Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia


IMPACT FACTOR 2018: 0.490

CiteScore 2018: 0.47

SCImago Journal Rank (SJR) 2018: 0.279
Source Normalized Impact per Paper (SNIP) 2018: 0.627

Mathematical Citation Quotient (MCQ) 2017: 0.26

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 59, Issue 4

Issues

Oscillation criteria for differential equations of second order

A. Nandakumaran / S. Panigrahi
Published Online: 2009-07-29 | DOI: https://doi.org/10.2478/s12175-009-0138-z

Abstract

In this article, we give sufficient condition in the form of integral inequalities to establish the oscillatory nature of non linear homogeneous differential equations of the form $$ (r(t)y')' + q(t)y' + p(t)f(y)g(y') = 0, t \geqslant t_0 , $$ where r, q, p, f and g are given data. We do this by separating the two cases f is monotonous and non monotonous.

MSC: Primary 34C10

Keywords: oscillation; homogeneous; non linear; disfocality; disconjugacy

  • [1] BURTON, T. A.— GRIMER, R.: Stability properties of (ru′)′+ af(u)g(u′) = 0, Monatsh. Math. 74 (1970), 211–222. http://dx.doi.org/10.1007/BF01303441Google Scholar

  • [2] CASSELL, J.S.: The assymptotic behaviour of a class of linear oscillatore, Quart. J. Math. Oxford Ser. (3) 32 (1981), 287–302. http://dx.doi.org/10.1093/qmath/32.3.287CrossrefGoogle Scholar

  • [3] COPPEL, W.A.: Stability and Asymptotic Behaviour of Differential Equations, Heath, Boston, 1965. Google Scholar

  • [4] EL-SAYED, M.A. An oscillation criterion for a forced second order linear differential equations, Proc. Amer. Math. Soc. 118 (1993), 813–817. http://dx.doi.org/10.2307/2160125CrossrefGoogle Scholar

  • [5] GRACE, S. R.— LALLI, B. S.: An oscillation criterion for second order strongly sublinear differential equations, J. Math. Anal. Appl. 123 (1987), 584–588. http://dx.doi.org/10.1016/0022-247X(87)90333-7CrossrefGoogle Scholar

  • [6] GRAEF, J. R.— SPIKES, P. W.: Asymptotic behaviour of solution of a second order nonlinear differential equations, J. Differential Equations 17 (1975), 451–476. http://dx.doi.org/10.1016/0022-0396(75)90056-XCrossrefGoogle Scholar

  • [7] GRAEF, J. R.— SPIKES, P. W.: Boundedness and convergence to zero of solutions of a forced second order nonlinear differential equations, J. Math. Anal. Appl. 62 (1978), 295–309. http://dx.doi.org/10.1016/0022-247X(78)90127-0CrossrefGoogle Scholar

  • [8] HARDY, G. H.— LITTLEWOOD, J. E.— POLYA, G.: Inequalities, Cambridge University Press, Cambridge, 1988. Google Scholar

  • [9] HUANG, C.C.: Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997), 712–723. http://dx.doi.org/10.1006/jmaa.1997.5428CrossrefGoogle Scholar

  • [10] KARTSATOUS, A.G.: Maintenance of oscillations under the effect of periodic forcing term, Proc. Amer. Math. Soc. 33 (1972), 377–383. http://dx.doi.org/10.2307/2038064CrossrefGoogle Scholar

  • [11] KEENER, M.S.: Solutions of a certain linear homogeneous second order differential equations, Appl. Anal. 1 (1971), 57–63. http://dx.doi.org/10.1080/00036817108839006CrossrefGoogle Scholar

  • [12] KONG, Q.: Interval criteria for oscillation of second order linear ordinary differential equations, J. Math. Anal. Appl. 229 (1999), 258–270. http://dx.doi.org/10.1006/jmaa.1998.6159CrossrefGoogle Scholar

  • [13] LALLI, B.S.: On boundedness of solutions of certain second order differential equations, J. Math. Anal. Appl. 25 (1969), 182–188. http://dx.doi.org/10.1016/0022-247X(69)90221-2CrossrefGoogle Scholar

  • [14] LEIGHTON, W.: Comparison theorems for linear differential equations of second order, Proc. Amer. Math. Soc. 13 (1962), 603–610. http://dx.doi.org/10.2307/2034834CrossrefGoogle Scholar

  • [15] LI, W.T.: Oscillation of certain second-order nonlinear differential equations, J. Math. Anal. Appl. 217 (1998), 1–14. http://dx.doi.org/10.1006/jmaa.1997.5680CrossrefGoogle Scholar

  • [16] LI, W. T.— AGARWAL, R. P.: Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations, J. Math. Anal. Appl. 245 (2000), 171–188. http://dx.doi.org/10.1006/jmaa.2000.6749CrossrefGoogle Scholar

  • [17] LI, W. T.— AGARWAL, R. P.: Interval oscillation criteria for second order nonlinear differential equations with damping, Comput. Math. Appl. 40 (2000), 217–230. http://dx.doi.org/10.1016/S0898-1221(00)00155-3CrossrefGoogle Scholar

  • [18] LI, W. T.— AGARWAL, R. P.: Interval oscillation criteria for a forced nonlinear ordinary differential equations, Appl. Anal. 75 (2000), 341–347. http://dx.doi.org/10.1080/00036810008840853CrossrefGoogle Scholar

  • [19] LI, W. T.— AGARWAL, R. P.: Interval oscillation criteria for second order forced nonlinear differential equations with damping, Panamer. Math. J. 11 (2001), 109–117. Google Scholar

  • [20] LI, W. T.— ZHANG, M. Y.— FEI, X. L.: Oscillation criteria for a second order nonlinear differential equation with damping term, Indian J. Pure Appl. Math. 30 (1999), 1017–1029. Google Scholar

  • [21] PARHI, N.— PANIGRAHI, S.: Disfocality and Liapunov type inequalies for third order equations, Appl. Math. Lett. 16 (2003), 227–233. http://dx.doi.org/10.1016/S0893-9659(03)80036-8CrossrefGoogle Scholar

  • [22] RAGOVCHENKO, Y. V.: Oscillation criteria for certain nonlinear differential equations, J. Math. Anal. Appl. 229 (1999), 399–416. http://dx.doi.org/10.1006/jmaa.1998.6148CrossrefGoogle Scholar

  • [23] RAINKEIN, S. M.: Oscillation theorems for second order nonhomogeneous linear differential equations, J. Math. Anal. Appl. 53 (1976), 550–553. http://dx.doi.org/10.1016/0022-247X(76)90091-3CrossrefGoogle Scholar

  • [24] SKIDMORE, A.— LEIGHTON, W.: On the equation y″ + p(x)y = f(x), J. Math. Anal. Appl. 43 (1973), 46–55. http://dx.doi.org/10.1016/0022-247X(73)90256-4CrossrefGoogle Scholar

  • [25] SKIDMORE, A.— BOWERS, J. J.: Oscillatory behaviour of solutions of y″ + p(x)y = f(x), J. Math. Anal. Appl. 49 (1975), 317–323. http://dx.doi.org/10.1016/0022-247X(75)90183-3CrossrefGoogle Scholar

  • [26] TEUFEL, H.: Forced second order nonlinear oscillations, J. Math. Anal. Appl. 40 (1972), 148–152. http://dx.doi.org/10.1016/0022-247X(72)90037-6CrossrefGoogle Scholar

  • [27] WONG, J. S.: Oscillation criteria for a forced second order linear differential eqations, J. Math. Anal. Appl. 231 (1999), 235–240. http://dx.doi.org/10.1006/jmaa.1998.6259CrossrefGoogle Scholar

  • [28] WONG, J. S.— BURTON, T. A.: Some properties of solution of u″ + a(t)f(u)g(u′) = 0, Monatsh. Math. 69 (1965), 364–374. http://dx.doi.org/10.1007/BF01297623Google Scholar

  • [29] UTZ, W. R.: Properties of solutions of u″ + g(t)u 2n−1 = 0, Monatsh. Math. 66 (1962), 56–60. http://dx.doi.org/10.1007/BF01418878Google Scholar

About the article

Published Online: 2009-07-29

Published in Print: 2009-08-01


Citation Information: Mathematica Slovaca, Volume 59, Issue 4, Pages 433–454, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.2478/s12175-009-0138-z.

Export Citation

© 2009 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Vadivel Sadhasivam and Jayapal Kavitha
Applied Mathematics, 2016, Volume 07, Number 03, Page 272

Comments (0)

Please log in or register to comment.
Log in