In the present paper, we prove some fixed point theorems by using non-decreasing mapping \(\kappa :\mathbb{R}^+\to \mathbb{R}^+\) known as altering distance function or control function, in context of \(S\)-metric space. Further, explore property \(P\) for these contractive mappings.

A Riemannian metric is said to be Einstein if the Ricci curvature is a constant multiple of the metric. Given a manifold M , one can ask whether M carries an Einstein metric, and if so, how many. This fundamental question in Riemannian geometry is for the most part unsolved (cf. [Bes]). As a global PDE or a variational problem, the question is intractible. It becomes more manageable in the homo...

We study the metric s-t path Traveling Salesman Problem (TSP). [An, Kleinberg, and Shmoys, STOC 2012] improved on the long standing 5 3 -approximation factor and presented an algorithm that achieves an approximation factor of 1+ √ 5 2 ≈ 1.61803. Later [Sebő, IPCO 2013] further improved the approximation factor to 85 . We present a simple, self-contained analysis that unifies both results; our m...