Existence and uniqueness of fixed point on closed ball in multiplicative ${\mathbf{G}}_{\mathcal{M}}-$metric space

Abstract

In this article, we investigate some fixed point results satisfying a new generalized $\Delta$-implicit contractive condition in ordered complete multiplicative ${\mathbf{G}}_{\mathcal{M}}-$metric space. Also, some new definitions and fixed point theorems are presented in ordered complete multiplicative ${\mathbf{G}}_{\mathcal{M}}-$metric space. Furthermore, some nontrivial and illustrative examples are given to validate our obtained results.