Best Proximity Points for Geraghty-Type Non-Self Mappings

We study Geraghty-type non-self mappings within the framework of best proximity point theory. By introducing auxiliary functions with subsequential convergence, we establish general conditions ensuring the existence and uniqueness of best proximity points. Our results extend and unify earlier work on proximal and Kannan-type contractions under a Geraghty setting, and provide counterexamples showing that the auxiliary assumptions are essential. As an illustration, we construct an explicit non-self alignment mapping on subsets of ${\mathbb{R}}^2$ for which all hypotheses can be verified and the unique best proximity point, as well as the convergence of the associated proximal iteration, can be computed in closed form.