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On periodic solutions of functional differential equations with infinite delay

On periodic solutions of functional differential equations with infinite delay In this paper, using Mawhin's continuation theorem in the theory of coincidence degree, we first prove the general existence theorem of periodic solutions for F.D.Es with infinite delay: $$\frac{{dx(t)}}{{dt}} = f(t,x_t ), x(t) \in R^n ,$$ which is an extension of Mawhin's existence theorem of periodic solutions of F.D.Es with finite delay. Second, as an application of it, we obtain the existence theorem of positive periodic solutions of the Lotka-Volterra equations: $$\begin{gathered} \frac{{dx(t)}}{{dt}} = x(t)(a - kx(t) - by(t)), \hfill \\ \frac{{dy(t)}}{{dt}} = - cy(t) + d\smallint _0^{ + \infty } x(t - s)y(t - s)d\mu (s) + p(t). \hfill \\ \end{gathered} $$ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

On periodic solutions of functional differential equations with infinite delay

Acta Mathematicae Applicatae Sinica , Volume 9 (4) – Jul 13, 2005

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Publisher
Springer Journals
Copyright
Copyright © 1993 by Science Press, Beijing, China and Allerton Press, Inc. New York, U.S.A.
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02005921
Publisher site
See Article on Publisher Site

Abstract

In this paper, using Mawhin's continuation theorem in the theory of coincidence degree, we first prove the general existence theorem of periodic solutions for F.D.Es with infinite delay: $$\frac{{dx(t)}}{{dt}} = f(t,x_t ), x(t) \in R^n ,$$ which is an extension of Mawhin's existence theorem of periodic solutions of F.D.Es with finite delay. Second, as an application of it, we obtain the existence theorem of positive periodic solutions of the Lotka-Volterra equations: $$\begin{gathered} \frac{{dx(t)}}{{dt}} = x(t)(a - kx(t) - by(t)), \hfill \\ \frac{{dy(t)}}{{dt}} = - cy(t) + d\smallint _0^{ + \infty } x(t - s)y(t - s)d\mu (s) + p(t). \hfill \\ \end{gathered} $$

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jul 13, 2005

References