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 note on periodic solutions for delay-differential systems


Authors: G. B. Gustafson and K. Schmitt
Journal: Proc. Amer. Math. Soc. 42 (1974), 161-166
MSC: Primary 34K15
DOI: https://doi.org/10.1090/S0002-9939-1974-0326109-3
MathSciNet review: 0326109
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f(t,x,y):[0,\infty ) \times {R^n} \times {R^n} \to {R^n}$ be continuous and 1-periodic in t, $ \tau (t):[0,\infty ) \to [0,h](0 < h \leqq 1)$ continuous and 1-periodic. A simple geometric condition (Theorem 1) is given for the existence of 1-periodic solutions $ x(t)$ of the nonlinear delay-differential system $ x'(t) = f(t,x(t),x(t - \tau (t)))$, with $ x(t)$ in a given bounded convex open set G in $ {R^n}$. The addition of a Lipschitz condition in x and monotonicity in y allows one to calculate $ x(t)$ by a monotone sequence of successive approximations (Theorem 2). Extensions to a more general functional differential equation $ x'(t) = g(t,x(t),{x_t})$ are given.


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

  • [1] G. S. Ladde and S. G. Deo, Some integral and differential inequalities, Bull. Calcutta Math. Soc. 61 (1969), 47–53. MR 0296453
  • [2] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
  • [3] Z. Mikołajska, Sur l’existence d’une solution périodique d’une équation différentielle du premier ordre avec le paramètre retardé, Ann. Polon. Math. 23 (1970/1971), 25–36 (French). MR 0265718
  • [4] J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher; Notes on Mathematics and its Applications. MR 0433481
  • [5] V. V. Strygin, A certain theorem on the existence of periodic solutions of systems of differential equations with retarded argument, Mat. Zametki 8 (1970), 229–234 (Russian). MR 0276583

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/S0002-9939-1974-0326109-3
Keywords: Periodic solutions, delay equations
Article copyright: © Copyright 1974 American Mathematical Society