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Fixed point results in generalized metric spaces without Hausdorff property
Last modified: 2014-02-28
Abstract
It is well-known that generalized metric spaces in the sense of
Branciari [A. Branciari, A fixed point theorem of
Banach-Caccioppoli type on a class of generalized metric spaces,
Publ. Math. Debrecen 57 (2000) 31--37] might not be Hausdorff and, hence, there may exist sequences in them having more than one limit. Thus, in most of the fixed point results obtained recently in such spaces, Hausdorffness was additionally assumed. We show in this note that, nevertheless, most of these results remain valid without this assumption.
Branciari [A. Branciari, A fixed point theorem of
Banach-Caccioppoli type on a class of generalized metric spaces,
Publ. Math. Debrecen 57 (2000) 31--37] might not be Hausdorff and, hence, there may exist sequences in them having more than one limit. Thus, in most of the fixed point results obtained recently in such spaces, Hausdorffness was additionally assumed. We show in this note that, nevertheless, most of these results remain valid without this assumption.
Keywords
Generalized metric space; rectangular space; Hausdorff property.