Abbas, Ali and Salvador [Fixed and periodic points of generalized contractions in metric spaces, Fixed Point Theory Appl. 2013, 2013:243] extended the concept of $F$-contraction mapping introduced in [21], to two mappings. The aim of this paper is to introduce the notion of a generalized $F_g$-weak contraction mapping and to study sufficient conditions for the existence of common fixed points for such class of mappings. As applications, related invariant approximation results are derived. The results obtained herein unify, generalize and complement various known results in the literature.