Abstract
It is well known that the Sturmian the ory is an important tool in solving numerous problems of mathematical physics. Usually, eigenvalue parameter appears linearly only in the differential equation of the classic Sturm–Liouville problems. However, in mathematical physics there are also problems, which contain eigenvalue parameter not only in differential equation, but also in the boundary conditions. In this paper, we consider a Sturm–Liouville equation with the eigenparameter dependent boundary condition and with transmission conditions at two points of discontinuity. The aim of this paper is to investigate the completeness, minimality and basis properties of rootfunctions for the considered boundary value problem.
Highlights
In [16], the asymptotic formulas for the eigenvalues and eigenfunctions of problem (1)–(7) are obtained
In this work, we consider (u) : u'' q(x)u = u, (1)L1(u) := 1u( 1) 2u' ( 1) = 0, (2) (3)and the transmission conditions (4) (5) (6) (7)where 1 < 1 < 2 < 1, q(x) is a real-valued function, which is continuous on 0)
The goal of this work is to investigate the problem of completeness, minimality and basis property of the eigenfunctions of boundary value problem (1)–(7)
Summary
In [16], the asymptotic formulas for the eigenvalues and eigenfunctions of problem (1)–(7) are obtained. Problems on eigenvalue for the second order equation with spectral parameter in the boundary conditions are considered in [5, 7, 8, 11,12,13,14,15, 18]. The goal of this work is to investigate the problem of completeness, minimality and basis property of the eigenfunctions of boundary value problem (1)–(7). Математика product in a special Hilbert space and construct a linear operator A in the space such that problem (1)–(7) can be interpreted as the eigenvalue problem for A .
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