On a class of nonlinear singular Riemann-Liouville fractional differential equations
type: Item
instance of: scholarly article
title: On a class of nonlinear singular Riemann-Liouville fractional differential equations
zbMATH Open document ID: 1403.34006
published in: Results in Mathematics
publication date: 2018-10-25T00:00:00Z
review text: The main objective of this paper is to study existence and multiplicity of positive solutions for the fractional boundary value problem \[ D_{0^{+}}^{\alpha}u(t)+f(t,u(t))=0,\quad t\in(0,1),\tag{1} \] \[ u(0)=u^{'}(0)=\dots =u^{(n-2)}(0)=0,\quad D_{0^{+}}^{p}u(1)=\int_{0}^{1}D_{0^{+}}^{q}u(t)dH(t)\tag{2} \] in which \(\alpha\in(n-1,n],~n\in\mathbb{N}_{\geq3},~p\in[1,n-2]\) and \(q\in[0,p]\). \(D_{0^{+}}^{\beta}\) stands for the Riemann-Liouville fractional derivative of order \(\beta\). In the boundary conditions (2), the integral boundary condition is a Riemann-Stieltjes integral with \(H\) a function of bounded variation. Furthermore, the nonlinearity \(f\) potentially may change sign and can be singular at the end points and/or in the space variable \(u\). The author, using the Green function technique and the Guo-Krasnoselskii fixed point theorem together with some controls on the Green function, the function \(H\) and the nonlinearity \(f\), has been obtained some existence and multiplicity criteria for solutions of the fractional boundary value problem (1)--(2) on a positive cone in the Banach space \(C[0,1]\).
reviewed by: Yousef Gholami
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34B16
zbMATH DE Number: 6965864
zbMATH Keywords: fractional boundary value problems
zbMATH Keywords: Green function
zbMATH Keywords: fixed point theorem
zbMATH Keywords: existence
zbMATH Keywords: multiplicity
zbMATH Keywords: positive solutions
author: Rodica Luca
MaRDI profile type: Publication
full work available at URL: s00025-018-0887-5
OpenAlex ID: W2889298235
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Wikidata QID: Q129326028
DOI: 10.1007/S00025-018-0887-5
description: scientific article; zbMATH DE number 6965864