Journal of the
Korean Mathematical Society JKMS

In this paper, we define the concepts of commute proximally, dominate proximally, weakly dominate proximally, proximal generalized $\varphi$-contraction and common best proximity point in probabilistic Menger space. We prove some common best proximity point and common fixed point theorems for dominate proximally and weakly dominate proximally mappings in probabilistic Menger space under certain conditions. Finally we show that proximal generalized $\varphi$-contractions have best proximity point in probabilistic Menger space. Our results generalize many known results in metric space.

Keywords: commute proximally, dominate proximally, weakly dominate proximally, common best proximity point, probabilistic Menger space

MSC numbers: Primary 47H10, 47H09