Some results on fractional Hahn difference boundary value problems

Abstract

Abstract Fractional Hahn boundary value problems are significant tools to describe mathematical and physical phenomena depending on non-differentiable functions. In this work, we develop certain aspects of the theory of fractional Hahn boundary value problems involving fractional Hahn derivatives of the Caputo type. First, we construct the Green function for an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mi mathvariant="normal">th</m:mi> </m:math> \alpha {\rm{th}} -order fractional boundary value problem, with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>α</m:mi> <m:mo>&lt;</m:mo> <m:mn>2</m:mn> </m:math> 1\lt \alpha \lt 2 , and discuss some important properties of the Green function. The solutions to the proposed problems are obtained in terms of the Green function. The uniqueness of the solutions is proved by various fixed point theorems. The Banach’s contraction mapping theorem, the Schauder’s theorem, and the Browder’s theorem are used.