This paper explores certain fixed point results for multivalued mapping in a metric space endowed with an arbitrary binary relation $\mathrm{R}$, briefly written as $\mathrm{R}$-metric space. The fixed point results proved  are subjected to contraction conditions corresponding to the multivalued counterpart of $F$-contraction and $F$-weak contraction in $\mathrm{R}$-metric space. The main results unify, extend and generalize the results on multivalued and single-valued mapping in the literature. To support the conclusion, several examples have been provided.