Fixed points of contractive maps on dcpo's

Abstract

In this paper we study approach structures on dcpo's. A dcpo (X, 6) will be endowed with several other structures: the Scott topology; an approach structure generated by a collection of weightable quasi metrics on X; and a collection W of weights corresponding to the quasi metrics. Understanding the interaction between these structures on X will eventually lead to some fixed-point theorems for the morphisms in the category of approach spaces, which are called contractions. Existing fixed-point theorems on both monotone and non-monotone maps are obtained as special cases.

The notions of closedness and D-connectedness in quantale-valued approach spaces

“…Approach spaces have been introduced by Lowen [30,31] to generalize metric and topological concepts, have many applications in almost all areas of mathematics including probability theory [15], convergence theory [16], domain theory [17], and fixed point theory [18]. Due to its huge importance, several generalizations of approach spaces appeared recently such as probabilistic approach spaces [22], quantale-valued gauge spaces [23], quantale-valued approach system [24], and quantale-valued approach with respect to (w.r.t.)…”

Normality in terms of distances and contractions

“…Such isometric settings get more and more attention like for instance in the study of approximation by Lipschitz functions in [12], of cofinal completeness and UC-property in [1], in investigations on hyperconvexity in [15] and on the non-symmetric analogue of the Urysohn metric space in [16] and [17]. For other applications the larger context of approach spaces with contractions is even more suitable as was recently shown in the context of probability m easures [2], [3] and [4], or complexity analysis [6] and [7].…”

The notion of closedness and D-connectedness in Quantale-valued approach spaces

“…Approach spaces have been introduced by Lowen [25,26] to generalize metric and topological concepts, have many applications in almost all areas of mathematics including probability theory [13], convergence theory [14], domain theory [15] and fixed point theory [16]. Due to its huge importance, several generalization of approach spaces appeared recently such as probabilistic approach spaces [18], quantale-valued gauge spaces [19], quantale-valued approach system [20] and quantale-valued approach w.r.t closure operators [24].…”