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Search: MSC category 34B24 ( Sturm-Liouville theory [See also 34Lxx] )

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1. CJM 2003 (vol 55 pp. 724)

Cao, Xifang; Kong, Qingkai; Wu, Hongyou; Zettl, Anton
Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign
For a given Sturm-Liouville equation whose leading coefficient function changes sign, we establish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint boundary condition can be numbered in terms of the Pr\"ufer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also relate this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.

Categories:34B24, 34C10, 34L05, 34L15, 34L20

2. CJM 2000 (vol 52 pp. 248)

Binding, Paul A.; Browne, Patrick J.; Watson, Bruce A.
Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions
The nonlinear Sturm-Liouville equation $$ -(py')' + qy = \lambda(1 - f)ry \text{ on } [0,1] $$ is considered subject to the boundary conditions $$ (a_j\lambda + b_j) y(j) = (c_j\lambda + d_j) (py') (j), \quad j = 0,1. $$ Here $a_0 = 0 = c_0$ and $p,r > 0$ and $q$ are functions depending on the independent variable $x$ alone, while $f$ depends on $x$, $y$ and $y'$. Results are given on existence and location of sets of $(\lambda,y)$ bifurcating from the linearized eigenvalues, and for which $y$ has prescribed oscillation count, and on completeness of the $y$ in an appropriate sense.

Categories:34B24, 34C23, 34L30

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