The spectral theory of fractional Sturm-Liouville problems

This thesis consists of …ve chapters. The first chapter is devoted to fractional calculus, general information for spectral theory, historical background and applications of the subject. In the second chapter, fundamental definitions and theorems on fractional calculus and Sturm-Liouvile theory are given. In the third chapter, Sturm -Liouville theory is introduced and spectral properties of Sturm-Liouville problem are examined. The last two chapters include original part of the thesis. In the fourth chapter, fractional Sturm-Liouville operator is defined and its spectral properties are examined. Fractional Sturm-Liouville operator with Coulomb potential is introduced, spectral properties are examined and theorems related to this subject are proved in detail. In addition, similar results are obtained for fractional Sturm-Liouville operator having hydrogen atom potential. The fifth chapter consists of two parts. In the first part, the existence of solution of fractional Sturm-Liouville operator for di¤usion operator under impulsive condition is proved by the help of Schaefer fixed point theorem. In the second part, the existence of the solution of p-Laplacian fractional Sturm-Liouville problem for diffiusion operator under impulsive conditions is demonstrated by using Schaefer fixed point theorem.