A Best Proximity Point Result in Modular Spaces with the Fatou Property

Abstract

Consider a nonself-mapping $T:A\to B$, where $(A,B)$ is a pair of nonempty subsets of a modular space $X_{\rho }$. A best proximity point of $T$ is a point $z\in A$ satisfying the condition: $\rho (z-Tz)=\text{inf}\{\rho (x-y):(x,y)\in A\times B\}$. In this paper, we introduce the class of proximal quasicontraction nonself-mappings in modular spaces with the Fatou property. For such mappings, we provide sufficient conditions assuring the existence and uniqueness of best proximity points.