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 L u = N u and the nonlinear time dependent problem u ′ = ( L + N ) u . Under appropriate assumptions on L and N 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.