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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
MathSciNet review: 0326109
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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.

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Keywords: Periodic solutions, delay equations
Article copyright: © Copyright 1974 American Mathematical Society