Oscillation of second-order nonlinear dynamic equations with positive and negative coefficients
© Chen et al.; licensee Springer 2013
Received: 18 January 2013
Accepted: 27 May 2013
Published: 13 June 2013
The paper considers the oscillation of a second-order nonlinear dynamic equation with positive and negative coefficients of the form
on an arbitrary time scale . We obtain some oscillation criteria for the equation by developing a generalized Riccati substitution technique. Our results extend and improve some known results in the literature. Several examples are given to illustrate our main results.
1 Introduction and preliminaries
on an arbitrary time scale with , subject to the following conditions:
(C1) and is a time scale interval in ;
(C2) and ;
(C4) , , δ has the inverse function , , , for , and , where and ;
(C5) , there exist positive constants , and M such that , and for , and for ;
(C6) for every sufficiently large .
Recall that a solution of (1) is a nontrivial real function x such that and for a certain and satisfying (1) for . Our attention is restricted to those solutions of (1) which exist on and satisfy for any . A solution x of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.
A time scale is an arbitrary nonempty closed subset of the real numbers ℝ. We assume throughout that has the topology that it inherits from the standard topology on the real numbers ℝ. Some examples of time scales are as follows: the real numbers ℝ, the integers ℤ, the positive integers ℕ, the nonnegative integers , , , and . But the rational numbers ℚ, the complex numbers ℂ and the open interval are not time scales. Many other interesting time scales exist, and they give rise to plenty of applications (see ).
Open time scale intervals and half-open time scale intervals etc. are defined accordingly.
In this case, we say that is the (delta) derivative of f at t and that f is (delta) differentiable at t.
where and .
The calculus on time scales was introduced by Hilger  with the motivation of providing a unified approach to continuous and discrete calculus. The theory of dynamic equations on time scales not only unifies the theories of differential equations and difference equations, but it is also able to extend these classical cases to cases ‘in between’, e.g., to the so-called q-difference equations. Dynamic equations on time scales have an enormous potential for modeling a variety of applications; see, for example, the monograph by Bohner and Peterson . For advances in dynamic equations on time scales, one can see the book by Bohner and Peterson .
where σ is the forward jump operator on , , , r and p are rd-continuous functions and at all left-dense and right-scattered points. They established a necessary and sufficient condition for the oscillation of (10) by using the so-called trigonometric transformation.
Medico and Kong [5, 6] also investigated the oscillation of (10). They supposed that with . Medico and Kong  gave some Kamenev-type and interval criteria for the oscillation of (10). Their results covered those for differential equations and offered new oscillation criteria for difference equations. Medico and Kong  extended the work in  by modifying the class of kernel functions and deriving new criteria of Sun type (see ).
on time scales in terms of the coefficients and the graininess function, where r and p are positive real-valued rd-continuous functions, or , and such that and for .
Erbe and Peterson  also discussed the oscillation of (11), where . When no explicit sign assumptions are made with respect to the coefficient p, they established some sufficient conditions for the oscillation of (11) when for sufficiently large T.
on a time scale , where , , is a real-valued rd-continuous function, is continuous, is nondecreasing, for , and for . They established the equivalence of the oscillation of (12) and (13), from which they obtained some oscillation criteria and a comparison theorem for (12).
on a time scale interval , where is a positive function, is an increasing function such that and , and satisfies for a certain positive constant L and for all .
where r and p are real rd-continuous positive functions defined on , the so-called delay function ξ satisfies is rd-continuous, for , , and is a continuous function satisfying for all and . The authors obtained some new oscillation criteria which improved the results established by Zhang and Zhu  and Şahiner .
Jia et al.  also dealt with the oscillation of (13), where , is a time scale, and is continuously differentiable and satisfies and for . Jia et al.  obtained several Kiguradze-type oscillation theorems for (13).
where is allowed to oscillate.
where , , , , and are strictly increasing and unbounded functions. The authors weakened the assumptions on the coefficients that are assumed to hold in the literature and improved some known results by providing necessary and sufficient conditions for the solutions of the equation to oscillate or to converge to zero. The coefficient associated with the neutral part was considered in three distinct ranges, in one of which the coefficient is allowed to oscillate.
to oscillate or to tend to zero as t tends to infinity, where is an integer, and . Both bounded and unbounded solutions were considered in this paper.
For some recent other results on the oscillation, nonoscillation and asymptotic behavior of solutions of different types of dynamic equations, we refer the reader to the papers [17–44] and the references cited therein.
The results in [4–6, 8–16] are very valuable. But these results also have some disadvantages. For example, the oscillation criteria of Došlý and Hilger  are unsatisfactory since additional assumptions have to be imposed on the unknown solutions.
is oscillatory if , but (17) does not give this result.
The restriction for is required in . This condition does not hold and cannot be applied in the case when , since changes sign four times.
Besides the above-mentioned disadvantages, it is clear that (10)-(15) are some special cases of (1), and that the results in [4–6, 8–13] cannot be applied to general cases of (1). Therefore, it is of great interest to investigate the oscillation of (1). To the best of our knowledge, nothing is known regarding the oscillatory behavior of (1) on time scales up to now. Following the trend shown in [4–6, 8–16], in this paper we deal with the oscillation of (1). We obtain some oscillation criteria for (1) by developing a generalized Riccati substitution technique. Our results are essentially new and extend and improve some results in [4–6, 8–13]. We also illustrate our main results with several examples.
In what follows, for convenience, when we write a functional inequality or equality without specifying its domain of validity, we assume that it holds for all sufficiently large t.
Lemma 2.1 (Substitution [, Theorem 1.98])
where is the inverse function of η and denotes the derivative on .
Lemma 2.2 (Existence of antiderivatives [, Theorem 1.74])
is an antiderivative of f.
Lemma 2.3 (Chain rule [, Theorem 1.93])
where denotes the derivative on .
3 Main results
where . Then every solution of (1) is oscillatory.
which contradicts (18). Thus, the proof is complete. □
where is an arbitrary constant and . Then all the solutions of (1) are oscillatory.
Thus, we find , which contradicts (40). Hence, the proof is complete. □
for every sufficiently large T, where and . Then all the solutions of (1) are oscillatory.
which contradicts (44). Therefore, this completes the proof. □
Remark 3.1 The results in this paper are of higher degree of generality. From Theorems 3.1-3.3, we can obtain many different sufficient conditions for the oscillation of (1) with different choices of the functions a, G and g. For instance, let , then we derive the following result from Theorem 3.1 or Theorem 3.2.
for every sufficiently large T. Then all the solutions of (1) are oscillatory.
Let , then from Theorem 3.1 we have the following corollary.
for every sufficiently large T. Then every solution of (1) is oscillatory.
Let for , where is a constant, then for (see Remark 3.3 in ). Take and let satisfy (43), then and for . In this case, Theorem 3.3 implies the following result.
Then all the solutions of (1) are oscillatory.
where is a constant, , are positive integers, , , σ is the forward jump operator on , and satisfies and for .
we get , which implies that (49) holds. Therefore, by Corollary 3.1 every solution of (52) is oscillatory.
for , where is a constant, , σ is the forward jump operator on , and .
which yields that (50) holds. Hence, by Corollary 3.2 every solution of (53) is oscillatory.
which implies that (49) does not hold. Therefore, Corollary 3.1 cannot be applied to (53).
we find that (51) holds. Thus, by Corollary 3.3 every solution of (54) is oscillatory.
The authors would like to express their deep gratitude to the anonymous referees for their valuable suggestions and comments, which helped the authors to improve the previous manuscript of the article. This work was supported by the National Natural Science Foundation of P.R. China (Grants No. 11271311 and No. 61104072) and the Natural Science Foundation of Hunan of P.R. China (Grant No. 11JJ3010).
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