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

Zhang, Xinguang; Liu, Lishan; Wu, Yonghong

Appl. Math. Comput. 218, No. 17, 8526-8536 (2012).

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

Keywords:

fractional differential equation; positive solution; Green function; eigenvalue problem

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