Multi-Point Transmission Problems for Sturm-Liouville Equation with an Abstract linear Operator

Abstract

In this paper, we consider the spectral problem for the equation -u ''(x) + (A + lambda I)u(x) = f(x) on the two disjoint intervals (-1, 0) and (0, 1) together with multi-point boundary conditions and supplementary transmission conditions at the point of interaction x = 0, where A is an abstract linear operator. So, our problem is not a pure differential boundary-value one. Starting with the analysis of the principal part of the problem, the coercive estimates, the Fredholmness and isomorphism are established for the main problem. The obtained results are new even in the case of boundary conditions without internal points.