An existence theorem for Volterra-type integral inclusion is establish in b-metric spaces. We first introduce two new F-contractions of Hardy–Rogers type and then establish fixed point theorems for these contractions in the setting of b-metric spaces. Finally, we apply our fixed point theorem to prove the existence theorem for Volterra-type integral inclusion. We also provide an example to show that our fixed point theorem is a proper generalization of a recent fixed point theorem by Cosentino et al.