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

摘要

In this paper we present Lyapunov inequality for the following fractional boundary value problem [ egin{cases} rac{d}{dt}(rac{1}{2} _aD_t^{-eta}u'(t)+rac{1}{2} _tD_b^{-eta}u'(t))+omega(t)u(t)=0,,,,,, quad atb, u(a)=u(b)=0. end{cases} ] where ( _aD_t^{-eta}) and ( _tD_b^{-eta}) are the left and right Riemann-Liouville fractional integrals of order (0leqeta1), respectively, and (omegain 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.