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A classical approach to eigenvalue problems associated with a pair of mixed regular Sturm-Liouville equations. I. (English) Zbl 0977.34022
The authors consider a class of problems of the type \[ L_1y_1 \equiv -y_1''+ q_1(x)y_1= \lambda y_1,\;x\in [0,h], \]
\[ L_2y_2\equiv-y_2''+ q_2 (x)y_2= \lambda y_1,\;x\in [h,1], \] with the matching conditions at the interface point \(x=h\): \[ y_1(h)=y_2(h),\;w_1y_1'(h)= w_2y_2'(h), \] with \(w_1,w_2 \in\mathbb{R} \setminus\{0\}\) and \(\lambda\in \mathbb{C}\). This problem appears in a variety of applications, hence there are many authors considering some problems of this type.
The authors here construct first a fundamental system for the pair \((L_1,L_2)\) and derive certain asymptotic estimates. These results under initial data and boundary-type data are applied to derive some corresponding spectral analysis results.
The method used is based on integral equation methods and the analysis involved is most classical. The proofs for few results are omitted and a reference is made to previously published papers. No examples or applications are mentioned.
34B24 Sturm-Liouville theory
34A30 Linear ordinary differential equations and systems
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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