  In this article, we derive a common fixed point result for a pair of single
valued and set-valued mappings on a metric space having graphical structure. In
this case, the set-valued map is assumed to be closed valued instead of closed
and bounded valued. Several results regarding common fixed points and fixed
points follow from the main theorem of this article. By applying our theorem,
we deduce the convergence of the iterates for a nonlinear $q$-analogue
Bernstein operator. Furthermore, we establish sufficient criteria for the
occurrence of a solution to a fractional differential equation.
