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Little vibrations of an abstract mechanical system and corresponding eigenvalue problem. (English) Zbl 1041.34050
Summary: The boundary value problem \[ u''(x)+ \int^l_0 u(y) d_yr(x,y)= f(x),\quad u(0)= 0,\;u(l)= 0, \] can be considered as a mechanical model under a certain condition of symmetry. An approach to abstract mechanical system is presented and applied to a boundary value problem. The mechanical interpretation of the functional-differential equation allows one to obtain base properties for the eigenvalue problem. The present article is a continuation of the investigation of [Russ. Math. 40, 50–56 (1996); translation from Vyssh. Uchebn. Zaved. Mat. 1996, 48–53 (1996; Zbl 0909.34070].

34K10 Boundary value problems for functional-differential equations
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
47A75 Eigenvalue problems for linear operators
74H45 Vibrations in dynamical problems in solid mechanics