Noninstantaneous Impulsive Fractional Quantum Hahn Integro-Difference Boundary Value Problems

Surang Sitho,Jessada Tariboon,Sotiris K Ntouyas,Chayapat Sudprasert

doi:10.3390/math8050671

Abstract

In this paper, we study the existence and uniqueness results for noninstantaneous impulsive fractional quantum Hahn integro-difference boundary value problems with integral boundary conditions, by using Banach contraction mapping principle and Leray–Schauder nonlinear alternative. Examples are included illustrating the obtained results. To the best of our knowledge, no work has reported on the existence of solutions to the Hahn-difference equation with noninstantaneous impulses.

Highlights

We present some properties of fractional calculus on any interval [si, ti+1 ), i = 0, 1, 2, . . . , m
The first result concerns the existence of a unique solution of the problem (1) and (2), and will be proved by using the Banach contraction mapping principle, involving the following constants: Λ1 =
We obtain ( L1 + L2 Λ2 )Λ7 ≈ 0.96235 < 1, which implies by the conclusion of Theorem 3 that the problem (26) and (27) has a unique solution on [0, 7]

Summary

Introduction and Preliminaries

We investigate the existence and uniqueness of solutions for noninstantaneous impulsive fractional quantum Hahn integro-difference equation of the form: αi. The fractional quantum Hahn difference operator of a Riemann–Liouville type of order αi ≥ 0 on interval [si , ti+1 ) is defined by The fractional quantum Hahn integral operator of Riemann–Liouville type is given by (si Iq0i ,ωi f )(t) = f (t) and (si Iqαii,ωi f )(t) =. The fractional quantum Hahn difference operator of Caputo type αi ≥ 0 on interval [si , ti+1 ) is defined by (C si Dqi ,ωi f )( t ) = f ( t ) and (Csi Dqαii,ωi f )(t) =. The problem (1) and (2) is based on fractional quantum Hahn calculus in Definitions 2 and 3.

Main Results

Examples

Conclusions