A new type Sturm Liouville problem with an abstract linear operator contained in the equation

Abstract

In this study we shall introduce to the consideration a new type Sturm-Liouville problem, which differs from the standard problems in that it contain non-differential term in the equation. Namely, the equation contains an abstract linear operator B , which may be non-selfadjoint. Moreover, the problem contains an additional transmission conditions at the finite number interior points of discontinuity and the spectral parameter λ appears not only in the equation, but also in the boundary conditions. It is not clear how to extend the known methods and results of classic Sturmian theory to our problem. The major difficulty lies in the existence and behavior the of eigenvalues in the complex plane.
We developed an our own approaches for investigation of some important spectral properties of such type non-classical boundary value problems. At first we defined a new Hilbert spaces for operator-treatment of the considered problem. Then we established an isomorphism and coerciveness. We also established some important properties of the resolvent operator. Finally, we proved discreteness of the spectrum and derived asymptotic formulas for the eigenvalues. Note that, the obtained results are new even in the continuous case and/or when the boundary conditions do not contain the spectral parameter λ.