A note on the periodicity in difference equations. (English) Zbl 1090.34582

Ronto, Andrei

Arch. Math., Brno 36, Suppl., 575-584 (2000).

Sufficient conditions are obtained for the existence of a unique periodic solution in a Banach space of the linear first order difference equation \[ (\Delta x)(n)=\sum _{\nu =0}^{N-1}L_{n,\nu }x(\nu )+f(n),\quad 0\leq n\leq N-1 \] with \((\Delta x)(n):=x(n+1)-x(n)\) for \(0\leq n\leq N-1\), \((\Delta x)(N-1):=x(0)-x(N-1)\) where \((L_{n,\nu })_{n,\nu =0}^N\) are certain linear bounded operators such that \(L_{N,\nu }=L_{0,\nu }\) for all \(\nu \in \{0,1,\dots ,N-1\}\) and \(f(n)\) is a summable sequence.

Reviewer: Josef Diblík (Brno)

MSC:

34K06	Linear functional-differential equations
34K13	Periodic solutions to functional-differential equations

Keywords:

difference equation; periodic solution