The existence of positive solutions for boundary value problem of the fractional Sturm-Liouville functional differential equation
type: Item
instance of: scholarly article
title: The existence of positive solutions for boundary value problem of the fractional Sturm-Liouville functional differential equation
zbMATH Open document ID: 1295.34086
DOI: 10.1155/2013/301560
author: Shurong Sun
author: Zhenlai Han
author: Hongling Lu
author: Yanan Li
published in: Abstract and Applied Analysis
publication date: 2014-06-23T00:00:00Z
review text: Summary: We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative \[ \begin{gathered} {^CD^\beta}(p(t)^\beta(p(t)^C D^\alpha u(t))+ f(t,u(t-\tau), u(t+\beta))= 0,\;t\in (0,1),\\ {^CD^\alpha} u(1)= ({^CD^\alpha} u(0))''= 0,\\ au(t)- bu'(t)= \eta(t),\;t\in [-\tau,0]\;cu(t)+ du'(t)= \xi(t),\;t\in [1,1+\theta],\end{gathered} \] where \({^CD^\alpha}\), \({^CD^\beta}\) denote the Caputo fractional derivatives, \(f\) is a nonnegative continuous functional defined on \(C([-\tau,1+\theta],\mathbb{R})\), \(1<\alpha\leq 2\), \(2< \beta\leq 3\), \(0<\tau\), \(\theta< 1/4\) are suitably small, \(a,b,c,d>0\), and \(\eta\in C([-\tau,0], [0,\infty))\), \(\xi\in C([1,1+\theta], [0,\infty))\). By means of the Guo-Krasnoselskii fixed point theorem and the fixed point index theorem, some positive solutions are obtained, respectively. As an application, an example is presented to illustrate our main results.
Mathematics Subject Classification ID: 34K37
Mathematics Subject Classification ID: 34K10
zbMATH DE Number: 6306634
zbMATH Keywords: Guo-Krasnoselskii fixed point theorem
zbMATH Keywords: fixed point index theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/301560
OpenAlex ID: W1968349713
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Wikidata QID: Q58915871
description: scientific article; zbMATH DE number 6306634