FRACTIONAL DISSIPATIVE STURM-LIOUVILLE PROBLEMS WITH DISCONTINUITY AND EIGEN-DEPENDENT BOUNDARY CONDITIONS

Abstract

This paper studies two main fractional discontinuous dissipative Sturm-Liouville type boundary-value problems with boundary conditions and transmission conditions. In both types of research, with the aid of the operator theory, we define different classes of new inner products by combining the parameters in the boundary and transmission conditions, then the boundary value problems are transferred to operators in the Hilbert spaces such that the eigenvalues and eigenfunctions of the main problem coincide with those of operators. And we prove those of operators are dissipative. Moreover, the fundamental solutions are constructed and the uniqueness of the solutions of the problem is also given.