A generalization of the concept of distance based on the simplex inequality

We introduce and discuss the concept of n-distance, a generalization to n elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality d(x1,…,xn)≤K∑i=1nd(x1,…,xn)zi,x1,…,xn,z∈X, where K=1. Here d(x1,…,xn)zi is obtained from the function d(x1,…,xn) by setting its ith variable to z. We provide several examples of n-distances, and for each of them we investigate the infimum of the set of real numbers K∈]0,1] for which the inequality above holds. We also introduce a generalization of the concept of n-distance obtained by replacing in the simplex inequality the sum function with an arbitrary symmetric function

Generalized coupled common fixed point results in partially ordered A-metric spaces

Sedghi et al. (Mat. Vesn. 64(3):258-266, 2012) introduced the notion of a S-metric as a generalized metric in 3-tuples S : X3→[0,∞), where X is a nonempty set. The aim of this paper is to introduce the concept of an n-tuple metric A : Xn→[0,∞) and to study its basic topological properties. We also prove some generalized coupled common fixed point theorems for mixed weakly monotone maps in partially ordered A-metric spaces. Some examples are presented to support the results proved herein. Our results generalize and extend various results in the existing literature.http://link.springer.com/journal/11784am201