The Research On Monotone Iterative Method For Nonlinear Fractional Differential Equations

Fractional calculus is an extremely useful mathematical tool for solving many engineering and physical problems.It has important applications in the fields of fractal and porous media dispersion,electrolytic chemistry,condensed matter physics,viscoelastic systems,biomathematics and so on.This dissertation is devoted to the study of the monotone iterative methods and existence of solutions for nonlinear fractional differential equations,and some new results are obtained.Three problems are mainly considered: one is the monotone iterative methods for the initial value problem of nonlinear fractional differential equations of Riemann-Liouville type;the other is the monotone iterative methods for the initial value problem of nonlinear fractional differential equations of Caputo type;the third is the monotone iterative methods for the nonlocal problem of nonlinear Caputo fractional differential equations with delayed arguments.In order to solve the above problems,we firstly apply the properties of fractional integral and derivative to prove the uniqueness of solutions of the related linear fractional differential equation in the appropriate weight function space,and establish a comparison principle for the problem.Then we construct a set of monotone sequences by using the method of upper and lower solutions.Thus,the existence of solutions of the nonlinear fractional differential equations is proved and the conditions are obtained.Finally,examples are given to illustrate it.