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

2006/07/03 by Chiara Zanini, Zanini, Chiara, Fabio Zanolin +1
Mathematics · #34C25 #37E40 #92C20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:34C25 #msc:37E40 #msc:92C20

paper · pdf · doi:10.48550/arxiv.math/0607042

18 pages, 2 figures

arxiv created 2006/07/03 · arxiv updated 2009/12/01

Abstract

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation v_xx - g v + n(x) F(v) = 0, previously considered by Grindrod and Sleeman and 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). Here we show that the number of positive periodic solutions may be very large for some special choices of a (large) weight n. We also obtain the existence of subharmonic solutions of any order. The proofs are based on the Poincaré - Bikhoff fixed point theorem.

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