On Fixed Points of Generalized α-φ Contractive Type Mappings in Partial Metric Spaces

Article Sidebar

PDF
Cite as:
Priya Shahi, Jatinderdeep Kaur, S. S. Bhatia, On Fixed Points of Generalized α-φ Contractive Type Mappings in Partial Metric Spaces, Int. J. Anal. Appl., 12 (1) (2016), 38-48.

Main Article Content

Priya Shahi, Jatinderdeep Kaur, S. S. Bhatia

Abstract

Recently, Samet et al. (B. Samet, C. Vetro and P. Vetro, Fixed point theorem for $\alpha$-$\psi$ contractive type mappings, Nonlinear Anal. 75 (2012), 2154--2165) introduced a very interesting new category of contractive type mappings known as $\alpha$-$\psi$ contractive type mappings. The results obtained by Samet et al. generalize the existing fixed point results in the literature, in particular the Banach contraction principle. Further, Karapinar and Samet (E. Karapinar and B. Samet, Generalized $\alpha$-$\psi$-contractive type mappings and related fixed point theorems with applications, Abstract and Applied Analysis 2012 Article ID 793486, 17 pages doi:10.1155/2012/793486) generalized the $\alpha$-$\psi$ contractive type mappings and established some fixed point theorems for this generalized class of contractive mappings. In (G. S. Matthews, Partial metric topology, Ann. New York Acad. Sci. 728 (1994), 183--197), the author 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 establish the fixed point theorems for generalized $\alpha$-$\psi$ contractive mappings in the context of partial metric spaces. As consequences of our main results, we obtain fixed point theorems on partial metric spaces endowed with a partial order and that for cyclic contractive mappings. Our results extend and strengthen various known results. Some examples are also given to show that our generalization from metric spaces to partial metric spaces is real.

Article Details

References

  1. M. Abbas and T. Nazir, Fixed points of generalized weakly contractive mappings in ordered partial metric spaces, Fixed Point Theory Appl. 2012 (2012), Art. ID 1.
  2. T. Abedelljawad, E. Karapinar and K.Tas, Existence and uniqueness of common fixed point on partial metric spaces, Appl. Math. Lett. 24 (2011), 1894-1899.
  3. R. P. Agarwal, M. A. Alghamdi and N. Shahzad, Fixed point theory for cyclic generalized contractions in partial metric spaces, Fixed Point Theory Appl. (2012) (2012), Art. ID 40.
  4. R. P. Agarwal, M. A. El-Gebeily and D. O' Regan, Generalized contractions in partially ordered metric spaces, Applicable Analysis 87 (2008), 1-8.
  5. I. Altun and A. Erduran, Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory Appl. 2011 (2011), Art. ID 508730.
  6. H. Aydi, Some fixed point results in ordered partial metric spaces,J. Nonlinear Sci. Appl. 4 (2011), 210217.
  7. H. Aydi, H.K. Nashine, B. Samet and H. Yazidi, Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations, Nonlinear Analysis 74 (2011), 6814-6825.
  8. T. G. Bhaskar and V. Lakshmikantham, Fixed Point Theory in partially ordered metric spaces and applications, Nonlinear Analysis 65 (2006), 1379-1393.
  9. D. Ili ´ c, V. Pavlovi ´ c and V. Rakocevi ´ c, Some new extensions of Banach's contraction principle to partial metric space, Appl. Math. Lett. 24 (2011), 1326-1330.
  10. E. Karapinar, Fixed point theory for cyclic weak φ-contraction, Appl. Math. Lett. 24 (2011), 822-825.
  11. E. Karapinar and K. Sadaranagni, Fixed point theory for cyclic (φ-ψ)-contractions, Fixed point theory Appl. 2011 (2011), Art. ID 69.
  12. E. Karapinar and B. Samet, Generalized α-ψ-contractive type mappings and related fixed point theorems with applications, Abstract and Applied Analysis 2012(2012), Article ID 793486.
  13. V. Lakshmikantham and L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Analysis 70 (2009), 4341-4349.
  14. G. S. Matthews, Partial metric topology, Research Report 212, Dept. of Computer Science University of Warwick, 1992.
  15. G. S. Matthews, Partial metric topology, Ann. New York Acad. Sci. 728 (1994), 183-197.
  16. H. K. Nashine, Z. Kadelburg and S. Radenovi ´ c, Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces, Math. Comput. Modelling (2012), doi:10.1016/j.mcm.2011.12.019.
  17. J. J. Nieto and R. R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), 223-239.
  18. M. Pacurar and I. A. Rus, Fixed point theory for cyclic φ-contractions, Nonlinear Anal. 72 (2010), 1181-1187.
  19. M. A. Petric, Some results concerning cyclic contractive mappings, General Mathematics 18 (2010), 213-226.
  20. A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435-1443.
  21. I. A. Rus, Cyclic representations and fixed points, Ann. T. Popovicin. Seminar Funct. Eq. Approx. Convexity 3 (2005), 171-178.
  22. B. Samet, M. Rajovi ´ c, R. Lazovi ´ c and R. Stoiljkovi ´ c, Common fixed point results for nonlinear contractions in ordered partial metric spaces, Fixed Point Theory Appl. 2011 (2011), Art. ID 71.
  23. B. Samet, C. Vetro and P. Vetro, Fixed point theorem for α-ψ contractive type mappings, Nonlinear Anal. 75 (2012), 2154-2165.
  24. M. Turinici, Abstract comparison principles and multivariable Gronwall-Bellman inequalities, J. Math. Anal. Appl. 117 (1986), 100-127.