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Existence of positive solutions for eigenvalue problem of nonlinear fractional differential equations

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In this article, by using the fixed point theorem, existence of positive solutions for eigenvalue problem of nonlinear fractional differential equations D 0 + α u ( t ) + λ a ( t ) f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = 0 is considered, where 1 < α < 2 is a real number, D 0 + α is the standard Riemann-Liouville derivate, λ is a positive parameter and a ( t ) ∈ C ([0, 1], [0, ∞)), f ( t , u ) ∈ C ([0, 1] × [0, ∞), [0, ∞)). MSC(2010): 34B18.

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Published 01 January 2012
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Han and GaoAdvances in Difference Equations2012,2012:66 http://www.advancesindifferenceequations.com/content/2012/1/66
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Existence of positive solutions for eigenvalue problem of nonlinear fractional differential equations * Xiaoling Han and Hongliang Gao
* Correspondence: hanxiaoling@nwnu.edu.cn Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P. R. China
Abstract In this article, by using the fixed point theorem, existence of positive solutions for eigenvalue problem of nonlinear fractional differential equations α D u(t) +λa(t)f(t,u(t0)) = 0, <t<1, 0+ u(0) =u(1) = 0
α D is considered, where 1<a<2 is a real number,0+is the standard Riemann Liouville derivate,lis a positive parameter anda(t)ÎC([0, 1], [0,)),f(t,u)ÎC([0, 1] × [0,), [0,)). MSC(2010):34B18. Keywords:fractional differential equation, positive solution, eigenvalue problem, fixed point
1 Introduction Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry, engineering, etc. For details, see [16] and references therein. Recently, many results were obtained dealing with the existence and multiplicity of solutions of nonlinear fractional differential equations by the use of techniques of non linear analysis, see [722] and the reference therein. Bai and Lu [7] studied the exis tence of positive solutions of nonlinear fractional differential equation α D u(t) +f(t,u(t0)) = 0, <t<1, 0+ (1:1) u(0) =u(1) = 0,
α where 1<a2 is a real number,Dis the standard RiemannLiouville differentia 0+ tion, andf: [0, 1] × [0,)®[0,) is continuous. They derived the corresponding Green function and obtained some properties as follows. Proposition 1The Green functionG(t,s) satisfies the following conditions: (R1)G(t,s)ÎC([0,1] × [0,1]), andG(t,s)>0 fort,sÎ(0, 1);
© 2012 Han and Gao; 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.