Properties Of Solutions Of Two Kinds Of Functional Equations

The self composition of functions is called iteration,which is one of the core operations in functional equations.It is also a basic concept in the theory of dynamical system.In this paper,we will focus on two important problems of iterative functional equations,which are the existence of iterative roots and the periodic solutions of iterative functional differential equations.Iterative roots can be regarded as a special kind of iteration,called fractional iteration.Because it involves embedded flow and interpolation calculation,the problem of iterative roots attracts the attention of many scholars.They give plentiful results for the iterative roots of different types of mappings,especially for delete monotone functions,which is proved by the piecewise definition method.However,the existence of iterative roots of set-valued functions is more complex,and it may have no roots.This is a very interesting phenomenon,and will be one of the key research contents of this paper.On the other hand,by adding differential we obtain the iterative functional differential equations.This kind of equation plays an important role in the theory of cybernetics and Biomathematics,its periodic solution often means the periodic oscillation law of the system and the periodic variation law of the population number.Therefore,more and more attention has been paid to the study of periodic solutions of iterative functional differential equations.In this paper,we will study the existence,uniqueness and stability of periodic solutions of a class of iterative functional differential equations.The main contents are as follows:In the first part,we focus on the nonexistence of iterative roots of set-valued functions.A set-valued function is a function that allows one to many and its value can be a set.The classical results show that,unlike ordinary functions,the iterative roots of set-valued functions may not exist even in the monotone case.This is a very interesting phenomenon,enabling us to find more set-valued functions without iterative roots.Therefore,based on the previous discussion of one set-valued point case,we will study the iterative roots of functions with two set-valued points in this chapter.Compared with previous work,two set-valued points will bring essential difficulties since the potential of a function value at a single set-valued point is?1,no contradiction can be derived in the case of two set-valued points.This requires us to find more conditions for the contradiction,and then prove the nonexistence of iterative roots of functions with two set-valued points.In the second part,we focus on the existence,uniqueness and stability of periodic solutions for a class of nonlinear iterative functional differential equations.In 2006,Burton proposed the idea of using Krasnoselskii fixed point theorem to study the periodic solutions of iterative functional differential equations.Then Zhao solved the non-homogeneous iterative functional differential equation with variable coefficients x'(t)=c1(t)x(t)+c2(t)x2(t)+…cn(t)xn(t)+F(t).The existence and uniqueness of periodic solutions are discussed.In this chapter,based on Zhao's results,we use Krasnoselskii's fixed point theorem to further solve the equation x'(t)=c1(t)x(t)+C2f(x2(t))+F(t),and discuss the existence,uniqueness and stability of the periodic solution.It is noted that when the quadratic iteration of the unknown function x(t)is implicit in the function f,we call it a nonlinear equation.Since it is more difficult to estimate the integral in the proof process,which enable us to find new conditions to ensure the effective estimation of the integral.