Weak eigenfunctions of two-interval Sturm-Liouville problems together with interaction conditions

Abstract

In this study, we consider a new type boundary value problem consisting of a Sturm-Liouville equation on two disjoint intervals together with interaction conditions and with eigenvalue parameter in the boundary conditions. We suggest a special technique to reduce the considered problem into an integral equation by the use of which we define a new concept, the so-called weak eigenfunction for the considered problem. Then we construct some Hilbert spaces and define some self-adjoint compact operators in these spaces in such a way that the considered problem can be interpreted as a self-adjoint operator-pencil equation. Finally, it is shown that the spectrum is discrete and the set of weak eigenfunctions form a Riesz basis of the suitable Hilbert space.