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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 $$\align -{\Cal D}^\alpha x(t)=\lambda f(x(t),{\Cal D}^{\mu_1} x(t),{\Cal D}^{\mu_2} x(t),\dots,{\Cal D}^{\mu_{n-1}}x(t)),\quad & 0< t< 1,\\ x(0)= 0,\quad{\Cal D}^{\mu_i}x(0)= 0,\quad{\Cal D}^{\mu}x(1)= \sum^{p-2}_{j=1} a_j{\Cal D}^\mu x(\xi_j),\quad & 1\le i\le n-1,\endalign$$ where $n\ge 3$, $n\in\bbfN$, $n-1<\alpha\le n$, $n-1-1<\alpha- \mu_l< n-l$, for $l= 1,2,\dots,n-2$, and $\mu- \mu_{n-1}> 0$, $\alpha-\mu_{n-1}\le 2$, $\alpha-\mu> 1$, $a_j\in [0,+\infty)$, $0<\xi_1< \xi_2<\cdots< \xi_{p-2}< 1$, $$0<\sum^{p-2}_{j=1} a_j \xi^{\alpha-\mu-1}_j< 1,$$ ${\Cal D}^\alpha$ 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 $i= 1,2,\dots, n$.

34A08Fractional differential equations
34B09Boundary eigenvalue problems for ODE
34B27Green functions
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
Full Text: DOI
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