The Class of \((\alpha ,\psi )\)-Contractions and Related Fixed Point Theorems

  • Praveen AgarwalEmail author
  • Mohamed Jleli
  • Bessem Samet


The class of \((\alpha ,\psi )\)-contractions was introduced by Samet et al. [26]. In this chapter, we prove three fixed point theorems for this class of mappings. The presented results are extensions of those obtained in [26]. Moreover, we show that the class of \((\alpha ,\psi )\)-contractions includes as special cases several types of contraction-type mappings, whose fixed points can be obtained by means of Picard iteration. As an application, the existence and uniqueness of solutions to a certain class of quadratic integral equations is discussed. The main references of this chapter are the papers [24, 26].


  1. 1.
    Argyros, I.K.: Quadratic equations and applications to Chandrasekhars and related equations. Bull. Aust. Math. Soc. 32, 275–292 (1985)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Argyros, I.K.: On a class of quadratic integral equations with perturbations. Funct. Approx. 20, 51–63 (1992)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Berinde, V.: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 9, 43–53 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Berinde, V.: Iterative Approximation of Fixed Points. Lecture Notes in Mathematics. Springer, Berlin (2007)Google Scholar
  5. 5.
    Chatterjee, S.K.: Fixed point theorems. Comptes Rendus Acad. Bulg. Sci. 25, 727–730 (1972)Google Scholar
  6. 6.
    Ćirić, L.J.: On some maps with a nonunique fixed point. Publ. Inst. Math. 17, 52–58 (1974)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dass, B.K., Gupta, S.: An extension of Banach contraction principle through rational expressions. Indian J. Pure Appl. Math. 6, 1455–1458 (1975)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)CrossRefGoogle Scholar
  9. 9.
    Edelstein, M.: An extension of Banach’s contraction principle. Proc. Am. Math. Soc. 12, 7–10 (1961)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hardy, G.E., Rogers, T.D.: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 16, 201–206 (1973)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136(4), 1359–1373 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jaggi, D.S.: Some unique fxed point theorems. Indian J. Pure Appl. Math. 8, 223–230 (1977)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kannan, R.: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71–76 (1968)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Karapinar, E.: Discussion on contractions on generalized metric spaces. Abstr. Appl. Anal. 2014, Article ID 962784, 7 (2014)Google Scholar
  15. 15.
    Karapinar, E., Samet, B.: Generalized \(\alpha \)-\(\psi \) contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012, Article ID 793486, 17 (2014)Google Scholar
  16. 16.
    Karapinar, E., Shahi, P., Tas, P.: Generalized \(\alpha \)-\(\psi \)-contractive type mappings of integral type and related fixed point theorems. J. Inequal. Appl. 2014, 16 (2014)Google Scholar
  17. 17.
    Kirk, W.A., Srinivasan, P.S., Veeramany, P.: Fixed poits for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4(1), 79–89 (2003)MathSciNetGoogle Scholar
  18. 18.
    Miandaragh, M.A., Postolache, M., Rezapour, S.H.: Some approximate fixed point results for generalized \(\alpha \)-contractive mappings. Univ. Politeh. Buchar. Ser. A 75(2), 3–10 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Nieto, J.J., Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22(3), 223–239 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Păcurar, M., Rus, I.A.: Fixed point theory for cyclic \(\phi \)-contractions. Nonlinear Anal. 72, 1181–1187 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rezapour, S.H., Samei, M.E.: Some fixed point results for \(\alpha \)-\(\psi \)-contractive type mappings on intruitionistic fuzzy metric spaces. J. Adv. Math. Stud. 7(1), 176–181 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Rus, I.A.: Cyclic representation and fixed points. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 3, 171–178 (2005)Google Scholar
  24. 24.
    Samet, B.: Fixed point for \(\alpha \)-\(\psi \) contractive mappings with an application to quadratic integral equations. Electron. J. Differ. Equ. 2014, 152 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Samet, B.: The class of \((\alpha ,\psi )\)-type contractions in b-metric spaces and fixed point theorems. Fixed Point Theory Appl. 2015, (2015)Google Scholar
  26. 26.
    Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for \(\alpha \)-\(\psi \)-contractive type mappings. Nonlinear Anal. 75(4), 2154–2165 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Suzuki, T.: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 136, 1861–1869 (2008)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Suzuki, T.: Some similarity between contractions and Kannan mappings. Fixed Point Theory Appl. 2008, Article ID 649749, 1–8 (2008)Google Scholar
  29. 29.
    Zamfirescu, T.: Fix point theorems in metric spaces. Arch. Math. (Basel) 23, 292–298 (1972)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