New concepts of Hahn calculus and impulsive Hahn difference equations

Jessada Tariboon,Sotiris K Ntouyas,Weerawat Sudsutad

doi:10.1186/s13662-016-0982-4

Abstract

In this paper, we introduce new concepts of Hahn difference operator, the $q_{k},\omega_{k}$ -Hahn difference operator. We aim to establish a calculus of differences based on the $q_{k},\omega_{k}$ -Hahn difference operator. We construct a right inverse of the $q_{k},\omega_{k}$ -Hahn operator and study some of its properties. As applications, we establish existence and uniqueness results for first- and second-order impulsive $q_{k},\omega_{k}$ -Hahn difference equations.

Highlights

1 Introduction and preliminaries Many physical phenomena are described by equations involving nondifferentiable functions, e.g., generic trajectories of quantum mechanics [ ]
A quantum calculus substitutes the classical derivative by a difference operator, which allows one to deal with sets of nondifferentiable functions
The Hahn difference operator is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems

Summary

Introduction

Introduction and preliminariesMany physical phenomena are described by equations involving nondifferentiable functions, e.g., generic trajectories of quantum mechanics [ ]. As applications of the qk, ωk-Hahn difference operator we establish existence and uniqueness results for first- and second-order impulsive fractional differential equations. Let f be a function defined on Jk. The qk, ωk-Hahn difference operator is given by tk Dqk ,ωk f (t)

Objectives

Results

Conclusion