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In this paper we are concerned with the spectral problem ( y 1 y 2 ) x = ( − 1 2 1 2 λ m − 1 2 λ m 1 2 ) ( y 1 y 2 ) 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 L 1 0 , T. 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. [ABSTRACT FROM AUTHOR]

In this paper we prove that the eigenvalues of a vibrating beam have a strongly continuous dependence on the elastic destructive force, i.e., the eigenvalues, as nonlinear functionals of the elastic destructive force, are continuous in the elastic destructive force with respect to the weak topologies in the Lebesgue spaces L p . In virtue of the minimax characterization for eigenvalues, we prove first the continuity of the lowest eigenvalue and then all the eigenvalues by the induction principle. [ABSTRACT FROM AUTHOR]

In this paper we will prove that the eigenvalues of nonhomogeneous hinged vibrating rods have a strongly continuous dependence on weights, i.e., as nonlinear functionals of weights, eigenvalues are continuous in weights with respect to the weak topologies in the Lebesgue spaces L p . [ABSTRACT FROM AUTHOR]

Abstract: For an integrable potential q on the unit interval, let be the zeroth Neumann eigenvalue of the Sturm–Liouville operator with the potential q. In this paper we will solve the minimization problem , where potentials q have mean value zero and norm r. The final result is . The approach is a combination of variational method and limiting process, with the help of continuity results of solutions and eigenvalues of linear equations in potentials and in measures with weak topologies. These extremal values can yield optimal estimates on the zeroth Neumann eigenvalues. [Copyright &y& Elsevier]

We will study the dependence of λ(a, b), half-eigenvalues of the one-dimensional p-Laplacian, on potentials $(a, b)\in ({\mathcal L}^\gamma, {\mathcal L}^\gamma)$, 1 ≤ γ ≤ ∞, where ${\mathcal L}^\gamma = L^\gamma([0, 1], {\mathbb R})$. Two results are obtained. One is the continuity of half-eigenvalues in $(a, b)\in({\mathcal L}^\gamma, w_\gamma)^2$, where wγ is the weak topology in ${\mathcal L}^\gamma$ space. The other is the continuous differentiability of half-eigenvalues in $(a, b)\in ({\mathcal L}^\gamma, \|\cdot\|_\gamma)^2$, where | ⋅ |γ is the Lγ norm of ${\mathcal L}^\gamma$. These results will be used to study extremal problems of half-eigenvalues in future work. [ABSTRACT FROM AUTHOR]

In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual L p topologies, but also with respect to the weak topologies of the L p spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way. [ABSTRACT FROM AUTHOR]

Abstract: Given a 1-periodic real potential . We use to denote the smallest 1-periodic eigenvalue of the Hill''s equation . Let be the ball centered at 0 of radius r in the space . It is trivial that for all . Based on continuity of in q with the weak topology and continuous differentiability of in q with the norm , we will apply scaling technique, variational approach to extremal values in balls, singular integrals and the limiting approach as to obtain (i) is bounded for q in any bounded set of , and (ii) the minimal value is simply , where , . The extremal values of the smallest Neumann eigenvalues for potentials in balls are also found explicitly. [Copyright &y& Elsevier]