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.
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References
M. Eshaghi Gordji, M. Ramezani, M. De La Sen, and Y. J. Cho, On orthogonal sets and Banach fixed Point theorem, Fixed Point Theory, 18 (2017), 569–578.
R. P. Pant, Discontinuity and Fixed Points, J. Math. Anal. Appl., 240 (1999), 284–289.
M. Ramezani and H. Baghani, The Meir-Keeler fixed point theorem in incomplete modular spaces with application, J. Fixed Point Theory Appl., 19 (2017), 2369–2382.
B. E. Rhoades, Contractive definitions and continuity, Contemporary Mathematics, 72 (1988), 233–245.
A. Smajdor and J. Szczawińska, Selections of set-valued functions satisfying the general linear inclusion, J. Fixed Point Theory Appl., 16 (2016), 133–145.
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The author would like to thank the referees for their careful reading of the paper and for several important suggestions, leading to the papers significant improvement.
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Baghani, H. A new contractive condition related to Rhoades’s open question. Indian J Pure Appl Math 51, 565–578 (2020). https://doi.org/10.1007/s13226-020-0417-5
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DOI: https://doi.org/10.1007/s13226-020-0417-5