Abstract.
We consider the general regular Sturm–Liouville problems on time scales with separated boundary conditions. By extending the result by Agarwal, Bohner, and Wong [2] on the existence of eigenvalues, we study the dependence of the eigenvalues on the boundary condition. We show that the n-th eigenvalue λ n depends continuously on the boundary condition except at the generalized “Dirichlet” conditions, where certain jump-discontinuities may occur. Furthermore, λ n as a function of the boundary condition angles is continuously differentiable wherever it is continuous. Formulas for such derivatives are obtained which reveal the monotone properties of λ n in terms of the boundary condition angles. Our results not only unify the Sturm–Liouville problems for differential equations and difference equations, but are also new for the discrete case.
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Received: May 11, 2007. Revised: September 24, 2007.
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Kong, Q. Sturm–Liouville Problems on Time Scales with Separated Boundary Conditions. Result. Math. 52, 111–121 (2008). https://doi.org/10.1007/s00025-007-0277-x
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DOI: https://doi.org/10.1007/s00025-007-0277-x