| Fractional order differential equation can better study and describe various complex physical phenomena and memory characteristics.Therefore,fractional differential equations have attracted the attention of many researchers.In this paper,monotone iteration methods for some generalized Hilfer fractional differential equations are studied.The structure of the paper is divided into six chapters.In chapter 1,the introduction introduces the historical research background of fractional calculus,the development status at home and abroad and the research content of this paper.In chapter 2,preliminary knowledge introduces the basic definition,lemma,theorem and properties of generalized Hilfer fractional calculus.In chapter 3,the monotone iteration method for a class of 0 < p < 1 order nonlinear generalized Hilfer fractional order differential equations with time delay is studied.Firstly,the existence and uniqueness of the approximate solution sequence are proved by using the quasilinear technique and the upper and lower solution method,and then the quadratic convergence of the approximate solution sequence to the unique solution of the initial value problem is proved.In chapter 4,we study the monotone iteration method for a class of nonlinear generalized 1 < p < 2 order Hilfer fractional differential equations.In this chapter,we will establish a new comparison principle based on the studied problem,prove the quadratic convergence of the sequential solutions and give an example to verify the theoretical results.In chapter 5,we study the monotone iteration method for a class of 1 < p < 2 order nonlinear generalized Hilfer fractional order differential equations with time delay.Based on the results in Chapter 4,a new monotone iteration method for generalized Hilfer fractional differential equations is obtained by adding time delay term to the studied problem.In chapter 6,we give a conclusion of our present work and make a plan for further research. |
