Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations. (English) Zbl 1244.34009

Zhang, Xinguang; Han, Yuefeng

Appl. Math. Lett. 25, No. 3, 555-560 (2012).

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

Keywords:

monotone iterative technique; fractional differential equation; existence and uniqueness; positive solution

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