Lyapunov Inequalities For Three Boundary Value Problems Of Fractional Differential Equations

Fractional differential equation models are widely used in many scientific fields,such as image processing,neural networks,signal analysis and processing,which can be described by differential equation models.Therefore,a solvable fractional differential equation model in subject research can play a certain role in social life.In this paper,we mainly investigate the Lyapunov inequalities for three types of nonlinear fractional differential equations with different orders.Firstly,a Lyapunov inequalities is obtained for a fractional differential equation with Caputo derivatives boundary value problem:q:[a,b]→R is Lebesgue integrable function,f:R→R is a continuous function.It is concluded that if the above problem has a non-zero solution,the following Lyapunov inequality holds.whereSecondly,Lyapunov inequalities for a fractional p-Laplacian differential equation with Riemann-Liouville derivative is gotten:where n∈N,n≥>3,n-1<α≤n,2<β≤3,Φp(s)=|s|p-2,p>1 and χ:[a,b]→R is a continuous function.It is concluded that if the above problem has a non-zero solution,the following Lyapunov-type inequality holds.Thirdly,a Lyapunov inequalities is obtained for a fractional differential equation with mixture derivatives boundary value problem:Where aCDα,aCDβ pare Caputo fractional derivatives with order of α,β,2<α≤3,q:[a,b]→R is Lebesgue integrable function,f:R→R is a continuous function.It is concluded that if the above problem has a non-zero solution,the following Lyapunov-type inequality holds.