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Zbl 1178.34006
Xu, Xiaojie; Jiang, Daqing; Yuan, Chengjun
Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 10, A, 4676-4688 (2009). ISSN 0362-546X

Summary: In this paper, we consider the properties of Green's function for the nonlinear fractional differential equation boundary value problem $${\bold D}^\alpha_{0+}u(t)=f(t,u(t)),\quad 0<t<1,\qquad u(0)=u(1)=u'(0)=u'(1)=0,$$ where $3<\alpha\le 4$ is a real number, and ${\bold D}^\alpha_{0+}$ is the standard Riemann-Liouville differentiation. As an application of Green's function, we give some multiple positive solutions for singular and nonsingular boundary value problems, and we also give the uniqueness of solution for a singular problem by means of the Leray-Schauder nonlinear alternative, a fixed-point theorem on cones and a mixed monotone method.

MSC 2000:: *34A08
34B18 Positive solutions of nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations
34B27 Green functions

Keywords: fractional differential equation; boundary value problem; positive solution; fractional Green's function; fixed-point theorem

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