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The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives. (English) Zbl 1254.34016

Summary: We study the following singular eigenvalue problem for a higher-orer fractional differential equation \[ \begin{aligned} -{\mathcal D}^\alpha x(t)=\lambda f(x(t),{\mathcal D}^{\mu_1} x(t),{\mathcal D}^{\mu_2} x(t),\dots,{\mathcal D}^{\mu_{n-1}}x(t)),\quad & 0< t< 1,\\ x(0)= 0,\quad{\mathcal D}^{\mu_i}x(0)= 0,\quad{\mathcal D}^{\mu}x(1)= \sum^{p-2}_{j=1} a_j{\mathcal D}^\mu x(\xi_j),\quad & 1\leq i\leq n-1,\end{aligned} \] where \(n\geq 3\), \(n\in\mathbb{N}\), \(n-1<\alpha\leq n\), \(n-1-1<\alpha- \mu_l< n-l\), for \(l= 1,2,\dots,n-2\), and \(\mu- \mu_{n-1}> 0\), \(\alpha-\mu_{n-1}\leq 2\), \(\alpha-\mu> 1\), \(a_j\in [0,+\infty)\), \(0<\xi_1< \xi_2<\cdots< \xi_{p-2}< 1\), \[ 0<\sum^{p-2}_{j=1} a_j \xi^{\alpha-\mu-1}_j< 1, \] \({\mathcal D}^\alpha\) is the standard Riemann-Liouville derivative, and\(f: (0,+\infty)^n\to [0,+\infty)\) is continuous. Firstly, we give the Green function and its properties. Then we established an eigenvalue interval for the existence of positive solutions from Schauder’s fixed point theorem and the lower and upper solutions method. The interesting point of this paper is that \(f\) may be singular at \(x_i= 0\), for \(i= 1,2,\dots, n\).

MSC:

34A08 Fractional ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
34B27 Green’s functions 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
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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References:

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