Multiple solutions with prescribed minimal period for second order odd Newtonian systems with symmetries

Abstract

For an orthogonal Γ-representation V (Γ is a finite group) and for an even, we consider the odd Newtonian systemẍ(t) = −∇f (x(t)) and establish the existence of multiple periodic solutions with a minimal period p (for any given p > 0). As an example, we prove the existence of arbitrarily many periodic solutions with minimal period p for a specific Dn-symmetric Newtonian system.

Periodic solutions with prescribed minimal period to Hamiltonian systems

“…[25]) proved that, if V satisfies the conditions (V 1 ), (V 2 ) and (V 4), system (1) admits a non-constant T-periodic solution with minimal period T. Under the same assumptions as in [25], Krawcewicz, Lv and Xiao (cf. [9]) showed the existence of multiple T-periodic solutions with common minimal period T. Recently, a variant of (V 2 ), which is the so-called ARS condition, was introduced by Souissi in [18]. The ARS condition was used to study the existence of periodic solutions to Hamiltonian systems, when the nonlinearity satisfies a local one (cf.…”