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

Wu Jianwu, Zhang Xinguang, Liu Lishan, Wu, Wu Yonghong

Description

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 .

Sujets

• Fractional calculus

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 3
• Langue: English

Extrait

Wuet al.Advances in Difference Equations2012,2012:71 http://www.advancesindifferenceequations.com/content/2012/1/71

R E S E A R C HOpen Access Positive solutions of higherorder nonlinear fractional differential equations with changing sign measure 1 2*3,4 4 Jianwu Wu , Xinguang Zhang, Lishan Liuand Yonghong Wu

* Correspondence: [e-mail] 2 School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, Shandong, People’s Republic of China Full list of author information is available at the end of the article

Abstract In this article, we consider the existence of positive solutions of the (n 1, 1) conjugatetype nonlocal fractional differential equation  α D x(t) +f(t,x(t0)) = 0,<t<1,n−1< α≤n, 0+  1 (k) x(0) = 0,0≤k≤n−2,x(1) =x(s)dA(s), 0  1 α wherea≥2, D 0+ is the standard RiemannLiouville derivative, x(s)d A(s) is a 0 linear functional given by the Stieltjes integral,Ais a function of bounded variation, anddAmay be a changingsign measure, namely the value of the linear functional is not assumed to be positive for all positivex. 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 nonlinearitiesf(t, x) are allowed to have the singularities att= 0 and (or) 1 and also atx= 0. AMS (MOS) Subject Classification:34B15; 34B25. Keywords:upper and lower solutions, fractional differential equation, Schauder’s fixed point theorem, positive solution.

1 Introduction In this article, we are studying the existence of positive solutions for the following sin gular nonlinear (n1, 1) conjugatetype fractional differential equations with a nonlocal term  α D x (t) +f(t,x(t0)) = 0,<t<1,n−1< α≤n, 0+  1 (1:1) (k) x0(0) = 0,≤k≤n−2,x(1) =x(s)dA(s), 0 α wherea≥2, D 0+ is the standard RiemannLiouville derivative,f: (0, 1) × (0, +∞)® [0, +∞) is continuous andfmay be singular atx= 0 andt= 0, 1.  1 In the BVP (1.1), x(s)d A(s) denotes the RiemannStieltijes integral, whereAis a 0 function of bounded variation, that isdAcan be a signed measure. In this work we do  1 not suppose that x(s)dA(s)≥0 for allx≥0, and hence the BVP (1.1) has a wider 0 © 2012 Wu et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.