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In this dissertation we develop certain aspects of the theory of discrete fractional calculus. The author begins with an introduction to the discrete delta calculus together with the fractional delta calculus which is used throughout this dissertation. The Cauchy function, the Green's function and some of their important properties for a fractional boundary value problem for are developed. This dissertation is comprised of four chapters. In the first chapter we introduce the delta fractional calculus. In the second chapter we give some preliminary definitions, properties and theorems for the fractional delta calculus and derive the appropriate Green's function and give some of its important properties. This allows us to prove some important theorems by using well-known fixed point theorems. In the third chapter we study and prove various results regarding the generalized fractional boundary value problem for the self-adjoint equation with Sturm-Liouville type boundary conditions. In the fourth chapter we prove some theorems regarding the existence and uniqueness of positive solution of a forced fractional equation with finite limit.
Adviser: Lynn Erbe and Allan Peterson