Oscillation of fourth-order neutral differential equations with p-Laplacian like operators
© Li et al.; licensee Springer. 2014
Received: 3 January 2014
Accepted: 11 February 2014
Published: 14 March 2014
We study oscillatory behavior of a class of fourth-order neutral differential equations with a p-Laplacian like operator using the Riccati transformation and integral averaging technique. A Kamenev-type oscillation criterion is presented assuming that the noncanonical case is satisfied. This new theorem complements and improves a number of results reported in the literature. An illustrative example is provided.
It is interesting to study equation (1.1) since the p-Laplace differential equations have applications in continuum mechanics as seen from . Throughout, we assume that is a constant, , , , , , , , , , there exists a function such that for , , , and .
We use the notation . By a solution of (1.1), we mean a function which has the property and satisfies (1.1) on . We consider only those solutions x of (1.1) which satisfy for all and tacitly assume that (1.1) possesses such solutions. A solution x of (1.1) is called oscillatory if it has arbitrarily large zeros on ; otherwise, it is said to be nonoscillatory. Equation (1.1) is termed oscillatory if all its solutions oscillate.
has been studied by Li et al. . Note that [, Theorem 2.2] ensures that every solution x of the studied equation is either oscillatory or tends to zero as and, unfortunately, cannot distinguish solutions with different behaviors.
It should be noted that research in this paper is strongly motivated by the recent paper . The purpose of this paper is to establish a Kamenev-type theorem which guarantees that all solutions of equation (1.1) are oscillatory in the case where (1.3) holds and without requiring conditions (1.4). In the sequel, all functional inequalities are assumed to hold for all t large enough.
2 Main results
We begin with the following lemma.
Lemma 2.1 (See )
for all .
Lemma 2.2 (See [, Lemma 2.2.3])
for all .
then equation (1.1) is oscillatory.
, , , , ;
, , , , ;
, , , , .
We consider each of these cases separately.
which contradicts (2.2).
which contradicts (2.3).
which contradicts (2.4). This completes the proof. □
Remark 2.4 Choosing different combinations of functions H, ρ, K, and δ, one can derive from Theorem 2.3 a variety of efficient tests for oscillation of equation (1.1) and its particular cases.
3 Example and discussion
The following example illustrates applications of Theorem 2.3.
Let , , , and . It is not difficult to verify that all assumptions of Theorem 2.3 are satisfied, and hence equation (3.1) is oscillatory. As a matter of fact, one such solution is .
Remark 3.2 Oscillation theorem established in this paper for equation (1.1) complements, on one hand, results reported by Baculíková and Džurina , Karpuz , and Li et al.  because we use assumption (1.3) rather than (1.2) and, on the other hand, those by Li et al.  and Zhang et al. [16, 17, 19–21] since our theorem can be applied to the case where .
Remark 3.3 We point out that, contrary to [, Theorem 2.2], Theorem 2.3 does not need restrictive conditions (1.4) and can ensure that all solutions of equation (1.1) oscillate, which, in a certain sense, is a significant improvement compared to [, Theorem 2.2] for fourth-order neutral differential equations.
for future research.
The authors express their sincere gratitude to the 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 research is supported by the National Key Basic Research Program of P.R. China (2013CB035604) and NNSF of P.R. China (Grant Nos. 61034007, 51277116, 51107069).
- Aronsson G, Janfalk U: On Hele-Shaw flow of power-law fluids. Eur. J. Appl. Math. 1992, 3: 343-366. 10.1017/S0956792500000905MathSciNetView ArticleGoogle Scholar
- Bartušek M, Cecchi M, Došlá Z, Marini M: Fourth-order differential equation with deviating argument. Abstr. Appl. Anal. 2012., 2012: Article ID 185242Google Scholar
- Agarwal RP, Bohner M, Li W-T: Nonoscillation and Oscillation: Theory for Functional Differential Equations. Dekker, New York; 2004.View ArticleGoogle Scholar
- Agarwal RP, Grace SR, O’Regan D: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht; 2000.View ArticleGoogle Scholar
- Agarwal RP, Grace SR, O’Regan D: Oscillation criteria for certain n th order differential equations with deviating arguments. J. Math. Anal. Appl. 2001, 262: 601-622. 10.1006/jmaa.2001.7571MathSciNetView ArticleGoogle Scholar
- Baculíková B, Džurina J: Oscillation theorems for higher order neutral differential equations. Appl. Math. Comput. 2012, 219: 3769-3778. 10.1016/j.amc.2012.10.006MathSciNetView ArticleGoogle Scholar
- Karpuz B: Sufficient conditions for the oscillation and asymptotic behaviour of higher-order dynamic equations of neutral type. Appl. Math. Comput. 2013, 221: 453-462.MathSciNetView ArticleGoogle Scholar
- Kiguradze IT, Chanturia TA: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic, Dordrecht; 1993. Translated from the 1985 Russian originalView ArticleGoogle Scholar
- Li T, Han Z, Zhao P, Sun S: Oscillation of even-order neutral delay differential equations. Adv. Differ. Equ. 2010., 2010: Article ID 184180Google Scholar
- Li T, Thandapani E, Tang S: Oscillation theorems for fourth-order delay dynamic equations on time scales. Bull. Math. Anal. Appl. 2011, 3: 190-199.MathSciNetGoogle Scholar
- Li T, Zhang C, Thandapani E: Asymptotic behavior of fourth-order neutral dynamic equations with noncanonical operators. Taiwan. J. Math. 2014. 10.11650/tjm.18.2014.2678Google Scholar
- Parhi N, Tripathy AK: On oscillatory fourth order nonlinear neutral differential equations. I. Math. Slovaca 2004, 54: 389-410.MathSciNetGoogle Scholar
- Parhi N, Tripathy AK: On oscillatory fourth order nonlinear neutral differential equations. II. Math. Slovaca 2005, 55: 183-202.MathSciNetGoogle Scholar
- Philos ChG: A new criterion for the oscillatory and asymptotic behavior of delay differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. 1981, 39: 61-64.Google Scholar
- Thandapani E, Savitri R: Oscillation and nonoscillation of fourth-order nonlinear neutral differential equations. Indian J. Pure Appl. Math. 2001, 32: 1631-1642.MathSciNetGoogle Scholar
- Zhang C, Agarwal RP, Bohner M, Li T: New results for oscillatory behavior of even-order half-linear delay differential equations. Appl. Math. Lett. 2013, 26: 179-183. 10.1016/j.aml.2012.08.004MathSciNetView ArticleGoogle Scholar
- Zhang C, Agarwal RP, Bohner M, Li T: Properties of higher-order half-linear functional differential equations with noncanonical operators. Adv. Differ. Equ. 2013., 2013: Article ID 54Google Scholar
- Zhang C, Agarwal RP, Li T: Oscillation and asymptotic behavior of higher-order delay differential equations with p -Laplacian like operators. J. Math. Anal. Appl. 2014, 409: 1093-1106. 10.1016/j.jmaa.2013.07.066MathSciNetView ArticleGoogle Scholar
- Zhang C, Li T, Agarwal RP, Bohner M: Oscillation results for fourth-order nonlinear dynamic equations. Appl. Math. Lett. 2012, 25: 2058-2065. 10.1016/j.aml.2012.04.018MathSciNetView ArticleGoogle Scholar
- Zhang C, Li T, Saker SH: Oscillation of fourth-order delay differential equations. Nonlinear Oscil. 2013, 16: 322-335.MathSciNetGoogle Scholar
- Zhang C, Li T, Sun B, Thandapani E: On the oscillation of higher-order half-linear delay differential equations. Appl. Math. Lett. 2011, 24: 1618-1621. 10.1016/j.aml.2011.04.015MathSciNetView ArticleGoogle Scholar
- Hardy GH, Littlewood JE, Polya G: Inequalities. Cambridge University Press, Cambridge; 1988.Google Scholar
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