tailieunhanh - POSITIVE PERIODIC SOLUTIONS FOR NONLINEAR DIFFERENCE EQUATIONS VIA A CONTINUATION THEOREM GEN-QIANG

POSITIVE PERIODIC SOLUTIONS FOR NONLINEAR DIFFERENCE EQUATIONS VIA A CONTINUATION THEOREM GEN-QIANG WANG AND SUI SUN CHENG Received 29 August 2003 and in revised form 4 February 2004 Based on a continuation theorem of Mawhin, positive periodic solutions are found for difference equations of the form yn+1 = yn exp( f (n, yn , yn−1 ,., yn−k )), n ∈ Z. 1. Introduction There are several reasons for studying nonlinear difference equations of the form yn+1 = yn exp f n, yn , yn−1 ,. , yn−k , n ∈ Z = {0, ±1, ±2,. }, () where f = f (t,u0 ,u1. | POSITIVE PERIODIC SOLUTIONS FOR NONLINEAR DIFFERENCE EQUATIONS VIA A CONTINUATION THEOREM GEN-QIANG WANG AND SUI SUN CHENG Received 29 August 2003 and in revised form 4 February 2004 Based on a continuation theorem of Mawhin positive periodic solutions are found for difference equations of the form yn 1 yn exp f n yn yn-1 . yn-k n e Z. 1. Introduction There are several reasons for studying nonlinear difference equations of the form yn 1 ynexp f n yn yn-1 . yn-k n e Z 0 1 2 . where f f t u0 u1 . Uk is a real continuous function defined on Rk 2 such that f t w u0 . uk f t u0 . Uk t u0 . uk e Rk 2 and w is a positive integer. For one reason the well-known equations yn 1 Ayn yn 1 Wn 1 - y yn 1 yn exp Ị 1 K- yn Ị K 0 are particular cases of . As another reason is intimately related to delay differential equations with piecewise constant independent arguments. To be more precise let us recall that a solution of is a real sequence of the form yn neZ which renders into an identity after substitution. It is not difficult to see that solutions can be found when an appropriate function f is given. However one interesting question is whether there are any solutions which are positive and w-periodic where a sequence yn neZ is said to be w-periodic if yn w yn for n e Z. Positive w-periodic solutions of are related to those of delay differential equations involving piecewise constant independent Copyright 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004 4 2004 311-320 2000 Mathematics Subject Classification 39A11 URL http S1687183904308113 312 Periodic solutions of difference equations arguments y t y t f t y t y t- 1 y t- 2 . y t- k t e R where x is the greatest-integer function. Such equations have been studied by several authors including Cooke and Wiener 5 6 Shah and Wiener 9 Aftabizadeh et al. 1 Busenberg and Cooke 2 and so forth. Studies of such equations were motivated by the fact that they .