  This paper addresses the study of novel constructions of variational analysis
and generalized differentiation that are appropriate for characterizing robust
stability properties of constrained set-valued mappings/multifunctions between
Banach spaces important in optimization theory and its applications. Our tools
of generalized differentiation revolves around the newly introduced concept of
$\varepsilon$-regular normal cone to sets and associated coderivative notions
for set-valued mappings. Based on these constructions, we establish several
characterizations of the central stability notion known as the relative
Lipschitz-like property of set-valued mappings in infinite dimensions. Applying
a new version of the constrained extremal principle of variational analysis, we
develop comprehensive sum and chain rules for our major constructions of conic
contingent coderivatives for multifunctions between appropriate classes of
Banach spaces.
