First, in the setting of Busemann convex spaces, we establish best proximity pair existence theorems for generalized pointwise noncyclic contractions. This provides a significant generalization of Kirk and Royalty’s fixed point theorem by relaxing the continuity assumptions. Also, we demonstrate the existence of best proximity pairs for generalized pointwise noncyclic relatively nonexpansive mappings by utilizing the key geometric concept of proximal normal structure. Furthermore, as a key advancement, we consider complete \(\mathrm {CAT(0)}\) spaces equipped with the weak topology introduced by Ahmadi Kakavandi (Proc Amer Math Soc 141:1029–1039, 2013). In this setting, we introduce the notion of weak proximal normal structure and employ it to investigate the existence of best proximity pairs for generalized pointwise noncyclic relatively nonexpansive mappings. Additionally, within the framework of Busemann convex geodesic spaces, we extend Edelstein’s theorem and a key result of Park (Rocky Mountain J Math 8:743–75, 1978) on fixed points of f-contractive maps. This leads to a new existence result of best proximity points for cyclic f-contractive mappings in such spaces. Finally, in the framework of complete \(\mathrm {CAT(0)}\) spaces, we derive the existence results of best proximity points for asymptotically relatively nonexpansive mappings.
