Interval Metric with Diameter Distance and Its Application to Fixed Point Theory

Abstract

The concept of interval metric is defined in 2010. In the general interval metric, the domain is any set of points and range is one -dimensional interval space $ IR $. The tool used here to prove its metric properties is the Moore interval module. In this paper, we extend the concept of interval metric to the function space $ C[a,b] $. We are giving a detailed extension of the concept of metric from a single value to an interval by taking all the possible values from the minimum-maximum interval, and the range of all such possible values is considered as the diameter distance. We also prove the completeness property and Banach fixed point theorem for the new metric in $ C[a,b] $.