We introduce the notion of \((\alpha ,\nu )\) -relaxed polygonal metric spaces, where \(\alpha >0\) and \(\nu : [0,\infty )\rightarrow [0,\infty )\) is a function satisfying certain conditions. This notion generalizes the concept of s-relaxed \(_p\) metric spaces proposed by Fagin et al. (SIAM J Discrete Math 17(1): 134–160, 2003). The originality of the introduced notion is justified by examples of \((\alpha ,\nu )\) -relaxed polygonal metric spaces that are not s-relaxed \(_p\) metric spaces. We establish several properties of \((\alpha ,\nu )\) -relaxed polygonal metric spaces. In particular, we introduce the concept of \((\alpha ,\nu )\) -metric bounded distance, and show that, if \(\delta : Z\times Z\rightarrow [0,\infty )\) is symmetric, then \(\delta \) is a \((\alpha ,\nu )\) -relaxed polygonal metric, if and only if \(\delta \) is a \((\alpha ,\nu )\) -metric bounded distance. We next define the convergence of sequences, Cauchy sequences, and completeness. We also establish some fixed point results in \((\alpha ,\nu )\) -relaxed polygonal metric spaces. Namely, we extend the Banach contraction principle and the Kannan fixed point theorem from metric spaces to \((\alpha ,\nu )\) -relaxed polygonal metric spaces. The novelty of our fixed point results are supported by examples in which the standard Banach and Kannan fixed point theorems are not applicable.
