Fixed point results for cyclic contractions in Menger PM-spaces and generalized Menger PM-spaces

Original Paper


In this paper, we introduce the concepts of cyclic \(\varphi \)-contractions in Menger PM-spaces and cyclic weak \(\phi \)-contractions in generalized Menger PM-spaces. Based on these concepts, some fixed point results for cyclic \(\varphi \)-contractions in Menger PM-spaces are obtained. Furthermore, some fixed point theorems for cyclic weak \(\phi \)-contractions and cyclic \(\varphi \)-contractions in generalized Menger PM-spaces are obtained by assuming that the number of cyclic sets is odd. An example is also given to show the validity of our main results.


Menger PM-space Generalized Menger PM-space Fixed point Cyclic \(\varphi \)-contraction Cyclic weak \(\phi \)-contraction 

Mathematics Subject Classification

47H10 46S50 47S50 



This work is supported by the Natural Science Foundation of China (11461045, 11361042, 11461043), the Natural Science Foundation of Jiangxi Province of China (20142BAB211016, 20132BAB201001), and the Scientific Program of the Provincial Education Department of Jiangxi (GJJ150008).

Author contributions

All authors contributed equally. All authors read and approved the final manuscript.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.


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Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.Department of MathematicsNanchang UniversityNanchangPeople’s Republic of China
  2. 2.Numerical Simulation and High-Performance Computing Laboratory, School of SciencesNanchang UniversityNanchangPeople’s Republic of China
  3. 3.Department of MathematicsSwansea UniversitySingleton ParkUK

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