Equivalence between the Existence of Best Proximity Points and Fixed Points for Some Classes of Proximal Contractions

Abstract

In the year 2014, Almeida et al. introduced a new class of mappings, namely, contractions of Geraghty type. Additionally, in the year 2021, Beg et al. introduced the concept of generalized F-proximal contraction of the first kind and generalized F-proximal contraction of the second kind, respectively. After developing these concepts, authors mainly studied the best proximity points for these classes of mappings. In this short note, we prove that the problem of the existence of the best proximity points for the said classes of proximal contractions is equivalent to the corresponding fixed points problems.

Keywords

• best proximity point
• F-proximal contraction
• F-contraction of Hardy-Rogers-type
• generalized F-proximal contraction of the first kind
• generalized F-proximal contraction of the second kind
• Geraghty contraction