Fixed Point in a Non-metrizable Space
Abdalla Tallafha2016, Springer Proceedings in Mathematics & Statistics
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Abstract
In this paper, we shall define Lipschitz condition for functions and contraction functions on non-metrizable spaces. Finally, we ask the natural question: "Does every contraction have a unique fixed point?". 5.1 Introduction A uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence. The notion of uniformity has been investigated by several mathematicians such as Weil [10-12], Cohen [3, 4] and Graves [6]. The theory of uniform spaces was given by Bourbaki in [2]. Also Weil's booklet [12] defines uniformly continuous mapping. Contraction functions on complete metric spaces played an important role in the theory of fixed point (Banach fixed point theory). Lipschitz condition and contractions are usually discussed in metric and normed spaces, and have never been studied in a non-metrizable space. The object of this paper is to define Lipschitz condition, and contraction mapping on semi-linear uniform spaces, which enables us to study fixed point for such functions. We believe that the structure of semilinear uniform spaces is very rich, and all the known results on fixed point theory can be generalized. Let X be a non-empty set and D X be a collection of all subsets of X X, such that each element V of D X contains the diagonal D f.x; x/ W x 2 Xgand V D V 1 D f.y; x/ W .x; y/ 2 Vg for all V 2 D X. D X is called the family of all entourages of the diagonal. Let be a sub-collection of D X. Then we have the following definition.
