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Properties of higher-order half-linear functional differential equations with noncanonical operators
Advances in Difference Equations volume 2013, Article number: 54 (2013)
Abstract
Some new results are presented for the oscillatory and asymptotic behavior of higher-order half-linear differential equations with a noncanonical operator. We study the delayed and advanced equations subject to various conditions.
MSC:34C10, 34K11.
1 Introduction
Over the past few years, there has been much research activity concerning the oscillation and asymptotic behavior of various classes of differential equations; we refer the reader to [1–36] and the references cited therein. Half-linear differential equations occur in a variety of real world problems such as in the study of p-Laplace equations, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium; see the related background details reported in [5]. Many authors have studied the properties of solutions of the higher-order differential equation
The operator Lx is said to be in canonical form if ; otherwise, it is called noncanonical. Throughout the paper, we assume that α and β are ratios of odd positive integers, , , , , , and .
Agarwal et al. [6] established a criterion for the existence of bounded solutions of (1.1) under the assumptions that n is even, , and
Zhang et al. [34, 36] obtained some results on asymptotic behavior of (1.1) in the case where (1.2) holds, , and . In [34, 36], an unsolved problem can be formulated as follows.
-
(P)
Is it possible to establish asymptotic criteria for equation (1.1) in the case where ?
As a special case when and , equation (1.1) becomes
Li et al. [24] established the following criterion for (1.3).
Theorem 1.1 (See [[24], Theorem 2.1])
Let (1.2) hold with , , , and for all . Assume that there exists a function with and such that, for all sufficiently large and for all positive constants M and L,
and
where and . Then (1.3) is oscillatory.
The purpose of this paper is to solve question (P) and to improve Theorem 1.1. By a solution of equation (1.1) we mean a function , , which has the property and satisfies (1.1) on . We consider only the solutions satisfying for all and tacitly assume that (1.1) possesses such solutions. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on ; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
2 Main results
In the sequel, all functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough. We use the notation and .
Theorem 2.1 Assume (1.2) and let , , and for all . Further, assume that the differential equation
is oscillatory for some constant . If
holds for some constant and for all constants , then every solution of (1.1) is oscillatory or tends to zero as .
Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that x is eventually positive. Moreover, suppose that . It follows from (1.1) that there exist two possible cases:
-
(1)
, , , ;
-
(2)
, , ,
for , where is large enough.
Assume that case (1) holds. From [[36], Lemma 2.1], we have
for every and for all sufficiently large t. Hence by (1.1), we see that is a positive solution of the differential inequality
Using [[28], Theorem 1], we see that equation (2.1) also has a positive solution, which is a contradiction.
Assume that case (2) holds. Define the function w by
Then for . Noting that is decreasing, we have
Dividing the above inequality by and integrating the resulting inequality from t to l, we obtain
Letting , we get
which yields
Thus, by (2.4), we see that
Differentiating (2.4), we have
It follows from (1.1) and (2.4) that
By virtue of (2.5), we have
On the other hand, by [[36], Lemma 2.1], we get
for every and for all sufficiently large t. Then from (2.7), (2.8), and (2.9), there exists a constant such that
Multiplying (2.10) by and integrating the resulting inequality from to t, we have
Set , , and . Using (2.6) and the inequality
we have
which contradicts (2.2). This completes the proof. □
Applying the result of [30] to equation (2.1), we have the following result due to Theorem 2.1.
Corollary 2.2 Assume (1.2) and let , , , and for all . Moreover, assume that there exists a continuously differentiable function φ such that
and
If (2.2) holds for some constant and for all constants , then every solution of (1.1) is oscillatory or tends to zero as .
In the following, we establish some results for (1.1) when is even.
Theorem 2.3 Assume (1.2) and let be even, , , and for all . Further, assume that there exists a function such that
holds for all constants and . If (2.2) holds for some constant and for all constants , then every solution of (1.1) is oscillatory or tends to zero as .
Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that x is eventually positive. Moreover, suppose that . It follows from (1.1) that there exist two possible cases (1) and (2) (as those of the proof of Theorem 2.1).
Assume that case (1) holds. From [[4], Lemma 2.1], we see that for . Define the function u by
Then for and
From [[4], Lemma 2.2], there exist a and a constant with such that
for all . It follows from (1.1), (2.16), (2.17), and (2.18) that
Using and (2.19), we get
for some constant . Set
Using inequality (2.11), we obtain
Substituting the last inequality into (2.20), we get
Integrating (2.21) from to t, we have
which contradicts (2.15). Assume that case (2) holds. Proceeding as in the proof of Theorem 2.1, we can obtain a contradiction to (2.2). This completes the proof. □
Next we establish a result for (1.1) when .
Theorem 2.4 Assume (1.2) and let , , , and for . Further, assume that there exists a function such that
for all constants . If
holds for all constants , then (1.1) is oscillatory.
Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that x is eventually positive. It follows from (1.1) that there exist two possible cases (1) and (2) with (as those of the proof of Theorem 2.1). Assume that case (1) holds. Define
The rest of the proof is similar to that of Theorem 2.3, and so is omitted. Assume that case (2) holds. Similar as in the proof of Theorem 2.1, we can obtain a contradiction to (2.23). This completes the proof. □
Next we establish some oscillation criteria for (1.1) when is even and for all .
Theorem 2.5 Assume (1.2) and let be even, , and for all . Further, assume that there exists a function such that
holds for all constants and . If
then every solution of (1.1) is oscillatory or tends to zero as .
Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that x is eventually positive. Moreover, suppose that . It follows from (1.1) that there exist two possible cases (1) and (2) (as those of the proof of Theorem 2.1).
Assume that case (1) holds. From [[4], Lemma 2.1], we see that for . Define the function u by
Then for and
From [[4], Lemma 2.2], there exist a and a constant with such that
for all . Thus
Similar as in the proof of Theorem 2.3, we can get a contradiction to (2.24). Assume that case (2) holds. We have (2.5) and (2.9) for every and for all sufficiently large t. Thus, we get by (1.1), (2.5), and (2.9) that
Let . Then is a solution of the advanced inequality
It follows from [[8], Lemma 2.3] that the corresponding advanced differential equation
has an eventually positive solution. Using condition (2.25) and [[22], Theorem 1], one can obtain a contradiction. This completes the proof. □
Finally, we establish a result for (1.1) when .
Theorem 2.6 Assume (1.2) and let , , and for . Further, assume that there exists a function such that
for all constants . If
then (1.1) is oscillatory.
Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that x is eventually positive. It follows from (1.1) that there exist two possible cases (1) and (2) with (as those of the proof of Theorem 2.1). Assume that case (1) holds. Define
The rest of the proof is similar to that of Theorem 2.3, and so is omitted. Assume that case (2) holds. Similar as in the proof of Theorem 2.5, we can obtain a contradiction to (2.27). This completes the proof. □
3 Examples and discussions
In the following, we illustrate possible applications with two examples.
Example 3.1 For , consider the second-order delay differential equation
Let , , and . Note that . Using Theorem 2.4, equation (3.1) is oscillatory. It is not difficult to see that Theorem 1.1 fails to apply due to condition (1.4).
Example 3.2 For , consider the second-order advanced differential equation
Let , , and . Note that . Using Theorem 2.6, equation (3.2) is oscillatory.
In this paper, we suggested some new results on the oscillation and asymptotic behavior of differential equation (1.1). Theorem 2.1 can be applied in the odd-order and even-order equations.
We stress that the study of equation (1.1) in the case (1.2) brings additional difficulties. Since the sign of is not known, our criteria include a pair of assumptions; see, e.g., (2.2) and (2.15). We utilized two different methods (Riccati substitution and comparison method) to deal with the cases and .
References
Agarwal RP, Bohner M, Li W Monographs and Textbooks in Pure and Applied Mathematics 267. In Nonoscillation and Oscillation: Theory for Functional Differential Equations. Marcel Dekker, New York; 2004.
Agarwal RP, Grace SR: Oscillation of certain functional differential equations. Comput. Math. Appl. 1999, 38: 143-153.
Agarwal RP, Grace SR, O’Regan D: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht; 2000.
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.2001.7571
Agarwal RP, Grace SR, O’Regan D: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic, Dordrecht; 2002.
Agarwal RP, Grace SR, O’Regan D: The oscillation of certain higher-order functional differential equations. Math. Comput. Model. 2003, 37: 705-728. 10.1016/S0895-7177(03)00079-7
Agarwal RP, Shieh S-L, Yeh C-C: Oscillation criteria for second-order retarded differential equations. Math. Comput. Model. 1997, 26: 1-11.
Baculíková B: Properties of third-order nonlinear functional differential equations with mixed arguments. Abstr. Appl. Anal. 2011, 2011: 1-15.
Baculíková B, Džurina J: Oscillation of third-order nonlinear differential equations. Appl. Math. Lett. 2011, 24: 466-470. 10.1016/j.aml.2010.10.043
Baculíková B, Džurina J: Oscillation of third-order functional differential equations. Electron. J. Qual. Theory Differ. Equ. 2010, 43: 1-10.
Baculíková B, Džurina J, Graef JR: On the oscillation of higher order delay differential equations. Nonlinear Oscil. 2012, 15: 13-24.
Dahiya RS: Oscillation criteria of even-order nonlinear delay differential equations. J. Math. Anal. Appl. 1976, 54: 653-665. 10.1016/0022-247X(76)90184-0
Džurina J, Baculíková B: Oscillation and asymptotic behavior of higher-order nonlinear differential equations. Int. J. Math. Math. Sci. 2012, 2012: 1-9.
Džurina J, Stavroulakis IP: Oscillation criteria for second order delay differential equations. Appl. Math. Comput. 2003, 140: 445-453. 10.1016/S0096-3003(02)00243-6
Erbe L, Kong Q, Zhang B: Oscillation Theory for Functional Differential Equations. Marcel Dekker, New York; 1995.
Grace SR: Oscillation theorems for n th-order differential equations with deviating arguments. J. Math. Anal. Appl. 1984, 101: 268-296. 10.1016/0022-247X(84)90066-0
Grace SR: Oscillation theorems for certain functional differential equations. J. Math. Anal. Appl. 1994, 184: 100-111. 10.1006/jmaa.1994.1187
Grace SR, Agarwal RP, Pavani R, Thandapani E: On the oscillation of certain third order nonlinear functional differential equations. Appl. Math. Comput. 2008, 202: 102-112. 10.1016/j.amc.2008.01.025
Grace SR, Lalli BS: Oscillation theorems for n th-order delay differential equations. J. Math. Anal. Appl. 1983, 91: 352-366. 10.1016/0022-247X(83)90157-9
Grace SR, Lalli BS: Oscillation of even order differential equations with deviating arguments. J. Math. Anal. Appl. 1990, 147: 569-579. 10.1016/0022-247X(90)90371-L
Kartsatos AG: On oscillation of solutions of even order nonlinear differential equations. J. Differ. Equ. 1969, 6: 232-237. 10.1016/0022-0396(69)90014-X
Kitamura Y, Kusano T: Oscillation of first-order nonlinear differential equations with deviating arguments. Proc. Am. Math. Soc. 1980, 78: 64-68. 10.1090/S0002-9939-1980-0548086-5
Ladde GS, Lakshmikantham V, Zhang BG: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker, New York; 1987.
Li T, Han Z, Zhang C, Sun S: On the oscillation of second-order Emden-Fowler neutral differential equations. J. Appl. Math. Comput. 2011, 37: 601-610. 10.1007/s12190-010-0453-0
Li T, Thandapani E: Oscillation of solutions to odd-order nonlinear neutral functional differential equations. Electron. J. Differ. Equ. 2011, 23: 1-12. 10.1007/s10884-010-9200-3
Mahfoud WE: Oscillation and asymptotic behavior of solutions of n th order nonlinear delay differential equations. J. Differ. Equ. 1977, 24: 75-98. 10.1016/0022-0396(77)90171-1
Philos CG: A new criterion for the oscillatory and asymptotic behavior of delay differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. 1981, 39: 61-64.
Philos CG: On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Arch. Math. 1981, 36: 168-178. 10.1007/BF01223686
Rogovchenko YV, Tuncay F: Oscillation theorems for a class of second order nonlinear differential equations with damping. Taiwan. J. Math. 2009, 13: 1909-1928.
Tang X: Oscillation for first order superlinear delay differential equations. J. Lond. Math. Soc. 2002, 65: 115-122. 10.1112/S0024610701002678
Xu Z, Xia Y: Integral averaging technique and oscillation of certain even order delay differential equations. J. Math. Anal. Appl. 2004, 292: 238-246. 10.1016/j.jmaa.2003.11.054
Yildiz MK, Öcalan Ö: Oscillation results of higher order nonlinear neutral delay differential equations. Selçuk J. Appl. Math. 2010, 11: 55-62.
Zhang B: Oscillation of even order delay differential equations. J. Math. Anal. Appl. 1987, 127: 140-150. 10.1016/0022-247X(87)90146-6
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.2012.08.004
Zhang C, Agarwal RP, Bohner M, Li T: Oscillation of third-order nonlinear delay differential equations. Taiwan. J. Math. 2013, 17(2):545-558.
Zhang C, Li T, Sun Bo, Thandapani E: On the oscillation of higher-order half-linear delay differential equations. Appl. Math. Lett. 2011, 24: 1618-1621. 10.1016/j.aml.2011.04.015
Acknowledgements
This research is supported by NNSF of P.R. China (Grant Nos. 61034007, 51277116, 50977054).
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Zhang, C., Agarwal, R.P., Bohner, M. et al. Properties of higher-order half-linear functional differential equations with noncanonical operators. Adv Differ Equ 2013, 54 (2013). https://doi.org/10.1186/1687-1847-2013-54
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DOI: https://doi.org/10.1186/1687-1847-2013-54