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Oscillation criteria for second-order nonlinear dynamic equations
Advances in Difference Equations volume 2012, Article number: 171 (2012)
Abstract
This paper concerns the oscillation of solutions to the second-order dynamic equation
on a time scale which is unbounded above. No sign conditions are imposed on , , and . The function is assumed to satisfy and for . In addition, there is no need to assume certain restrictive conditions and also the both cases
are considered. Our results will improve and extend results in (Baoguo et al. in Can. Math. Bull. 54:580-592, 2011; Bohner et al. in J. Math. Anal. Appl. 301:491-507, 2005; Hassan et al. in Comput. Math. Anal. 59:550-558, 2010; Hassan et al. in J. Differ. Equ. Appl. 17:505-523, 2011) and many known results on nonlinear oscillation. These results have significant importance to the study of oscillation criteria on discrete time scales such as , , , or and the space of harmonic numbers . Some examples illustrating the importance of our results are also included.
MSC:34K11, 39A10, 39A99.
1 Introduction
The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD dissertation written under the direction of Bernd Aulbach (see [1]). Since then a rapidly expanding body of literature has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus. Recall that a time scale is a nonempty closed subset of the reals, and the cases when this time scale is the reals or the integers represent the classical theories of differential and of difference equations. Not only does the new theory of the so-called ‘dynamic equations’ unify the theories of differential equations and difference equations, but it also extends these classical cases to cases ‘in between’, e.g., to the so-called q-difference equations when , and can be applied to different types of time scales like , , and the space of harmonic numbers . In this work, knowledge and understanding of time scales and time scale notation is assumed. For an excellent introduction to the calculus on time scales, see Bohner and Peterson [2, 3]. We recall the following concepts related to the notion of time scales. A time scale is an arbitrary nonempty closed subset of the real numbers ℝ. For , we define the forward and backward jump operators and by
where and , where ∅ denotes the empty set. A point , , is said to be left-dense if , right-dense if and , left-scattered if , and right-scattered if . A function is said to be right-dense continuous (rd-continuous) provided g is continuous at right-dense points and at left-dense points in , left-hand limits exist and are finite. The set of all such rd-continuous functions is denoted by .
We are concerned with the oscillatory behavior of the following second-order dynamic equation
on a time scale which is unbounded above, where r, p, and q are real-valued, right-dense continuous functions on . The function is assumed to satisfy and for .
By a solution of (1.1), we mean a nontrivial real-valued function , which has the property that and satisfies equation (1.1) on , where 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 said to be nonoscillatory. There has been a great deal of research into obtaining criteria for oscillation of all solutions of dynamic equations on time scales. It is usually assumed that r, p, q are nonnegative functions. We refer the reader to the papers [4–11, 22–33] and the references cited therein. On the other hand, very little is known about equations when no explicit sign assumptions are made with respect to the coefficient functions p, q, and r. In the papers [12–14], it is shown that one may relate oscillation and boundedness of solutions of the nonlinear equation (1.1) to a related linear equation, which, in the case and , reduces to
where , for which many oscillation criteria are known. However, it was assumed that the nonlinearity has the property
Bohner, Erbe, and Peterson [15] studied the second-order nonlinear dynamic equation
where p is a positively regressive function and satisfies condition (A), that is,
for all large T and
where and . Oscillation criterion for equation (1.2) is shown in [15] when satisfied condition (B), if for each there exists such that provided . We say satisfies condition (C) if there is an such that , , where χ is the characteristic function of the set . We note that if satisfies condition (C), then the subset of defined by
is necessarily countable and . Then we can rewrite by
and so
where A is the set of all integers for which the real open interval is contained in . To be precise, we have
Baoguo, Erbe, and Peterson established in [16] some oscillation criteria of Kiguradze-type in particular, for the second-order superlinear dynamic equation
where satisfies condition (C) and satisfies the superlinearity condition
Hassan, Erbe, and Peterson [17] improved these results and generalized these to the superlinear dynamic equation (1.1), where the coefficient functions r, p, q are allowed to change sign for large t, and satisfies condition (C). Also, Hassan, Erbe, and Peterson [18] applied these results to the sublinear dynamic equation (1.1), where the functions r, p, q are also allowed to change sign for large t, and satisfies condition (C), and satisfies the sublinearity condition
A number of sufficient conditions for oscillation were obtained in [15, 16] for the case when
and in [17, 18] for the case when
where ϕ is a rd-continuous function. Therefore, it will be of great interest to establish oscillation criteria for (1.1) for both of the cases
and
where the function ϕ will be defined in the next section. We will still assume that the functions , , and change sign for arbitrarily large values of t, and conditions (A), (B), and (C) are not needed. Hence, our results will improve and extend results in [15–18] and many known results on nonlinear oscillation. In addition, linear, sublinear, and superlinear results will be presented.
These results have significant importance to the study of oscillation criteria on discrete time scales such as , , , or and the space of harmonic numbers . In particular, we give examples where the coefficient function r changes sign for large t as well as p and q and without conditions (1.3) and (1.4).
2 Main results
Before stating our main results, we begin with the following lemma (Second Mean Value Theorem) which will play an important role in the proof of our main results.
Lemma 2.1 ([[3], Theorem 5.45])
Let h be a bounded function that is integrable on . Let and be the infimum and supremum of the function on respectively. Suppose that g is a nonnegative and nonincreasing function on . Then there is some number Λ with such that
Our first result is stated in terms of an auxiliary function .
Theorem 2.1 Assume that f satisfies (1.3). If there exists a function ϕ such that
where and
then every solution of equation (1.1) is oscillatory.
Proof Assume (1.1) has a nonoscillatory solution x on . Then, without loss of generality, there is a solution x of (1.1) and a such that on . Define
Then from the product and quotient rules and the Pötzsche chain rule [[2], Theorem 1.90], we get
where for , . Therefore, from (1.1), we have
Since , , for all and , for all , we get
Integrating the above inequality from T to t (≥T), we obtain
We claim that is bounded above for all . Since and , we have from Lemma 2.1 that for each ,
where , and where m and M denote the infimum and supremum, respectively, of the function for . Define
and so
For a fixed point , we have
and so
Then
And so, from (2.6), we have
Also,
Then, from (1.3), we get
Hence, it follows that
and from (2.5), for all , we have
From (2.4) and (2.7), we get for ,
In view of condition (2.2), it follows from the last inequality that there exists a sufficiently large such that
Also, from (2.2), there exists such that
Indeed, (1.1) yields on integration

Now by the integration by parts, we have

and by the integration by parts and the Pötzsche chain rule and then by (2.8), we get
since and for all and for . Using (2.9) and (2.10) in (2.11), we get
and so
Since , we conclude from (2.2) that , which is a contradiction. This completes the proof. □
In the case , , , and , , Theorem 2.1 is due to Kiguradze [19].
If , , , , , , and , then Theorem 2.1 includes Theorem 4.1 in Hooker and Patula [[20], Theorem 4.1] and Mingarelli [21].
Suppose that there exists a function ϕ such that, for ,
Then, in this case, we have , and so we do not need to assume the superlinearity conditions (1.3), and so the result applies to linear, sublinear, and superlinear case.
Corollary 2.1 If there exists a function ϕ such that (2.2) and (2.12) hold, then every solution of equation (1.1) is oscillatory.
If we do not assume superlinearity condition (1.3) and condition (2.12), then we can conclude that all bounded solutions are oscillatory.
Corollary 2.2 If there exists a function ϕ such that (2.1) and (2.2) hold, then every bounded solution of equation (1.1) is oscillatory.
In the following, we assume that there exists a function ϕ such that
holds and establish some sufficient conditions for equation (1.1).
Theorem 2.2 Assume that f satisfies (1.3). If there exists a function ϕ such that (2.1) and (2.13) hold and if
and
then every solution of equation (1.1) is either oscillatory or tends to zero.
Proof Assume (1.1) has a nonoscillatory solution x on . Then, without loss of generality, there is a solution x of (1.1) and a such that on . As in the proof of Theorem 2.1, by using (2.14), we have that there exists a sufficiently large such that
and so
which yields
That is,
so integration from to t () yields
Define
Then
Since , we have . If we assume , then
so
and let to get a contradiction to (2.15). This completes the proof. □
Corollary 2.3 If there exists a function ϕ such that (2.12), (2.13), (2.14), and (2.15) hold, then every solution of equation (1.1) is either oscillatory or tends to zero.
Corollary 2.4 If there exists a function ϕ such that (2.1), (2.13), (2.14), and (2.15) hold, then every bounded solution of equation (1.1) is either oscillatory or tends to zero.
Next we present oscillation criteria for equation (1.1) where f satisfies sublinearity condition (1.4).
Theorem 2.3 Assume that f satisfies (1.4). If there exists a function ϕ such that (2.12), (2.13), (2.14), and (2.15), then every solution of equation (1.1) is oscillatory.
Proof Assume (1.1) has a nonoscillatory solution x on . Then, without loss of generality, there is a solution x of (1.1) and a such that on . As in the proof of Theorems 2.1 and 2.2, we have that there exists a sufficiently large such that
and, from (2.16) and (2.17), we get
Since , then and so
Then
which is a contradiction to (2.15). This completes the proof. □
Theorem 2.4 Assume that f satisfies (1.4). If there exists a function ϕ such that (2.1), (2.13), (2.14), and (2.15) hold, then every bounded solution of equation (1.1) is oscillatory.
The results are very general. With appropriate choices of , we can obtain several sufficient conditions for the oscillation of equation (1.1).
3 Examples
In this section, we give two examples to illustrate our main results. We note that in the first example, all of the coefficient functions , , change sign for arbitrarily large values of t, and the function f may be linear, superlinear, or sublinear, since we do not assume nonlinearity conditions (1.3) and (1.4).
Example 3.1 Let and be a discrete time scale satisfying on . Consider the second-order nonlinear dynamic equation with damping
where f satisfies and for . Define, for , ,
and
where , such that and . Therefore,
so (2.12) is satisfied. Also,
and
by Example 5.60 in [3]. Then, by Theorem 2.1, every solution of equation (3.1) is oscillatory.
Example 3.2 Let and consider the difference equation
where f satisfies and for and nonlinearity condition (1.3). Define
and
where and . It is easy to see that (2.1) and hold. Note that
If , we get
Then, by Theorem 2.1, every solution of equation (3.2) is oscillatory.
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Hassan, T.S. Oscillation criteria for second-order nonlinear dynamic equations. Adv Differ Equ 2012, 171 (2012). https://doi.org/10.1186/1687-1847-2012-171
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DOI: https://doi.org/10.1186/1687-1847-2012-171
Keywords
- oscillation
- second order
- dynamic equations
- time scales