Extensions of the Zamfirescu theorem to partial metric spaces. (English) Zbl 1255.54022

Summary: T. Zamfirescu [Arch. Math. 23, 292–298 (1972; Zbl 0239.54030)] obtained a very interesting fixed point theorem on complete metric spaces by combining the results of S.Banach, R.Kannan and S.K.Chatterjea. S. G. Matthews [Ann. N. Y. Acad. Sci. 728, 183–197 (1994; Zbl 0911.54025)] introduced and studied the concept of partial metric spaces, and obtained a Banach type fixed point theorem on complete partial metric spaces. In this paper, we study new extensions of the Zamfirescu theorem to the context of partial metric spaces, and among other things, we give some generalized versions of the fixed point theorem of Matthews. The theory is illustrated by some examples.


54H25 Fixed-point and coincidence theorems (topological aspects)
54E35 Metric spaces, metrizability
Full Text: DOI


[1] Zamfirescu, T., Fixed point theorems in metric spaces, Arch. Math. (Basel), 23, 292-298 (1972) · Zbl 0239.54030
[2] Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. J., 3, 133-181 (1922) · JFM 48.0201.01
[3] Kannan, R., Some results on fixed points-II, Amer. Math. Monthly, 76, 405-408 (1969) · Zbl 0179.28203
[4] Chatterjea, S. K., Fixed-point theorems, C.R. Acad. Bulgare Sci., 25, 727-730 (1972) · Zbl 0274.54033
[5] Matthews, S. G., Partial metric topology, Proc. 8th Summer Conference on General Topology and Applications. Proc. 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci., 728, 183-197 (1994) · Zbl 0911.54025
[6] Abdeljawad, T., Fixed points for generalized weakly contractive mappings in partial metric spaces, Math. Comput. Modelling (2011) · Zbl 1237.54038
[7] Altun, I.; Erduran, A., Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory Appl., 10 (2011), Article ID 508730 · Zbl 1207.54051
[8] Altun, I.; Simsek, H., Some fixed point theorems on dualistic partial metric spaces, J. Adv. Math. Stud., 1, 01-08 (2008)
[9] Altun, I.; Sola, F.; Simsek, H., Generalized contractions on partial metric spaces, Topology Appl., 157, 18, 2778-2785 (2010) · Zbl 1207.54052
[10] Bukatin, M.; Kopperman, R.; Matthews, S.; Pajoohesh, H., Partial metric spaces, Amer. Math. Monthly, 116, 708-718 (2009) · Zbl 1229.54037
[11] Ilić, D.; Pavlović, V.; Rakočević, V., Some new extensions of Banach’s contraction principle to partial metric space, Appl. Math. Lett., 24, 1326-1330 (2011) · Zbl 1292.54025
[12] Karapinar, E., Generalizations of Caristi Kirk’s theorem on partial metric spaces, Fixed Point Theory Appl., 2011, 4 (2011) · Zbl 1281.54027
[13] Oltra, S.; Valero, O., Banach’s fixed theorem for partial metric spaces, Rend. Istit. Mat. Univ. Trieste, 36, 17-26 (2004) · Zbl 1080.54030
[14] Rabarison, A. F., Partial Metrics, Supervised by Hans-Peter A. Künzi (2007), African Institute for Mathematical Sciences
[16] Valero, O., On Banach’s fixed point theorems for partial metric spaces, Appl. Gen. Topol., 6, 229-240 (2005) · Zbl 1087.54020
[17] Bukatin, M. A.; Shorina, S. Yu., Partial metrics and co-continuous valuations, (Nivat, M., Foundations of Software Science and Computation Structure. Foundations of Software Science and Computation Structure, Lecture Notes in Computer Science, vol.1378 (1998), Springer), 125-139 · Zbl 0945.06006
[18] Matthews, S. G., An extensional treatment of lazy data flow deadlock, Theoret. Comput. Sci., 151, 195-205 (1995) · Zbl 0872.68110
[19] Schellekens, M. P., The correspondence between partial metrics and semivaluations, Theoret. Comput. Sci., 315, 135-149 (2004) · Zbl 1052.54026
[20] Valero, O., On Banach’s fixed point theorem and formal balls, Appl. Sci., 10, 256-258 (2008) · Zbl 1158.06300
[21] Berinde, V., On the convergence of Ishikawa iteration for a class of quasi contractive operators, Acta Math. Univ. Comenian., 73, 119-126 (2004) · Zbl 1100.47054
[22] Berinde, V., A convergence theorem for Mann iteration in the class of Zamfirescu operators, An. Univ. Vest Timi. Ser. Mat.-Inform., 45, 33-41 (2007) · Zbl 1164.47065
[23] Berinde, V., Iterative Approximation of Fixed Points (2007), Springer-Verlag: Springer-Verlag Berlin Heidelberg New York · Zbl 1165.47047
[24] Berinde, V.; Berinde, M., On Zamfirescu’s fixed point theorem, Rev. Roumaine Math. Pures Appl., 50, 443-453 (2005) · Zbl 1104.47049
[25] Rhoades, B. E., Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc., 196, 161-176 (1974) · Zbl 0285.47038
[26] Rhoades, B. E., A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226, 257-290 (1977) · Zbl 0365.54023
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