in <t2n, t2n+2), we get /p ^

x(t) max(X2w, X2łl+2) {t2n+2 t2n).

Pi Thus, if t2n < t < t2n+2, then

(2.7) P 2

|*®(^)l ^ max(X2w, X 2n+2) (^2n+2 • Pi

Combining (2.6), (2.7) and g e J 3, we get

*2w+2 (2?l+2

m(t)dt > J* g[<> д?(<)3 x(t) dt >

P 2 Lw+2 ^2

for 71 = 0,1,2, . . . , and therefore if the integral (2.2) is convergent, so is the series (2.3).

§ 3 . Let J4 = {q : qeJ23,q(t, —x) = — д(/, ж) in ^}, and let {^}, {Xn} , { Уи} be as above.

Lem ma 3. S u p p o s e t h a t x ( t ) i s a n S - o s c i l l a t o r y s o l u t i o n o f (0.1) wh e r e р е £ Р , geJ4, <md t h a t

(3.1) ^{i f a t s}, t h e n \ j p(^)g(^? «r) ^p (^)^{q ^ t}, ж) J 0.

T h e n

x 2n+l x 2n- bl

(3.2) Pi^zn) J g(Lw? £)d£ ^ ^^(Lw+i) J" g(Lft+u £) d£,

о 0

^2W+1 *2n+l

(3.3) p(l2n+i) J g(Ln+u ^{£ ) d £} ^ |X2n+2 ^^(^гя+г) J ” (/(Lw+2> ^{£ ) d £ .}

о о

Proof. We have shown that x(t)x'(t) > 0 for / in <ź2№, <2ri+1>; hence, by (3.1),

P(hn)q[t2n, x(t)]x'(t) <p{t )q[ t, x{t )] x’{t) < p(t2n+1)q[t2n+1, x{t)]x'{t) in <[t2n, #2n+i>. Substituting x ( t ) = £ { x ( t ) being monotone in this in­

terval), we get

x62w+i) h n+i x(hn+i>

p(kn) f q(t2n, £ ) d £ <

J

J f(f2)l+i,f)^,

Since

у -у

J q(t, x)dx = J q{t, —x)dx,

о о

we get (3.2). The proof of (3.3) is similar.

Lemma 4. Suppose that pefP, g<ri?4, that

(3.4) if a <51 < s, then [p{s)q(s, x ) —p{t)q{t, x ^ x ^ 0, and that x(t) is an S-oscillatory solution of (0.1). Then

X%n+1 -^2«+l

(3.5) p (^2w+1) J q{t%n-\-15 £)d£ ^ \ Y 2n^ p { t 2n) J q(t2n, £)d£j

о о

x 2n+l x 2n+l

(3.6) p(t2n+2) J q{t'2n+2i £)d£ jjY2n+2 ^ V (^277+2) J* q (^271+11 £)d£'

0 0

The proof is similar to that of Lemma 3.

Theorem 3. Let р*&, q e l x, and let x(t) be an S-oscillatory solution of (0.1). I f (3.1) is satisfied, then

-3^27i—i ^ -3l2ti+i and Y 2n ^ 3^2^1+25 if (3.4) is satisfied, then

Ха„_1 ^ -ЗГ2тг+1 and Y 2n> Y 2n+2.

Proof. If and (3.1) holds, then from (3.2) and (3.3) it follows that Y 2n < Y2n+2 for w = 0 , 1 , 2 , . . . On the other hand, from (3.5) and (3.6) it follows that if g eJ4 and (3.4) holds, then Y 2n > Y2„,+2. Sub­

stituting n —1 instead of n in (3.3), we get

*2n~ 1

h Y 2n ^ p(I21г) J q(t2nj £)df; for n — 1, 2, ...

0

Combining this inequality with (3.2) we infer that

"^■277-f-1 Х2П— 1

V (^27г) J q{t2n, £)di; ^ p (t2n) J q(t2n, £)d£ for n = 1,2, . . .

о 0

In viitue of the properties of the functions p and q we get X 2n_ x > X2n+1 if (3.1) is valid and, analogously, X 2n_1 < X 2n+l if (3.4) is valid, n — 1, 2, ... This concludes the proof.

Corollary 2. Assume successively that pet?, p is nondecreasing, ge J 4, (3.4) is valid, and that q(t, x) > qxx > 0 for x ^ 0 and t in J , qx being a positive constant. Then every S-oscillatory solution of (0.1) has the

following properties:'

X 2n X2n+2 (indi X 2n_^x Х2п^3 ^ — 0 , 1 , 2 , . . . and

\x{t)\^ X0Vp{t0)/q1 for t in J .

Indeed, from Theorem 3 it follows that the sequence {X2n} is non­

increasing. From (3.5) it also follows that X2n-{-l

J <z(^2n+ij £)d£ ^ * j » ( * o ) X 0.

о

Hence X \ n+l < p(t0)Xl/q1 and we get the conclusion. A similar argument may applied if the inequalities are reversed.

Corollary 3. Suppose that реЗР, p is nonincreasing, qe£4, and q(t, x) < q2x < oo for x > 0 and t in J (q2 being a constant), and (3.1) holds. Then any S-oscillatory solution of (0.1) has the following properties:

X2n^ X 2n+2, X2n+i ^ Х2П+31 and X 2n+i ^ X0\/p(t0)lq2 for n = 0,1,2, ...

Theorem 4. Let pe^>, g e J 4, let

(3.7) 0 < Рг <p (t ) ^ p 2 = Xpx,

where 1 < 1 < 00 and p x and p2 are constants, and te J , let (3.1) holds, and let

(3.8) 0 qxx q(t, x) cC q2x = pqxx,

where 1 < p < 00, qx and q2 are constants, qx > 0, t e J and x ^ 0. Then any solution x(t) of (0.1) is S-oscillatory, {X 2n+1} is nonincreasing lim X 2n+1 > 0, and

71—>0O

^X. ^{n yi} 1

(3.9) 0 < log —--- <logA//, n = 1 , 2 , 3 , . . . х 2п+%

This yields an estimation of the logarithmic decrement of {X2n+l}.

Proof. The function p(t) = 1 fulfils the hypotheses of Corollary 1;

in particular,

/ [ p w r1 inf Г h t A l

|ac|<oo |_ X

J

Consequently, equation (0.1) is $-oscillatory. Moreover, by Theorem 3, {X2n+i} is nonincreasing and bounded away from zero. In turn, by (3.2) and (3.4) it follows that

Combining this inequality with the hypotheses of the theorem we get X 2n_ 1 < Vhy < X 2n+1 and hence (3.9). This concludes the proof.

Theorem 5. Assume that peSP, and assume (3.4), (3.7) and (3.8). Then any solution x(t) of (0.1) is 8 -oscillatory and bounded for t in J , {X2n_i} is noninereasing, and the logarithmic -increment of {X2n-i}

satisfies the inequalities

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о о

А-2п-1

The proof is similar to that of Theorem 4.

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