The purpose of this paper is to introduce and analyze a new idea of proximally compatible mappings and we extend some results of Jungck via proximally compatible mappings. Furthermore, we obtain common best proximity point theorems for proximally compatible mappings through two different ways of construction of sequences. In addition, we provide an example to support our main result.

In this paper, we develop the proximal iterated function systems $(PIFS)$ ( P I F S ) using cyclic Meir-Keeler contractions and hence give an application in the fractal theory by constructing fractals using the new $IFS$ I F S . Furthermore, we extend the results to countable $PIFS$ P I F S .

In this paper, first we introduce the notion of proximally weakly compatible mappings and we extend the CLRg property (CLRg-Common limit in the range of g) to the case of non-self mappings. Then we prove the existence of proximally coincident point for this new class of mappings with the assumption of proximal CLRg property in fuzzy metric space. Further, we establish new common best proximity point result for proximally weakly compatible mappings in the setting of fuzzy metric space.