Subharmonic solutions in reversible non-autonomous differential equations

摘要

We study the existence of subharmonic solutions in the system u(t) = f (t, u(t)) with u(t) is an element of V := R-k, where f (t, u) is a continuous map that is p-periodic and even with respect to t and odd and Gamma-equivariant with respect to u (where V is a representation of a finite group Gamma). The problem of finding mp-periodic solutions is reformulated, in an appropriate functional space, as a nonlinear Gamma x Z(2) x 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 alpha.

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