On Some Initial Value Problems Of Fractional Evolution Equations

In recent years,fractional calculus and fractional differential equations have been the hotspots of many scholars' research,and they have penetrated from pure mathematical theory to many fields of science and engineering.This paper mainly studies several types of Cauchy problems of fractional differential equations,and the conclusions obtained are verified by examples.The full text consists of five parts.the main content is as follows:Chapter 1 mainly introduces the relevant research background,significance and development at domestic and foreign,as well as the relevant definitions and basic theories used in this article.Chapter 2 using the Sadovskii fixed point theorem and the monotone iterative sequence,we study the initial value problem of fractional differential evolution equations of the following form the existence of mild solutions and positive mild solutions are given,and gives examples to illustrate the results.Chapter 3 mainly studies the Cauchy problem of uniform fractional differential equations,gives the concept of fractional semigroups,and obtains some related properties.It is proved that the Cauchy problem of homogeneous fractional differential equations is well-posed if and only if the coefficients the coefficient operator generates a fractional semigroup,discusses and obtained the following the homogeneous fractional Cauchy problem the existence and uniqueness of mild solutions and strong solutions,and examples to prove the conclusions.Chapter 4 is based on the newly defined concept of fractional semigroups in Chapter 3,and using the Schauder's fixed point theorem and Banach's fixed point theorem,we study the following nonlocal Cauchy problems for fractional differential equations the existence and uniqueness of mild solutions,and examples are given to illustrate the results.Chapter 5 is a summary of the thesis and prospect for future research.