The last free coexistence-like problem
Gaetano Zampieri · Differential and Integral Equations · 1990
This paper deals with the Liapunov stability of the origin for the systemIf there exists s(x, i:) such that ys-ys is a first integral, and some smoothness and nondegeneracy conditions hold, then the stability is equivalent to "coexistence" of periodic solutions of every Hill's equation in a certain family.Given the functions s and /, there exists at most one function w such that the system ( *) admits ys -ys as first integral, but generally no such w exists.Certain special functions s have the property that w can be found in connection with each f so that ( *) has the first integral ys-ys (an example is s(x, i:) = x where we can choose w = f).Each of these special functions s generates the following problem: determine all the functions f such that the origin is a stable equilibrium for ( *) with w defined by s and f.We call such problems free coexistence-like.Some previous papers solved all the free coexistence-like problems except the one generated by s(x,i:) =xi: which is solved in this paper.