On oscillation of solutions of forced nonlinear neutral differential equations of higher order

N. Parhi; Radhanath N. Rath

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 4, page 805-825
  • ISSN: 0011-4642

Abstract

top
In this paper, necessary and sufficient conditions are obtained for every bounded solution of [ y ( t ) - p ( t ) y ( t - τ ) ] ( n ) + Q ( t ) G y ( t - σ ) = f ( t ) , t 0 , ( * ) to oscillate or tend to zero as t for different ranges of p ( t ) . It is shown, under some stronger conditions, that every solution of ( * ) oscillates or tends to zero as t . Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B 27 (1986), 502–511, and generalize some known results.

How to cite

top

Parhi, N., and Rath, Radhanath N.. "On oscillation of solutions of forced nonlinear neutral differential equations of higher order." Czechoslovak Mathematical Journal 53.4 (2003): 805-825. <http://eudml.org/doc/30817>.

@article{Parhi2003,
abstract = {In this paper, necessary and sufficient conditions are obtained for every bounded solution of \[ [y (t) - p (t) y (t - \tau )]^\{(n)\} + Q (t) G \bigl (y (t - \sigma )\bigr ) = f (t), \quad t \ge 0, \qquad \mathrm \{(*)\}\] to oscillate or tend to zero as $t \rightarrow \infty $ for different ranges of $p (t)$. It is shown, under some stronger conditions, that every solution of $(*)$ oscillates or tends to zero as $t \rightarrow \infty $. Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B 27 (1986), 502–511, and generalize some known results.},
author = {Parhi, N., Rath, Radhanath N.},
journal = {Czechoslovak Mathematical Journal},
keywords = {oscillation; nonoscillation; neutral equations; asymptotic behaviour; oscillation; nonoscillation; neutral equations; asymptotic behaviour},
language = {eng},
number = {4},
pages = {805-825},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On oscillation of solutions of forced nonlinear neutral differential equations of higher order},
url = {http://eudml.org/doc/30817},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Parhi, N.
AU - Rath, Radhanath N.
TI - On oscillation of solutions of forced nonlinear neutral differential equations of higher order
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 4
SP - 805
EP - 825
AB - In this paper, necessary and sufficient conditions are obtained for every bounded solution of \[ [y (t) - p (t) y (t - \tau )]^{(n)} + Q (t) G \bigl (y (t - \sigma )\bigr ) = f (t), \quad t \ge 0, \qquad \mathrm {(*)}\] to oscillate or tend to zero as $t \rightarrow \infty $ for different ranges of $p (t)$. It is shown, under some stronger conditions, that every solution of $(*)$ oscillates or tends to zero as $t \rightarrow \infty $. Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B 27 (1986), 502–511, and generalize some known results.
LA - eng
KW - oscillation; nonoscillation; neutral equations; asymptotic behaviour; oscillation; nonoscillation; neutral equations; asymptotic behaviour
UR - http://eudml.org/doc/30817
ER -

References

top
  1. Oscillation and asymptotic behaviour of higher order neutral differential equations, Bull. Inst. Math. Acad. Sinica 22 (1994), 203–217. (1994) MR1297358
  2. 10.1002/mana.19911500103, Math. Nachr. 150 (1991), 15–24. (1991) MR1109642DOI10.1002/mana.19911500103
  3. 10.1155/S0161171291000923, Internat. J.  Math. Math. Sci. 14 (1991), 689–696. (1991) MR1125418DOI10.1155/S0161171291000923
  4. Oscillation in odd order neutral differential equations, Czechoslovak Math.  J. 42 (1992), 313–323. (1992) MR1179502
  5. Oscillation of even order neutral differential equations, Indian J.  Math. 35 (1993), 9–25. (1993) MR1249639
  6. 10.1006/jmaa.1996.0334, J.  Math. Anal. Appl. 202 (1996), 555–577. (1996) MR1406248DOI10.1006/jmaa.1996.0334
  7. Oscialltion Theory of Delay-Differential Equations with Applications, Clarendon Press, Oxford, 1991. (1991) MR1168471
  8. Introduction to the Theory of Integration, Academic Press, New York, 1963. (1963) Zbl0112.28302MR0154957
  9. On the oscillation of solutions of the equation d m u d t m + a ( t ) u m u = 0 , Mat. Sb. 65 (1964), 172–187. (1964) MR0173060
  10. 10.1017/S0334270000005105, Austral. Math. Soc. Ser.  B 27 (1986), 502–511. (1986) MR0836222DOI10.1017/S0334270000005105
  11. Oscillations of higher order neutral differential equations, Portugal. Math. 48 (1991), 291–307. (1991) MR1127127
  12. Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker INC., New York, 1987. (1987) MR1017244
  13. Oscillation and nonoscillation for a class of neutral differential equations, Differential Equations Dynam. Systems 1 (1993), 197–204. (1993) MR1258897
  14. Oscillation of solutions of forced neutral differential equations of n -th order, Czechoslovak Math.  J. 45 (1995), 413–433. (1995) MR1344507
  15. Maintenance of oscillation of neutral differential equations under the effect of a forcing term, Indian J.  Pure Appl. Math. 26 (1995), 909–919. (1995) MR1347539
  16. 10.4064/ap-65-1-1-10, Ann. Polon. Math. 65 (1996), 1–10. (1996) MR1414747DOI10.4064/ap-65-1-1-10
  17. Oscillations of neutral differential equations of higher order, Bull. Inst. Math. Acad. Sinica 24 (1996), 139–150. (1996) MR1398241
  18. 10.1023/A:1022405707349, Czechoslovak Math.  J. 50 (2000), 155–173. (2000) DOI10.1023/A:1022405707349
  19. On oscillation criteria for a forced neutral differential equation, Bull. Inst. Math. Acad. Sinica 28 (2000), 59–70. (2000) MR1750329
  20. 10.1006/jmaa.2000.7315, J.  Math. Anal. Appl. 256 (2001), 525–541. (2001) MR1821755DOI10.1006/jmaa.2000.7315
  21. On oscillation and asymptotic behaviour of solutions of forced first order neutral differential equations, Proc. Indian. Acad. Sci. (Math. Sci.), Vol.  111, 2001, pp. 337–350. (2001) MR1851095
  22. Real Analysis, 3rd edition, MacMillan Publ. Co., New York, 1989. (1989) MR1013117
  23. 10.1006/jmaa.1996.0262, J.  Math. Anal. Appl. 201 (1996), 387–395. (1996) Zbl0860.34040MR1396907DOI10.1006/jmaa.1996.0262
  24. Oscillation of higher order nonlinear neutral functional differential equation, Ann. Differential Equations 12 (1996), 83–88. (1996) Zbl0849.34059MR1394017
  25. Oscillation of higher order neutral differential equations, Rocky Mountain J.  Math (to appear). (to appear) MR1340027
  26. Oscillations and nonoscillations in higher order neutral equations, J.  Math. Phys. Sci. 25 (1991), 152–165. (1991) 

Citations in EuDML Documents

top
  1. N. Parhi, Arun Kumar Tripathy, On oscillatory fourth order nonlinear neutral differential equations. I.
  2. N. Parhi, Radhanath N. Rath, Oscillatory behaviour of solutions of nonlinear higher order neutral differential equations
  3. N. Parhi, Arun Kumar Tripathy, On oscillatory fourth order nonlinear neutral differential equations. II.
  4. N. Parhi, Radhanath N. Rath, On oscillation criteria for forced nonlinear higher order neutral differential equations
  5. R.N. Rath, K.C. Panda, S.K. Rath, Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator
  6. Julio G. Dix, Dillip Kumar Ghose, Radhanath Rath, Oscillation of a higher order neutral differential equation with a sub-linear delay term and positive and negative coefficients

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.