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)

 
 

 

A comparison theorem for certain functional differential equations


Author: Ernest D. True
Journal: Proc. Amer. Math. Soc. 47 (1975), 127-132
DOI: https://doi.org/10.1090/S0002-9939-1975-0352657-7
MathSciNet review: 0352657
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: The oscillatory character of solutions to the functional differential equation $ {x^{(n)}}(t) + a(t)f(x(g(t))) = Q(t)$ is investigated, by comparison with the oscillatory character of solutions to $ {x^{(n)}}(t) + s(t)f(x(t)) = 0$ where $ s(t) \geqslant \gamma a(t),0 < \gamma < 1$. Here, $ Q(t)$ represents a bounded, oscillatory forcing function, and $ g(t)$ tends to $ \infty $ as $ t \to \infty $ or $ g(t) \geqslant t - c$ for large $ t$ but is otherwise arbitrary.


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

  • [1] G. Grefsrud, Existence and oscillation of solutions of certain functional differential equations, Ph. D. Thesis, Montana State University, Bozeman, 1971.
  • [2] Myron S. Henry, Approximate solutions of functional differential equations, Lecture Notes in Math., vol. 333, Springer-Verlag, Berlin and New York, 1972, pp. 144-152. MR 0431704 (55:4699)
  • [3] A. G. Kartsatos, Maintenance of oscillations under the effect of a periodic forcing term, Proc. Amer. Math. Soc. 33 (1972), 377-383. MR 0330622 (48:8959)
  • [4] R. J. Oberg, On the local existence of solutions of certain functional-differential equations, Proc. Amer. Math. Soc. 20 (1969), 295-302. MR 38 #2413. MR 0234094 (38:2413)
  • [5] Muril L. Robertson, The equation $ y'(t) = F(t,y(g(t)))$. Pacific J. Math. 43 (1972), 483-491. MR 0318623 (47:7170)
  • [6] Gerald H. Ryder, Solutions of a functional differential equation, Amer. Math. Monthly 76 (1969), 1031-1033. MR 40 #502. MR 0247233 (40:502)
  • [7] Gerald H. Ryder and David 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)
  • [8] P. Waltman, A note on an oscillation criterion for an equation with a functional argument, Canad. Math. Bull. 11 (1968), 593-595. MR 38 #6193. MR 0237916 (38:6193)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0352657-7
Keywords: Functional differential equation, oscillation, comparison, forcing function
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society