Semi-metric spaces and fixed points of \(\alpha-\varphi\)-contractive maps

Volume 9, Issue 5, pp 3147--3156 http://dx.doi.org/10.22436/jnsa.009.05.104

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1840 Downloads
• 2801 Views

Authors

Naseer Shahzad - Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Mohammed Ali Alghamdi - Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Sarah Alshehri - Department of Mathematics, King Abdulaziz University, Science Faculty for Girls, P. O. Box 4087, Jeddah 21491, Saudi Arabia. Ivan Arandelovic - Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16., 11000 Beograd, Serbia.

Abstract

A negative answer to an open problem is provided. Fixed point results for \(\alpha-\varphi\)-contractive mappings in semi-metric spaces are proved. To show the generality of our results, examples are given. Finally, an application of our result to probabilistic spaces is derived.

Share and Cite

Keywords

• Semi-metric space
• \(\alpha-\varphi\)-contractive mapping
• fixed point
• probabilistic space.

MSC

•  47H10
•  54H25.

References

• [1] M. A. Alghamdi, N. Shahzad, O. Valero, On fixed point theory in partial metric spaces, Fixed Point Theory Appl., 2012 (2012), 25 pages.


• [2] M. U. Ali, Q. Kiran, N. Shahzad, Fixed point theorems for multivalued mappings involving \(\alpha\)-function, Abstr. Appl. Anal., 2014 (2014), 6 pages.


• [3] R. P. Agarwal, M. A. Alghamdi, N. Shahzad, Fixed point theory for cyclic generalized contractions in partial metric spaces , Fixed Point Theory Appl., 2012 (2012), 11 pages.


• [4] R. P. Agarwal, D. O’Regan, N. Shahzad, Fixed point theory for generalized contractive maps of Meir-Keeler type, Math. Nachr., 276 (2004), 3–22.


• [5] S. Alshehri, I. Arandelovic, N. Shahzad , Symmetric spaces and fixed points of generalized contractions , Abstr. Appl. Anal., 2014 (2014), 8 pages.


• [6] I. D. Arandelovic, D. J. Keckic , Symmetric spaces approach to some fixed point results , Nonlinear Anal., 75 (2012), 5157–5168.


• [7] I. D. Arandelovic, D. S. Petkovic , On some topological properties of semi-metric spaces related to fixed-point theory , Int. Math. Forum, 4 (2009), 2159–2160.


• [8] J. H. Asl, S. Rezapour, N. Shahzad , On fixed points of \(\alpha-\psi\)-contractive multifunctions, Fixed Point Theory Appl., 2012 (2012), 6 pages.


• [9] C. J. R. Borges, On continuously semimetrizable and stratifiable spaces, Proc. Amer. Math. Soc., 24 (1970), 193–196.


• [10] S.-H. Cho, G.-Y. Lee, J.-S. Bae , On coincidence and fixed-point theorems in symmetric spaces, Fixed Point Theory Appl., 2008 (2008), 9 pages.


• [11] M. Cicchese, Questioni di completezza e contrazioni in spazi metrici generalizzati, (Italian) Boll. Un. Mat. Ital., 13A (1976), 175–179.


• [12] G. Constantin, I. Istrateseu, Elements of Probabilistic Analysis, Kluwer Academic Publisher, Dordrecht (1989)


• [13] R. H. Haghi, Sh. Rezapour, N. Shahzad , Some fixed point generalizations are not real generalizations, Nonlinear Anal., 74 (2011), 1799–1803.


• [14] R. H. Haghi, Sh. Rezapour, N. Shahzad, Be careful on partial metric fixed point results, Topology Appl., 160 (2013), 450–454.


• [15] T. L. Hicks, B. E. Rhoades , Fixed point theory in symmetric spaces with applications to probabilistic spaces, Nonlinear Anal., 36 (1999), 331–334.


• [16] J. Jachymski, J. Matkowski, T. Swiatkowski , Nonlinear contractions on semimetric spaces, J. Appl. Anal., 1 (1995), 125–134.


• [17] E. Karapinar, Discussion on -contractions on generalized metric spaces, Abstr. Appl. Anal., 2014 (2014), 7 pages


• [18] E. Karapinar, P. Kumam, P. Salimi , On \(\alpha-\psi\)-Meir-Keeler contractive mappings, Fixed Point Theory Appl., 2013 (2013), 12 pages


• [19] W. A. Kirk, Contraction mappings and extensions , Handbook of metric fixed point theory, 1–34, Kluwer Acad. Publ., Dordrecht (2001)


• [20] W. A. Kirk, N. Shahzad , Generalized metrics and Caristi’s theorem , Fixed Point Theory Appl., 2013 (2013), 9 pages.


• [21] W. A. Kirk, N. Shahzad, Fixed point theory in distance spaces , Springer, Cham, (2014),


• [22] D. Mihet, A note on a paper of Hicks and Rhoades, Nonlinear Anal., 65 , 1411–1413. (2006)


• [23] B. Mohammadi, S. Rezapour, N. Shahzad, Some results on fixed points of \(\alpha-\psi\)-Ciric generalized multifunctions, Fixed Point Theory Appl., 2013 (2013), 10 pages.


• [24] J. J. Nieto, R. Rodrígues-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223–239.


• [25] Sh. Rezapour R. H. Haghi, N. Shahzad, Some notes on fixed points of quasi-contraction maps, Appl. Math. Lett., 23 (2010), 498–502.


• [26] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2003), 1435–1443.


• [27] I. A. Rus, A. Petrusel, G. Petrusel, Fixed point theory , Cluj University Press, Cluj-Napoca (2008)


• [28] B. Samet, C. Vetro, B. Vetro, Fixed point theorem for \(\alpha-\psi\)-contractive type mappings, Nonlinear Anal., 75 (2012), 2154 – 2165.


• [29] W. A. Wilson , On semi-metric spaces , Amer. J. Math., 53 (1931), 361–373.