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Journal of Convex Analysis 17 (2010), No. 3&4, 737--763
Copyright Heldermann Verlag 2010

Infimal Convolutions and Lipschitzian Properties of Subdifferentials for Prox-Regular Functions in Hilbert Spaces

Miroslav Bacak
School of Mathematical and Physical Sciences, University of Newcastle, Newcastle -- NSW 2308, Australia

Jonathan M. Borwein
Centre for Computer Assisted Research Mathematics and its Applications, University of Newcastle, Callaghan NSW 2308, Australia

Andrew Eberhard
School of Mathematical and Geospatial Sciences, RMIT - GPO Box 2476V, Melbourne - Victoria, Australia 3001

Boris S. Mordukhovich
Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.

We study infimal convolutions of extended-real-valued functions in Hilbert spaces paying a special attention to the rather broad and remarkable class of prox-regular functions. Such functions have been well recognized as highly important in many aspects of variational analysis and its applications in both finite-dimensional and infinite-dimensional settings. Based on advanced variational techniques, we discover some new subdifferential properties of infimal convolutions and apply them to the study of Lipschitzian behavior of subdifferentials for prox-regular functions in Hilbert spaces. It is shown, in particular, that the fulfillment of a natural Lipschitz-like property for (set-valued) subdifferentials of prox-regular functions forces such functions, under weak assumptions, actually to be locally smooth with single-valued subdifferentials reduced to Lipschitz continuous gradient mappings.

Keywords: Subdifferentials, Lipschitz continuity, infimal convolutions, prox-regular functions, prox-bounded functions, set-valued mappings.

MSC: 49J52, 46C05

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