论文信息 - Generalized metrics and Caristi’s theorem

Generalized metrics and Caristi’s theorem

A ‘generalized metric space’ is a semimetric space which does not satisfy the triangle inequality, but which satisfies a weaker assumption called the quadrilateral inequality. After reviewing various related axioms, it is shown that Caristi’s theorem holds in complete generalized metric spaces without further assumptions. This is noteworthy because Banach’s fixed point theorem seems to require more than the quadrilateral inequality, and because standard proofs of Caristi’s theorem require the triangle inequality.MSC:54H25, 47H10.

William A. Kirk | Naseer Shahzad | W. A. Kirk | N. Shahzad

[1] Leonard M. Blumenthal,et al. Theory and applications of distance geometry , 1954 .

[2] E. W. Chittenden. On the equivalence of Écart and voisinage , 1917 .

[3] K. Menger. Untersuchungen über allgemeine Metrik , 1928 .

[4] M. Turinici. Functional contractions in local Branciari metric spaces , 2012, 1208.4610.

[5] J. Jachymski,et al. Nonlinear Contractions on Semimetric Spaces , 1995 .

[6] A Branciari,et al. A fixed point theorem of Banach--Caccioppoli type on a class of generalized metric spaces , 2000, Publicationes Mathematicae Debrecen.

[7] F. Browder,et al. A general principle on ordered sets in nonlinear functional analysis , 1976 .

[8] Troy L. Hicks,et al. Fixed point theory in symmetric spaces with applications to probabilistic spaces , 1999 .

[9] William A. Kirk,et al. A FIXED POINT THEOREM REVISITED , 1997 .

[10] J. Caristi,et al. Fixed point theorems for mapping satisfying inwardness conditions , 1976 .

[11] I. Ramabhadra Sarma,et al. CONTRACTIONS OVER GENERALIZED METRIC SPACES , 2009 .

[12] F. Hausdorff. Grundzüge der Mengenlehre , 1914 .

[13] On a Fixed Point Theorem of Contractive Type , 1983 .

[14] D. Burke. On $K$-semimetric spaces , 1979 .

[15] Bessem Samet,et al. Discussion on "A fixed point theorem of Banach--Caccioppoli type on a class of generalized metric spaces" by A. Branciari , 2010, Publicationes Mathematicae Debrecen.

[16] Wallace Alvin Wilson,et al. On Semi-Metric Spaces , 1931 .