We consider a relatively new hybrid generalized F-contraction involving a pair of mappings and use this contraction to prove a common fixed-point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying the generalized (F, 𝜑)-contraction condition with the common limit range property in complete metric spaces. A similar result involving a hybrid pair of mappings satisfying the rational-type Hardy–Rogers (F, 𝜑)-contractive condition is also proved. We also generalize and improve several results available from the existing literature. As applications of our results, we prove two theorems on the existence of solutions of certain systems of functional equations encountered in dynamic programming and the Volterra integral inclusion. Moreover, we present an illustrative example. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.