Abstract
In this paper we are concerned with the spectral problem(y1y2)x=(−1212λm−12λm12)(y1y2) for the periodic generalized Camassa-Holm equations. The first aim is to study the structure of eigenvalues and prove the continuous dependence of eigenvalues on potentials. We prove that as nonlinear functionals of potentials, eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space L1[0,T]. The second aim is to find the estimates of the eigenvalues when the L1 norm of potentials is given, based on an application of trace formulas.
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