Two point fractional boundary value problems with a fractional boundary condition
type: Item
instance of: scholarly article
title: Two point fractional boundary value problems with a fractional boundary condition
zbMATH Open document ID: 1404.34009
author: J. W. Lyons
author: J. T. Neugebauer
publication date: 2018-10-19T00:00:00Z
review text: Suppose \(n\in \mathbb{N}\) such that \(n\geq 5\), \(\alpha\in (n-1, n]\) and \(\lambda>0\) is a parameter. Consider \(f:[0, \infty)\rightarrow [0, \infty)\) and \(g:[0, 1]\rightarrow [0, \infty)\) two continuous functions such that \(\displaystyle\int_0^1g(t)dt>0\). Then the existence and non-existence of the positive solution \(u\) of the following fractional differential equation is considered: \[ D_{0+}^{\alpha}u = \lambda g(t) f(u),\quad t\in (0, 1)\tag{1} \] with respect to the boundary conditions: \[ u^{(i)}(0) = 0, \quad i= 0, 1,\dots, n-4,\quad D_{0+}^{\alpha-2}u(0)=0, \quad D_{0+}^{\alpha-2}u(1)=0,\quad D_{0+}^{\beta}u(1)=0\tag{2} \] or \[ u^{(i)}(0) = 0,\quad i= 0, 1,\dots, n-4,\quad D_{0+}^{\alpha-2}u(0)=0,\quad D_{0+}^{\alpha-1}u(1)=0,\quad D_{0+}^{\beta}u(1)=0,\tag{3} \] where \(\beta\in[1, n-3]\) and \(D_{0+}^{\alpha}, D_{0+}^{\beta}\) are the Riemann-Liouville fractional derivatives of order \(\alpha\), \(\beta\) respectively. Further, in another section the problem of existence and non-existence of positive solution \(u\) of the following fractional differential equation is discussed: \[ D_{0+}^{\mu}u + \lambda g(t) f(u)=0,\quad t\in (0, 1)\tag{4} \] with the boundary conditions \[ u^{(i)}(0)= 0,\quad i= 0, 1,\dots, m-2, \quad D_{0+}^{\beta}u(1)=0,\tag{5} \] where \(\beta\in [1, m-1]\), \(\mu\in (m-1, m]\), \(m\in \mathbb{N}\), \(m\geq 3\). Throughout this work the authors establish the existence and non-existence of the positive solution \(u\) for the pairs \((1, 2)\), \((1,3)\) and \((4, 5)\) of fractional differential equations with boundary conditions. The authors apply Krasnosel'skii's fixed point theorem to obtain these results.
reviewed by: Atanu Manna
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 6958685
zbMATH Keywords: fractional differential equation
zbMATH Keywords: positive solution
zbMATH Keywords: Krasnosel'skii's fixed point theorem
zbMATH Keywords: existence of solution
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/fca-2018-0025
OpenAlex ID: W2808183779
cites work: Positive solutions for boundary value problem of nonlinear fractional differential equation
cites work: The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type
cites work: Q2954119
cites work: Q3431882
cites work: Q3635288
cites work: Uniqueness and parameter dependence of positive solutions of a discrete fourth-order problem
cites work: Positive solutions of a nonlinear higher order boundary-value problem
cites work: Positive solutions for a semipositone fractional boundary value problem with a forcing term
cites work: First order impulsive differential inclusions with periodic conditions
cites work: Positive solutions to a three point fourth order focal boundary value problem
cites work: Positive solutions for a system of nonlocal fractional boundary value problems
cites work: Computing the positive solutions of the discrete third-order three-point right focal boundary-value problems
cites work: The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application
cites work: Positive solutions of a boundary value problem for a nonlinear fractional differential equation
cites work: Q5699297
cites work: Q3847203
cites work: Positive solutions of a singular fractional boundary value problem with a fractional boundary condition
cites work: Q3141902
cites work: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
cites work: Q3111280
cites work: Upper estimate for positive solutions of the \((p,n-p)\) conjugate boundary value problem
cites work: Q3378412
DOI: 10.1515/FCA-2018-0025
published in: Fractional Calculus \ Applied Analysis
description: scientific article; zbMATH DE number 6958685