The Study Of Lyapunov Inequality For Two Kinds Of Fractional Differential Equations

Fractional differential equation model has a wide range of applications in many disciplines,such as signal control and processing,polymer material melting,automatic control system theory,biomedicine and so on.Therefore,the existence of solutions is a very important research topic in the field of fractional differential equation researchIn this paper,we study the boundary value problems of two classes of fractional differential equations and obtain their Lyapunov inequalities.Firstly,a Lyapunov inequalities is obtained for a fractional differential equation with Riemann-Liouville derivatives boundary value problem:(?)Where n?N,n?3,??(n-1,n],??[0,?-1],0+RD? and 0+RD? are ? order and ? order Riemann-liouville fractional derivative.It is concluded that if the above problem has a non-zero solution,the following Lyapunov inequality holds.When ? ?[1,?-1],(?).When (?).The second type investigates boundary value problems with Caputo derivatives.(?)Where (?)is the fractional derivative of Caputo,? and ? are two real Numbers.It is concluded that if the above problem has a non-zero solution,the following Lyapunov-type inequality holds.(?)If ??(0,1),??(0,1),then(?);(?)If ??(1,+?),? ?(0,1),then(?);(?)If ??(0,1),??(1,1+(v-1)?/2-v],then(?);(?)If ??(1,+?),??(1,1/2-v],then(?).