Fixed points of convex and generalized convex contractions

  • Ravindra K. Bisht
  • Vladimir Rakočević


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.


Fixed point Convex contractions k-Continuity 

Mathematics Subject Classification

Primary 47H09 Secondary 47H10 



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


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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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