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Zbl 1096.34016
Zhang, Shuqin
Positive solutions for boundary-value problems of nonlinear fractional differential equations.
(English)
[J] Electron. J. Differ. Equ. 2006, Paper No. 36, 12 p., electronic only (2006). ISSN 1072-6691/e

Summary: We consider the existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem $$\bold{D}_{0+}^\alpha u(t)=f(t,u(t)),\quad 0<t<1,\quad u(0)+u'(0)=0,\quad u(1)+u'(1)=0,$$ where $1<\alpha\leq 2$ is a real number, and $\bold{D}_{0+}^\alpha$ is the Caputo fractional derivative, and $f:[0,1]\times[0,+\infty)\to [0,+\infty)$ is continuous. By means of a fixed-point theorem on cones, some existence and multiplicity results on positive solutions are obtained.
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B15 Nonlinear boundary value problems of ODE
26A99 Functions of one real variable

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

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