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Zbl 1220.34006
Wang, Yongqing; Liu, Lishan; Wu, Yonghong
Positive solutions for a nonlocal fractional differential equation.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 11, 3599-3605 (2011). ISSN 0362-546X

Summary: We study the following singular boundary value problem of a nonlocal fractional differential equation $$\cases D^\alpha_{0^+}u(t)+q(t)f(t,u(t))=0,\quad & 0<t<1,\ n-1<\alpha\le n,\\ u(0)=u'(0)=\cdots=u^{(n-2)}(0)=0,& u(1)=\int^1_0u(s)\,dA(s),\endcases$$ where $\alpha \geq 2$, $D^\alpha_{0^+}$ is the standard Riemann-Liouville derivative, $\int^1_0u(s)\,dA(s)$ is given by the Riemann-Stieltjes integral with a signed measure, $q$ may be singular at $t=0$ and/or $t=1,f(t,x)$ may also have a singularity at $x=0$. Existence and multiplicity of positive solutions are obtained by means of fixed point index theory in cones.
MSC 2000:
*34A08
34B16 Singular nonlinear boundary value problems
34B18 Positive solutions of nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations

Keywords: fractional differential equation; positive solution; singular problem; fixed point index

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