Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure

Springer Science and Business Media LLC - 2012
Jianwu Wu1, Xinguang Zhang2, Lishan Liu3, Yonghong Wu4
1School of Politics Law and Public Administration, Hubei University, Wuhan, 430062, Hubei, People’s Republic of China
2School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, Shandong, People’s Republic of China
3School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, People's Republic of China
4Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Tóm tắt

Abstract

In this article, we consider the existence of positive solutions of the (n - 1, 1) conjugate-type nonlocal fractional differential equation

D 0 + α x ( t ) + f ( t , x ( t ) ) = 0 , 0 < t < 1 , n - 1 < α ≤ n , x ( k ) ( 0 ) = 0 , 0 ≤ k ≤ n - 2 , x ( 1 ) = ∫ 0 1 x ( s ) d A ( s ) ,

where α ≥ 2, D 0 + α is the standard Riemann-Liouville derivative, ∫ 0 1 x ( s ) d A ( s ) is a linear functional given by the Stieltjes integral, A is a function of bounded variation, and dA may be a changing-sign measure, namely the value of the linear functional is not assumed to be positive for all positive x. By constructing upper and lower solutions, some sufficient conditions for the existence of positive solutions to the problem are established utilizing Schauder's fixed point theorem in the case in which the nonlinearities f(t, x) are allowed to have the singularities at t = 0 and (or) 1 and also at x = 0.

AMS (MOS) Subject Classification: 34B15; 34B25.

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