Fixed point theorems in constructive mathematics

This paper gives the beginnings of a development of the theory of fixed point theorems within Bishop’s constructive analysis. We begin with a constructive proof of a result, due to Borwein, which characterises when some sets have the contraction mapping property. A review of the constructive content of Brouwer’s fixed point theorem follows, before we turn our attention to Schauder’s generalisation of Brouwer’s fixed point theorem. As an application of our constructive Schauder’s fixed point theorem we give an approximate version of Peano’s theorem on the existence of solutions of differential equations. Other fixed point theorems are mentioned in passing. 2010 Mathematics Subject Classification 03F55, 03F60, 46S30 (primary); 34A30, 47H10 (secondary)