Abstract and Applied Analysis

Fixed Points for Weak α - ψ -Contractions in Partial Metric Spaces

Poom Kumam, Calogero Vetro, and Francesca Vetro

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Abstract

Recently, Samet et al. (2012) introduced the notion of α - ψ -contractive mappings and established some fixed point results in the setting of complete metric spaces. In this paper, we introduce the notion of weak α - ψ -contractive mappings and give fixed point results for this class of mappings in the setting of partial metric spaces. Also, we deduce fixed point results in ordered partial metric spaces. Our results extend and generalize the results of Samet et al.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 986028, 9 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511952

Digital Object Identifier
doi:10.1155/2013/986028

Mathematical Reviews number (MathSciNet)
MR3073469

Citation

Kumam, Poom; Vetro, Calogero; Vetro, Francesca. Fixed Points for Weak $\alpha $ - $\psi $ -Contractions in Partial Metric Spaces. Abstr. Appl. Anal. 2013 (2013), Article ID 986028, 9 pages. doi:10.1155/2013/986028. https://projecteuclid.org/euclid.aaa/1393511952


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References

  • S. G. Matthews, “Partial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 728 of Annals of the New York Academy of Sciences, pp. 183–197, 1994.
  • M. A. Bukatin and J. S. Scott, “Towards computing distances between programs via Scott domains,” in Logical Foundations of Computer Science, S. Adian and A. Nerode, Eds., vol. 1234 of Lecture Notes in Computer Science, pp. 33–43, Springer, Berlin, Germany, 1997.
  • M. A. Bukatin and S. Yu. Shorina, “Partial metrics and co-continuous valuations,” in Foundations of Software Science and Computation Structures, M. Nivat, Ed., vol. 1378 of Lecture Notes in Computer Science, pp. 33–43, Springer, Berlin, Germany, 1998.
  • P. Chaipunya, W. Sintunavarat, and P. Kumam, “On $P$-contractions in ordered metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 219, 2012.
  • S. J. O'Neill, “Partial metrics, valuations and domain theory,” in Proceedings of the 11th Summer Conference on General Topology and Applications, vol. 806 of Annals of the New York Academy of Sciences, pp. 304–315, 1996.
  • S. Romaguera and M. Schellekens, “Partial metric monoids and semivaluation spaces,” Topology and Its Applications, vol. 153, no. 5-6, pp. 948–962, 2005.
  • W. Sintunavarat and P. Kumam, “Weak condition for generalized multi-valued $(f,\alpha ,\beta )$-weak contraction mappings,” Applied Mathematics Letters, vol. 24, no. 4, pp. 460–465, 2011.
  • W. Sintunavarat, J. K. Kim, and P. Kumam, “Fixed point theorems for a generalized almost $(\phi ,\varphi )$-contraction with respect to $S$ in ordered metric spaces,” Journal of Inequalities and Applications, vol. 2012, article 263, 11 pages, 2012.
  • 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.
  • S. Romaguera, “A Kirk type characterization of completeness for partial metric spaces,” Fixed Point Theory and Applications, Article ID 493298, 6 pages, 2010.
  • W. A. Kirk, “Caristi's fixed point theorem and metric convexity,” Colloquium Mathematicum, vol. 36, no. 1, pp. 81–86, 1976.
  • S. Park, “Characterizations of metric completeness,” Colloquium Mathematicum, vol. 49, no. 1, pp. 21–26, 1984.
  • P. V. Subrahmanyam, “Completeness and fixed-points,” vol. 80, no. 4, pp. 325–330, 1975.
  • T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008.
  • B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for $\alpha $-$\psi $-contractive type mappings,” Nonlinear Analysis, vol. 75, no. 4, pp. 2154–2165, 2012.
  • 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.
  • O. Valero, “On Banach fixed point theorems for partial metric spaces,” Applied General Topology, vol. 6, no. 2, pp. 229–240, 2005.
  • A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2003.
  • R. P. Agarwal, M. A. El-Gebeily, and D. O'Regan, “Generalized contractions in partially ordered metric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008.
  • I. Altun and G. Durmaz, “Some fixed point theorems on ordered cone metric spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 58, no. 2, pp. 319–325, 2009.
  • H. K. Nashine and B. Samet, “Fixed point results for mappings satisfying $(\psi ,\varphi )$-weakly contractive condition in partially ordered metric spaces,” Nonlinear Analysis, vol. 74, no. 6, pp. 2201–2209, 2011.
  • J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005.
  • J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, vol. 23, no. 12, pp. 2205–2212, 2007.
  • J. J. Nieto, R. L. Pouso, and R. Rodríguez-López, “Fixed point theorems in ordered abstract spaces,” Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2505–2517, 2007.
  • A. Amini-Harandi and H. Emami, “A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations,” Nonlinear Analysis, vol. 72, no. 5, pp. 2238–2242, 2010.
  • J. Chu and P. J. Torres, “Applications of Schauder's fixed point theorem to singular differential equations,” Bulletin of the London Mathematical Society, vol. 39, no. 4, pp. 653–660, 2007.