Abstract
In this paper, the authors introduce the notion of neutrosophic double controlled metric spaces as a generalization of neutrosophic metric spaces. For this purpose, two non-comparable functions, xi and Gamma, are used in triangle inequalities. The authors prove several interesting results for contraction mappings with non-trivial examples. At the end of the paper, the authors prove the existence, and the uniqueness, of the integral equation to support the main result.