The Research About Lyapunov Inequality For Fractional Differential Equations And Properties Of Solutions

In this paper,we mainly study the Lyapunov type inequalities and the properties of the solutions of the fractional differential equations.We obtain some Lyapunov type inequalities for a class of fractional differential equations,and the existence and attractiv-ity of solutions of fractional differential equations with Caputo-Katugampola fractional calculus.This paper is divided into four chapters.The first chapter mainly summarizes the research background of Lyapunov type inequalities and fractional calculus theory,the some related lemmas and definitions which will be used in this paper.The second chapter establishes some new Lyapunov type inequalities for fractional(p,q)-Laplacian operators in an open bounded set Ω RN,under homogeneous Dirichlet boundary conditions.We mainly use Holder’s inequality,mean-value inequality and other inequalities to obtain some related conclusions.Next,we use the obtained inequalities to derive some geometric properties of the generalized spectrum associated to the considered problem.The third chapter studies the existence of solutions for fractional differential equa-tions with integral boundary value conditions (?).By using the definition of Caputo-Katugampola fractional derivative and its related prop-erties,the differential equation is transformed into integral equation,the Green function and its related properties are derived,and the existence results of solutions are obtained by using the fixed point theorem.Three examples are given to demonstrate the applica-tions of the theorems obtained.The fourth chapter consideres the following fractional differential equation (?).It’s obtained that the global attractivity of the solution by using the definition of Caputo-Katugampola fractional derivative and its related properties,the replacement of variable and Schauder’ fixed point theorem.