×

A generalized contraction principle with control functions on partial metric spaces. (English) Zbl 1238.54017

Summary: Partial metric spaces were introduced by S. G. Matthews [in: Papers on general topology and applications. Papers from the 8th summer conference at Queens College, New York, NY, USA, June 18–20, 1992. New York, NY: The New York Academy of Sciences. Ann. N. Y. Acad. Sci. 728, 183–197 (1994; Zbl 0911.54025)] as a part of the study of denotational semantics of data flow networks. We prove a generalized contraction principle with control functions \(\varphi \) and \(\psi \) on partial metric spaces. The theorems we prove generalize many previously obtained results. We also give some examples showing that our theorems are indeed proper extensions.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems

Citations:

Zbl 0911.54025
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fréchet, M., Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo, 22, 1-74 (1906) · JFM 37.0348.02
[2] S.G. Matthews, Partial metric topology, Research Report 212, Dept. of Computer Science, University of Warwick, 1992.; S.G. Matthews, Partial metric topology, Research Report 212, Dept. of Computer Science, University of Warwick, 1992. · Zbl 0911.54025
[3] S.G. Matthews, Partial metric topology, in: General Topology and its Applications, Proc. 8th Summer Conf., Queen’s College, 1992. Annals of the New York Academy of Sciences, vol. 728, 1994, pp. 183-197.; S.G. Matthews, Partial metric topology, in: General Topology and its Applications, Proc. 8th Summer Conf., Queen’s College, 1992. Annals of the New York Academy of Sciences, vol. 728, 1994, pp. 183-197. · Zbl 0911.54025
[4] Kahn, G., The semantics of a simple language for parallel processing, (Proc. IFIP Congress (1974), Elsevier, North Holland: Elsevier, North Holland Amsterdam), 471-475 · Zbl 0299.68007
[5] Waszkiewicz, P., Distance and measurement in domain theory, Electronic Notes in Theoretical Computer Science, 40, 448-462 (2001) · Zbl 1260.68222
[6] P. Waszkiewicz, Quantitative continuous domains, School of Computer Scicence, University of Birmingham, UK, 2002, pp. 41-67.; P. Waszkiewicz, Quantitative continuous domains, School of Computer Scicence, University of Birmingham, UK, 2002, pp. 41-67. · Zbl 1030.06005
[7] Waszkiewicz, P., The local triangle axiom in topology and domain theory, Applied General Topology, 4, 1, 47-70 (2003) · Zbl 1052.54024
[8] Waszkiewicz, P., Quantitative continuous domains, Applied Categorical Structures, 11, 1, 41-67 (2003) · Zbl 1030.06005
[9] Schellekens, M., A characterization of partial metrizability, Theoretical Computer Science, 305, 409-432 (2003) · Zbl 1043.54011
[10] Romaguera, S.; Schellekens, M., Weightable quasi-metric semigroups and semilattices, Electronic Notes in Theoretical Computer Science, 40, 347-358 (2001) · Zbl 1264.54052
[11] Schellekens, M., The correspondence between partial metrics and semivaluations, Theoretical Computer Science, 315, 135-149 (2004) · Zbl 1052.54026
[12] Rus, A. I., Fixed point theory in partial metric spaces, Anale Universtatii de Vest Timişoara Seria Matematică—Informatică, XLVI, 2, 149-160 (2008) · Zbl 1199.54244
[13] Abdeljawad, T.; Karapınar, E.; Taş, K., Existence and uniqueness of a common fixed point on partial metric spaces, Applied Mathematics Letters, 24, 11, 1900-1904 (2011) · Zbl 1230.54032
[14] Karapınar, E.; Erhan, I. M., Fixed point theorems for operators on partial metric spaces, Applied Mathematics Letters, 24, 11, 1894-1899 (2011) · Zbl 1229.54056
[15] Karapınar, E., Weak \(\phi \)-contraction on partial contraction and existence of fixed points in partially ordered sets, Mathematica Aeterna, 1, 4, 237-244 (2011) · Zbl 1291.54060
[16] Karapınar, E., Generalizations of Caristi Kirk’s theorem on partial metric spaces, Fixed Point Theory and Applications, 4 (2011) · Zbl 1281.54027
[17] Abdeljawad, T., Fixed points for generalized weakly contractive mappings in partial metric spaces, Mathematical and Computer Modelling, 54, 11-12, 2923-2927 (2011) · Zbl 1237.54038
[18] Altun, I.; Erduran, A., Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory and Applications, 2011 (2011), Article ID 508730, 10 pages · Zbl 1207.54051
[19] Altun, I.; Sola, F.; Simsek, H., Generalized contractions on partial metric spaces, Topology and its Applications, 157, 18, 2778-2785 (2010) · Zbl 1207.54052
[20] Valero, O., On Banach fixed point theorems for partial metric spaces, Applied General Topology, 6, 2, 229-240 (2005) · Zbl 1087.54020
[21] Oltra, S.; Valero, O., Banach’s fixed point theorem for partial metric spaces, Rendiconti dell’Instituto di Matematica dell’Universitá di Trieste, 36, 1-2, 17-26 (2004) · Zbl 1080.54030
[22] Boyd, D. W.; Wong, S. W., On nonlinear contractions, Proceedings of the American Mathematical Society, 20, 458-464 (1969) · Zbl 0175.44903
[23] Khan, M. S.; Sweleh, M.; Sessa, S., Fixed point theorems by alternating distance between the points, Bulletin of the Australian Mathematical Society, 30, 1, 1-9 (1984) · Zbl 0553.54023
[24] Rhoades, B. E., Some theorems on weakly contractive maps, Nonlinear Analysis: Theory Methods and Applications, 47, 4, 2283-2693 (2001) · Zbl 1042.47521
[25] Dutta, P. N.; Choudhury, B. S., A generalization of contraction principle in metric spaces, Fixed Point Theory and Applications (2008), Article ID 406368, 8 pages · Zbl 1177.54024
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.