Iterative bounds for the stable solutions of convex non-linear boundary value problems
J.W. Mooney, Gary F. Roach · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1977
Synopsis We consider a class of convex non-linear boundary value problems of the form where L is a linear, uniformly elliptic, self-adjoint differential expression, f is a given non-linear function, B is a boundary differential expression of either Dirichlet or Neumann type and D is a bounded open domain with boundary ∂D. Particular problems of this class arise in the process of thermal combustion [8]. In this paper we show that stable solutions of this class can be bounded from below (above) by a monotonically increasing (decreasing) sequence of Newton (Picard) iterates. The possibility of using these schemes to construct unstable solutions is also considered.