Best proximity results in metric spaces endowed with a hyperconvex structure

In the current article, we focus on hyperconvex metric spaces and survey the existence of best proximity points and optimal pair of fixed points for cyclic and noncyclic relatively u-continuous mappings which are r-condensing by applying a suitable measure of noncompactness. The method of the proof of our main results relies on the fact that every hyperconvex metric space $(\mathcal {M}, d)$ can be isometrically embedded into the Banach space $\ell ^{\infty}(\mathcal {M})$ . Another important tool which will be used in the proof of the existence theorems is to show that the proximal pair of every nonempty and admissible pair in a hyperconvex metric space $\mathcal {M}$ is also nonempty and admissible.