The Research About Fractional Lyapunov Inequality And Oscillation Of Solutions

In this paper,we study oscillation criteria of differential equations and Lyapunov-type inequalities of fractional order differential equations.The known results of Lyapunov inequalities of fractional differential equation are improved.This paper is divided into four chapters.In part one,we summarize the research background of fractional order calculus and Lyapunov inequalities as well as the related definitions used in this paper.In part two,we research oscillation criteria for nonlinear fractional differential equa-tion with forced term: where(?)is the modified Riemann-Lionville derivative.(?)and F(t,x)· x>0 for all x ? 0.Based on the definition of the modified Riemann-Liouville derivative and some properties,using generalized Riccati transformation,the interval oscillation criterion of the nonlinear fractional differential equation with forced term are obtained,and some known results in the literature are extended and improved.In part three,we consider the oscillation of following second-order differential equa-tions with non-monotone arguments:and where y(t):= x(t)+ p(t)x(?(t)),(?),1?i?m are nonmonotonic.The second order differential equation with non-monotonic term is transformed into first order differential equation by means of descend?ing order method and related inequality knowledge,and then the oscillation criterion of'this kind of equation is obtained by using the correlation results of first order differential equation,and the oscillation criterion of the solution of second order differential equation is improved.At the same time,the corresponding examples are given.In part four,we establish the following Lyapunov-type inequalities for differential equations:where 0<?k<2,1?k? n,??(23],??E(1,2].The main methods are to apply algebraic inequalities and some known results to obtain new conclusions.And the existing Lyapunov inequalities are extended.