Advances in Difference Equations
Volume 2005 (2005), Issue 2, Pages 173-192

Periodic solutions of nonlinear second-order difference equations

Jesús Rodriguez1 and Debra Lynn Etheridge2

1Department of Mathematics, North Carolina State University, P.O. Box 8205, Raleigh 27695-8205, NC, USA
2Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill 27599, NC, USA

Received 6 August 2004

Copyright © 2005 Jesús Rodriguez and Debra Lynn Etheridge. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We establish conditions for the existence of periodic solutions of nonlinear, second-order difference equations of the form y(t+2)+by(t+1)+cy(t)=f(y(t)), where c0 and f: is continuous. In our main result we assume that f exhibits sublinear growth and that there is a constant β>0 such that uf(u)>0 whenever |u|β. For such an equation we prove that if N is an odd integer larger than one, then there exists at least one N-periodic solution unless all of the following conditions are simultaneously satisfied: c=1, |b|<2, and N across-1(b/2) is an even multiple of π.