We are concerned with the spectral problem 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. The second aim is to find the estimates of the eigenvalues when the L^1 norm of potentials is given, based on an application of trace formulas.

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