O-metrics: new metric-types, polygon inequalities and fixed point theorems from binary operations

The class of O-metric spaces generalizes several existing metric-type spaces in literature including metric spaces, b-metric spaces, and ultra metric spaces. In this paper, we discuss the properties of the topology induced by an O-metric and establish in the setting, fixed point theorems for contractions and generalized contractions. The proofs of the theorems rely heavily on polygon \({{\textbf {o}}}\)-inequalities which are a natural generalization of the triangle inequality, and the construction of which leads to the notion of \({{\textbf {o}}}\)-series following a pattern of functions. As application, conditions for the existence of solutions of initial value problems are discussed and a generalization of Lebesgue spaces is introduced.