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Some oscillation results for second-order nonlinear delay dynamic equations
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Some oscillation results for second-order nonlinear delay dynamic equations Chenghui Zhang ∗ , Tongxing Li School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, PR China
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info
Article history: Received 15 April 2013 Received in revised form 29 May 2013 Accepted 29 May 2013 Keywords: Oscillation Delay dynamic equation Second-order nonlinear equation Time scale
abstract This paper is concerned with oscillatory behavior of a class of second-order delay dynamic equations on a time scale. Two new oscillation criteria are presented that improve some known results in the literature. The results obtained are sharp even for the second-order ordinary differential equations. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we are concerned with oscillation of a second-order nonlinear delay dynamic equation
rx∆
∆
(t ) + q(t )f (x(δ(t ))) = 0,
(1.1)
where t ∈ [t0 , ∞)T := [t0 , ∞) ∩ T, and (H1 ) r , q ∈ Crd ([t0 , ∞)T , R), r (t ) > 0, q(t ) > 0; (H2 ) δ ∈ Crd ([t0 , ∞)T , T), δ(t ) ≤ t , limt →∞ δ(t ) = ∞; (H3 ) f ∈ C(R, R) such that yf (y) > 0, f (y)/y ≥ K > 0 for y ̸= 0, where K is a constant. Throughout this paper, we assume that solutions of (1.1) exist for any t ∈ [t0 , ∞)T . A solution x of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, we call it nonoscillatory. Dynamic equation (1.1) is said to be oscillatory if all its solutions oscillate. A time scale T is an arbitrary nonempty closed subset of the real numbers R. Since we are interested in oscillatory behavior, we suppose that the time scale under consideration is not bounded above and is a time scale interval of the form [t0 , ∞)T . On any time scale we define the forward and backward jump operators by σ (t ) := inf{s ∈ T|s > t } and ρ(t ) := sup{s ∈ T|s < t }, where inf ∅ := sup T and sup ∅ := inf T, ∅ denotes the empty set. A point t ∈ T is said to be left-dense if ρ(t ) = t and t > inf T, right-dense if σ (t ) = t and t < sup T, left-scattered if ρ(t ) < t, and right-scattered if σ (t ) > t. Points that are right-scattered and left-scattered at the same time are called isolated. There are many time scales that consist of only isolated points; see, for example, T = Z, T = hZ, T = qN , and T = 2N , etc. The graininess function µ : T → [0, ∞) is defined by µ(t ) := σ (t ) − t, and for any function f : T → R the notation f σ (t ) := f (σ (t )). Some concepts related to the notion of time scales; see Bohner and Peterson [1]. In recent years, there has been an increasing interest in obtaining sufficient conditions for oscillatory or nonoscillatory behavior of different classes of dynamic equations on time scales, we refer the reader to [1–24]. In what follows, we present
∗
Corresponding author. Tel.: +86 0531 88395047. E-mail addresses: [email protected] (C. Zhang), [email protected] (T. Li).
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some details that motivate the contents of this paper. Regarding oscillation of second-order delay equations, Zhang and Zhu [22] studied a dynamic equation x∆∆ (t ) + q(t )f (x(t − δ)) = 0. Agarwal et al. [2] considered an equation x∆∆ (t ) + q(t )x(δ(t )) = 0. Şahiner [18] investigated a nonlinear equation x∆∆ (t ) + q(t )f (x(δ(t ))) = 0. On the basis of condition ∞
r (t )
t0
t
1
q(s)1s1t = ∞, t0
Erbe et al. [13] obtained a sufficient condition which ensures that the solution x of the delay dynamic equation (1.1) is either oscillatory or satisfies limt →∞ x(t ) = 0. Zhang [24] established some oscillation criteria for (1.1) in the case where ∞
r (t )
t0
t
1
q(s) t0
∞
s
1u 1s1t = ∞. r ( u)
It is well known (see [20]) that the Euler differential equations x′′ (t ) +
q0
x(t ) = 0,
t2
q0 > 0 is a constant
(1.2)
and
(t 2 x′ (t ))′ + q0 x(t ) = 0,
q0 > 0 is a constant
(1.3)
are oscillatory if q0 > 1/4. However, results obtained in [13,24] cannot give this conclusion for (1.3). The natural question now is: Can one obtain new oscillation results for (1.1), which may cover (1.2) or (1.3)? The aim of this paper is to give an affirmative answer to this question. The results reported improve those by [13,24]. 2. Main results In this section, we will establish two new Philos-type oscillation criteria for (1.1). For the sake of convenience, we use the notation
D ≡ {(t , s) : t0 ≤ s ≤ t , t , s ∈ [t0 , ∞)T } and D0 ≡ {(t , s) : t0 ≤ s < t , t , s ∈ [t0 , ∞)T }. All functional inequalities considered in this section are assumed to hold eventually, that is, they are satisfied for all t large enough. Theorem 2.1. Assume (H1 )–(H3 ) and
∞ t0
1t r (t )
= ∞.
(2.1)
Suppose further that there exist two functions η, a ∈ C1rd ([t0 , ∞)T , R) such that η(t ) > 0, a(t ) ≥ 0, and there exists a function H ∈ Crd (D, R) such that H (t , t ) = 0,
H (t , s) > 0,
t ≥ t0 ;
t > s ≥ t0 ,
(2.2)
(t , s) on D0 with respect to the second variable and satisfies (H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))2 H (σ (t ), σ (s))Q (s) − 1s = ∞ (2.3) 4H (σ (t ), σ (s))A(s)
and H has a nonpositive rd-continuous ∆-partial derivative H lim sup t →∞
∆s
t
1 H (σ (t ), t2 )
t2
for all sufficiently large t1 and for some t2 ≥ t1 , where
δ(s)
σ
1v
t1
r (v)
t1
r (v)
Q (s) := η (s) Kq(s) σ (s)
1v
s 1v η σ ( s) t1 r (v) A(s) := , r (s)η2 (s) σ (s) 1v t1
Then (1.1) is oscillatory.
r (v)
s
1v
t1 r (v)
+ r (s)a (s) σ (s) 2
t1
1v
− (r (s)a(s))
∆
,
r (v)
s 1v η∆ (s) 2ησ (s)a(s) t1 r (v) B(s) := + σ (s) 1v . η(s) η(s) t1
r (v)
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Proof. Assume that (1.1) has a nonoscillatory solution x on [t0 , ∞)T . Without loss of generality, suppose that it is an eventually positive solution. From (1.1) and (2.1), we have (see also [13, (2.7)]) x∆ (t ) > 0
and (rx∆ )∆ (t ) < 0
for t ∈ [t1 , ∞)T , where t1 ∈ [t0 , ∞)T is large enough. Hence we have ∆
x (t )r (t )
1s
t
r (s)
t1
≤ x(t ).
(2.4)
Define the function ω by
ω(t ) := η(t )
r (t )x∆ (t ) x(t )
+ r (t )a(t ) ,
t ≥ t1 .
(2.5)
Then ω(t ) > 0 and we have
ω∆ = η∆
∆ ∆ rx + ra + ησ + ra
rx∆ x
x
∆ η rx∆ ω + ησ (ra)∆ + ησ η x ∆ ∆ η∆ ( rx ) x − r ( x∆ ) 2 = ω + ησ (ra)∆ + ησ η xxσ ∆ ∆ ∆ ∆ 2 η (rx ) σ r (x ) = ω + ησ (ra)∆ + ησ − η η xσ xxσ 2 ∆ ∆ η∆ x∆ x σ ∆ σ (rx ) σ = ω + η (ra) + η −η r . η xσ x xσ ∆
=
(2.6)
By virtue of (2.4), we have
x(t )
∆ ≤ 0,
1s t1 r (s)
t
which yields x(δ(t )) xσ ( t )
δ(t ) t1
1s r (s)
t1
1s r (s)
≥ σ (t )
and
x(t ) xσ ( t )
1s r (s)
t t
≥ σ 1(t ) t1
1s r (s)
.
(2.7)
On the other hand, we have by (2.5) that
x∆
2
x
ω = −a rη
2
ω = rη
2
+ a2 − 2
ωa . rη
(2.8)
Putting (2.7) and (2.8) into (2.6) and using (1.1), we have
δ(t ) σ
∆
t1
1s r (s)
t1
1s r (s)
ω ≤ −Kqη σ (t ) = −Q +
−η
η∆ 2η σ a + η η
1s t1 r (s)
t
σ
2
ra σ (t ) t1
1s t1 r (s) σ (t ) 1s t1 r (s)
t
1s r (s)
− (ra)
∆
t 1s t 1s η∆ 2ησ a t1 r (s) ησ t1 r (s) 2 + + σ (t ) 1s ω − 2 σ (t ) 1s ω η η rη
t1
1s ησ t1 r (s) r η2 σ (t ) 1s t1 r (s)
r (s)
t1
r (s)
t
ω−
ω2
= −Q + Bω − Aω2 . Hence we have
σ (t )
H (σ (t ), σ (s))ω∆ (s)1s ≤ −
t2
σ (t )
t2
H (σ (t ), σ (s))Q (s)1s +
σ (t )
H (σ (t ), σ (s))B(s)ω(s)1s
t2
σ (t )
− t2
H (σ (t ), σ (s))A(s)ω2 (s)1s.
(2.9)
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By using integration by parts, we obtain σ (t )
σ (t )
H (σ (t ), σ (s))ω∆ (s)1s = −H (σ (t ), t2 )ω(t2 ) −
H ∆s (σ (t ), s)ω(s)1s.
(2.10)
t2
t2
Substituting (2.10) into (2.9), we have σ (t )
H (σ (t ), σ (s))Q (s)1s ≤ H (σ (t ), t2 )ω(t2 ) +
σ (t )
(H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))ω(s)1s
t2
t2
σ (t )
−
H (σ (t ), σ (s))A(s)ω2 (s)1s.
t2
Note that σ (t )
∆s
(H (σ (t ), σ (s))B(s) + H (σ (t ), s))ω(s)1s −
σ (t )
H (σ (t ), σ (s))A(s)ω2 (s)1s
t2
t2 t
∆s
(H (σ (t ), σ (s))B(s) + H (σ (t ), s))ω(s)1s −
=
t2
t
H (σ (t ), σ (s))A(s)ω2 (s)1s t2
+ µ(t )(H (σ (t ), σ (t ))B(t ) + H ∆s (σ (t ), t ))ω(t ) − µ(t )H (σ (t ), σ (t ))A(t )ω2 (t ). Using H (σ (t ), σ (t )) = 0, µ(t )H ∆s (σ (t ), t )ω(t ) ≤ 0, and the method of completing the square, we have σ (t )
(H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))ω(s)1s −
t2
σ (t )
H (σ (t ), σ (s))A(s)ω2 (s)1s
t2 t
≤
(H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))ω(s)1s −
t2
t
H (σ (t ), σ (s))A(s)ω2 (s)1s t2
(H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))2 1s. 4H (σ (t ), σ (s))A(s)
t
≤ t2
Thus, we get by H (σ (t ), σ (t )) = 0 that σ (t )
(H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))2 H (σ (t ), σ (s))Q (s)1s − 1s H (σ (t ), t2 ) 4H (σ (t ), σ (s))A(s) t2 t2 t 1 (H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))2 = H (σ (t ), σ (s))Q (s) − 1s ≤ ω(t2 ), H (σ (t ), t2 ) t2 4H (σ (t ), σ (s))A(s)
1
which contradicts (2.3). The proof is complete.
t
Example 2.1. Assume T = R and consider the second-order differential equation (1.2). In this equation, σ (t ) = t , K = 1, r (t ) = 1, q(t ) = q0 /t 2 , and δ(t ) = t. Let H (t , s) = (t − s)2 , η(t ) = t, and a(t ) = 0. Then we have lim sup t →∞
t
1 H (t , t2 )
= lim sup t →∞
t2
1
(t − t2 )2
(H (t , s)B(s) + H ∆s (t , s))2 1s 4H (t , s)A(s) 2 t s t −s s − 2 (t − s)2 q0 − ds = ∞,
H (t , s)Q (s) −
t2
s
4
if q0 > 1/4. Hence by Theorem 2.1, (1.2) is oscillatory when q0 > 1/4. In the following, we give an oscillation criterion for (1.1) when
∞ t0
1t r (t )
< ∞.
(2.11)
Theorem 2.2. Let (2.1) be replaced by (2.11) and all other assumptions of Theorem 2.1 hold. If there exist two functions ζ , b ∈ C1rd ([t0 , ∞)T , R) such that ζ (t ) > 0, b(t ) ≥
1 r (t )R(t )
,
(2.12)
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where R(t ) :=
1s , r (s)
∞
t
and there exists a function G ∈ Crd (D, R) such that G(t , t ) = 0,
G(t , s) > 0,
t ≥ t0 ;
t > s ≥ t0 ,
(2.13)
and G has a nonpositive rd-continuous ∆-partial derivative G∆s (t , s) on D0 with respect to the second variable and satisfies
t
G(σ (t ), σ (s))Q∗ (s) −
t0
lim sup
(G(σ (t ),σ (s))B∗ (s)+G∆s (σ (t ),s))2 4G(σ (t ),σ (s))A∗ (s)
G(σ (t ), t0 )
t →∞
1s
= ∞,
(2.14)
where
Q∗ (s) := ζ σ (s) Kq(s) + r (s)b2 (s) − (r (s)b(s))∆ , ζ σ (s) , r (s)ζ 2 (s)
A∗ (s) :=
B∗ (s) :=
ζ ∆ (s) 2ζ σ (s)b(s) + , ζ (s) ζ (s)
then (1.1) is oscillatory. Proof. Assume that (1.1) has a nonoscillatory solution x on [t0 , ∞)T . Without loss of generality, suppose that it is an eventually positive solution. From (1.1), there exists t1 ∈ [t0 , ∞)T such that x∆ (t ) > 0 or x∆ (t ) < 0 for t ∈ [t1 , ∞)T . The proof of the case where x∆ (t ) > 0 is the same as that of Theorem 2.1, and we can get a contradiction to (2.3). Assume now that x∆ (t ) < 0. We have x∆ (s) ≤
r (t ) r (s)
x∆ (t ),
s ≥ t ≥ t1 .
Hence we get x∆ ( t ) x(t )
≥−
1 r (t )R(t )
.
Define the function ω by
ω(t ) := ζ (t )
r (t )x∆ (t ) x( t )
+ r (t )b(t ) ,
t ≥ t1 .
Then ω(t ) ≥ 0 due to (2.12),
ω∆ =
(rx∆ )∆ ζ∆ − ζσr ω + ζ σ (rb)∆ + ζ σ ζ xσ
x∆ x
2
x xσ
,
(2.15)
and
x∆ x
2
ω = −b rζ
2
ω = rζ
2
+ b2 − 2
ωb . rζ
(2.16)
Note that x/xσ ≥ 1. Using (1.1), (2.15), and (2.16), we have
∆ 2 ζ 2ζ σ b ζσ σ ∆ − ζ rb − ( rb ) + + ω − 2 ω2 σ x ζ ζ rζ ∆ σ σ ζ 2 ζ b ζ ≤ −Kqζ σ − ζ σ rb2 − (rb)∆ + + ω − 2 ω2 ζ ζ rζ ∆ ζ 2ζ σ b ζσ 2 = −Q∗ + + ω − 2ω ζ ζ rζ
ω∆ ≤ ζ σ
(rx∆ )∆
= −Q∗ + B∗ ω − A∗ ω2 . The rest of the proof is similar to that of Theorem 2.1, and we can get a contradiction to (2.14). This completes the proof.
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Example 2.2. Assume T = R and consider the second-order differential equation (1.3). In this equation, σ (t ) = t , K = 1, r (t ) = t 2 , q(t ) = q0 , and δ(t ) = t. Let H (t , s) = (t − s)2 , η(t ) = 1/t, and a(t ) = 1/t. Then we have lim sup t →∞
t
1 H (t , t2 )
= lim sup t →∞
t2
1
(t − t2 )2
(H (t , s)B(s) + H ∆s (t , s))2 1s 4H (t , s)A(s) 2 t s t −s s − 2 (t − s)2 q0 − ds = ∞, if q0 > 1/4.
H (t , s)Q (s) −
t2
s
4
Let G(t , s) = (t − s)2 , ζ (t ) = 1/t, and b(t ) = 1/t. Then we obtain
(G(t , s)B∗ (s) + G∆s (t , s))2 G(t , s)Q∗ (s) − 1s lim sup 4G(t , s)A∗ (s) t →∞ G(t , t0 ) t0 2 t s t −s s − 2 1 (t − s)2 = lim sup q0 − ds = ∞, if q0 > 1/4. 2 s 4 t →∞ (t − t0 ) t0 1
t
Hence by Theorem 2.2, (1.3) is oscillatory when q0 > 1/4. Acknowledgments The authors express their sincere gratitude to the anonymous referees for careful reading of the original manuscript and useful comments that helped to improve presentation of results and accentuate important details. This research is supported by National Key Basic Research Program of China (2013CB035604) and NNSF of P.R. China (Grant Nos. 61034007, 51277116, 51107069). References [1] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. [2] R.P. Agarwal, M. Bohner, S.H. Saker, Oscillation of second order delay dynamic equations, Can. Appl. Math. Q. 13 (2005) 1–17. [3] R.P. Agarwal, D. O’Regan, S.H. Saker, Oscillation criteria for second-order nonlinear neutral delay dynamic equations, J. Math. Anal. Appl. 300 (2004) 203–217. [4] R.P. Agarwal, D. O’Regan, S.H. Saker, Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales, Appl. Anal. 86 (2007) 1–17. [5] R.P. Agarwal, D. O’Regan, S.H. Saker, Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales, Acta Math. Sin. 24 (2008) 1409–1432. [6] E. Akın-Bohner, M. Bohner, S.H. Saker, Oscillation criteria for a certain class of second order Emden–Fowler dynamic equations, Electron. Trans. Numer. Anal. 27 (2007) 1–12. [7] M. Bohner, S.H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain J. Math. 34 (2004) 1239–1254. [8] M. Bohner, S.H. Saker, Oscillation criteria for perturbed nonlinear dynamic equations, Math. Comput. Modelling 40 (2004) 249–260. [9] E. Braverman, B. Karpuz, Nonoscillation of second-order dynamic equations with several delays, Abstr. Appl. Anal. 2011 (2011) 1–34. [10] L. Erbe, B. Karpuz, A. Peterson, Kamenev-type oscillation criteria for higher-order neutral delay dynamic equations, Int. J. Difference Equ. 6 (2011) 1–16. [11] L. Erbe, A. Peterson, S.H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. Lond. Math. Soc. 67 (2003) 701–714. [12] L. Erbe, A. Peterson, S.H. Saker, Kamenev-type oscillation criteria for second-order linear delay dynamic equations, Dynam. Systems Appl. 15 (2006) 65–78. [13] L. Erbe, A. Peterson, S.H. Saker, Oscillation criteria for second-order nonlinear delay dynamic equations, J. Math. Anal. Appl. 333 (2007) 505–522. [14] S.R. Grace, R.P. Agarwal, M. Bohner, D. O’Regan, Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3463–3471. [15] S.R. Grace, M. Bohner, R.P. Agarwal, On the oscillation of second-order half-linear dynamic equations, J. Difference Equ. Appl. 15 (2009) 451–460. [16] Z. Han, T. Li, S. Sun, C. Zhang, Oscillation for second-order nonlinear delay dynamic equations on time scales, Adv. Difference Equ. 2009 (2009) 1–13. [17] T. Li, Z. Han, S. Sun, D. Yang, Existence of nonoscillatory solutions to second-order neutral delay dynamic equations on time scales, Adv. Difference Equ. 2009 (2009) 1–10. [18] Y. Şahiner, Oscillation of second-order delay differential equations on time scales, Nonlinear Anal. 63 (2005) 1073–1080. [19] S.H. Saker, Oscillation of nonlinear dynamic equations on time scales, Appl. Math. Comput. 148 (2004) 81–91. [20] S.H. Saker, Oscillation Theory of Dynamic Equations on Time Scales, Lambert Academic Publishing, Germany, 2010. [21] S.H. Saker, D. O’Regan, New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 423–434. [22] B.G. Zhang, S.L. Zhu, Oscillation of second-order nonlinear delay dynamic equations on time scales, Comput. Math. Appl. 49 (2005) 599–609. [23] C. Zhang, T. Li, R.P. Agarwal, M. Bohner, Oscillation results for fourth-order nonlinear dynamic equations, Appl. Math. Lett. 25 (2012) 2058–2065. [24] Q. Zhang, Oscillation of second-order half-linear delay dynamic equations with damping on time scales, J. Comput. Appl. Math. 235 (2011) 1180–1188.
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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Some oscillation results for second-order nonlinear delay dynamic equations Chenghui Zhang ∗ , Tongxing Li School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, PR China
article
info
Article history: Received 15 April 2013 Received in revised form 29 May 2013 Accepted 29 May 2013 Keywords: Oscillation Delay dynamic equation Second-order nonlinear equation Time scale
abstract This paper is concerned with oscillatory behavior of a class of second-order delay dynamic equations on a time scale. Two new oscillation criteria are presented that improve some known results in the literature. The results obtained are sharp even for the second-order ordinary differential equations. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we are concerned with oscillation of a second-order nonlinear delay dynamic equation
rx∆
∆
(t ) + q(t )f (x(δ(t ))) = 0,
(1.1)
where t ∈ [t0 , ∞)T := [t0 , ∞) ∩ T, and (H1 ) r , q ∈ Crd ([t0 , ∞)T , R), r (t ) > 0, q(t ) > 0; (H2 ) δ ∈ Crd ([t0 , ∞)T , T), δ(t ) ≤ t , limt →∞ δ(t ) = ∞; (H3 ) f ∈ C(R, R) such that yf (y) > 0, f (y)/y ≥ K > 0 for y ̸= 0, where K is a constant. Throughout this paper, we assume that solutions of (1.1) exist for any t ∈ [t0 , ∞)T . A solution x of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, we call it nonoscillatory. Dynamic equation (1.1) is said to be oscillatory if all its solutions oscillate. A time scale T is an arbitrary nonempty closed subset of the real numbers R. Since we are interested in oscillatory behavior, we suppose that the time scale under consideration is not bounded above and is a time scale interval of the form [t0 , ∞)T . On any time scale we define the forward and backward jump operators by σ (t ) := inf{s ∈ T|s > t } and ρ(t ) := sup{s ∈ T|s < t }, where inf ∅ := sup T and sup ∅ := inf T, ∅ denotes the empty set. A point t ∈ T is said to be left-dense if ρ(t ) = t and t > inf T, right-dense if σ (t ) = t and t < sup T, left-scattered if ρ(t ) < t, and right-scattered if σ (t ) > t. Points that are right-scattered and left-scattered at the same time are called isolated. There are many time scales that consist of only isolated points; see, for example, T = Z, T = hZ, T = qN , and T = 2N , etc. The graininess function µ : T → [0, ∞) is defined by µ(t ) := σ (t ) − t, and for any function f : T → R the notation f σ (t ) := f (σ (t )). Some concepts related to the notion of time scales; see Bohner and Peterson [1]. In recent years, there has been an increasing interest in obtaining sufficient conditions for oscillatory or nonoscillatory behavior of different classes of dynamic equations on time scales, we refer the reader to [1–24]. In what follows, we present
∗
Corresponding author. Tel.: +86 0531 88395047. E-mail addresses: [email protected] (C. Zhang), [email protected] (T. Li).
0893-9659/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.aml.2013.05.014
2
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)
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some details that motivate the contents of this paper. Regarding oscillation of second-order delay equations, Zhang and Zhu [22] studied a dynamic equation x∆∆ (t ) + q(t )f (x(t − δ)) = 0. Agarwal et al. [2] considered an equation x∆∆ (t ) + q(t )x(δ(t )) = 0. Şahiner [18] investigated a nonlinear equation x∆∆ (t ) + q(t )f (x(δ(t ))) = 0. On the basis of condition ∞
r (t )
t0
t
1
q(s)1s1t = ∞, t0
Erbe et al. [13] obtained a sufficient condition which ensures that the solution x of the delay dynamic equation (1.1) is either oscillatory or satisfies limt →∞ x(t ) = 0. Zhang [24] established some oscillation criteria for (1.1) in the case where ∞
r (t )
t0
t
1
q(s) t0
∞
s
1u 1s1t = ∞. r ( u)
It is well known (see [20]) that the Euler differential equations x′′ (t ) +
q0
x(t ) = 0,
t2
q0 > 0 is a constant
(1.2)
and
(t 2 x′ (t ))′ + q0 x(t ) = 0,
q0 > 0 is a constant
(1.3)
are oscillatory if q0 > 1/4. However, results obtained in [13,24] cannot give this conclusion for (1.3). The natural question now is: Can one obtain new oscillation results for (1.1), which may cover (1.2) or (1.3)? The aim of this paper is to give an affirmative answer to this question. The results reported improve those by [13,24]. 2. Main results In this section, we will establish two new Philos-type oscillation criteria for (1.1). For the sake of convenience, we use the notation
D ≡ {(t , s) : t0 ≤ s ≤ t , t , s ∈ [t0 , ∞)T } and D0 ≡ {(t , s) : t0 ≤ s < t , t , s ∈ [t0 , ∞)T }. All functional inequalities considered in this section are assumed to hold eventually, that is, they are satisfied for all t large enough. Theorem 2.1. Assume (H1 )–(H3 ) and
∞ t0
1t r (t )
= ∞.
(2.1)
Suppose further that there exist two functions η, a ∈ C1rd ([t0 , ∞)T , R) such that η(t ) > 0, a(t ) ≥ 0, and there exists a function H ∈ Crd (D, R) such that H (t , t ) = 0,
H (t , s) > 0,
t ≥ t0 ;
t > s ≥ t0 ,
(2.2)
(t , s) on D0 with respect to the second variable and satisfies (H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))2 H (σ (t ), σ (s))Q (s) − 1s = ∞ (2.3) 4H (σ (t ), σ (s))A(s)
and H has a nonpositive rd-continuous ∆-partial derivative H lim sup t →∞
∆s
t
1 H (σ (t ), t2 )
t2
for all sufficiently large t1 and for some t2 ≥ t1 , where
δ(s)
σ
1v
t1
r (v)
t1
r (v)
Q (s) := η (s) Kq(s) σ (s)
1v
s 1v η σ ( s) t1 r (v) A(s) := , r (s)η2 (s) σ (s) 1v t1
Then (1.1) is oscillatory.
r (v)
s
1v
t1 r (v)
+ r (s)a (s) σ (s) 2
t1
1v
− (r (s)a(s))
∆
,
r (v)
s 1v η∆ (s) 2ησ (s)a(s) t1 r (v) B(s) := + σ (s) 1v . η(s) η(s) t1
r (v)
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Proof. Assume that (1.1) has a nonoscillatory solution x on [t0 , ∞)T . Without loss of generality, suppose that it is an eventually positive solution. From (1.1) and (2.1), we have (see also [13, (2.7)]) x∆ (t ) > 0
and (rx∆ )∆ (t ) < 0
for t ∈ [t1 , ∞)T , where t1 ∈ [t0 , ∞)T is large enough. Hence we have ∆
x (t )r (t )
1s
t
r (s)
t1
≤ x(t ).
(2.4)
Define the function ω by
ω(t ) := η(t )
r (t )x∆ (t ) x(t )
+ r (t )a(t ) ,
t ≥ t1 .
(2.5)
Then ω(t ) > 0 and we have
ω∆ = η∆
∆ ∆ rx + ra + ησ + ra
rx∆ x
x
∆ η rx∆ ω + ησ (ra)∆ + ησ η x ∆ ∆ η∆ ( rx ) x − r ( x∆ ) 2 = ω + ησ (ra)∆ + ησ η xxσ ∆ ∆ ∆ ∆ 2 η (rx ) σ r (x ) = ω + ησ (ra)∆ + ησ − η η xσ xxσ 2 ∆ ∆ η∆ x∆ x σ ∆ σ (rx ) σ = ω + η (ra) + η −η r . η xσ x xσ ∆
=
(2.6)
By virtue of (2.4), we have
x(t )
∆ ≤ 0,
1s t1 r (s)
t
which yields x(δ(t )) xσ ( t )
δ(t ) t1
1s r (s)
t1
1s r (s)
≥ σ (t )
and
x(t ) xσ ( t )
1s r (s)
t t
≥ σ 1(t ) t1
1s r (s)
.
(2.7)
On the other hand, we have by (2.5) that
x∆
2
x
ω = −a rη
2
ω = rη
2
+ a2 − 2
ωa . rη
(2.8)
Putting (2.7) and (2.8) into (2.6) and using (1.1), we have
δ(t ) σ
∆
t1
1s r (s)
t1
1s r (s)
ω ≤ −Kqη σ (t ) = −Q +
−η
η∆ 2η σ a + η η
1s t1 r (s)
t
σ
2
ra σ (t ) t1
1s t1 r (s) σ (t ) 1s t1 r (s)
t
1s r (s)
− (ra)
∆
t 1s t 1s η∆ 2ησ a t1 r (s) ησ t1 r (s) 2 + + σ (t ) 1s ω − 2 σ (t ) 1s ω η η rη
t1
1s ησ t1 r (s) r η2 σ (t ) 1s t1 r (s)
r (s)
t1
r (s)
t
ω−
ω2
= −Q + Bω − Aω2 . Hence we have
σ (t )
H (σ (t ), σ (s))ω∆ (s)1s ≤ −
t2
σ (t )
t2
H (σ (t ), σ (s))Q (s)1s +
σ (t )
H (σ (t ), σ (s))B(s)ω(s)1s
t2
σ (t )
− t2
H (σ (t ), σ (s))A(s)ω2 (s)1s.
(2.9)
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By using integration by parts, we obtain σ (t )
σ (t )
H (σ (t ), σ (s))ω∆ (s)1s = −H (σ (t ), t2 )ω(t2 ) −
H ∆s (σ (t ), s)ω(s)1s.
(2.10)
t2
t2
Substituting (2.10) into (2.9), we have σ (t )
H (σ (t ), σ (s))Q (s)1s ≤ H (σ (t ), t2 )ω(t2 ) +
σ (t )
(H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))ω(s)1s
t2
t2
σ (t )
−
H (σ (t ), σ (s))A(s)ω2 (s)1s.
t2
Note that σ (t )
∆s
(H (σ (t ), σ (s))B(s) + H (σ (t ), s))ω(s)1s −
σ (t )
H (σ (t ), σ (s))A(s)ω2 (s)1s
t2
t2 t
∆s
(H (σ (t ), σ (s))B(s) + H (σ (t ), s))ω(s)1s −
=
t2
t
H (σ (t ), σ (s))A(s)ω2 (s)1s t2
+ µ(t )(H (σ (t ), σ (t ))B(t ) + H ∆s (σ (t ), t ))ω(t ) − µ(t )H (σ (t ), σ (t ))A(t )ω2 (t ). Using H (σ (t ), σ (t )) = 0, µ(t )H ∆s (σ (t ), t )ω(t ) ≤ 0, and the method of completing the square, we have σ (t )
(H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))ω(s)1s −
t2
σ (t )
H (σ (t ), σ (s))A(s)ω2 (s)1s
t2 t
≤
(H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))ω(s)1s −
t2
t
H (σ (t ), σ (s))A(s)ω2 (s)1s t2
(H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))2 1s. 4H (σ (t ), σ (s))A(s)
t
≤ t2
Thus, we get by H (σ (t ), σ (t )) = 0 that σ (t )
(H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))2 H (σ (t ), σ (s))Q (s)1s − 1s H (σ (t ), t2 ) 4H (σ (t ), σ (s))A(s) t2 t2 t 1 (H (σ (t ), σ (s))B(s) + H ∆s (σ (t ), s))2 = H (σ (t ), σ (s))Q (s) − 1s ≤ ω(t2 ), H (σ (t ), t2 ) t2 4H (σ (t ), σ (s))A(s)
1
which contradicts (2.3). The proof is complete.
t
Example 2.1. Assume T = R and consider the second-order differential equation (1.2). In this equation, σ (t ) = t , K = 1, r (t ) = 1, q(t ) = q0 /t 2 , and δ(t ) = t. Let H (t , s) = (t − s)2 , η(t ) = t, and a(t ) = 0. Then we have lim sup t →∞
t
1 H (t , t2 )
= lim sup t →∞
t2
1
(t − t2 )2
(H (t , s)B(s) + H ∆s (t , s))2 1s 4H (t , s)A(s) 2 t s t −s s − 2 (t − s)2 q0 − ds = ∞,
H (t , s)Q (s) −
t2
s
4
if q0 > 1/4. Hence by Theorem 2.1, (1.2) is oscillatory when q0 > 1/4. In the following, we give an oscillation criterion for (1.1) when
∞ t0
1t r (t )
< ∞.
(2.11)
Theorem 2.2. Let (2.1) be replaced by (2.11) and all other assumptions of Theorem 2.1 hold. If there exist two functions ζ , b ∈ C1rd ([t0 , ∞)T , R) such that ζ (t ) > 0, b(t ) ≥
1 r (t )R(t )
,
(2.12)
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where R(t ) :=
1s , r (s)
∞
t
and there exists a function G ∈ Crd (D, R) such that G(t , t ) = 0,
G(t , s) > 0,
t ≥ t0 ;
t > s ≥ t0 ,
(2.13)
and G has a nonpositive rd-continuous ∆-partial derivative G∆s (t , s) on D0 with respect to the second variable and satisfies
t
G(σ (t ), σ (s))Q∗ (s) −
t0
lim sup
(G(σ (t ),σ (s))B∗ (s)+G∆s (σ (t ),s))2 4G(σ (t ),σ (s))A∗ (s)
G(σ (t ), t0 )
t →∞
1s
= ∞,
(2.14)
where
Q∗ (s) := ζ σ (s) Kq(s) + r (s)b2 (s) − (r (s)b(s))∆ , ζ σ (s) , r (s)ζ 2 (s)
A∗ (s) :=
B∗ (s) :=
ζ ∆ (s) 2ζ σ (s)b(s) + , ζ (s) ζ (s)
then (1.1) is oscillatory. Proof. Assume that (1.1) has a nonoscillatory solution x on [t0 , ∞)T . Without loss of generality, suppose that it is an eventually positive solution. From (1.1), there exists t1 ∈ [t0 , ∞)T such that x∆ (t ) > 0 or x∆ (t ) < 0 for t ∈ [t1 , ∞)T . The proof of the case where x∆ (t ) > 0 is the same as that of Theorem 2.1, and we can get a contradiction to (2.3). Assume now that x∆ (t ) < 0. We have x∆ (s) ≤
r (t ) r (s)
x∆ (t ),
s ≥ t ≥ t1 .
Hence we get x∆ ( t ) x(t )
≥−
1 r (t )R(t )
.
Define the function ω by
ω(t ) := ζ (t )
r (t )x∆ (t ) x( t )
+ r (t )b(t ) ,
t ≥ t1 .
Then ω(t ) ≥ 0 due to (2.12),
ω∆ =
(rx∆ )∆ ζ∆ − ζσr ω + ζ σ (rb)∆ + ζ σ ζ xσ
x∆ x
2
x xσ
,
(2.15)
and
x∆ x
2
ω = −b rζ
2
ω = rζ
2
+ b2 − 2
ωb . rζ
(2.16)
Note that x/xσ ≥ 1. Using (1.1), (2.15), and (2.16), we have
∆ 2 ζ 2ζ σ b ζσ σ ∆ − ζ rb − ( rb ) + + ω − 2 ω2 σ x ζ ζ rζ ∆ σ σ ζ 2 ζ b ζ ≤ −Kqζ σ − ζ σ rb2 − (rb)∆ + + ω − 2 ω2 ζ ζ rζ ∆ ζ 2ζ σ b ζσ 2 = −Q∗ + + ω − 2ω ζ ζ rζ
ω∆ ≤ ζ σ
(rx∆ )∆
= −Q∗ + B∗ ω − A∗ ω2 . The rest of the proof is similar to that of Theorem 2.1, and we can get a contradiction to (2.14). This completes the proof.
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Example 2.2. Assume T = R and consider the second-order differential equation (1.3). In this equation, σ (t ) = t , K = 1, r (t ) = t 2 , q(t ) = q0 , and δ(t ) = t. Let H (t , s) = (t − s)2 , η(t ) = 1/t, and a(t ) = 1/t. Then we have lim sup t →∞
t
1 H (t , t2 )
= lim sup t →∞
t2
1
(t − t2 )2
(H (t , s)B(s) + H ∆s (t , s))2 1s 4H (t , s)A(s) 2 t s t −s s − 2 (t − s)2 q0 − ds = ∞, if q0 > 1/4.
H (t , s)Q (s) −
t2
s
4
Let G(t , s) = (t − s)2 , ζ (t ) = 1/t, and b(t ) = 1/t. Then we obtain
(G(t , s)B∗ (s) + G∆s (t , s))2 G(t , s)Q∗ (s) − 1s lim sup 4G(t , s)A∗ (s) t →∞ G(t , t0 ) t0 2 t s t −s s − 2 1 (t − s)2 = lim sup q0 − ds = ∞, if q0 > 1/4. 2 s 4 t →∞ (t − t0 ) t0 1
t
Hence by Theorem 2.2, (1.3) is oscillatory when q0 > 1/4. Acknowledgments The authors express their sincere gratitude to the anonymous referees for careful reading of the original manuscript and useful comments that helped to improve presentation of results and accentuate important details. This research is supported by National Key Basic Research Program of China (2013CB035604) and NNSF of P.R. China (Grant Nos. 61034007, 51277116, 51107069). References [1] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. [2] R.P. Agarwal, M. Bohner, S.H. Saker, Oscillation of second order delay dynamic equations, Can. Appl. Math. Q. 13 (2005) 1–17. [3] R.P. Agarwal, D. O’Regan, S.H. Saker, Oscillation criteria for second-order nonlinear neutral delay dynamic equations, J. Math. Anal. Appl. 300 (2004) 203–217. [4] R.P. Agarwal, D. O’Regan, S.H. Saker, Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales, Appl. Anal. 86 (2007) 1–17. [5] R.P. Agarwal, D. O’Regan, S.H. Saker, Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales, Acta Math. Sin. 24 (2008) 1409–1432. [6] E. Akın-Bohner, M. Bohner, S.H. Saker, Oscillation criteria for a certain class of second order Emden–Fowler dynamic equations, Electron. Trans. Numer. Anal. 27 (2007) 1–12. [7] M. Bohner, S.H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain J. Math. 34 (2004) 1239–1254. [8] M. Bohner, S.H. Saker, Oscillation criteria for perturbed nonlinear dynamic equations, Math. Comput. Modelling 40 (2004) 249–260. [9] E. Braverman, B. Karpuz, Nonoscillation of second-order dynamic equations with several delays, Abstr. Appl. Anal. 2011 (2011) 1–34. [10] L. Erbe, B. Karpuz, A. Peterson, Kamenev-type oscillation criteria for higher-order neutral delay dynamic equations, Int. J. Difference Equ. 6 (2011) 1–16. [11] L. Erbe, A. Peterson, S.H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. Lond. Math. Soc. 67 (2003) 701–714. [12] L. Erbe, A. Peterson, S.H. Saker, Kamenev-type oscillation criteria for second-order linear delay dynamic equations, Dynam. Systems Appl. 15 (2006) 65–78. [13] L. Erbe, A. Peterson, S.H. Saker, Oscillation criteria for second-order nonlinear delay dynamic equations, J. Math. Anal. Appl. 333 (2007) 505–522. [14] S.R. Grace, R.P. Agarwal, M. Bohner, D. O’Regan, Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3463–3471. [15] S.R. Grace, M. Bohner, R.P. Agarwal, On the oscillation of second-order half-linear dynamic equations, J. Difference Equ. Appl. 15 (2009) 451–460. [16] Z. Han, T. Li, S. Sun, C. Zhang, Oscillation for second-order nonlinear delay dynamic equations on time scales, Adv. Difference Equ. 2009 (2009) 1–13. [17] T. Li, Z. Han, S. Sun, D. Yang, Existence of nonoscillatory solutions to second-order neutral delay dynamic equations on time scales, Adv. Difference Equ. 2009 (2009) 1–10. [18] Y. Şahiner, Oscillation of second-order delay differential equations on time scales, Nonlinear Anal. 63 (2005) 1073–1080. [19] S.H. Saker, Oscillation of nonlinear dynamic equations on time scales, Appl. Math. Comput. 148 (2004) 81–91. [20] S.H. Saker, Oscillation Theory of Dynamic Equations on Time Scales, Lambert Academic Publishing, Germany, 2010. [21] S.H. Saker, D. O’Regan, New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 423–434. [22] B.G. Zhang, S.L. Zhu, Oscillation of second-order nonlinear delay dynamic equations on time scales, Comput. Math. Appl. 49 (2005) 599–609. [23] C. Zhang, T. Li, R.P. Agarwal, M. Bohner, Oscillation results for fourth-order nonlinear dynamic equations, Appl. Math. Lett. 25 (2012) 2058–2065. [24] Q. Zhang, Oscillation of second-order half-linear delay dynamic equations with damping on time scales, J. Comput. Appl. Math. 235 (2011) 1180–1188.