Optimal existence results for \(n\)th order periodic boundary value difference equations
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.
scientific article; zbMATH DE number 1497289
scientific article