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

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.

### MSC:

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

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