A note on the periods of periodic solutions of some autonomous functional differential equations (original) (raw)

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A problem of finding lower bounds for periods of periodic solutions of a Lipschitzian differential equation, expressed in the supremum Lipschitz constant, is considered. Such known results are obtained for systems with inner product norms. However, utilizing the supremum norm requires development of a new technique, which is presented in this paper. Consequently, sharp bounds for equations of even order, both without delay and with arbitrary time-varying delay, are found. For both classes of system, the obtained bounds are attained in linear differential equations.

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0. Introduction. Recently, certain arguments which originated in asymptotic fixed point theory have been used with remarkable success in the study of functional differential equations by G. S. Jones [10], S. N. Chow [3] and R. D. Nussbaum [14, 15, 16]. Our observations here are in this line and are directly motivated by results of Nussbaum in [ 14]. He considers differential-delay equations which can be transformed to the form x '(t) =-\f(x(t-1)). Our approach here will be to obtain an abstract existence result providing a continuum of non-trivial solutions for a nonlinear eigenvalue problem F(x, A) = x and apply it to obtain a continuum of nontrivial periodic solutions for certain differential-delay equations. Our abstract theorem is based on a kind of asymptotic version of Krasnosel'skiï's results in [11,12] on expansions and compressions of a cone in a Banach space and this is due to G. Fournier and the author. The application then substantially relies on earlier work of E. M. Wright [20] and R. D. Nussbaum [14].

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