The concept of \(k\) -intuitionistic fuzzy metric spaces was recently introduced by associating the degree of nearness between two points with \(k\) parameters \((\mathfrak {N}_1, \mathfrak {N}_2, \mathfrak {N}_3, \ldots , \mathfrak {N}_k)\) . However, the results in this framework were confined to continuous mappings, and no findings were established for discontinuous mappings in \(k\) -intuitionistic fuzzy metric spaces. In this paper, we address this gap by deriving fixed-point theorems for mappings without requiring continuity in \(k\) -intuitionistic fuzzy metric spaces. Specifically, we present a general fixed-point theorem for single-valued \(k\) -intuitionistic fuzzy Kannan-type contractions. Furthermore, as an application, we employ one of these fixed-point results to establish the existence of solutions to fractional differential equations.
