分数阶微分方程边值问题的Lyapunov不等式

Lyapunov inequality for a fractional differential equation with boundary value problem

对于一类非线性分数阶边值问题,求解了相应的Lyapunov不等式,并给出证明。首先,把非线性分数阶微分方程转化为等价的积分方程,结合边值条件,求解出相应的格林函数。然后,通过对格林函数求导发现其不具有正定性,利用分析学技巧,对其绝对值上界进行放缩,求得一个较小值作为格林函数绝对值的上界。最后,根据解的表达式、Cheyshev范数定义和该上界,求出Lyapunov不等式。对于一类分数阶微分方程特征值问题和一类MittagLeffler函数零解不存在区间问题,给出了相关应用。

A Lyapunov inequality for a nonlinear fractional boundary value problem is established and corresponding proof is given. Firstly, a nonlinear fractional differential equation is transferred into its equivalent integral equation,, and combining with the boundary value conditions, the corresponding Green function is gotten. By taking the derivative of Green function, we find that positive qualitative of Green function can＇ t be insured. Then we use skills of analysis to reduce the upper bound of its absolute value, and then gets a smaller value as the upper bound of its absolute value. Finally, According to the expression of solution, the definition of Cheyshev norm, and the upper bound of the Green function, the Lyapunov inequality is obtained. For a fractional differential equation eigenvalue problem and certain Mittag-Leffler function of an interval where has no real zeros, the relevant applications are gotten.