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

Minimal periods of periodic solutions of some Lipschitzian differential equations

Applied Mathematics Letters, 2009

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.

On continua of periodic solutions for functional differential equations

Rocky Mountain Journal of Mathematics, 1977

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].

Minimal periods of solutions to higher-order functional differential equations

Russian Mathematics, 2013

We show that a problem on minimal periods of solutions of Lipschitz functional differential equations is closely related to the unique solvability of the periodic problem for linear functional differential equations. Sharp bounds for minimal periods of non-constant solutions of higher order functional differential equations with different Lipschitz nonlinearities are obtained.

Bounds for periodic solutions of difference and differentia equations

1986

W ojciech Slo m c z y n s k i (Krakôw) Introduction. In this note we consider the bounds for periods of periodic solutions of difference equations in normed linear spaces with Lipschitz continuous right-hand sides. As a consequence of an obtained result we get a theorem on bounds for periods of periodic solutions of differential equations in Banach spaces, giving thus a generalization of the result which has been obtained in a different way by Lasota and Yorke in [2]. An approach applied in the note, consisting in considering first the discrete cases and then passing to the limit, has been used previously in similar situations by Ky Fan, Taussky and Todd in . The existence of a lower bound for the periods of periodic solutions of differential equations has also been studied by Li [4], Vidossich [5] and Yorke . I wish to thank Professor Stanislaw Sçdziwy for his kind advice and guidance.

Periodic solutions of first order functional differential equations

2011

We study the existence of T-periodic solutions of some first order functional differential equations. Several existence criteria are established for our problems; in particular, we obtain conditions for the existence of multiple (even infinitely many) T-periodic solutions of one of the problems. Examples are also included to illustrate our results.

Periodic Perturbations of a Class of Functional Differential Equations

Journal of Dynamics and Differential Equations, 2021

We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations of a particular class of functional differential equations on differentiable manifolds. Our result is obtained by a combination of degree-theoretic methods and a technique that allows to associate the bounded solutions of the functional equation to bounded solutions of a suitable ordinary differential equation.

Periods of solutions of periodic differential equations

Differential and Integral Equations, 2016

Smooth non-autonomous T-periodic differential equations x'(t)=f(t,x(t)) defined in \R\K^n, where \K is \R or \C and n 2 can have periodic solutions with any arbitrary period~S. We show that this is not the case when n=1. We prove that in the real C^1-setting the period of a non-constant periodic solution of the scalar differential equation is a divisor of the period of the equation, that is T/S\N. Moreover, we characterize the structure of the set of the periods of all the periodic solutions of a given equation. We also prove similar results in the one-dimensional holomorphic setting. In this situation the period of any non-constant periodic solution is commensurable with the period of the equation, that is T/S\Q.