Oscillation Results for Emden–Fowler Type Differential Equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

205, 406]422 Ž1997.

AY975206

Oscillation Results for Emden]Fowler Type Differential Equations M. Cecchi and M. Marini Department of Electronic Engineering, Uni¨ ersity of Florence, Via di S. Marta, 3, 50139 Firenze, Italy Submitted by Zhi¨ ko S. Athanasso¨ Received January 26, 1996

The third order nonlinear differential equation Z

X

x q aŽ t . x q b Ž t . f Ž x . s 0

Ž).

is considered. We present oscillation and nonoscillation criteria which extend and improve previous results existing in the literature, in particular some results recently stated by M. Gregusˇ and M. Gregus, ˇ Jr., Ž J. Math. Anal. Appl. 181, 1994, 575]585.. In addition, contributions to the classification of solutions are given. The techniques used are based on a transformation which reduces Ž). to a suitable disconjugate form. To this aim auxiliary results on the asymptotic behavior of solutions of a second order linear differential equation associated to Ž). are stated. They are presented in an independent form because they may be applied also to simplify and improve other qualitative problems concerning differential equations with quasiderivatives. Q 1997 Academic Press

INTRODUCTION The aim of this paper is to study the oscillatory and nonoscillatory behavior of the nonlinear differential equation xZ q a Ž t . xX q b Ž t . f Ž x . s 0, where a, b g C Ž J . , J s 0, ` . , b Ž t . ) 0 except at isolated points, f g C Ž R . , u ? f Ž u . ) 0 for u / 0. 406 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

Ž 1.

OSCILLATION FOR DIFFERENTIAL EQUATIONS

407

In some cases, concerning the nonlinearity, the following hypotheses will be also assumed Žnot necessarily all together.: f nondecreasing for < u < large enough; lim

f Ž u. u

uª0

su,

0 F u - `.

Ž H1 . ŽH2 .

We recall that a nontrivial continuable solution of Eq. Ž1. is said to be oscillatory if it has infinitely large zeros, nonoscillatory otherwise. Equation Ž1. is said to be nonoscillatory if all its solutions are nonoscillatory, oscillatory otherwise. A prototype of Eq. Ž1. is the Emden]Fowler equation xZ q a Ž t . xX q b Ž t . < x < a sgn x s 0

Ž a ) 0. .

Ž 2.

Oscillation and nonoscillation of Eq. Ž1. or Ž2. has been considered by many authors with some additional assumptions on the function f w2]4, 10, 11, 14]17, 21, 23]25x. By a suitable transformation preserving zeros of solutions Žsee, e.g., w27x., the complete equation zZ q a1 Ž t . zY q a2 Ž t . zX q a3 Ž t . f Ž z . s 0

Ž 3.

may be written in the form of Eq. Ž1.. Hence Eq. Ž1. is not much less general than Eq. Ž3. as regards the oscillation and nonoscillation. A classical approach in the study of the qualitative behavior of solutions of Ž1. is based on a suitable transformation, associated to a disconjugate differential operator, which reduces Ž1. to an equation of the type 1

1

pŽ t .

rŽ t.

ž ž

X X

x

X

//

q b Ž t . f Ž x . s 0,

Ž 4.

where p g C 1 Ž J ., r g C 2 Ž J ., pŽ t . ) 0, r Ž t . ) 0. If `

Ht

0

p Ž t . dt s

`

Ht r Ž t . dt s `,

Ž 5.

0

then Ž4. is said to be in the canonical form w28x. The divergence of the integrals of the functions p and r plays an important role in the study of nonoscillation of Eq. Ž4.. Indeed if Ž5. is satisfied, then Eq. Ž4. has very interesting properties. For example it is possible to classify the nonoscillatory solutions of Ž1. in a very simple way. In the linear case we can also give necessary and sufficient criteria for the nonoscillation which are useful in the study of the nonlinear oscillation via a linearization device. A discussion on these topics is given at the beginning of Section 3.

408

CECCHI AND MARINI

Equation Ž1. may be written in the disconjugate form Ž4. if the second order comparison equation yY q a Ž t . y s 0

Ž 6.

is nonoscillatory. Nevertheless, the question whether Ž1. may be written in the canonical form is still open. A partial answer is given in w3x by assuming, in addition to other assumptions, that lim t ª` aŽ t . s c - 0. In this paper we give other sufficient conditions in order for Ž1. to be written in the canonical form, which extend those quoted in w3x. Such a result is employed to improve and generalize some recent oscillatory and nonoscillatory results in w17x. Indeed in w17x Gregusˇ and Gregus, ˇ using a technique already employed in w4x, have given some oscillation and nonoscillation results for Eqs. Ž1. and Ž2.. For example, the following results are proved: THEOREM A w17, THEOREM 5x.

Assume ŽH 2 . and

Ži. a g C 1 ŽŽ0, `.., aŽ t . G 0, aX Ž t . F 0; b g C ŽŽ0, `.., bŽ t . ) 0, H` tbŽ t . dt s `; Žii. the linear differential Eq. Ž6. is disconjugate on Ž0, `., that is, e¨ ery nontri¨ ial solution of Eq. Ž6. has at most one zero on Ž0, `.. Then e¨ ery bounded continuable solution of Eq. Ž1. with a zero at some point t 1 ) 0 is oscillatory. THEOREM B w17, THEOREM 3x.

Assume

Ži. a g C 1 ŽŽ0, `.., aŽ t . G 0, aX Ž t . F 0; b g C ŽŽ0, `.., bŽ t . ) 0, H t bŽ t . y aX Ž t .x dt - `; Žii. the linear differential Eq. Ž6. is disconjugate on Ž0, `.. ` 2w

Then each bounded continuable solution of Eq. Ž2. with a ) 1 is nonoscillatory. In this paper we consider Eqs. Ž1. and Ž2. without sign or monotonicity or regularity conditions on the function a as assumed in w17x. We obtain oscillatory and nonoscillatory criteria that extend and improve previous ones stated in w17x, in particular Theorems A and B quoted above. In addition, contributions to the classification of solutions are given, which are related with some results contained in the book w20x, as well as in the papers w2, 24, 26x. For a wide bibliography on this last argument we refer the reader in particular to the quoted paper w2x. Finally the obtained results are extended to the nonlinear general equation Ž4.. We also note that our results do not require that the perturbation f satisfies hypotheses on superlinearity andror sublinearity in the whole domain R. Relationships and comparisons with other known results Žin particular w2]4, 23x. will be pointed out throughout the paper.

OSCILLATION FOR DIFFERENTIAL EQUATIONS

409

The approach used is based on a suitable transformation and a linearization device. In particular Eq. Ž1. is transformed into the equation

ž ž h2 Ž t .

X X

1 hŽ t .

xX

//

q h Ž t . b Ž t . f Ž x . s 0,

Ž 1X .

where h is a positive nonoscillatory solution of Eq. Ž6. satisfying 1

`

H0

2

h Ž t.

`

dt s `,

H0 h Ž t . dt s `, lim

t ª` h

Ž t . ) 0.

Ž 7.

To prove the existence of such a solution of Ž6., we need to state some auxiliary results on the asymptotic behavior of solutions of Eq. Ž6. when the function a does not exhibit fixed sign. Such results are given in Section 1. They are related to certain asymptotic properties of the principal solutions of Eq. Ž6., and are presented as independent results because, in our opinion, they may be applied also to problems different from those considered in this paper, in particular to ones concerning differential equations with quasiderivatives.

1. NOTATION AND AUXILIARY RESULTS In this section we present some preliminary results on second order linear differential equations that we will use in the proof of the main results. Consider the equation

ž

X

1 rŽ t.

y

X

/

q q Ž t . y s 0,

Ž 8.

where r g C 1 Ž J ., q g C Ž J ., r Ž t . ) 0. In the study of the nonoscillation of Eq. Ž8., an important role Žsee, e.g., w18, 27x. is played by principal solutions Žat infinity., that is, by solutions y 0 of Eq. Ž8. such that `

H

rŽ t. y 02 Ž t .

dt s `.

Ž 9.

If Eq. Ž8. is nonoscillatory, then Eq. Ž8. has a solution y 0 satisfying Ž9. which is uniquely determined up to a constant factor Žsee, e.g., w18x.. In addition an arbitrary solution y 1 of Eq. Ž8., linearly independent of y 0 ,

410

CECCHI AND MARINI

satisfies `

H

rŽ t. y 12 Ž t .

dt - `,

and it is called nonprincipal Žat infinity.. Henceforward, for sake of simplicity, we will denote by a principal wnonprincipalx solution a principal wnonprincipalx solution at infinity. We recall also that if Eq. Ž8. is nonoscillatory, then Eq. Ž8. is said to be disconjugate on J if each nontrivial solution of Eq. Ž8. has at most one zero on J. Moreover Eq. Ž8. is disconjugate on J if and only if Eq. Ž8. has a positive solution on Ž0, `. Žsee, e.g., again w18x.. Consider now the linear binomial differential equation Ž6. where a g Ž C J ., and let aqŽ t . s max t g J  aŽ t ., 04 , ayŽ t . s min t g J  aŽ t ., 04 . Clearly aŽ t . s aqŽ t . q ayŽ t .. The following holds: Assume the following conditions

PROPOSITION 1. Ži. Žii.

H0` tay Ž t . dt s yK ) y`; the equation yY q ey2 K aq Ž t . y s 0

Ž 10 .

is disconjugate on J. Then Eq. Ž6. is disconjugate on J and there exists a Ž principal . solution h of Eq. Ž6., hŽ t . ) 0 on Ž0, `., such that 1

`

H0

2

h Ž t.

dt s `,

`

H0 h Ž t . dt s `,

lim t ª` h Ž t . ) 0.

Ž 7.

If, in addition, the following Žiii.

H0` taq Ž t . dt - `,

holds, then the principal solution h satisfying Ž7. is bounded on J. Roughly speaking Proposition 1 guarantees the nonoscillatoriness of Eq. Ž6. when the negative part of a is small in some sense and the corresponding equation associated to the positive part of a is nonoscillatory. The crucial point of this proposition is the fact that Eq. Ž6. has principal solutions which verify conditions Ž7., and are bounded if Žiii. is verified. It is easy to give an example which illustrates the necessity of assumption Ži.. To this aim it is sufficient to consider the equation yY y y s 0. Clearly Ži. is not satisfied, and there is no solution satisfying Ž7. since hŽ t . s eyt .

OSCILLATION FOR DIFFERENTIAL EQUATIONS

411

Remark 1. Assumption Žiii. of Proposition 1 implies that Eq. Ž10. is eventually disconjugate. Indeed, from Theorem 1 in w5x, Eq. Ž10. is nonoscillatory. Then, from a result in w8x, Eq. Ž10. is eventually disconjugate. We remark that assumption Žii. requires, in addition, that Eq. Ž10. be disconjugate on the whole half real line. The proof of Proposition 1 depends on the following lemmas concerning Eq. Ž8. with ‘‘q Ž t . F 0 on J ’’ and ‘‘q Ž t . G 0 on J,’’ respectively. As it is well known, ‘‘q Ž t . F 0 on J ’’ is sufficient for Eq. Ž8. to be disconjugate on J. The following hold: LEMMA 1. Consider Eq. Ž8. with q Ž t . F 0. If the following conditions Ži. Žii.

H0` r Ž t . dt s `, H0` < q Ž t .< H0t r Ž s . ds dt - `,

are satisfied, then any positi¨ e principal solution u 0 of Eq. Ž8. is nonincreasing and such that u 0 Ž ` . s lim t ª` u 0 Ž t . ) 0 u Ž 0. u0 Ž `.

F exp

ž

`

Ž 111 .

t

H0 < q Ž t .
/

.

Ž 11 2 .

Proof. A classical result of Kneser w18, Exercise 6.7, p. 352x states the existence of positive nonincreasing principal solutions u 0 of Eq. Ž8. approaching a nonzero limit as t ª `. For these solutions, it is easy to prove that lim t ª`Ž1rr Ž t .. uX0 Ž t . s 0 Žsee, e.g., w22x.. Hence integrating Eq. Ž8. in Ž t, `. we obtain 1 rŽ t.

uX0 Ž t . s

`

Ht q Ž s . u Ž s . ds; 0

because u 0 is a positive nonincreasing function, we get uX0 Ž t . G r Ž t . u 0 Ž t .

`

Ht q Ž s . ds.

Dividing by u 0 and integrating again in J we obtain log

u0 Ž `. u0 Ž 0.

G

`

`

`

t

H0 r Ž t . Ht q Ž s . ds dt s H0 q Ž t . H0 r Ž s . ds dt ) y`

which implies Ž11 2 ..

412

CECCHI AND MARINI

LEMMA 2. Consider Eq. Ž8. with q Ž t . G 0. If the following conditions Ži. H0` r Ž t . dt s `, Žii. lim sup t ª` r Ž t . - `, Žiii. Eq. Ž8. is disconjugate on J, are satisfied, then Eq. Ž8. has principal solutions ¨ 0 satisfying X

¨ 0 Ž t . ) 0,

¨ 0 Ž t . G 0 on Ž 0, ` . ,

1

`

H0

¨ 02

Ž t.

dt s `.

Ž 12 1 . Ž 12 2 .

If, in addition, the following Živ.

H0`q Ž t .H0t r Ž s . ds dt - `,

holds, then ¨ 0 is bounded on J. Proof. With Eq. Ž8. being disconjugate on J, Eq. Ž8. has positive solutions on Ž0, `.. Then the existence of a principal solution ¨ 0 satisfying Ž12 1 . follows from two results of Hartman and Potter Žsee, e.g., w18, Chap. XI, Corollary 6.3; 27, Theorem 2.39x.. Hence `

H0

rŽ t. ¨ 02 Ž t .

dt s `,

and, taking into account Žii., also Ž12 2 . is satisfied. In order to complete the proof it remains to show that if Ži. ] Živ. holds, then ¨ 0 is bounded on J. Two cases are possible: Ža. q is eventually positive; Žb. there exists a sequence  t k 4 , t k ª `, such that q Ž t k . s 0. Case Ža.. The assertion follows from Proposition 2 in w5x. Case Žb.. In this case the assertion follows by using the Sturm Comparison Theorem. Consider the linear perturbed equation

ž

X

1 rŽ t.

yX q Ž q Ž t . q q1 Ž t . . y s 0,

/

Ž 13 .

where q1 g C Ž J ., q1Ž t . ) 0 for t g J and H0`q1Ž t .H0t r Ž s . ds dt - `. Thus Eq. Ž13. is nonoscillatory and has bounded solutions ¨ b Žsee, e.g., again w5, Proposition 2x.. In addition from the quoted result in w8x, Eq. Ž13. is also eventually disconjugate. With Eq. Ž13. being a Sturm majorant of Eq. Ž8., from a result of Hartman and Wintner Žsee, e.g., w18, Chap. XI, Corollary 6.5x. we have for t large enough 1

X

¨0Ž t.

rŽ t. ¨0Ž t.

F

1

X

¨bŽ t.

r Ž t. ¨bŽ t.

OSCILLATION FOR DIFFERENTIAL EQUATIONS

413

or ¨0Ž t. F

¨ 0 Ž t0 . ¨ b Ž t0 .

¨bŽ t. .

This implies that ¨ 0 is bounded. The proof is now complete. We remark that as already noted in Remark 1, assumption Živ. of Lemma 2 implies that Eq. Ž8. is eventually disconjugate. Proof of Proposition 1. Consider the differential equation yY q ay Ž t . y s 0.

Ž 14 .

From Lemma 1, Eq. Ž14. has a positive principal nonincreasing solution u ˜0 satisfying u ˜0 Ž ` . ) 0,

1

`

H0

u 02

˜ Ž t.

u ˜0 Ž 0 .

dt s `,

u ˜0 Ž `.

F eyK ,

Ž 15 .

where K s yH0` tay Ž t . dt. Let m s u ˜20 Ž`., M s u˜20 Ž0., and consider the linear differential equation X

Ž myX . q Maq Ž t . y s 0

Ž 16 1 .

or yY q

M m

aq Ž t . y s 0.

Ž 16 2 .

With Mrm - ey2 K , the Sturm Comparison Theorem implies that Eq. Ž16 2 . is disconjugate on J. Consider now the linear differential equation X

Ž u˜20 Ž t . yX . q u˜20 Ž t . aq Ž t . y s 0.

Ž 17 .

With M G u ˜20 Ž t . G m, again from the Sturm Comparison Theorem we get that Eq. Ž17. is disconjugate on J. Hence assumptions Ži., Žii., Žiii. of Lemma 2 are satisfied and so Eq. Ž17. has a principal solution ¨˜0 verifying on Ž0, `. ¨˜0 Ž t . ) 0,

X

¨˜0 Ž t . G 0,

1

`

H0

¨ 02

˜ Ž t.

dt s `.

Ž 18 .

In order to complete the proof it is sufficient to consider the function h given by hŽ t . s u ˜0 Ž t . ¨˜0 Ž t . .

414

CECCHI AND MARINI

It is easy to show, by standard calculations, that h is a solution of Eq. Ž6.. With h being positive on Ž0, `. Eq. Ž6. is then disconjugate on J. Taking into account Ž15. and Ž18., we obtain that h satisfies also conditions Ž7.. Finally if also condition Žiii. holds, then we have `

q

2 0

H0 u˜ Ž t . a

1

t

Ž t.H

0

u ˜02 Ž s .

ds dt F

M

`

q

H ta m 0

Ž t . dt - `,

which implies that condition Živ. of Lemma 2 is satisfied. Then, from Lemma 2, ¨˜0 is bounded on J. With u ˜0 being positive nonincreasing, also h is bounded on J. This completes the proof of Proposition 1.

2. MAIN RESULTS Consider the nonlinear differential Eq. Ž1.. If the second order linear Eq. Ž6. is nonoscillatory, that is, Eq. Ž6. does not have oscillatory solutions, then, by standard computations, Eq. Ž1. may be transformed, for t G t 0 ) 0, in the disconjugate form

ž ž 2

h Ž t.

1 hŽ t .

X X

x

X

//

Ž 1X .

q h Ž t . b Ž t . f Ž x . s 0,

where h is a solution of Eq. Ž6.. This equation is a prototype of the more general equation Ž4.. The divergence of the integrals of the functions p and r plays an important role in the study of nonoscillation of Eq. Ž4.. Indeed in w26x ˇ Svec, generalizing a Lemma of Kiguradze and Elias Žsee, e.g., w8; 9; 20, Lemma 1.1, p. 2, Lemma 2.1, p. 43x. states that if the conditions Ž5. are satisfied, then every nonoscillatory solution x of Eq. Ž4. satisfies, for t large enough, either < x Ž t . < ) 0, x Ž t . ? x w1x Ž t . - 0, x Ž t . ? x w2x Ž t . ) 0

Ž 19 .

or < x Ž t . < ) 0, x Ž t . ? x w1x Ž t . ) 0, x Ž t . ? x w2x Ž t . ) 0,

Ž 20 . X

where x w1x, x w2x are the quasideri¨ ati¨ es of x, that is, x w1xŽ t . s x Ž t .rr Ž t ., x w2xŽ t . s Ž x w1xŽ t ..XrpŽ t .. Solutions satisfying Ž19. wŽ20.x are said to be solutions of degree zero w two x Žsee, e.g., w12x.. Solutions satisfying Ž19. are known also as Kneser solutions Žsee, e.g., w20x.. If we denote by N the set of all nonoscillatory solutions of Eq. Ž4. and by N0 w N2 x the set of solutions of degree zero wtwox, then Ž5. implies that N s N0 j N2 . A simple consequence of Proposition 1 gives us sufficient conditions in order for the same classification of nonoscillatory solutions

415

OSCILLATION FOR DIFFERENTIAL EQUATIONS

to occur for Eq. Ž1X .. The following holds: Assume

PROPOSITION 2. Ži. Žii.

H0` tay Ž t . dt s yK ) y`; Equation Ž10. is disconjugate on J.

Then Eq. Ž1. can be written for t ) 0 in the disconjugate form Ž1X . and e¨ ery nonoscillatory solution of Eq. Ž1X . is either in the class N0 or in the class N2 . Proof. The assertion follows immediately from Proposition 1 choosing as function h a principal solution of Eq. Ž6.. We can state now our theorems which improve the quoted results in w17x. Assume

THEOREM 1. Ži. Žii.

H0` tay Ž t . dt s yK ) y`; Equation Ž10. is disconjugate on J.

Then e¨ ery bounded continuable solution of Eq. Ž1. with a zero at some point t 1 G 0 is oscillatory. Proof. Let x be a continuable solution of Eq. Ž1. such that x Ž t 1 . s 0, t 1 G 0. Without loss of generality we may suppose x Ž t . ) 0 for t large enough. From Proposition 2, Eq. Ž1. may be transformed, for t ) 0 into Ž1X . where h satisfies Ž7.. Assume x nonoscillatory. Then, from Proposition 2, x is either in the class N0 or in the class N2 . Assume that x g N0 . Without loss of generality suppose x Ž t . ) 0, x w1xŽ t . - 0, x w2xŽ t . ) 0 for t ) T. We assert first that x does not have positive maxima. Let t 1 , t x - t 1 - T, be the last point of maximum for x. Then x Ž t . ) 0 for t G t 1 , which implies that the quasiderivative x w2x is decreasing on w t 1 , `.. Hence we have x w2x Ž t . s h 2 Ž t .

ž

1 rŽ t.

X

xX Ž t . - h 2 Ž t 1 .

/

ž

1 hŽ t .

X

xX Ž t .

/

s tst 1

1 h Ž t1 .

xY Ž t 1 . F 0,

which is a contradiction. Then x does not have positive maxima, and so x does not have zeros on its existence interval. This is again a contradiction since x Ž t 1 . s 0. Hence x f N0 . Assume now that x g N2 and x Ž t . ) 0, x w1xŽ t . ) 0, x w2xŽ t . ) 0 for t G T ) t 1. With x w1x being an increasing function, we have for t G T, xX Ž t . ) hŽ t .Ž xX ŽT .rhŽT .. or x Ž t . ) x ŽT . q Ž xX ŽT .rhŽT ..HTt hŽ s . ds. Since H` hŽ t . dt s ` we get that x is unbounded, which is a contradiction. The proof is now complete.

416

CECCHI AND MARINI

Remark 2. In the first part of the above proof we have shown that solutions of Eq. Ž1X . in the class N0 cannot have zeros on its existence interval. An alternative proof of this assertion is given in w2x. Remark 3. Theorem 1, as well as Theorems 2]4 below, requires the continuability of solutions of Eq. Ž1. with a zero. On this topic we refer the reader to the books w1, 20x and to the papers w2, 7, 19x. Theorem 1 improves the quoted Theorem A. Observe that Theorem 1 does not require monotonicity and regularity assumptions on the function a nor does it require hypotheses on the function f of superlinearity andror sublinearity at zero or at infinity. With an additional assumption on the nonlinearity in a neighborhood of infinity, we may state the following result which guarantees the oscillatory behavior of all continuable solutions, possibly unbounded, vanishing at some point t 1 G 0. THEOREM 2. Ži. Žii. Žiii.

Assume condition ŽH 1 . and

H0` tay Ž t . dt s yK ) y`; Equation Ž10. is disconjugate on J; H0` f Ž kt . bŽ t . dt s yH0` f Žykt . bŽ t . ds s ` for e¨ ery k g Ž0, 1..

Then e¨ ery continuable solution of Eq. Ž1. with a zero at some point t 1 G 0 is oscillatory. Proof. Let x be a continuable solution of Eq. Ž1. such that x Ž t 1 . s 0, t 1 G 0. Without loss of generality we may suppose x Ž t . ) 0 for t large enough. From Proposition 2, Eq. Ž1. may be transformed, for t ) 0, into Ž1X . where h satisfies Ž7.. Assume x is nonoscillatory. From Proposition 2, we have that x g N0 j N2 . Reasoning again as in the proof of Theorem 1, we get that x is not in the class N0 . Suppose that x g N2 and x Ž t . ) 0, x w1xŽ t . ) 0, x w2xŽ t . ) 0 for t G T ) t 1. Denote m h s inf t g wT , `. hŽ t .. With T ) 0, from Proposition 1 we get that m h ) 0. Integrating Eq. Ž1. in ŽT, t ., t ) T, we obtain x w2x Ž t . y x w2x Ž T . q

t

HT h Ž s . b Ž s . f Ž x Ž s . . ds s 0,

which implies x w2x Ž T . )

t

HT h Ž s . b Ž s . f Ž x Ž s . . ds.

Ž 21 .

OSCILLATION FOR DIFFERENTIAL EQUATIONS

417

With x w1x being a positive increasing function, we have for t G T x Ž t . ) x Ž T . q x w1x Ž T .

t

HT h Ž s . ds ) x

w1x

t

Ž T . H h Ž s . ds T

w1x

) x Ž T . ? mh ? Ž t y T . .

Ž 22 .

Let k be a constant such that 0 - k - minw1, x w1xŽT . ? m h x. Then from Ž22. we have for all t sufficiently large, x Ž t . ) k ? t. With f eventually increasing, there exists T1 such that for t ) T1 , f Ž x Ž t .. ) f Ž k ? t ., and, from Ž21., we obtain x w2x Ž T . ) G

T1

HT

T1

HT

h Ž s . b Ž s . f Ž x Ž s . . ds q

t

HT h Ž s . b Ž s . f Ž k ? s . ds 1

h Ž s . b Ž s . f Ž x Ž s . . ds q m h

t

HT b Ž s . f Ž k ? s . ds.

Ž 23.

1

Taking into account Žiii., the right side of Ž23. tends to infinity as t ª `, which is a contradiction. Then x is oscillatory and the proof is complete. Theorem 1 and 2 are related to a result in w3, Corollary 1x in which the case aqŽ t . ' 0, H0`aŽ t . dt s y`, is considered. Remark 4. When the perturbation f is superlinear at infinity, that is, lim inf < u < ª`Ž f Ž u.ru. ) 0, condition Žiii. of Theorem 2 is satisfied if H0` tbŽ t . dt s `. Moreover in this case it is easy to prove that monotonicity assumption ŽH 1 . is unnecessary. For the Emden]Fowler equation Ž Na ., assumption Žiii. becomes H0`s a bŽ s . ds s `. The following examples show that assumptions Ži. and Žiii. cannot be dropped without violating the validity of Theorem 2. EXAMPLE 1. Consider the sublinear differential equation 3 3 xZ y Ž t y 1 . xX q 2 Ž t y 1 . < x < 1r2 sgn x s 0.

Ž E1 .

The function x given by x Ž t . s Ž t y 1. 2 is a solution of Eq. ŽE 1 . with a zero at t 1 s 1. For this equation the assumption Ži. does not hold, since ayŽ t . s aŽ t . s y2Ž t y 1. 3 , while condition Žiii. is verified. As regards assumption Žii., equation yY q ey2 K aq Ž t . y s 0 is not defined since K s `, but Žii. is satisfied for the ‘‘limit equation,’’ that is, for the equation yY s 0. EXAMPLE 2. Consider the sublinear differential equation xZ q

1

Ž t q 1.

3

< x < a sgn x s 0

0 - a - 1.

ŽE2 .

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CECCHI AND MARINI

Since ay' aq' 0, the assumptions Ži. and Žii. are satisfied, but Žiii. does not hold since H0`s a bŽ s . ds - `. Moreover every solution of Eq. ŽE 2 . is nonoscillatory as it follows from a result in w21, Corollary 5x. Assuming that the function f is superlinear in a neighborhood of zero, we can give nonoscillatory results which generalize some criteria obtained in w4, 17x. Assume ŽH 2 . and

THEOREM 3. Ži. Žii.

H0` tay Ž t . H0` taq Ž t .

dt s yK ) y`; dt - `.

Let c be a positi¨ e function defined on J such that H0` t 2 bŽ t . c Ž t . dt - `. If x is a continuable solution of Eq. Ž1. such that, for t large enough, < f Ž xŽ t. . < F < xŽ t.
¡ ~ ¢u

if x Ž t . s 0.

From Remark 1 we have that Eq. Ž10. is eventually disconjugate, that is, there exists t 0 G 0 such that Eq. Ž10. is disconjugate on Ž t 0 , `.. Hence, from Proposition 2, we have that Eq. Ž25. may be transformed, for t G T ) t 0 , into

ž ž 2

h Ž t.

1 hŽ t .

X X

w

X

//

q h Ž t . b Ž t . F Ž t . w s 0,

where h satisfies Ž7.. Define mh s

inf

tg w T , ` .

hŽ t . ,

Mh s sup h Ž t . . tg w T , ` .

As already denoted in the proof of Theorem 2, because T ) 0, we have m h ) 0. With h bounded, we get also Mh - `. Thus `

ž

t

ž

1

s

HT h Ž t . b Ž t . F Ž t . HT h Ž s . HT h Ž l. F Ch

`

HT b Ž t . F Ž t . Ž t y T .

2

2

dt,

d l ds dt

/ /

OSCILLATION FOR DIFFERENTIAL EQUATIONS

419

where Ch s Ž1r2.Ž Mhrm h . 2 . Taking into account Ž24., we obtain `

ž

t

ž

1

s

HT h Ž t . b Ž t . F Ž t . HT h Ž s . HT h Ž l. F Ch

`

HT b Ž t . c Ž t . Ž t y T .

2

2

d l ds dt

/ /

dt - `.

By a slight modification of a result in w6, Theorem 5x we obtain that Eq. Ž25. is nonoscillatory, which is a contradiction because x is an oscillatory solution. Remark 5. It is easy to show that Theorem 3 improves Theorem B given in w17x, for Eq. Ž2. with a ) 1. To this end choose aŽ t . G 0 and c ' 1: it is sufficient to prove that the assumptions of Theorem B imply H0` taq Ž t . dt s H0` taŽ t . dt - `. Assume lim t ª ` HTt saŽ s . ds s `. With HTt saŽ s . ds s Ž t 2 r2. aŽ t . y ŽT 2r2. aŽT . y HTt s 2 aX Ž s . ds, we obtain lim t ª`Ž t 2r2. aŽ t . s `. Hence a classical result of Kneser Žsee, e.g., w27, p. 45x. implies that Eq. Ž6. is oscillatory, which is a contradiction. A suitable choice of c gives the following: COROLLARY 1. Assume condition ŽH 2 . and suppose that conditions Ži., Žii. of Theorem 3 hold. If H0` t 2 bŽ t . dt - `, then Eq. Ž1. does not ha¨ e bounded oscillatory solutions. Proof. The assertion follows from Theorem 3 choosing c Ž t . s c, c constant. When Eq. Ž1. is sublinear in a neighborhood of infinity, we have the following Žsee also w4, Corollary 3; 13, Theorem 2x.. COROLLARY 2. Assume condition ŽH 2 . and suppose that conditions Ži., Žii. of Theorem 3 hold. If H0` t 2 bŽ t . dt - ` and Žiii. lim sup < u < ª`Ž f Ž u.ru. - `, then Eq. Ž1. does not ha¨ e oscillatory solutions. Proof. Taking into account ŽH 2 . and Žiii., there exists a constant k such that 0 F f Ž u.ru F k. Then the assertion follows by reasoning as in the proof of Corollary 1. We conclude this section with some applications of the previous results to the nonlinear equation xZ q a Ž t . xX q b Ž t . < x < a sgn x s 0

Ž a ) 0. .

Ž 2.

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The following holds: THEOREM 4. ŽA. Assume Ži. H0` tay Ž t . dt s yK ) y`; Žii. Equation Ž10. is disconjugate on J; Žiii. H0` t a bŽ t . dt s `. Then e¨ ery continuable solution of Eq. Ž2. with a zero at some point t 1 G 0 is oscillatory. ŽB. Let a G 1 and assume Ž i ., H0` taq Ž t . dt - ` and ` nŽ a y1.q2 Ž . H0 t b t dt - `. Then Eq. Ž2. does not ha¨ e continuable oscillatory solutions x such that < x Ž t .< F t n. In particular Ž n s 2. if H0` t 2 a bŽ t . dt - `, then Eq. Ž2. does not ha¨ e continuable oscillatory solutions x such that < x Ž t .< F t 2 . Proof. Claim ŽA. follows from Theorem 2. Claim ŽB. follows from Theorem 3 with c Ž t . s t nŽ ay1.. Theorem 4 extends to Eq. Ž2. an analogous result stated in w4x for the binomial equation xZ q b Ž t . < x < a sgn x s 0 Ž a ) 0. . Ž 26 . Ž . w x Part A of Theorem 4 is also related with some results in 23 , in which the oscillation of solutions with a zero is considered. Part ŽB. of Theorem 4 is related with a problem settled in w11x. Indeed in w11x the authors conjecture that if H0` t 2 a bŽ t . dt - `, then the binomial Eq. Ž26. with a ) 1 does not have oscillatory solutions. Other conditions assuring that Eq. Ž26. does not have oscillatory solutions have been given recently in w7x. 3. SOME EXTENSIONS Consider now Eq. Ž4.. It is easy to extend to Eq. Ž4. all results stated in the previous section by assuming the condition Ž5.. For example the following holds: THEOREM 5. Assume condition Ž5.. Then e¨ ery bounded continuable solution of Eq. Ž4. with a zero is oscillatory. Proof. ŽSketch.. The argument is similar to that given in the proof of Theorem 1. Let x be a continuable solution of Eq. Ž4. defined on w t x , `., t x G 0, such that x Ž t 1 . s 0, t 1 G t x . Assume x g N , that is, x nonoscillatory. From the quoted result w26x, x is either in the class N0 or in the class N2 . The assertion follows by showing that: Ža. solutions in N0 cannot have zeros in their existence interval; Žb. H0` r Ž t . dt s ` implies that solutions in N2 are unbounded.

421

OSCILLATION FOR DIFFERENTIAL EQUATIONS

Assume conditions ŽH 1 ., Ž5., and

THEOREM 6. `

H0 b Ž t . f

ž

k?

t

H0 r Ž s . ds

/

dt s y

`

H0 b Ž t . f

ž

yk ?

t

H0 r Ž s . ds

/

dt s `

for e¨ ery k g Ž 0, 1 . .

Ž 27 .

Then e¨ ery continuable solution of Eq. Ž4. with a zero is oscillatory. THEOREM 7. Assume conditions ŽH 2 . and Ž5.. Let c be a positi¨ e function defined on J such that `

s

t

H0 b Ž t . c Ž t . H0 r Ž s . H0 p Ž l. d l ds dt - `. If x is a continuable solution of Eq. Ž4. such that, for t large enough, < f Ž xŽ t. . < F < xŽ t.
Ž 24 .

then x is nonoscillatory. The proofs of Theorem 6 and 7 are similar to those of Theorems 2 and 3, respectively, and are omitted. As already noted in Remark 4, when the perturbation f is superlinear at infinity, that is, lim inf < u < ª`Ž f Ž u.ru. ) 0, if `

t

H0 b Ž t . H0 r Ž s . ds dt s `, then Ž27. is satisfied. Also in this case the monotonicity assumption ŽH 1 . is unnecessary. Finally, extensions of the above results to the equation 1

1

pŽ t .

rŽ t.

ž ž

X X

x

X

//

q b Ž t . < x < a sgn x s 0

Ž a ) 0. .

are left to the reader.

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