A new contractive condition related to Rhoades’s open question

Hamid Baghani ()
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Hamid Baghani: University of Sistan and Baluchestan

Indian Journal of Pure and Applied Mathematics, 2020, vol. 51, issue 2, 565-578

Abstract: 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.

Keywords: Orthogonal set; fixed point; multivalued mapping; selection; Picard operator; 47H10; 54C65; 31A10 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s13226-020-0417-5

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