Generalized Gradient Equivariant Multivalued Maps, Approximation and Degree

Zdzisław Dzedzej,Tomasz Gzella

doi:10.3390/math8081262

Abstract

Consider the Euclidean space Rn with the orthogonal action of a compact Lie group G. We prove that a locally Lipschitz G-invariant mapping f from Rn to R can be uniformly approximated by G-invariant smooth mappings g in such a way that the gradient of g is a graph approximation of Clarkés generalized gradient of f. This result enables a proper development of equivariant gradient degree theory for a class of set-valued gradient mappings.

Highlights

It is well known that various versions of degree theory are very useful in nonlinear analysis; see the books [1,2,3] and their extensive references
In order to extend the degrees to set-valued maps, a graph approximation method in the spirit of [5] appeared to be very successful
In [6], a series of selection and graph approximation results for convex-valued mappings was obtained in the presence of symmetries given by a compact group action as extensions of many classical results

Summary

Introduction

It is well known that various versions of degree theory are very useful in nonlinear analysis; see the books [1,2,3] and their extensive references. A powerful special degree for maps commuting with an action of a Lie group gives many multiplicity results; see the book [4]. In [6], a series of selection and graph approximation results for convex-valued mappings was obtained in the presence of symmetries given by a compact group action as extensions of many classical results. An equivariant version of the Cellina approximation theorem for convex-valued upper semicontinuous mappings was used in [7] to define the equivariant degree, which extended the one from [4,8]. See [9,10] for further applications of that degree to obtain multiple solutions to some implicit functional differential equations

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Conclusion