Multiplicity of periodic solutions for differential equations arising in the study of a nerve fiber model

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo-type equation vxx-gv+n(x)F(v)=0vxx-gv+n(x)F(v)=0, previously considered by Chen and Bell in the study of the model of a nerve fiber with excitable spines. In a recent work we proved a result of nonexistence of nontrivial solutions as well as a result of existence of two positive solutions, the different situations depending by a threshold parameter related to the integral of the weight function n(x)n(x). Here we show that the number of positive periodic solutions may be very large for some special choices of a (large) weight nn. We also obtain the existence of subharmonic solutions of any order. The proofs are based on the Poincaré–Bikhoff fixed point theorem.