Generalized common best proximity point results in fuzzy multiplicative metric spaces

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 $X:\mathcal{P}\to \mathcal{G}$ and $Z:\mathcal{P}\to \mathcal{G}$ and the mappings do not necessarily have a common fixed-point. In a complete fuzzy multiplicative metric space, if $\phi$ satisfy the condition $\phi \left(b,Zb,\varsigma \right)=\phi \left(\mathcal{P},\mathcal{G},\varsigma \right)=\phi \left(b,Xb,\varsigma \right)$, then $b$ is a common best proximity point. Further, we obtain the common best proximity point for the real valued functions $\mathcal{L},\mathcal{M}:(0,1]\to \mathbb{R}$ 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.