Fixed point theory in symmetric spaces with applications to probabilistic spaces

✍ Scribed by Troy L. Hicks; B.E. Rhoades

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 110 KB
• Volume: 36
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(98)00002-9

✦ Synopsis

Deÿnitions 2-5. Let (X; d) be a symmetric space.

(2) (X; d) is S-complete if for every d-Cauchy sequence {x n }, there exists x in X with lim d(x n ; x) = 0. (3) (X; d) is d-Cauchy complete if for every d-Cauchy sequence {x n }, there exists x in X with lim x n = x with respect to t(d).

Deÿnition 6. Let (X; t) be a topological space and d :

) ¡ ∞ implies there exists x in X with lim x n = x with respect to the topology t.

The following two axioms were given by Wilson [17]. Let (X; d) be a symmetric space. (W.3) Given {x n }; x and y in X; d(x n ; x) → 0 and d(x n ; y) → 0 imply that x = y. (W.4) Given {x n }; {y n } and an x in X; d(x n ; x) → 0 and d(x n ; y n ) → 0 imply that d(y n ; x) → 0.

Note that for a semi-metric d, if t(d) is Hausdor , then (W.3) holds. In [3-5] many metric space ÿxed point theorems were extended to d-complete topological spaces. The idea of completeness in d-complete topological spaces generalizes completeness in metric and quasi-metric spaces. In [9], Jachymski et al. state that, in a semi-metric setting, this concept of completeness is rather strong. They choose to use the above concept of d-Cauchy complete. They give an example of a ÿxed point free Banach contraction on a Hausdor d-Cauchy complete semi-metric space. They show that Banach's ÿxed point theorem holds if d is also bounded. Their work clearly motivates the following theorem. They also prove that a d-Cauchy complete semi-metric space that satisÿes (W.4) is a d-complete topological space. This gives another class of dcomplete topological spaces. In [6], it is shown that the F-complete topological spaces of Choudhury [1] are d-complete topological spaces. Thus, many ÿxed points theorems already given for d-complete topological spaces hold for F-complete topological spaces.