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Boundary value problems with regular singularities and singular boundary conditions. (English) Zbl 1093.34049

Considered is a class of singular differential equations which can be transformed to \[ y''(x)+q(x)y(x)=\lambda y(x), \quad x\in (0,T),\tag{1} \] where \(q\) is continuous and has second-order singularities at the endpoints. For a certain class of boundary value problems associated with (1), it is shown that there is a countable set of eigenvalues, which has an explicitly stated asymptotic behaviour. Then, it is shown that the set of eigenfunctions and associated functions is complete in the Banach space \(B_{\beta ,\xi ,s}=\{f:f(x)x^{-\beta }(T-x)^{-\xi }\in L_s(0,T)\}\) for \(1\leq s\leq \infty \), \(\beta <\theta +1/s\), \(\xi <\theta _1+1/s\), where \(\beta \), \(\xi \), \(\theta \), \(\theta _1\) depend on \(q\).

MSC:

34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B05 Linear boundary value problems for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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