Research in Best Approximation for Fixed Points in Generalized Metric Spaces

Takahashi (Kodai Math Sem Rep 22:142–149, 1970) demonstrated convex structures in metric spaces that were not embedded in Banach spaces in 1970, sparking additional investigation. Extending to metric spaces, our work develops and extends Branciari’s theorem (Branciari in Int J Math Math Sci 29:531–536, 2002) to show common fixed point theorems for mappings meeting a specific integral condition. We address the best approximation method for exhibiting practical significance, emphasizing its relevance in both theoretical and practical situations as indicated by the authors (Manav Tatar and Agarwal in Mathematics 11(21):4455, 2023; Manav Tatar in Best approximation of Fixed Point results in generalized metric spaces, 2023). This study presents a concept that is consistent across several metric spaces. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.