Existence Solutions For Interval-valued Fractional Differential Equations With Time Delay

In the recent years,since it could be better to describe and remember many phenomena of physical,biological,mechanical and aeronautical engineering,researchers are more and more focus about fractional differential equations.And the result concerning fuzzy functions theory also has triggered wide public concern.The interval-valued functions analysis also become a hot issue,because it is an importance branch of fuzzy function theory.According to this circumstance,interval-valued analysis and interval-valued calculation have developed rapidly.By using the fixed point method,we discussed fractional differential equations and the existence and uniqueness of solutions for interval-valued fractional differential equations with time delay.Our results extend and improve the previous results,meanwhile,obtain the conditions for the study of the interval-valued fractional differential equations.This thesis consists of five chapters:In chapter 1,we present the background,the current situation and the future tendency of the research on fractional differential equations,interval-valued functions and the conception of time delay.In chapter 2,we introduce the preliminaries which will be used in this thesis,including some significant definitions and conclusions regarding interval-valued functions and traditional fractional calculus.we present the basic notations on the fractional integral and differential calculus for interval-valued functions and then study the fractional integral and derivative for interval-valued functions,as well as some fixed point theorems which are helpful in our study.In chapter 3,we studied with The properties and applications of fractional differential equations were studied.By analyzing the properties and conceptions of the Riemann-Liouville fractional derivative,the Riemann-Liouville fractional integral and Caputo fractional derivative,the conclusions are achieved.On the other hand,the Erdelyi-Kober type fractional integral was used as boundary value conditions,by exploring the conceptions of the Erdelyi-Kober type fractional integral,to show the existence and uniqueness of solutions for Caputo fractional differential equations with the Erdelyi-Kober type fractional integral boundary value conditions,which is reached by using the classical Banach fixed point theorem.In chapter 4,The properties and applications of a class of interval-valued fractional integro-differential equations with time delay are studied.By analyzing the theories of fractional integro-differential equations with time delay and the interval-valued functions,the conclusions are achieved.On the other hand,to show the existence and uniqueness of solutions for a class of Caputo interval-valued fractional differential equations with time delay,which is reached by applying the classical Banach fixed point theorem and exploring the relationship of the interval-valued fractional integro-differential equations with time delay under certain conditions.In chapter 5,we sum up the main conclusions of the paper and presented our main work in the future.