Abstract and Applied Analysis

Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations

Abstract

We study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type ${}^{c}\mathcal{D}{}_{0+}^{\delta }u(t)+f(t,u(t))=0$, $t\in (0,1)$, $0\lt t\lt 1$. $u(1)=\beta u(\eta )+{\lambda }_{2}$, ${u}^{\prime }(0)=\alpha {u}^{\prime }(\eta )-{\lambda }_{1}$, ${u}^{\prime \prime }(0)=0$, ${u}^{\prime \prime \prime }(0)=0\cdots {u}^{(n-1)}(0)=0$, where, $n-1\lt \delta \lt n$, $n(\ge 3)\in \mathbb{N}$, $0\lt \eta ,\alpha ,\beta \lt 1$, the boundary parameters ${\lambda }_{1},{\lambda }_{2}\in {\mathbb{R}}^{+}$ and ${}^{c}D{}_{0+}^{\delta}$ is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results.

Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 501230, 15 pages.

Dates
First available in Project Euclid: 12 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1313170919

Digital Object Identifier
doi:10.1155/2010/501230

Mathematical Reviews number (MathSciNet)
MR2739683

Zentralblatt MATH identifier
1208.34002

Citation

Rehman, Mujeeb Ur; Ali Khan, Rahmat. Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations. Abstr. Appl. Anal. 2010 (2010), Article ID 501230, 15 pages. doi:10.1155/2010/501230. https://projecteuclid.org/euclid.aaa/1313170919