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 D 4 -symmetries.

Solutions of fixed period in the nonlinear wave equation on networks

“…3 Reduction to Fixed-Point Subspace of H We consider the system (5) where f : V → R is differentiable at 0 satisfying the assumptions (A1)-(A3). We do not exclude the case that 0 ∈ σ(A ) (which means that zero is degenerate solution to (5)). Notice that Ker (A ) is finite dimensional because A : H → H is a Fredholm operator.…”

“…ω9 < λ < 2 ω8 then there exist three p periodic solutions such that the solutions x(t) = u(λt) to(5) have the S 4 × Z 2 -isotropy groups (D d 4 ), (D d 2 )and (Dd 4 ).A Appendix: Equivariant Brouwer DegreeW assume G is a compact Lie group. For a subgroup H ⊂ G (which is always assumed to be closed), denote by N (H) the normalizer of H in G, by W (H) = N (H)/H the Weyl group of H in G, and by (H) the conjugacy class of H in G. The set Φ(G) of all conjugacy classes in G admits a partial order defined as follows: (H) ≤ (K) if and only if gHg −1 ⊂ K for some g ∈ G. We will also put Φ k (G) := {(H) ∈ Φ(G) : dim W (H) = k}.…”

Generalized Gradient Equivariant Multivalued Maps, Approximation and Degree

“…The degree theory was applied in [7] to obtain nontrivial solutions to multivalued boundary value problems. See also [9,10] for further applications of that degree to obtain multiple solutions to some implicit functional differential equations.…”

Periodic Solutions to Reversible Second Order Autonomous DDEs in Prescribed Symmetric Nonconvex Domains

“…Our method is based on the usage of the Brouwer equivariant degree theory; for the detailed exposition of this theory, we refer to the monographs [8,21,20,23] and survey [7] (see also [5,6,4]). In short, the equivariant degree is a topological tool allowing "counting" orbits of solutions to symmetric equations in the same way as the usual Brouwer degree does, but according to their symmetry properties.…”

“…We only prove Lemma 2.1 assuming that f satisfies (A 5 ) and (A 6 ). The case when f satisfies (A 5 ) and (A 6 ) was treated in [5] (see also Remark 2.2).…”