A monotone iterative technique for stationary and time dependent problems in Banach spaces

An abstract monotone iterative method is developed for operators between partially ordered Banach spaces for the nonlinear problem Lu=NuLu=Nu and the nonlinear time dependent problem u′=(L+N)uu′=(L+N)u. Under appropriate assumptions on LL and NN we obtain maximal and minimal solutions as limits of monotone sequences of solutions of linear problems. The results are illustrated by means of concrete examples.