Journal of Advanced Research (2013) 4, 201–204

Cairo University

Journal of Advanced Research

SHORT COMMUNICATION

On the oscillation of higher order dynamic equations Said R. Grace

*

Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt Received 4 March 2012; revised 2 April 2012; accepted 23 April 2012 Available online 2 July 2012

KEYWORDS

Abstract

Oscillation; Higher Order; Dynamic equations



aðtÞðx

We present some new criteria for the oscillation of even order dynamic equation

Dn1

ðtÞÞa

D

þ qðtÞðxðtÞÞa ¼ 0;

on time scale T, where a is the ratio of positive odd integers a and q is a real valued positive rdcontinuous functions defined on T. ª 2012 Cairo University. Production and hosting by Elsevier B.V. All rights reserved.

(ii) a, q: T fi R+ = (0, 1) is a real-valued rd-continuous functions, aD(t) P 0 for t 2 [t0, 1)T and

Introduction This paper is concerned with the oscillatory behavior of all solutions of the even order dynamic equation 

n1

aðtÞðxD ðtÞÞa

D

þ qðtÞðxðtÞÞa ¼ 0;

ð1:1Þ

on an arbitrary time-scale T ˝ R with Sup T = 1 and n P 2 is an even integer. We shall assume that: (i) a P 1 is the ratio of positive odd integers,

* Tel.: +20 2 35876998. E-mail address: [email protected]. Peer review under responsibility of Cairo University.

Production and hosting by Elsevier

Z

1

a1=a ðsÞDs ¼ 1:

ð1:2Þ

We recall that a solution x of Eq. (1.1) is said to be nonoscillatory if there exists a t0 2 T. Such that x(t)x(r(t)) > 0 for all t 2 [t0, 1)T; otherwise, it is said to be oscillatory. Eq. (1.1) is said to be oscillatory if all its solutions are oscillatory. The study of dynamic equations on time-scales which goes back to its founder Hilger [1] as an area of mathematics that has received a lot of attention. It has been created in order to unify the study of differential and difference equations. Recently, there has been an increasing interest in studying the oscillatory behavior of first and second order dynamic equations on time-scales, see [2–7]. With respect to dynamic equations on time scales it is fairly new topic and for general basic ideas and background, we refer to [8,9]. It appears that very little is known regarding the oscillation of higher order dynamic equations [10–15] and our purpose here to establish some new criteria for the oscillation criteria for such equations. The obtained results are new even for the special cases when T = R and T = Z.

2090-1232 ª 2012 Cairo University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jare.2012.04.003

202

S.R. Grace Theorem 2.2. Let conditions (i), (ii) and (1.2) hold and

Main results

Z

We shall employ the following well-known lemma.

 Z rðsÞ ðaðsÞÞ1

1

t0

Lemma 2.1. [6, Corollary 1]. Assume that n 2 N, s, t 2 T and f 2 Crd(T, R). Then Z

t

s

Z

t

 gn þ1

¼ ð1Þn

Z

Z

Z t!1

t

ð2:1Þ

  gðsÞqðsÞ  aðsÞgD ðsÞha n1 ðs; t0 Þ Ds ¼ 1;

ð2:4Þ

t1

t

hn ðs; rðgÞÞfðgÞDg:

ð2:3Þ

If there exists a positive non-decreasing delta-differentiable function g such that for every t1 2 [t0, 1)T. lim sup

gn

1=a qðuÞDu Ds ¼ 1:

s

t

fðg1 ÞDg1 Dg2    Dgnþ1

1

then Eq. (1.1) is oscillatory.

s

We shall employ the Kiguradze’s following well-known lemma. Theorem 2.1. (Kiguarde’s Lemma [8, Theorem 5]),Let n 2 N; f 2 Cnrd ðT; RÞ and sup T = 1. Suppose that f is either n positive or negative and fD is not identically zero and is either nonnegative or nonpositive on [t0, 1)T for some t0 2 T. Then, there exist t1 2 [t0, 1)Tm 2 [0, n)Z such that n ð1Þnm fðtÞfD ðtÞ P 0 holds for all t 2 [t0, 1)T with j

(I) f ðtÞf D ðtÞ > 0 holds for all t 2 [t0, 1)T and all j 2 [0, m)z, j (II) ð1Þmþj f ðtÞf D ðtÞ > 0 holds for all t 2 [t0, 1)T and all j 2 [m, n)z

Proof. Let x(t) be a nonoscillatory solution of Eq. (1.1) on [t0, 1)T. It suffices to discuss the case x is eventually positive (as – x also solves (1.1) if x does), say x(t) > 0 for t P t1 P t0. Now, we see that  D n1 aðtÞðxD ðtÞÞa 6 0 for t P t1 : n1

It is easy to see that xD ðtÞ > 0 for t P t1 for otherwise, and by using condition (1.2) we obtain a contradiction to the fact that x(t) > 0 for t P t1. Now, aD(t) P 0 for t 2 [t0, 1)T. We have  D  n1 D  n1 a n1 aðtÞðxD ðtÞÞa ¼ aD ðtÞ xD ðtÞ þ ar ðtÞ ðxD ðtÞÞa 6 0:

The following result will be used to prove the next corollary. Lemma 2.2. [12, Lemma 2.8].Let supT = 1 and f 2 Cnrd ðT; RÞðn P 2Þ. Moreover, suppose that Kiguradze’s theon rem holds with m 2 [1, n)N and fD 6 0 on T. Then there exists a sufficiently large t1 2 T such that m

fD ðtÞ P hm1 ðt; t1 ÞfD ðtÞ for all t 2 ½t1 ; 1ÞT :

This implies n1

ððxD ðtÞÞa ÞD 6 0 for t P t1 : n1

Next, we let y ¼ xD rem 1.90] we see that

ð2:2Þ

Z

on [t1, 1)T. From Ref. [9, Theo-

1

ðy þ hlyD Þa1 dh P ayD

Z

1

The proof of the following corollary follows by an integration of (2.2).

0 P ðya ÞD ¼ ayD

Corollary 2.1. Assume that the conditions of Lemma 2.2 hold. Then

Thus we have yD ¼ xD 6 0 on [t1, 1)T and from Theorem 2.1, there exists an integer m 2 {1, 3, . . . , n  1}. Such that (I) and (II) hold on [t1, 1)T Clearly xD(t) > 0 for t P t1 and hence, there exists a constant c > 0 such that

m

fðtÞ P hm ðt; t1 ÞfD ðtÞ for all t 2 ½t1 ; 1ÞT : Next, we need the following lemma see [16]. Lemma 2.3. If X and Y are nonnegative and k > 1, then

0

ya1 dh

0

¼ aya1 yD : n

xðtÞ P c

ð2:5Þ

for t P t1 :

First, we claim that m = n  1. To this end, we assume that

Xk  kXYk1 þ ðk  1ÞYk P 0;

xD ðtÞ < 0

where equality holds if and only if X = Y,

Integrating Eq. (1.1) from t P t1 to u P t, letting u fi 1 we have

It will be convenient to employ the Taylor monomials (see [9, Section 1.6]) fhn ðt; sÞ1 n¼0 g which are defined recursively by: Z t hn ðs; sÞDs; t; s 2 T and n P 1; hnþ1 ðt; sÞ ¼

n2

n3

and xD ðtÞ > 0

 Z n1 xD ðtÞ P c ðaðtÞÞ1

for t P t1 :

1=a

1

for t P t1 :

qðsÞDs t

s

where it follows that h1(t, s) = t  s but simple formulas in general do not hold for n P 2. Now we present the following oscillation results for Eq. (1.1).

Integrating (2.6) from to and letting fi 1 we get n2

0 < xD ðtÞ 6 

Z t

1

 Z ðaðuÞÞ1 u

1=a

1

qðsÞDs

Du:

ð2:6Þ

On the oscillation of higher order dynamic equations

203

Integrating this inequality from t and using condition (2.3) after Lemma 2.1 we arrive at the desired contradiction. It follows from Lemma 2.2 with m = n  1 that Dn1

D

x ðtÞ P hn2 ðt; t1 Þx

ðtÞ for t P t1 ;

ð2:7Þ

and by applying Corollary 2.1 with m = n  1 instead of Lemma 2.2, we get n1

xðtÞ P hn1 ðt; t1 ÞxD ðtÞ

 D   D  g g x wD 6 gq þ r wr  a r wr : g g x Using Lemma 2.2 with m = n  1 in (2.15), we find !  D   n1 g g xD r wr hn2 w 6 gq þ r w  a r g g x D

n1

Dn1 a

aðx Þ xa

on ½t1 ; 1ÞT :

ð2:9Þ

 g D

g

n1

axD x

ð2:10Þ

Now set ð2:11Þ

a

¼ t P t1 :

Integrating this inequality from t2 > t1 to t P t2, we have t

½gðsÞqðsÞ  aðsÞgD ðsÞha n1 ðs; t1 ÞDs:

ð2:12Þ

t2

Taking upper limit of both sides of the inequality (2.12) as t fi 1 and using (2.4) we obtain a contradiction to the fact that w(t) > 0 on [t1, 1)T. This completes the proof. h Next, we establish the following result.

t!1

 hn2 ðs; t0 ÞgðsÞ

Ds ¼ 1:

x þ lhxD

a1

a ða þ 1Þaþ1



1 ghn2

a

ðgD Þaþ1

on ½t2 ; 1ÞT :

ð2:18Þ

Integrating (2.18) from t2 to t, we get Z t"

gðsÞqðsÞ 

aðsÞ



1

a

ða þ 1Þaþ1 gðsÞhn2 ðs;t1 Þ

# ðgD ðsÞÞaþ1 Ds: ð2:19Þ

ð2:13Þ

Taking upper limit of both sides of (2.19) as t fi 1 and using (2.13), we obtain a contradiction to the fact that w(t) > 0 for t P t1 This completes the proof. h Finally, we present the following interesting result.

Proceeding as in the proof of Theorem 2.1, we obtain m = n  1 and (2.7) and (2.8). Define w as in (2.9) and obtain (2.10). Now from Ref. [3, Theorem 1.90], ðxa ÞD ¼ ðaxD Þ

and therefore, we find

t2

Proof. Let x be a nonoscillatory solution of Eq. (1.1), say x(t) > 0 for t P t1 P t0.

1

P 0;

aðsÞðgD ðsÞÞaþ1 aþ1

then Eq. (1.1) is oscillatory.

Z

in Lemma 2.3 with k ¼ aþ1 > 1 to conclude that a !   D g 1 g a ðgD Þaþ1 r 1þ1=a ðw Þ  r wr þ a aþ1 1þ1=a hn2 g ga han2 ða þ 1Þ ðgr Þ

wðtÞ 6 wðt2 Þ 

ða þ 1Þ !a #

t1

a a ðgD Þa aþ1

wD 6 gq þ

Theorem 2.3. Let conditions (i), (ii) (1.2) and (2.3) hold. If there exists a a positive non-decreasing delta-differentiable function g such that for every t1 2 [t0, 1)T

 r w and Y g a !a  1=a aþ1 a ; aghn2

X ¼ ðaða1=a Þghn2 Þaþ1 

wD ðtÞ 6 gðtÞqðtÞ þ aðtÞgD ðtÞðhn1 ðt; t1 ÞÞa ;

1

ð2:17Þ

for t P t2 P t1 ;

Using (2.8) in (2.11), we get

Z t" lim sup gðsÞqðsÞ 

on ½t2 ; 1ÞT ;

!  D g ðaÞ1=a g r ðhn2 Þðwr Þ1þ1=a w 6 gq þ r w  a g ðgr Þ1þ1=a

ðaðxD Þa Þr þ a ðaðxD Þa ÞD " x # D a a D Dn1 a r g x  gðx Þ ; ¼ gq þ ðaðx Þ Þ xa ðxr Þa !a Dn1 D x 6 gq þ ag : x

Z

 1=a  r 1=a w w P g g

D

n1

xa

wðtÞ 6 wðt2 Þ 

¼

and thus,

Then on [t1, 1)T, we have wD ¼

ð2:16Þ

where hn2 = hn2(t, t1). Now we see that

Now, we let w :¼ g

for t

P t2 P t1 ;

ð2:8Þ

for t P t1 :

ð2:15Þ

dh P axD

0

Z

Theorem 2.4. Let conditions (i), (ii), (1.2) and (2.3) hold. If there exists a a positive, delta-differentiable function g such that for every t1 2 [t0, 1)T Z t" lim sup t!1

1

xa1 dh ¼ axa1 xD :

t1

#   1 ðaðsÞÞr ðgD ðsÞÞ2 gðsÞqðsÞ  Ds 4a gðsÞðhn1 ðs; t1 ÞÞa1 hn2 ðs; t1 Þ

¼ 1;

0

ð2:20Þ

ð2:14Þ

Using (2.14) in (2.10), we have

then Eq. (1.1) is oscillatory.

204

S.R. Grace

Proof. Let x be a nonoscillatoy solution of Eq. (1.1), say x(t) > 0 for t P t1 P t0 Proceeding as in the proof of Theorem 2.3, we obtain (2.17) which cam be rewritten as !  D g ðaÞ1=a g r w 6 gq þ r w  a ðhn2 Þ g ðgr Þ1þ1=a

2. We may also employ other types of the time-scales [8,9] e.g., T = hZ with h > 0; qN o ; q > 1, T ¼ N 20 , etc. The detail are left to the reader.

References

D

r 1=a1

 ðw Þ

r 2

ðw Þ

on ½t2 ; 1ÞT :

ð2:21Þ

Now, using Corollary 2.1 with m = n  1 we have x n1 xD

P hn1 ;

implies on [t2, 1)T that (2.22) !1a Dn1 1=a1 1=a1 1=a1 x w ¼a g x  a1 x ¼ a1=a1 g1=a1 Dn1 P a1=a1 g1=a1 ha1 n1 : x

ð2:22Þ

Using (2.22) in (2.21) we have on [t2, 1)T that wD 6 gq þ

 D g gr

¼ gq  þ

 a1 r  gðh Þ h wr  a ðaÞn1r ðgr Þn2 ðwr Þ2 : 2

r ah1 n2 ðhn1 Þ

a1

1=2

g



ððaÞr Þ1=2 gr r

ðaÞ

ðgD Þ2

4aðhrn1 Þ

6 gq þ

a1

hn2



1 4a

!2

ððaÞr Þ1=2 gD a1

2ðahn2 ðhrn1 Þ

gÞ1=2

: ðaÞr ðgD Þ2

gðhrn1 Þ

a2

hn2

 : ð2:23Þ

Integrating this inequality from t2 to t, taking upper limit of the resulting inequality as t fi 1, and applying condition (2.20) we obtain a contradiction to the fact that w(t) > 0 for t P t1. This completes the proof.

h

Remarks 1. The results of this paper are presented in a form which is essentially new and of high degree of generality. Also, we can easily formulate the above conditions which are new sufficient for the oscillation of Eq. (1.1) on different time-scales e.g., T = R and T = Z. The details are left to the reader.

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