Advertisement

On Fixed Points That Belong to the Zero Set of a Certain Function

  • Praveen Agarwal
  • Mohamed Jleli
  • Bessem Samet
Chapter

Abstract

Let \(T: X\rightarrow X\) be a given mapping. The set \({\text {Fix}}(T)\) is said to be \(\varphi \)-admissible with respect to a certain mapping \(\varphi : X\rightarrow [0,\infty )\), if \(\emptyset \ne \text{ Fix }(T)\subseteq Z_\varphi \), where \(Z_\varphi \) denotes the zero set of \(\varphi \), i.e., \(Z_\varphi =\{x\in X: \varphi (x)=0\}\). In this chapter, we present the class of extended simulation functions recently introduced by Roldán and Samet [13], which is more large than the class of simulation functions, introduced by Khojasteh et al. [8]. We obtain a \(\varphi \)-admissibility result involving extended simulation functions, for a new class of mappings \(T: X\rightarrow X\), with respect to a lower semi-continuous function \(\varphi : X\rightarrow [0,\infty )\), where X is a set equipped with a certain metric d. From the obtained results, some fixed point theorems in partial metric spaces are derived, including Matthews fixed point theorem [9]. Moreover, we answer to three open problems posed by Ioan A. Rus in [16].The main references for this chapter are the papers [7, 13, 17].

References

  1. 1.
    Alghamdi, M.A., Shahzad, N., Valero, O.: On fixed point theory in partial metric spaces. Fixed Point Theory Appl. 2012, 175 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Altun, I., Sola, F., Simsek, H.: Generalized contractions on partial metric spaces. Topol. Appl. 157, 2778–2785 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berzig, M., Karapınar, E., Roldán, A.: Discussion on generalized-\((\alpha \psi ,\beta \varphi )\)-contractive mappings via generalized altering distance function and related fixed point theorems. Abstr. Appl. Anal. 2014, Article ID 259768 (2014)Google Scholar
  4. 4.
    Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ćirić, L.J., Samet, B., Aydi, H., Vetro, C.: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput. 218, 2398–2406 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Haghi, R.H., Rezapour, S., Shahzad, N.: Be careful on partial metric fixed point results. Topol. Appl. 160(3), 450–454 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Karapınar, E., O’Regan, D., Samet, B.: On the existence of fixed points that belong to the zero set of a certain function. Fixed Point Theory Appl. 2015, 152 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Khojasteh, F., Shukla, S., Radenović, S.: A new approach to the study of fixed point theory for simulation functions. Filomat 29(6), 1189–1194 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Matthews, S.G.: Partial metric topology. In: Proceeding of the 8th Summer Conference on General Topology and Applications. Annals of the New York Academy of Sciences, vol. 728, pp. 183–197 (1994)Google Scholar
  10. 10.
    Oltra, S., Valero, O.: Banach’s fixed point theorem for partial metric spaces. Rend. Istit. Mat. Univ. Trieste. 36(1–2), 17–26 (2004)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Paesano, D., Vetro, P.: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl. 159, 911–920 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Roldán, A., Roldán, C., Karapınar, E.: Multidimensional fixed-point theorems in partially ordered complete partial metric spaces under \((\psi ,\varphi )\)-contractivity conditions. Abstr. Appl. Anal. 2013, Article ID 634371 (2013)Google Scholar
  13. 13.
    Roldán, A., Samet, B.: \(\varphi \)-admissibility results via extended simulation functions. J. Fixed Point Theory Appl. 19, 1197–2015 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Romaguera, S.: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. 2010, Article ID 493298 (2010)Google Scholar
  15. 15.
    Romaguera, S.: Fixed point theorems for generalized contractions on partial metric spaces. Topol. Appl. 159, 194–199 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rus. I.A.: Fixed point theory in partial metric spaces. Anal. Univ. de Vest, Timisoara, Seria Matematica-Informatica 46(2) 141–160 (2008)Google Scholar
  17. 17.
    Samet, B.: Existence and uniqueness of solutions to a system of functional equations and applications to partial metric spaces. Fixed Point Theory 14(2), 473–482 (2013)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Samet, B., Rajović, M., Lazović, R., Stoiljković, R.: Common fixed point results for nonlinear contractions in ordered partial metric spaces. Fixed Point Theory Appl. 2011, 71 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Valero, O.: On Banach fixed point theorems for partial metric spaces. Appl. Gen. Topol. 6(2), 229–240 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsAnand International College of EngineeringJaipurIndia
  2. 2.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia

Personalised recommendations