Subharmonic solutions in reversible non-autonomous differential equations. (March 2022) - Eze, Izuchukwu; García-Azpeitia, Carlos; Krawcewicz, Wieslaw; Lv, Yanli
Abstract: We study the existence of subharmonic solutions in the system u ̈ ( t ) = f ( t, u ( t ) ) with u ( t ) ∈ V : = R k, where f ( t, u ) is a continuous map that is p -periodic and even with respect to t and odd and Γ -equivariant with respect to u (where V is a representation of a finite group Γ ). The problem of finding m p -periodic solutions is reformulated, in an appropriate functional space, as a nonlinear Γ × Z 2 × D m -equivariant equation. Under certain hypotheses on the linearization of f at zero and Nagumo growth condition on f at infinity, we prove the existence of an infinite number of subharmonic solutions by means of the Brouwer equivariant degree. In addition, we discuss the bifurcation problem of subharmonic solutions in the case of a system depending on an extra parameter α .