On nonlocal fractional sum-difference boundary value problems for Caputo fractional functional difference equations with delay

Thanin Sitthiwirattham,Bunjong Kaewwisetkul

doi:10.1186/s13662-017-1283-2

Abstract

In this article, we study the existence and uniqueness results for a nonlocal fractional sum-difference boundary value problem for a Caputo fractional functional difference equation with delay, by using the Schauder fixed point theorem and the Banach contraction principle. Finally, we present some examples to display the importance of these results.

Highlights

In this paper, we consider a nonlocal fractional sum boundary value problem for a Caputo fractional functional difference equation with delay of the form α C u(t) =F t + α, ut+α, β C u(t +α – β), t ∈ N,T := {, . . . , T}, ( . ) γ C u(α γ ), u(T + α) = ρ –ωu(η + ω), and uα– = ψ, where ρ =(ω)((α– )[(α– )( –γ )+ ]+(T+α)[(T–α+ )( –γ )– ]) ηs=α– (η+ω–σ (s))ω– ((α– )[(α– )( –γ )+ ]+(η+ω)[(η+ω– α+ )( –γ
For r ∈ N,T+ we denote Cr is the Banach space of all continuous functions ψ : Nα–r,α– → R endowed with the norm ψ max s∈Nα–r,α
In Section, we prove existence and uniqueness results of the problem ( . ) by using the Schauder fixed point theorem and the Banach contraction principle

Summary

Introduction

1 Introduction In this paper, we consider a nonlocal fractional sum boundary value problem for a Caputo fractional functional difference equation with delay of the form α C For r ∈ N ,T+ we denote Cr is the Banach space of all continuous functions ψ : Nα–r– ,α– → R endowed with the norm ψ max s∈Nα–r– ,α– The development of boundary value problems for fractional difference equations which show an operation of the investigative function. Goodrich [ ] considered the discrete fractional boundary value problem

Results

Conclusion