Multiple solutions to symmetric boundary value problems for second order ODEs: Equivariant degree approach

Abstract

For a long time, a vector nonlinear pendulum equation (VNPE) stands as a paradigmatic example of nonlinear analysis. Under the so-called Hartman–Nagumo conditions, the solvability of the Dirichlet boundary value problem for the VNPE was established by Hartman (using the Schauder Fixed Point Theorem). In this paper, we establish a general framework for studying the impact of symmetries to multiple symmetric solutions to Dirichlet/Neumann BVP for VNPE based on the usage of the equivariant degree method. Abstract results are supported by concrete computations related to VNPE respecting dihedral group symmetries.