- Email: [email protected]
Oscillation theorems for second-order equations with damping
Nonlinear Analysis 41 (2000) 1005 – 1028
www.elsevier.nl/locate/na
Oscillation theorems for second-order equations with damping 1 Yuri V. Rogovchenkoa;b; ∗ a Institute
of Mathematics, National Academy of Sciences, Tereshchenkivs’ka Str. 3, 252601 Kyiv, Ukraine b Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey Received 5 February 1997; accepted 14 August 1998
Keywords: Second order; Nonlinear di erential equation; Damping; Oscillation
1. Introduction In this paper, we study the problem of oscillation of the nonlinear second-order di erential equation with damping [r(t)x0 (t)]0 + p(t)x0 (t) + q(t)f(x(t)) = 0;
(1)
where r(t)∈C 1 ([t0 ; ∞); (0; ∞)); p(t); q(t) ∈ C([t0 ; ∞); (−∞; ∞)); f(x) ∈ C((−∞; ∞); (−∞; ∞)) and xf(x) ¿ 0 for x 6= 0; t0 ≥ 0. We recall that a function x : [t0 ; t1 ) → (−∞; ∞); t1 ¿ t0 is called a solution of Eq. (1) if x(t) satis es Eq. (1) for all t ∈ [t0 ; t1 ). In what follows, it will be always assumed that solutions of Eq. (1) exist for any t0 ≥ 0. Furthermore, a solution x(t) of Eq. (1) is called continuable if x(t) exists for all t ≥ t0 . A continuable solution x(t) of Eq. (1) is called oscillatory if it has arbitrarily large zeroes, otherwise it is called nonoscillatory. Finally, it is said that Eq. (1) is oscillatory if all its solutions are oscillatory. ∗ Correspondence address: Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey. E-mail address: Rogovch [email protected] (Y.V. Rogovchenko) 1 This research was supported by a fellowship of the Italian Consiglio Nazionale delle Ricerche.
0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 3 2 4 - 1
1006
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
In the last two decades there has been an increasing interest in obtaining sucient conditions for the oscillation and/or nonoscillation of solutions for di erent classes of second-order di erential equations [1–25]. In the absence of damping, there is a great number of papers (see, for example, [12–17, 20–22] and the references quoted therein) devoted to the particular cases of Eq. (1) such as the linear equation x00 (t) + q(t)x(t) = 0
(2)
and the more general nonlinear equation [r(t)x0 (t)]0 + q(t)f(x(t)) = 0:
(3)
An important tool in the study of oscillatory behavior of solutions for Eqs. (1) – (3) is the averaging technique. This goes back as far as to the classical results of Wintner [21] giving a sucient condition for oscillation of Eq. (2), namely, Z Z 1 t s lim q( ) d ds = ∞ (C1 ) t→∞ t t t0 0 and Hartman [13] who showed that the above limit cannot be replaced by the upper limit and proved that the condition Z Z Z Z 1 t s 1 t s − ∞ ¡ lim inf q( ) d ds ¡ lim sup q( ) d ds ≤ ∞ (C2 ) t→∞ t t t→∞ t t0 t0 t0 0 implies that Eq. (2) is oscillatory. The result of Wintner was improved by Kamenev [14] who proved that the condition Z t 1−n lim sup t (t − s)n−1 q(s) ds = ∞ for some n ¿ 2 (C3 ) t→∞
t0
is sucient for the oscillation of Eq. (2). Recently, Yan [24] presented new oscillation criteria for Eq. (1) with f(x) = x involving the Kamenev’s type condition. Theorem A (Yan [24, Theorem 2]). Suppose that there exist a positive continuously di erentiable function h(t) on [t0 ; ∞) and ∈ (1; ∞) such that Z t lim sup t − (C4 ) (t − ) h( )q( ) d ¡ ∞; t→∞
t0
there exists a continuous function (t) on [t0 ; ∞) such that Z t" r( ) 1 − lim inf t (t − ) h( )q( ) − (t − ) −2 (C5 ) t→∞ 4 h( ) s 2 # h( )p( ) 0 + h( ) − (t − )h ( ) d ≥ (s) × (t − ) r( )
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
and
Z
(C6 )
lim
t→∞
t
t0
1007
2+ ( ) d = ∞; h( )r( )
where + (t) = max( (t); 0). Then the equation [r(t)x0 (t)]0 + p(t)x0 (t) + q(t)x(t) = 0
(4)
is oscillatory. The above result of Yan was extended further to Eq. (2) by Philos [17] and to Eq. (3) by Li [16] (it should be noted that the original results of Li need some correction; see [18] for the details). In the presence of damping, a number of oscillation criteria were obtained for di erent classes of nonlinear equations by Baker [1], Bobisud [2], Butler [3], Grace [4 – 6], Grace and Lalli [7–9], Grace et al. [10], Yan [23, 24], Yeh [25] and others. Very recently, the extension of the results of Philos [17] both for Eq. (1) and for the more general equation [r(t) (x(t))x0 (t)]0 + p(t)x0 (t) + q(t)f(x(t)) = 0 was obtained by Grace [6]. In order to present the oscillation criterion of Grace we rst introduce, following Philos [17], the class of functions P which will be extensively used in the sequel. Namely, let D0 = {(t; s): t ¿ s ≥ t0 } and D = {(t; s): t ≥ s ≥ t0 }. We will say that the function H ∈C(D; (−∞; ∞)) belongs to the class P if it satis es the following two conditions: 1. H (t; t) = 0 for t ≥ t0 , H (t; s) ¿ 0 in D0 ; 2. H has a continuous and nonpositive partial derivative in D0 with respect to the second variable. Theorem B (Grace [6, Theorem 6]). Suppose that f0 (x) ≥ K ¿ 0 for x 6= 0 and assume that there exist the function a ∈ C 1 ([t0 ; ∞); (0; ∞)) and the functions h; H ∈ C(D; (−∞; ∞)) such that H belongs to the class P and −
@H (t; s) = h(t; s)(H (t; s))1=2 @s
for all (t; s) ∈ D0 :
Suppose; moreover; that H (t; s) ≤∞ 0 ¡ inf lim inf (C7 ) t→∞ H (t; t0 ) s≥t0 and (C8 )
lim sup t→∞
1 H (t; t0 ) 0
Z
t
t0
r(s)a(s)[h(t; s) − (s)(H (t; s))1=2 ]2 ds ¡ ∞;
where (t) = (r(t)a (t) − p(t)a(t))=(r(t)a(t)).
1008
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
If there exists a function ∈ C([t0 ; ∞); (−∞; ∞)) such that Z t r(s)a(s) 1 lim sup H (t; s)a(s)q(s) − (C9 ) 4K t→∞ H (t; T ) T ×(h(t; s) − (s)(H (t; s))1=2 )2 ds ≥ (T ) for every T ≥ t0 and (C6 ) holds; then Eq. (1) is oscillatory. The purpose of this paper is to improve Theorem B, as well as other related results in [4 –9, 23–25] concerning the oscillation of Eq. (1) by making use of techniques similar to those used by Grace [6], Philos [17], Rogovchenko [19], and Yan [24]. It is worth pointing out that, in contrast to the recent results due to Grace [6] and Grace and Lalli [8], we do not impose any additional condition on the damping coecient p(t). 2. Oscillation theorems Theorem 1. Suppose that f0 (x) exists and f0 (x) ≥ K ¿ 0
(5)
for some constant K and for all x 6= 0. Suppose; further; that the functions h; H ∈ C (D; (−∞; ∞)) are such that H belongs to the class P and −
@H (t; s) = h(t; s)(H (t; s))1=2 @s
for all (t; s) ∈ D0 :
(6)
If there exists a function g ∈ C 1 ([t0 ; ∞); (0; ∞)) such that Z t" 1 a(s)r(s) H (t; s) (s) − lim sup H (t; t ) 4K t→∞ 0 t0 # 2 p(s) (H (t; s))1=2 ds = ∞; (7) × h(t; s) + r(s) Rs where a(s) = exp(−2 g(u) du) and (s) = K −1 a(s)(Kq(s) − p(s)g(s) − [r(s)g(s)]0 + r(s)g2 (s)); then Eq. (1) is oscillatory. Proof. Let x(t) be a nonoscillatory solution of the di erential equation (1) and let T0 ≥ t0 be such that x(t) 6= 0 for all t ≥ T0 . Without loss of generality, we may assume that x(t) ¿ 0 for all t ≥ T0 since the similar argument holds for the case when x(t) is eventually negative. Following Harris [12], we use a generalized Riccati transformation and let 0 g(t) x (t) : (8) + v(t) = a(t)r(t) f(x(t)) K
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1009
Then di erentiating (8) and making use of Eq. (1) we obtain # " 2 0 [r(t)x0 (t)]0 x (t) [r(t)g(t)]0 0 0 + − r(t)f (x(t)) v (t) = −2g(t)v(t) + a(t) f(x(t)) f(x(t)) K " = −2g(t)v(t) + a(t) −q(t) − p(t)
0
−r(t)f (x(t))
x0 (t) f(x(t))
#
2 +K
for all t ≥ T0 . In view of (5), (9) yields " v0 (t) ≤ −2g(t)v(t) − a(t) q(t) + p(t) +Kr(t)
x0 (t) f(x(t))
2
x0 (t) f(x(t))
−1
0
[r(t)g(t)]
(9)
x0 (t) f(x(t)) #
− K −1 [r(t)g(t)]0 :
(10)
Now, from (8) and (10), we obtain v0 (t) ≤ − (t) −
K p(t) v(t) − v2 (t) r(t) a(t)r(t)
(11)
for all t ≥ T0 with (t) de ned as above. Hence, by (11) and (6), for all t ≥ T ≥ T0 , we have Z t Z t Z t p(s) 0 v(s) ds H (t; s) (s) ds ≤ − H (t; s)v (s) ds − H (t; s) r(s) T T T Z t K v2 (s) ds H (t; s) − a(s)r(s) T Z t p(s) @H t (t; s)v(s) + H (t; s) v(s) = − H (t; s)v(s)|T − − @s r(s) T K + H (t; s) v2 (s) ds a(s)r(s) Z t p(s) v(s) h(t; s)(H (t; s))1=2 v(s) + H (t; s) = H (t; T )v(T ) − r(s) T K v2 (s) ds + H (t; s) a(s)r(s) 1=2 Z t " KH (t; s) v(s) = H (t; T )v(T ) − a(s)r(s) T
1010
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1 + 2
Z +
t T
a(s)r(s) K
a(s)r(s) 4K
1=2
p(s) (H (t; s))1=2 h(t; s) + r(s)
h(t; s) +
p(s) (H (t; s))1=2 r(s)
#2 ds
2 ds:
Therefore, for all t ≥ T ≥ T0 , we have Z t" T
a(s)r(s) H (t; s) (s) − 4K
≤ H (t; T )v(T ) −
Z t " T
p(s) h(t; s) + (H (t; s))1=2 r(s)
KH (t; s) a(s)r(s)
1=2 v(s) +
1 2
2 # ds
a(s)r(s) K
1=2
#2 p(s) 1=2 (H (t; s)) × h(t; s) + ds: r(s)
(12)
It follows from (12) and property (2) of the function H (t; s) that for every t ≥ T0 Z t" T0
a(s)r(s) H (t; s) (s) − 4K
p(s) h(t; s) + (H (t; s))1=2 r(s)
2 # ds
≤ H (t; T0 )|v(T0 )| ≤ H (t; t0 )|v(T0 )|:
(13)
Thus, from (13) and property (2) of the function H (t; s), we obtain Z t" t0
a(s)r(s) H (t; s) (s) − 4K Z
=
T0
"
t0
+
p(s) h(t; s) + (H (t; s))1=2 r(s)
a(s)r(s) H (t; s) (s) − 4K
Z t" T0
a(s)r(s) H (t; s) (s) − 4K Z
≤ H (t; t0 )
T0
t0
Z =H (t; t0 )
ds
p(s) h(t; s) + (H (t; s))1=2 r(s)
2 #
p(s) h(t; s) + (H (t; s))1=2 r(s)
ds 2 # ds
| (s)| ds + H (t; t0 )|v(T0 )|
T0
t0
2 #
| (s)| ds + |v(T0 )| :
(14)
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1011
Inequality (14) implies that Z t" a(s)r(s) 1 H (t; s) (s) − lim sup 4K t→∞ H (t; t0 ) t0 ×
p(s) (H (t; s))1=2 h(t; s) + r(s)
2 #
Z ds ≤
T0
t0
| (s)| ds + |v(T0 )|;
which contradicts assumption (7) of the theorem. Hence, all solutions of Eq. (1) are oscillatory. As immediate consequences of Theorem 1 we obtain the following corollaries. Corollary 1. Let the assumptions of Theorem 1 hold except that Eq. (7) is replaced by Z t 1 H (t; s) (s) ds = ∞ lim sup t→∞ H (t; t0 ) t0 and lim sup t→∞
1 H (t; t0 )
Z
t
t0
2 p(s) a(s)r(s) h(t; s) + (H (t; s))1=2 ds ¡ ∞: r(s)
Then Eq. (1) is oscillatory. Corollary 2 (cf. Li [16, Theorem 2.1]). Let f(x) = x; p(t) ≡ 0; and let the functions h; H be as in Theorem 1 with (6) holding. If there exists a function g ∈ C 1 ([t0 ; ∞); (0; ∞)) such that Z t 1 1 2 H (t; s) (s) − a(s)r(s)h (t; s) ds = ∞; lim sup 4 t→∞ H (t; t0 ) t0 (s) = a(s){q(s) − [r(s)g(s)]0 + r(s)g2 (s)}; then
where a(s) is the same as above and Eq. (3) is oscillatory.
In the same way as it was done in [16], with an appropriate choice of the functions H and h, we can derive from Theorem 1 a number of oscillation criteria for Eqs. (2) and (3). Let us consider, for example, the function H (t; s) de ned by H (t; s) = (t − s)n−1 ;
(t; s) ∈ D;
where n ¿ 2 is an integer. Clearly, H belongs to the class P. Furthermore, the function h(t; s) = (n − 1)(t − s)(n−3)=2 ;
(t; s) ∈ D
is continuous on [t0 ; ∞) and satis es condition (6). Then, by Theorem 1, we obtain the following oscillation criteria.
1012
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Corollary 3. Let assumption (5) hold. Suppose also that there exists a function g ∈ C 1 ([t0 ; ∞); (0; ∞)) such that Z t" a(s)r(s) (t − s)n−3 lim sup t 1−n (t − s)n−1 (s) − 4K t→∞ t0 2 # p(s) × n−1+ (t − s) ds = ∞ r(s) for some integer n ¿ 2; where a(s) and Eq. (1) is oscillatory.
(15)
(s) are the same as in Theorem 1. Then
Corollary 4 (cf. Grace and Lalli [8, Corollary 3]). Let assumption (5) hold and suppose also that there exist functions b ∈ C([t0 ; ∞); (0; ∞)); g ∈ C 1 ([t0 ; ∞); (0; ∞)) such that for some ¿ 1 Z t" a(s)r(s) − (B(t) − B(s))−2 (B(t) − B(s)) (s) − lim sup B (t) 4K t→∞ t0 2 # p(s) (B(t) − B(s)) × b(t) + ds = ∞; (16) r(s) Rt where B(t) = t0 b(s) ds; and a(s); (s) are the same as in Theorem 1. Then Eq. (1) is oscillatory. Proof. Let us put H (t; s) = [B(t) − B(s)] ;
(t; s) ∈ D;
then with the choice h(t; s) = b(t)[B(t) − B(s)](−2)=2 ;
(t; s) ∈ D;
the conclusion follows directly from Theorem 1. Example 1. Consider the nonlinear di erential equation ((1 + sin2 t)x0 (t))0 − 3sin t cos t x0 (t) +
1 x(t)(1 + x4 (t)) = 0; 1 + cos4 t
(17)
where x ∈ (−∞; ∞), and t ∈ [1; ∞). Clearly, f0 (x) exists and f0 (x) = 5x4 + 1 ≥ 1
for all x ∈ (−∞; ∞);
so assumption (5) holds. Let us apply Corollary 3 with n = 3 and g(t) = −1=t, so that a(t) = t 2 . A straightforward computation yields !0 " Z 2 3sin s cos s 1 + sin 1 1 t s − + (t − s)2 s2 lim sup 2 1 + cos4 s s s t→∞ t T
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1 + sin2 s + − (1 + sin2 s) s2 1 = lim sup 2 t→∞ t
Z T
t
2 2
(t − s) s
3sin s cos s 1 − t − s 2(1 + sin2 s)
1013
2 #
9sin2 s cos2 s 1 − 1 + cos4 s 4(1 + sin2 s) !
ds !
+ s(t − s)(4s − t)sin s cos s − s2 (1 + sin2 s) ds = ∞; since 9sin2 s cos2 s 1 ¿0 − 4 1 + cos s 4(1 + sin2 s) for any s ∈ (−∞; +∞). Thus, assumption (15) holds, and we conclude by Corollary 3 that all continuable solutions of Eq. (17) are oscillatory. Observe that x(t) = cos t is such a solution. The important point to note here is that the recent results due to Grace [6] and Grace and Lalli [8] do not apply to Eq. (17). A simple inspection with the same choice (t) = t n as in [6, 8] shows that assumptions p(t) ≤ 0; (p(t)(t))0 ≥ 0 for all t ≥ t0 [6, Theorem 1]; (r(t)(t))0 ≤ 0 for all t ≥ t0 [6, Theorem 2]; r(t)0 (t) − p(t)(t)0 ≥ 0 for all t ≥ t0 [8, Theorem 1, Corollary 1]; r(t)0 (t) − p(t)(t)0 ≥ 0 and (r(t)0 (t) − p(t)(t))0 ≤ 0 for all t ≥ t0 [6, Theorem 8; 8, Theorems 5; 7]; ( 5 ) p(t) ≤ 0 for all t ≥ t0 [8, Theorems 2; 4; 6] all fail for Eq. (17) since all the functions in ( 1 ) – ( 5 ) are oscillatory for any n ≥ 1 and for t ≥ t0 : ( 1 ) ( 2 ) ( 3 ) ( 4 )
3 p(t) = − sin 2t; 2 3 (p(t)(t))0 = − t n−1 (2t cos 2t + n sin 2t); 2 (r(t)0 (t))0 = nt n−2 (t sin 2t + (n − 1)(1 + sin2 t)); 1 r(t)0 (t) − p(t)(t) = t n−1 (2n(1 + sin2 t) + 3t sin 2t); 2 5n 2 0 0 n−2 2 t sin 2t + n(n − 1)(1 + sin t) + 3t cos 2t : (r(t) (t) − p(t)(t)) = t 2 It is not dicult to check that assumptions ( 1 ) – ( 5 ) all fail also for n = 0 and 0 ¡ n ¡ 1.
1014
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Theorem 2. Let the functions H and h be deÿned as in Theorem 1; the function f satisfy (5); and suppose that H (t; s) ≤ ∞: (18) 0 ¡ inf lim inf t→∞ H (t; t0 ) s≥t0 Assume that there exist functions g ∈ C 1 ([t0 ; ∞); (0; ∞)) and ∈ C ([t0 ; ∞); (−∞; ∞)) such that for all t ¿ t0 2 Z t 1 p(s) 1=2 (H (t; s)) lim sup a(s)r(s) h(t; s) + ds ¡ ∞ (19) r(s) t→∞ H (t; t0 ) t0 and
Z lim sup t→∞
t
t0
2+ (s) = ∞; a(s)r(s)
(20)
and suppose that for any T ≥ t0 Z t" a(s)r(s) 1 lim sup H (t; s) (s) − H (t; T ) 4K t→∞ T ×
p(s) (H (t; s))1=2 h(t; s) + r(s)
2 # ds ≥ (T );
(21)
where a(s) and (s) are the same as in Theorem 1 and + (t) = max( (t); 0). Then Eq. (1) is oscillatory. Proof. Without loss of generality, we may assume that there exists a solution x(t) of Eq. (1) such that x(t) ¿ 0 on [T0 ; ∞) for some T0 ≥ t0 ; a similar argument holds when x(t) ¡ 0 on [T0 ; ∞). De ne the function v(t) by (8); as in Theorem 1, we obtain inequality (12). It follows from (12) that 2 # Z t" p(s) a(s)r(s) 1 h(t; s) + (H (t; s))1=2 H (t; s) (s) − ds H (t; T ) T 4K r(s) 1 ≤ v(T ) − H (t; T )
Z t " T
KH (t; s) a(s)r(s)
1=2
1 v(s) + 2
a(s)r(s) K
#2 p(s) 1=2 × h(t; s) + ds (H (t; s)) r(s) for t ¿ T ≥ T0 , and therefore Z t p(s) a(s)r(s) 1 h(t; s) + H (t; s) (s) − lim sup 4K r(s) t→∞ H (t; T ) T
1=2
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1=2 2
× (H (t; s)) 1 + 2
ds ≤ v(T ) − lim inf t→∞
a(s)r(s) K
1=2
1 H (t; T )
Z t " T
p(s) (H (t; s))1=2 h(t; s) + r(s)
KH (t; s) a(s)r(s)
1015
1=2 v(s)
#2 ds
for all T ≥ T0 . It follows from (21) that 1 v(T ) ≥ (T ) + lim inf t→∞ H (t; T ) 1 + 2
a(s)r(s) K
1=2
Z t " T
KH (t; s) a(s)r(s)
1=2 v(s)
p(s) (H (t; s))1=2 h(t; s) + r(s)
#2 ds
for T ≥ T0 , so v(T ) ≥ (T )
(22)
for all T ≥ T0
and 1 lim inf t→∞ H (t; T0 )
Z t " T0
KH (t; s) a(s)r(s)
1=2 v(s) +
1 2
a(s)r(s) K
1=2
# 2 p(s) (H (t; s))1=2 × h(t; s) + ds ≤ v(T0 ) − (T0 ) = M ¡ ∞: r(s) Thus, for all t ≥ T0 , 1 ∞ ¿ lim inf t→∞ H (t; T0 ) ×
Z t " T0
KH (t; s) a(s)r(s)
p(s) (H (t; s))1=2 h(t; s) + r(s)
1 ≥ lim inf t→∞ H (t; T0 )
Z t T0
1=2 v(s) +
1 2
a(s)r(s) K
1=2
# 2 ds
KH (t; s) 2 v (s) a(s)r(s)
p(s) (H (t; s))1=2 v(s) ds: + (H (t; s))1=2 h(t; s) + r(s)
(23)
1016
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Let us de ne for t ¿ T0 the functions (t) and (t) by (t) =
1 H (t; T0 )
Z
t
KH (t; s) 2 v (s) ds a(s)r(s)
T0
and 1 (t) = H (t; T0 )
Z
t
T0
1=2
(H (t; s))
p(s) 1=2 (H (t; s)) h(t; s) + v(s) ds: r(s)
Then (23) may be rewritten as lim inf [ (t) + (t)] ¡ ∞: t→∞
Let us show now that Z ∞ 2 v (s) ds ¡ ∞: a(s)r(s) T0
(24)
(25)
To this end, suppose that Z
∞
T0
v2 (s) ds = ∞: a(s)r(s)
It follows from (18) that there exists a positive constant such that H (t; s) ¿ ¿ 0: inf lim inf t→∞ H (t; t0 ) s≥t0
(26)
(27)
Furthermore, by (26) for any positive number there exists T1 ¿ T0 such that Z
t T0
v2 (s) ds ≥ a(s)r(s)
for all t ≥ T1 :
Thus, for all t ≥ T1 , we have Z s 2 Z t v (u) 1 KH (t; s) d (t) = du H (t; T0 ) T0 T0 a(u)r(u) Z s 2 Z t v (u) @H (t; s) 1 du ds K − = H (t; T0 ) T0 @s T0 a(u)r(u) Z s 2 Z t v (u) @H (t; s) 1 K − ≥ du ds H (t; T0 ) T1 @s T0 a(u)r(u) Z t 1 H (t; T1 ) @H (t; s) ds = : K − ≥ H (t; T0 ) T1 @s H (t; T0 )
(28)
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1017
It follows from (27) that lim inf t→∞
H (t; s) ¿ ¿ 0; H (t; t0 )
so there exists T2 ≥ T1 such that H (t; T1 ) ≥ H (t; t0 )
for all t ≥ T2 ;
and therefore by (28), (t) ≥
for all t ≥ T2 :
Since is an arbitrary constant, we have lim (t) = ∞:
(29)
t→∞
Consider now a sequence {tn }∞ n=1 in the interval (T0 ; ∞) such that lim tn = ∞
t→∞
and lim [ (tn ) + (tn )] = lim inf [ (t) + (t)]:
n→∞
t→∞
It follows from (24) that there exists a natural number N such that (tn ) + (tn ) ≤ M
for all n ¿ N
(30)
with the same constant M as de ned above. Obviously by (29) lim (tn ) = ∞;
n→∞
and thus (30) implies that lim (tn ) = −∞:
n→∞
(31)
Furthermore, (30) and (31) lead to the inequality (tn ) + 1¡ (tn ) for large values of n, where ∈ (0; 1) is a constant. Hence, for n large enough, (tn ) ¡ − 1 ¡ 0: (tn )
(32)
It follows from (31) and (32) that lim
n→∞
(tn ) (tn ) = ∞: (tn )
(33)
1018
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
On the other hand, by the Schwarz inequality, we have for any natural number n Z tn 2 1 p(s) 2 1=2 1=2 (H (tn ; s)) (H (tn ; s)) h(tn ; s) + v(s) ds (tn ) = 2 H (tn ; T0 ) r(s) T0 "Z
1 = 2 H (tn ; T0 )
tn
T0
KH (tn ; s) a(s)r(s)
1=2
v(s)
a(s)r(s) K
1=2
#2 p(s) 1=2 (H (tn ; s)) × h(tn ; s) + ds r(s) ≤
1 H (tn ; T0 )
Z
tn
KH (tn ; s)v2 (s) ds a(s)r(s)
T0
"
Z
1 × H (tn ; T0 )
1 ≤ (tn ) H (tn ; T0 )
1 1 2 (tn ) ≤ (tn ) H (tn ; t0 )
a(s)r(s) K
T0
"
and therefore
tn
Z
tn
t0
Z
tn
T0
p(s) h(tn ; s) + (H (tn ; s))1=2 r(s)
a(s)r(s) K
a(s)r(s) K
ds
p(s) h(tn ; s) + (H (tn ; s))1=2 r(s)
p(s) h(tn ; s) + (H (tn ; s))1=2 r(s)
#
2
lim sup t→∞
1 H (t; t0 )
Z
t
t0
ds
2 ds
for all n large enough. By (33) we obtain 2 Z tn 1 p(s) (H (tn ; s))1=2 a(s)r(s) h(tn ; s) + ds = ∞; lim n→∞ H (tn ; t0 ) t r(s) 0 so
#
2
(34)
2 p(s) (H (t; s))1=2 a(s)r(s) h(t; s) + ds = ∞; r(s)
and the last equality contradicts assumption (19). Therefore, we have proved that (26) fails, and so inequality (25) holds true. Furthermore, using (22), we obtain Z ∞ 2 Z ∞ 2 + (s) v (s) ds ≤ ds ¡ ∞ a(s)r(s) a(s)r(s) T0 T0 which contradicts condition (20), so Eq. (1) is oscillatory. The following result is the direct consequence of Theorem 2 and uses the same choice of the functions H and h as in Corollary 3 above.
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1019
Corollary 5. Let assumption (5) hold. Suppose there exist functions g ∈ C 1 ([t0 ; ∞); (0; ∞)) and ∈ C([t0 ; ∞); (−∞; ∞)) such that (20) holds along with 2 Z t p(s) (t − s) ds ¡ ∞ a(s)r(s)(t − s)n−3 n − 1 + lim sup t 1−n r(s) t→∞ t0 and lim sup t 1−n t→∞
Z t" t0
(t − s)n−1 (s) −
a(s)r(s) (t − s)n−3 4K
2 # p(s) × n−1+ (t − s) ds ≥ (T ) r(s) for all T ≥ t0 and for some integer n ¿ 2; where a(s) and Theorem 1. Then Eq. (1) is oscillatory.
(s) are the same as in
Proof. The only thing to be checked is condition (18). With the above choice of the functions H and h, this is ful lled automatically since H (t; s) (t − s)n−1 =1 = lim t→∞ H (t; t0 ) t→∞ (t − t0 )n−1 lim
for any s ≥ t0 . Example 2. Consider the following second-order nonlinear di erential equation: 1 00 (35) x (t) + 2x0 (t) + t cos t(x(t) + x3 (t)) = 0; t where x ∈ (−∞; ∞); t ∈ [1; ∞) and ≥ −1. With the functions r(t); p(t), and q(t) de ned by 1 1 exp(t 2 ); p(t) = 2 exp(t 2 ); q(t) = t cos t exp(t 2 ); t t Eq. (35) can be written as 0 1 1 2 0 exp(t )x (t) + 2 exp(t 2 )x0 (t) + t cos t exp(t 2 )(x(t) + x3 (t)) = 0: t t r(t) =
(36)
Clearly, f0 (x) exists and f0 (x) = 3x2 + 1 ≥ 1
for all x ∈ (−∞; ∞);
so assumption (5) holds. Furthermore, for the sake of simplicity, let H (t; s) = (t − s)2 ; h(t; s) = 2, and g(t) = 2t, so that a(t) = exp(−t 2 ). Then we have 2 Z Z t−s 1 t1 1 t (t + s)2 1 2+ ds = lim sup 2 ds = ; lim sup 2 3 s s 2 t→∞ t t→∞ t 1 s 1 so assumption (19) holds.
1020
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Furthermore, for any T ¿ 1 1 t2
lim sup t→∞
Z t (t + s)2 2 − (t − s)2 s cos s − ds ≥ −T sin T − K; 4 s 4s T
where K is a positive constant. Now set (T ) = −T sin T − K. Then there exists an integer N such that (2N + 1) + =4 ¿ 1 and for all n ≥ N (2n + 1) +
≤ T ≤ 2(n + 1) − ; 4 4
(T ) = −T sin T − K ≥ T ; where is a small constant. Thus, taking into account that ≥ −1, we have Z lim
t→∞
1
t
2+ (s) ds = lim t→∞ a(s)r(s) ≥
∞ X
Z
2
n=N
≥
∞ X n=N
2
t
1
2+ (s) ds s−1
Z
2(n+1)−=4
(2n+1)+=4
Z
2(n+1)−=4
(2n+1)+=4
s2 +1 ds
s−1 ds = ∞;
so condition (20) is also ful lled. Thus, by Corollary 5, we conclude that Eq. (36) is oscillatory, though none of the known criteria (see [4 –10, 23–25]) can cover this result. We can easily see that condition (C8 ) of Theorem B [6, Theorem 6] fails since 1 lim sup 2 t t→∞
Z
t
1
2 (t − s)(2s2 + 1) 1 2+ ds s s Z
1 t2
≥ 4 lim sup t→∞
t
1
s(t − s)2 ds
1 2 1 t2 − + − 2 = ∞: = 4 lim sup 12 2 3t 4t t→∞ Theorem 3. Let the functions H and h be deÿned as in Theorem 1; the function f to satisfy (5); and suppose that (18) holds. Assume that there exist functions g ∈ C 1 ([t0 ; ∞); (0; ∞)) and ∈ C([t0 ; ∞); (−∞; ∞)) such that (20) holds; lim inf t→∞
1 H (t; t0 )
Z
t
t0
H (t; s) (s) ds ¡ ∞
(37)
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
and 1 lim inf t→∞ H (t; T ) ×
Z t" T
H (t; s) (s) −
p(s) (H (t; s))1=2 h(t; s) r(s)
for all T ≥ t0 ; where a(s); Eq. (1) is oscillatory.
1021
a(s)r(s) 4K
2 # ds ≥ (T )
(38)
(s) and + (s) are the same as in Theorem 2. Then
Proof. Without loss of generality, we may assume that there exists a solution x(t) of Eq. (1) such that x(t) ¿ 0 on [T0 ; ∞) for some T0 ≥ t0 . De ne the function v(t) by (8). Then as in the proof of Theorem 1 inequality (12) holds for all t ≥ T ≥ T0 , and thus 2 # Z t" p(s) a(s)r(s) 1 1=2 h(t; s) + (H (t; s)) H (t; s) (s) − ds lim inf t→∞ H (t; T ) T 4K r(s) 1 ≤ v(T ) − lim sup t→∞ H (t; T ) 1 + 2
a(s)r(s) K
1=2
Z t " T
KH (t; s) a(s)r(s)
1=2 v(s)
p(s) (H (t; s))1=2 h(t; s) + r(s)
#2 ds
for all T ≥ T0 . It follows from (38) that 1 v(T ) ≥ (T ) + lim sup t→∞ H (t; T ) 1 + 2
a(s)r(s) K
1=2
Z t " T
KH (t; s) a(s)r(s)
1=2 v(s)
p(s) (H (t; s))1=2 h(t; s) + r(s)
#2 ds
for all T ≥ T0 . Consequently, for all T ≥ T0 , inequality (22) holds, and 1=2 1=2 Z t " KH (t; s) 1 1 a(s)r(s) v(s) + lim sup a(s)r(s) 2 K t→∞ H (t; T0 ) T0 ×
p(s) h(t; s) + (H (t; s))1=2 r(s)
with the same M as in Theorem 2.
2 ds ≤ M ¡ ∞
1022
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Hence, we have lim sup [ (t) + (t)] t→∞
1 ≤ lim sup t→∞ H (t; T0 ) 1 + 2
a(s)r(s) K
Z t " T0
1=2
KH (t; s) a(s)r(s)
1=2 v(s)
p(s) (H (t; s))1=2 h(t; s) + r(s)
#2 ds ≤ M ¡ ∞
(39)
with (t) and (t) the same as in the proof of Theorem 2. Making use of assumption (38), we obtain Z t" a(s)r(s) 1 H (t; s) (s) − (t0 ) ≤ lim inf t→∞ H (t; t0 ) t 4K 0 ×
p(s) (H (t; s))1=2 h(t; s) + r(s)
≤ lim inf t→∞
−
1 H (t; t0 )
Z
t
t0
2 # ds
H (t; s) (s) ds
1 1 lim inf 4K t→∞ H (t; t0 )
Z
t
t0
2 p(s) a(s)r(s) h(t; s) + ds: (H (t; s))1=2 r(s)
It follows from (37) and (40) that 2 Z t p(s) 1 (H (t; s))1=2 a(s)r(s) h(t; s) + ds ¡ ∞: lim inf t→∞ H (t; t0 ) t r(s) 0
(40)
(41)
By (41) there exists a sequence {tn }∞ n=1 in the interval (T0 ; ∞) such that lim tn = ∞
t→∞
and 1 lim n→∞ H (tn ; t0 )
Z
tn
t0
= lim inf t→∞
p(s) (H (tn ; s))1=2 a(s)r(s) h(tn ; s) + r(s)
1 H (t; t0 )
Z
t
t0
2 ds
2 p(s) a(s)r(s) h(t; s) + ds ¡ ∞: (42) (H (t; s))1=2 r(s)
Suppose now that (26) holds. With the same argument as in Theorem 2, we conclude that (29) holds. It follows from (39) that there exists a natural number N such that (30) is ful lled. Proceeding as in the proof of Theorem 2, we obtain (34) which contradicts
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1023
(42), and hence (26) fails. Making use of (22), (25) yields Z
∞
T0
2+ (s) ds ≤ a(s)r(s)
Z
∞
T0
v2 (s) ds ¡ ∞ a(s)r(s)
which contradicts assumption (20), so Eq. (1) is oscillatory. This completes the proof of the theorem.
3. Asymptotics of the forced equation In this section, we study the asymptotic behavior of solutions of the forced nonlinear equation with damping [r(t)x0 (t)]0 + p(t)x0 (t) + q(t)f(x(t)) = e(t):
(43)
The main result of this section is the following. Theorem 4. Let the assumptions of Theorem 1 hold and suppose that the function e ∈ C([t0 ; ∞); (−∞; ∞)) satisÿes Z ∞ a(s)|e(s)| ds = N ¡ ∞: (44) Then every solution x(t) of Eq. (43) satisÿes lim inf |x(t)| = 0: t→∞
Proof Let x(t) be a solution of Eq. (43) and suppose that lim inf |x(t)| = C ¿ 0; t→∞
so x(t) is nonoscillatory. Without loss of generality, we may assume that x(t) ¿ 0 on [T0 ; ∞) for some T0 ≥ t0 . Di erentiating the function v(t) de ned by (8), we obtain " e(t) x0 (t) 0 − p(t) v (t) = −2g(t)v(t) + a(t) −q(t) + f(x(t)) f(x(t)) 0
−r(t)f (x(t))
x0 (t) f(x(t))
#
2 +K
−1
0
[r(t)g(t)]
for all t ≥ T0 and thus v0 (t) ≤ −
p(t) 1 K v(t) − v2 (t) + a(t)|e(t)| − (t): r(t) a(t)r(t) f(C)
1024
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Hence, for all t ≥ T ≥ T0 , we have Z t Z t Z t p(s) v(s) ds H (t; s) (s) ds ≤ − H (t; s)v0 (s) ds − H (t; s) r(s) T T T Z t Z t K 1 2 H (t; s) H (t; s) − v (s) ds + a(s)|e(s)| ds a(s)r(s) f(C) T T and consequently 2 # Z t" p(s) a(s)r(s) h(t; s) + H (t; s) (s) − ds (H (t; s))1=2 4K r(s) T ≤ H (t; T )v(T ) −
Z t " T
KH (t; s) a(s)r(s)
1=2 v(s) +
1 2
a(s)r(s) K
1=2
# 2 p(s) (H (t; s))1=2 ) × h(t; s) + ds r(s) 1 + f(C)
Z T
t
H (t; s)a(s)|e(s)| ds:
(45)
It follows from (44), (45), and property (2) of the function H (t; s) that for every t ≥ T0 2 # Z t" p(s) a(s)r(s) 1=2 h(t; s) + (H (t; s)) H (t; s) (s) − ds 4K r(s) T0 1 H (t; T0 ) ≤ H (t; T0 )|v(T0 )| + f(C)
Z T
t
a(s)|e(s)| ds:
Now the proof proceeds in the same way as in Theorem 1. Example 3. Consider the nonlinear di erential equation 0 1 1 13 1 1 0 x (t) + 2 x0 (t) + 3 (x(t) + x3 (t)) = 6 + 12 ; t t t t t
(46)
where x ∈ (−∞; ∞); and t ∈ [1; ∞). Clearly, f0 (x) exists and f0 (x) = 3x2 + 1 ≥ 1
for all
x ∈ (−∞; ∞);
so assumption (5) holds. Now, let H (t; s) = (t − s)2 ; h(t; s) = 2, and g(t) = −1=t, so that a(t) = t 2 . Then, by a straightforward computation, we obtain 2 ! Z s t−s 1 t (t − s)2 − 2+ ds lim sup 2 s 4 s t→∞ t T
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
= lim sup t→∞
= lim sup t→∞
1 t2
1025
(t + s)2 (t − s)2 − ds s 4s T 17 5 3 3 ln t − + − 2 = ∞; 4 8 2t 8t Z t
so assumption (6) holds. Thus, by Theorem 4, we conclude that all solutions of Eq. (46) satisfy lim inf t→∞ |x(t)| = 0. Observe that x(t) = t −3 is such a solution.
4. Discussion and ÿnal remarks In this section, we would like to establish connections between our results and those known in the literature. It is natural to expect that, Eqs. (2) and (3) being particular cases of Eq. (1), Theorems 1–3 will coincide with theorems of Philos [17] (for Eq. (2)) and Li [16] (for Eq. (3)). Indeed, with f(x) = x and p(t) ≡ 0, our Theorems 1–3 reduce correspondingly to Theorems 2:1; 2:6 and 2:8 obtained by Li [16] with the following correction: condition (C1 ) in the latter three theorems should be omitted as a super uous one (see [18] for the details). If, in addition, we set g(t) ≡ 0 and, consequently, a(t) ≡ 1, then Theorems 1–3 reduce Theorems to 1–3 obtained by Philos [17]. Furthermore, we note that with g(t) ≡ 0 (and hence with a(t) ≡ 1) and r(t) ≡ 1 Corollary 3 reduces to the results of Yan [24; Corollary 3] and Yeh [25; Theorem 2]; and with p(t) ≡ 0, r(t) ≡ 1, and g(t) ≡ 0, it reduces to the Kamenev’s criterion [14]. In order to make a comparison between our results and those derived very recently for nonlinear equations with damping by Grace [6; Theorems 6–8], let us examine assumptions (C7 )–(C9 ) of Theorem B given in the Introduction and the corresponding assumptions (18)–(20) of our Theorem 2. Clearly, (C7 ) coincides R t with (18). Condition (C8 ) is more restrictive than (19) since letting a(t) = exp(−2 g(u) du) in (C8 ) we have 2 Z t 1 p(s) 1=2 (H (t; s)) a(s)r(s) h(t; s) + ds lim sup r(s) t→∞ H (t; t0 ) t0 " 2 Z t 1 p(s) (H (t; s))1=2 a(s)r(s) h(t; s) + ≤ lim sup r(s) t→∞ H (t; t0 ) t0 1=2
+4g(s)(H (t; s))
p(s) h(t; s) + (H (t; s))1=2 r(s)
# 2
+ 4g (s)H (t; s) ds:
Hence, if (C8 ) holds, then (19) always holds, but the converse is not true. Naturally, the di erent structure of assumptions (C9 ) and (20) makes their comparison a little bit complicated, though a simple but lengthy computation shows that,
1026
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
at least for the case when h(t; s)g(s)r(s) − [r(s)g(s)]0 ≥ 0; if (C8 ) holds, then so does (20). In summary, we may conclude that the conditions of Theorem 2 are less restrictive than those of Theorem 6 due to Grace [6]. The same reasoning is also valid for other results presented in this paper. We also stress that our results do not involve any particular assumptions on the damping coecient p(t) and thus they are more exible than the recent results due to Grace [6] and Grace and Lalli [8]. We conclude with the following remarks which also indicate the directions for the possible further investigation. Remark 1. The above results hold true if we replace condition (5) with the following one: f(x) ≥ K ¿0 x
for x 6= 0;
but the function q(t) should be nonnegative (we refer to [15] for details): Remark 2. Some of the results of this paper can be extended to the more general equations [r(t) (x(t))x0 (t)]0 + p(t)x0 (t) + q(t)f(x(t)) = 0 and [r(t) (x(t))x0 (t)]0 + p(t)k(t; x(t); x0 (t)) + q(t)f(x(t)) = 0; where is a positive continuous function and k is a nonnegative continuous function; as well as to the di erential equation with a deviating argument [r(t) (x(t))x0 (t)]0 + p(t)x0 (t) + q(t)f(x[(t)]) = 0: Remark 3. The results in this paper are presented in a form with a high degree of generality, and thus they give many possibilities for oscillation criteria with an appropriate choice of the functions H and h. For example; there are interesting perspectives to apply Theorems 1–4 with n−1 Z t dz ; (t; s) ∈ D; H (t; s) = s (z) where n ¿ 2 is a constant and is a positive continuous function on [t0 ; ∞) such that Z ∞ dz =∞ (z) t0 (an important case to be considered is (z) = z with real).
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1027
Acknowledgements This research was done during the author’s visit to the University of Florence which he thanks for its extremely warm hospitality with a special gratitude to Professors Roberto Conti and Gabriele Villari for permanent support and encouragement. The author is indebted to the referee for a careful reading of the manuscript and for kindly prompting improvements in presentation.
References [1] J.W. Baker, Oscillation theorems for a second order damped nonlinear di erential equation, SIAM J. Math. Anal. 25 (1973) 37– 40. [2] L.E. Bobisud, Oscillation of solutions of damped nonlinear di erential equations, SIAM J. Math. Anal. 18 (1970) 601– 606. [3] G.J. Butler, The oscillatory behavior of a second order nonlinear di erential equation with damping, J. Math. Anal. Appl. 57 (1977) 273–289. [4] S.R. Grace, Oscillation theorems for second order nonlinear di erential equations with damping, Math. Nachr. 141 (1989) 117–127. [5] S.R. Grace, Oscillation criteria for second order nonlinear di erential equations with damping, J. Austral. Math. Soc. Ser. A 49 (1990) 43–54. [6] S.R. Grace, Oscillation theorems for nonlinear di erential equations of second order, J. Math. Anal. Appl. 171 (1992) 220 –241. [7] S.R. Grace, B.S. Lalli, Oscillation theorems for certain second order perturbed nonlinear di erential equations, J. Math. Anal. Appl. 77 (1980) 205–214. [8] S.R. Grace, B.S. Lalli, Integral averaging technique for the oscillation of second order nonlinear di erential equations, J. Math. Anal. Appl. 149 (1990) 277–311. [9] S.R. Grace, B.S. Lalli, Oscillation theorems for second order superlinear di erential equations with damping, J. Austral. Math. Soc. Ser. A 53 (1992) 156–165. [10] S.R. Grace, B.S. Lalli, C.C. Yeh, Oscillation theorems for nonlinear second order di erential equations with a nonlinear damping term, SIAM J. Math. Anal. 15 (1984) 1082–1093. [11] S.R. Grace, B.S. Lalli, C.C. Yeh, Addendum: oscillation theorems for nonlinear second order di erential equations with a nonlinear damping term, SIAM J. Math. Anal. 19 (1988) 1252–1253. [12] B.J. Harris, On the oscillation of solutions of linear di erential equations, Mathematika 31 (1984) 214 –226. [13] P. Hartman, On nonoscillatory linear di erential equations of second order, Amer. J. Math. 74 (1952) 389–400. [14] I.V. Kamenev, An integral criterion for oscillation of linear di erential equations, Mat. Zametki 23 (1978) 249–251. [15] M. Kirane, Yu.V. Rogovchenko, On oscillation of nonlinear second order di erential equation with damping term, preprint. [16] H.J. Li, Oscillation criteria for second order linear di erential equations, J. Math. Anal. Appl. 194 (1995) 217–234. [17] Ch.G. Philos, Oscillation theorems for linear di erential equations of second order, Arch. Math. 53 (1989) 483–492. [18] Yu.V. Rogovchenko, Note on ‘Oscillation criteria for second order linear di erential equations’, J. Math. Anal. Appl. 203 (1996) 560–563. [19] Yu.V. Rogovchenko, Oscillation criteria for second order nonlinear perturbed di erential equations, J. Math. Anal. Appl. 215 (1997) 334–357.
1028
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
[20] C.A. Swanson, Comparison and Oscillation Theory of Linear Di erential Equations, Academic Press, New York, 1968. [21] A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949) 115–117. [22] J.S.W. Wong, A second order nonlinear oscillation, Funkcial. Ekvac. 11 (1968) 207–234. [23] J. Yan, A note on an oscillation criterion for an equation with damped term, Proc. Amer. Math. Soc. 90 (1984) 277–280. [24] J. Yan, Oscillation theorems for second order linear di erential equations with damping, Proc. Amer. Math. Soc. 98 (1986) 276–282. [25] C.C. Yeh, Oscillation theorems for nonlinear second order di erential equations with damping term, Proc. Amer. Math. Soc. 84 (1982) 397–402.
www.elsevier.nl/locate/na
Oscillation theorems for second-order equations with damping 1 Yuri V. Rogovchenkoa;b; ∗ a Institute
of Mathematics, National Academy of Sciences, Tereshchenkivs’ka Str. 3, 252601 Kyiv, Ukraine b Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey Received 5 February 1997; accepted 14 August 1998
Keywords: Second order; Nonlinear di erential equation; Damping; Oscillation
1. Introduction In this paper, we study the problem of oscillation of the nonlinear second-order di erential equation with damping [r(t)x0 (t)]0 + p(t)x0 (t) + q(t)f(x(t)) = 0;
(1)
where r(t)∈C 1 ([t0 ; ∞); (0; ∞)); p(t); q(t) ∈ C([t0 ; ∞); (−∞; ∞)); f(x) ∈ C((−∞; ∞); (−∞; ∞)) and xf(x) ¿ 0 for x 6= 0; t0 ≥ 0. We recall that a function x : [t0 ; t1 ) → (−∞; ∞); t1 ¿ t0 is called a solution of Eq. (1) if x(t) satis es Eq. (1) for all t ∈ [t0 ; t1 ). In what follows, it will be always assumed that solutions of Eq. (1) exist for any t0 ≥ 0. Furthermore, a solution x(t) of Eq. (1) is called continuable if x(t) exists for all t ≥ t0 . A continuable solution x(t) of Eq. (1) is called oscillatory if it has arbitrarily large zeroes, otherwise it is called nonoscillatory. Finally, it is said that Eq. (1) is oscillatory if all its solutions are oscillatory. ∗ Correspondence address: Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey. E-mail address: Rogovch [email protected] (Y.V. Rogovchenko) 1 This research was supported by a fellowship of the Italian Consiglio Nazionale delle Ricerche.
0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 3 2 4 - 1
1006
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
In the last two decades there has been an increasing interest in obtaining sucient conditions for the oscillation and/or nonoscillation of solutions for di erent classes of second-order di erential equations [1–25]. In the absence of damping, there is a great number of papers (see, for example, [12–17, 20–22] and the references quoted therein) devoted to the particular cases of Eq. (1) such as the linear equation x00 (t) + q(t)x(t) = 0
(2)
and the more general nonlinear equation [r(t)x0 (t)]0 + q(t)f(x(t)) = 0:
(3)
An important tool in the study of oscillatory behavior of solutions for Eqs. (1) – (3) is the averaging technique. This goes back as far as to the classical results of Wintner [21] giving a sucient condition for oscillation of Eq. (2), namely, Z Z 1 t s lim q( ) d ds = ∞ (C1 ) t→∞ t t t0 0 and Hartman [13] who showed that the above limit cannot be replaced by the upper limit and proved that the condition Z Z Z Z 1 t s 1 t s − ∞ ¡ lim inf q( ) d ds ¡ lim sup q( ) d ds ≤ ∞ (C2 ) t→∞ t t t→∞ t t0 t0 t0 0 implies that Eq. (2) is oscillatory. The result of Wintner was improved by Kamenev [14] who proved that the condition Z t 1−n lim sup t (t − s)n−1 q(s) ds = ∞ for some n ¿ 2 (C3 ) t→∞
t0
is sucient for the oscillation of Eq. (2). Recently, Yan [24] presented new oscillation criteria for Eq. (1) with f(x) = x involving the Kamenev’s type condition. Theorem A (Yan [24, Theorem 2]). Suppose that there exist a positive continuously di erentiable function h(t) on [t0 ; ∞) and ∈ (1; ∞) such that Z t lim sup t − (C4 ) (t − ) h( )q( ) d ¡ ∞; t→∞
t0
there exists a continuous function (t) on [t0 ; ∞) such that Z t" r( ) 1 − lim inf t (t − ) h( )q( ) − (t − ) −2 (C5 ) t→∞ 4 h( ) s 2 # h( )p( ) 0 + h( ) − (t − )h ( ) d ≥ (s) × (t − ) r( )
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
and
Z
(C6 )
lim
t→∞
t
t0
1007
2+ ( ) d = ∞; h( )r( )
where + (t) = max( (t); 0). Then the equation [r(t)x0 (t)]0 + p(t)x0 (t) + q(t)x(t) = 0
(4)
is oscillatory. The above result of Yan was extended further to Eq. (2) by Philos [17] and to Eq. (3) by Li [16] (it should be noted that the original results of Li need some correction; see [18] for the details). In the presence of damping, a number of oscillation criteria were obtained for di erent classes of nonlinear equations by Baker [1], Bobisud [2], Butler [3], Grace [4 – 6], Grace and Lalli [7–9], Grace et al. [10], Yan [23, 24], Yeh [25] and others. Very recently, the extension of the results of Philos [17] both for Eq. (1) and for the more general equation [r(t) (x(t))x0 (t)]0 + p(t)x0 (t) + q(t)f(x(t)) = 0 was obtained by Grace [6]. In order to present the oscillation criterion of Grace we rst introduce, following Philos [17], the class of functions P which will be extensively used in the sequel. Namely, let D0 = {(t; s): t ¿ s ≥ t0 } and D = {(t; s): t ≥ s ≥ t0 }. We will say that the function H ∈C(D; (−∞; ∞)) belongs to the class P if it satis es the following two conditions: 1. H (t; t) = 0 for t ≥ t0 , H (t; s) ¿ 0 in D0 ; 2. H has a continuous and nonpositive partial derivative in D0 with respect to the second variable. Theorem B (Grace [6, Theorem 6]). Suppose that f0 (x) ≥ K ¿ 0 for x 6= 0 and assume that there exist the function a ∈ C 1 ([t0 ; ∞); (0; ∞)) and the functions h; H ∈ C(D; (−∞; ∞)) such that H belongs to the class P and −
@H (t; s) = h(t; s)(H (t; s))1=2 @s
for all (t; s) ∈ D0 :
Suppose; moreover; that H (t; s) ≤∞ 0 ¡ inf lim inf (C7 ) t→∞ H (t; t0 ) s≥t0 and (C8 )
lim sup t→∞
1 H (t; t0 ) 0
Z
t
t0
r(s)a(s)[h(t; s) − (s)(H (t; s))1=2 ]2 ds ¡ ∞;
where (t) = (r(t)a (t) − p(t)a(t))=(r(t)a(t)).
1008
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
If there exists a function ∈ C([t0 ; ∞); (−∞; ∞)) such that Z t r(s)a(s) 1 lim sup H (t; s)a(s)q(s) − (C9 ) 4K t→∞ H (t; T ) T ×(h(t; s) − (s)(H (t; s))1=2 )2 ds ≥ (T ) for every T ≥ t0 and (C6 ) holds; then Eq. (1) is oscillatory. The purpose of this paper is to improve Theorem B, as well as other related results in [4 –9, 23–25] concerning the oscillation of Eq. (1) by making use of techniques similar to those used by Grace [6], Philos [17], Rogovchenko [19], and Yan [24]. It is worth pointing out that, in contrast to the recent results due to Grace [6] and Grace and Lalli [8], we do not impose any additional condition on the damping coecient p(t). 2. Oscillation theorems Theorem 1. Suppose that f0 (x) exists and f0 (x) ≥ K ¿ 0
(5)
for some constant K and for all x 6= 0. Suppose; further; that the functions h; H ∈ C (D; (−∞; ∞)) are such that H belongs to the class P and −
@H (t; s) = h(t; s)(H (t; s))1=2 @s
for all (t; s) ∈ D0 :
(6)
If there exists a function g ∈ C 1 ([t0 ; ∞); (0; ∞)) such that Z t" 1 a(s)r(s) H (t; s) (s) − lim sup H (t; t ) 4K t→∞ 0 t0 # 2 p(s) (H (t; s))1=2 ds = ∞; (7) × h(t; s) + r(s) Rs where a(s) = exp(−2 g(u) du) and (s) = K −1 a(s)(Kq(s) − p(s)g(s) − [r(s)g(s)]0 + r(s)g2 (s)); then Eq. (1) is oscillatory. Proof. Let x(t) be a nonoscillatory solution of the di erential equation (1) and let T0 ≥ t0 be such that x(t) 6= 0 for all t ≥ T0 . Without loss of generality, we may assume that x(t) ¿ 0 for all t ≥ T0 since the similar argument holds for the case when x(t) is eventually negative. Following Harris [12], we use a generalized Riccati transformation and let 0 g(t) x (t) : (8) + v(t) = a(t)r(t) f(x(t)) K
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1009
Then di erentiating (8) and making use of Eq. (1) we obtain # " 2 0 [r(t)x0 (t)]0 x (t) [r(t)g(t)]0 0 0 + − r(t)f (x(t)) v (t) = −2g(t)v(t) + a(t) f(x(t)) f(x(t)) K " = −2g(t)v(t) + a(t) −q(t) − p(t)
0
−r(t)f (x(t))
x0 (t) f(x(t))
#
2 +K
for all t ≥ T0 . In view of (5), (9) yields " v0 (t) ≤ −2g(t)v(t) − a(t) q(t) + p(t) +Kr(t)
x0 (t) f(x(t))
2
x0 (t) f(x(t))
−1
0
[r(t)g(t)]
(9)
x0 (t) f(x(t)) #
− K −1 [r(t)g(t)]0 :
(10)
Now, from (8) and (10), we obtain v0 (t) ≤ − (t) −
K p(t) v(t) − v2 (t) r(t) a(t)r(t)
(11)
for all t ≥ T0 with (t) de ned as above. Hence, by (11) and (6), for all t ≥ T ≥ T0 , we have Z t Z t Z t p(s) 0 v(s) ds H (t; s) (s) ds ≤ − H (t; s)v (s) ds − H (t; s) r(s) T T T Z t K v2 (s) ds H (t; s) − a(s)r(s) T Z t p(s) @H t (t; s)v(s) + H (t; s) v(s) = − H (t; s)v(s)|T − − @s r(s) T K + H (t; s) v2 (s) ds a(s)r(s) Z t p(s) v(s) h(t; s)(H (t; s))1=2 v(s) + H (t; s) = H (t; T )v(T ) − r(s) T K v2 (s) ds + H (t; s) a(s)r(s) 1=2 Z t " KH (t; s) v(s) = H (t; T )v(T ) − a(s)r(s) T
1010
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1 + 2
Z +
t T
a(s)r(s) K
a(s)r(s) 4K
1=2
p(s) (H (t; s))1=2 h(t; s) + r(s)
h(t; s) +
p(s) (H (t; s))1=2 r(s)
#2 ds
2 ds:
Therefore, for all t ≥ T ≥ T0 , we have Z t" T
a(s)r(s) H (t; s) (s) − 4K
≤ H (t; T )v(T ) −
Z t " T
p(s) h(t; s) + (H (t; s))1=2 r(s)
KH (t; s) a(s)r(s)
1=2 v(s) +
1 2
2 # ds
a(s)r(s) K
1=2
#2 p(s) 1=2 (H (t; s)) × h(t; s) + ds: r(s)
(12)
It follows from (12) and property (2) of the function H (t; s) that for every t ≥ T0 Z t" T0
a(s)r(s) H (t; s) (s) − 4K
p(s) h(t; s) + (H (t; s))1=2 r(s)
2 # ds
≤ H (t; T0 )|v(T0 )| ≤ H (t; t0 )|v(T0 )|:
(13)
Thus, from (13) and property (2) of the function H (t; s), we obtain Z t" t0
a(s)r(s) H (t; s) (s) − 4K Z
=
T0
"
t0
+
p(s) h(t; s) + (H (t; s))1=2 r(s)
a(s)r(s) H (t; s) (s) − 4K
Z t" T0
a(s)r(s) H (t; s) (s) − 4K Z
≤ H (t; t0 )
T0
t0
Z =H (t; t0 )
ds
p(s) h(t; s) + (H (t; s))1=2 r(s)
2 #
p(s) h(t; s) + (H (t; s))1=2 r(s)
ds 2 # ds
| (s)| ds + H (t; t0 )|v(T0 )|
T0
t0
2 #
| (s)| ds + |v(T0 )| :
(14)
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1011
Inequality (14) implies that Z t" a(s)r(s) 1 H (t; s) (s) − lim sup 4K t→∞ H (t; t0 ) t0 ×
p(s) (H (t; s))1=2 h(t; s) + r(s)
2 #
Z ds ≤
T0
t0
| (s)| ds + |v(T0 )|;
which contradicts assumption (7) of the theorem. Hence, all solutions of Eq. (1) are oscillatory. As immediate consequences of Theorem 1 we obtain the following corollaries. Corollary 1. Let the assumptions of Theorem 1 hold except that Eq. (7) is replaced by Z t 1 H (t; s) (s) ds = ∞ lim sup t→∞ H (t; t0 ) t0 and lim sup t→∞
1 H (t; t0 )
Z
t
t0
2 p(s) a(s)r(s) h(t; s) + (H (t; s))1=2 ds ¡ ∞: r(s)
Then Eq. (1) is oscillatory. Corollary 2 (cf. Li [16, Theorem 2.1]). Let f(x) = x; p(t) ≡ 0; and let the functions h; H be as in Theorem 1 with (6) holding. If there exists a function g ∈ C 1 ([t0 ; ∞); (0; ∞)) such that Z t 1 1 2 H (t; s) (s) − a(s)r(s)h (t; s) ds = ∞; lim sup 4 t→∞ H (t; t0 ) t0 (s) = a(s){q(s) − [r(s)g(s)]0 + r(s)g2 (s)}; then
where a(s) is the same as above and Eq. (3) is oscillatory.
In the same way as it was done in [16], with an appropriate choice of the functions H and h, we can derive from Theorem 1 a number of oscillation criteria for Eqs. (2) and (3). Let us consider, for example, the function H (t; s) de ned by H (t; s) = (t − s)n−1 ;
(t; s) ∈ D;
where n ¿ 2 is an integer. Clearly, H belongs to the class P. Furthermore, the function h(t; s) = (n − 1)(t − s)(n−3)=2 ;
(t; s) ∈ D
is continuous on [t0 ; ∞) and satis es condition (6). Then, by Theorem 1, we obtain the following oscillation criteria.
1012
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Corollary 3. Let assumption (5) hold. Suppose also that there exists a function g ∈ C 1 ([t0 ; ∞); (0; ∞)) such that Z t" a(s)r(s) (t − s)n−3 lim sup t 1−n (t − s)n−1 (s) − 4K t→∞ t0 2 # p(s) × n−1+ (t − s) ds = ∞ r(s) for some integer n ¿ 2; where a(s) and Eq. (1) is oscillatory.
(15)
(s) are the same as in Theorem 1. Then
Corollary 4 (cf. Grace and Lalli [8, Corollary 3]). Let assumption (5) hold and suppose also that there exist functions b ∈ C([t0 ; ∞); (0; ∞)); g ∈ C 1 ([t0 ; ∞); (0; ∞)) such that for some ¿ 1 Z t" a(s)r(s) − (B(t) − B(s))−2 (B(t) − B(s)) (s) − lim sup B (t) 4K t→∞ t0 2 # p(s) (B(t) − B(s)) × b(t) + ds = ∞; (16) r(s) Rt where B(t) = t0 b(s) ds; and a(s); (s) are the same as in Theorem 1. Then Eq. (1) is oscillatory. Proof. Let us put H (t; s) = [B(t) − B(s)] ;
(t; s) ∈ D;
then with the choice h(t; s) = b(t)[B(t) − B(s)](−2)=2 ;
(t; s) ∈ D;
the conclusion follows directly from Theorem 1. Example 1. Consider the nonlinear di erential equation ((1 + sin2 t)x0 (t))0 − 3sin t cos t x0 (t) +
1 x(t)(1 + x4 (t)) = 0; 1 + cos4 t
(17)
where x ∈ (−∞; ∞), and t ∈ [1; ∞). Clearly, f0 (x) exists and f0 (x) = 5x4 + 1 ≥ 1
for all x ∈ (−∞; ∞);
so assumption (5) holds. Let us apply Corollary 3 with n = 3 and g(t) = −1=t, so that a(t) = t 2 . A straightforward computation yields !0 " Z 2 3sin s cos s 1 + sin 1 1 t s − + (t − s)2 s2 lim sup 2 1 + cos4 s s s t→∞ t T
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1 + sin2 s + − (1 + sin2 s) s2 1 = lim sup 2 t→∞ t
Z T
t
2 2
(t − s) s
3sin s cos s 1 − t − s 2(1 + sin2 s)
1013
2 #
9sin2 s cos2 s 1 − 1 + cos4 s 4(1 + sin2 s) !
ds !
+ s(t − s)(4s − t)sin s cos s − s2 (1 + sin2 s) ds = ∞; since 9sin2 s cos2 s 1 ¿0 − 4 1 + cos s 4(1 + sin2 s) for any s ∈ (−∞; +∞). Thus, assumption (15) holds, and we conclude by Corollary 3 that all continuable solutions of Eq. (17) are oscillatory. Observe that x(t) = cos t is such a solution. The important point to note here is that the recent results due to Grace [6] and Grace and Lalli [8] do not apply to Eq. (17). A simple inspection with the same choice (t) = t n as in [6, 8] shows that assumptions p(t) ≤ 0; (p(t)(t))0 ≥ 0 for all t ≥ t0 [6, Theorem 1]; (r(t)(t))0 ≤ 0 for all t ≥ t0 [6, Theorem 2]; r(t)0 (t) − p(t)(t)0 ≥ 0 for all t ≥ t0 [8, Theorem 1, Corollary 1]; r(t)0 (t) − p(t)(t)0 ≥ 0 and (r(t)0 (t) − p(t)(t))0 ≤ 0 for all t ≥ t0 [6, Theorem 8; 8, Theorems 5; 7]; ( 5 ) p(t) ≤ 0 for all t ≥ t0 [8, Theorems 2; 4; 6] all fail for Eq. (17) since all the functions in ( 1 ) – ( 5 ) are oscillatory for any n ≥ 1 and for t ≥ t0 : ( 1 ) ( 2 ) ( 3 ) ( 4 )
3 p(t) = − sin 2t; 2 3 (p(t)(t))0 = − t n−1 (2t cos 2t + n sin 2t); 2 (r(t)0 (t))0 = nt n−2 (t sin 2t + (n − 1)(1 + sin2 t)); 1 r(t)0 (t) − p(t)(t) = t n−1 (2n(1 + sin2 t) + 3t sin 2t); 2 5n 2 0 0 n−2 2 t sin 2t + n(n − 1)(1 + sin t) + 3t cos 2t : (r(t) (t) − p(t)(t)) = t 2 It is not dicult to check that assumptions ( 1 ) – ( 5 ) all fail also for n = 0 and 0 ¡ n ¡ 1.
1014
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Theorem 2. Let the functions H and h be deÿned as in Theorem 1; the function f satisfy (5); and suppose that H (t; s) ≤ ∞: (18) 0 ¡ inf lim inf t→∞ H (t; t0 ) s≥t0 Assume that there exist functions g ∈ C 1 ([t0 ; ∞); (0; ∞)) and ∈ C ([t0 ; ∞); (−∞; ∞)) such that for all t ¿ t0 2 Z t 1 p(s) 1=2 (H (t; s)) lim sup a(s)r(s) h(t; s) + ds ¡ ∞ (19) r(s) t→∞ H (t; t0 ) t0 and
Z lim sup t→∞
t
t0
2+ (s) = ∞; a(s)r(s)
(20)
and suppose that for any T ≥ t0 Z t" a(s)r(s) 1 lim sup H (t; s) (s) − H (t; T ) 4K t→∞ T ×
p(s) (H (t; s))1=2 h(t; s) + r(s)
2 # ds ≥ (T );
(21)
where a(s) and (s) are the same as in Theorem 1 and + (t) = max( (t); 0). Then Eq. (1) is oscillatory. Proof. Without loss of generality, we may assume that there exists a solution x(t) of Eq. (1) such that x(t) ¿ 0 on [T0 ; ∞) for some T0 ≥ t0 ; a similar argument holds when x(t) ¡ 0 on [T0 ; ∞). De ne the function v(t) by (8); as in Theorem 1, we obtain inequality (12). It follows from (12) that 2 # Z t" p(s) a(s)r(s) 1 h(t; s) + (H (t; s))1=2 H (t; s) (s) − ds H (t; T ) T 4K r(s) 1 ≤ v(T ) − H (t; T )
Z t " T
KH (t; s) a(s)r(s)
1=2
1 v(s) + 2
a(s)r(s) K
#2 p(s) 1=2 × h(t; s) + ds (H (t; s)) r(s) for t ¿ T ≥ T0 , and therefore Z t p(s) a(s)r(s) 1 h(t; s) + H (t; s) (s) − lim sup 4K r(s) t→∞ H (t; T ) T
1=2
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1=2 2
× (H (t; s)) 1 + 2
ds ≤ v(T ) − lim inf t→∞
a(s)r(s) K
1=2
1 H (t; T )
Z t " T
p(s) (H (t; s))1=2 h(t; s) + r(s)
KH (t; s) a(s)r(s)
1015
1=2 v(s)
#2 ds
for all T ≥ T0 . It follows from (21) that 1 v(T ) ≥ (T ) + lim inf t→∞ H (t; T ) 1 + 2
a(s)r(s) K
1=2
Z t " T
KH (t; s) a(s)r(s)
1=2 v(s)
p(s) (H (t; s))1=2 h(t; s) + r(s)
#2 ds
for T ≥ T0 , so v(T ) ≥ (T )
(22)
for all T ≥ T0
and 1 lim inf t→∞ H (t; T0 )
Z t " T0
KH (t; s) a(s)r(s)
1=2 v(s) +
1 2
a(s)r(s) K
1=2
# 2 p(s) (H (t; s))1=2 × h(t; s) + ds ≤ v(T0 ) − (T0 ) = M ¡ ∞: r(s) Thus, for all t ≥ T0 , 1 ∞ ¿ lim inf t→∞ H (t; T0 ) ×
Z t " T0
KH (t; s) a(s)r(s)
p(s) (H (t; s))1=2 h(t; s) + r(s)
1 ≥ lim inf t→∞ H (t; T0 )
Z t T0
1=2 v(s) +
1 2
a(s)r(s) K
1=2
# 2 ds
KH (t; s) 2 v (s) a(s)r(s)
p(s) (H (t; s))1=2 v(s) ds: + (H (t; s))1=2 h(t; s) + r(s)
(23)
1016
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Let us de ne for t ¿ T0 the functions (t) and (t) by (t) =
1 H (t; T0 )
Z
t
KH (t; s) 2 v (s) ds a(s)r(s)
T0
and 1 (t) = H (t; T0 )
Z
t
T0
1=2
(H (t; s))
p(s) 1=2 (H (t; s)) h(t; s) + v(s) ds: r(s)
Then (23) may be rewritten as lim inf [ (t) + (t)] ¡ ∞: t→∞
Let us show now that Z ∞ 2 v (s) ds ¡ ∞: a(s)r(s) T0
(24)
(25)
To this end, suppose that Z
∞
T0
v2 (s) ds = ∞: a(s)r(s)
It follows from (18) that there exists a positive constant such that H (t; s) ¿ ¿ 0: inf lim inf t→∞ H (t; t0 ) s≥t0
(26)
(27)
Furthermore, by (26) for any positive number there exists T1 ¿ T0 such that Z
t T0
v2 (s) ds ≥ a(s)r(s)
for all t ≥ T1 :
Thus, for all t ≥ T1 , we have Z s 2 Z t v (u) 1 KH (t; s) d (t) = du H (t; T0 ) T0 T0 a(u)r(u) Z s 2 Z t v (u) @H (t; s) 1 du ds K − = H (t; T0 ) T0 @s T0 a(u)r(u) Z s 2 Z t v (u) @H (t; s) 1 K − ≥ du ds H (t; T0 ) T1 @s T0 a(u)r(u) Z t 1 H (t; T1 ) @H (t; s) ds = : K − ≥ H (t; T0 ) T1 @s H (t; T0 )
(28)
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1017
It follows from (27) that lim inf t→∞
H (t; s) ¿ ¿ 0; H (t; t0 )
so there exists T2 ≥ T1 such that H (t; T1 ) ≥ H (t; t0 )
for all t ≥ T2 ;
and therefore by (28), (t) ≥
for all t ≥ T2 :
Since is an arbitrary constant, we have lim (t) = ∞:
(29)
t→∞
Consider now a sequence {tn }∞ n=1 in the interval (T0 ; ∞) such that lim tn = ∞
t→∞
and lim [ (tn ) + (tn )] = lim inf [ (t) + (t)]:
n→∞
t→∞
It follows from (24) that there exists a natural number N such that (tn ) + (tn ) ≤ M
for all n ¿ N
(30)
with the same constant M as de ned above. Obviously by (29) lim (tn ) = ∞;
n→∞
and thus (30) implies that lim (tn ) = −∞:
n→∞
(31)
Furthermore, (30) and (31) lead to the inequality (tn ) + 1¡ (tn ) for large values of n, where ∈ (0; 1) is a constant. Hence, for n large enough, (tn ) ¡ − 1 ¡ 0: (tn )
(32)
It follows from (31) and (32) that lim
n→∞
(tn ) (tn ) = ∞: (tn )
(33)
1018
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
On the other hand, by the Schwarz inequality, we have for any natural number n Z tn 2 1 p(s) 2 1=2 1=2 (H (tn ; s)) (H (tn ; s)) h(tn ; s) + v(s) ds (tn ) = 2 H (tn ; T0 ) r(s) T0 "Z
1 = 2 H (tn ; T0 )
tn
T0
KH (tn ; s) a(s)r(s)
1=2
v(s)
a(s)r(s) K
1=2
#2 p(s) 1=2 (H (tn ; s)) × h(tn ; s) + ds r(s) ≤
1 H (tn ; T0 )
Z
tn
KH (tn ; s)v2 (s) ds a(s)r(s)
T0
"
Z
1 × H (tn ; T0 )
1 ≤ (tn ) H (tn ; T0 )
1 1 2 (tn ) ≤ (tn ) H (tn ; t0 )
a(s)r(s) K
T0
"
and therefore
tn
Z
tn
t0
Z
tn
T0
p(s) h(tn ; s) + (H (tn ; s))1=2 r(s)
a(s)r(s) K
a(s)r(s) K
ds
p(s) h(tn ; s) + (H (tn ; s))1=2 r(s)
p(s) h(tn ; s) + (H (tn ; s))1=2 r(s)
#
2
lim sup t→∞
1 H (t; t0 )
Z
t
t0
ds
2 ds
for all n large enough. By (33) we obtain 2 Z tn 1 p(s) (H (tn ; s))1=2 a(s)r(s) h(tn ; s) + ds = ∞; lim n→∞ H (tn ; t0 ) t r(s) 0 so
#
2
(34)
2 p(s) (H (t; s))1=2 a(s)r(s) h(t; s) + ds = ∞; r(s)
and the last equality contradicts assumption (19). Therefore, we have proved that (26) fails, and so inequality (25) holds true. Furthermore, using (22), we obtain Z ∞ 2 Z ∞ 2 + (s) v (s) ds ≤ ds ¡ ∞ a(s)r(s) a(s)r(s) T0 T0 which contradicts condition (20), so Eq. (1) is oscillatory. The following result is the direct consequence of Theorem 2 and uses the same choice of the functions H and h as in Corollary 3 above.
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1019
Corollary 5. Let assumption (5) hold. Suppose there exist functions g ∈ C 1 ([t0 ; ∞); (0; ∞)) and ∈ C([t0 ; ∞); (−∞; ∞)) such that (20) holds along with 2 Z t p(s) (t − s) ds ¡ ∞ a(s)r(s)(t − s)n−3 n − 1 + lim sup t 1−n r(s) t→∞ t0 and lim sup t 1−n t→∞
Z t" t0
(t − s)n−1 (s) −
a(s)r(s) (t − s)n−3 4K
2 # p(s) × n−1+ (t − s) ds ≥ (T ) r(s) for all T ≥ t0 and for some integer n ¿ 2; where a(s) and Theorem 1. Then Eq. (1) is oscillatory.
(s) are the same as in
Proof. The only thing to be checked is condition (18). With the above choice of the functions H and h, this is ful lled automatically since H (t; s) (t − s)n−1 =1 = lim t→∞ H (t; t0 ) t→∞ (t − t0 )n−1 lim
for any s ≥ t0 . Example 2. Consider the following second-order nonlinear di erential equation: 1 00 (35) x (t) + 2x0 (t) + t cos t(x(t) + x3 (t)) = 0; t where x ∈ (−∞; ∞); t ∈ [1; ∞) and ≥ −1. With the functions r(t); p(t), and q(t) de ned by 1 1 exp(t 2 ); p(t) = 2 exp(t 2 ); q(t) = t cos t exp(t 2 ); t t Eq. (35) can be written as 0 1 1 2 0 exp(t )x (t) + 2 exp(t 2 )x0 (t) + t cos t exp(t 2 )(x(t) + x3 (t)) = 0: t t r(t) =
(36)
Clearly, f0 (x) exists and f0 (x) = 3x2 + 1 ≥ 1
for all x ∈ (−∞; ∞);
so assumption (5) holds. Furthermore, for the sake of simplicity, let H (t; s) = (t − s)2 ; h(t; s) = 2, and g(t) = 2t, so that a(t) = exp(−t 2 ). Then we have 2 Z Z t−s 1 t1 1 t (t + s)2 1 2+ ds = lim sup 2 ds = ; lim sup 2 3 s s 2 t→∞ t t→∞ t 1 s 1 so assumption (19) holds.
1020
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Furthermore, for any T ¿ 1 1 t2
lim sup t→∞
Z t (t + s)2 2 − (t − s)2 s cos s − ds ≥ −T sin T − K; 4 s 4s T
where K is a positive constant. Now set (T ) = −T sin T − K. Then there exists an integer N such that (2N + 1) + =4 ¿ 1 and for all n ≥ N (2n + 1) +
≤ T ≤ 2(n + 1) − ; 4 4
(T ) = −T sin T − K ≥ T ; where is a small constant. Thus, taking into account that ≥ −1, we have Z lim
t→∞
1
t
2+ (s) ds = lim t→∞ a(s)r(s) ≥
∞ X
Z
2
n=N
≥
∞ X n=N
2
t
1
2+ (s) ds s−1
Z
2(n+1)−=4
(2n+1)+=4
Z
2(n+1)−=4
(2n+1)+=4
s2 +1 ds
s−1 ds = ∞;
so condition (20) is also ful lled. Thus, by Corollary 5, we conclude that Eq. (36) is oscillatory, though none of the known criteria (see [4 –10, 23–25]) can cover this result. We can easily see that condition (C8 ) of Theorem B [6, Theorem 6] fails since 1 lim sup 2 t t→∞
Z
t
1
2 (t − s)(2s2 + 1) 1 2+ ds s s Z
1 t2
≥ 4 lim sup t→∞
t
1
s(t − s)2 ds
1 2 1 t2 − + − 2 = ∞: = 4 lim sup 12 2 3t 4t t→∞ Theorem 3. Let the functions H and h be deÿned as in Theorem 1; the function f to satisfy (5); and suppose that (18) holds. Assume that there exist functions g ∈ C 1 ([t0 ; ∞); (0; ∞)) and ∈ C([t0 ; ∞); (−∞; ∞)) such that (20) holds; lim inf t→∞
1 H (t; t0 )
Z
t
t0
H (t; s) (s) ds ¡ ∞
(37)
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
and 1 lim inf t→∞ H (t; T ) ×
Z t" T
H (t; s) (s) −
p(s) (H (t; s))1=2 h(t; s) r(s)
for all T ≥ t0 ; where a(s); Eq. (1) is oscillatory.
1021
a(s)r(s) 4K
2 # ds ≥ (T )
(38)
(s) and + (s) are the same as in Theorem 2. Then
Proof. Without loss of generality, we may assume that there exists a solution x(t) of Eq. (1) such that x(t) ¿ 0 on [T0 ; ∞) for some T0 ≥ t0 . De ne the function v(t) by (8). Then as in the proof of Theorem 1 inequality (12) holds for all t ≥ T ≥ T0 , and thus 2 # Z t" p(s) a(s)r(s) 1 1=2 h(t; s) + (H (t; s)) H (t; s) (s) − ds lim inf t→∞ H (t; T ) T 4K r(s) 1 ≤ v(T ) − lim sup t→∞ H (t; T ) 1 + 2
a(s)r(s) K
1=2
Z t " T
KH (t; s) a(s)r(s)
1=2 v(s)
p(s) (H (t; s))1=2 h(t; s) + r(s)
#2 ds
for all T ≥ T0 . It follows from (38) that 1 v(T ) ≥ (T ) + lim sup t→∞ H (t; T ) 1 + 2
a(s)r(s) K
1=2
Z t " T
KH (t; s) a(s)r(s)
1=2 v(s)
p(s) (H (t; s))1=2 h(t; s) + r(s)
#2 ds
for all T ≥ T0 . Consequently, for all T ≥ T0 , inequality (22) holds, and 1=2 1=2 Z t " KH (t; s) 1 1 a(s)r(s) v(s) + lim sup a(s)r(s) 2 K t→∞ H (t; T0 ) T0 ×
p(s) h(t; s) + (H (t; s))1=2 r(s)
with the same M as in Theorem 2.
2 ds ≤ M ¡ ∞
1022
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Hence, we have lim sup [ (t) + (t)] t→∞
1 ≤ lim sup t→∞ H (t; T0 ) 1 + 2
a(s)r(s) K
Z t " T0
1=2
KH (t; s) a(s)r(s)
1=2 v(s)
p(s) (H (t; s))1=2 h(t; s) + r(s)
#2 ds ≤ M ¡ ∞
(39)
with (t) and (t) the same as in the proof of Theorem 2. Making use of assumption (38), we obtain Z t" a(s)r(s) 1 H (t; s) (s) − (t0 ) ≤ lim inf t→∞ H (t; t0 ) t 4K 0 ×
p(s) (H (t; s))1=2 h(t; s) + r(s)
≤ lim inf t→∞
−
1 H (t; t0 )
Z
t
t0
2 # ds
H (t; s) (s) ds
1 1 lim inf 4K t→∞ H (t; t0 )
Z
t
t0
2 p(s) a(s)r(s) h(t; s) + ds: (H (t; s))1=2 r(s)
It follows from (37) and (40) that 2 Z t p(s) 1 (H (t; s))1=2 a(s)r(s) h(t; s) + ds ¡ ∞: lim inf t→∞ H (t; t0 ) t r(s) 0
(40)
(41)
By (41) there exists a sequence {tn }∞ n=1 in the interval (T0 ; ∞) such that lim tn = ∞
t→∞
and 1 lim n→∞ H (tn ; t0 )
Z
tn
t0
= lim inf t→∞
p(s) (H (tn ; s))1=2 a(s)r(s) h(tn ; s) + r(s)
1 H (t; t0 )
Z
t
t0
2 ds
2 p(s) a(s)r(s) h(t; s) + ds ¡ ∞: (42) (H (t; s))1=2 r(s)
Suppose now that (26) holds. With the same argument as in Theorem 2, we conclude that (29) holds. It follows from (39) that there exists a natural number N such that (30) is ful lled. Proceeding as in the proof of Theorem 2, we obtain (34) which contradicts
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1023
(42), and hence (26) fails. Making use of (22), (25) yields Z
∞
T0
2+ (s) ds ≤ a(s)r(s)
Z
∞
T0
v2 (s) ds ¡ ∞ a(s)r(s)
which contradicts assumption (20), so Eq. (1) is oscillatory. This completes the proof of the theorem.
3. Asymptotics of the forced equation In this section, we study the asymptotic behavior of solutions of the forced nonlinear equation with damping [r(t)x0 (t)]0 + p(t)x0 (t) + q(t)f(x(t)) = e(t):
(43)
The main result of this section is the following. Theorem 4. Let the assumptions of Theorem 1 hold and suppose that the function e ∈ C([t0 ; ∞); (−∞; ∞)) satisÿes Z ∞ a(s)|e(s)| ds = N ¡ ∞: (44) Then every solution x(t) of Eq. (43) satisÿes lim inf |x(t)| = 0: t→∞
Proof Let x(t) be a solution of Eq. (43) and suppose that lim inf |x(t)| = C ¿ 0; t→∞
so x(t) is nonoscillatory. Without loss of generality, we may assume that x(t) ¿ 0 on [T0 ; ∞) for some T0 ≥ t0 . Di erentiating the function v(t) de ned by (8), we obtain " e(t) x0 (t) 0 − p(t) v (t) = −2g(t)v(t) + a(t) −q(t) + f(x(t)) f(x(t)) 0
−r(t)f (x(t))
x0 (t) f(x(t))
#
2 +K
−1
0
[r(t)g(t)]
for all t ≥ T0 and thus v0 (t) ≤ −
p(t) 1 K v(t) − v2 (t) + a(t)|e(t)| − (t): r(t) a(t)r(t) f(C)
1024
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
Hence, for all t ≥ T ≥ T0 , we have Z t Z t Z t p(s) v(s) ds H (t; s) (s) ds ≤ − H (t; s)v0 (s) ds − H (t; s) r(s) T T T Z t Z t K 1 2 H (t; s) H (t; s) − v (s) ds + a(s)|e(s)| ds a(s)r(s) f(C) T T and consequently 2 # Z t" p(s) a(s)r(s) h(t; s) + H (t; s) (s) − ds (H (t; s))1=2 4K r(s) T ≤ H (t; T )v(T ) −
Z t " T
KH (t; s) a(s)r(s)
1=2 v(s) +
1 2
a(s)r(s) K
1=2
# 2 p(s) (H (t; s))1=2 ) × h(t; s) + ds r(s) 1 + f(C)
Z T
t
H (t; s)a(s)|e(s)| ds:
(45)
It follows from (44), (45), and property (2) of the function H (t; s) that for every t ≥ T0 2 # Z t" p(s) a(s)r(s) 1=2 h(t; s) + (H (t; s)) H (t; s) (s) − ds 4K r(s) T0 1 H (t; T0 ) ≤ H (t; T0 )|v(T0 )| + f(C)
Z T
t
a(s)|e(s)| ds:
Now the proof proceeds in the same way as in Theorem 1. Example 3. Consider the nonlinear di erential equation 0 1 1 13 1 1 0 x (t) + 2 x0 (t) + 3 (x(t) + x3 (t)) = 6 + 12 ; t t t t t
(46)
where x ∈ (−∞; ∞); and t ∈ [1; ∞). Clearly, f0 (x) exists and f0 (x) = 3x2 + 1 ≥ 1
for all
x ∈ (−∞; ∞);
so assumption (5) holds. Now, let H (t; s) = (t − s)2 ; h(t; s) = 2, and g(t) = −1=t, so that a(t) = t 2 . Then, by a straightforward computation, we obtain 2 ! Z s t−s 1 t (t − s)2 − 2+ ds lim sup 2 s 4 s t→∞ t T
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
= lim sup t→∞
= lim sup t→∞
1 t2
1025
(t + s)2 (t − s)2 − ds s 4s T 17 5 3 3 ln t − + − 2 = ∞; 4 8 2t 8t Z t
so assumption (6) holds. Thus, by Theorem 4, we conclude that all solutions of Eq. (46) satisfy lim inf t→∞ |x(t)| = 0. Observe that x(t) = t −3 is such a solution.
4. Discussion and ÿnal remarks In this section, we would like to establish connections between our results and those known in the literature. It is natural to expect that, Eqs. (2) and (3) being particular cases of Eq. (1), Theorems 1–3 will coincide with theorems of Philos [17] (for Eq. (2)) and Li [16] (for Eq. (3)). Indeed, with f(x) = x and p(t) ≡ 0, our Theorems 1–3 reduce correspondingly to Theorems 2:1; 2:6 and 2:8 obtained by Li [16] with the following correction: condition (C1 ) in the latter three theorems should be omitted as a super uous one (see [18] for the details). If, in addition, we set g(t) ≡ 0 and, consequently, a(t) ≡ 1, then Theorems 1–3 reduce Theorems to 1–3 obtained by Philos [17]. Furthermore, we note that with g(t) ≡ 0 (and hence with a(t) ≡ 1) and r(t) ≡ 1 Corollary 3 reduces to the results of Yan [24; Corollary 3] and Yeh [25; Theorem 2]; and with p(t) ≡ 0, r(t) ≡ 1, and g(t) ≡ 0, it reduces to the Kamenev’s criterion [14]. In order to make a comparison between our results and those derived very recently for nonlinear equations with damping by Grace [6; Theorems 6–8], let us examine assumptions (C7 )–(C9 ) of Theorem B given in the Introduction and the corresponding assumptions (18)–(20) of our Theorem 2. Clearly, (C7 ) coincides R t with (18). Condition (C8 ) is more restrictive than (19) since letting a(t) = exp(−2 g(u) du) in (C8 ) we have 2 Z t 1 p(s) 1=2 (H (t; s)) a(s)r(s) h(t; s) + ds lim sup r(s) t→∞ H (t; t0 ) t0 " 2 Z t 1 p(s) (H (t; s))1=2 a(s)r(s) h(t; s) + ≤ lim sup r(s) t→∞ H (t; t0 ) t0 1=2
+4g(s)(H (t; s))
p(s) h(t; s) + (H (t; s))1=2 r(s)
# 2
+ 4g (s)H (t; s) ds:
Hence, if (C8 ) holds, then (19) always holds, but the converse is not true. Naturally, the di erent structure of assumptions (C9 ) and (20) makes their comparison a little bit complicated, though a simple but lengthy computation shows that,
1026
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
at least for the case when h(t; s)g(s)r(s) − [r(s)g(s)]0 ≥ 0; if (C8 ) holds, then so does (20). In summary, we may conclude that the conditions of Theorem 2 are less restrictive than those of Theorem 6 due to Grace [6]. The same reasoning is also valid for other results presented in this paper. We also stress that our results do not involve any particular assumptions on the damping coecient p(t) and thus they are more exible than the recent results due to Grace [6] and Grace and Lalli [8]. We conclude with the following remarks which also indicate the directions for the possible further investigation. Remark 1. The above results hold true if we replace condition (5) with the following one: f(x) ≥ K ¿0 x
for x 6= 0;
but the function q(t) should be nonnegative (we refer to [15] for details): Remark 2. Some of the results of this paper can be extended to the more general equations [r(t) (x(t))x0 (t)]0 + p(t)x0 (t) + q(t)f(x(t)) = 0 and [r(t) (x(t))x0 (t)]0 + p(t)k(t; x(t); x0 (t)) + q(t)f(x(t)) = 0; where is a positive continuous function and k is a nonnegative continuous function; as well as to the di erential equation with a deviating argument [r(t) (x(t))x0 (t)]0 + p(t)x0 (t) + q(t)f(x[(t)]) = 0: Remark 3. The results in this paper are presented in a form with a high degree of generality, and thus they give many possibilities for oscillation criteria with an appropriate choice of the functions H and h. For example; there are interesting perspectives to apply Theorems 1–4 with n−1 Z t dz ; (t; s) ∈ D; H (t; s) = s (z) where n ¿ 2 is a constant and is a positive continuous function on [t0 ; ∞) such that Z ∞ dz =∞ (z) t0 (an important case to be considered is (z) = z with real).
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
1027
Acknowledgements This research was done during the author’s visit to the University of Florence which he thanks for its extremely warm hospitality with a special gratitude to Professors Roberto Conti and Gabriele Villari for permanent support and encouragement. The author is indebted to the referee for a careful reading of the manuscript and for kindly prompting improvements in presentation.
References [1] J.W. Baker, Oscillation theorems for a second order damped nonlinear di erential equation, SIAM J. Math. Anal. 25 (1973) 37– 40. [2] L.E. Bobisud, Oscillation of solutions of damped nonlinear di erential equations, SIAM J. Math. Anal. 18 (1970) 601– 606. [3] G.J. Butler, The oscillatory behavior of a second order nonlinear di erential equation with damping, J. Math. Anal. Appl. 57 (1977) 273–289. [4] S.R. Grace, Oscillation theorems for second order nonlinear di erential equations with damping, Math. Nachr. 141 (1989) 117–127. [5] S.R. Grace, Oscillation criteria for second order nonlinear di erential equations with damping, J. Austral. Math. Soc. Ser. A 49 (1990) 43–54. [6] S.R. Grace, Oscillation theorems for nonlinear di erential equations of second order, J. Math. Anal. Appl. 171 (1992) 220 –241. [7] S.R. Grace, B.S. Lalli, Oscillation theorems for certain second order perturbed nonlinear di erential equations, J. Math. Anal. Appl. 77 (1980) 205–214. [8] S.R. Grace, B.S. Lalli, Integral averaging technique for the oscillation of second order nonlinear di erential equations, J. Math. Anal. Appl. 149 (1990) 277–311. [9] S.R. Grace, B.S. Lalli, Oscillation theorems for second order superlinear di erential equations with damping, J. Austral. Math. Soc. Ser. A 53 (1992) 156–165. [10] S.R. Grace, B.S. Lalli, C.C. Yeh, Oscillation theorems for nonlinear second order di erential equations with a nonlinear damping term, SIAM J. Math. Anal. 15 (1984) 1082–1093. [11] S.R. Grace, B.S. Lalli, C.C. Yeh, Addendum: oscillation theorems for nonlinear second order di erential equations with a nonlinear damping term, SIAM J. Math. Anal. 19 (1988) 1252–1253. [12] B.J. Harris, On the oscillation of solutions of linear di erential equations, Mathematika 31 (1984) 214 –226. [13] P. Hartman, On nonoscillatory linear di erential equations of second order, Amer. J. Math. 74 (1952) 389–400. [14] I.V. Kamenev, An integral criterion for oscillation of linear di erential equations, Mat. Zametki 23 (1978) 249–251. [15] M. Kirane, Yu.V. Rogovchenko, On oscillation of nonlinear second order di erential equation with damping term, preprint. [16] H.J. Li, Oscillation criteria for second order linear di erential equations, J. Math. Anal. Appl. 194 (1995) 217–234. [17] Ch.G. Philos, Oscillation theorems for linear di erential equations of second order, Arch. Math. 53 (1989) 483–492. [18] Yu.V. Rogovchenko, Note on ‘Oscillation criteria for second order linear di erential equations’, J. Math. Anal. Appl. 203 (1996) 560–563. [19] Yu.V. Rogovchenko, Oscillation criteria for second order nonlinear perturbed di erential equations, J. Math. Anal. Appl. 215 (1997) 334–357.
1028
Y.V. Rogovchenko / Nonlinear Analysis 41 (2000) 1005 – 1028
[20] C.A. Swanson, Comparison and Oscillation Theory of Linear Di erential Equations, Academic Press, New York, 1968. [21] A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949) 115–117. [22] J.S.W. Wong, A second order nonlinear oscillation, Funkcial. Ekvac. 11 (1968) 207–234. [23] J. Yan, A note on an oscillation criterion for an equation with damped term, Proc. Amer. Math. Soc. 90 (1984) 277–280. [24] J. Yan, Oscillation theorems for second order linear di erential equations with damping, Proc. Amer. Math. Soc. 98 (1986) 276–282. [25] C.C. Yeh, Oscillation theorems for nonlinear second order di erential equations with damping term, Proc. Amer. Math. Soc. 84 (1982) 397–402.