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A new contractive condition related to Rhoades’s open question

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  • Hamid Baghani

    (University of Sistan and Baluchestan)

Abstract

An open problem proposed by Rhoades is the following. Is there a contractive condition which guarantees the existence of a fixed point, but does not require the mapping to be continuous at the point? In this paper, we generalize a celebrated result of Eshaghi et al., [On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 18 (2017), 569–578], which allows us to find a new solution to this open problem. Furthermore we show that a claim of the aforementioned paper, that Banach’s fixed point theorem cannot be applied in their application, is incorrect. Finally, as an application, we prove that a multivalued function satisfying a general linear functional inclusion admits a unique selection fulfilling the corresponding functional equation.

Suggested Citation

  • Hamid Baghani, 2020. "A new contractive condition related to Rhoades’s open question," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(2), pages 565-578, June.
    • Handle: RePEc:spr:indpam:v:51:y:2020:i:2:d:10.1007_s13226-020-0417-5
    DOI: 10.1007/s13226-020-0417-5
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