Abstract and Applied Analysis

Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations

Jing Wu and Xinguang Zhang

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Abstract

We study the eigenvalue interval for the existence of positive solutions to a semipositone higher order fractional differential equation - 𝒟 t μ x ( t ) = λ f ( t , x ( t ) , 𝒟 t μ 1 x ( t ) , 𝒟 t μ 2 x ( t ) , , 𝒟 t μ n - 1 x ( t ) ) 𝒟 t μ i x ( 0 ) = 0 , 1 i n - 1 , 𝒟 t μ n - 1 + 1 x ( 0 ) = 0 , 𝒟 t μ n - 1 x ( 1 ) = j = 1 m - 2 a j 𝒟 t μ n - 1 x ( ξ j ) ,   where n - 1 < μ n ,   n 3 , 0 < μ 1 < μ 2 < < μ n - 2 < μ n - 1 , n - 3 < μ n - 1 < μ - 2 , a j , 0 < ξ 1 < ξ 2 < < ξ m - 2 < 1 satisfying 0 < j = 1 m - 2 a j ξ j μ - μ n - 1 - 1 < 1 , 𝒟 t μ is the standard Riemann-Liouville derivative, f C ( ( 0,1 ) × n , ( - , + ) ) , and f is allowed to be changing-sign. By using reducing order method, the eigenvalue interval of existence for positive solutions is obtained.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 740760, 14 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365168876

Digital Object Identifier
doi:10.1155/2012/740760

Mathematical Reviews number (MathSciNet)
MR3004914

Zentralblatt MATH identifier
1260.34015

Citation

Wu, Jing; Zhang, Xinguang. Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 740760, 14 pages. doi:10.1155/2012/740760. https://projecteuclid.org/euclid.aaa/1365168876


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