Multiplicity of periodic solutions to symmetric delay differential equations

Abstract

By applying the method based on the usage of the equivariant gradient degree introduced by Geba (1997) and the cohomological equivariant Conley index introduced by Izydorek (2001), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant \({\omega(\nabla \Phi)}\), expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of \({\Phi}\) exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation $$x\prime = - \nabla f \big ({x \big (t - \frac{\pi}{2} \big )} \big ), \quad x \in V \qquad \quad (*)$$ (here \({f : V \rightarrow \mathbb{R}}\) is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (2005). The equivariant invariant \({\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}\) in the case nk ≠ 0 (for maximal twisted orbit types (Hk)) guarantees the existence of at least |nk| different G-orbits of periodic solutions with symmetries at least (Hk). This result generalizes the result by Guo and Yu (2005) obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of \({\omega(\,\nabla \Phi)}\) evaluated.