New directions in fixed point theory in multiplicative metric spaces

The notion of a multiplicative metric space was introduced in 2008 by Bashirov et al. In a such space, the usual triangular inequality is replaced by a multiplicative triangle inequality. The literature includes numerous fixed point results in multiplicative metric spaces. Unfortunately, it was shown that the most obtained fixed point results in such spaces are equivalent to the corresponding fixed point results in metric spaces. In this paper, we open new directions in fixed point theory in multiplicative metric spaces by investigating new contractions on such spaces that cannot be reduced to contractions on metric spaces. We first establish a new multiplicative version of Banach?s fixed point theorem. Next, a new multiplicative version of Kannan?s fixed point theorem is proved. Unlike Kannan?s contraction in metric spaces, we show that a multiplicative contraction of Kannan-type may have more than one fixed point. We also provide sufficient conditions under which any multiplicative contraction of Kannan-type possesses one and only one fixed point. Some examples are provided to illustrate the validity of our obtained results.