Journal of Fixed Point Theory and Applications

, Volume 19, Issue 4, pp 2437–2439 | Cite as

A remark on the result of Radu Miculescu and Alexandru Mihail

Article
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Abstract

In this note, we show that the assumption of continuity considered in the recent result of Miculescu and Mihail (J Fixed Point Theory Appl, doi: 10.1007/s11784-017-0411-7, 2017) can be relaxed further. We also observe that the power convex contraction introduced by Miculescu and Mihail (see condition (1.1)) provides one more solution to the open question of Rhoades (Contemp Math 72:233–245, 1988] regarding existence of a contractive definition which is strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point.

Keywords

Fixed point convex contraction mappings 

Mathematics subject classification

Primary 47H09 Secondary 47H10 

References

  1. 1.
    Miculescu, R., Mihail, A.: A generalization of Matkowski s fixed point theorem and Istrăţescu s fixed point theorem concerning convex contractions. J. Fixed Point Theory Appl. (2017). doi: 10.1007/s11784-017-0411-7 MATHGoogle Scholar
  2. 2.
    Pant, R.P.: Discontinuity and fixed points, J. Math. Anal. Appl. \(\bf {240}\), 284–289 (1999)Google Scholar
  3. 3.
    Rhoades,B.E.: Contractive definitions and continuity, Contemp. Math. \(\bf {72}\), 233–245 (1988)Google Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of MathematicsNational Defence AcademyPuneIndia

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