Research On The Solutions Of Several Kinds Of Nonlinear Fractional Differential Equations

As a generalization of integer-order calculus in order,fractional-order calculus has attracted wide attention of many scholars at home and abroad.Due to its powerful ad-vantage and broad application in many fields such as enginee.ring,chemistry,gene and network,fractional-order calculus has become gradually research focus.Because the the-ory of fractional calculus has been successfully applied in various fields in recent years,people gradually find that fractional calculus can describe some non-classical phenomena in the field of natural science and engineering applications.Therefore,studying fraction-al differential equations has important prac:tical significance.This paper will discuss the existence and uniqueness of the solution of the nonlinear fractional differential equations.Furthermore,we will give the Holder continuity of the obtained mild solution.The article structure is as follows.In the first chapter,the relevant background of fractional differential equations and the research progress of related fractional differential equat.ions are int.roduced briefly,then some preliminary knowledge needed in this paper are presented.In the second chapter,we consider the mild solutions of a class of nonlinear integro-different.ial evolution equations in Bana.ch space..By using the fractional t.heory of power operator,analyt,ic semigroup and fixed point theorem,we prove the existence and unique-ness of mild solution of the given problem.Furthermore,we give the Holder continuity of the obtained mild solution.In the third cha.pter,we consider the global existence of the mild solutions for a class of nonlinear differential evolution equat.ions in Banach space.By restricting the nonlinear terms,we give the existence of global solutions.In the fourth chapter,we mainly consider the existence and uniqueness of solutions of a class of Hilfer fractional differential equations.By using the fixed point theorem,we prove the existence of solutions.And under different conditions,we give the existence and uniqueness of the solutions.Finally,several examples are given to show the application of our theorems.