Impact Factor(JCC): 1.8207 - This article can be downloaded from www.impactjournals.us
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)
82 M. M. A. El-Sheikh, R. Sallam & Nahed A. Mohamady
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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
84 M. M. A. El-Sheikh, R. Sallam & Nahed A. Mohamady
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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.
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