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Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations. (English) Zbl 1244.34009
Summary: We are concerned with the existence and uniqueness of positive solutions for the following singular nonlinear \((n-1,1)\) conjugate-type fractional differential equation with a nonlocal term \[ \begin{cases} D^\alpha_{0+}x(t)+f(t,x(t))=0,\;0<t<1,\;n-1<\alpha\leq n,\\ x^{(k)}(0) =0,\;0\leq k\leq n-2,\;x(1)=\int^1_0x(s)dA(s),\end{cases} \] where \(\alpha \geq 2\), \(D^\alpha_{0+}\) is the standard Riemann-Liouville derivative, \(A\) is a function of bounded variation and \(\int^1_0u(s)dA(s)\) denotes the Riemann-Stieltjes integral of \(u\) with respect to \(A\), \(dA\) can be a signed measure.

MSC:
34A08 Fractional ordinary differential equations and fractional differential inclusions
47N20 Applications of operator theory to differential and integral equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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