Optimal existence results for \(n\)th order periodic boundary value difference equations. (English) Zbl 0962.39006

Cabada, Alberto; Otero-Espinar, Victoria

J. Math. Anal. Appl. 247, No. 1, 67-86 (2000).

The authors study the \(n\)-th order nonlinear difference equation \[ u(k+n)= f\bigl(k,u(k), \dots,u(k+n) \bigr),\;k\in\{0, \dots, N-1\}, \] with periodic boundary conditions. Supposing that there exist ordered lower and upper solutions, they obtain a solution for this problem.
The existence results are equivalent to finding the values of \(K_i\), \(i=1, \dots,n\), for which the linear operator \[ u(k+n)+ \sum^n_{i=0} K_iu(k+i) \] in the space of periodic functions is an inverse positive operator.
Moreover an expression of the Green function is obtained. Applications to the first and second order equations are included.

Reviewer: Dobiesław Bobrowski (Poznań)

MSC:

39A11	Stability of difference equations (MSC2000)
39A70	Difference operators

Keywords:

difference operator; periodic solutions; nonlinear difference equation; periodic boundary conditions; lower and upper solutions; inverse positive operator; Green function

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