Relation-theoretic contraction principle
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In this paper, we present yet another new and novel variant of classical Banach contraction principle on a complete metric space endowed with a binary relation which, under universal relation, reduces to Banach contraction principle. In process, we observe that various kinds of binary relations, such as partial order, preorder, transitive relation, tolerance, strict order, symmetric closure, etc., utilized by earlier authors in several well-known metrical fixed point theorems can be weakened to the extent of an arbitrary binary relation.
KeywordsComplete metric spaces binary relations contraction mappings
Mathematics Subject Classification47H10 54H25
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- 3.P. Charoensawan, Tripled coincidence point theorems for a φ-contractive mapping in a complete metric space without the mixed g-monotone property. Fixed Point Theory Appl. 2013 (2013), doi: 10.1186/1687-1812-2013-252, 16 pp.
- 4.V. Flaška, J. Ježek, T. Kepka and J. Kortelainen, Transitive closures of binary relations. I. Acta Univ. Carolin. Math. Phys. 48 (2007), 55–69.Google Scholar
- 5.S. Ghods, M. E. Gordji, M. Ghods and M. Hadian, Comment on “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces” [Lakshmikantham and Ćirić, Nonlinear Anal. TMA 70 (2009) 4341–4349]. J. Comput. Anal. Appl. 14 (2012), 958–966.Google Scholar
- 6.B. Kolman, R. C. Busby and S. Ross, Discrete Mathematical Structures. 3rd ed., PHI Pvt. Ltd., New Delhi, 2000.Google Scholar
- 7.M. A. Kutbi, A. Roldan, W. Sintunavarat, J. Martinez-Moreno and C. Roldan, F-closed sets and coupled fixed point theorems without the mixed monotone property. Fixed Point Theory Appl. 2013 (2013), doi: 10.1186/1687-1812-2013-330, 11 pp.
- 8.S. Lipschutz, Schaum’s Outlines of Theory and Problems of Set Theory and Related Topics. McGraw–Hill, New York, 1964.Google Scholar
- 9.R. D. Maddux, Relation Algebras. Stud. Logic Found. Math. 150, Elsevier B. V., Amsterdam, 2006.Google Scholar
- 11.J. J. Nieto and R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. (Engl. Ser.) 23 (2007), 2205–2212.Google Scholar
- 12.A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc. 132 (2004), 1435–1443.Google Scholar
- 13.B. Samet and M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications. Commun. Math. Anal. 13 (2012), 82–97.Google Scholar
- 16.M. Turinici, Product fixed points in ordered metric spaces. arXiv:1110.3079v1, 2011.