A note on periodic solutions for delay-differential systems
HTML articles powered by AMS MathViewer
- by G. B. Gustafson and K. Schmitt PDF
- Proc. Amer. Math. Soc. 42 (1974), 161-166 Request permission
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
- G. S. Ladde and S. G. Deo, Some integral and differential inequalities, Bull. Calcutta Math. Soc. 61 (1969), 47–53. MR 296453
- Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
- Z. Mikołajska, Sur l’existence d’une solution périodique d’une équation différentielle du premier ordre avec le paramètre retardé, Ann. Polon. Math. 23 (1970/71), 25–36 (French). MR 265718, DOI 10.4064/ap-23-1-25-36
- J. T. Schwartz, Nonlinear functional analysis, Notes on Mathematics and its Applications, 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. MR 0433481
- 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 276583
Additional Information
- © Copyright 1974 American Mathematical Society
- 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