Monotone Iterative Methods

The basic notion in this chapter is that of an ordered Banach space. We try to localize solutions of an operator equation u = T (u) in an ordered interval [u₀, v₀] of an ordered Banach space X. In addition we look for solutions which are limits of increasing or decreasing sequences of elements of X. The basic property of the operator T is monotonicity. This combined with certain properties of the ordered Banach space X guarantees the convergence of monotone sequences. Thus we may say that this chapter explores the contribution of monotonicity to compactness.