Abstract and Applied Analysis

Existence of Positive Solutions to Nonlinear Fractional Boundary Value Problem with Changing Sign Nonlinearity and Advanced Arguments

Abstract

We discuss the existence of positive solutions to a class of fractional boundary value problem with changing sign nonlinearity and advanced arguments ${D}^{\alpha }x(t)+\mu h(t)f(x(a(t)))=\mathrm{0},\mathrm{}\mathrm{}\mathrm{}\mathrm{}t\in (\mathrm{0,1}),\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{2}<\alpha \le \mathrm{3},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mu >\mathrm{0},$ $x(\mathrm{0})={\mathrm{x}}^{\mathrm{\prime }}(\mathrm{0})=\mathrm{0},$ $x(\mathrm{1})=\beta x(\eta )+\lambda [x],\mathrm{}\mathrm{}\mathrm{}\mathrm{}\beta >\mathrm{0}, and \mathrm{}\mathrm{}\mathrm{}\mathrm{}\eta \in (\mathrm{0,1}),$ where ${D}^{\alpha }$ is the standard Riemann-Liouville derivative, $f:[\mathrm{0},\mathrm{\infty })\to [\mathrm{0},\mathrm{\infty })$ is continuous, $f(\mathrm{0})>\mathrm{0}$,$\mathrm{}\mathrm{}\mathrm{ h }:[\mathrm{0,1}]\to (-\mathrm{\infty },+\mathrm{\infty })$, and $\mathrm{}\mathrm{}a(t)$ is the advanced argument. Our analysis relies on a nonlinear alternative of Leray-Schauder type. An example is given to illustrate our results.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 158436, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412279715

Digital Object Identifier
doi:10.1155/2014/158436

Mathematical Reviews number (MathSciNet)
MR3232822

Zentralblatt MATH identifier
07021833

Citation

Hao, Zhaocai; Huang, Yubo. Existence of Positive Solutions to Nonlinear Fractional Boundary Value Problem with Changing Sign Nonlinearity and Advanced Arguments. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 158436, 7 pages. doi:10.1155/2014/158436. https://projecteuclid.org/euclid.aaa/1412279715