Langevin equation involving two fractional orders with three-point boundary conditions.
type: Item
instance of: scholarly article
title: Langevin equation involving two fractional orders with three-point boundary conditions.
zbMATH Open document ID: 1488.34062
author: Ahmed Salem
author: Faris Alzahrani
author: Balqees Alghamdi
published in: Differential and Integral Equations
publication date: 2020-07-02T00:00:00Z
review text: The authors give existence and uniqueness results for the equation \[D^{\beta}(D^{\alpha}+\lambda)u(t)=f(t,u(t))\] where \(t\in[0,1]\) and the boundary conditions are \(u(0)=0\), \(D^{\alpha}u(0)=0\) and \(u(1)=\mu u(\eta)\). \(D^{\alpha}\) and \(D^{\beta}\) are fractional derivatives of orders \(0<\alpha\leq 1\) and \(1<\beta\leq 2\), \(\mu\), \(\lambda\) are real numbers and \(0<\eta<1\) with \(1-\mu\eta^{\alpha+1}\) not zero. First, the authors give some elementary properties of fractional integrals and fractional derivatives. In the first result it is assumed that \(f\) is globally Lipschitz in the second variable. Under certain smallness conditions on the Lipschiz constant and \(\lambda\) the contraction principle can be applied to the equation. If the function \(f\) is assumed to be uniformly bounded in the second variable the Krasnoselskii fixed point theorem can be applied. Under rather strong assumptions on the nonlinearity \(f\) the Leray-Schauder degree theory can be applied. All these applications give existence results (some give also uniqueness) for the equation above.
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34B08
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 7217168
zbMATH Keywords: fractional differential equations
zbMATH Keywords: boundary value problems
zbMATH Keywords: fixed point theorems
reviewed by: Stig-Olof Londen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 7217168