Generalized n-metric spaces and fixed point theorems

G\"ahler ([3],[4]) introduced the concept of 2-metric as a possible generalization of usual notion of a metric space. In many cases the results obtained in the usual metric spaces and 2-metric spaces are found to be unrelated (see [5]). Mustafa and Sims [8] took a different approach and introduced the notion of $G$-metric. The author [6] generalized the notion of G-metric to more than three variables and introduced the concept of $K$-metric as a function $K\colon X^n \to \mathbb{R}^+$, $(n\ge 3)$. In this paper, We improve the definition of $K$-metric by making symmetry condition more general. This improved metric denoted by $G_n$ is called the Generalized $n$-metric. We develop the theory for generalized $n$-metric spaces and obtain some fixed point theorems.