Abstract
In this paper, we prove some fixed point theorems under a convex combination of generalized (\(\epsilon -\delta \)) type rational contractions in which the fixed point may or may not be a point of discontinuity. As a by-product we explore some new answers to the open question posed by Rhoades (Contemp Math 72:233–245, 1988). Furthermore, we consider geometric properties of the fixed point set of a self-mapping on a metric space. We define a new kind of contractive mapping and prove that the fixed point set of this kind of contraction contains a circle (resp. a disc). Several non-trivial examples are given to illustrate our results. Apart from these, an application of discontinuous activation functions, frequently used in neural networks is also given.
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The authors are thankful to the learned referees for suggesting some improvements in the presentation of the paper.
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Bisht, R.K., Özgür, N. Geometric properties of discontinuous fixed point set of (\(\varvec{\epsilon -\delta }\)) contractions and applications to neural networks. Aequat. Math. 94, 847–863 (2020). https://doi.org/10.1007/s00010-019-00680-7
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DOI: https://doi.org/10.1007/s00010-019-00680-7
Keywords
- Fixed point
- \((\epsilon -\delta )\) rational contraction
- Discontinuous mappings
- k-continuity
- Fixed circle
- Fixed disc