Comparison theorems for noncanonical third order nonlinear differential equations

Ivan Mojsej; Ján Ohriska

Open Mathematics (2007)

  • Volume: 5, Issue: 1, page 154-163
  • ISSN: 2391-5455

Abstract

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The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with quasiderivatives. We prove comparison theorems on property A between linear and nonlinear equations. Some integral criteria ensuring property A for nonlinear equations are also given. Our assumptions on the nonlinearity of f are restricted to its behavior only in a neighborhood of zero and a neighborhood of infinity.

How to cite

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Ivan Mojsej, and Ján Ohriska. "Comparison theorems for noncanonical third order nonlinear differential equations." Open Mathematics 5.1 (2007): 154-163. <http://eudml.org/doc/269181>.

@article{IvanMojsej2007,
abstract = {The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with quasiderivatives. We prove comparison theorems on property A between linear and nonlinear equations. Some integral criteria ensuring property A for nonlinear equations are also given. Our assumptions on the nonlinearity of f are restricted to its behavior only in a neighborhood of zero and a neighborhood of infinity.},
author = {Ivan Mojsej, Ján Ohriska},
journal = {Open Mathematics},
keywords = {Comparison theorem; property A; quasiderivative; noncanonical form; nonlinear equation},
language = {eng},
number = {1},
pages = {154-163},
title = {Comparison theorems for noncanonical third order nonlinear differential equations},
url = {http://eudml.org/doc/269181},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Ivan Mojsej
AU - Ján Ohriska
TI - Comparison theorems for noncanonical third order nonlinear differential equations
JO - Open Mathematics
PY - 2007
VL - 5
IS - 1
SP - 154
EP - 163
AB - The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with quasiderivatives. We prove comparison theorems on property A between linear and nonlinear equations. Some integral criteria ensuring property A for nonlinear equations are also given. Our assumptions on the nonlinearity of f are restricted to its behavior only in a neighborhood of zero and a neighborhood of infinity.
LA - eng
KW - Comparison theorem; property A; quasiderivative; noncanonical form; nonlinear equation
UR - http://eudml.org/doc/269181
ER -

References

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  1. [1] M. Cecchi, Z. Došlá and M. Marini: “On nonlinear oscillations for equations associated to disconjugate operators”, Nonlinear Analysis, Theory, Methods & Applications, Vol. 30(3), (1997), pp. 1583–1594. http://dx.doi.org/10.1016/S0362-546X(97)00028-X Zbl0892.34032
  2. [2] M. Cecchi, Z. Došlá and M. Marini: “Comparison theorems for third order differential equations”, Proceeding of Dynamic Systems and Applications, Vol. 2, (1996), pp. 99–106. Zbl0873.34021
  3. [3] M. Cecchi, Z. Došlá and M. Marini: “Asymptotic behavior of solutions of third order delay differential equations”, Archivum Mathematicum(Brno), Vol. 33, (1997), pp. 99–108. Zbl0916.34059
  4. [4] M. Cecchi, Z. Došlá and M. Marini: “Some properties of third order differential operators”, Czech. Math. J., Vol. 47(122), (1997), pp. 729–748. http://dx.doi.org/10.1023/A:1022878804065 Zbl0903.34032
  5. [5] M. Cecchi, Z. Došlá and M. Marini: “An Equivalence Theorem on Properties A, B for Third Order Differential Equations”, Annali di Matematica pura ed applicata (IV), Vol. CLXXIII, (1997), pp. 373–389. http://dx.doi.org/10.1007/BF01783478 Zbl0937.34029
  6. [6] M. Cecchi, Z. Došlá, M. Marini and Gab. Villari: “On the qualitative behavior of solutions of third order differential equations”, J. Math. Anal. Appl., Vol. 197, (1996), pp. 749–766. http://dx.doi.org/10.1006/jmaa.1996.0050 Zbl0856.34034
  7. [7] J. Džurina: “Property (A) of n-th order ODE’s”, Mathematica Bohemica, Vol. 122(4), (1997), pp. 349–356. Zbl0903.34031
  8. [8] T. Kusano and M. Naito: “Comparison theorems for functional differential equations with deviating arguments”, J. Math. Soc. Japan, Vol. 33(3), (1981), pp. 509–532. http://dx.doi.org/10.2969/jmsj/03330509 Zbl0494.34049
  9. [9] I. Mojsej and J. Ohriska: “On solutions of third order nonlinear differential equations”, CEJM, Vol. 4(1), (2006), pp. 46–63. Zbl1104.34048
  10. [10] J. Ohriska: “Oscillatory and asymptotic properties of third and fourth order linear differential equations”, Czech. Math. J., Vol. 39(114), (1989), pp. 215–224. Zbl0688.34018
  11. [11] J. Ohriska: “Adjoint differential equations and oscillation”, J. Math. Anal. Appl., Vol. 195, (1995), pp. 778–796. http://dx.doi.org/10.1006/jmaa.1995.1389 Zbl0847.34037
  12. [12] V. Šeda: “Nonoscillatory solutions of differential equations with deviating argument”, Czech. Math. J., Vol. 36(111), (1986), pp. 93–107. Zbl0603.34064

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