Generalized common best proximity point results in fuzzy multiplicative metric spaces

Abstract

In this manuscript, we prove the existence and uniqueness of a common best proximity point for a pair of non-self mappings satisfying the iterative mappings in a complete fuzzy multiplicative metric space. We consider the pair of non-self mappings $ X:\mathcal{P}\rightarrow \mathcal{G} $ and $ Z:\mathcal{P }\rightarrow \mathcal{G} $ and the mappings do not necessarily have a common fixed-point. In a complete fuzzy multiplicative metric space, if $ \mathcal{\varphi } $ satisfy the condition $ \mathcal{\varphi } \left(b, Zb, \varsigma \right) = \mathcal{\varphi }\left(\mathcal{P}, \mathcal{ G}, \varsigma \right) = \mathcal{\varphi }\left(b, Xb, \varsigma \right) $, then $ b $ is a common best proximity point. Further, we obtain the common best proximity point for the real valued functions $ \mathcal{L}, \mathcal{M}:(0, 1]\rightarrow \mathbb{R} $ by using a generalized fuzzy multiplicative metric space in the setting of $ (\mathcal{L}, \mathcal{M}) $-iterative mappings. Furthermore, we utilize fuzzy multiplicative versions of the $ (\mathcal{L}, \mathcal{M}) $-proximal contraction, $ (\mathcal{L}, \mathcal{M}) $-interpolative Reich-Rus-Ciric type proximal contractions, $ (\mathcal{L}, \mathcal{M}) $-Kannan type proximal contraction and $ (\mathcal{L}, \mathcal{M}) $-interpolative Hardy-Rogers type proximal contraction to examine the common best proximity points in fuzzy multiplicative metric space. Moreover, we provide differential non-trivial examples to support our results.

Keywords

• fuzzy multiplicative metric space
• fixed-point
• best proximity point theorems
• contraction mappings
• uniqueness