Positivity 4: 1–39, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Viscosity Coderivatives and Their Limiting
Behavior in Smooth Banach Spaces
BORIS S. MORDUKHOVICH
, YONGHENG SHAO
and QIJI ZHU
Department of Mathematics, Wayne State University, Detroit, Michigan 48202, U.S.A.
Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, Michigan
(Received: 14 July 1998; Accepted: 4 November 1998)
Abstract. This paper concerns with generalized differentiation of set-valued and nonsmooth map-
pings between Banach spaces. We study the so-called viscosity coderivatives of multifunctions and
their limiting behavior under certain geometric assumptions on spaces in question related to the
existence of smooth bump functions of any kind. The main results include various calculus rules for
viscosity coderivatives and their topological limits. They are important in applications to variational
analysis and optimization.
Mathematical Subject Classiﬁcations (1991): 49J52, 58C06, 58C20.
Key words: Banach spaces, smooth bump functions, set-valued mappings, viscosity coderivatives,
topological limits, calculus rules, variational analysis
The primary objects of this study are coderivatives of multifunctions (set-valued
mappings) : X ⇒ Y between Banach spaces. Such objects are natural general-
izations of the classical adjoint derivative operator to nonsmooth and set-valued
mappings. It is worth mentioning that the main motivation for introducing the ﬁrst
coderivative in Mordukhovich  was to describe a proper adjoint system in an
extension of the Pontryagin maximum principle to optimal control problems gov-
erned by differential inclusions. In contrast to the classical case, the coderivative
of , generated by a nonconvex normal cone  to the graph of , cannot be
treated as a dual object to tangentially generated graphical derivatives introduced
by Aubin [1, 2]. In spite of (actually due to) the nonconvexity of its values, this
coderivative enjoys a full calculus and important applications to some principal
aspects of variational analysis and optimization; see the books [30, 42], the survey
Research was partly supported by the National Science Foundation under grant DMS-9704751,
by the USA-Israel Binational Science Foundation under grant 94-00237, and by an NSERC Foreign
Research was partly supported by the National Science Foundation under grant DMS-9704203
and by the Faculty Research and Creative Activities Support Fund at Western Michigan University.