## Monotone Iterative Technique for Differential Equations in a Banach Space

##### Abstract

Let E be a real Banach space with norm [see pdf for notation].
Consider the initial value problem (1.1) [see pdf for notation],
where [see pdf for notation]. Generally speaking of approximate
solutions of (1.1) consist of three steps, namely,
(i) constructing a sequence of approximate solutions of some kinds for (1.1);
(ii) showing the convergence of the constructed sequence;
(iii) proving that the limit function is a solution.
If f is continous, steps (i) and (iii) are standard and straight
forward. It is a step (ii) that deserves attention. This in turn
leads to three possibilities; namely to show that the sequence of
approximate solutions is (a) a Cauchy sequence; (b) relatively
compact so that one can appeal to Ascoli's theorem; and (c) a monotone
sequence in a cone. The first two possibilities are well known and are
discussed in [2,3]. This paper is devoted to the investigation of (c)
which leads to the development of a monotone interative technique in
an arbitrary cone.