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Resolvent operator and spectrum of new type boundary value problems

Mukhtarov Oktay Sh. (Gaziosmanpaşa University, Faculty of Science-Art, Department of Mathematics, Tokat, Turkey + Azerbaijan National Academy of Sciences, Institute of Mathematics and Mechanics, Baku, Azerbaijan)
Olğar Hayati (Gaziosmanpaşa University, Faculty of Science-Art, Department of Mathematics, Tokat, Turkey)
Aydemir Kadriye (Gaziosmanpaşa University, Faculty of Science-Art, Department of Mathematics, Tokat, Turkey)

The aim of this study is to investigate a new type boundary value problems which consist of the equation -y''(x) + (By)(x) = αy(x) on two disjoint intervals (-1,0) and (0,1) together with transmission conditions at the point of interaction x = 0 and with eigenparameter dependent boundary conditions, where B is an abstract linear operator, unbounded in general, in the direct sum of Lebesgue spaces L2(-1,0)( L2(0,1). By suggesting an own approaches we introduce modified Hilbert space and linear operator in it such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. We establish such properties as isomorphism and coerciveness with respect to spectral parameter, maximal decreasing of the resolvent operator and discreteness of the spectrum. Further we examine asymptotic behaviour of the eigenvalues.

Keywords: Boundary-value problems, transmission conditions, eigenvalues, coerciveness, spectrum, resolvent