Characterizations of quasi-metric completeness in terms of Kannan-type fixed point theorems

Yıl 2017, Cilt: 46 Sayı: 1, 67 - 76, 01.02.2017

Carmen Alegre Hacer Dağ Salvador Romaguera Pedro Tirado

Öz

We obtain quasi-metric versions of Kannan's fixed point theorem for self-mappings and multivalued mappings, respectively, which are used to deduce characterizations of d-sequentially complete and of left K-sequentially complete quasi-metric spaces, respectively.

Anahtar Kelimeler

Quasi-metric space, complete, Kannan mapping, fixed point

Kaynakça

• C. Alegre and J. Marín, Modified w-distances on quasi-metric spaces and a fixed point theorem on complete quasi-metric spaces, Top. Appl. 203 (2016), 32-41.
• C. Alegre, J. Marín and S. Romaguera, A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces, Fixed Point Theory Appl. 2014, 2014:40.
• S. Al-Homidan, Q.H. Ansari and J.C. Yao, Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory, Nonlinear Anal. TM&A 69 (2008), 126-139.
• M. Ali-Akbari, B. Honari, M. Pourmahdian and M.M. Rezaii, The space of formal balls and models of quasi-metric spaces, Math. Struct. Comput. Sci. 19 (2009), 337-355.
• I. Altun, N. Al Ari, M. Jleli, A. Lashin and B. Samet, A new concept of $(\alpha, F_d)$-contraction on quasi metric space, J. Nonlinear Sci. Appl. 9 (2016), 3354-3361.
• J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251.
• S. Cobzaş, Completeness in quasi-metric spaces and Ekeland Variational Principle, Top. Appl. 158 (2011), 1073-1084.
• S. Cobzaş, Functional Analysis in Asymmetric Normed Spaces, Birkhäuser, Springer Basel, 2013.
• H. Dağ, G. Minak and I. Altun, Some fixed point results for multivalued F-contractions on quasi metric spaces, RACSAM, DOI: 10.1007/s13398-016-0285-3, to appear.
• T.K. Hu, On a fixed point theorem for metric spaces, Amer. Math. Monthly 74 (1967), 436-437.
• R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76.
• E. Karapinar and S. Romaguera, On the weak form of Ekeland's Variational Principle in quasi-metric spaces, Top. Appl. 184 (2015), 54-60.
• A.W. Kirk, Caristi's fixed point theorem and metric convexity, Colloq. Math. 36 (1976), 81-86.
• H.P.A. Künzi, Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology, in: C.E. Aull, R. Lowen (Eds.), Handbook of the History of General Topology, vol. 3, Kluwer, Dordrecht, 2001, pp. 853-968.
• A. Latif, and S.A. Al-Mezel, Fixed point results in quasimetric spaces, Fixed Point Theory Appl. 2011 (2011), Article ID 178306, 8 pages.
• J. Marín, S. Romaguera and P. Tirado, Q-functions on quasi-metric spaces and fixed points for multivalued maps, Fixed Point Theory Appl. 2011 (2011), Article ID 603861, 10 pages.
• J. Marín, S. Romaguera and P. Tirado, Generalized contractive set-valued maps on complete preordered quasi-metric spaces, J. Funct. Spaces Appl. 2013 (2013), Article ID 269246, 6 pages.
• S. Park, Characterizations of metric completeness, Colloquium Mathematicum 49 (1984), 21-26.
• I.L. Reilly, P.V. Subrahmanyam and M.K. Vamanamurthy, Cauchy sequences in quasipseudo- metric spaces, Mh. Math. 93 (1982), 127-140.
• S. Romaguera, M.P. Schellekens and O. Valero, Complexity spaces as quantitative domains of computation, Top. Appl. 158 (2011), 853-860.
• S. Romaguera and P. Tirado, The complexity probabilistic quasi-metric space, J. Math. Anal. Appl. 376 (2011), 732-740.
• S. Romaguera and P Tirado, A characterization of Smyth complete quasi-metric spaces via Caristi's fixed point theorem, Fixed Point Theory Appl. 2015, 2015:183.
• S. Romaguera and O. Valero, Domain theoretic characterisations of quasi-metric completeness in terms of formal balls, Math. Struct. Comput. Sci. 20 (2010), 453-472.
• M.P. Schellekens, A characterization of partial metrizability: domains are quantiable, Theor. Comput. Sci. 305 (2003), 409-432.
• N. Shioji, T. Suzuki, W. Takahashi, Contractive mappings, Kannan mappings and metric completeness, Proc. Amer. Math. Soc. 126 (1998), 3117-3124.
• P.V. Subrahmanyam, Completeness and fixed-points, Mh. Math. 80 (1975), 325-330.
• T. Suzuki, W. Takahashi, Fixed point theorems and characterizations of metric completeness, Top. Methods Nonlinear Anal. 8 (1996), 371-382.
• T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., 136 (2008), 1861-1869.