We establish a relation theoretic version of the main result of Aydi et al.
[H. Aydi, M. Abbas, C. Vetro, Partial Hausdorff metric and Nadler?s fixed
point theorem on partial metric space, Topol. Appl. (159), 2012, 3234-3242]
and extend the results of Alam and Imdad [A. Alam, M. Imdad,
Relation-theoretic contraction priciple, J. Fixed Point Theory Appl., 17(4),
2015, 693-702.] for a set-valued map in a partial Pompeiu-Hausdorff metric
space. Numerical examples are presented to validate the theoretical finding
and to demonstrate that our results generalize, improve and extend the
recent results in different spaces equipped with binary relations to their
set-valued variant exploiting weaker conditions. Our results provide a new
answer, in the setting of relation theoretic contractions, to the open
question posed by Rhoades on continuity at fixed point. We also show that,
under the assumption of k-continuity, the solution to the Rhoades? problem
given by Bisht and Rakocevic characterizes completeness of the metric
space. As an application of our main result, we solve an integral inclusion
of Fredholm type.