University of Southern California Dissertations and Theses

Linear differential difference equations

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Linear differential difference equations

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Linear Differential Difference Equations
by
Yuqi Wang

A Thesis Presented to the
FACULTY OF THE USC DORNSIFE
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
APPLIED MATHEMATICS

December 2021

Copyright 2021 Yuqi Wang
ii

TABLE OF CONTENTS
Abstract…………………………………………………………………iii
Chapter 1: Differential and difference equations…..……………….........1
Chapter 2: Retarded differential difference equations…………..……….4
Chapter 3: Exponential estimates of 𝑥 (𝜙 , 𝑓 )……………………..……..7
Chapter 4: The characteristic equation……………..……………………9
Chapter 5: The fundamental solution……………..........……………….11
Chapter 6: The variation-of-constants formula……………………...….16
Chapter 7: Neutral differential difference equations………………...….18
Chapter 8: Stability and periodic solutions of first order delay
equations……………………….………………………..…27
Bibliography………………………..………………………….………31
Appendices………………………..……………………………………32
Appendix A:Figure I…...……...…………....……………32
Appendix B:Figure II.……………………………………35

iii

Abstract
The paper introduces the subject through linear differential difference equations of retarded
and neutral type with constant coefficients. There is a series of theories of linear equations with
constant coefficients, the simplest possible differential difference equations, are based on such
fundamental methods. In addition, this paper introduces the characteristic equation, the
fundamental solution, the role of fundamental solution in determining the exact exponential bound
of the solution of homogeneous equation and the behavior of the solution of non-homogeneous
equation. Finally, it is found that the solution of the first order delay equation is unstable under
some conditions and that the first order delay equation has stability and periodicity under certain
conditions.

1

Chapter 1: Differential and difference equations
Let 𝑅 = (−∞, ∞) , 𝑅 𝑛 be any real n-dimensional normed vector space. Consider the scalar
differential equation
(1.1) 𝑥 ̇ = 𝐴𝑥 .
For A is a constant, given c is an arbitrary constant, all solutions are given by (exp𝐴𝑡 )𝑐 .
Of course A can be an n×n matrix and x is an n-vector, then c is an n-vector.
Each column of exp𝐴𝑡 has the form ∑ 𝑝 𝑗 (𝑡 )exp𝜆 𝑗 𝑡 , where 𝑝 𝑗 (𝑡 ) is an 𝑛 − 1 vector
polynomial in t and λj is an eigenvalue of the matrix A, so each λj satisfies the characteristic
equation
𝑑𝑒𝑡 (𝜆𝐼 − 𝐴 ) = 0.
The complete information of the solution of Equation (1.1) is obtained from the eigenvalues
and eigenvectors of matrix A.
The nonhomogeneous equation
(1.2) 𝑥 ̇ = 𝛢𝑥 + 𝑓 (𝑡 ),
where f is a given continuous function from 𝑅 to 𝑅 𝑛 .
With 𝑥 (0) = 𝑐 , the solution of Equation (1.2) is given by
(1.3) 𝑥 (𝑡 ) = 𝑒 𝐴𝑡
𝑐 + ∫ 𝑒 𝐴 (𝑡 −𝑠 )
𝑓 (𝑠 )𝑑𝑠 𝑡 0
.
The delayed difference equation
(1.4) 𝑥 (𝑡 ) = 𝐴𝑥 (𝑡 − 1) + 𝐵𝑥 (𝑡 − 2)
where A and B are constants. Let 𝑦 (𝑡 ) = 𝑥 (𝑡 − 1), the scalar equation (1.4) can be regarded as
the two-dimensional equation
2

(1.5) 𝑧 (𝑡 ) = 𝐶𝑧 (𝑡 − 1), 𝑧 = [
𝑥 𝑦 ] , 𝐶 = [
𝐴 𝐵 1 0
].
For all 𝑡 ≥ 0, the solution of Equation (1.5) must specify a 2-vector function 𝜙 ∈ [−1,0], so
the solution z of (1.5) for any 𝜃 ∈ [−1,0] is
𝑧 (𝑡 ) = 𝐶 𝑡 −𝜃 𝜙 (𝜃 ), 𝑡 = 𝜃 , 𝜃 + 1, … , 𝜃 + 𝑘 , ….
Therefore, we need get the eigenvalues and eigenvectors of the matrix C. The root of the
characteristic equation
(1.6) 𝜌 2
− 𝐴𝜌 − 𝐵 = 0
are the eigenvalues of C.
For c is a nonzero constant, the nontrivial solutions of Equation (1.4) are the form 𝑥 (𝑡 ) =
𝑝 𝑡 𝑐 , where p is a solution of the characteristic equation (1.6).
It seems that Equation (1.4) is no more complicated than Equation (1.1), but if we consider
the equation
(1.7) 𝑥 (𝑡 ) = 𝐴𝑥 (𝑡 − 𝑟 ) + 𝐵𝑥 (𝑡 − 𝑠 ),
where 𝑟 ∕ 𝑠 is irrational, 𝑠 > 𝑟 > 0 . For 𝑡 ≥ 0 , Equation (1.7) of a reasonable initial-value
problem is to specify an initial function on [−𝑠 , 0], then use Equation (1.7) to determine a solution.
According to the previous method, by the form 𝑥 (𝑡 ) = 𝑝 𝑡 𝑐 , where 𝑐 ≠ 0 is constant, the
new form of Equation (1.7) is
(1.8) 𝜌 𝑠 − 𝐴 𝜌 𝑠 −𝑟 − 𝐵 = 0.
This equation for 𝑟 ∕ 𝑠 irrational has an infinite number of solutions, so it is not obvious that
by the linear combinations of the characteristic functions can get the solutions of Equation (1.7).
However, it is obvious for Equation (1.8). Therefore, one method is through the Laplace transform.
3

Equation (1.7) would be the equation
𝑥 (𝑡 ) = ∫ 𝑑 [𝜇 (𝜃 )]𝑥 (𝑡 + 𝜃 )
0
−∞
, 𝜇 (𝜃 ) = {
0 , 𝜃 < −𝑠 𝐵 , −𝑠 ≤ 𝜃 < −𝑟 𝐴 + 𝐵 , 𝑟 ≤ 𝜃 < 0

where μ is a function of bounded variation.

4

Chapter 2: Retarded differential difference equations
The simplest linear retarded differential difference equation has the form
(2.1) 𝑥 ̇(𝑡 ) = 𝐴𝑥 (𝑡 ) + 𝐵𝑥 (𝑡 − 𝑟 ) + 𝑓 (𝑡 )
Theorem 2.1. If x is a solution of Equation (2.1) which coincides with 𝜙 and if 𝜙 is a given
continuous function on [−𝑟 , 0], then there is a unique function 𝑥 (𝜙 , 𝑡 ) defined on (−𝑟 ,∞] that
coincides with 𝜙 on [−𝑟 , 0] and satisfies Equation (2.1) where 𝑥 ̇ represents the right-hand
derivative.
Proof. If x is a solution of Equation (2.1) that coincides with 𝜙 on [−𝑟 , 0], because 𝑥 (𝑡 ) =
𝑒 𝐴𝑡
𝑐 + ∫ 𝑒 𝐴 (𝑡 −𝑠 )
𝐹 (𝑠 )𝑑𝑠 𝑡 0
, where 𝐹 = 𝑓 − 𝐵𝑥 (𝑡 − 𝑟 ), then formula (1.3) deduces x satisfies
(2.2) {
𝑥 (𝑡 ) = 𝜙 (𝑡 ), 𝑡 ∈ [−𝑟 , 0]
𝑥 (𝑡 ) = 𝑒 𝐴𝑡
𝜙 (0) + ∫ 𝑒 𝐴 (𝑡 −𝑠 )
[𝐵𝑥 (𝑠 − 𝑟 ) + 𝑓 (𝑠 )]𝑑𝑠 , 𝑡 ≥ 0.
𝑡 0

Besides, x satisfies Equation (2.1), so x is unique, and x is continuous is obvious from Equation
(2.2). □
Theorem 2.2. If 𝑥 (𝜙 , 𝑓 ) is the solution of Equation (2.1) defined by Theorem 2.1, then the
following assertions are valid.
(i) 𝑥 (𝜙 , 𝑓 )(𝑡 ) has a continuous first derivative for all 𝑡 > 0 and has a continuous derivative at
𝑡 = 0 if and only if 𝜙 (𝜃 ) has a derivative at 𝜃 = 0 with
(2.3) 𝜙 ̇ (0) = 𝐴𝜙 (0) + 𝐵𝜙 (−𝑟 ) + 𝑓 (0).
If f has derivatives of all orders, then 𝑥 (𝜙 , 𝑓 ) becomes smoother with increasing values of t.
(ii) If 𝐵 ≠ 0, then 𝑥 (𝜙 , 𝑓 ) can be extended as a solution of Equation (2.1) on [−𝑟 −
𝜖 , ∞), 0 < 𝜖 ≤ 𝑟 , if and only if 𝜙 has a continuous first derivative on [−𝜖 , 0] and Equation (2.3)
5

is satisfied. Extension further to the left requires more smoothness of 𝜙 and f and additional
boundary conditions similar to Condition (2.3).
Proof. Part(i) and the necessity of Part (ii) are obvious, so the sufficiency of part (ii) needs to be
proved.
The expansion to the left of – r can be accomplished by using the formula
(2.4) 𝑥 (𝑡 − 𝑟 ) =
1
𝐵 [𝑥 ̇(𝑡 ) − 𝐴𝑥 (𝑡 ) − 𝑓 (𝑡 )].

If 𝜙 satisfies the stated conditions, then the right-hand side is known explicitly for 𝑡 ∈ [−𝜖 , 0],
therefore, 𝑥 (𝑠 ) is known for 𝑠 ∈ [−𝑟 − 𝜖 , −𝑟 ] . If 𝜙 has a first derivative on [−𝑟 , 0] , then
Relation (2.4) defines the solution on [−2𝑟 , −𝑟 ], therefore the solution is extended to [−2𝑟 ,∞).
In order to extend the solution to the interval [−2𝑟 − 𝜖 , ∞], 0 < 𝜖 ≤ 𝑟 , it requires that the
function 𝑥 (𝑠 ) , 𝑠 ∈ [−𝑟 − 𝜖 , −𝑟 ] , defined by oormula (2.4) is continuously differentiable and
satisfies
(2.5) 𝑥 ̇(−𝑟 ) = 𝐴𝑥 (−𝑟 ) + 𝐵𝑥 (−2𝑟 ) + 𝑓 (−𝑟 ).
It requires that the right-hand side of oormula (2.4) must to be continuously differentiable on
[−𝜖 , 0], so the conditions are that it imposes boundary conditions at 0, f needs to be differentiable
on [−𝜖 , 0], and 𝜙 needs to have two continuous derivatives on [−𝜖 , 0]. ourthermore, 𝜙 needs
to satisfy some additional boundary conditions, which can be obtained by oormula (2.4) , oormula
(2,5), and the relation
6

𝜙 ̇ (−𝑟 ) =
1
𝐵 [𝜙 ̈ (0) − 𝐴 𝜙 ̇ (0) − 𝑓 ̇ (0)]. □
So many results of ordinary differential equations are valid for retarded equations.

7

Chapter 3: Exponential estimates of 𝑥 (𝜙 , 𝑓 )
If one know 𝜙 and f, the solution 𝑥 (𝜙 , 𝑓 ) of Equation (2.1) can be estimated by the application
of the Laplace transform and acquire the analog of the variation-of-constants formula.
Consider the Gronwall Inequality, let u and α be real-valued continuous functions on [𝑎 , 𝑏 ],
and 𝛽 ≥ 0 is integrable on [𝑎 , 𝑏 ] with
𝑢 (𝑡 ) ≤ 𝛼 (𝑡 ) + ∫ 𝛽 (𝑠 )𝑢 (𝑠 )𝑑𝑠 𝑡 𝑎 , 𝑎 ≤ 𝑡 ≤ 𝑏 ,
and 𝑅 (𝑡 ) = ∫ 𝛽 (𝑠 )𝑢 (𝑠 )𝑑𝑠 𝑡 𝑎 . Then
𝑑𝑅 𝑑𝑡 = 𝛽𝑢 ≤ 𝛽𝛼 + 𝛽 𝑅
and
𝑑 𝑑𝑠
[𝑅 (𝑠 )𝑒𝑥𝑝 (− ∫ 𝛽 𝑠 𝑎 )] ≤ 𝛽 (𝑠 )𝛼 (𝑠 )𝑒𝑥 𝑝 (− ∫ 𝛽 𝑠 𝑎 ),
so
𝑅 (𝑡 ) ≤ ∫ 𝛽 (𝑠 )𝛼 (𝑠 )𝑒𝑥𝑝 (∫ 𝛽 𝑠 𝑎 )𝑑𝑠 𝑡 𝑎 .
Then
𝑢 (𝑡 ) ≤ 𝛼 (𝑡 ) + ∫ 𝛽 (𝑠 )𝛼 (𝑠 )[𝑒𝑥𝑝 ∫ 𝛽 (𝜏 )𝑑𝜏 𝑡 𝑠 ] 𝑑𝑠 𝑡 𝑎 , 𝑎 ≤ 𝑡 ≤ 𝑏 .
If, in addition, α is nondecreasing, then
𝑢 (𝑡 ) ≤ 𝛼 (𝑡 )𝑒𝑥𝑝 ∫ 𝛽 (𝑠 )𝑑𝑠 𝑡 𝑎 , 𝑎 ≤ 𝑡 ≤ 𝑏 .
Theorem 3.1. Suppose 𝑥 (𝜙 , 𝑓 ) is the solution of Equation (2.1) defined by Theorem 2.1. Then
there are positive constants a and b such that
(3.1) |𝑥 (𝜙 , 𝑓 )(𝑡 )| ≤ 𝑎 𝑒 𝑏𝑡
(|𝜙 | + ∫
|𝑓 (𝑠 )|𝑑𝑠 𝑡 0
) , 𝑡 ≥ 0
8

where |𝜙 | = 𝑠𝑢𝑝 −𝑟 ≤ 𝜃 ≤ 0
|𝜙 (𝜃 )|.
Proof. oor 𝑡 ≥ 0, 𝑥 = 𝑥 (𝜙 , 𝑡 ) satisfies Equation (2.1), then
{
𝑥 (𝑡 ) = 𝜙 (0) + ∫
[𝐴𝑥 (𝑠 ) + 𝐵𝑥 (𝑠 − 𝑟 ) + 𝑓 (𝑠 )]𝑑𝑠 𝑡 0
, 𝑡 ≥ 0
𝑥 (𝑡 ) = 𝜙 (0), − 𝑟 ≤ 𝑡 ≤ 0
.
Thus,
|𝑥 (𝑡 )| ≤ |𝜙 | + ∫ |𝑓 (𝑠 )|𝑑𝑠 + ∫ |𝐴 ||𝑥 (𝑠 )|𝑑𝑠 + ∫ |𝐵 ||𝑥 (𝑠 )|𝑑 𝑠 𝑡 −𝑟 𝑡 0
𝑡 0

≤ (1 + |𝐵 |𝑟 )|𝜙 | + ∫
|𝑓 (𝑠 )|𝑑𝑠 𝑡 0
+ ∫
(|𝐴 | + |𝐵 |)|𝑥 (𝑠 )|𝑑𝑠 𝑡 0

≤ (1 + |𝐵 |𝑟 )|𝜙 | + ∫
|𝑓 (𝑠 )|𝑑𝑠 𝑡 0
𝑒𝑥𝑝 (|𝐴 | + |𝐵 |)𝑡 .
Then one can deduce the Inequality (3.1) with 𝑎 = 1 + |𝐵 |𝑟 and 𝑏 = (|𝐴 | + |𝐵 |). □
Equation (2.1) is linear and solutions are uniquely defined by 𝜙 , so for any continuous
functions 𝜙 and 𝜓 on [−𝑟 , 0] and any scalar 𝑎 , the solution 𝑥 (𝜙 , 0) of the homogeneous
equation
𝑥 ̇(𝑡 ) = 𝐴𝑥 (𝑡 ) + 𝐵𝑥 (𝑡 − 𝑟 )
satisfies both 𝑥 (𝜙 + 𝜓 , 0) = 𝑥 (𝜙 , 0) + 𝑥 (𝜓 , 0) and 𝑥 (𝑎𝜙 , 0) = 𝑎𝑥 (𝜙 , 0) . When 𝑓 = 0 in
inequality (3.1), 𝑥 (∙ ,0)(𝑡 ) on [−𝑟 , 0] is a continuous linear functional on the space of
continuous functions.
The solution of Equation (2.1) with zero initial data is 𝑥 (0, 𝜙 ), and 𝑥 (0,∙)(𝑡 ) is a continuous
linear functional on the locally integral functions. Therefore, there exists an integral for 𝑥 (0, 𝑓 )(𝑡 ).
However, for this simple equation, the method of Laplace transform is more convenient.

9

Chapter 4: The characteristic equation
Given a constant c, if one knows the nontrivial solutions of the form 𝑒 𝜆𝑡
𝑐 for a homogeneous
linear differential difference equation with constant coefficients, its characteristic equation can be
calculated.
For example, the scalar equation
(4.1) 𝑥 ̇(𝑡 ) = 𝐴𝑥 (𝑡 ) + 𝐵𝑥 (𝑡 − 𝑟 )
has a nontrivial solution 𝑒 𝜆𝑡
𝑐 if and only if
(4.2) ℎ(𝜆 ) ≝ 𝜆 − 𝐴 − 𝐵 𝑒 −𝜆𝑟
= 0.
For any solution 𝜆 of Equation (4.2), |𝜆 − 𝐴 | = |𝐵 |𝑒 −𝑟𝑅𝑒𝜆 , so if |𝜆 | → ∞ , then
𝑒𝑥𝑝 (−𝑟𝑅𝑒𝜆 ) → ∞. Therefore, if there is a sequence { 𝜆 𝑗 } of solutions of Equation (4.2) such that
|𝜆 𝑗 | → ∞ as 𝑗 → ∞, then 𝑅𝑒 𝜆 𝑗 → −∞ as 𝑗 → ∞.
Because ℎ(𝜆 ) is an entire function, in any compact set, there is only a finite number of zeros
of ℎ(𝜆 ). Thus, there is a real number α such that all solutions of Equation (4.2) satisfy Re𝜆 < 𝛼
and there are only a finite number of solutions in any vertical strip in the complex plane.
Theorem 4.1. Suppose 𝜆 is a root of multiplicity m of characteristic equation (4.2). Then each of
the functions 𝑡 𝑘 𝑒𝑥𝑝𝜆 𝑡 , k=0, 1, 2, … , m-1, is a solution of Equation (4.1). Since Equation (4.1) is
linear, any finite sum of such solution is also a solution. Infinite sums are also solutions under
suitable conditions to ensure convergence.
Proof. If 𝑥 (𝑡 ) = 𝑡 𝑘 𝑒 𝜆𝑡
, then
𝑒 𝜆𝑡
[𝑥 ̇ (𝑡 ) − 𝐴𝑥 (𝑡 ) + 𝐵𝑥 (𝑡 − 𝑟 )] = 𝑡 𝑘 𝜆 + 𝑘 𝑡 𝑘 −1
− 𝐴 𝑡 𝑘 − 𝐵 (𝑡 − 𝑟 )
𝑘 𝑒 −𝜆𝑟

10

= ∑ (
𝑘 𝑗 ) 𝑡 𝑘 −𝑗 ℎ
(𝑗 )
(𝜆 )
𝑘 𝑗 =0
.
If 𝜆 is a zero of ℎ(𝜆 ) of multiplicity m, then
ℎ(𝜆 ) = ℎ
(1)
(𝜆 ) = ⋯ = ℎ
(𝑚 −1)
(𝜆 ) = 0.
Thus, for 𝑘 = 0, 1, 2, … , 𝑚 − 1, one solution of Equation (4.1) is 𝑥 (𝑡 ) = 𝑡 𝑘 𝑒 𝜆𝑡
□

11

Chapter 5: The fundamental solution
To find a solution of the homogeneous equation (4.1), the characteristic equation (4.2) arises
naturally as does the need for a function whose Laplace transform is ℎ
−1
(𝜆 ).
Let 𝑋 (𝑡 ) be the solution that satisfies Equation (4.1), and satisfies
(5.1) 𝑋 (𝑡 ) = {
0, 𝑡 < 0,
1, 𝑡 = 0.

Our basic existence theorem does not apply to the initial data (5.1), but existence can be proved.
𝑋 (𝑡 ) is bounded variation. To apply the Laplace transform to 𝑋 (𝑡 ), it needs the “Existence
and Convolution of Laplace Transform” and the “Inversion Theorem".
Existence and Convolution of Laplace Transform: if 𝑓 : [0, ∞) → 𝑅 is integrable on every
compact interval and satisfies
|𝑓 (𝑡 )| ≤ 𝑎 𝑒 𝑏𝑡
, 𝑡 ∈ [0, ∞),
for some constants a and b, then the Laplace transform ℒ(f) defined by
ℒ(𝑓 )(𝜆 ) = ∫ 𝑒 −𝜆𝑡
𝑓 (𝑡 )𝑑𝑡 ∞
0

exists and is an analytic function of 𝜆 for 𝑅𝑒𝜆 > 𝑏 . If the function 𝑓 ∗ 𝑔 is defined by 𝑓 ∗
𝑔 (𝑡 ) = ∫ 𝑓 (𝑡 − 𝑠 )𝑔 (𝑠 )𝑑𝑠 𝑡 0
, then ℒ(𝑓 ∗ 𝑔 ) = ℒ(𝑓 )ℒ(𝑔 ).
The following notation is used
∫ = 𝑙𝑖𝑚 𝑇 →∞
1
2𝜋𝑖
∫
𝑐 +𝑖𝑇
𝑐 −𝑖𝑇 (𝑐 )
,
where c is a real number.
Inversion Theorem: suppose 𝑓 : [0, ∞) → 𝑅 is a given function, 𝑏 > 0 is a given constant
such that f is of bounded variation on any compact set, and 𝑡 ↦ 𝑓 (𝑡 )𝑒𝑥𝑝 (−𝑏𝑡 ) is Lebesgue
12

integrable on [0, ∞). Then, for any 𝑐 > 𝑏 ,
(5.2) ∫ ℒ(𝑓 )(𝜆 )𝑒 𝜆𝑡
𝑑𝜆 =
(𝑐 )
{
1
2
[𝑓 (𝑡 +) + 𝑓 (𝑡 −)], 𝑡 > 0
1
2
𝑓 (0 +), 𝑡 = 0

Theorem 5.1. The solution 𝑋 (𝑡 ) of Equation (4.1) with initial data (5.1) is the fundamental
solution; that is,
(5.3) ℒ(𝑋 )(𝜆 ) = ℎ
−1
(𝜆 ).
Also, for any 𝑐 > 𝑏 ,
(5.4) 𝑋 (𝑡 ) = ∫ 𝑒 𝜆𝑡
ℎ
−1
(𝜆 )𝑑𝜆 (𝑐 )
, 𝑡 > 0
where b is the exponent associated with the bound on 𝑋 (𝑡 ) in Theorem 3.1,
|𝑋 (𝑡 )| ≤ 𝑎 𝑒 𝑏𝑡
, 𝑡 ≥ 0.
Proof. 𝑋 (𝑡 ) satisfies the exponential bounds in Theorem 3.1, so ℒ(𝑋 ) exists. Let both sides of
Equation (4.1) be multiplied by 𝑒 −𝜆𝑡
, and integrate from 0 to ∞ to get the first part we get
Equation (5.3). Then 𝑋 (𝑡 ) is bounded and continuous for 𝑡 ≥ 0, so relation (5.4) follows from
the inversion formula (5.2). □
The next goal is to obtain a very precise exponential bound on 𝑋 (𝑡 ), based on the maximum
value of the real part of the solutions to the characteristic equation.
Theorem 5.2. If 𝛼 0
= 𝑚𝑎𝑥 { 𝑅𝑒𝜆 ∶ ℎ(𝜆 ) = 0}, then, for any 𝛼 > 𝛼 0
, there is a constant
𝑘 = 𝑘 (𝛼 ) such that the fundamental solution 𝑋 (𝑡 ) satisfies the inequality
|𝑋 (𝑡 )| ≤ 𝑘 𝑒 𝛼𝑡
, 𝑡 ≥ 0.
Proof. oirst of all, we need to prove
(5.5) 𝑋 (𝑡 ) = ∫ 𝑒 𝜆𝑡
ℎ
−1
(𝜆 )𝑑𝜆 𝛼 .
13

Assume there is a large real number c such that 𝑐 > 𝛼 , then from Theorem 5.1,
we get 𝑋 (𝑡 ) = ∫ 𝑒 𝜆𝑡
ℎ
−1
(𝜆 )𝑑𝜆 𝑐 . Integrate the function 𝑒 𝜆𝑡
ℎ
−1
(𝜆 ) around the boundary
of the box 𝛤 in the plane with boundary 𝐿 1
𝑀 1
𝐿 2
𝑀 2
, where 𝐿 1
is set { 𝑐 + 𝑖𝜏 : − 𝑇 ≤
𝜏 ≤ 𝑇 } , 𝐿 2
is set { 𝛼 + 𝑖𝜏 : − 𝑇 ≤ 𝜏 ≤ 𝑇 } , 𝑀 1
is set { 𝜎 + 𝑖𝑇 : 𝛼 ≤ 𝜎 ≤ 𝑐 } , 𝑀 2
is set
{ 𝜎 − 𝑖𝑇 : 𝛼 ≤ 𝜎 ≤ 𝑐 }.
Therefore, the function (5.5) follows if we let 𝑇 → ∞,
∫ 𝑒 𝜆𝑡
ℎ
−1
(𝜆 )𝑑𝜆 → 0
𝑀 1
, ∫ 𝑒 𝜆𝑡
ℎ
−1
(𝜆 )𝑑𝜆 → 0
𝑀 2
.
Take 𝑇 0
for all 𝑇 ≥ 𝑇 0
, then
(1 +
𝛼 2
𝑇 2
)
1/2
−
1
𝑇 (|𝐴 | + |𝐵 |𝑒 −𝛼𝑟
) ≥
1
2
.
Besides, if 𝜆 ∈ 𝑀 1
, then
|ℎ
−1
(𝜆 )| ≤
1
(𝜎 2
+𝑇 2
)
1/2
−|𝐴 |−|𝐵 |𝑒 𝜎𝑟
≤
2
𝑇 ,
so we can get
∫ 𝑒 𝜆𝑡
ℎ
−1
(𝜆 )𝑑𝜆 ≤
2
𝑇 𝑒 𝑐𝑡
(𝑐 − 𝛼 ) → 0
𝑀 1
as 𝑇 → ∞.
In the same way, we can prove 𝑀 2
→ 0 as 𝑇 → ∞, so Equation (5.5) is proved.
oor all 𝑇 ≥ 𝑇 0
and 𝜆 = 𝛼 + 𝑖𝑇 , if 𝑔 (𝜆 ) = ℎ
−1
(𝜆 ) − (𝜆 − 𝛼 0
)
−1
, then
𝑔 (𝜆 ) = |
𝐴 +𝐵 𝑒 −𝜆𝑟
−𝛼 0
𝜆 −𝛼 0
ℎ
−1
(𝜆 )| ≤
2
𝑇 2
(|𝐴 | + |𝐵 |𝑒 −𝛼𝑟
+ |𝛼 0
|),
so let 𝐾 1
, 𝐾 2
be constant,
∫
|𝑔 (𝜆 )|
(𝛼 )
𝑑𝜆 < ∞ and |∫ 𝑒 𝜆𝑡
𝑔 (𝜆 )𝑑𝜆 (𝛼 )
| ≤ 𝐾 1
𝑒 𝛼𝑡
, 𝑡 > 0.
Therefore,
|∫ 𝑒 𝜆𝑡
(𝜆 − 𝛼 0
)
−1
𝑑𝜆 (𝛼 )
| ≤ 𝐾 2
𝑒 𝛼𝑡
, 𝑡 > 0.
Based on the Equation (5.5), let 𝑘 = 𝐾 1
+ 𝐾 2
, the theorem is proved. □
14

We proceed as follows to accomplish. The characteristic equation has no roots in the region
{ 𝜆 ∈ 𝐶 : |𝜆 − 𝐴 | > |𝐵 |𝑒 −𝑅𝑒𝜆𝑟 }, so
(5.6) (𝛼 − 𝐴 )
2
+ 𝑇 2
> 𝐵 2
𝑒 −2𝛼𝑟
.
Then, along the segments 𝑀 1
, 𝑀 2
, Estimate (5.6) holds, so
|∫ ℎ
−1
(𝜆 )𝑒 𝜆𝑡
𝑑𝜆 𝑀 1
| ≤
2
𝑇 𝑒 𝑐𝑡
(𝑐 − 𝛼 ) → 0 𝑎𝑠 𝑇 → ∞.
In the same way, the integral over𝑀 2
→ 0 as 𝑇 → ∞.
Let 𝑚 = 1, 2, …, and define a sequence 𝛼 𝑚 so that the line Re𝜆 = 𝛼 𝑚 does not contain a
root of the characteristic equation. Than the Cauchy theorem of residues shows
𝑋 (𝑡 ) = ∫ 𝑒 𝜆𝑡
ℎ
−1
(𝜆 )𝑑𝜆 (𝑐 )
= ∫ 𝑒 𝜆𝑡
ℎ
−1
(𝜆 )𝑑𝜆 + ∑ 𝑅 𝑒𝑠
𝜆 =𝜆 𝑗 𝑘 𝑚 𝑗 =1
(𝛼 𝑚 )
𝑒 𝜆𝑡
ℎ
−1
(𝜆 ),
where 𝜆 1
, 𝜆 2
, … , 𝜆 𝑘 𝑚 are the roots of the characteristic equation such that Re𝜆 > 𝛼 𝑚 , so
Re𝑠 𝜆 = 𝜆 𝑗 𝑒 𝜆𝑡
ℎ
−1
(𝜆 ) = 𝑒 𝜆 𝑗 𝑡 𝑃 𝑗 (𝑡 )

and that 𝑒 𝜆 𝑗 𝑡 𝑃 𝑗 (𝑡 ) is a solution of Equation (4.1) with initial function
𝜙 𝑗 (𝜃 ) = 𝑃 𝑗 (𝜃 )𝑒 𝜆 𝑗 𝜃 , −𝑟 ≤ 𝜃 ≤ 0, 𝑗 = 1, 2, …,
so
𝑌 𝛼 (𝑡 ) = ∫ 𝑒 𝜆𝑡
ℎ
−1
(𝜆 )𝑑𝜆 (𝛼 )

is a solution of Equation (4.1). In addition,
|𝑌 𝛼 𝑚 (𝑡 )| ≤ 𝐾 𝑒 𝛼 𝑚 𝑡 .
As 𝛼 → −∞, one can get the solution
𝑌 0
(𝑡 ) = 𝑙𝑖𝑚 𝑚 →∞
𝑌 𝛼 𝑚 (𝑡 ), 𝑡 ≥ 0.
15

In general, 𝑌 0
(𝑡 ) ≠ 0

16

Chapter 6: The variation-of-constants formula
Based on the fundamental solution 𝑋 (𝑡 ), we can get the solution 𝑥 (𝜙 , 𝑓 ) of the nonhomogeneous
equation (2.1) is given by the variation-of-constants formula
(6.1) x(𝜙 , 𝑓 )(𝑡 ) = x(𝜙 , 𝑓 )(𝑡 ) + ∫ 𝑋 (𝑡 − 𝑠 )𝑓 (𝑠 )𝑑𝑠 𝑡 0
, 𝑡 ≥ 0.
Furthermore, using the fundamental solution 𝑋 (𝑡 ), one can obtain the solution 𝑥 (𝜙 , 0) of
the homogeneous equation. Then
Theorem 6.1. The solution 𝑥 (𝜙 , 𝑓 ) of the nonhomogenous equation (2.1) can be represented in
the form (6.1). ourthermore, for 𝑡 ≥ 0,
𝑥 (𝜙 , 0)(𝑡 ) = 𝑋 (𝑡 )𝜙 (0) + 𝐵 ∫ 𝑋 (𝑡 − 𝜃 − 𝑟 )𝜙 (𝜃 )𝑑𝜃 0
−𝑟 .
One of the implications for Theorem 6.1 is that the characteristic equation (4.2)
determines the exponential behavior of the solution of the homogeneous equation
(4.1).
Theorem 6.2. Suppose 𝛼 0
= 𝑚𝑎 𝑥 { 𝑅𝑒𝜆 : ℎ(𝜆 ) = 0} and 𝑥 (𝜙 ) is the solution of the homogeneous
equation (4.1), which coincides with 𝜙 on [−𝑟 , 0]. Then, for any 𝛼 > 𝛼 0
, there os a constant
𝐾 = 𝐾 (𝛼 ) such that
|𝑥 (𝜙 )(𝑡 )| ≤ 𝐾 𝑒 𝛼𝑡
|𝜙 |, 𝑡 ≥ 0, |𝜙 | = 𝑠𝑢𝑝 −𝑟 ≤ 𝜃 ≤ 0
|𝜙 (𝜃 )|.
In particular, if 𝛼 0
< 0, then one can choose 𝛼 0
< 𝛼 < 0 to obtain the fact that all solutions
approach zero exponentially as 𝑡 → ∞.
The asymptotic behavior of perturbed nonlinear systems of the form
(6.2) 𝑥 ̇(𝑡 ) = 𝐴𝑥 (𝑡 ) + 𝐵𝑥 (𝑡 − 𝑟 ) + 𝑓 (𝑥 (𝑡 ), 𝑥 (𝑡 − 𝑟 ))
can be determined by use of the variation-of-constants formula as follows.
17

For 𝑦 = 𝑦 (𝜙 ) and X is the solution and the fundamental solution of the homogeneous
equation (4.1), respectively, the solution 𝑥 = 𝑥 (𝜙 ) of Equation (6.2) with initial data 𝜙 on
[−𝑟 , 0] is given by
𝑥 (𝑡 ) = 𝑦 (𝑡 ) + ∫ 𝑋 (𝑡 − 𝑠 )𝑓 (𝑥 (𝑠 ), 𝑥 (𝑠 − 𝑟 ))𝑑𝑠 𝑡 0
.
The analogue of the theorem on stability with respect to the first approximation can be proved
if 𝑓 (𝑥 , 𝑦 ) satisfies 𝑓 (0,0) = 0,
𝜕𝑓 (0,0)
𝜕 (𝑥 ,𝑦 )
= 0 and 𝛼 0
< 0 in Theorem 6.2. In a similar
way, we can also get other perturbations.

18

Chapter 7: Neutral differential difference equations
Neutral differential difference equations depends on past and present values, and involve delay
derivatives and the function itself.
The model of nonhomogeneous equation is
(7.1) 𝑥 ̇(𝑡 ) − 𝐶 𝑥 ̇ (𝑡 − 𝑟 ) = 𝐴𝑥 (𝑡 ) + 𝐵𝑥 (𝑡 − 𝑟 ) + 𝑓 (𝑡 ),
where A, B, C, and r are constants with 𝑟 > 0, 𝐶 ≠ 0 and f is a continuous function on R. The
corresponding homogeneous equation is
𝑥 ̇(𝑡 ) − 𝐶 𝑥 ̇ (𝑡 − 𝑟 ) = 𝐴𝑥 (𝑡 ) + 𝐵𝑥 (𝑡 − 𝑟 ).
For Equation (7.1) with increasing t, if 𝐶 ≠ 0, the smoothing of the solution does not occur.
For Equation (7.1), t ∈ [−𝑟 , 0], suppose 𝜙 is a given continuously differentiable function on
[−𝑟 , 0] such that we have a solution 𝑥 = 𝑥 (𝜙 , 𝑓 ) that is a continuous function of x with 𝑥 (𝑡 ) =
𝜙 (𝑡 ) except at the points 𝑘𝑟 , 𝑘 = 0, 1, 2, … .There always exists a solution of Equation (7.1)
through 𝜙 . Because 𝐶 ≠ 0, the solution does not gain more derivatives than the initial function
𝜙 has. Besides, the solution 𝑥 = 𝑥 (𝜙 , 𝑓 ) has a discontinuous derivative at 𝑘𝑟 . Therefore, the
Equation (7.1) can be written
(7.2) 𝜙 ̇ (0) = 𝐶 𝜙 ̇ (−𝑟 ) + 𝐴𝜙 (0) + 𝐵𝜙 (−𝑟 ) + 𝑓 (0).
If the solution 𝑥 (𝑡 ) has a discontinuous derivative at 𝑡 = 0, than 𝑥 (𝑡 ) is discontinuous at
𝑡 = 𝑟 since 𝐶 ≠ 0.
If the solution 𝑥 (𝑡 ) has a continuous derivative at 𝑡 = 0. Equation (7.1) shows that 𝑥 (𝑡 )
has a continuous derivative for all 𝑡 ≥ −𝑟 .
Since 𝐶 ≠ 0, we can also get a unique solution of Equation (7.1) on (−∞, 𝑟 ]. If 𝑦 (𝑡 ) =
19

𝑥 (𝑡 − 𝑟 ), then y satisfies the differential equation
𝑦 ̇ (𝑡 ) = −
𝐵 𝐶 𝑦 (𝑡 ) −
𝐴 𝐶 𝑥 (𝑡 ) +
1
𝐶 𝑥 ̇(𝑡 ) −
1
𝐶 𝑓 (𝑡 ).
Theorem 7.1. If 𝐶 ≠ 0 and 𝜙 is a continuously differentiable function on [−𝑟 , 0], then there
exists a unique function 𝑥 : (−∞, ∞) → 𝐼𝑅 that coincides with 𝜙 on [−𝑟 , 0] , is continuously
differentiable and satisfies Equation (7.1) except maybe at the points 𝑘𝑟 , 𝑘 = 0, ±1, ±2, …. This
solution x can have no more derivatives than 𝜙 and is continuously differentiable if and only if
relation (7.2) is satisfied.
For Equation (7.1), it is very difficult if there are many delays. Furthermore, for more general
functional equations are hardly to describe the exceptional set where the derivative is not required
to exist. One method to overcome this is to rewrite Equation (7.1) as
(7.3)
𝑑 𝑑𝑡
[𝑥 (𝑡 ) − 𝐶𝑥 (𝑡 − 𝑟 )] = 𝐴𝑥 (𝑡 ) + 𝐵𝑥 (𝑡 − 𝑟 ) + 𝑓 (𝑡 ).
It is meaningful to consider the initial-value problem. Suppose 𝜙 is a continuous function on
[−𝑟 , 0], the solution of Equation (7.3) through 𝜙 is a continuous function on [−𝑟 ,∞). Therefore,
for 𝑡 ≥ 0, the difference for 𝑥 (𝑡 ) − 𝐶𝑥 (𝑡 − 𝑟 ) is differentiable and satisfies Equation (7.3) with
𝜙 .
Let 𝑥 (𝑡 ) = e
𝐴𝑡
𝑦 (𝑡 ), so y satisfies the equation
𝑑 𝑑𝑡
[𝑦 (𝑡 ) − 𝑒 −𝐴𝑟
𝐶𝑦 (𝑡 − 𝑟 )] = 𝑒 −𝐴𝑟
(𝐴 𝐶 + 𝐵 )𝑦 (𝑡 − 𝑟 ) + 𝑒 −𝐴𝑡
𝑓 (𝑡 ).
Theorem 7.2. If 𝜙 is a continuous function on [−𝑟 , 0], then there is a unique solution of Equation
(7.3) on [−𝑟 ,∞) through 𝜙 . If 𝐶 ≠ 0, this solution exists on (−∞, ∞) and is unique.
Equation (7.3) can be considered as a generalization of the retarded equations(𝐶 = 0) and a
generalization of difference equations[𝐴 = 𝐵 = 𝑓 = 0, 𝜙 (0) = 𝐶𝜙 (−𝑟 )].
20

We will consider only Equation (7.3) with continuous initial data. The corresponding
homogeneous equation is
(7.4)
𝑑 𝑑𝑡
[𝑥 (𝑡 ) − 𝐶𝑥 (𝑡 − 𝑟 )] = 𝐴𝑥 (𝑡 ) + 𝐵𝑥 (𝑡 − 𝑟 ).
As the theory evolved, retarded equations are very similar to ordinary differential equations
and parabolic partial differential equations, and neutral equations are very similar to difference
equations and hyperbolic partial differential equations.
Theorem 7.3. Let 𝑥 (𝜙 , 𝑓 ) be the solution of Equation (7.3) given in Theorem 7.2. Then there are
positive constants a and b such that
|𝑥 (𝜙 , 𝑓 )(𝑡 )| ≤ 𝑎 𝑒 𝑏𝑡
[|𝜙 | + ∫
|𝑓 (𝑠 )|𝑑𝑠 𝑡 0
] , 𝑡 ≥ 0,
where |𝜙 | = 𝑠𝑢𝑝 −𝑟 ≤ 𝜃 ≤ 0
|𝜙 (𝜃 )|.
Proof. If x is a solution of Equation (7.3), then
𝑥 (𝑡 ) = 𝜙 (0) − 𝐶𝜙 (−𝑟 ) + 𝐶𝑥 (𝑡 − 𝑟 ) + ∫ [𝐴𝑥 (𝑠 ) + 𝐵𝑥 (𝑠 − 𝑟 )]𝑑𝑠 + ∫ 𝑓 (𝑠 )𝑑𝑠 𝑡 0
𝑡 0

for 𝑡 ≥ 0. Let 𝛼 = (1 + 2𝐶 ) and 𝛽 = |𝐴 | + |𝐵 |, then
𝑥 (𝑡 ) ≤ 𝛼 (𝜙 ) + ∫ 𝑓 (𝑠 )𝑑𝑠 + 𝛽 ∫ 𝑦 (𝑠 )𝑑𝑠 𝑡 0
𝑡 0

if |𝜙 | = 𝑠𝑢𝑝 −𝑟 ≤ 𝜃 ≤ 0
|𝜙 (𝜃 )|, y(𝑡 ) = 𝑠𝑢𝑝 −𝑟 ≤ 𝜃 ≤ 0
|𝑥 (𝑡 + 𝜃 )| for 0 ≤ 𝑡 ≤
𝑟 2
.
Obviously, 𝛼 ≥ 0 and 𝑥 (𝑡 ) = 𝜙 (𝑡 ) for 𝑡 ≤ 0, so
y(𝑡 ) ≤ 𝛼 |𝜙 | + ∫ 𝑓 (𝑠 )𝑑𝑠 + 𝛽 ∫ 𝑦 (𝑠 )𝑑𝑠 𝑡 0
𝑡 0

for 0 ≤ 𝑡 ≤
𝑟 2
.
According Lemma 3.1, we get
21

𝑦 (𝑡 ) ≤ [𝛼 |𝜙 | + ∫ 𝑓 (𝑠 )𝑑𝑠 𝑡 0
] 𝑒𝑥𝑝 (𝛽𝑡 )
for 0 ≤ 𝑡 ≤
𝑟 2
.
Similarly,
𝑦 (𝑡 ) ≤ [𝛼𝑦 |𝜏 | + ∫ |𝑓 (𝑠 )|𝑑𝑠 𝑡 𝜏 ] 𝑒𝑥𝑝 [𝛽 (𝑡 − 𝜏 )]
for 0 ≤ 𝜏 ≤ 𝑡 ≤ 𝜏 +
𝑟 2
.
Take γ > β such that αe
(𝛽 −𝛾 )r/2
< 1, so we need to prove that
𝑦 (𝑡 ) ≤ [𝛼 |𝜙 | + ∫ 𝑓 (𝑠 )𝑑𝑠 𝑡 0
] 𝑒𝑥𝑝 (𝛾 𝑡 ), 0 ≤ 𝑡 ≤
𝑘𝑟
2
,
where k is an integer.
oor 𝑘 = 1, it is obviously true.
Assume that it is true for 𝑘 ≥ 1, then
𝑦 (𝑡 ) ≤ [𝛼𝑦 (𝑡 −
𝑟 2
) + ∫ 𝑓 (𝑠 )𝑑𝑠 𝑡 𝑡 −𝑟 /2
] 𝑒𝑥𝑝 𝛽𝑟
2

if r/2 ≤ t ≤ (𝑘 + 1)𝑟 /2. Therefore,
y(𝑡 ) ≤ [𝛼 { 𝛼 |𝜙 | + ∫ |𝑓 (𝑠 )|𝑑𝑠
𝑡 −𝑟 /2
0
} 𝑒 𝛾 (𝑡 −𝑟 /2)
+ ∫ |𝑓 (𝑠 )|𝑑𝑠
𝑡 𝑡 −𝑟 /2
] 𝑒 𝛽𝑟 /2

≤ [α|ϕ| + ∫ |f(s)|ds
t−r/2
0
] 𝑒 𝛾𝑡
+ 𝑒 𝛽𝑟 /2
∫ |𝑓 (𝑠 )|𝑑𝑠
𝑡 𝑡 −𝑟 /2
.
Therefore, we have
𝑦 (𝑡 ) ≤ [𝛼 |𝜙 | + ∫
|𝑓 (𝑠 )|𝑑𝑠 𝑡 0
] exp (𝛾𝑡 ),
so the theorem is proved. □
If Equation (7.4) has a solution 𝑒 𝛾𝑡
, then λ must satisfy the characteristic equation
(7.5) 𝐻 (𝜆 ) ≝ 𝜆 (1 − 𝐶 𝑒 −𝜆𝑟
) − 𝐴 − 𝐵 𝑒 −𝜆𝑟
= 0.
22

Lemma 7.1. There is a real number 𝛼 such that all solutions of Equation (7.5) satisfy 𝑅𝑒𝜆 <
𝛼 . If 𝐶 ≠ 0, then all solutions of Equation (7.5) lie in a vertical strip { 𝛽 < 𝑅𝑒𝜆 < 𝛼 } in the
complex plane. If 𝐶 ≠ 0 and there is a sequence { 𝜆 𝑗 } of solutions such that |𝜆 𝑗 | → ∞ as 𝑗 →
∞, then there is a sequence { 𝜆 ′
𝑗 } of zeros of
(7.6) 1 − 𝐶 𝑒 −𝜆𝑟
= 0
such that 𝜆 𝑗 − 𝜆 ′
𝑗 → 0 as 𝑗 → ∞. Also, one can show there always exists such a sequence { 𝜆 𝑗 }
when 𝐶 ≠ 0.
For 𝐶 ≠ 0, the roots of Equation (7.6) are given by
𝜆 =
𝑙𝑛 𝐶 𝑟 +
2𝑘𝜋𝑖 4
, 𝑘 = 0, ±1, ±2, ….
If ln 𝐶 > 0 , then there are many solutions of Equation (7.4) which approach ∞ at an
exponential rate, so Equation (7.4) can never have stability.
Suppose r is a perturbation term of small value of Equation (7.4), then Equation (7.4) can be
written as
(7.7)
𝑑 𝑑𝑡
(1 − 𝐶 )𝑥 (𝑡 ) = (𝐴 + 𝐵 )𝑥 (𝑡 ).
If r is small enough, then for |𝐶 | < 1, all roots of Equation (7.5) satisfy Re𝜆 < −𝛿 < 0 for
some 𝛿 > 0.
For Equation (7.4) with 𝐶 = 0, there are finite number of λ with Re𝜆 > 0.
Corollary 7.1. If there is a 𝛿 > 0 such that every solution of Equation (7.7) satisfies
𝑥 (𝑡 )𝑒𝑥𝑝 𝛿𝑡 → 0 as 𝑡 → ∞, then it is necessary that |𝐶 | < 1. If |𝐶 | < 1 and every solution of
the ordinary differential equation (7.7) approaches zero, then there are 𝛿 > 0 and 𝑟 0
> 0 such
that every solution satisfies 𝑥 (𝑡 )𝑒𝑥𝑝 𝛿𝑡 → 0 as 𝑡 → ∞ for 0 < 𝑟 < 𝑟 0
.
23

The function 𝑋 (𝑡 ) will actually have a continuous first derivative on each interval
(𝑘𝑟 , (𝑘 + 1)𝑟 ), 𝑘 = 0, 1, 2, …, the right- and left-hand limits of 𝑋 (𝑡 ) exist at each of the points
𝑘𝑟 , 𝑘 = 0, 1, 2, …. Therefore, 𝑋 (𝑡 ) is of bounded variation on each compact interval and satisfies
(7.8) 𝑋 ̇ (𝑡 ) − 𝐶𝑋 (𝑡 − 𝑟 ) = 𝐴 𝑋 ̇ (𝑡 ) + 𝐵𝑋 (𝑡 − 𝑟 )
for 𝑡 ≠ 𝑘𝑟 , 𝑘 = 0, 1, 2, …. These assertions are proved easily from the fact that 𝑋 (𝑡 ) satisfies the
integral equation
𝑋 (𝑡 ) = 1 + 𝐶𝑋 (𝑡 − 𝑟 ) + ∫
[𝐴𝑋 (𝑠 ) + 𝐵𝑋 (𝑠 − 𝑟 )]𝑑𝑠 𝑡 0
, 𝑡 ≥ 0,
where 𝑋 (𝑡 ) satisfies the inequality
|𝑋 (𝑡 )| ≤ 𝑎 𝑒 𝑏𝑡
, 𝑡 ∈ 𝐼𝑅 .
It is easy to obtain ℒ(𝑋 ) = 𝐻 −1
(𝜆 ) in the following theorem.
Theorem 7.4. The solution 𝑋 (𝑡 ) of Equation (7.8) with initial data 𝑋 (𝑡 ) = 0, 𝑡 < 0, 𝑋 (0) = 1,
is the fundamental solution of Equation (7.3); that is, ℒ(𝑋 ) = 𝐻 −1
(𝜆 ).
The variation-of-constants formula
(7.9) 𝑥 (𝜙 , 𝑓 )(𝑡 ) = 𝑥 (𝜙 , 0)(𝑡 ) + ∫ 𝑋 (𝑡 − 𝑠 )𝑓 (𝑠 )𝑑𝑠 𝜏 0

can obtain the solution 𝑥 (𝜙 , 𝑓 ) of Equation (7.1).
Based on the fundamental solution X, it derives
𝑥 (𝜙 , 𝑓 )(𝑡 ) = 𝑋 (𝑡 )[𝜙 (0) − 𝐶𝜙 (−𝑟 )] + 𝐵 ∫ 𝑋 (𝑡 − 𝜃 − 𝑟 )𝜙 (𝜃 )𝑑𝜃 0
−𝑟 + 𝐶𝜙 (−𝑟 + 𝑡 )𝜔 (−𝑟 + 𝑡 )
+ 𝐶 ∫ 𝑋 ̇ (𝑡 − 𝜃 − 𝑟 )𝜙 (𝜃 )𝑑𝜃 0
−𝑟
where 𝜃 ≥ 0, 𝜔 (𝜃 ) = 0, otherwise, 𝜔 (𝜃 ) = 1.
If 𝑡 ≥ 𝑟 , the term contains 𝜔 does not exist. If 𝑡 < 𝑟 , this term is 𝐶𝜙 (−𝑟 + 𝑡 ), which is
24

equal to the Stieltjes integral − ∫ 𝑑 [𝑋 (𝑡 − 𝜃 − 𝑟 )]𝜙 (𝜃 )
𝑡 −𝑟 −𝑟 .
Using the Stieltjes integral, we get
(7.10) 𝑥 (𝜙 , 0)(𝑡 ) = 𝑋 (𝑡 )[𝜙 (0) − 𝐶𝜙 (−𝑟 )] + 𝐵 ∫ 𝑋 (𝑡 − 𝜃 − 𝑟 )𝜙 (𝜃 )𝑑𝜃 0
−𝑟
−𝐶 ∫ 𝑑 [𝑋 (𝑡 − 𝜃 − 𝑟 )]𝜙 (𝜃 )
0
−𝑟
Theorem 7.5. If 𝑋 (𝑡 ) is the fundamental solution of Equation (7.4), then the solution 𝑥 (𝜙 , 𝑓 ) of
Equation (7.1) is obtained from the variation-of- constants formula (7.9) with 𝑥 (𝜙 , 0) as given
in oormula (7.10 ).
For (7.9) and (7.10), the solutions of the characteristic equation determines the asymptotic
behavior of the fundamental solution 𝑋 (𝑡 ). For the retarded equation, the Laplace transform of X
was ℎ(𝜆 ), so ℒ(𝑋 ) = 𝐻 −1
(𝜆 ). From Equation (7.4), it can get similar estimates.
Theorem 7.6. If 𝛼 0
= 𝑠𝑢𝑝 { 𝑅𝑒𝜆 : 𝐻 (𝜆 ) = 0}, then, for any 𝛼 > 𝛼 0
, there is a constant 𝑘 =
𝑘 (𝛼 ) such that the fundamental solution X of Equation (7.4) satisfies the inequality
|𝑋 (𝑡 )| ≤ 𝑘 𝑒 𝛼𝑡
, 𝑉𝑎𝑟 [𝑡 =𝑟 ,𝑡 ]
𝑋 ≤ 𝑘 𝑒 𝛼𝑡
, 𝑡 ≥ 0,
where 𝑉𝑎𝑟 [𝑡 =𝑟 ,𝑡 ]
𝑋 denotes the total variation of X on [𝑡 − 𝑟 , 𝑡 ].
Proof. orom Lemma 7.1, we know that αr > ln|C| since 𝛼 > 𝛼 0
, so there is an interval I
containing α such that 1 − Ce
−𝜆𝑟
is bounded away from zero in the strip S =
{ 𝜆 ∈ 𝐶 : 𝑅𝑒𝜆 ∈ 𝐼 }. Therefore, we can conclude
X(𝑡 ) = ∫ 𝑒 𝜆𝑟
𝐻 −1
(𝜆 )𝑑𝜆 (𝛼 )

through the similar proof of Theorem 5.2. To estimate the value of this integral, observe that
1
𝐻 (𝜆 )
=
1
𝜆 (1−𝐶 𝑒 −𝜆𝑟
)
+
𝐴 +𝐵 𝑒 −𝜆𝑟
𝜆 (1−𝐶 𝑒 −𝜆𝑟
)𝐻 (𝜆 )
.
25

On the line Reλ = α, the integral
∫
𝐴 + 𝐵 𝑒 −𝜆𝑟
𝜆 (1 − 𝐶 𝑒 −𝜆𝑟
)𝐻 (𝜆 )
𝑒 𝜆𝑡
(𝛼 )

is convergent since the kernel is like λ
−2
. Hence, this part of inverse Laplace transform of
𝐻 −1
(𝜆 ) has an estimate of the type stated in the theorem.
oor the term involving 𝜆 −1
(1 − 𝐶 𝑒 −𝜆𝑟
) , in the strip S, if 𝜆 = 𝛽 + 𝑖𝜔 , 𝛽 ∈ 𝐼 , (1 −
𝐶 𝑒 −𝜆𝑟
)
−1
is analytic, then this function is periodic of period 2𝜋 /𝑟 . Thus, the function has
absolutely convergent oourier series
(1 − 𝐶 𝑒 −𝜆𝑟
)
−1
= ∑ ℎ
𝑘 𝑒 𝜆𝑘𝑟 ∞
𝑘 =−∞
, 𝜆 ∈ 𝑆
∑ |ℎ
𝑘 |𝑒 𝛽𝑘𝑟 ∞
𝑘 =−∞
< ∞, 𝛽 ∈ 𝐼 .
Therefore,
∫
1
𝜆 (1 − 𝐶 𝑒 −𝜆𝑟
)
𝑒 𝜆𝑡
𝑑𝜆 =
(𝛼 )
∑ ℎ
𝑘 ∫ 𝑒 𝜆 (𝑡 +𝑘𝑟 )
𝜆 −1
𝑑𝜆 (𝛼 )
= ∑ ℎ
𝑘 𝛼 (𝑡 +𝑘𝑟 )>0
∞
𝑘 =−∞

If 𝑡 > 0, 𝑡 + 𝑘𝑟 ≠ 0, so
| ∑ ℎ
𝑘 𝛼 (𝑡 +𝑘𝑟 )>0
| ≤ 𝑒 𝛼𝑡
∑ |ℎ
𝑘 |𝑒 𝛼𝑘𝑟 𝛼 (𝑡 +𝑘𝑟 )>0
≤ 𝑒 𝛼𝑡
∑ |ℎ
𝑘 |𝑒 𝛼𝑘𝑟 𝑘 =−∞
,
and 𝑋 (𝑡 ) has an estimate of the type stated in the theorem.
Next, we need to prove that 𝑋 ̇ (𝑡 ) is bounded.
If y(𝑡 ) = 𝑋 ̇ (𝑡 ), then y(𝑡 ) − 𝐶 y(𝑡 − 𝑟 ) = 𝑝 (𝑡 ), where 𝑝 (𝑡 ) = 𝐴𝑋 (𝑡 ) + 𝐵𝑋 (𝑡 − 𝑟 ) satisfies
|𝑝 (𝑡 )| ≤ 𝐷 𝑒 𝛼𝑡
, where D is a constant. If y(𝑡 ) = 𝑧 (𝑡 )e
𝛼𝑡
, q(𝑡 ) =
26

𝑝 (𝑡 )𝑒 −𝛼𝑡
, |𝑞 (𝑡 )| 𝑖𝑠 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 , then z(𝑡 ) − 𝐶 ′
𝑧 (𝑡 − 𝑟 ) = 𝑞 (𝑡 ), where |𝐶 ′| = |𝐶 𝑒 −𝛼𝑟
| < 1.
Therefore, all solutions of this equation are bounded. o ourthermore,
X(𝑘 𝑟 +
) − X(𝑘 𝑟 −
) = 𝐶 (X((𝑘 − 1)𝑟 +
) − X((𝑘 − 1)𝑟 −
)).
Thus, we know that the jumps in 𝑋 (𝑡 ) at kr are bounded by a constant times e
𝛼𝑡
. The theorem
is proved. □
There is a corollary for the asymptotic behavior of the solution of Equation (7.4).
Corollary 7.2. If 𝛼 0
= 𝑠𝑢𝑝 { 𝑅𝑒𝜆 : 𝜆 (1 − 𝐶 𝑒 −𝜆𝑟
) = 𝐴 + 𝐵 𝑒 −𝜆𝑟
} and 𝑥 (𝜙 ) is the solution of
Equation (7.4), which coincides with 𝜙 on [−𝑟 , 0], then, for any 𝛼 > 𝛼 0
, there is a constant
𝐾 = 𝐾 (𝛼 ) such that
|𝑥 (𝜙 )(𝑡 )| ≤ 𝐾 𝑒 𝛼𝑡
|𝜙 |, 𝑡 ≥ 0, |𝜙 | = 𝑠𝑢𝑝 −𝑟 ≤ 𝜃 ≤ 0
|𝜙 (𝜃 )|.
In particular, if 𝛼 0
< 0, then all solutions of Equation (7.4) approach zero exponentially.
It shows that the Perturbation results are similar to those of the retarded equations.

27

Chapter 8: Stability and periodic solutions of first order delay
equations
For the following initial value problem of first-order linear ordinary differential equation
(8.1) 𝑥 ̇(𝑡 ) = −𝑎𝑥 (𝑡 ), 𝑥 (0) = 1.
Therefore, the solution is 𝑥 (𝑡 ) = 𝑒 −𝑎𝑡
, the figure below as shown
,
so the solution 𝑥 (𝑡 ) of Equation (8.1) converges to the t axis.
Forthermore, for the first-order linear differential difference equation
(8.2) 𝑥 ̇(𝑡 ) = −𝑎𝑥 (𝑡 − 𝑟 ),
where 𝑎 ∈ 𝑅 , 𝑟 ＞0.
If 𝑟 = 0, the differential difference equation (8.2) and the ordinary differential equation (8.1)
are the same equations.
If 𝑟 ≠ 0, we choose the initial time 𝑡 0
= 0 and the initial function is 𝜑 (𝑡 ) = 1. We first take
𝑎 = 2. If 𝑟 = 1, the figure below as shown
28

We see that when 𝑟 = 0, the solution 𝑥 (𝑡 ) of the equation converges to the t axis; When
𝑟 = 1, the solution 𝑥 (𝑡 ) of the equation diverges around t axis. In order to explore whether there
is a point that can make 𝑥 (𝑡 ) vibrate with a certain amplitude, we take the value of r in interval
[0,1]. Let R be 0.1, 0.2, 0.3,..., 0.8 and 0.9 respectively. The figures are shown in Appendix A.
Therefore, it takes 𝑎 = 2, we can roughly draw the following conclusions: As the value of r
increases, the vibration of the solution 𝑥 (𝑡 ) increases and the amplitude increases gradually.
When 𝑟 < 0.2, the solution of Equation (8.2) monotonically approaches 0. When 0.2 ≤ 𝑟 < 0.8,
the solution oscillates close to 0. When 𝑟 ≈ 0.8 the solution vibrates with a certain amplitude.
When 𝑟 > 0.8, the solution tends to diverge.
Therefore when 𝑟 ≈ 0.8, the curve of solution 𝑥 (𝑡 ) is similar to cosine curve.
Similarly, We take 𝑟 = 1, and take different values of a, we have reached similar conclusions.
As the value of a increases, the vibration of the solution 𝑥 (𝑡 ) increases and the amplitude
increases gradually. When 𝑎 < 0.4, the solution of Equation (8.2) monotonically approaches 0.
When 0.4 ≤ 𝑎 < 1.5, the solution oscillates close to 0. There is a value of a is between 1.5 and
29

1.6, the solution vibrates with a certain amplitude. When 𝑎 > 1.6, the solution tends to diverge.
The figures are shown in Appendix B.
Therefore, there is a value of a is between 1.5 and 1.6, the curve of the solution 𝑥 (𝑡 ) is similar
to the cosine curve.
By observation, assume that Equation (8.2) has a solution similar to cosine curve, then
Equation (8.2) has a solution like x(𝑡 ) = cos 𝜔𝑡 . Substitute x(𝑡 ) = cos 𝜔𝑡 into Equation (8.2),
it has
(8.3) −ω sin 𝜔𝑡 = −𝑎 cos(𝜔𝑡 − 𝜔𝑟 ).
Therefore, the sufficient condition for Equation (8.3) is 𝑎 = 𝜔 , ωr =
𝜋 2
.Thus, Equation (8.2) has
a periodic solution 𝑥 (𝑡 ) = 𝑐𝑜𝑠 𝜋 2𝑟 𝑡 .
However, we find that when a or r is very small, the solution of Equation (8.2) will not vibrate,
but monotonically tends to 0. At this time, the solution 𝑥 (𝑡 ) of Equation (8.2) is nonoscillatory
and does not have periodic solution.
Assume existence 𝛽 > 0 so that
(8.4) 𝑥 (𝑡 ) = 𝑒 −𝛽𝑡

is the solution of Equation (8.2). Substitute Equation (8.4) into Equation (8.2), it has
−𝛽 𝑒 −𝛽𝑡
= −𝑎 𝑒 −𝛽𝑡
𝑒 𝛽𝑡
.
Therefore,
(8.5) 𝛽 = 𝑎 𝑒 𝛽𝑟
.
Support 𝛽 0
is the solution of Equation (8.5), then
(8.6) 𝑎 𝑒 𝛽 0
𝑟 − 𝛽 0
= 0.
30

Let 𝑓 (𝛽 ) = 𝑎 𝑒 𝛽 0
𝑟 − 𝛽 0
, the necessary condition for the establishment of Equation (8.6) is
𝑓 ′
(𝛽 ) = 0, that is, 𝑎𝑟𝑒 𝛽 0
𝑟 − 1 = 0. Thus, 𝛽 0
=
1
𝑟 𝑙𝑜𝑔 1
𝑎𝑟
, and
𝑓 (𝛽 0
) = 𝑎 𝑒 𝛽 0
𝑟 − 𝛽 0
=
1
𝑟 −
1
𝑟 𝑙𝑜𝑔 1
𝑎𝑟
=
1
𝑟 (1 − 𝑙𝑜𝑔 1
𝑎𝑟
) ≤ 0,
Finally we have 𝑎𝑟 ≤
1
𝑒 , so when 0 < 𝑎𝑟 ≤
1
𝑒 , Equation (8.2) has a solution with nonoscillatory
solution.

31

Bibliography
Bellman, R. and K. Cooke [1] Differential Difference Equations. Academic Press, 1963.

Datko, R. Remarks concerning the asymptotic stability and stabilization of linear delay differential
equations. J. Math. Anal. Appl. 111 (1985), 571-584.

Hadeler, K. P. and J. Tomiuk [1] Periodic solutions of difference differential equations. Arch. Rat.
Mech. Anal. 65 (1977), 87-95.

Halanay, A. [1] Differential Equations, Stability, Oscillations, Time Lags. Academic Press, 1966.

Hale, J. K., Infante, E. F. and F.-S. P. Tsen [1] Stability in linear delay equations. J. Math. Anal.
Appl. 105 (1985), 533-555.

Kappel, F. Laplace-transform methods and linear autonomous functional differential equations.
Ber. math.-stat. Sektion Forschungszentrum Graz. No. 64 (1976).

Wright, E. M. Linear differential difference equations. Proc. Camb. Phil. Soc. 44 (1948), 179-185.

Widder, D.V . [1] The Laplace Transform. Princeton University Press, Princeton, 1946.

32

Appendices
Appendix A: Figure I

33

34

35

Appendix B: Figure II

36

37

Abstract (if available)

Abstract The paper introduces the subject through linear differential difference equations of retarded and neutral type with constant coefficients. There is a series of theories of linear equations with constant coefficients, the simplest possible differential difference equations, are based on such fundamental methods. In addition, this paper introduces the characteristic equation, the fundamental solution, the role of fundamental solution in determining the exact exponential bound of the solution of homogeneous equation and the behavior of the solution of non-homogeneous equation. Finally, it is found that the solution of the first order delay equation is unstable under some conditions and that the first order delay equation has stability and periodicity under certain conditions.

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Publication Date 11/19/2021

Defense Date 11/17/2021

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