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A Philos-type theorem for second-order neutral delay dynamic equations with damping
Advances in Difference Equations volume 2016, Article number: 44 (2016)
Abstract
We establish a Philos-type oscillation theorem for a class of nonlinear second-order neutral delay dynamic equations with damping on a time scale by using the Riccati transformation and integral averaging technique. An illustrative example is provided to show that our theorem has practicability and maneuverability.
1 Introduction
Oscillation, as a kind of physical phenomena, widely exists in the natural sciences and engineering. For instance, self-excited vibration in control system, beam vibration in synchrotron accelerator, the complicated oscillation in chemical reaction, and so forth. The assorted phenomena can be unified into oscillation theory of equations; see [1]. On the basis of these background details, we are concerned with the oscillation of a nonlinear second-order damped delay dynamic equation of neutral type
where \(t \in \left.[t_{0}, + \infty )\right._{\mathbf {T}}: = [t_{0}, + \infty ) \cap \mathbf {T}\), T is a time scale satisfying \(\sup \mathbf {T} = + \infty\), \(t_{0} \in \mathbf {T}\), \(\varphi (u): = |u|^{\lambda - 1}u\), \(0 < \lambda \le 1\), \(y(t): = x(t) + B(t)g(x(\tau (t)))\), and
- (A1):
-
\(A, B, b, P \in C_{rd}(\left.[t_{0}, + \infty )\right._{\mathbf {T}},\mathbf {R})\), \(A(t) >0\), \(0 \le B(t) \le b_{0} < + \infty\), \(b(t) \ge 0\), \(- b/A \in \Re^{ +}\) (i.e., \(A(t) - \mu (t)b(t) > 0\)), \(\int_{t_{0}}^{ + \infty} [ e_{ - b/A}(t,t_{0}) / A(t) ]^{1/\lambda} \Delta t = + \infty\), \(P(t) >0\), \(F \in C(\mathbf {R},\mathbf {R})\), \(uF(u) > 0\), and \(F(u) / u \ge L > 0\) for all \(u \ne 0\), where \(b_{0}\) and L are constants;
- (A2):
-
\(\tau \in C_{rd}^{1}(\left.[t_{0}, + \infty )\right._{\mathbf {T}},\mathbf {T})\), \(\delta \in C_{rd}(\left.[t_{0}, + \infty )\right._{\mathbf {T}},\mathbf {T})\), \(\tau (t) \le t\), \(\delta (t) \le t\), \(\tau (\left.[t_{0}, + \infty )\right._{\mathbf {T}}) = \left.[\tau (t_{0}), + \infty )\right._{\mathbf {T}}\), \(\lim_{t \to + \infty} \delta (t) = + \infty\), \(\tau^{\Delta} (t) \ge \tau_{0} > 0\), \(\tau \circ \delta = \delta \circ \tau\), and \(\delta (t) \ge \tau (t)\), where \(\tau_{0}\) is a constant;
- (A3):
-
\(g \in C(\mathbf {R},\mathbf {R})\), \(ug(u) > 0\), and \(g(u) / u \le \eta \le 1\) for all \(u \ne 0\), where \(\eta > 0\) is a constant.
By a solution to equation (1.1) we mean a nontrivial real-valued function x satisfying (1.1) for \(t \in \left.[t_{0}, + \infty )\right._{\mathbf {T}}\). A solution of (1.1) is said to be oscillatory if it has arbitrarily large generalized zeros on \(\left.[t_{0}, + \infty )\right._{\mathbf {T}}\); otherwise, it is termed nonoscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory. It should be noted that the solutions vanishing in some neighborhood of infinity will be excluded from our consideration.
The theory of oscillation is an important branch of the qualitative theory of dynamic equations. In recent years, there has been a great deal of interest in studying oscillatory behavior of solutions to dynamic equations on various classes of time scales; see, for example, [1–26] and the references therein. To establish sufficient conditions for oscillation of dynamic equations, one usually uses either an integral averaging technique involving integrals and weighted integrals of coefficients of a given dynamic equation (see, e.g., [1–4, 6, 8–22, 24–26]) or comparison methods and linearization techniques; see, for instance, the papers by Agarwal et al. [5], Zhang et al. [23], and the references therein. For some concepts related to the notion of time scales, see [27–29].
Let us briefly comment on a number of closely related background details, which strongly motivated our research in this paper. Assuming that \(B(t) = 0\), Agarwal et al. [2], Qiu and Wang [14], Åženel [16], Zhang [25], and Zhang and Gao [26] investigated oscillatory properties of (1.1), whereas Bohner and Li [8] and Erbe et al. [9] considered oscillation of solutions to a nonlinear second-order dynamic equation
under the assumption that γ is a ratio of two odd positive integers. In a special case where \(g(u) = u\), \(b(t) = 0\), and λ is the quotient of two odd positive integers, oscillatory behavior of (1.1) was studied by Li and Saker [12] and Saker and O’Regan [15]. Agarwal et al. [5] and Zhang et al. [22] established several oscillation results for a second-order linear neutral dynamic equation
assuming that
and
respectively. Very recently, assuming that (1.2) is satisfied and using the double Riccati substitutions, Li and Rogovchenko [11] obtained two oscillation theorems for the second-order neutral delay differential equation
Our principal goal is to analyze oscillation of (1.1) assuming that conditions (A1)-(A3) are satisfied. It should be noted that the topic of this paper is new for dynamic equations on time scales due to the fact that the results reported in [2, 5, 8, 9, 12, 14–16, 22, 25, 26] cannot be applied to a general equation (1.1) in the case where \(B(t) \ne 0\) and \(b(t) \ne 0\).
2 Philos-type oscillation result
In the section, we employ the integral averaging technique to establish a Philos-type (see [13]) oscillation theorem for (1.1). To this end, let
We say that a function \(H \in C_{rd}(D,\mathbf {R})\) belongs to the class Ω (denoted by \(H \in \Omega\)) if
and it has a nonpositive rd-continuous Δ-partial derivative \(H^{\Delta_{s}}(t,s)\) with respect to the second variable satisfying the condition
for some function \(h \in C_{rd}(D_{0},\mathbf {R})\), where the meaning of Ï• will be explained later.
We need the following auxiliary results.
Lemma 2.1
(see [27], Theorem 1.93)
Assume that \(\nu: \mathbf {T} \to \mathbf {R}\) is strictly increasing, \(\tilde{\mathbf {T}}: = \nu (\mathbf {T})\) is a time scale, and let \(w:\tilde{\mathbf {T}} \to \mathbf {R}\). If \(\nu^{\Delta} (t)\) and \(w^{\tilde{\Delta}} (\nu (t))\) exist for \(t \in \mathbf {T}^{k}\), then
Lemma 2.2
(see [18], Lemma 2.3)
If \(0 < \lambda \le 1\) and \(x_{1}, x_{2} \in [0, + \infty)\), then \(x_{1}^{\lambda} + x_{2}^{\lambda} \ge (x_{1} + x_{2})^{\lambda}\).
In what follows, all functional inequalities are tacitly supposed to hold for all t large enough.
Theorem 2.1
Assume that conditions (A1)-(A3) are satisfied. If there exist two functions \(\phi \in C_{rd}^{1}(\left.[t_{0}, + \infty )\right._{\mathbf {T}},(0, + \infty ))\) and \(H \in \Omega\) such that
for all sufficiently large \(t_{1} \in \left.[t_{0}, + \infty )\right._{\mathbf {T}}\), where
and
then (1.1) is oscillatory.
Proof
Suppose to the contrary that x is a nonoscillatory solution of (1.1) on \(\left.[t_{0}, + \infty )\right._{\mathbf {T}}\). Without loss of generality, we may assume that there is a \(t_{1} \in \left.[t_{0}, + \infty )\right._{\mathbf {T}}\) such that \(x(t) > 0\), \(x(\tau (t)) > 0\), and \(x(\delta (t)) > 0\) for \(t \in \left.[t_{1}, + \infty )\right._{\mathbf {T}}\) (since the proof of the case where x is eventually negative is similar). Then \(y(t) > 0\). An application of (1.1) implies that, for \(t \in \left.[t_{1}, + \infty )\right._{\mathbf {T}}\),
Thus, using ([27], Theorem 2.33), we conclude that
and so \(A\varphi (y^{\Delta} ) / e_{ - b/A}( \cdot,t_{0})\) is decreasing and eventually of one sign. That is, \(y^{\Delta}\) is either eventually positive or eventually negative. We assert that
If (2.4) does not hold, then there exists a \(t_{2} \in \left.[t_{1}, + \infty )\right._{\mathbf {T}}\) such that \(y^{\Delta} (t) < 0\) for \(t \in \left.[t_{2}, + \infty )\right._{\mathbf {T}}\). From (2.3), for \(t \in \left.[t_{2}, + \infty )\right._{\mathbf {T}}\), we obtain
where
By virtue of (2.5),
An integration of the latter inequality yields
which contradicts the fact that \(y(t) > 0\). Using Lemma 2.1 and the condition \(\tau (\left.[t_{0}, + \infty )\right._{\mathbf {T}}) = \left.[\tau (t_{0}), + \infty )\right._{\mathbf {T}}\), we get \([A(\tau (t))\varphi (y^{\Delta} (\tau (t)))]^{\Delta} = [A\varphi (y^{\Delta} )]^{\Delta} (\tau (t))\tau^{\Delta} (t)\). Hence, by (2.2), we have
Combining (2.2) and (2.6), we deduce that
Note that \(\tau^{\Delta} \ge \tau_{0} > 0\), \(\tau \circ \delta = \delta \circ \tau\), and \(y(t) \le x(t) + b_{0}\eta x(\tau (t))\). It follows from the definition of ξ, (2.4), \(\delta (t) \ge \tau (t)\), and Lemma 2.2 that
An application of \([A\varphi (y^{\Delta} )]^{\Delta} < 0\) implies that
Using these inequalities in (2.7), we conclude that
Define the function W by
Then \(W > 0\). From (2.4) and ([27], Theorems 1.90 and 1.93), we see that
By virtue of (2.9),
The inequality \([A\varphi (y^{\Delta} )]^{\Delta} < 0\) yields \(A(\tau (t))(y^{\Delta} (\tau (t)))^{\lambda} \ge A(\sigma (t))(y^{\Delta} (\sigma (t)))^{\lambda}\), and hence
Then, using the condition \(y(\tau (t)) \le y(\tau (\sigma (t)))\), we have
Define another function V by
Then \(V > 0\). With a similar proof as before, we conclude that
Applications of (2.8), (2.10), (2.12), and the condition \(y(\tau (t)) \le y(\tau (\sigma (t)))\) imply that
which yields, for some \(t_{3} \in \left.[t_{1}, + \infty )\right._{\mathbf {T}}\),
Multiplying both sides of this inequality by \(H(t,s)\) and integrating the resulting inequality from \(t_{3}\) to t, we obtain
Applying the inequality (see [11])
we have
and
Using (2.14) and (2.15) in (2.13), we conclude that
and hence
which contradicts (2.1). This completes the proof. □
Remark 2.1
One can deduce from Theorem 2.1 a great number of oscillation criteria for (1.1) with different choices of the functions ϕ and H.
Remark 2.2
Results reported in this paper can be easily extended to (1.1) for \(\lambda > 1\) when using the inequality
As a matter of fact, we have to replace \(\xi (t): = \min \{ P(t),P(\tau (t))\}\) with
3 Example
The following example is given to illustrate possible applications of theoretical results obtained in Section 2.
Example 3.1
For \(\mathbf {T} = \overline{2^{\mathbf {Z}}}\) and \(t \in \left.[2, + \infty )\right._{\mathbf {T}}\), consider the second-order 2-difference equation
where \(p_{0} > 0\) is a constant. Let \(A(t) = 1\), \(B(t) = b_{0} = 5\), \(b(t) = t^{ - 5 / 2}\), \(P(t) = p_{0}\), \(\tau^{\Delta} = 1 / 2\), \(\eta = L = 1\), \(\phi (t) = 1\), and \(H(t,s) = (t - s)^{2}\). By virtue of ([7], Lemma 2),
It is not difficult to verify that all assumptions of Theorem 2.1 are satisfied. Therefore, equation (3.1) is oscillatory.
Remark 3.1
Using the double Riccati substitutions (2.9) and (2.11), Theorem 2.1 complements and improves the results reported in [2, 5, 8, 9, 12, 14–16, 22, 25, 26] because our criteria can be applied to the case where \(B(t) \ne 0\) and \(b(t) \ne 0\). However, as in the paper by Zhang et al. [22], such flexibility is achieved at the cost of imposing conditions \(\tau (\left.[t_{0}, + \infty )\right._{\mathbf {T}}) = \left.[\tau (t_{0}), + \infty )\right._{\mathbf {T}}\), \(\tau^{\Delta} (t) \ge \tau_{0} > 0\), and \(\tau \circ \delta = \delta \circ \tau\) on the function τ. It would be of interest to analyze the oscillatory behavior of (1.1) with other methods that do not require these restrictive assumptions.
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Acknowledgements
The authors express their sincere gratitude to the editors and two anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details. This work was supported by NNSF of P.R. China (Grant Nos. 61503171, 61403061, and 11447005), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2012FL06), Scientific Research Fund of Education Department of Guangxi Zhuang Autonomous Region (Grant No. 2013YB223), DSRF of Linyi University (Grant No. LYDX2015BS001), and the AMEP of Linyi University, P.R. China.
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Yang, J., Li, T. A Philos-type theorem for second-order neutral delay dynamic equations with damping. Adv Differ Equ 2016, 44 (2016). https://doi.org/10.1186/s13662-016-0767-9
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DOI: https://doi.org/10.1186/s13662-016-0767-9
MSC
- 34K11
- 34N05
- 39A10
Keywords
- oscillation
- neutral delay dynamic equation
- second-order
- damping term
- time scale