×

Nonlinear contractions in \(0\)-complete partial metric spaces. (English) Zbl 1247.54047

Summary: Using the setting of 0-complete partial metric spaces, some common fixed point results of maps that satisfy nonlinear contractive conditions are obtained. These results generalize and improve the existing fixed point results in the literature in the sense that weaker conditions are used. An example shows how our result can be used when the corresponding result in standard metric spaces cannot.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. G. Matthews, “Partial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 728, pp. 183-197, Annals of the New York Academy of Sciences. · Zbl 0911.54025
[2] S. G. Matthews, “Partial metric topology,” Research Report 212, Department of Computer Science, University of Warwick, 1992. · Zbl 0911.54025
[3] S. Oltra and O. Valero, “Banach’s fixed point theorem for partial metric spaces,” Rendiconti dell’Istituto di Matematica dell’Università di Trieste, vol. 36, no. 1-2, pp. 17-26, 2004. · Zbl 1080.54030
[4] D. Ilić, V. Pavlović, and V. Rako, “Some new extensions of Banach’s contraction principle to partial metric space,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1326-1330, 2011. · Zbl 1292.54025 · doi:10.1016/j.aml.2011.02.025
[5] D. Ilić, V. Pavlović, and V. Rako, “Extensions of the Zamfirescu theorem to partial metric spaces,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 801-809, 2012. · Zbl 1255.54022 · doi:10.1016/j.mcm.2011.09.005
[6] T. Abdeljawad, E. Karapinar, and K. Ta\cs, “Existence and uniqueness of a common fixed point on partial metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1900-1904, 2011. · Zbl 1230.54032 · doi:10.1016/j.aml.2011.05.014
[7] M. Abbas, T. Nazir, and S. Romaguera, “Fixed pooint results for generalized cyclic contrac-tion mappings in partial metric spaces,” Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. In press. · Zbl 1256.54064 · doi:10.1007/s13398-011-0051-5
[8] T. Abdeljawad, “Fixed points for generalized weakly contractive mappings in partial metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2923-2927, 2011. · Zbl 1237.54038 · doi:10.1016/j.mcm.2011.07.013
[9] T. Abdeljawad, E. Karapinar, and K. Ta\cs, “A generalized contraction principle with control functions on partial metric spaces,” Computers & Mathematics with Applications. An International Journal, vol. 63, no. 3, pp. 716-719, 2012. · Zbl 1238.54017 · doi:10.1016/j.camwa.2011.11.035
[10] R. P. Agarwal, M. A. Alghamdi, and N. Shahzad, “Fixed point theory for cyclic generalized contractions in partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 40, 2012. · Zbl 1477.54033 · doi:10.1186/1687-1812-2012-40
[11] I. Altun and A. Erduran, “Fixed point theorems for monotone mappings on partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 508730, 10 pages, 2011. · Zbl 1207.54051 · doi:10.1155/2011/508730
[12] I. Altun, F. Sola, and H. Simsek, “Generalized contractions on partial metric spaces,” Topology and its Applications, vol. 157, no. 18, pp. 2778-2785, 2010. · Zbl 1207.54052 · doi:10.1016/j.topol.2010.08.017
[13] L. Ćirić, B. Samet, H. Aydi, and C. Vetro, “Common fixed points of generalized contractions on partial metric spaces and an application,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2398-2406, 2011. · Zbl 1244.54090 · doi:10.1016/j.amc.2011.07.005
[14] D. {\Dj}ukić, Z. Kadelburg, and S. Radenović, “Fixed points of Geraghty-type mappings in various generalized metric spaces,” Abstract and Applied Analysis, vol. 2011, Article ID 561245, 13 pages, 2011. · Zbl 1231.54030 · doi:10.1155/2011/561245
[15] N. Hussain, Z. Kadelburg, and S. Radenović, “Comparison functions and fixed point results in partial metric spaces,” Abstrac and Applied Analysis, vol. 2012, Article ID 605781, 15 pages. · Zbl 1242.54023 · doi:10.1155/2012/605781
[16] Z. M. Fadail and A. G. B. Ahmad, “Coupled fixed point theorems of single-valued mapping for c-distance in cone metric spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 246516, 20 pages, 2012. · Zbl 1251.54043 · doi:10.1155/2012/246516
[17] Z. M. Fadail, A. G. B. Ahmad, and Z. Golubovic, “Fixed Point theorems of single-valued mapping for c-distance in cone metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 826815, 10 pages, 2012. · Zbl 1252.54031 · doi:10.1155/2012/826815
[18] E. Karapinar and M. Erhan, “Fixed point theorems for operators on partial metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1894-1899, 2011. · Zbl 1229.54056 · doi:10.1016/j.aml.2011.05.013
[19] J. Matkowski, “Fixed point theorems for mappings with a contractive iterate at a point,” Proceedings of the American Mathematical Society, vol. 62, no. 2, pp. 344-348, 1977. · Zbl 0349.54032 · doi:10.2307/2041041
[20] H. K. Nashine, Z. Kadelburg, and S. Radenović, “Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces,” Mathematical and Computer Modelling. In press. · Zbl 1286.54051 · doi:10.1016/j.mcm.2011.12.019
[21] S. J. O’Neill, “Partial metrics, valuations and domain theory,” in Proceedings of the 11th Summer Conference on General Topology and Applications, vol. 806, pp. 304-315, Annals of the New York Academy of Sciences, 1996. · Zbl 0889.54018
[22] S. J. ONeill, “Two topologies are better than one,” Tech. Rep., University of Warwick, Conventry, UK, 1995, http://www.dcs.warwick.ac.uk/reports/283.html.
[23] S. Oltra, S. Romaguera, and E. A. Sánchez-Pérez, “Bicompleting weightable quasi-metric spaces and partial metric spaces,” Rendiconti del Circolo Matematico di Palermo. Serie II, vol. 51, no. 1, pp. 151-162, 2002. · Zbl 1098.54027 · doi:10.1007/BF02871458
[24] D. Paesano and P. Vetro, “Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces,” Topology and its Applications, vol. 159, no. 3, pp. 911-920, 2012. · Zbl 1241.54035 · doi:10.1016/j.topol.2011.12.008
[25] S. Romaguera, “Fixed point theorems for generalized contractions on partial metric spaces,” Topology and its Applications, vol. 159, no. 1, pp. 194-199, 2012. · Zbl 1232.54039 · doi:10.1016/j.topol.2011.08.026
[26] S. Romaguera, “A Kirk type characterization of completeness for partial metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 493298, 6 pages, 2010. · Zbl 1193.54047 · doi:10.1155/2010/493298
[27] B. Samet, M. Rajovi, R. Lazovi, and R. Stoiljkovi, “Common fixed point results for nonlinear contractions in ordered partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 71, 2011. · Zbl 1271.54086 · doi:10.1186/1687-1812-2011-71
[28] S. Radenović, Z. Kadelburg, D. Jandrli, and A. Jandrli, “Some results on weakly con-tractive maps,” Bulletin of the Iranian Mathematical Society. In press. · Zbl 1391.54036
[29] M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 416-420, 2008. · Zbl 1147.54022 · doi:10.1016/j.jmaa.2007.09.070
[30] I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, Romania, 2001. · Zbl 0968.54029
[31] G. Jungck, “Commuting mappings and fixed points,” The American Mathematical Monthly, vol. 83, no. 4, pp. 261-263, 1976. · Zbl 0321.54025 · doi:10.2307/2318216
[32] D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American Mathematical Society, vol. 20, pp. 458-464, 1969. · Zbl 0175.44903 · doi:10.2307/2035677
[33] B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257-290, 1977. · Zbl 0365.54023 · doi:10.2307/1997954
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.