延遲微分方程長期被研究是由於他們的實際應用。對延遲係數不是一個恆定數字，許多學者研究這係數是一個和本身狀態有關的變數，所以我們處理未知合成函數的微分和泛函方程，即疊代微分方程和疊代泛函方程。這篇論文主要目的是研究此類微分方程，包括解析解，數值解，還有定性的性質。

首先，我們收集整理這類常見微分方程的解析解，然後古典方法，例如未定係數被用於計算一階微分方程解析解在第一章，二階是第二章，泛函方程則在第三章，泰勒級數法第四章。我們有一套系統的方法找出固定點，因此可以使用這些方法計算解析解的係數，此外，我們在第五章也建立解析解的存在唯一定理。

第二，我們收集整理已知，具有代表性的疊代微分方程和泛函方程解的存在唯一性，然後我們利用Schauder固定點定理，建立更一般性的疊代微分方程連續解的存在性，在某些條件下，局部解還可擴充到一般解。

第八章是處理最簡單的疊代微分方程 ，我們對Eder 的例子給予定性及解不唯一證明，第八章我們使用Euler方法得到疊代微分方程的數值解。在某種條件下，我們有疊代微分方程的誤差分析，第九章我們使用未定係數，泰勒級數，Picard's疊代和Si 他們的方法得到的解析解，這些方法的好處和缺點也被討論。

Functional differential equations with delay have long been studied due to their practical applications. For the delay term is not a constant number, many researches study the case when this deviating argument depends on the state variable. So we deal with the differential and functional equations involving with the compositions of the unknown function, i.e. the iterative functional differential equations (IFDEs) and iterative functional equations (IFEs) without derivative. The main purpose of this dissertation is to investigate the solutions of such equations, including their analytic solutions, numerical solutions and qualitative behaviors.

First, we survey some well known differential equations of this type which possess analytic solutions. Then the classical method of undetermined coefficients is used to compute these power series solutions for the first order IFDEs in Chapter 1, the second order IFDEs in Chapter 2 and FDEs in Chapter 3. Taylor series method is also used to get these analytic solutions in Chapter 4. Systematical method is found to locate the fixed point in generalized sense, so we can use these methods to calculate the coefficients of their analytic solutions. Furthermore, we also establish the existence and uniqueness theorem for analytic solution in Chapter 5.

Second, we survey the known existence and uniqueness theorems of solutions for these IFDEs and FDEs in Chapter 6. Then we apply Schauder fixed point theorem to establish new existence theorems of local solutions for general IFDEs. Under certain conditions, these local solutions can be extended to global solutions.

Chapter 7 deals with the simplest IFDEs the Eder's equation. We extend the qualitative properties of this case and find its solution is not unique. In Chapter 8, we use Euler method to get the numerical solution of IFDEs. Under some conditions, we have the error analysis on these equations. In Chapter 9, we employ the method of undetermined coefficients, Taylor series, Picard's iteration and Si's methods to get their analytic solutions. Their comparisons, the advantage and disadvantage of these methods are also discussed.

1. Method of Undetermined Coefficients for Analytic Solutions of Iterative Functional Differential Equations of First Order . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 introduction. . . . . . . . . . . . . . . . . . . . . . . .1
1.2 The m-th iterative. . . . . . . . . . . . . . . . . .3
1.3 StepfanType . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Neutral Type . . . . . . . . . . . . . . . . . . . . . .12
1.5 Combination Type of Iterates. . . . . . . 16
1.6 Reciprocal of Iterates. . . . . . . . . . . . . . .19
1.7 General Form. . . . . . . . . . . . . . . . . . . . . 22
1.8 Conclusion. . . . . . . . . . . . . . . . . . . .. . . .24

2. Method of Undetermined Coefficients for Analytic Solutions of Iterative Functional Differential Equations of Second Order . . . . . . . . . . . . . . . . . . . . . .25
2.1 Introduction . . . .. . . . . . . . . . . . . . . . . . . 25
2.2 Square of Iterates . . . . . . . . . . . . . . . . . 27
2.3 terates with Derivatives . . . . . . . . . . . . 32
2.4 Combination of Iterates . . . . . . . . . . . 34
2.5 General Form . . . . . . . . . . . . . . . . . . . .40

3. Method of Undetermined Coefficients for Analytic Solutions of Iterative Functional Equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . .45
3.2 Polynomial-like Type . . . . . . . . . . . . . .47
3.3 Curve Type . . . . . .. . . . . . . . . . . . . . . . 52
3.4 nvariant Curve Type . . . . . . . . . . . . . . .55
3.5 Neutral Type . . . . . . . . . .. . . . . . . . . . .57
3.6 General Type . . . . . . . . . .. . . . . . . . . . .58
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . 61

4. Taylor Series Method for Analytic Solutions of Iterative Functional Equations62
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 62
4.2 Combination of Iterates . . . . . . . . . . . 64
4.3 General Form . . . . . . . .. . . . . . . . . . . . 68
4.4 Curve Type . . . . . . . . . . . . . . . . . . . . . .71
4.5 Invariant Curve Type . . . . . . .. . . . . . . .73
4.6 Si’s Results . . . . . . . . . . . . . . . . . . ..75
4.7 Conclusion . . . . . . . . . .. . . . . . . . . . . . 78

5. Existence and Uniqueness of Undetermined Coefficients for Analytic Solutions
of Iterative Functional Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . .. 79
5.1 Introduction . . . . . . . . . . . . . . .. . . . . . .79
5.2 Existence and Uniqueness . .. . . . . .80
5.3 First Order . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Second Order. . . . . . . . . . . . . . . . . . . .91
5.5 Analytic Solution does not exists . .. 95
5.6 Conclusion . . . . . . . . . .. . . . . . . . . . .98

6. Existence of Solutions for Iterative Functional Differential Equations . . . . . . . . . . . . . . 100
6.1 Introduction. . . . . . . . . . . . . . . . . . . . 100
6.2 Monotonic Properties . . . . . . . . . . . 102
6.3 Existence Theorem . . . . . . . . . . . . 106
6.4 Local Solutions . . . . . . . . . . . . . . . ..114
6.5 Conclusion . . . . . . . .. . . . . . . . . . . . .116

7. Qualitative Behavior of an Iterative Functional Differential Equation . . . . . . . . . . . . . . 117
7.1 Introduction . . . . . . . . . . . . . . . . . . . .117
7.2 Eder’s Results . . . .. . . . . . . . .. . . 118
7.3 Qualitative Properties . . . . . . . . . . .121
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . 124

8. Numerical Solutions of Iterative Functional Differential Equations Using Euler
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.1 Introduction . . . . . . . . . . . . . . . . . . .125
8.2 Test Model . . . . .. . . . . . . . . . . . . . . .126
8.3 Iterative Differential Equations . . . 127
8.4 Stepfan Type . . . . . . . . . . . . . . . . . . 133
8.5 Error Analysis . . .. . . . . . . . . . . . . . .135
8.6 Conclusion. . . . . . . . . . . . . . . . .. . . 139

9. Four Symbolic Methods for the Analytic
Solutions of Iterative Functional
Differential Equations . . . . . . . . . . . . .140
9.1 introduction. . . . . . . . .. . . . . . . . . . .140
9.2 Methods of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . 141
9.3 Picard’s Iteration . . . . . . . . . . . .142
9.4 Taylor Series Method . . . . . . . . . 143
9.5 Method of Si’s Transformation 145
9.6 Comparison. . . .. . . . . . . . . . . . . . 146

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