Nonlinear Fractionalq-Difference Equation with Fractional Hadamard and Quantum Integral Nonlocal Conditions

Athasit Wongcharoen,Arisa Thatsatian,Sotiris K Ntouyas,Jessada Tariboon

doi:10.1155/2020/9831752

Abstract

In this paper, we establish existence and uniqueness results for a boundary value problem consisting by a nonlinear fractionalq-difference equation subject to a new type of boundary condition, combining the fractional Hadamard and quantum integrals. Our analysis is based on Banach’s fixed point theorem, a fixed point theorem for nonlinear contractions, Krasnosel’skii˘’s fixed point theorem, and Leray-Schauder nonlinear alternative. Examples are given to illustrate our results.

Highlights

The aim of this paper is to investigate the existence and uniqueness of solutions for a nonlinear fractional q-difference equation subject to fractional Hadamard and quantum integral condition of the form: 0 DαqxðtÞ = f ðt, xðtÞÞ, 1 < α ≤ 2, t ∈ ð0, TÞ, BB@ xð0Þ
We prove two existence and uniqueness results with the help of the Banach contraction mapping principle and a fixed point theorem on nonlinear contractions due to Boyd and Wong
We investigated the existence and uniqueness of solutions for a nonlocal boundary value problem involving a q-difference equation, supplemented with a new type of boundary condition, including both Hadamard fractional and quantum integrals

Summary

Introduction

The aim of this paper is to investigate the existence and uniqueness of solutions for a nonlinear fractional q-difference equation subject to fractional Hadamard and quantum integral condition of the form: BB@ xð0Þ = 0, n 〠 γi I μi pi xðξi Þ m. ΒjJσjx ηj , ð1Þ i=1 j=1 where Dαq is the fractional q-derivative of order α, with a quantum number q ∈ ð0, 1Þ, f : 1⁄20, T × R ⟶ R is a nonlinear continuous function, μi pi denotes the fractional quantum integral of order μi > 0, with quantum number 0 < pi < 1, Jσj is the Hadamard fractional integral of order σj > 0, γi and βj are given constants, and ξi, ηj ∈ ð0, TÞ are fixed points, for i = 1, ⋯, n and j = 1, ⋯, m. The Hadamard-type fractional derivative differs from the preceding ones in the sense that the kernel of the integral and derivative contain logarithmic function of arbitrary exponent.

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Conclusion