Equivariant degree of convex-valued maps applied to set-valued BVP

Abstract

An equivariant degree is defined for equivariant completely continuous multivalued vector fields with compact convex values. Then it is applied to obtain a result on existence of solutions to a second order BVP for differential inclusions carrying some symmetries. MSC:47H11, 34A60, 34B99

Multiple Solutions to Implicit Symmetric Boundary Value Problems for Second Order Ordinary Differential Equations (ODEs): Equivariant Degree Approach

“…More precisely: Finally, as in Subsection 3.2, we assume that V is a coordinate permutation G-representation given by (12), and condition (13) is satisfied. Furthermore, assume that: such that:…”

“…To some extent, our recent paper [10] opened a door to a systematic usage of the equivariant degree theory for analysis of multiple solutions to symmetric (1) and its generalizations. The starting point for our discussion was Example 6.1 from [11] in which a particular case of BVP (1) in the presence of D 4 -symmetries was considered (see also [12] in which a "multivalued perturbation of this example" was discussed).…”

“…Up to several standard steps, such an extension is very simple (see, for example, [12]). Therefore, below, we will only outline the key steps of the construction (we refer the reader to Appendix 1 of the present paper, where the axiomatic definition for single-valued fields is presented).…”

“…In light of [15] (see also (12) and (13), conditions (A6) and (A7)), we need to check only condition (H4), i.e. :…”

Generalized Gradient Equivariant Multivalued Maps, Approximation and Degree

“…Some of them can be basic tools in the construction of equivariant degree theory. 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]. The degree theory was applied in [7] to obtain nontrivial solutions to multivalued boundary value problems.…”