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Positive solutions for a nonlocal fractional differential equation. (English) Zbl 1220.34006
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.

34A08Fractional differential equations
34B16Singular nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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