Existence Of Solutions For Boundary Value Problems Of Fractional Differential Equations With Delay

Fractional order differential model can be well used to describe the memory and genetic properties.It overcomes the defect that the theory of classical integer order differential model is not in good agreement with the experimental results,and it can achieve better simulation effect with fewer parameters in the modeling.At present,fractional differential equations have been applied in the fields of fluid mechanics,neural model,intelligent control,circuit system analysis and so on.As an important part of fractional differential system theory,the boundary value problems of fractional differential equations have made a lot of theoretical achievements in recent years.However,there are few related achievements on the fractional boundary value problems with time delay.Therefore,it is meaningful for us to consider the time delay in our research.In this paper,the existence of solutions for several kinds of boundary value problems of fractional delay differential equations with different boundary conditions are studied.These boundary conditions involve integral boundary conditions and multi-point boundary conditions.The research methods include e-norm,nonlinear contraction principle,Leray-Schauder degree theory,cone extension and cone compression fixed point theorem and Avery-Peterson fixed point theory,etc.The full text mainly includes the following chapters:In the first chapter,the relevant research background of fractional calculus and boundary value problems for fractional differential equations is briefly introduced.The main research contents of this paper are summarized and the related preparatory knowledge is given.In the second chapter,by constructing an e-norm,the uniqueness and existence of solutions for a class of fractional delay differential equations are discussed based upon the Banach contraction mapping principle and Schauder fixed point theorem.In the third chapter,we study the positive solutions of a fractional Langevin equation with delay and three-point boundary conditions.Based on the nonlinear contraction,the uniquencess of positive solution of problem is obtained.The existence of positive solution is derived by means of Leray-Schauder degree theory.In the fourth chapter,with the help of Green function and its properties,a class of positive solutions of fractional delay differential equations with integral boundary value conditions are studied.Different from the previous works,we use the fixed point theorems of cone stretch and cone compression at the same time,and obtain the sufficient conditions to ensure the existence of at least two positive solutions for the problem.In the fifth chapter,by using Avery-Peterson fixed point theory,the existence of at least three positive solutions for a class of four point boundary value problems of fractional delay differential equations with p-Laplacian operators is considered.