Fixed point theorems for nonexpansive mappings

Some fixed point theorems

“…Specifically, our main object is to show that results of Edelstein [4] and [5], and Reich [8] can be generalized to the non-metric situation afforded by gauge and quasi-gauge spaces. In this sense, our work is a continuation and extension of that of Tan [13].…”

A development of contraction mapping principles on Hausdorff uniform spaces

“…It is sufficient to assume that L(x0, T) is nonempty instead of the conditions involving completeness and densifyingness. Doing this, the theorem involving (II)w has been obtained by Tan [10] and that involving (HI)w is just an extension of Theorem 2.2 in [2] by Belluce and Kirk. The part of the proof involving (IV)w is trivial, while the other parts follow immediately from the following result, which can be deduced easily from Theorem 1 in [4] by Edelstein.…”

“…In condition (BRS), if N(x) = 1 for all x E X and Xp = a constant for eachp G 9, then Theorem 1 reduces to Theorem 2.3 by Tan [10] (cf. also Theorem 1.1 by Tarafdar [11]).…”

“…The well-known Banach's contraction mapping principle asserts that if X with a metric d is complete and if F is a ¿-contraction, i.e., there exists X E [0,1) (B) such that d(T(x), T(y)) < Xd(x,y) for all x, y E X, then T has a unique fixed point in X which can be realized as the limit of the Picard iterates {T"(x)} for each x in X. An extension of the principle to the general setting for X a Hausdorff uniform space has been recently given by Tan [10,Theorem 2.3] and by Tarafdar [11,Theorem 1.1]. On the other hand, in the metric space setting, many authors have obtained the same conclusions or parts of the conclusions under various conditions which are somehow weaker than condition (B).…”

A Solution to the Completion Problem for Quasi-Pseudometric Spaces

“…More precisely, this problem has been studied in [3], where seven different notions of Cauchy sequence are presented. (The definitions are that of left and right -Cauchy sequence ( is the quasimetric) defined by Reilly [5] and Subrahmanyam [6], respectively, -Cauchy sequence defined by Tan [7], right -and leftCauchy sequence defined by Kelly [2], and weakly right -and weakly left -Cauchy sequence defined in [3].) By combining the seven notions of Cauchy sequences with the topologies , −1 , and , we may reach a total of fourteen different definitions of "complete space" (considering the symmetry of using the −1 instead of ).…”