Abstract
In this paper we shall generalize the definition given in [1] for Lipschitz condition and contractions for functions on a non-metrizable space, besides we shall give more properties of semi-linear uniform spaces.
Highlights
The notion of uniformity has been investigated by several mathematician as Weil [2]-[4], Cohen [5] [6], and Graves [7].The theory of uniform spaces was given by Burbaki in [8]
The object of this paper is to generalize the definition of Lipschitz condition, and contraction mapping on semi-linear uniform spaces given by Tallafha [12]
Let ( X, Γ) be a semi-linear uniform space and τΓ the topology on X indused by Γ
Summary
The notion of uniformity has been investigated by several mathematician as Weil [2]-[4], Cohen [5] [6], and Graves [7]. [10], defined a new type of uniform spaces, namely semi-linear uniform spaces and they gave example of semi-linear space which was not metrizable They defined a set valued map ρ on X × X , by which they studied some cases of best approximation in such spaces. In [11], Tallafha, A. defined another set valued map δ on X × X , and gave some properties of semi-linear uniform spaces using the maps ρ and δ. In [1] [12], Tallafha defined Lipschitz condition and contractions for functions on semi-linear uniform spaces, which enabled us to study fixed point for such functions. The object of this paper is to generalize the definition of Lipschitz condition, and contraction mapping on semi-linear uniform spaces given by Tallafha [12]. We shall give a new topopological properties and more properties of semi-linear uniform spaces
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