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Periodic solutions for functional differential equations with sign-changing nonlinearities

Published online by Cambridge University Press:  21 May 2010

John R. Graef  and
Lingju Kong
John R. Graef
Affiliation:
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA ([email protected]; [email protected])
Lingju Kong
Affiliation:
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA ([email protected]; [email protected])
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Abstract

We establish some eigenvalue criteria for the existence of non-trivial T-periodic solutions of a class of first-order functional differential equations with a nonlinearity f(x). The nonlinear term f(x) can take negative values and may be unbounded from below. Conditions are determined by the relationship between the behaviour of the quotient f(x)/x for x near 0 and ±∞ and the smallest positive characteristic value of an associated linear integral operator. This linear operator plays a key role in the proofs of the results and its construction is non-trivial. Applications to related eigenvalue problems are also discussed. The analysis mainly relies on the topological degree theory.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 140 , Issue 3 , June 2010 , pp. 597 - 616
Copyright
Copyright © Royal Society of Edinburgh 2010

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