Solutions of a max-type system of difference equations

摘要

We show, in an elegant way, that the general solution to the following max-type system of difference equationsxn+1=maxAxn,ynxn,yn+1=maxAyn,xnyn,n∈N0,for the case y0,x0⩾A>0,y0/x0⩾max{A,1/A}, is given by:xn=Afk(n)-1-α(n)x0fk(n)y0fk(n)(-1)n,n∈N,y1=x0/y0, andyn=y0fk(n-1)+1Afk(n-1)+α(n)-1x0fk(n-1)+1(-1)n,n⩾2,where k(n)=n, for n even, k(n)=n+1, for n odd, n∈N0,α(n)≡n(mod2), for n∈N0, and where the sequence (fn)n∈N is the Fibonacci sequence, correcting and giving a proof of a wrong result in the paper [D. Simsek, B. Demir, C. Cinar, On the solutions of the system of difference equations xn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn}, Discrete Dyn. Nat. Soc. 2009 (2009), 11 pp. (Article ID 325296)].