Research On Lyapunov Inequalities For Boundary Value Problems Of Fractional Differential Equations

With the application of fractional differential equations in fields such as biology,chemistry,and engineering,various definitions of fractional order derivatives have emerged,leading to the research on Lyapunov inequalities for boundary value problems of fractional differential equations becoming a hot topic.In order to expand the fundamental theory of boundary value problems for fractional differential equations,this paper investigates two types of boundary value problems for fractional order derivative equations and obtains Lyapunov and Lyapunov-type inequalities for them.The first type of research is the existence of Lyapunov and Lyapunov-type inequalities for boundary value problems of fractional differential equations with CFC-fractional derivatives and boundary conditions with parameters.The acquisition of Green’s function and the discussion of its properties are the key.As an application,the eigenvalue problem is discussed.Finally,the existence of a unique solution to the corresponding nonlinear problem is studied by using the principle of contractive mapping,and an example is given to illustrate it.The second type of research is the existence of Lyapunov and Lyapunov-type inequalities for boundary value problems of fractional differential equations with Hadamard fractional derivative and boundary conditions with parameters.The acquisition of Green’s function and the discussion of its properties are the key.As an application,the eigenvalue problem is discussed.Finally,the Guo-krasnoselskii fixed point theorem and Leggett-Williams fixed point theorem are used to study the existence of the first solution and the third solution of the corresponding nonlinear problem,and illustrate with examples.