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Interval Oscillation Criteria Related to Integral Averaging Technique for Certain Nonlinear Differential Equations
Journal of Mathematical Analysis and Applications 245, 171᎐188 Ž2000. doi:10.1006rjmaa.2000.6749, available online at http:rrwww.idealibrary.com on
Interval Oscillation Criteria Related to Integral Averaging Technique for Certain Nonlinear Differential Equations Wan-Tong Li 1 Department of Mathematics, Lanzhou Uni¨ ersity, Lanzhou, Gansu, 730000, People’s Republic of China
and Ravi P. Agarwal Department of Mathematics, National Uni¨ ersity of Singapore, 10 Kent Ridge Crescent, Singapore, 119260, Singapore Submitted by George Leitmann Received December 23, 1999
We present new interval oscillation criteria related to integral averaging technique for certain classes of second order nonlinear differential equations that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of w t 0 , ⬁., rather than on the whole half-line. Our results are sharper than some previous results and handle the cases which are not covered by known criteria. In particular, several examples that dwell upon the sharp conditions of our results are also included. 䊚 2000 Academic Press Key Words: interval criteria; oscillation; second order; nonlinear differential equations; integral averaging technique.
1. INTRODUCTION We consider the oscillation behavior of solutions of the second-order nonlinear differential equation y⬙ Ž t . q q Ž t . f Ž y Ž t . . g Ž y⬘ Ž t . . s 0,
Ž 1.1.
1 Research was supported by the NSF of Gansu Province and the NSF of Gansu University of Technology.
171 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
172
LI AND AGARWAL
where t G t 0 , and the functions q, f, and g are to be specified in the following text. We recall that a function y: w t 0 , t 1 . ª Žy⬁, ⬁., t 1 ) t 0 is called a solution of Eq. Ž1.1. if y Ž t . satisfies Eq. Ž1.1. for all t g w t 0 , t 1 .. In the sequel it will be always assumed that solutions of Eq. Ž1.1. exist for any t 0 G 0. A solution y Ž t . of Eq. Ž1.1. is called oscillatory if it has arbitrary large zeros, otherwise it is called nonoscillatory. Equation Ž1.1. has been studied by Grace and Lalli w7x. They mentioned that, though stability, boundedness, and convergence to zero of all solutions of Eq. Ž1.1. have been investigated in the papers of Burton and Grimmer w2x, Graef and Spikes w5, 6x, Lalli w12x, and Wong and Burton w18x. Nothing much has been known regarding the oscillatory behavior of Eq. Ž1.1. except for the result by Wong and Burton w18, Theorem 4x regarding the oscillatory behavior of Eq. Ž1.1. in connection with that of the corresponding linear equation y⬙ Ž t . q q Ž t . y Ž t . s 0. Ž 1.2. Recently, Rogovchenko w17x presented new sufficient conditions which ensure the oscillatory character of Eq. Ž1.1.. They are different from those of Grace and Lalli w7x and are applicable to other classes of equations which are not covered by the results of Grace and Lalli w7x. However, all the above mentioned oscillation results involve the interval of q and hence require the information of q on the entire half-line w t 0 , ⬁.. From the Sturm Separation Theorem, we see that oscillation is only an interval property, i.e., if there exists a sequence of subintervals w a i , bi x of w t 0 , ⬁., as a i ª ⬁, such that for each i there exists a solution of Eq. Ž1.2. that has at least two zeros in w a i , bi x, then every solution of Eq. Ž1.2. is oscillatory. Ei-Sayed w4x established an interval criterion for oscillation of a forced second-order equation, but the result is not very sharp, because a comparison with equations of constant coefficient is used in the proof. In 1997, Huang w9x presented the following interval criteria for oscillation and nonoscillation of the second order linear differential equation Ž1.2., where q Ž t . G 0, t g w t 0 , ⬁.. THEOREM A. Ži.
If there exists t 0 ) 0 such that for e¨ ery n g N, ␣0 2 nq1 t 0 q Ž s . ds F nq 1 , Ž 1.3. n 2 t0 2 t0
H
then e¨ ery solution of Eq. Ž1.2. is nonoscillatory, where ␣ 0 s 3 y 2'2 . Žii. If there exist t 0 ) 0 and ␣ ) ␣ 0 such that for e¨ ery n g N, ␣ 2 nq1 t 0 q Ž s . ds G nq 1 , Ž 1.4. n 2 t0 2 t0
H
then e¨ ery solution of Eq. Ž1.2. is oscillatory, where ␣ 0 s 3 y 2'2 .
INTERVAL OSCILLATION CRITERIA
173
As an application, Huang w9x obtained the following corollary. If
COROLLARY A. Ži.
lim t
2t
H tª⬁ t
q Ž s . ds s ␣ -
␣0
,
Ž 1.5.
q Ž s . ds s ␣ ) ␣ 0 ,
Ž 1.6.
2
then e¨ ery solution of Eq. Ž1.2. is nonoscillatory. Žii. If lim t
2t
H tª⬁ t
then e¨ ery solution of Eq. Ž1.2. is oscillatory, where ␣ 0 s 3 y 2'2 . We note that the above result seems surprisingly interesting because the interval Ž ␣ 0r2 nq 1 t 0 , ␣ 0r2 n t 0 . is not covered by the conditions Ž1.3. and Ž1.4.. In particular, if q Ž t . s ␥rt 2 , where ␥ ) 0 is constant, then lim t
2t
H tª⬁ t
␥ s
2
ds s
␥ 2
-
␣0 2
ª ␥ - 3 y 2'2 - 1r4
and lim t
2t
H tª⬁ t
␥ s
2
ds s
␥ 2
s ␣ ) ␣ 0 ª ␥ ) 6 y 4'2 ) 1r4.
That implies that Huang’s result remains open for ␥ g Ž3 y 2'2 , 6 y 4'2 .. That is to say, Huang’s oscillation criterion is not sharp. In fact, the Euler equation y⬙ Ž t . q
␥ t2
yŽ t. s 0
is oscillatory if ␥ ) 1r4, and nonoscillatory if ␥ F 1r4 w13, 15x. We remark that Kong w11x employed the technique in the work of Philos w16x and obtained several interval oscillation results for the second order linear equation Ž1.2.. However, the results cannot be applied to the nonlinear differential equation Ž1.1.. Motivated by the ideas of Li w15x and Rogovchenko w17x, in this paper we obtain, by using a generalized Riccati technique, several new interval criteria for oscillation; that is, criteria given by the behavior of Eq. Ž1.1. Žor of r, q, and f . only on a sequence of subintervals of w t 0 , ⬁.. Our results involve the Kamenev’s type condition and improve and extend the results of Huang w9x, Kamenev w10x, and Philos w16x. Finally, several examples that dwell upon the sharp conditions of our results are also included. For other related oscillation results one can refer to w1, 3, 6, 10, 14, and 19x.
174
LI AND AGARWAL
Hereafter, we assume that ŽH1. The function q: w t 0 , ⬁. ª w0, ⬁. is continuous and q Ž t . k 0 on any ray w T, ⬁. for some T G t 0 . ŽH2. The function f : R ª R is continuous and yf Ž y . ) 0 for y / 0. ŽH3. The function g: R ª R is continuous and g Ž y . G K ) 0 for y / 0. We say that a function H s H Ž t, s . belongs to a function class X, denoted by H g X, if H g C Ž D, Rq ., where D s Ž t, s .: y⬁ - s F t ⬁4 , which satisfies H Ž t , t . s 0,
H Ž t , s . ) 0, for t ) s,
Ž 1.7.
and has partial derivatives ⭸ Hr⭸ t and ⭸ Hr⭸ s on D such that
⭸H ⭸t
s h1 Ž t , s . H Ž t , s .
1r2
and
⭸H ⭸s
s yh 2 Ž t , s . H Ž t , s .
1r2
, Ž 1.8.
where h1 , h 2 g L loc Ž D, R .. 2. OSCILLATION RESULTS FOR f Ž x . WITH MONOTONICITY In this section we always assume the following condition holds. ŽH4. There exists f ⬘Ž y . for y g R and f ⬘Ž y . G ) 0 for y / 0. Ž2.1. First, we establish two interesting lemmas, which will be useful for establishing oscillation criteria for Eq. Ž1.1.. LEMMA 2.1. Let assumptions Ž H 1. ᎐ Ž H4. hold. If y is a solution of Eq. Ž1.1. such that y Ž t . ) 0 on w c, b .. For any ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., let uŽ t . s ¨ Ž t .
y⬘ Ž t .
Ž 2.2.
f Ž yŽ t. .
on w c, b .. Then for any H g X, b
Hc H Ž b, s . K¨ Ž s . q Ž s . ds F H Ž b, c . u Ž c . q
1 4
b
Hc ¨ Ž s .
h 2 Ž b, s . y
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž b, s .
ds.
Ž 2.3.
175
INTERVAL OSCILLATION CRITERIA
Proof. From Ž1.1. and Ž2.2. we have for s g w c, b ., u⬘ Ž t . s y¨ Ž t . q Ž t . g Ž y⬘ Ž t . . y
f ⬘Ž y Ž t . . ¨ Ž t.
u2 Ž t . q
¨ ⬘Ž t . ¨ Ž t.
.
Ž 2.4.
In view of f ⬘Ž y . G ) 0 and g Ž y⬘. G K ) 0, we obtain by the above equality
u⬘ Ž t . q K¨ Ž t . q Ž t . q
¨ Ž t.
u2 Ž t . y
¨ ⬘Ž t . ¨ Ž t.
u Ž t . F 0.
Ž 2.5.
Multiplying Ž2.5. by H Ž t, s ., integrating it with respect to s from c to t for t g w c, b ., and using Ž1.6. and Ž1.7., we get t
Hc H Ž t , s . K¨ Ž s . q Ž s . ds t
Fy
Hc
H Ž t , s . u⬘ Ž s . ds y ¨ ⬘Ž s .
t
q
Hc H Ž t , s . ¨ Ž s .
t
Hc
H Ž t, s.
u2 Ž s . ¨ Ž s.
ds
u Ž s . ds
s H Ž t , c . uŽ c . y
t
Hc
½
'
h2 Ž t , s . H Ž t , s . uŽ s . yH Ž t , s .
¨ ⬘Ž s . ¨ Ž s.
uŽ s . q H Ž t , s .
u2 Ž s . ¨ Ž s.
5
ds
s H Ž t , c . uŽ c . y
t
Hc
½(
1
q
q
1 4
H Ž t, s.
2
¨ Ž s.
(
¨ Ž s.
uŽ s .
h2 Ž t , s . y
t
Hc
¨ Ž s . h2 Ž t , s . y
F H Ž t , c . uŽ c . q
1 4
t
¨ ⬘Ž s . ¨ Ž s.
Hc ¨ Ž s .
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž t , s .
5
ds
2
'H Ž t , s .
h2 Ž t , s . y
ds
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž t , s .
Letting t ª by in the above, we obtain Ž2.3.. The proof is complete.
ds.
176
LI AND AGARWAL
LEMMA 2.2. Let assumptions Ž H 1. ᎐ Ž H4. hold and suppose that y is a solution of Eq. Ž1.1. such that y Ž t . ) 0 on Ž a, c x. For any ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., let uŽ t . be defined by Ž2.2. on Ž a, c x. Then for any H g X, c
Ha H Ž s, a. K¨ Ž s . q Ž s . ds F yH Ž c, a . u Ž c . q
1 4
c
Ha
¨ Ž s . h1 Ž s, a . q
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž s, a.
ds.
Ž 2.6.
Proof. Similar to the proof of Lemma 2.1, we multiply Ž2.5. by H Ž s, t ., integrate it with respect to s from c for t g Ž a, c x, and use Ž1.6. and Ž1.7., then we get c
Ht H Ž s, t . K¨ Ž s . q Ž s . ds c
Fy
u2 Ž s .
c
Ht H Ž s, t . u⬘ Ž s . ds y Ht H Ž s, t . ¨ Ž s . r Ž s . ¨ ⬘Ž s .
c
q
Ht H Ž t , s . ¨ Ž s .
ds
u Ž s . ds
s yH Ž c, t . u Ž c . q
c
Ht
½
'
h1 Ž s, t . H Ž s, t . u Ž s . yH Ž s, t .
s yH Ž c, t . u Ž c . y
u2 Ž s . ¨ Ž s. c
Ht
1 ¨ Ž s.
q H Ž t, s.
¨ ⬘Ž s . ¨ Ž s.
5
u Ž s . ds
½'
H Ž s, t . u Ž s .
2
yh1 Ž s, t . H Ž s, t . u Ž s . ¨ Ž s .
'
q
q
1 4 1 4
¨
2
Ž s . h1 Ž s, t . q
c
Ht ¨ Ž s .
¨ ⬘Ž s . ¨ Ž s.
h1 Ž s, t . q
s yH Ž c, t . u Ž c .
2
'H Ž s, t .
¨ ⬘Ž s . ¨ Ž s.
5
ds
2
'H Ž s, t .
ds
177
INTERVAL OSCILLATION CRITERIA
1
c
y
Ht
¨ Ž s.
½'
H Ž s, t . u Ž s .
y
q
1
1 2
'
¨ Ž s . h1 Ž s, t . q
c
4
Ht
¨ Ž s . h1 Ž s, t . q
F yH Ž c, t . u Ž c . q
1 4
¨ ⬘Ž s . ¨ Ž s.
c
Ht ¨ Ž s .
¨ ⬘Ž s . ¨ Ž s.
2
5
'H Ž s, t .
ds
2
'H Ž s, t .
h1 Ž s, t . q
ds ¨ ⬘Ž s . ¨ Ž s.
2
'H Ž s, t .
ds.
Letting t ª ay in the above, we obtain Ž2.6.. The proof is complete. The following theorem is an immediate result from Lemmas 2.1 and 2.2. THEOREM 2.1. Assume that Ž H 1. ᎐ Ž H4. hold and that for some c g Ž a, b . and for some H g X, ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., 1
1
c
b
H H Ž s, a. K¨ Ž s . q Ž s . ds q H Ž b, c . Hc H Ž b, s . K¨ Ž s . q Ž s . ds H Ž c, a . a )
1
c
4 H Ž c, a . q
Ha ¨ Ž s .
1 4 H Ž b, c .
Hc
h1 Ž s, a . q
¨ ⬘Ž s . ¨ Ž s.
b
¨ Ž s . h 2 Ž b, s . y
2
'H Ž s, a.
¨ ⬘Ž s . ¨ Ž s.
ds 2
'H Ž b, s .
ds.
Ž 2.7.
Then e¨ ery solution of Eq. Ž1.1. has at least one zero in Ž a, b .. Proof. Suppose the contrary. Then without loss of generality we may assume that there is a solution y Ž t . of Eq. Ž1.1. such that y Ž t . ) 0 for t g Ž a, b .. From Lemmas 2.1 and 2.2 we see that both Ž2.3. and Ž2.6. hold. By dividing Ž2.3. and Ž2.6. by H Ž b, c . and H Ž c, a., respectively, and then adding them, we have that 1
1
c
b
H H Ž s, a. K¨ Ž s . q Ž s . ds q H Ž b, c . Hc H Ž b, s . K¨ Ž s . q Ž s . ds H Ž c, a . a F
1
c
4 H Ž c, a . q
Ha ¨ Ž s .
1 4 H Ž b, c .
Hc
b
h1 Ž s, a . q
¨ ⬘Ž s . ¨ Ž s.
¨ Ž s . h 2 Ž b, s . y
2
'H Ž s, a.
¨ ⬘Ž s . ¨ Ž s.
ds 2
'H Ž b, s .
ds,
which contradicts the assumption Ž2.7. and completes the proof.
178
LI AND AGARWAL
THEOREM 2.2. Assume that Ž H 1. ᎐ Ž H4. holds. If, for each T G t 0 , there exist H g X, ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. and a, b, c g R such that T F a - c - b and Ž2.7. holds, then e¨ ery solution of Eq. Ž1.1. is oscillatory. Proof. Pick up a sequence Ti 4 ; w t 0 , ⬁. such that Ti ª ⬁ as i ª ⬁. By the assumption, for each i g N, there exist a i , bi , c i g R such that Ti F a i - c i - bi , and Ž2.7. holds, where a, b, c are replaced by a i , bi , c i , respectively. From Theorem 2.1, every solution y Ž t . has at least one zero, t i g Ž a i , bi .. Noting that t i ) a i G Ti , i g N, we see that every solution has arbitrary large zeros. Thus, every solution of Eq. Ž1.1. is oscillatory. The proof is complete. THEOREM 2.3.
lim sup tª⬁
t
Hl
Assume that Ž H 1. ᎐ Ž H4. hold. If
H Ž s, l . K¨ Ž s . q Ž s .
y
1 4
¨ Ž s . h1 Ž s, l . q
ž
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž s, l .
/
ds ) 0
Ž 2.8.
and lim sup tª⬁
t
Hl
H Ž t , s . K¨ Ž s . q Ž s .
1
¨ ⬘Ž s .
y ¨ Ž s . h2 Ž t , s . y 4 ¨ Ž s.
ž
2
'H Ž t , s .
/
ds ) 0, Ž 2.9.
for some H g X, ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. and for each l G t 0 , then e¨ ery solution of Eq. Ž1.1. is oscillatory. Proof. For any T G t 0 , let a s T. In Ž2.8. we choose l s a. Then there exists c ) a such that
c
Ha
H Ž s, a . K¨ Ž s . q Ž s . y
1 4
¨ Ž s . h1 Ž s, a . q
ž
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž s, a.
/
ds)0.
Ž 2.10.
179
INTERVAL OSCILLATION CRITERIA
In Ž2.9. we choose l s c. Then there exists b ) c such that
Hc
b
H Ž b, s . K¨ Ž s . q Ž s . y
1 4
¨ Ž s . h 2 Ž b, s . y
ž
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž b, s .
/
ds)0.
Ž 2.11. Combining Ž2.10. and Ž2.11. we obtain Ž2.7.. The conclusion thus comes from Theorem 2.2. The proof is complete. For the case where H [ H Ž t y s . g X, we have that h1Ž t y s . s h 2 Ž t y s . and denote them by hŽ t y s .. The subclass of X containing such H Ž t y s . is denoted by X 0 . Applying Theorem 2.2 to X 0 , we obtain THEOREM 2.4. Assume that Ž H 1. ᎐ Ž H4. hold. If for each T G t 0 , there exist H g X 0 , ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., and a, c g R such that T F a - c and c
Ha H Ž s y a. K )
1
c
H 4 a q q
¨ Ž s . q Ž s . q ¨ Ž 2 c y s . q Ž 2 c y s . ds ¨ Ž s . q ¨ Ž 2 c y s . h 2 Ž s y a . ds
1
c
1
c
¨ Ž 2 c y s . y ¨ Ž s . h Ž s y a . H Ž s y a . ds
'
H 2 a H 4 a
¨ ⬘ Ž s . y ¨ ⬘ Ž 2 c y s . H Ž s y a . ds,
Ž 2.12.
then e¨ ery solution of Eq. Ž1.1. is oscillatory. Proof. Let b s 2 c y a. Then H Ž b y c . s H Ž c y a. s H ŽŽ b y a.r2., and for any w g Lw a, b x, we have c
b
Hc w Ž s . ds s Ha w Ž 2 c y s . ds. Hence c
b
Hc H Ž b y s . K¨ Ž s . q Ž s . ds s Ha H Ž s y a. K¨ Ž 2 c y s . q Ž 2 c y s . ds and b
c
Hc ¨ Ž s . h Ž b y s . ds s Ha ¨ Ž 2 c y s . h Ž s y a. ds. 2
2
180
LI AND AGARWAL
Thus that Ž2.12. holds implies that Ž2.7. holds for H g X 0 , ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁. and therefore every solution of Eq. Ž1.1. is oscillatory by Theorem 2.2. The proof is complete. From above oscillation criteria, we can obtain different sufficient conditions for oscillation of all solutions of Eq. Ž1.1. by different choices of H Ž t, s .. Let
H Ž t, s. s Ž t y s. ,
t G s G t0 ,
where ) 1 is a constant. COROLLARY 2.1. Assume that Ž H 1. ᎐ Ž H4. hold. Then e¨ ery solution of Eq. Ž1.1. is oscillatory pro¨ ided that for each l G t 0 and for some ) 1, there exists a function ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. such that the following two inequalities hold lim sup tª⬁
1 t
t
y1
Hl Ž s y l .
K¨ Ž s . q Ž s .
y
1
¨ Ž s.
ž
Hl Ž t y s .
4
Ž s y l.
y
¨ ⬘Ž s . ¨ Ž s.
2
ds ) 0,
/
Ž 2.13.
and lim sup tª⬁
1 t
t
y1
1
y
4
¨ Ž s.
K¨ Ž s . q Ž s .
ž
Ž t y s.
q
¨ ⬘Ž s . ¨ Ž s.
2
/
ds ) 0. Ž 2.14.
The proof is similar to that of Theorem 2.3, we omit it here. If we choose ¨ Ž t . s 1, then we have following result. THEOREM 2.5. Assume that Ž H 1. ᎐ Ž H4. hold. Then e¨ ery solution of Eq. Ž1.1. is oscillatory pro¨ ided that for each l G t 0 and for some ) 1, the following two inequalities hold lim sup tª⬁
t y1
t
Hl Ž s y l .
Kq Ž s . ds )
2
Ž 2.15.
4 Ž y 1.
and lim sup tª⬁
t
t
t y s . Kq Ž s . ds ) y1 H Ž
l
2 4 Ž y 1.
.
Ž 2.16.
181
INTERVAL OSCILLATION CRITERIA
Proof. Since H Ž t y s . s Ž t y s . , then h1 Ž t y s . s h 2 Ž t y s . s Ž t y s .
Ž y2 .r2
.
Noting that t 2
t 2
Hl h Ž s y l . ds s Hl Ž s y l .
y2
1
ds s
2 y1
Ž t y l.
y1
Ž t y l.
y1
,
and t 2
Hl h
2
t
y2 2 ds s 2 Ž t y s . ds s H Ž t y s .
l
y1
.
Hence, lim tª⬁
1 4 t y1
2
t 2
h1 Ž s y l . ds s
Hl
4 Ž y 1.
,
Ž 2.17.
.
Ž 2.18.
and lim tª⬁
1 4 t y1
2
t 2
Hl
h 2 Ž t y s . ds s
4 Ž y 1.
From Ž2.15. and Ž2.17. we have that lim sup tª⬁
t
y1
t
Hl
s lim sup tª⬁
s lim sup tª⬁
½
Ž s y l . Kq Ž s . y
t
y1
t y1
t
Hl Ž s y l . t
Hl Ž s y l .
1 4
5
h12 Ž s y l . ds
Kq Ž s . ds y lim
tª⬁
Kq Ž s . ds y
1 t
y1
2 4 Ž y 1.
t
1
Hl 4 h Ž s y l . ds 2 1
) 0,
i.e., Ž2.8. holds. Similarly, Ž2.16. implies that Ž2.9. holds. By Theorem 2.3, every solution of Eq. Ž1.1. is oscillatory. The proof is complete.
3. OSCILLATION RESULTS FOR f Ž x . WITHOUT MONOTONICITY In this section we consider the oscillation of Eq. Ž1.1. when the function f Ž y . is not monotonous. In this case we always assume that the following
182
LI AND AGARWAL
condition holds f Ž y . ry G 0 ) 0 for y / 0, where 0 is a constant. Ž 3.1.
Ž H4⬘.
LEMMA 3.1. Let assumptions Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold, and suppose y is a solution of Eq. Ž1.1. such that y Ž t . ) 0 on w c, b .. For any ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., let wŽ t. s ¨ Ž t.
y⬘ Ž t .
Ž 3.2.
yŽ t.
on w c, b .. Then for any H g X, b
Hc H Ž b, s . K ¨ Ž s . q Ž s . ds 0
F H Ž b, c . w Ž c . q
1 4
Hc
b
¨ Ž s . h 2 Ž b, s . y
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž b, s .
ds.
Ž 3.3.
Proof. From Ž1.1. and Ž3.2. we have for s g w c, b . w⬘ Ž t . s y¨ Ž t . q Ž t .
f Ž yŽ t. . yŽ t.
g Ž y⬘ Ž t . . y
1 ¨ Ž t.
u2 Ž t . q
¨ ⬘Ž t . ¨ Ž t.
wŽ t. .
Ž 3.4. In view of f Ž y .ry G 0 ) 0 and g Ž y⬘. G K ) 0, we obtain by the above equality w⬘ Ž t . q K 0¨ Ž t . q Ž t . q
1 ¨ Ž t.
u2 Ž t . y
¨ ⬘Ž t . ¨ Ž t.
u Ž t . F 0.
Ž 3.5.
The rest of the proof is similar to that of Lemma 2.1. LEMMA 3.2. Let assumptions Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold and suppose that y is a solution of Eq. Ž1.1. such that y Ž t . ) 0 on Ž a, c x. For any ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., let w Ž t . be defined by Ž3.2. on Ž a, c x. Then for any H g X, c
Ha H Ž s, a. K ¨ Ž s . q Ž s . ds 0
F yH Ž c, a . w Ž c .
183
INTERVAL OSCILLATION CRITERIA
q
1
c
Ha ¨ Ž s .
4
h1 Ž s, a . q
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž s, a.
ds.
The following theorem is an immediate result from Lemmas 3.1 and 3.2. THEOREM 3.1. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold and that for some c g Ž a, b . and for some H g X, ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., 1
c
H H Ž s, a. K ¨ Ž s . q Ž s . ds H Ž c, a . a 0
q )
1
b
H Ž b, c .
Hc H Ž b, s . K ¨ Ž s . q Ž s . ds
1
c
4 H Ž c, a . q
0
Ha ¨ Ž s .
1
b
4 H Ž b, c .
h1 Ž s, a . q
Hc ¨ Ž s .
¨ ⬘Ž s . ¨ Ž s.
h 2 Ž b, s . y
2
'H Ž s, a.
¨ ⬘Ž s . ¨ Ž s.
ds 2
'H Ž b, s .
ds. Ž 3.6.
Then e¨ ery solution of Eq. Ž1.1. has at least one zero in Ž a, b .. THEOREM 3.2. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. holds. If, for each T G t 0 , there exist H g X, ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. and a, b, c g R such that T F a - c - b and Ž3.6. holds, then e¨ ery solution of Eq. Ž1.1. is oscillatory. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold. If
THEOREM 3.3. lim sup tª⬁
t
Hl
H Ž s, l . K 0¨ Ž s . q Ž s . 1
¨ ⬘Ž s .
y ¨ Ž s . h1 Ž s, l . q 4 ¨ Ž s.
ž
2
'H Ž s, l .
ds ) 0,
/
and lim sup tª⬁
t
Hl
H Ž t , s . K 0¨ Ž s . q Ž s . 1 ¨ ⬘Ž s . y ¨ Ž s . h2 Ž t , s . y 4 ¨ Ž s.
ž
2
'H Ž t , s .
/
ds ) 0,
for some H g X, g g C 1 Žw t 0 , ⬁., Ž0, ⬁.. and for each l G t 0 , then e¨ ery solution of Eq. Ž1.1. is oscillatory.
184
LI AND AGARWAL
THEOREM 3.4. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold. If for each T G t 0 , there exist H g X 0 , ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. and a, c g R such that T F a - c and c
Ha H Ž s y a. K 1
)
c
q
¨ Ž s . q Ž s . q ¨ Ž 2 c y s . q Ž 2 c y s . ds
¨ Ž s . q ¨ Ž 2 c y s . h 2 Ž s y a . ds
H 4 a q
0
1
c
1
c
¨ Ž 2 c y s . y ¨ Ž s . h Ž s y a . H Ž s y a . ds
'
H 2 a
¨ ⬘ Ž s . y ¨ ⬘ Ž 2 c y s . H Ž s y a . ds,
H 4 a
then e¨ ery solution of Eq. Ž1.1. is oscillatory. COROLLARY 3.1. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold. Then e¨ ery solution of Eq. Ž1.1. is oscillatory pro¨ ided that for each l G t 0 and for some ) 1, there exists a function ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. such that the following two inequalities hold lim sup tª⬁
1 t
t
y1
Hl Ž s y l .
K 0¨ Ž s . q Ž s . 1
¨ ⬘Ž s .
y ¨ Ž s. y 4 ¨ Ž s. Ž s y l.
ž
2
/
ds ) 0,
and lim sup tª⬁
1 t
y1
t
Hl Ž t y s .
K 0¨ Ž s . q Ž s . 1
¨ ⬘Ž s .
y ¨ Ž s. q 4 ¨ Ž s. Ž t y s.
ž
2
/
ds ) 0.
THEOREM 3.5. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold. Then e¨ ery solution of Eq. Ž1.1. is oscillatory pro¨ ided that for each l G t 0 and for some ) 1, the following two inequalities hold lim sup tª⬁
1
t
t
s y l . K 0 q Ž s . ds ) y1 H Ž
l
2 4 Ž y 1.
185
INTERVAL OSCILLATION CRITERIA
and lim sup tª⬁
1
t
2
t
t y s . K 0 q Ž s . ds ) y1 H Ž
4 Ž y 1.
l
.
4. EXAMPLES In this section we will show the applications of our oscillation criteria by two examples. We will see that the equations in the examples are oscillatory based on the results in Sections 2 and 3, though the oscillation cannot be demonstrated by the results of Huang w9x and Kong w11x and most other known criteria. EXAMPLE 1. Consider the nonlinear differential equation y⬙ Ž t . q
␥
y Ž t . Ž 1 q y 2 Ž t . . 1 q Ž y⬘ Ž t . .
t2
2
s 0,
t G 1.
Ž 4.1.
Let f Ž y . s y Ž1 q y 2 . and g Ž y . s 1 q y 2 . Then f ⬘Ž y . s 1 q 3 y 2 G 1 s ,
g Ž y . s 1 q y 2 G 1.
Note that for ) 1, lim tª⬁
1
t
t
s y l. y1 H Ž
␥
s
l
2
ds s
␥ y1
Ž t y l.
lim
t
tª⬁
s
␥ y1
. Ž 4.2.
Next, we will prove that t
Hl Ž t y s .
␥ s
2
ds G
t
Hl Ž s y l .
␥ s2
ds.
Ž 4.3.
Let FŽ t. s
t
Hl
Ž t y s. y Ž s y l .
␥ s2
ds.
Then F Ž l . s 0, and for t G l, F⬘ Ž t . s G s
t
Hl Ž t y s . ␥ 2
t ␥ t
2
t
y1
Hl Ž t y s .
␥ s
2
y1
ds y Ž t y l . ds y Ž t y l .
Ž t y l. y Ž t y l.
␥ t2
s 0.
␥ t2
␥ t2
186
LI AND AGARWAL
Hence F Ž t . G F Ž l . s 0 for t G l, i.e., Ž4.3. holds. By Ž4.2. and Ž4.3., for any ␥ ) 1r4 there exists ) 1 such that ␥rŽ y 1. ) 2r4Ž y 1.. This means that Ž2.15. and Ž2.16. hold for the same . Applying Theorem 2.5, we find that Ž4.1. is oscillatory for ␥ ) 1r4. However, the results of Huang w9x and Kong w11x fail since f Ž y . is a nonlinear function and g Ž y . k 1. We remark that if f Ž y . s y and g Ž y . s 1, then Eq. Ž4.1. reduces to the Euler equation ␥ y⬙ Ž t . q 2 y Ž t . s 0, Ž 4.4. t where ␥ ) 0 is a constant. It is well known w12, 14x that every solution of Eq. Ž4.4. is oscillatory if ␥ ) 1r4. However, Theorem A of Huang only applies to the case ␥ ) 2Ž3 y 2'2 ., but for the case 1r4 - ␥ F 2Ž3 y 2'2 . it remains open. This implies that our results are sharp. EXAMPLE 2. Consider the nonlinear differential equation y⬙ Ž t . q
2 Ž 1 q cos 2 t .
Ž 3 q cos t . Ž 1 q sin t . 2
2
yŽ t.
ž
1
1
q
2
/Ž
1 q y2 Ž t.
1 q Ž y⬘ Ž t . .
s 0,
2
.
Ž 4.5.
where t G 1. Observe that f Ž y. s y
ž
1 2
q
1 1qy
2
/
f ⬘Ž y . s y
and
2y 1 q y2
.
Clearly, Theorem 2.5 cannot apply to Eq. Ž4.5.. In spite of this, with
¨ Ž t . s t 2 and s 2, we can prove the oscillatory character of Eq. Ž4.5. by
Corollary 3.1. Because, for all y g R _ 04 , f Ž y. y
s
1 2
1
q
1qy
2
G
1 2
s 0 ,
and g Ž y . s 1 q y 2 G 1 s K. Thus, lim tª⬁
1 t
t
Hl
2
Ž s y l.
s lim
tª⬁
G lim
tª⬁
1 t 1
1 2
t
¨ Ž s. qŽ s. y
H Ž s y l . 2 s2 l
t
H t l
1 3
1
1 4
¨ Ž s.
ž
2 syl
y
¨ ⬘Ž s . ¨ Ž s.
2 Ž 1 q cos 2 t .
2 Ž 3 q cos 2 t . Ž 1 q sin 2 t .
Ž s y l . s 2 y l 2 ds s ⬁,
2
/
y
ds l2 s2 Ž s y l .
2
ds
187
INTERVAL OSCILLATION CRITERIA
and 1
lim tª⬁
t
t
2
Hl Ž t y s .
s lim
tª⬁
G lim
tª⬁
1 t 1
1 2
t
¨ Ž s. qŽ s. y
Hl Ž s y l . t
H t l
1 3
2 2
s
1 4
1
¨ Ž s.
ž
2 tys
q
¨ ⬘Ž s . ¨ Ž s.
2 Ž 1 q cos 2 t .
2 Ž 3 q cos 2 t . Ž 1 q sin 2 t .
2
/
y
ds t2 s2 Ž t y s.
2
ds
2 Ž t y s . s 2 y t 2 ds s ⬁.
Thereby, Eq. Ž4.5. is oscillatory by Corollary 3.1. Observe that y Ž t . s cost is an oscillatory solution of Eq. Ž4.5.. Remark 1. The oscillation criteria presented in this paper require the assumption ŽH4. or ŽH4⬘. on f Ž y . and thus the results obtained here are not applicable to equation of type Ž1.1. where, for example, f Ž y . s < y < sgny, ) 0, / 1. This problem is related to the question posed by Kamenev w10x whether condition ŽH4. can be relaxed to
Ž H4⬙ . f ⬘Ž y . G 0 for y / 0, or whether it can be omitted. For the case of equation
Ž aŽ t . y⬘ Ž t . . ⬘ q p Ž t . y⬘ Ž t . q q Ž t . < y Ž t . < sgn l y Ž t . s 0,
)0
an affirmative answer to the preceding question has been given by Grace and Lalli w8, Theorem 8x. The study of this problem for Eq. Ž1.1. will form the subject of one of our forthcoming papers.
REFERENCES 1. G. J. Butler, L. H. Erbe, and A. B. Mingarelli, Riccati techniques and variational principles in oscillation theory for linear systems, Trans. Amer. Math. Soc. 303 Ž1987., 263᎐282. 2. T. A. Burton and R. C. Grimmer, Stability properties of Ž ru⬘.⬘ q af Ž u. g Ž u⬘. s 0, Monatsh. Math. 74 Ž1970., 211᎐222. 3. R. Byers, B. J. Harri, and M. K. Kwong, Weighted means and oscillation conditions for second order matrix differential equations, J. Differential Equations 61 Ž1986., 164᎐177. 4. M. A. Ei-Sayed, An oscillation criterion for a forced second order linear differential equation, Proc. Amer. Math. Soc. 118 Ž1993., 813᎐817. 5. J. R. Graef and P. W. Spikes, Asymptotic behavior of solutions of a second order nonlinear differential equation, J. Differential Equations 17 Ž1975., 451᎐476. 6. J. R. Graef and P. W. Spikes, Boundedness and convergence to zeros of solutions of a forced second order nonlinear differential equation, J. Math. Anal. Appl. 62 Ž1978., 295᎐309.
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7. S. R. Grace and B. S. Lalli, An oscillation criterion for second order strongly sublinear differential equations, J. Math. Anal. Appl. 123 Ž1987., 584᎐588. 8. S. R. Grace and B. S. Lalli, Integral averaging technique for the oscillation of second order nonlinear differential equations, J. Math. Anal. Appl. 149 Ž1990., 277᎐311. 9. C. C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 Ž1997., 712᎐723. 10. I. V. Kamenev, Oscillation criteria related to averaging of solutions of second order differential equations, Differentsial’nye Ura¨ nenyia 10 Ž1974., 246᎐252 Žin Russian.. 11. Q. Kong, Interval criteria for oscillation of second-order linear ordinary differential equations, J. Math. Anal. Appl. 229 Ž1999., 258᎐270. 12. B. S. Lalli, On boundedness of solutions of certain second order differential equations, J. Math. Anal. Appl. 25 Ž1969., 182᎐188. 13. H. J. Li, Oscillation criteria for second order linear differential equations, J. Math. Anal. Appl. 194 Ž1995., 217᎐234. 14. W. T. Li and R. P. Agarwal, Interval oscillation criteria for second order nonlinear differential equations with damping, Comput. Math. Appl. Žto appear.. 15. W. T. Li, Oscillation of certain second-order nonlinear differential equations, J. Math. Anal. Appl. 217 Ž1998., 1᎐14. 16. CH. G. Philos, Oscillation theorems for linear differential equations of second order, Arch. Math. Ž Basel . 53 Ž1989., 482᎐492. 17. Yuri V. Rogovchenko, Oscillation criteria for certain nonlinear differential equations, J. Math. Anal. Appl. 229 Ž1999., 399᎐416. 18. J. S. W. Wong and T. A. Burton, Some properties of solution of u⬙ q aŽ t . f Ž u. g Ž u⬘. s 0, Monatsh. Math. 69 Ž1965., 364᎐374. 19. P. J. Y. Wong and R. P. Agarwal, Oscillatory behavior of solutions of certain second order nonlinear differential equations, J. Math. Anal. Appl. 198 Ž1996., 337᎐354.
Interval Oscillation Criteria Related to Integral Averaging Technique for Certain Nonlinear Differential Equations Wan-Tong Li 1 Department of Mathematics, Lanzhou Uni¨ ersity, Lanzhou, Gansu, 730000, People’s Republic of China
and Ravi P. Agarwal Department of Mathematics, National Uni¨ ersity of Singapore, 10 Kent Ridge Crescent, Singapore, 119260, Singapore Submitted by George Leitmann Received December 23, 1999
We present new interval oscillation criteria related to integral averaging technique for certain classes of second order nonlinear differential equations that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of w t 0 , ⬁., rather than on the whole half-line. Our results are sharper than some previous results and handle the cases which are not covered by known criteria. In particular, several examples that dwell upon the sharp conditions of our results are also included. 䊚 2000 Academic Press Key Words: interval criteria; oscillation; second order; nonlinear differential equations; integral averaging technique.
1. INTRODUCTION We consider the oscillation behavior of solutions of the second-order nonlinear differential equation y⬙ Ž t . q q Ž t . f Ž y Ž t . . g Ž y⬘ Ž t . . s 0,
Ž 1.1.
1 Research was supported by the NSF of Gansu Province and the NSF of Gansu University of Technology.
171 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
172
LI AND AGARWAL
where t G t 0 , and the functions q, f, and g are to be specified in the following text. We recall that a function y: w t 0 , t 1 . ª Žy⬁, ⬁., t 1 ) t 0 is called a solution of Eq. Ž1.1. if y Ž t . satisfies Eq. Ž1.1. for all t g w t 0 , t 1 .. In the sequel it will be always assumed that solutions of Eq. Ž1.1. exist for any t 0 G 0. A solution y Ž t . of Eq. Ž1.1. is called oscillatory if it has arbitrary large zeros, otherwise it is called nonoscillatory. Equation Ž1.1. has been studied by Grace and Lalli w7x. They mentioned that, though stability, boundedness, and convergence to zero of all solutions of Eq. Ž1.1. have been investigated in the papers of Burton and Grimmer w2x, Graef and Spikes w5, 6x, Lalli w12x, and Wong and Burton w18x. Nothing much has been known regarding the oscillatory behavior of Eq. Ž1.1. except for the result by Wong and Burton w18, Theorem 4x regarding the oscillatory behavior of Eq. Ž1.1. in connection with that of the corresponding linear equation y⬙ Ž t . q q Ž t . y Ž t . s 0. Ž 1.2. Recently, Rogovchenko w17x presented new sufficient conditions which ensure the oscillatory character of Eq. Ž1.1.. They are different from those of Grace and Lalli w7x and are applicable to other classes of equations which are not covered by the results of Grace and Lalli w7x. However, all the above mentioned oscillation results involve the interval of q and hence require the information of q on the entire half-line w t 0 , ⬁.. From the Sturm Separation Theorem, we see that oscillation is only an interval property, i.e., if there exists a sequence of subintervals w a i , bi x of w t 0 , ⬁., as a i ª ⬁, such that for each i there exists a solution of Eq. Ž1.2. that has at least two zeros in w a i , bi x, then every solution of Eq. Ž1.2. is oscillatory. Ei-Sayed w4x established an interval criterion for oscillation of a forced second-order equation, but the result is not very sharp, because a comparison with equations of constant coefficient is used in the proof. In 1997, Huang w9x presented the following interval criteria for oscillation and nonoscillation of the second order linear differential equation Ž1.2., where q Ž t . G 0, t g w t 0 , ⬁.. THEOREM A. Ži.
If there exists t 0 ) 0 such that for e¨ ery n g N, ␣0 2 nq1 t 0 q Ž s . ds F nq 1 , Ž 1.3. n 2 t0 2 t0
H
then e¨ ery solution of Eq. Ž1.2. is nonoscillatory, where ␣ 0 s 3 y 2'2 . Žii. If there exist t 0 ) 0 and ␣ ) ␣ 0 such that for e¨ ery n g N, ␣ 2 nq1 t 0 q Ž s . ds G nq 1 , Ž 1.4. n 2 t0 2 t0
H
then e¨ ery solution of Eq. Ž1.2. is oscillatory, where ␣ 0 s 3 y 2'2 .
INTERVAL OSCILLATION CRITERIA
173
As an application, Huang w9x obtained the following corollary. If
COROLLARY A. Ži.
lim t
2t
H tª⬁ t
q Ž s . ds s ␣ -
␣0
,
Ž 1.5.
q Ž s . ds s ␣ ) ␣ 0 ,
Ž 1.6.
2
then e¨ ery solution of Eq. Ž1.2. is nonoscillatory. Žii. If lim t
2t
H tª⬁ t
then e¨ ery solution of Eq. Ž1.2. is oscillatory, where ␣ 0 s 3 y 2'2 . We note that the above result seems surprisingly interesting because the interval Ž ␣ 0r2 nq 1 t 0 , ␣ 0r2 n t 0 . is not covered by the conditions Ž1.3. and Ž1.4.. In particular, if q Ž t . s ␥rt 2 , where ␥ ) 0 is constant, then lim t
2t
H tª⬁ t
␥ s
2
ds s
␥ 2
-
␣0 2
ª ␥ - 3 y 2'2 - 1r4
and lim t
2t
H tª⬁ t
␥ s
2
ds s
␥ 2
s ␣ ) ␣ 0 ª ␥ ) 6 y 4'2 ) 1r4.
That implies that Huang’s result remains open for ␥ g Ž3 y 2'2 , 6 y 4'2 .. That is to say, Huang’s oscillation criterion is not sharp. In fact, the Euler equation y⬙ Ž t . q
␥ t2
yŽ t. s 0
is oscillatory if ␥ ) 1r4, and nonoscillatory if ␥ F 1r4 w13, 15x. We remark that Kong w11x employed the technique in the work of Philos w16x and obtained several interval oscillation results for the second order linear equation Ž1.2.. However, the results cannot be applied to the nonlinear differential equation Ž1.1.. Motivated by the ideas of Li w15x and Rogovchenko w17x, in this paper we obtain, by using a generalized Riccati technique, several new interval criteria for oscillation; that is, criteria given by the behavior of Eq. Ž1.1. Žor of r, q, and f . only on a sequence of subintervals of w t 0 , ⬁.. Our results involve the Kamenev’s type condition and improve and extend the results of Huang w9x, Kamenev w10x, and Philos w16x. Finally, several examples that dwell upon the sharp conditions of our results are also included. For other related oscillation results one can refer to w1, 3, 6, 10, 14, and 19x.
174
LI AND AGARWAL
Hereafter, we assume that ŽH1. The function q: w t 0 , ⬁. ª w0, ⬁. is continuous and q Ž t . k 0 on any ray w T, ⬁. for some T G t 0 . ŽH2. The function f : R ª R is continuous and yf Ž y . ) 0 for y / 0. ŽH3. The function g: R ª R is continuous and g Ž y . G K ) 0 for y / 0. We say that a function H s H Ž t, s . belongs to a function class X, denoted by H g X, if H g C Ž D, Rq ., where D s Ž t, s .: y⬁ - s F t ⬁4 , which satisfies H Ž t , t . s 0,
H Ž t , s . ) 0, for t ) s,
Ž 1.7.
and has partial derivatives ⭸ Hr⭸ t and ⭸ Hr⭸ s on D such that
⭸H ⭸t
s h1 Ž t , s . H Ž t , s .
1r2
and
⭸H ⭸s
s yh 2 Ž t , s . H Ž t , s .
1r2
, Ž 1.8.
where h1 , h 2 g L loc Ž D, R .. 2. OSCILLATION RESULTS FOR f Ž x . WITH MONOTONICITY In this section we always assume the following condition holds. ŽH4. There exists f ⬘Ž y . for y g R and f ⬘Ž y . G ) 0 for y / 0. Ž2.1. First, we establish two interesting lemmas, which will be useful for establishing oscillation criteria for Eq. Ž1.1.. LEMMA 2.1. Let assumptions Ž H 1. ᎐ Ž H4. hold. If y is a solution of Eq. Ž1.1. such that y Ž t . ) 0 on w c, b .. For any ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., let uŽ t . s ¨ Ž t .
y⬘ Ž t .
Ž 2.2.
f Ž yŽ t. .
on w c, b .. Then for any H g X, b
Hc H Ž b, s . K¨ Ž s . q Ž s . ds F H Ž b, c . u Ž c . q
1 4
b
Hc ¨ Ž s .
h 2 Ž b, s . y
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž b, s .
ds.
Ž 2.3.
175
INTERVAL OSCILLATION CRITERIA
Proof. From Ž1.1. and Ž2.2. we have for s g w c, b ., u⬘ Ž t . s y¨ Ž t . q Ž t . g Ž y⬘ Ž t . . y
f ⬘Ž y Ž t . . ¨ Ž t.
u2 Ž t . q
¨ ⬘Ž t . ¨ Ž t.
.
Ž 2.4.
In view of f ⬘Ž y . G ) 0 and g Ž y⬘. G K ) 0, we obtain by the above equality
u⬘ Ž t . q K¨ Ž t . q Ž t . q
¨ Ž t.
u2 Ž t . y
¨ ⬘Ž t . ¨ Ž t.
u Ž t . F 0.
Ž 2.5.
Multiplying Ž2.5. by H Ž t, s ., integrating it with respect to s from c to t for t g w c, b ., and using Ž1.6. and Ž1.7., we get t
Hc H Ž t , s . K¨ Ž s . q Ž s . ds t
Fy
Hc
H Ž t , s . u⬘ Ž s . ds y ¨ ⬘Ž s .
t
q
Hc H Ž t , s . ¨ Ž s .
t
Hc
H Ž t, s.
u2 Ž s . ¨ Ž s.
ds
u Ž s . ds
s H Ž t , c . uŽ c . y
t
Hc
½
'
h2 Ž t , s . H Ž t , s . uŽ s . yH Ž t , s .
¨ ⬘Ž s . ¨ Ž s.
uŽ s . q H Ž t , s .
u2 Ž s . ¨ Ž s.
5
ds
s H Ž t , c . uŽ c . y
t
Hc
½(
1
q
q
1 4
H Ž t, s.
2
¨ Ž s.
(
¨ Ž s.
uŽ s .
h2 Ž t , s . y
t
Hc
¨ Ž s . h2 Ž t , s . y
F H Ž t , c . uŽ c . q
1 4
t
¨ ⬘Ž s . ¨ Ž s.
Hc ¨ Ž s .
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž t , s .
5
ds
2
'H Ž t , s .
h2 Ž t , s . y
ds
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž t , s .
Letting t ª by in the above, we obtain Ž2.3.. The proof is complete.
ds.
176
LI AND AGARWAL
LEMMA 2.2. Let assumptions Ž H 1. ᎐ Ž H4. hold and suppose that y is a solution of Eq. Ž1.1. such that y Ž t . ) 0 on Ž a, c x. For any ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., let uŽ t . be defined by Ž2.2. on Ž a, c x. Then for any H g X, c
Ha H Ž s, a. K¨ Ž s . q Ž s . ds F yH Ž c, a . u Ž c . q
1 4
c
Ha
¨ Ž s . h1 Ž s, a . q
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž s, a.
ds.
Ž 2.6.
Proof. Similar to the proof of Lemma 2.1, we multiply Ž2.5. by H Ž s, t ., integrate it with respect to s from c for t g Ž a, c x, and use Ž1.6. and Ž1.7., then we get c
Ht H Ž s, t . K¨ Ž s . q Ž s . ds c
Fy
u2 Ž s .
c
Ht H Ž s, t . u⬘ Ž s . ds y Ht H Ž s, t . ¨ Ž s . r Ž s . ¨ ⬘Ž s .
c
q
Ht H Ž t , s . ¨ Ž s .
ds
u Ž s . ds
s yH Ž c, t . u Ž c . q
c
Ht
½
'
h1 Ž s, t . H Ž s, t . u Ž s . yH Ž s, t .
s yH Ž c, t . u Ž c . y
u2 Ž s . ¨ Ž s. c
Ht
1 ¨ Ž s.
q H Ž t, s.
¨ ⬘Ž s . ¨ Ž s.
5
u Ž s . ds
½'
H Ž s, t . u Ž s .
2
yh1 Ž s, t . H Ž s, t . u Ž s . ¨ Ž s .
'
q
q
1 4 1 4
¨
2
Ž s . h1 Ž s, t . q
c
Ht ¨ Ž s .
¨ ⬘Ž s . ¨ Ž s.
h1 Ž s, t . q
s yH Ž c, t . u Ž c .
2
'H Ž s, t .
¨ ⬘Ž s . ¨ Ž s.
5
ds
2
'H Ž s, t .
ds
177
INTERVAL OSCILLATION CRITERIA
1
c
y
Ht
¨ Ž s.
½'
H Ž s, t . u Ž s .
y
q
1
1 2
'
¨ Ž s . h1 Ž s, t . q
c
4
Ht
¨ Ž s . h1 Ž s, t . q
F yH Ž c, t . u Ž c . q
1 4
¨ ⬘Ž s . ¨ Ž s.
c
Ht ¨ Ž s .
¨ ⬘Ž s . ¨ Ž s.
2
5
'H Ž s, t .
ds
2
'H Ž s, t .
h1 Ž s, t . q
ds ¨ ⬘Ž s . ¨ Ž s.
2
'H Ž s, t .
ds.
Letting t ª ay in the above, we obtain Ž2.6.. The proof is complete. The following theorem is an immediate result from Lemmas 2.1 and 2.2. THEOREM 2.1. Assume that Ž H 1. ᎐ Ž H4. hold and that for some c g Ž a, b . and for some H g X, ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., 1
1
c
b
H H Ž s, a. K¨ Ž s . q Ž s . ds q H Ž b, c . Hc H Ž b, s . K¨ Ž s . q Ž s . ds H Ž c, a . a )
1
c
4 H Ž c, a . q
Ha ¨ Ž s .
1 4 H Ž b, c .
Hc
h1 Ž s, a . q
¨ ⬘Ž s . ¨ Ž s.
b
¨ Ž s . h 2 Ž b, s . y
2
'H Ž s, a.
¨ ⬘Ž s . ¨ Ž s.
ds 2
'H Ž b, s .
ds.
Ž 2.7.
Then e¨ ery solution of Eq. Ž1.1. has at least one zero in Ž a, b .. Proof. Suppose the contrary. Then without loss of generality we may assume that there is a solution y Ž t . of Eq. Ž1.1. such that y Ž t . ) 0 for t g Ž a, b .. From Lemmas 2.1 and 2.2 we see that both Ž2.3. and Ž2.6. hold. By dividing Ž2.3. and Ž2.6. by H Ž b, c . and H Ž c, a., respectively, and then adding them, we have that 1
1
c
b
H H Ž s, a. K¨ Ž s . q Ž s . ds q H Ž b, c . Hc H Ž b, s . K¨ Ž s . q Ž s . ds H Ž c, a . a F
1
c
4 H Ž c, a . q
Ha ¨ Ž s .
1 4 H Ž b, c .
Hc
b
h1 Ž s, a . q
¨ ⬘Ž s . ¨ Ž s.
¨ Ž s . h 2 Ž b, s . y
2
'H Ž s, a.
¨ ⬘Ž s . ¨ Ž s.
ds 2
'H Ž b, s .
ds,
which contradicts the assumption Ž2.7. and completes the proof.
178
LI AND AGARWAL
THEOREM 2.2. Assume that Ž H 1. ᎐ Ž H4. holds. If, for each T G t 0 , there exist H g X, ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. and a, b, c g R such that T F a - c - b and Ž2.7. holds, then e¨ ery solution of Eq. Ž1.1. is oscillatory. Proof. Pick up a sequence Ti 4 ; w t 0 , ⬁. such that Ti ª ⬁ as i ª ⬁. By the assumption, for each i g N, there exist a i , bi , c i g R such that Ti F a i - c i - bi , and Ž2.7. holds, where a, b, c are replaced by a i , bi , c i , respectively. From Theorem 2.1, every solution y Ž t . has at least one zero, t i g Ž a i , bi .. Noting that t i ) a i G Ti , i g N, we see that every solution has arbitrary large zeros. Thus, every solution of Eq. Ž1.1. is oscillatory. The proof is complete. THEOREM 2.3.
lim sup tª⬁
t
Hl
Assume that Ž H 1. ᎐ Ž H4. hold. If
H Ž s, l . K¨ Ž s . q Ž s .
y
1 4
¨ Ž s . h1 Ž s, l . q
ž
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž s, l .
/
ds ) 0
Ž 2.8.
and lim sup tª⬁
t
Hl
H Ž t , s . K¨ Ž s . q Ž s .
1
¨ ⬘Ž s .
y ¨ Ž s . h2 Ž t , s . y 4 ¨ Ž s.
ž
2
'H Ž t , s .
/
ds ) 0, Ž 2.9.
for some H g X, ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. and for each l G t 0 , then e¨ ery solution of Eq. Ž1.1. is oscillatory. Proof. For any T G t 0 , let a s T. In Ž2.8. we choose l s a. Then there exists c ) a such that
c
Ha
H Ž s, a . K¨ Ž s . q Ž s . y
1 4
¨ Ž s . h1 Ž s, a . q
ž
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž s, a.
/
ds)0.
Ž 2.10.
179
INTERVAL OSCILLATION CRITERIA
In Ž2.9. we choose l s c. Then there exists b ) c such that
Hc
b
H Ž b, s . K¨ Ž s . q Ž s . y
1 4
¨ Ž s . h 2 Ž b, s . y
ž
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž b, s .
/
ds)0.
Ž 2.11. Combining Ž2.10. and Ž2.11. we obtain Ž2.7.. The conclusion thus comes from Theorem 2.2. The proof is complete. For the case where H [ H Ž t y s . g X, we have that h1Ž t y s . s h 2 Ž t y s . and denote them by hŽ t y s .. The subclass of X containing such H Ž t y s . is denoted by X 0 . Applying Theorem 2.2 to X 0 , we obtain THEOREM 2.4. Assume that Ž H 1. ᎐ Ž H4. hold. If for each T G t 0 , there exist H g X 0 , ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., and a, c g R such that T F a - c and c
Ha H Ž s y a. K )
1
c
H 4 a q q
¨ Ž s . q Ž s . q ¨ Ž 2 c y s . q Ž 2 c y s . ds ¨ Ž s . q ¨ Ž 2 c y s . h 2 Ž s y a . ds
1
c
1
c
¨ Ž 2 c y s . y ¨ Ž s . h Ž s y a . H Ž s y a . ds
'
H 2 a H 4 a
¨ ⬘ Ž s . y ¨ ⬘ Ž 2 c y s . H Ž s y a . ds,
Ž 2.12.
then e¨ ery solution of Eq. Ž1.1. is oscillatory. Proof. Let b s 2 c y a. Then H Ž b y c . s H Ž c y a. s H ŽŽ b y a.r2., and for any w g Lw a, b x, we have c
b
Hc w Ž s . ds s Ha w Ž 2 c y s . ds. Hence c
b
Hc H Ž b y s . K¨ Ž s . q Ž s . ds s Ha H Ž s y a. K¨ Ž 2 c y s . q Ž 2 c y s . ds and b
c
Hc ¨ Ž s . h Ž b y s . ds s Ha ¨ Ž 2 c y s . h Ž s y a. ds. 2
2
180
LI AND AGARWAL
Thus that Ž2.12. holds implies that Ž2.7. holds for H g X 0 , ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁. and therefore every solution of Eq. Ž1.1. is oscillatory by Theorem 2.2. The proof is complete. From above oscillation criteria, we can obtain different sufficient conditions for oscillation of all solutions of Eq. Ž1.1. by different choices of H Ž t, s .. Let
H Ž t, s. s Ž t y s. ,
t G s G t0 ,
where ) 1 is a constant. COROLLARY 2.1. Assume that Ž H 1. ᎐ Ž H4. hold. Then e¨ ery solution of Eq. Ž1.1. is oscillatory pro¨ ided that for each l G t 0 and for some ) 1, there exists a function ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. such that the following two inequalities hold lim sup tª⬁
1 t
t
y1
Hl Ž s y l .
K¨ Ž s . q Ž s .
y
1
¨ Ž s.
ž
Hl Ž t y s .
4
Ž s y l.
y
¨ ⬘Ž s . ¨ Ž s.
2
ds ) 0,
/
Ž 2.13.
and lim sup tª⬁
1 t
t
y1
1
y
4
¨ Ž s.
K¨ Ž s . q Ž s .
ž
Ž t y s.
q
¨ ⬘Ž s . ¨ Ž s.
2
/
ds ) 0. Ž 2.14.
The proof is similar to that of Theorem 2.3, we omit it here. If we choose ¨ Ž t . s 1, then we have following result. THEOREM 2.5. Assume that Ž H 1. ᎐ Ž H4. hold. Then e¨ ery solution of Eq. Ž1.1. is oscillatory pro¨ ided that for each l G t 0 and for some ) 1, the following two inequalities hold lim sup tª⬁
t y1
t
Hl Ž s y l .
Kq Ž s . ds )
2
Ž 2.15.
4 Ž y 1.
and lim sup tª⬁
t
t
t y s . Kq Ž s . ds ) y1 H Ž
l
2 4 Ž y 1.
.
Ž 2.16.
181
INTERVAL OSCILLATION CRITERIA
Proof. Since H Ž t y s . s Ž t y s . , then h1 Ž t y s . s h 2 Ž t y s . s Ž t y s .
Ž y2 .r2
.
Noting that t 2
t 2
Hl h Ž s y l . ds s Hl Ž s y l .
y2
1
ds s
2 y1
Ž t y l.
y1
Ž t y l.
y1
,
and t 2
Hl h
2
t
y2 2 ds s 2 Ž t y s . ds s H Ž t y s .
l
y1
.
Hence, lim tª⬁
1 4 t y1
2
t 2
h1 Ž s y l . ds s
Hl
4 Ž y 1.
,
Ž 2.17.
.
Ž 2.18.
and lim tª⬁
1 4 t y1
2
t 2
Hl
h 2 Ž t y s . ds s
4 Ž y 1.
From Ž2.15. and Ž2.17. we have that lim sup tª⬁
t
y1
t
Hl
s lim sup tª⬁
s lim sup tª⬁
½
Ž s y l . Kq Ž s . y
t
y1
t y1
t
Hl Ž s y l . t
Hl Ž s y l .
1 4
5
h12 Ž s y l . ds
Kq Ž s . ds y lim
tª⬁
Kq Ž s . ds y
1 t
y1
2 4 Ž y 1.
t
1
Hl 4 h Ž s y l . ds 2 1
) 0,
i.e., Ž2.8. holds. Similarly, Ž2.16. implies that Ž2.9. holds. By Theorem 2.3, every solution of Eq. Ž1.1. is oscillatory. The proof is complete.
3. OSCILLATION RESULTS FOR f Ž x . WITHOUT MONOTONICITY In this section we consider the oscillation of Eq. Ž1.1. when the function f Ž y . is not monotonous. In this case we always assume that the following
182
LI AND AGARWAL
condition holds f Ž y . ry G 0 ) 0 for y / 0, where 0 is a constant. Ž 3.1.
Ž H4⬘.
LEMMA 3.1. Let assumptions Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold, and suppose y is a solution of Eq. Ž1.1. such that y Ž t . ) 0 on w c, b .. For any ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., let wŽ t. s ¨ Ž t.
y⬘ Ž t .
Ž 3.2.
yŽ t.
on w c, b .. Then for any H g X, b
Hc H Ž b, s . K ¨ Ž s . q Ž s . ds 0
F H Ž b, c . w Ž c . q
1 4
Hc
b
¨ Ž s . h 2 Ž b, s . y
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž b, s .
ds.
Ž 3.3.
Proof. From Ž1.1. and Ž3.2. we have for s g w c, b . w⬘ Ž t . s y¨ Ž t . q Ž t .
f Ž yŽ t. . yŽ t.
g Ž y⬘ Ž t . . y
1 ¨ Ž t.
u2 Ž t . q
¨ ⬘Ž t . ¨ Ž t.
wŽ t. .
Ž 3.4. In view of f Ž y .ry G 0 ) 0 and g Ž y⬘. G K ) 0, we obtain by the above equality w⬘ Ž t . q K 0¨ Ž t . q Ž t . q
1 ¨ Ž t.
u2 Ž t . y
¨ ⬘Ž t . ¨ Ž t.
u Ž t . F 0.
Ž 3.5.
The rest of the proof is similar to that of Lemma 2.1. LEMMA 3.2. Let assumptions Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold and suppose that y is a solution of Eq. Ž1.1. such that y Ž t . ) 0 on Ž a, c x. For any ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., let w Ž t . be defined by Ž3.2. on Ž a, c x. Then for any H g X, c
Ha H Ž s, a. K ¨ Ž s . q Ž s . ds 0
F yH Ž c, a . w Ž c .
183
INTERVAL OSCILLATION CRITERIA
q
1
c
Ha ¨ Ž s .
4
h1 Ž s, a . q
¨ ⬘Ž s . ¨ Ž s.
2
'H Ž s, a.
ds.
The following theorem is an immediate result from Lemmas 3.1 and 3.2. THEOREM 3.1. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold and that for some c g Ž a, b . and for some H g X, ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.., 1
c
H H Ž s, a. K ¨ Ž s . q Ž s . ds H Ž c, a . a 0
q )
1
b
H Ž b, c .
Hc H Ž b, s . K ¨ Ž s . q Ž s . ds
1
c
4 H Ž c, a . q
0
Ha ¨ Ž s .
1
b
4 H Ž b, c .
h1 Ž s, a . q
Hc ¨ Ž s .
¨ ⬘Ž s . ¨ Ž s.
h 2 Ž b, s . y
2
'H Ž s, a.
¨ ⬘Ž s . ¨ Ž s.
ds 2
'H Ž b, s .
ds. Ž 3.6.
Then e¨ ery solution of Eq. Ž1.1. has at least one zero in Ž a, b .. THEOREM 3.2. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. holds. If, for each T G t 0 , there exist H g X, ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. and a, b, c g R such that T F a - c - b and Ž3.6. holds, then e¨ ery solution of Eq. Ž1.1. is oscillatory. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold. If
THEOREM 3.3. lim sup tª⬁
t
Hl
H Ž s, l . K 0¨ Ž s . q Ž s . 1
¨ ⬘Ž s .
y ¨ Ž s . h1 Ž s, l . q 4 ¨ Ž s.
ž
2
'H Ž s, l .
ds ) 0,
/
and lim sup tª⬁
t
Hl
H Ž t , s . K 0¨ Ž s . q Ž s . 1 ¨ ⬘Ž s . y ¨ Ž s . h2 Ž t , s . y 4 ¨ Ž s.
ž
2
'H Ž t , s .
/
ds ) 0,
for some H g X, g g C 1 Žw t 0 , ⬁., Ž0, ⬁.. and for each l G t 0 , then e¨ ery solution of Eq. Ž1.1. is oscillatory.
184
LI AND AGARWAL
THEOREM 3.4. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold. If for each T G t 0 , there exist H g X 0 , ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. and a, c g R such that T F a - c and c
Ha H Ž s y a. K 1
)
c
q
¨ Ž s . q Ž s . q ¨ Ž 2 c y s . q Ž 2 c y s . ds
¨ Ž s . q ¨ Ž 2 c y s . h 2 Ž s y a . ds
H 4 a q
0
1
c
1
c
¨ Ž 2 c y s . y ¨ Ž s . h Ž s y a . H Ž s y a . ds
'
H 2 a
¨ ⬘ Ž s . y ¨ ⬘ Ž 2 c y s . H Ž s y a . ds,
H 4 a
then e¨ ery solution of Eq. Ž1.1. is oscillatory. COROLLARY 3.1. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold. Then e¨ ery solution of Eq. Ž1.1. is oscillatory pro¨ ided that for each l G t 0 and for some ) 1, there exists a function ¨ g C 1 Žw t 0 , ⬁., Ž0, ⬁.. such that the following two inequalities hold lim sup tª⬁
1 t
t
y1
Hl Ž s y l .
K 0¨ Ž s . q Ž s . 1
¨ ⬘Ž s .
y ¨ Ž s. y 4 ¨ Ž s. Ž s y l.
ž
2
/
ds ) 0,
and lim sup tª⬁
1 t
y1
t
Hl Ž t y s .
K 0¨ Ž s . q Ž s . 1
¨ ⬘Ž s .
y ¨ Ž s. q 4 ¨ Ž s. Ž t y s.
ž
2
/
ds ) 0.
THEOREM 3.5. Assume that Ž H 1. ᎐ Ž H 3. and Ž H4⬘. hold. Then e¨ ery solution of Eq. Ž1.1. is oscillatory pro¨ ided that for each l G t 0 and for some ) 1, the following two inequalities hold lim sup tª⬁
1
t
t
s y l . K 0 q Ž s . ds ) y1 H Ž
l
2 4 Ž y 1.
185
INTERVAL OSCILLATION CRITERIA
and lim sup tª⬁
1
t
2
t
t y s . K 0 q Ž s . ds ) y1 H Ž
4 Ž y 1.
l
.
4. EXAMPLES In this section we will show the applications of our oscillation criteria by two examples. We will see that the equations in the examples are oscillatory based on the results in Sections 2 and 3, though the oscillation cannot be demonstrated by the results of Huang w9x and Kong w11x and most other known criteria. EXAMPLE 1. Consider the nonlinear differential equation y⬙ Ž t . q
␥
y Ž t . Ž 1 q y 2 Ž t . . 1 q Ž y⬘ Ž t . .
t2
2
s 0,
t G 1.
Ž 4.1.
Let f Ž y . s y Ž1 q y 2 . and g Ž y . s 1 q y 2 . Then f ⬘Ž y . s 1 q 3 y 2 G 1 s ,
g Ž y . s 1 q y 2 G 1.
Note that for ) 1, lim tª⬁
1
t
t
s y l. y1 H Ž
␥
s
l
2
ds s
␥ y1
Ž t y l.
lim
t
tª⬁
s
␥ y1
. Ž 4.2.
Next, we will prove that t
Hl Ž t y s .
␥ s
2
ds G
t
Hl Ž s y l .
␥ s2
ds.
Ž 4.3.
Let FŽ t. s
t
Hl
Ž t y s. y Ž s y l .
␥ s2
ds.
Then F Ž l . s 0, and for t G l, F⬘ Ž t . s G s
t
Hl Ž t y s . ␥ 2
t ␥ t
2
t
y1
Hl Ž t y s .
␥ s
2
y1
ds y Ž t y l . ds y Ž t y l .
Ž t y l. y Ž t y l.
␥ t2
s 0.
␥ t2
␥ t2
186
LI AND AGARWAL
Hence F Ž t . G F Ž l . s 0 for t G l, i.e., Ž4.3. holds. By Ž4.2. and Ž4.3., for any ␥ ) 1r4 there exists ) 1 such that ␥rŽ y 1. ) 2r4Ž y 1.. This means that Ž2.15. and Ž2.16. hold for the same . Applying Theorem 2.5, we find that Ž4.1. is oscillatory for ␥ ) 1r4. However, the results of Huang w9x and Kong w11x fail since f Ž y . is a nonlinear function and g Ž y . k 1. We remark that if f Ž y . s y and g Ž y . s 1, then Eq. Ž4.1. reduces to the Euler equation ␥ y⬙ Ž t . q 2 y Ž t . s 0, Ž 4.4. t where ␥ ) 0 is a constant. It is well known w12, 14x that every solution of Eq. Ž4.4. is oscillatory if ␥ ) 1r4. However, Theorem A of Huang only applies to the case ␥ ) 2Ž3 y 2'2 ., but for the case 1r4 - ␥ F 2Ž3 y 2'2 . it remains open. This implies that our results are sharp. EXAMPLE 2. Consider the nonlinear differential equation y⬙ Ž t . q
2 Ž 1 q cos 2 t .
Ž 3 q cos t . Ž 1 q sin t . 2
2
yŽ t.
ž
1
1
q
2
/Ž
1 q y2 Ž t.
1 q Ž y⬘ Ž t . .
s 0,
2
.
Ž 4.5.
where t G 1. Observe that f Ž y. s y
ž
1 2
q
1 1qy
2
/
f ⬘Ž y . s y
and
2y 1 q y2
.
Clearly, Theorem 2.5 cannot apply to Eq. Ž4.5.. In spite of this, with
¨ Ž t . s t 2 and s 2, we can prove the oscillatory character of Eq. Ž4.5. by
Corollary 3.1. Because, for all y g R _ 04 , f Ž y. y
s
1 2
1
q
1qy
2
G
1 2
s 0 ,
and g Ž y . s 1 q y 2 G 1 s K. Thus, lim tª⬁
1 t
t
Hl
2
Ž s y l.
s lim
tª⬁
G lim
tª⬁
1 t 1
1 2
t
¨ Ž s. qŽ s. y
H Ž s y l . 2 s2 l
t
H t l
1 3
1
1 4
¨ Ž s.
ž
2 syl
y
¨ ⬘Ž s . ¨ Ž s.
2 Ž 1 q cos 2 t .
2 Ž 3 q cos 2 t . Ž 1 q sin 2 t .
Ž s y l . s 2 y l 2 ds s ⬁,
2
/
y
ds l2 s2 Ž s y l .
2
ds
187
INTERVAL OSCILLATION CRITERIA
and 1
lim tª⬁
t
t
2
Hl Ž t y s .
s lim
tª⬁
G lim
tª⬁
1 t 1
1 2
t
¨ Ž s. qŽ s. y
Hl Ž s y l . t
H t l
1 3
2 2
s
1 4
1
¨ Ž s.
ž
2 tys
q
¨ ⬘Ž s . ¨ Ž s.
2 Ž 1 q cos 2 t .
2 Ž 3 q cos 2 t . Ž 1 q sin 2 t .
2
/
y
ds t2 s2 Ž t y s.
2
ds
2 Ž t y s . s 2 y t 2 ds s ⬁.
Thereby, Eq. Ž4.5. is oscillatory by Corollary 3.1. Observe that y Ž t . s cost is an oscillatory solution of Eq. Ž4.5.. Remark 1. The oscillation criteria presented in this paper require the assumption ŽH4. or ŽH4⬘. on f Ž y . and thus the results obtained here are not applicable to equation of type Ž1.1. where, for example, f Ž y . s < y < sgny, ) 0, / 1. This problem is related to the question posed by Kamenev w10x whether condition ŽH4. can be relaxed to
Ž H4⬙ . f ⬘Ž y . G 0 for y / 0, or whether it can be omitted. For the case of equation
Ž aŽ t . y⬘ Ž t . . ⬘ q p Ž t . y⬘ Ž t . q q Ž t . < y Ž t . < sgn l y Ž t . s 0,
)0
an affirmative answer to the preceding question has been given by Grace and Lalli w8, Theorem 8x. The study of this problem for Eq. Ž1.1. will form the subject of one of our forthcoming papers.
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LI AND AGARWAL
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