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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
\[ \begin{cases} D^\alpha_{0^+}u(t)+q(t)f(t,u(t))=0,\quad & 0<t<1,\;n-1<\alpha\leq n,\\ u(0)=u'(0)=\cdots=u^{(n-2)}(0)=0,& u(1)=\int^1_0u(s)\,dA(s),\end{cases} \]
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:

34A08 Fractional ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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