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An analog of a theorem of Keldysh for a multipoint problem of de la Vallée Poussin. (English. Russian original) Zbl 0789.34023
Russ. Acad. Sci., Dokl., Math. 46, No. 2, 307-310 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 326, No. 4, 587-591 (1992).
The authors study the problem (1) $$x^{(n)} +p_ 1(t)x^{(n-1)} +\cdots+ p_ n(t) x=\lambda x$$, $$a\leq t\leq b$$, (2) $$x^{(i)} (a_ k)=0$$, $$i=0,\dots,\nu_ k-1$$, $$k=1,\dots,N$$, $$\nu_ 1+ \cdots+\nu_ N=n$$, where $$a=a_ 1<a_ 2<\cdots<a_ N=b$$. They establish a representation of the residue of the Green function of problem (1), (2) in a neighborhood of the eigenvalue $$\lambda_ 0$$. Both a simple eigenvalue and the case of multiplicity are considered.
##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators