Lyapunov inequality for a class of fractional differential equations with Dirichlet boundary conditions

Volume 10, Issue 6, pp 3357--3363 http://dx.doi.org/10.22436/jnsa.010.06.44

Publication Date: June 25, 2017 Submission Date: May 01, 2017

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Authors

Yasong Chen - Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China. Yeong-Cheng Liou - Department of Healthcare Administration and Medical Informatics, Center for Big Data Analytics & Intelligent Healthcare and Research Center of Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan. - Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 807, Taiwan. Ching-Hua Lo - Department of Management, Yango University, Fujian 350015, China.

Abstract

In this paper we present Lyapunov inequality for the following fractional boundary value problem \[ \begin{cases} \frac{d}{dt}(\frac{1}{2} _aD_t^{-\beta}u'(t)+\frac{1}{2} _tD_b^{-\beta}u'(t))+\omega(t)u(t)=0,\,\,\,\,\, \quad a<t<b,\\ u(a)=u(b)=0. \end{cases} \] where \( _aD_t^{-\beta}\) and \( _tD_b^{-\beta}\) are the left and right Riemann-Liouville fractional integrals of order \(0\leq\beta<1\), respectively, and \(\omega\in L^1([a,b],\mathbb{R})\). Using the obtained inequality, we provide lower bounds for the first eigenvalue of the fractional differential equations with homogeneous Dirichlet boundary problem.

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Keywords

• Lyapunov type inequality
• fractional differential equations
• boundary value problem
• eigenvalue.

MSC

•  15A42
•  34A08
•  35P15

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