The Research About Lyapunov Inequalities For Boundary Value Problem Of Nonlinear Fractional Differential Equation

The theory research about fractional differential equations has been gotten intensive development of the fractional calculus theory.Some fractional equation model have been widely used in the scientific fields,such as image processing,neutral network,and so on.Thus,the solvable fractional equation model from the special academic research can make a great difference in the modern society.In this paper,we mainly investigate the Lyapunov inequalities for three types of fractional differential equations.Firstly,a Lyapunov inequalities is obtained for a fractional differential equation with Rimann-Liouville derivatives boundary value problem:q:[a,b]→R is a Lebesgue integral function,f:R→R is a continuous function.The conclusion is∫ba|q(s)|ds≥Γ(v)/△(b-a)v-1N,where△=max{(2/v-1)2/v-3(3-v/v-1),(v-2)v-2/(v-1)v-1}.Secondly,Lyapunov inequalities for a fractional p-Laplacian equation is gotten:where 2<α≤3,2＜β≤3,Daa+,Daa+ are Riemann-Liouville fractional derivatives with orders of α,β,Φp(s)=|s|p-2s,p>1,and q:[a,b]→R is a continuous function.The conclusion is∫ba(b-s)β-2(s-a)|q(s)|ds≥[Γ(α)]p-1Γ(β)(b-a)(p-1)(α-1)(∫ba(b-t)α-1(t-a)α-1dt)1-pThirdly,a Lyapunov inequalities is obtained for a fractional differential equation with mixture derivatives boundary value problem:{(acDay)(t)+q(t)f(y(t))=0,a<t<b,y(a)=y(a)=y(a)=0,acDβy(b)=0,where 3<α ≤ 4,1<β≤2,q:[a,b]→R is a continuous function.The conclusion is∫ba(b-s)α-β-1|q(s)|ds≥Γ(α-β)/Γ(3-β)(b-a)βN.