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Spectral properties of a nonlocal problem for a second-order differential equation with an involution

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Abstract

For the equation αu″(-x) - u″(x) = λu(x), ™1 < x < 1, where α ∈ (™1, 1), we study the problem with the nonlocal conditions u(™1) = 0, u′(™1) = u′(1). We show that if \(r = \sqrt {\left( {1 - \alpha } \right)/\left( {1 + \alpha } \right)} \) is irrational, then the system of eigenfunctions is complete and minimal in L 2(™1, 1) but is not a basis. For rational r, we indicate a method for choosing associated functions for which the system of root functions of the problem is an unconditional basis in L 2(™1, 1).

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    L. V. Kritskov & A. M. Sarsenbi

  2. M. Auezov South Kazakhstan State University, Shymkent, Kazakhstan

    L. V. Kritskov & A. M. Sarsenbi

Authors
  1. L. V. Kritskov
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  2. A. M. Sarsenbi
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Correspondence to L. V. Kritskov.

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Original Russian Text © L.V. Kritskov, A.M. Sarsenbi, 2015, published in Differentsial’nye Uravneniya, 2015, Vol. 51, No. 8, pp. 990–996.

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Kritskov, L.V., Sarsenbi, A.M. Spectral properties of a nonlocal problem for a second-order differential equation with an involution. Diff Equat 51, 984–990 (2015). https://doi.org/10.1134/S0012266115080029

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