On the geometry of fixed points and discontinuity

Rajendra Prasad Pant, Nihal Özgür, Bharti Joshi, Mangey Ram

Research output: Contribution to journalArticlepeer-review


Recently, there has been a considerable effort to obtain new solutions to the Rhoades’ open problem on the existence of contractive mappings that admit discontinuity at the fixed point. An extended version of this problem is also stated using a geometric approach. In this paper, we obtain new solutions to this extended version of the Rhoades’ open problem. A related problem, the fixed-circle problem (resp. fixed-disc problem) is also studied. Both of these problems are related to the geometric properties of the fixed point set of a self- mapping on a metric space. Furthermore, a new result about metric completeness and a short discussion on the activation functions used in the study of neural networks are given. By providing necessary examples, we show that our obtained results are effective.

Original languageEnglish
Pages (from-to)155-170
Number of pages16
JournalHacettepe Journal of Mathematics and Statistics
Issue number1
Publication statusPublished - 29 Feb 2024
Externally publishedYes


  • activation function
  • discontinuity
  • fixed circle
  • fixed disc
  • fixed point

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Statistics and Probability
  • Geometry and Topology


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