BASIS PROPERTY OF SYSTEMS OF EIGEN AND ASSOCIATED FUNCTIONS OF A SHTURM-LIOUVILLE MULTIPLE DIFFERENTIATION OPERATOR WITH NONLOCAL «PERTURBED» BOUNDARY CONDITIONS

This paper is devoted to the study of a spectral problem for a Sturm–Liouville multiple differentiation operator defined on an interval with nonlocal integrally “perturbed” boundary conditions. The considered boundary conditions are regular but not strongly regular, which leads to essential difficulties in the analysis of the spectral properties and the basis behavior of the corresponding systems of functions. The main objective of the study is to investigate the basis properties of systems of eigen and associated functions in the space of square-summable functions and to analyze their stability and instability under small perturbations of the boundary conditions. As a particular case, the Samarskii–Ionkin problem with integrally perturbed boundary conditions is examined in detail. It is proved that the system of eigen and associated functions of this problem forms a Riesz basis in the L2 space on the interval. The paper also shows that small changes in the integral kernel of the boundary conditions may lead either to the preservation or to the loss of the basis property. Moreover, it is established that the sets of root function systems forming a Riesz basis and the sets of eigen and associated function systems that do not form an ordinary basis are dense in the corresponding functional space L 1. The obtained results contribute to the development of the spectral theory of differential operators with nonlocal boundary conditions.

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