Monotone Iterative Technique for Differential Equations in a Banach Space
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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.