A unified monotone iterative technique for semilinear elliptic boundary value problems
The authors consider the elliptic problem \[ -\sum_{{i,j=1}}^n a_{i,j}(x)u_{x_{i}x_{j}} + \sum_{i=1}^{n} b_{i}(x) + c(x)u = F(x,u) \text{ in } \Omega \tag{a} \] subject to general boundary conditions in a bounded smooth domain \(\Omega\). The right-hand side is assumed to be splitted in the form \[ F(x,u) = f(x,u) + g(x,u), \] where for each \(x \in \Omega\) the function \(f(x,\cdot)\) is nonincreasing and \(g(x,\cdot)\) is nondecreasing. Starting from suitable upper and lower initial iterates of (a) it is shown with the aid of the maximum principle that the usual iterates converge monotonely to an upper and lower solution of (a), respectively. A similar result is obtained for the variational formulation of (a). Results of this type for operators that allow a monotone splitting are known since the late sixtieth. Surprisingly, the authors do not include any reference to the numerous existing publications in this field.
scientific article; zbMATH DE number 1810008