On self-adjoint Caputo-type fractional Hahn difference equations

Abstract

Abstract In this article, we investigate the existence and uniqueness of solutions for self-adjoint difference equations containing two Hahn difference operators. One is of the first order, and the second is of an $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -order with $$0&lt;\alpha \leqslant 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>⩽</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> under certain initial and boundary conditions. For obtaining solutions to fractional Hahn difference equations with boundary conditions, we use the Green function, which is defined by the Cauchy function. The basic and important properties of this function are discussed. The existence of solutions to the considered initial value problems is obtained in terms of the Cauchy function. The solutions to the boundary value problems are established in terms of the Green function. Also, the uniqueness of the solutions is proved by applying Banach’s fixed point theorem. An example is given to illustrate our main results.