Coerciveness and isomorphism of discontinuous Sturm-Liouville problems with transmission conditions

This study investigates a discontinuous Sturm-Liouville boundary value problem(BVP) on two intervals with functionals and transmission conditions in the direct sum ofSobolev spaces. Moreover, it presents the differential operator generated by the problem underinvestigation. The definition space of this operator is the direct sum of Sobolev spaces, andthe value space of the operator is the space obtained by adding the complex spaces where theboundary conditions are evaluated about the direct sum of Sobolev spaces. This paper establishesthe solvability of the problem and some important spectral properties of the operator, such asisomorphism, Fredholmness, and coerciveness concerning spectral parameters. In addition, theconclusion section discusses how different original problems can be produced.