The paper investigates a new type of contraction called -HR-Ćirić-Reich-Rus contractions, which extends the existing research on Ćirić-Reich-Rus contraction mappings. We prove fixed point theorems by enriching
-HR-Ćirić-Reich-Rus contractions with respect to the asymptotic regularity of generalized averaged mappings. These mappings are denoted as
, where
is the identity map and
is a function from a normed space to real numbers.