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

Abstract

In this paper, we develop a general framework for studying Dirichlet Boundary Value Problems (BVP) for second order symmetric implicit differential systems satisfying the Hartman-Nagumo conditions, as well as a certain non-expandability condition. The main result, obtained by means of the equivariant degree theory, establishes the existence of multiple solutions together with a complete description of their symmetric properties. The abstract result is supported by a concrete example of an implicit system respecting D4-symmetries.

Keywords

• symmetric BVP
• second order implicit Ordinary Differential Equation (ODE)
• multiple solutions
• a priori bounds
• equivariant degree
• multivalued maps
• dihedral group symmetries