RENDICONTI DEL SEMINARIO MATEMATICO

120 D. Papini - F. ZanolinFigure 5: An illustration of the possible behavior of another solution u(t) of (2) alongan interval made by two subintervals where q > 0 which are separated by a subintervalwhere q < 0. In this latter subinterval u > 0 and u ′ vanishes exactly once.results about the rich structure and the complex dynamics of the solutions of the nonlinearequation (2) with a sign changing weight q(t).In the case when q(t) is a periodic function such that its interval of periodicitycan be decomposed into a finite number of adjacent subintervals where q(t) alternatesits sign, a natural problem turns out to be that of the search of periodic (harmonic andsubharmonic) solutions to (2). Results about the existence of infinitely many periodicsolutions for the superlinear case were obtained by Butler in his pioneering work [6].For a nonlinearity having superlinear growth at infinity, Terracini and Verzini in [70]proved the existence of periodic solutions which have an arbitrarily large (but possiblyfixed in advance) number of zeros in the intervals when q > 0 and precisely onezero in the intervals when q < 0. At the best of our knowledge, this is the first resultgiving evidence of a very complicated behavior for the solutions of the nonlinear Hill’sequations with a sign changing weight.In view of the path-stretching property (H ± ) and the above quoted results forthe periodic problem, as a next step, one can raise the question whether it is possible toobtain fixed points (as well as periodic points) for a map like the ϕ considered in (H ± ).This goal was achieved in [56] where we obtained a fixed point theorem for planarmaps (subsequently reconsidered and generalized in [59, 60]) that, when applied tothe case of a periodic weight function, shows that the path-stretching condition (H ± )implies the existence of infinitely many periodic solutions (harmonic and subharmonic)as well as the presence of a chaotic-like dynamics for the solutions of (2).1.2. Fixed points and periodic points for planar mappingsIn order to present the results in [56, 59, 60], first of all we put in a more abstract formthe situation described in (H ± ). For sake of simplicity, we give here only some of the