In this paper, we utilize the family $\mathfrak{F}$ and the notion of $\omega$-distance in an ordered $\mathcal{G}$-metric space and introduce $(F,\omega)$-contractions in order to derive some fixed point results. We also discuss the problems of Ulam-Hyers stability, well-posedness and limit shadowing property. In order to illustrate the use of our results, we apply them to nonlinear integral equations, as well as to some three-point fractional integral boundary value problems, both with numerical examples.