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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与Rhoades的公开问题有关的新的收缩条件

Rhoades提出的一个开放问题如下。是否存在可收缩的条件来保证不动点的存在，但不要求映射在该点处是连续的？在本文中，我们推广了Eshaghi等人的著名结果[关于正交集和Banach不动点定理，不动点理论，18（2017），569–578]，这使我们能够找到这种开放性的新解决方案。问题。此外，我们证明上述论文的主张，即不能在其应用中应用Banach不动点定理是不正确的。最后，作为一种应用，我们证明了满足一般线性泛函的多值函数允许满足相应泛函方程的唯一选择。