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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,... },

(1.1)

where f = f (t,u0 ,u1 ,...,uk ) is a real continuous function defined on Rk+2 such that 











t,u0 ,...,uk ∈ Rk+2 ,

f t + ω,u0 ,...,uk = f t,u0 ,...,uk ,

(1.2)

and ω is a positive integer. For one reason, the well-known equations yn+1 = λyn ,





yn+1 = µyn 1 − yn ,  

µ 1 − yn yn+1 = yn exp K



,

(1.3) K > 0,

are particular cases of (1.1). As another reason, (1.1) is intimately related to delay differential equations with piecewise constant independent arguments. To be more precise, let us recall that a solution of (1.1) is a real sequence of the form { yn }n∈Z which renders (1.1) 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 ω-periodic, where a sequence { yn }n∈Z is said to be ω-periodic if yn+ω = yn , for n ∈ Z. Positive ω-periodic solutions of (1.1) 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://dx.doi.org/10.1155/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 ∈ R,

(1.4)

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 represent a hybrid of discrete and continuous dynamical systems and combine the properties of both differential and differential-difference equations. In particular, the following equation 





y  (t) = ay(t) 1 − y [t] ,

(1.5)

is in Carvalho and Cooke [3], where a is constant. By a solution of (1.4), we mean a function y(t) which is defined on R and which satisfies the following conditions [1]: (i) y(t) is continuous on R; (ii) the derivative y  (t) exists at each point t ∈ R with the possible exception of the points [t] ∈ R, where one-sided derivatives exist; and (iii) (1.4) is satisfied on each interval [n,n + 1) ⊂ R with integral endpoints. Theorem 1.1. Equation (1.1) has a positive ω-periodic solution if and only if (1.4) has a positive ω-periodic solution. Proof. Let y(t) be a positive ω-periodic solution of (1.4). It is easy to see that for any n ∈ Z, 



y  (t) = y(t) f n, y(n), y(n − 1),..., y(n − k) ,

n ≤ t < n + 1.

(1.6)

Integrating (1.6) from n to t, we have 





y(