Abstract
This paper is concerned with oscillations of the second-order delay nonlinear dynamic equation on a time scale , where a and q are real-valued rd-continuous positive functions on
, α and β are ratios of odd positive integers, , , , and . We establish some new sufficient conditions for this equation.
MSC:34K11, 39A10, 39A99.
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1 Introduction
The study of dynamic equations on time scales, which goes back to Hilger [1], provides a rapidly expanding body of literature where the main idea is to unify, extend, and generalise concepts from continuous, discrete and quantum calculus to arbitrary time scales analysis, where a time scale is simply any nonempty closed subset of the reals.
For detailed information regarding calculus on time scales, we refer the reader to Bohner and Peterson [2, 3].
Now we consider the second-order delay nonlinear dynamic equation
on an arbitrary time scale , where a and q are real-valued rd-continuous positive functions defined on
; α and are ratios of odd positive integers; , , and .
We shall also consider the case
Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume that , and we define the time scale interval by . By a solution of (1.1) we mean a nontrivial real-valued function , which has the property that and satisfies (1.1) on ; here is the space of rd-continuous functions. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory.
If then , , , and when (1.1) becomes the half-linear delay differential equation
When and , (1.3) becomes
Ohriska [4] proved that every solution of (1.4) oscillates if
holds. Agarwal et al. [5] considered (1.3) and extended the condition (1.5) and proved that if
then every solution of (1.3) oscillates.
If , then , , , , and (1.1) becomes the quasi-linear difference equation
When , (1.1) becomes the half-linear delay dynamic equation which has been considered by some authors and some oscillation and nonoscillation results have been obtained [2, 3, 5–10]. As a special case of (1.1) Agarwal et al. [11] considered the second-order delay dynamic equations on time scales
and established some sufficient conditions for oscillations of (1.8) when
Saker [12] studied (1.8) to extend the result of Lomtatidze [13]. Recently Higgins [14] proved oscillation results for (1.8). Saker [15] examined oscillations for the half-linear dynamic equation
on time scales, where is an odd positive integer and Agarwal et al. [6] and Grace et al. [10] studied oscillations for the same equation, (1.10), where is the quotient of odd positive integers which cannot be applied when . Han et al. [16] and Hassan [17] solved this problem and improved Agarwal’s and Saker’s results. Erbe et al. [7] considered the half-linear delay dynamic equation
on time scales, where is the quotient of odd positive integers and
and utilised a Riccati transformation technique and established some oscillation criteria for (1.11). Erbe et al. [8] considered the half-linear delay dynamic equation (1.11) on time scales, where is the quotient of odd positive integers and established some sufficient conditions for oscillations when (1.12) holds. Han et al. [18] considered (1.11) and followed the proof that has been used in [15] and established some sufficient conditions for oscillations when . For oscillations of quasi-linear dynamic equations, Grace et al. [19] considered the equation
and Saker and Grace [20] considered the delay equation
on an arbitrary time scale where or . The special case of (1.14) where has been studied in [9] by Erbe et al.
In this paper, we establish some new sufficient conditions for oscillations of (1.1).
2 Preliminary result
For a function , the (delta) derivative at is defined to be the number (if it exists) such that for all there is a neighborhood U of t with
for all . If the (delta) derivative exists for all , then we say that f is (delta) differentiable on . For two (delta) differentiable functions f and g, the derivative of the product fg and the quotient (where ) are given as in [[2], Theorem 1.20]
as well as of the chain rule [[2], Theorem 1.90] for the derivative of the composite function for a continuously differentiable function and a (delta) differentiable function
For and a differentiable function f, the Cauchy integral of is defined by
and the improper integral is defined as
Lemma 2.1 If A and B are nonnegative real numbers and , then
where the equality holds if and only if .
3 The main results
In this section, by employing a Riccati transformation technique, we establish oscillation criteria for (1.1). To prove our main result, we will use the formula
which is a simple consequence of the Pötzsche chain rule [2].
In the following theorems, we assume that (1.2) holds.
Theorem 3.1 Assume that there exists a positive nondecreasing delta differentiable function such that, for all sufficiently large , and for , we have
where ,
and
Then every solution of (1.1) is oscillatory on .
Proof Let x be a nonoscillatory solution of (1.1) on . Then, without loss of generality, there is a , sufficiently large, such that and on . Since is a strictly decreasing function, it is of one sign. We claim that on . If not, then there is a such that . Using the fact is strictly decreasing, we get
which implies that is eventually negative. This is a contradiction. Hence on . Consider the generalized Riccati substitution
then . By the product rule and then the quotient rule, we have
Using the fact that is increasing and is decreasing, we have
and so
for . Also, we see that
Therefore, (3.7) and (3.8) imply that
In view of (3.1), we get
From (3.9), (3.10), the fact that is an increasing function and the definition of , if , we have
whereas, if , we have
Using the fact that is strictly decreasing on , we find
Thus for , we have
We consider the following three cases:
Case (i). .
In this case, since , there exists such that . This implies that , where .
Case (ii). .
In this case, we see that .
Case (iii). .
Then there exists a positive constant . Using the decreasing of , we have
Integrating this inequality from to t, we have
Thus, there exist a constant and such that
and hence
where .
By (3.3), we get
for . Defining and by
and using Lemma 2.1 we get
where . Therefore, by (3.12), we have
Integrating (3.13) from to t, we get as
which contradicts (3.5). □
Theorem 3.2 Assume that there exists a positive nondecreasing delta differentiable function such that, for all sufficiently large , and for , we have
where
and θ is as in Theorem 3.1. Then (1.1) is oscillatory on .
Proof Let x be a nonoscillatory solution of (1.1) on . Then, without loss of generality, there is a , sufficiently large, such that and on . Proceeding as in the proof of Theorem 3.1, we obtain (3.11). Therefore
Using the definition of , it follows from (3.16) that
Now, from
it follows that
Using (3.18) in (3.17), we have
Next as in the proof of Theorem 3.1, we consider the Cases (i), (ii) and (iii).
Case (i). .
In this case, since , there exists such that . This implies that , where .
Case (ii). .
In this case, we see that .
Case (iii). .
Proceeding as in the proof of Theorem 3.1, there exist a constant and such that
and hence
where .
Using these three cases in (3.19) and the definition of , we get
for . Integrating the above inequality from to t, we have
which gives a contradiction using (3.14). □
We next state and prove a Philos-type oscillation criterion for (1.1).
Theorem 3.3 Assume that there exist functions H and h such that for each fixed t, and are rd-continuous with respect to s on such that
and H has a non-positive continuous Δ-partial derivative with respect to the second variable, and that it satisfies
and for all sufficiently large , and for , , we have
where is a positive Δ-differentiable function and . Then every solution of (1.1) is oscillatory on .
Proof Let x be a nonoscillatory solution of (1.1) on . Then, without loss of generality, there is a , sufficiently large, such that and on . Again we define function as in the proof of Theorem 3.1. Then, proceeding as in the proof of Theorem 3.1, we obtain (3.12). Multiplying both sides of inequality (3.12), with t replaced by s, by and integrating with respect to s from to t, we get
Integrating (3.23) by parts and using (3.20) and (3.21), we obtain
Again, defining and by
and using Lemma 2.1 where , we get
From (3.24) and (3.25), we have
which contradicts assumption (3.22).
Now we introduce the following notation for . For all sufficiently large T such that ,
Assume that . Note that . We assume that
□
Theorem 3.4 Let and assume is a delta differentiable function such that and (3.30) holds. Furthermore, assume and
or
for all large T. Then every solution of (1.1) is oscillatory on .
Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality we assume that there is a , sufficiently large, such that and on . Again we define as in the proof of Theorem 3.1 by putting and . Proceeding as in the proof of Theorem 3.1, we obtain from (3.12)
First, we assume that inequality (3.31) holds. It follows from (3.5) and being strictly decreasing that
and it follows that
and hence
Using (1.2), we have . Integrating (3.33) from to ∞ and using , we have
Multiplying (3.35) by we get
Let . Then by the definition of and we can pick sufficiently large, so we have
for . Using the Pötzsche chain rule we get
Then from (3.37) and (3.38), we have , and
Taking the lim inf of both sides as we get
Since is arbitrary, we get
where . By using Lemma 2.1, with
we get
It follows from (3.39) and (3.40) that
which contradicts (3.31). Next, we assume that (3.32) holds. Multiplying both sides of (3.33) by , and integrating from T to t () we get
Using integration by parts, the quotient rule and applying the Pötzsche chain rule, we obtain
Let be given. Then using the definition of l, we can assume without loss of generality that T is sufficiently large so that
It follows that
Therefore
It follows that
where and . Using the inequality of Lemma 2.1 with
we get
Then we have
Since and dividing the last inequality by t we have
Taking the lim sup of both sides as we obtain
Since is arbitrary, we get
Using (3.39), we have
Therefore
which contradicts (3.32). □
4 Examples
In this section, we give some examples to illustrate our main results. To obtain the conditions for oscillations, we will use the following fact:
Example 4.1 Consider the second-order delay half-linear dynamic equation on times scales
where , , and . We see that
Taking such that , we get , . Then
Then, taking by Theorem 3.1, we have
Then (4.1) is oscillatory.
Example 4.2 Consider the second-order delay half-linear dynamic equation on times scales
where T is sufficiently large, , , and .
If we choose , we get
When by Theorem 3.1, we have
Then (4.2) is oscillatory on .
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AFG conceived of the study, and participated in its design and coordination. FN carried out the mathematical studies and participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.
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Güvenilir, A.F., Nizigiyimana, F. Oscillation criteria for second-order quasi-linear delay dynamic equations on time scales. Adv Differ Equ 2014, 45 (2014). https://doi.org/10.1186/1687-1847-2014-45
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DOI: https://doi.org/10.1186/1687-1847-2014-45