PERIODIC SOLUTIONS OF NONLINEAR SECOND-ORDER DIFFERENCE EQUATIONS ´ JESUS RODRIGUEZ AND DEBRA LYNN ETHERIDGE Received 6 August 2004 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 c = 0 and f : R → R is continuous. In our main result we assume that f exhibits sublinear growth and that there is a constant β 0 such that u f (u) 0 whenever |u| ≥ β. For such an equation we prove that if N is an odd. | PERIODIC SOLUTIONS OF NONLINEAR SECOND-ORDER DIFFERENCE EQUATIONS JESUS RODRÍGUEZ AND DEBRA LYNN ETHERIDGE Received 6 August 2004 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 c 0 and f R R is continuous. In our main result we assume that f exhibits sublinear growth and that there is a constant ft 0 such that uf u 0 whenever u ft. 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 Narccos-1 -b 2 is an even multiple of n. 1. Introduction In this paper we study the existence of periodic solutions of nonlinear second-order discrete time equations of the form y t 2 by t 1 cy t f y t t 0 1 2 3 . where we assume that b and c are real constants c is different from zero and f is a realvalued continuous function defined on R. In our main result we consider equations where the following hold. i There are constants a1 a2 and s with 0 s 1 such that I f u I a1 u s a2 Vu in R. ii There is a constant ft 0 such that uf u 0 whenever u ft. We prove that if N is odd and larger than one then the difference equation will have a N-periodic solution unless all of the following conditions are satisfied c 1 b 2 and Narccos-1 -b 2 is an even multiple of n. Copyright 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005 2 2005 173-192 DOI 174 Periodic solutions of nonlinear second-order difference equations As a consequence of this result we prove that there is a countable subset s of -2 2 such that if b s then y t 2 by t 1 cy t f y t will have periodic solutions of every odd period larger than one. The results presented in this paper extend previous ones of Etheridge and Rodriguez 4 who studied the existence of periodic solutions of difference equations under significantly more
