Iteration procedures of shuttle iteration type in continuous non-monotone problems (Q1598250)

Let \(\Omega\subseteq \mathbb{R}^n\) be a closed bounded domain. The author considers nonlinear boundary value problems (BVPs) \(Lx=f(t,x)\), \(x=0\) on \(\partial \Omega\), with \(-L\) an elliptic differential operator, and \(f\) between functions \(h<g\), for which the corresponding BVPs for \(Lx=h(t,x)\) and \(Lx=g(t,x)\) are solvable. By means of the monotonicity of the operators \((L+\alpha I)^{-1} (\alpha \geq\alpha_0)\), monotone iteration processes in \(C(\Omega)\) are constructed, which lead to robust stable solutions or robust stable branches of solutions of the original problem. In difference to many results using monotone iteration processes, \(f\) is not supposed to be monotone. The results are discussed by some scalar examples.