A new contractive condition related to Rhoades’s open question

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.