9. Applied-New Criteria for Oscillation -Nahed Abd El Fattah Mohamady

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IMPACT: International Journal of Research in Applie d, Natural and Social Sciences (IMPACT: IJRANSS) ISSN(E): 2321-8851; ISSN(P): 2347-4580 Vol. 3, Issue 3, Mar 2015, 79-86 © Impact Journals

NEW CRITERIA FOR OSCILLATION OF SECOND ORDER NONLIN EAR DYNAMIC

EQUATIONS WITH DAMPING ON TIME SCALES

M. M. A. EL-SHEIKH 1, R. SALLAM 2 & NAHED A. MOHAMADY 3 1,2Department of Mathematics, Faculty of Science, Menofia University, Egypt

3Department of Mathematics, Faculty of Information Technology, University of Pannonia, Hungary

ABSTRACT

The oscillation of solutions of the second order nonlinear damped dynamic equation (�(�)�(�(�))�∆(�))∆ +(�)�∆(�) + (�, �(�(�))) = 0 on an arbitrary time scale � is investigated. A generalized Riccati transformation is

applied for the study of the Kamenev-type oscillation criteria for this nonlinear dynamic equation. Several new sufficient

conditions for the oscillation of solutions are obtained to extend some known results in the literature.

KEYWORDS: Damped Delay Dynamic Equations, Oscillation Criteria, Time Scales

INTRODUCTION

Much recent attention has been given to dynamic equations on time scales, we refer the reader to the landmark

paper of S. Hilger [1]. Since then, several authors have expounded on various aspects of this new theory; see the survey

paper by Agarwal, Bohner, O'Regan and Peterson [2]. A time scale � is an arbitrary nonempty closed subset of the real

numbers �. Thus, �; �; �;��, i.e., the real numbers, the integers, the natural numbers, and the nonnegative integers are

examples of time scales. On any time scale �, we define the forward and backward jump operators by

�(�) = inf �� ∈ �, � > �� , �(�) = sup�� ∈ �, � < ��. A point t � ∈ �, � > #$� is said to be left dense if �(�) = �, right dense if � < �%� and �(�) = �, left

scattered if �(�) < �, and right scattered if �(�) > �. A function ∶ � → �is called rd-continuous provided that it is

continuous at right dense points of T, and itsleft-sided limits exist (finite) at left-dense points of�. The set of rd-

continuous functionsis denoted by )*+(�, �). By )*+, (�, �), we mean the set of functions whose delta derivativebelongs to )*+(�, �). In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions

of various equations on time scales (see [5],[6],[7]). However, there are few results dealing the oscillation of solutions of

delay dynamic equations on time scales [8-13]. Following this trend, we are concerned in this paper with oscillation

for the second-order nonlinear delay dynamic equations of the type

(�(�)�(�(�))�∆(�))∆ + (�)�∆(�) + (�, �(�(�))) = 0(1.1) We assume that

(./)�, � and are real-valued positive rd-continuous functions defined on �, 0 ≤ (�) ≤ 1and there are two positive constants 1,, 12 such that 1, ≤ �(�(�)) ≤ 12.

80 M. M. A. El-Sheikh, R. Sallam & Nahed A. Mohamady

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(.3)� ∶ � → � is strictly increasing, and�(�) ≤ � and � → ∞ as � → ∞

(.4)(�, %) ∈ )*+(� × �, �) satisfies %(�, %) > 0, for % ≠ 0 and there exists a positive

rd-continuous function 7 defined on � such that89(:,;); 8 ≥ 7(�)for % ≠ 0.

2. MAIN RESULTS

We need the following lemma for the proof of main results.

Lemma 2.1. Assume that �(�) is a positive solution of Eq. (1.1) on [��,∞)>. If

? @AB(C)DEF(C)(:,:G)*(:) ∞� ∆� = ∞ (2.1)

and

? �(�)7(�)∞� ∆� = ∞ , (2.2)

then there exists �, ∈ [��,∞)>, such that

(I)�∆(�) > 0, (�(�)�(�(�))�∆(�))∆ < 0for �, ∈ [��,∞)>

(II) J(:): is decreasing.Proof. Assume that �(�) is a positive solution of Eq. (1.1) on [��,∞)>. Pick �2 ∈ [��,∞)> ,such that �(�) > 0 and �(�(�)) > 0 on [�2,∞)> . Then without loss of generality we can take �∆(�) < 0 for all � ≥ �2 ≥ �,. Now from (1.1)

we have

(�(�)�K�(�)L�∆(�))∆ + (�)�∆(�) = −K�, �N(�)L < 0. (2.3)

Putting O(�) = −�(�)�K�(�)L�∆(�), then we can write (2.3) in the form

P −O∆(�) − (�)�(�)�K�(�)L QO(�) < 0 Thus

O∆(�) > − (�)1,�(�) QO(�) Therefore

O(�) > O(�2)RST(:)UE*(:)(�, �2) i.e.

−�(�)�K�(�)L�∆(�) > −�(�2)�K�(�2)L�∆(�2)RST(:)UE*(:)(�, �2)

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�∆(�) < �(�2)�K�(�2)L�∆(�2)�K�(�)LRST(:)UE*(:)(�, �2)�(�)

Then

�(�) < �(�V) + �(�2)�K�(�2)L�∆(�2)1, W RST(X)UE*(X)(�, �2)�(�)::Y ∆�

By (2.1) we can see that �(�) → −∞ as � → ∞ which contradicts the assumption �(�) > 0so �∆(�) > 0and hence (�(�)�(�(�))�∆(�))∆ < 0 for all large �. Now to prove (II), we define Z(�) = �(�) − ��∆(�), if there is a �V ∈[�2,∞)> such that Z(�) > 0. Suppose this is false, then Z(�) < 0 on [�V,∞)>, thus

(�(�)� )∆ = ��∆(�) − P �(�)��(�) = −Z(�)��(�) > 0. Hence

J(:): is strictly increasing on [�V,∞)> .Pick �[ ∈ [�V,∞)> , so that �(�) ≥ �(�[) for all � ≥ �[. Then

�(�(�))�(�) ≥ �(�(�[))�(�[) ≔ ] > 0�K�(�)L ≥ ]�(�). (2.4)

But from (1.1) we have

(�(�)�K�(�)L�∆(�))∆ + (�)�∆(�) = −K�, �N(�)L < 0. i.e.

(�(�)�K�(�)L�∆(�))∆ ≤ −K�, �N(�)L ≤ −7(�)�N(�) ≤ −Q]7(�)�(�)�(�)�K�(�)L�∆(�) − �(�[)�K�(�[)L�∆(�[) ≤ −Q]W 7(�)�(�):

:^ ∆� i.e.

�(�[)�K�(�[)L�∆(�[) ≥ ]W 7(�)�(�)::^ ∆�

This contradicts (2.2). Therefore Z(�) > 0 for all � ∈ [�V,∞)> , and hence �(�) > ��∆(�).Also we see that

(�(�)� )∆ = ��∆(�) − P �(�)��(�) = −Z(�)��(�) < 0. So

J(:): is strictly decreasing on [�V,∞)>.

Theorem 2.2. Suppose that (2.1) holds. Furthermore, suppose that there exists a positive ∆ −differentiable function_(�) such that, for all sufficiently large �, ∈ [��,∞)> , one has

lim:→∞ �%W b_(�)7(�)�(�)�(�) − 12��(�)c2(�)4_(�)�(�) e ∆� = ∞::E (2.5)

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where c(�): = _∆(�) − i(:)T(:)Uj*(:) . Then Eq. (1.1) is oscillatory.

Proof. Suppose to the contrary that �(�) is a nonoscillatory solution of (1.1). We may assume that �(�) > 0, �K�(�)L >0∀� ∈ [�,,∞)>. The proof when �(�) is eventually negative is similar. From Lemma 2.1 and Eq. (1.1) it follows that

�K�(�)L > 0, �∆(�) > 0, (�(�)�(�(�))�∆(�))∆ < 0∀� ≥ �,. Define

l(�) = _(�) �(�)�K�(�)L�∆(�)�(�) Then

l∆(�) = _∆ m���∆� no + _ m���∆� n∆= _∆_o lo + (���∆)�o

∆ − _ ��(�∆)2��o From (1.1) and (.4) we have

l∆(�) ≤ _∆_o lo − _�� ���∆�o − _7 �N�o − _ ���∆�o ���∆���

Since� > �, then in view of (./), and (���∆)∆ < 0, we have

l∆(�) ≤ _∆_o lo − _12� (���∆)o�o − _7 �N�o − _12� (���

∆)o�o (���∆)o�o �o�

Since J: is strictly decreasing and � < �; � < �, then

JpJq ≥ No ,JqJ ≥ o: , thus

l∆(�) ≤ _∆_o lo − _12�_o lo − _7�� − _�12��(_o)2 (lo)2

= riqlo − ioUj*:(iq)j (lo)2 − isNo (2.6)

i.e.

l∆(�) ≤ t 1_o u _�12��_o lo − c2u12��_� v2+ 12��c24_� − _7��

therefore

l∆(�) ≤ 12��c24_� − _7��

Integrating this inequality from �2 to �, we get

W b_(�)7(�)�(�)�(�) − 12��(�)c2(�)4_(�)�(�) e ∆�::j ≤ l(�2) − l(�) ≤ l(�2)

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This contradicts with (2.5), hence the proof is completed. By choosing _(�) = 1,� ≥ �� in Theorem 2.2 we

have the following oscillation result.

Corollary 2.3. Assume that the assumptions of Theorem 2.2 hold and for all sufficiently large �,,

lim:→∞ �%W b7(�)�(�)�(�) − �2(�)412�(�)�(�)e ∆�::E = ∞

Then every solution of Eq. (1.1) is oscillatory on [��,∞)>. By choosing _(�) = �,� ≥ �� in Theorem 2.2 we

have the following oscillation result.

Corollary 2.4. Assume that the assumptions of Theorem 2.2 hold. Then every solution of Eq. (1.1) is oscillatory on [��,∞)> provided that

lim:→∞ �%W b�7(�)�(�)�(�) − 12�(�)c2(�)4�(�) e ∆�::E = ∞

where c(�) = 1 − :T(:)Uj*(:) . Now, we define the function space ℜ as follows: x ∈ ℜprovided xis defined for �� ≤ � ≤ �, �, � ∈[��,∞)> , x(�, �) = 0, x(�; �) ≥ 0 and xhas a nonpositive continuous ∆ −partial derivative x∆y(�, �) ≥ 0 with

respect to the second variable and satisfies for some ℎ ∈ ℜ.The following theorem extends Theorem 2.2 of [5].

x∆y(�, �) + x(�, �) c(�)_o(�) = −ℎ(�, �)_o(�) {x(�, �)Theorem 2.5. Suppose that the assumptions of Theorem 2.2 hold. If there exists a positive functions x, ℎ ∈ ℜ such that

for all sufficiently large �, ∈ [��,∞)> , one has then every solution of Eq. (1.1) is oscillatory on [��,∞)>.

lim:→∞ sup 1x(�, ��)W b_(�)7(�)�(�)�(�) x(�, �) − 12��(�)ℎ2(�, �)4_(�)�(�) e ∆�::G = ∞,(2.7)

Proof. Suppose to the contrary that �(�) is a nonoscillatory solution of (1.1). Then as in Theorem 2.2 we have �K�(�)L >0, �∆(�) > 0, (�(�)�(�(�))�∆(�))∆ < 0∀� ≥ �, ∈ [��,∞)>. We define l(�) as in Theorem 2.2, then from (2.6) we

have

_7�� ≤ −l∆ + c_o lo − _�12��(_o)2 (lo)2(2.8) Multiplying (2.8) by x(�, �) and integrating from �, to �, we get

W _(�)7(�)�(�)�(�) x(�, �)∆�::E ≤ W l∆(�)x(�, �)∆�:

:E +W c(�)_o(�) lo(�)x(�, �)∆�::E

−W _(�)�(�)12�(�)�K_o(�)L2 (lo)2(�)x(�, �)∆�::E

= l(�,)x(�, �,) + W x∆y(�, �)lo(�)∆�::E +W c(�)_o(�) lo(�)x(�, �)∆�:

:E

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−W _(�)�(�)12�(�)�K_o(�)L2 (lo)2(�)x(�, �)∆�::E

Thus

W _(�)7(�)�(�)�(�) x(�, �)∆�::E ≤ l(�,)x(�, �,)

−W ~ 1_o(�)u_(�)�(�)x(�, �)12�(�)� lo(�) + ℎ(�, �)2 u 12�(�)�_(�)�(�)�2∆�:

:E +W 12�(�)�ℎ2(�, �)4_(�)�(�) ∆�::E

≤ l(�,)x(�, �,) + W 12�(�)�ℎ2(�, �)4_(�)�(�) ∆�::E

i.e.

1x(�, �,)W b_(�)7(�)�(�)�(�) x(�, �) − 12��(�)ℎ2(�, �)4_(�)�(�) e ∆�::G ≤ l(�,)

which contradicts (2.7) and hence the proof is completed.

Theorem 2.6. Suppose that (2.1) holds. Furthermore, suppose that there exists a function �(�) such that �(�)�(�) is a ∆ − P differentiable function and there exists a positive real rd-function �(�) such that, for all sufficiently large �, ∈[��,∞)> , one has

lim:→∞ �%W ��(�) − �2(�)4�(�)�∆�::E = ∞,(2.9)

where �(�):= �q(:):Uj*(:)o(:)�j(:) ,�(�): = �∆(:)�(:) − :�q(:)T(:)Uj*(:)o(:)�(:) + 2:�q(:)�(:)Ujo(:)�(:) ,and �(�): = �q(:)s(:)N(:)o(:) +:*(:)�q(:)�j(:)Ujo(:) − :T(:)�q(:)Ujo(:) − �o(�)[�(�)�(�)]∆.Then Eq. (1.1) is oscillatory.

Proof. Suppose to the contrary that �(�) is a nonoscillatory solution of (1.1). We may assume that �(�) > 0, �K�(�)L >0∀� ∈ [�,,∞)>. The proof when �(�) is eventually negative is similar. From Lemma 2.1 and Eq. (1.1) it follows that

�K�(�)L > 0, �∆(�) > 0, (�(�)�(�(�))�∆(�))∆ < 0∀� ≥ �, ∈ [��,∞)>. Define

%(�) = �(�)[�(�)�K�(�)L�∆(�)�(�) + �(�)�(�)] Then

%∆(�) = �∆ m���∆� + ��n + �o m���∆� + ��n∆

=�∆� % + �o (���∆)�o∆ − �o ��(�∆)2��o +�o[��]∆

From (1.1) and (.4)we have

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%∆(�) ≤ �∆� % − �o�� ���∆� ��o − �o7 �N�o − �o�� ��o (%� − ��)2+�o[��]∆

Using Lemma 2.1 and (./), we get

%∆(�) ≤ �∆� % − �o�12�� �%� − ��� − �o7�� − �o�%212���2 + 2�o�%12�� −�o���212� + �o[��]∆

i.e.

%∆(�) ≤ �∆� % − �o�12��� % + �o��12� − �o7�� − �o�12���2 %2 + 2�o�%12�� −�o���212� + �o[��]∆

= −�%2 + �% − � = −[√�% − �2√�]2 + �24� − �

i.e.

%∆(�) ≤ �2(�)4�(�) − �(�) By integrating the above inequality, we obtain

W ��(�) − �2(�)4�(�)�∆�::j ≤ %(�2) − %(�) ≤ %(�2)

This contradicts (2.9), and the proof is completed. We can get the following result by choosing �(�) = 1 in

Theorem 2.6.

Corollary 2.7. Assume that the assumptions of Theorem 2.5 hold and for some �, sufficiently large . We have

lim:→∞ �%W ��(�) − �2(�)4�(�)�∆�::E = ∞,

where�(�): = :Uj*(:)o(:) , �(�): = − :T(:)Uj*(:)o(:) + 2:�(:)Ujo(:),and�(�): = s(:)N(:)o(:) + :*(:)�j(:)Ujo(:) − :T(:)�(:)Ujo(:) − [�(�)�(�)]∆. Then Eq. (1.1) is oscillatory.

REFERENCES

1. R. Agarwal, M. Bohner, D. ORegan, and A. Peterson, Dynamic equations on time scales: a survey, Journal of

Computational and Applied Mathematics, Vol. 141, No. 1-2(2002), pp. 1-26.

2. R. P. Agarwal, D. ORegan, and S. H. Saker, Philos-type oscillation criteria for second order halinear dynamic

equations on time scales, The Rocky Mountain Journal of Mathematics, Vol. 37, No. 4(2007), pp. 1085-1104.

3. M. Bohner and S. H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, The Rocky

Mountain Journal of Mathematics, Vol. 34, No. 4(2004), pp. 1239-1254.

4. L. Erbe, A. Peterson, and S. H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time

scales, Journal of the London Mathematical Society. Second Series, Vol. 67, No. 3 (2003), pp. 701-714.

86 M. M. A. El-Sheikh, R. Sallam & Nahed A. Mohamady

Index Copernicus Value: 3.0 - Articles can be sent to editor@impactjournals.us

5. L. Erbe, A. Peterson and S. H. Saker, Oscillation criteria for second-order nonlinear delay dynamic equations on

time scales, J. Math. Anal. Appl. 333(2007), 505-522.

6. L. Erbe, T.S. Hassan and A. Peterson, Oscillation criteria for nonlinear damped dynamic equations on time scales,

Applied Mathematics and Computation, 203 (2008), 343-357.

7. S. R. Grace, R. P. Agarwal, M. Bohner, and D. ORegan, Philos type criteria for second-order halinear dynamic

equations, Mathematical Inequalities and Applications, Vol. 14, No. 1(2011), pp. 211-222.

8. Z. Han, S. Sun, T. Li and C. Zhang, Oscillatory behavior of quasilinear neutral delay dynamic equations on time

scales, Adv. Di_. Equ. 2010 (2010) 1-24.

9. S. Hilger, Analysis on measure chainsa unified approach to continuous and discrete calculus, Results in

Mathematics, Vol. 18, No. 1-2(1990), pp. 18-56.

10. S. Sun, Z. Han, and C. Zhang, Oscillation of second-order delay dynamic equations on time scales, Journal of

Applied Mathematics and Computing, Vol. 30, No. 1-2 (2009), pp. 459-468.

11. Y. Sun, Z. Han and T. Li, Oscillation criteria for second-order quasilinear neutral delay equations, Journal of

University of Jinan. 24 (2010) 308-311.

12. B.G. Zhang and D. Xinghua, Oscillation of delay differential equations on time scales, Mathl. Comput. Modelling

36 (11-13)(2002), 1307-1318.

13. B. G. Zhang and S. Zhu, Oscillation of second-order nonlinear delay dynamic equations on time scales, Comp.

Math. Appl. 49 (2005) 599-609.