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COMPARISON THEOREMS ON THE OSCILLATION AND ASYMPTOTIC BEHAVIOUR OF HIGHER-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

Published online by Cambridge University Press:  31 July 2009

BAŞAK KARPUZ ,
ÖZKAN ÖCALAN  and
SERMIN ÖZTÜRK
BAŞAK KARPUZ
Affiliation:
Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey e-mail: bkarpuz@gmail.com, ozkan@aku.edu.tr, ssahin@aku.edu.tr URLs: http://www2.aku.edu.tr/~bkarpuz(B. Karpuz), http://www2.aku.edu.tr/~ozkan(Ö. Öcalan)
ÖZKAN ÖCALAN
Affiliation:
Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey e-mail: bkarpuz@gmail.com, ozkan@aku.edu.tr, ssahin@aku.edu.tr URLs: http://www2.aku.edu.tr/~bkarpuz(B. Karpuz), http://www2.aku.edu.tr/~ozkan(Ö. Öcalan)
SERMIN ÖZTÜRK
Affiliation:
Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey e-mail: bkarpuz@gmail.com, ozkan@aku.edu.tr, ssahin@aku.edu.tr URLs: http://www2.aku.edu.tr/~bkarpuz(B. Karpuz), http://www2.aku.edu.tr/~ozkan(Ö. Öcalan)
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Abstract

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In this work, oscillatory and asymptotic behaviours of all solutions of higher-order neutral differential equations are compared with first-order delay differential equations, depending on two different ranges of the coefficient associated with the neutral part. Some simple examples are given to compare our results with the existing results in the literature and to illustrate the significance and applicability of our new results. Our results generalise, improve and correct some of the existing results in the literature.

Keywords

34K1134C15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 1 , January 2010 , pp. 107 - 114
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Agarwal, R. P., Grace, S. R. and O'Regan, D., Oscillation theory for difference and functional differential equations (Kluwer Academic, Dordrecht, 2000).CrossRefGoogle Scholar
2.Candan, T. and Dahiya, R. S., Oscillatory and asymptotic behavior of odd order neutral differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 14 (6) (2007), 767–774.Google Scholar
3.Chen, M. and Xu, Z., Interval oscillation of second-order Emden–Fowler neutral delay differential equations, Electron. J. Diff. Eq. (58) (2007), 9.Google Scholar
4.Das, P., Oscillation in odd-order neutral delay differential equations, Proc. Indian Acad. Sci. Math. Sci. 105 (2) (1995), 219–225.CrossRefGoogle Scholar
5.Das, P., Mishra, B. B. and Dash, C. R., Oscillation theorems for neutral delay differential equations of odd order, Bull. Inst. Math. Acad. Sin. 1 (4) (2007) 557–568.Google Scholar
6.Džurina, J., Oscillation theorems for neutral differential equations of higher order, Czechoslovak Math. J. 54 (129) (2004), 185–195.CrossRefGoogle Scholar
7.Erbe, L. H., Kong, Q. and Zhang, B. G., Oscillation theory for functional-differential equations (Marcel Dekker, New York, 1995).Google Scholar
8.Gopalsamy, K., Lalli, B. S. and Zhang, B. G., Oscillation of odd order neutral differential equations, Czechoslovak Math. J. 42 (2) (1992), 313–323.CrossRefGoogle Scholar
9.Győri, I. and Ladas, G., Oscillation theory of delay differential equations: With applications (Oxford University Press, New York, 1991).CrossRefGoogle Scholar
10.Hale, J. K., Theory of functional differential equations (Springer, New York, 1977).CrossRefGoogle Scholar
11.Ladas, G., Laskhmikantham, V. and Papadakis, J. S., Oscillations of higher-order retarded differential equations generated by the retarded argument, in Delay and functional differential equations and their applications (Schmitt, K., Editor) (Academic, New York, 1972), 219–231.CrossRefGoogle Scholar
12.Ladde, G. S., Lakshmikantham, V. and Zhang, B. G., Oscillation theory of differential equations with deviating arguments (Marcel Dekker, New York, 1987).Google Scholar
13.Mallik, S. C. and Arora, S., Mathematical analysis (New Age International, New Delhi, 2001).Google Scholar
14.Parhi, N. and Rath, R. N., Oscillation criteria for forced first order neutral differential equations with variable coefficients, J. Math. Anal. Appl. 256 (2) (2001), 525–541.CrossRefGoogle Scholar
15.Parhi, N. and Rath, R. N., On oscillation of solutions of forced nonlinear neutral differential equations of higher order, Czechoslovak Math. J. 53 (128) (2003), 805–825.CrossRefGoogle Scholar
16.Parhi, N. and Rath, R. N., On oscillation of solutions of forced nonlinear neutral differential equations of higher order II, Ann. Polon. Math. 81 (2), (2003), 101–110.CrossRefGoogle Scholar
17.Rath, R. N., Oscillatory and asymptotic behaviour of solutions of higher order neutral equations, Bull. Inst. Math. Acad. Sinica 30 (3) (2002), 219–228.Google Scholar
18.Rath, R. N., Padhy, L. N. and Misra, N., Oscillation of solutions of non-linear neutral delay differential equations of higher order for p(t) = ±1, Arch. Math. (Brno) 40 (4) (2004), 359–366.Google Scholar
19.Şahiner, Y. and Zafer, A., Bounded oscillation of nonlinear neutral differential equations of neutral type, Czechoslovak Math. J. 51 (126) (2001) 185–195.Google Scholar
20.Shen, J. H., New oscillation criteria for odd order neutral equations, J. Math. Anal. Appl. 201 (2) (1996), 387–395.CrossRefGoogle Scholar
21.Shen, J. H. and Tang, X. H., New oscillation criteria for linear delay differential equations, Comput. Math. Appl. 36 (6) (1998), 53–61.CrossRefGoogle Scholar
22.Tang, X. H., Oscillation for first order superlinear delay differential equations, J. Lond. Math. Soc. 65 (2), no. 1 (2002), 115–122.CrossRefGoogle Scholar
23.Wang, L. W., Oscillation of first-order nonlinear neutral functional-differential equations, Acta Math. Appl. Sinica 14 (3) (1991), 348–359.Google Scholar
24.Wei, J. J., Oscillation of first-order sublinear differential equations with deviating arguments, Dongbei Shida Xuebao 3 (1991), 5–9.Google Scholar
25.Xu, Z. and Liu, X., Philos-type oscillation criteria for Emden-Fowler neutral delay differential equations, J. Comput. Appl. Math. 206 (2) (2007), 1116–1126.CrossRefGoogle Scholar
26.Yang, Q., Yang, L. and Zhu, S., Interval criteria for oscillation of second-order nonlinear neutral differential equations, Comput. Math. Appl. 46 (5–6) (2003), 903–918.CrossRefGoogle Scholar
27.Zhang, B. G. and Li, W. T., On the oscillation of odd order neutral differential equations. Fasc. Math. 29 (1999), 167–183.Google Scholar
28.Zhang, Q. and Yan, J., Oscillation behavior of even order neutral differential equations with variable coefficients, Appl. Math. Lett. 19 (11) (2006), 1202–1206.CrossRefGoogle Scholar