Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Nonlinear oscillation of a sublinear delay equation of arbitrary order


Authors: Takaŝi Kusano and Hiroshi Onose
Journal: Proc. Amer. Math. Soc. 40 (1973), 219-224
MSC: Primary 34K15
DOI: https://doi.org/10.1090/S0002-9939-1973-0324177-5
MathSciNet review: 0324177
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The equations considered generalize

$\displaystyle {x^{(n)}}(t) + p(t)\vert x(g(t)){\vert^\alpha }\operatorname{sgn} x(g(t)) = 0,\quad 0 < \alpha < 1.$

A necessary and sufficient condition is established that all solutions are oscillatory when $ n$ is even and are either oscillatory or strongly monotone when $ n$ is odd. The result makes clear a difference in oscillatory property between sublinear delay equations and the corresponding ordinary differential equations.

References [Enhancements On Off] (What's this?)

  • [1] F. Burkowski, Nonlinear oscillation of a second order sublinear functional differential equation, SIAM J. Appl. Math. 21 (1971), 486-490. MR 0308557 (46:7671)
  • [2] H. E. Gollwitzer, On nonlinear oscillation for a second order delay equation, J. Math. Anal. Appl. 26 (1969), 385-389. MR 39 #581. MR 0239224 (39:581)
  • [3] T. G. Hallam, Asymptotic behavior of the solutions of an $ n$th order nonhomogeneous ordinary differential equation, Trans. Amer. Math. Soc. 122 (1966), 177-194. MR 32 #6000. MR 0188562 (32:6000)
  • [4] I. T. Kiguradze, On the question of variability of solutions of nonlinear differential equations. Differenciarnye Uravnenija 1 (1965), 995-1006 = Differential Equations 1 (1965), 773-782. MR 33 #2896. MR 0194689 (33:2896)
  • [5] T. Kusano and H. Onose, Oscillation of solutions of nonlinear differential delay equations of arbitrary order, Hiroshima Math. J. 2 (1972), 1-13. MR 0324175 (48:2527)
  • [6] -, Oscillation theorems for delay equations of arbitrary order, Hiroshima Math. J. 2 (1972), 263-270. MR 0324176 (48:2528)
  • [7] G. Ladas, Oscillation and asymptotic behavior of solutions of differential equations with retarded argument, J. Differential Equations 10 (1971), 281-290. MR 0291590 (45:681)
  • [8] I. Ličko and M. Švec, Le caractère oscillatoire des solutions de l'équation $ {y^{(n)}} + f(x){y^\alpha } = 0,n > 1$, Czechoslovak Math. J. 13 (88) (1963), 481-491. MR 28 #4210. MR 0161001 (28:4210)
  • [9] G. H. Ryder and D. V. V. Wend, Oscillation of solutions of certain ordinary differential equations of $ n$th order, Proc. Amer. Math. Soc. 25 (1970), 463-469. MR 41 #5710. MR 0261091 (41:5710)
  • [10] V. N. Ševelo and O. N. Odarič, Certain questions of the theory of oscillation (nonoscillation) of solutions of second order differential equations with retarded argument, Ukrain. Mat. Ž. 23 (1971), 508-516. (Russian) MR 44 #7088. MR 0289901 (44:7088)
  • [11] V. N. Ševelo and N. V. Vareh, On oscillability of solutions of higher order linear differential equations with retarded argument, Ukrain. Mat. Ž. 24 (1972), 513-520. (Russian) MR 0308562 (46:7676)
  • [12] Y. P. Singh, The asymptotic behavior of solutions of a nonlinear $ n$th-order differential equation, Okayama Math. J. 15 (1971), 71-73. MR 0301312 (46:470)
  • [13] P. Waltman, On the asymptotic behavior of solutions of an $ n$th order equation, Monatsh. Math. 69 (1965), 427-430. MR 32 #2686. MR 0185217 (32:2686)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34K15

Retrieve articles in all journals with MSC: 34K15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0324177-5
Keywords: Oscillatory, nonoscillatory, sublinear, nonlinear delay equation
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society