Abstract
A number of generalizations of the well-known Banach contraction theorem are obtained in various directions. However one of them, stated in [5], shows that the Banach contraction theorem still holds for a class of non-metric spaces. This suggests that the notion of metric may not be essential in the Banach contraction theorem and some of its generalizations. The main purpose of this paper is to show that the Banach contraction theorem and its generalizations due to Diaz and Margolis [1], Luxemburg [7], [8], Maia [9], and the author [5] can be easily derived from a simple fixed point theorem in spaces of type L of Frechet, which we shall call separated L-spaces. Similar results in non-separated L-spaces will be also stated Moreover as an application to linear spaces, we shall derive a generalization of a theorem of Dotson [2].
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