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Fixed points of convex and generalized convex contractions

  • Ravindra K. Bisht
  • Vladimir Rakočević
Article

Abstract

Istr\(\check{a}\)tescu (Lib Math 1:151–163, 1981) introduced the notion of convex contraction. He proved that each convex contraction has a unique fixed point on a complete metric space. In this paper we study fixed points of convex contraction and generalized convex contractions. We show that the assumption of continuity condition in [11] can be replaced by a relatively weaker condition of k-continuity under various settings. On this way a new and distinct solution to the open problem of Rhoades (Contemp Math 72:233–245, 1988) is found. Several examples are given to illustrate our results.

Keywords

Fixed point Convex contractions k-Continuity 

Mathematics Subject Classification

Primary 47H09 Secondary 47H10 

Notes

Acknowledgements

The authors are thankful to the learned referees for suggesting some improvements in the presentation of the paper.

References

  1. 1.
    Alghamdi, M.A., Alnafei, S.H., Radenović, S., Shahzad, N.: Fixed point theorems for convex contraction mappings on cone metric spaces. Math. Comput. Model. 54, 2020–2026 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alizadeh, S., Moradlou, F., Salimi, P.: Some fixed point results for \((\alpha,\beta ), (\psi -\phi )\)-contractive mappings. Filomat 28(3), 635–647 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bisht, R.K., Pant, R.P.: A remark on discontinuity at fixed point. J. Math. Anal. Appl. 445, 1239–1242 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bisht, R.K., Rakočević, V.: Generalized Meir–Keeler type contractions and discontinuity at fixed point. Fixed Point Theory 19(1), 57–64 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bisht, R.K., Hussain, N.: A note on convex contraction mappings and discontinuity at fixed point. J. Math. Anal. 8(4), 90–96 (2017)MathSciNetGoogle Scholar
  6. 6.
    Bryant, V.W.: A remark on a fixed point theorem for iterated mappings. Am. Math. Mon. 75(4), 399–400 (1968)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ding X, Cao J, Zhao X, Alsaadi FE (2017) Mittag–Leffler synchronization of delayed fractional-order bidirectional associative memory neural networks with discontinuous activations: state feedback control and impulsive control schemes. Proc. R. Soc. A Math. Eng. Phys. Sci.  https://doi.org/10.1098/rspa.2017.0322 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Forti, M., Nistri, P.: Global convergence of neural networks with discontinuous neuron activations. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50(11), 1421–1435 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ghorbanian, V., Rezapour, S., Shahzad, N.: Some ordered fixed point results and the property (P). Comput. Math. Appl. 63, 1361–1368 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hussain, N., Salimi, P.: Fixed points for generalized \(\psi \)-contraction with application to intergral equations. J. Nonlinear Convex Anal. 16(4), 711–729 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Istr\(\check{a}\)tescu, V.I.: Some fixed point theorems for convex contraction mappings and convex nonexpansive mapping I. Lib. Math. 1, 151–163 (1981)Google Scholar
  12. 12.
    Istr\(\check{a}\)tescu, V.I.: Fixed Point Theory: An Introduction. Mathematics and Its Applications, vol. 7. D. Reidel Publishing Company, Dordrecht (1981)Google Scholar
  13. 13.
    Istr\(\check{a}\)tescu, V.I.: Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters—I. Ann. Mat. Pura Appl. 130(1), 89–104 (1982)Google Scholar
  14. 14.
    Kannan, R.: Some results on fixed points—II. Am. Math. Mon. 76, 405–408 (1969)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Latif, A., Ninsri, A., Sintunavarat, W.: The \((\alpha,\beta )\)-generalized convex contractive condition with approximate fixed point results and some consequence. Fixed Point Theory Appl. 2016, 58 (2016).  https://doi.org/10.1186/s13663-016-0546-z MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Latif, A., Sintunavarat, W., Ninsri, A.: Approximate fixed point theorems for partial generalized convex contraction in \(\alpha \)-complete metric spaces. Taiwan J. Math. 19(1), 315–333 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Miandaragh, M.A., Postolache, M., Rezapour, S.: Approximate fixed points of generalized convex contractions. Fixed Point Theory Appl. 2013, 255 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nie, X., Zheng, W.X.: On multistability of competitive neural networks with discontinuous activation functions. In: 4th Australian Control Conference (AUCC), pp. 245–250 (2014)Google Scholar
  19. 19.
    Nie, X., Zheng, W.X.: Multistability of neural networks with discontinuous non-monotonic Piecewise linear activation functions and time-varying delays. Neural Netw. 65, 65–79 (2015)CrossRefGoogle Scholar
  20. 20.
    Nie, X., Zheng, W.X.: Dynamical behaviors of multiple equilibria in competitive neural networks with discontinuous nonmonotonic piecewise linear activation functions. IEEE Trans. Cybern. 46(3), 679–693 (2016)CrossRefGoogle Scholar
  21. 21.
    Pant, R.P.: Discontinuity and fixed points. J. Math. Anal. Appl. 240, 284–289 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pant, A., Pant, R.P.: Fixed points and continuity of contractive maps. Filomat 31(11), 3501–3506 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rashid, M., Batool, I., Mehmood, N.: Discontinuous mappings at their fixed points and common fixed points with applications. J. Math. Anal. 9(1), 90–104 (2018)MathSciNetGoogle Scholar
  24. 24.
    Reich, S.: Some remarks concerning contraction mappings. Can. Math. Bull. 14, 121–124 (1971)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rhoades, B.E.: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rhoades, B.E.: Contractive definitions and continuity. Contemp. Math. 72, 233–245 (1988)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tas, N., Ozgur, N.Y.: A new contribution to discontinuity at fixed point. arXiv:1705.03699v2 (2018 )
  28. 28.
    Wu, H., Shan, C.: Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses. Appl. Math. Model. 33(6), 2564–2574 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Computational SciencesNational Defence AcademyPuneIndia
  2. 2.Faculty of Sciences and MathematicsUniversity of NisNisSerbia

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