Extremal solutions for the difference \(\phi\)-Laplacian problem with nonlinear functional boundary conditions. (English) Zbl 1001.39006

Cabada, A.

Comput. Math. Appl. 42, No. 3-5, 593-601 (2001).

The paper deals with the difference equation \[ -\Delta[\phi(\Delta u_k)]=f(k,u_{k+1}),\quad k\in \{ 0,1,\dots,N-1\}, \] where \(f\) is continuous and \(\phi\) is continuous and strictly increasing and \(\phi(\mathbb{R})=\mathbb{R}\). The equation is studied together with functional boundary conditions that include Dirichlet, Neumann and mixed type as well as periodic conditions. Under some hypotheses the problem under consideration has at least one solution (resp. a unique solution, resp. an extremal solution).
The construction of two monotone sequences that converge to the extremal solution is presented in details.

Reviewer: Dobiesław Bobrowski (Poznań)

MSC:

39A10

Additive difference equations

Keywords:

difference equation; \(\phi\)-Laplacian problem; nonlinear functional boundary conditions; upper and lower solutions; extremal solution

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