  In this paper we will establish some necessary condition and sufficient
condition respectively for a set-valued mapping to have the Lipschitz-like
property relative to a closed set by employing regular normal cone and limiting
normal cone of a restricted graph of the set-valued mapping. We will obtain a
complete characterization for a set-valued mapping to have the
Lipschitz-property relative to a closed and convex set by virtue of the
projection of the coderivative onto a tangent cone. Furthermore, by introducing
a projectional coderivative of set-valued mappings, we establish a verifiable
generalized Mordukhovich criterion for the Lipschitz-like property relative to
a closed and convex set. We will study the representation of the graphical
modulus of a set-valued mapping relative to a closed and convex set by using
the outer norm of the corresponding projectional coderivative value. For an
extended real-valued function, we will apply the obtained results to
investigate its Lipschitz continuity relative to a closed and convex set and
the Lipschitz-like property of a level-set mapping relative to a half line.
