A Study on Differential Equations with Deviating Arguments

A STUDY ON DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

THESIS SUBMITTED FOR THE AWARD OF THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS

BY

MAMTA KUMARI

RESEARCH GUIDE DR. Y.S. VALAULIKAR

DEPARTMENT OF MATHEMATICS GOA UNIVERSITY, GOA

AUGUST 2018

DEVIATING ARGUMENTS

THESIS SUBMITTED FOR THE AWARD OF THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS

BY

MAMTA KUMARI

RESEARCH GUIDE DR. Y.S. VALAULIKAR

DEPARTMENT OF MATHEMATICS GOA UNIVERSITY, GOA

AUGUST 2018

This thesis is dedicated to my parents.

This thesis entitled “A Study on Differential Equations with Deviating Arguments” submitted by me to the Goa University for the award of the degree of Doctor of Philosophy in Math- ematics is a research work carried out by me under the supervision and guidance of Dr.

Y.S.Valaulikar.

The research work embodied in it is original and has not been submitted earlier in part or full or in any other form to any university or institute, here or elsewhere, for the award of any degree or diploma.

Place: Taleigao Plateau

Mamta Kumari

Certificate

This is to certify that Ms Mamta Kumari has successfully completed the thesis entitled

“A Study on Differential Equations with Deviating Arguments” for the degree of Doctor of Philosophy in Mathematics under my guidance during the period 2012-2018 and to best of my knowledge, the research work embodied in it is original and has not been submitted earlier in part or full or in any other form to any university or institute, here or elsewhere, for the award of any degree or diploma.

Date: August- 2018

Y.S.Valaulikar Guide, Associate Professor, Department of Mathematics, Goa University

Head

Department of Mathematics Goa University

I express my heartfelt gratitude to the Almighty God for bestowing his grace upon me.

It gives me great pleasure to extend my sincerest gratitude and profound thanks to my guide Dr. Y. S. Valaulikar, Department of Mathematics, Goa University, for his benevolent and inspiring guidance throughout the course of this work.

I express my gratitude to the faculty members Dr. A. J. Jayanthan, Dr. A. N. Mohapatra Dr. M. Tamba, Dr. M. Kunhanandan for their valuable suggestions during the period of my study. I also acknowledge the help and goodwill of Dr. M. Kunhanandan during the course of my work. I wish to thank Mr. Nanda Gawas and the other non-teaching staff, for their support and co-operation. My heartfelt gratitude goes out to Dr. Prita D.

Mallya, Principal of VVM’s Shree Damodar College of Commerce & Economics, Margao and the staff members for the co-operation extended towards me.

I am indebted to my parents, Mr. Roshan Lal Verma and Mrs. Nirmala Verma without whose help and support this work would not have been possible. I also thank my brothers, sister-in laws and my loving nephew and niece for their support and encouragement.

During the course of my study, I have encountered support and encouragement from many people at varying stages and in various forms. I am greatly indebted to all these people whose names may not feature but their kindness, friendship, moral support and enthusiasm gave me the strength to reach here.

Finally, my sincere thanks to the authorities of Goa University, for providing the necessary facilities to carry out this research work.

iii

Contents

1. Introduction . . . 1

1.1. Classification . . . 2

1.2. Applications . . . 3

1.3. Research Objective . . . 5

1.4. Organization of Thesis . . . 6

2. Survey of Literature . . . 7

2.1. Review of literature . . . 7

2.2. Preliminaries . . . 11

3. Existence, Uniqueness and Continuous Dependence of Solutions . . . 13

3.1. Introduction . . . 13

3.2. Existence and Uniqueness of Solutions . . . 14

3.3. Continuous Dependence of Solutions . . . 20

3.4. The Method of Steps . . . 21

3.5. Chaplygin’s Method . . . 29

3.6. Positive solutions of Neutral Delay Differential Equation . . . 41

4. Stability . . . 47

4.1. Introduction . . . 47

4.2. Hyers-Ulam and Hyers-Ulam-Rassias Stability . . . 50

4.3. Stability using Fixed Point Method . . . 53

5. Periodic Solutions and Oscillation Criteria . . . 57

5.1. Introduction . . . 57

5.2. Periodic Solutions . . . 57

5.3. Oscillation Criteria . . . 73

v

Summary . . . 79 Publications and Presentations . . . 81 Bibliography . . . 85

Chapter

1

Introduction

The most interesting natural phenomenon involves change and is described mathemat- ically by equations which involve derivatives and integrals. These equations are called either differential equations or integral equations, as the case may be. In different areas such as physical, biological, social and engineering sciences various differential equa- tions arise which may be linear or nonlinear in nature. While studying a certain system if we obtain a differential equation, then it is known as modeling of the system using differential equation.

In modeling, most of the time, ordinary differential equations are used as tools. In ordi- nary differential equation (ODE) models, we assume that the systems are independent of the past state and future state depends only on the present state. In ODE models, there is an assumption that the interaction is instant and there is no delay between the system and its subsystem. But in realistic models, there is a certain amount of delay in the interaction. Hence, there is need to consider the past as well as present states and sometimes the derivative of past states to determine the future state. These mod- els are represented by functional differential equations (FDEs). Implicitly, we assume that the history of the system has an influence on the future state of the system and hence many models are better described by FDEs instead of ODEs. Delay differential equations (DDEs) are the simplest form of FDEs. They are also known as differential equation with retarded argument as the derivative of the unknown function depends on the past history. Neutral delay differential equations (NDDEs) are natural extensions of the DDEs which involve derivative of the unknown function at the delayed argument.

1

There are some special types of differential equations which have very interesting prop- erties and one of these types is differential equations with piecewise constant arguments or continuous argument (EPCA). EPCA were first studied by A.D. Myskhis [40], that had features of both difference equations and differential equations, that is, it sometimes represents the hybrid system with both discrete as well as continuous delays.

1.1. C

LASSIFICATION

We classify differential equations with deviating arguments into three categories as pro- posed by G.A. Kamenskii [33].

•

Delay Differential Equations (DDEs) or Retarded Functional Differential Equa- tions (RFDEs)

DDEs are equations involving present and past values of the unknown function.

Here the past dependence is through the state variable and not through the derivative of the state variable. A general form of a first order DDEs is

x⁰

(

t

) =

f

(

t,x

(

t

)

_,x

(

t

−

_τ

))

_, (1.1) whereτ>0and f :R

×

R

×

R

→

R^.

Example: x⁰

(

t

) = −

ax

(

t

−

2

)

.

•

Neutral Delay Differential Equations (NDDEs)

NDDEs are equations involving unknown function as well as its derivatives at present and past values of the argument. A general form of a first order NDDE is

x⁰

(

t

) =

g

(

t,x

(

t

)

,x

(

t

−

_τ

)

,x⁰

(

t

−

_τ

))

, (1.2) whereg:R

×

R

×

R

×

R

→

R^andτ>0.

Example: x⁰

(

t

) = −

ax

(

t

) +

x

(

t

−

1

) −

x⁰

(

t

−

1

)

.

•

Advanced Differential Equations(ADEs)

ADEs are equations involving unknown functions at present and future values of the arguments. A first order ADE is of the form:

x⁰

(

t

) =

h

(

t,x

(

t

)

,x

(

t

+

_τ

))

, (1.3) whereh

(

t

)

_:R

×

R

×

R

→

R^{and τ}>_0.

Example: x⁰

(

t

) =

x²

(

t

) +

x

(

t

+

₁₀

)

_.

1.2. APPLICATIONS 3

The usual way of studying these equations is, like ODEs, establishing the existence and uniqueness of their solutions. In the absence of a method to find explicit solutions, the qualitative properties of solutions are investigated. These properties include oscillations, periodicity, stability and so on and such study is of prime importance in the development of the theory of differential equations.

1.2. A

PPLICATIONS

Following are few of the applications.

(1) Engineering

•

Rocket Propulsion:

In rocket engines, liquid propellant is injected into the combustion chamber transforming eventually into a hot gas. There is a time gap between the injection and its conversion into a hot gas, which was studied by Tsien [48]

and he formulated the following FDE:

dp

dt

+ (

1

−

n

)

p

(

t

) +

np

(

t

−

a

) =

0, (1.4) where p is the dimensionless deviation from steady pressure, t is the di- mensionless time variable,ais the dimensionless constant time lag of com- bustion, andnis a constant.

(2) Biology

•

Predation:

Predation is the fundamental type of interaction in the biological world be- tween the predator and prey. Chen [12] studied a delayed predator-prey model with predator migration to describe the biological control. The model is given by

( x⁰

(

t

) =

x

(

t

)[

r

−

ay

(

t

)]

,

y⁰

(

t

) =

y

(

t

)[−

d

+

bx

(

t

−

τ

) −

cy

(

t

)] +

m

[

x

(

t

) −

py

(

t

)]

, ^(1.5) where x and y are the biomasses of the prey and predator respectively. In the absence of predators, the prey species follows the exponential equa- tions,x⁰

(

t

) =

rx

(

t

)

, whereris the intrinsic growth rate. ay

(

t

)

is the hunting term in the presence of predators. The positive feedback bx

(

t

−

τ

)

has a positive delay τ, which is the time due to converting prey biomass into

predator biomass. p is the consumption rate of the predator on prey per predator per unit time, m is the migration rate of predators, c is the self- limitation constant of the predator,d is the death rate of the predator.

(3) Economics

•

Finance

In today’s world everyone is concerned with the financial security. The stock market is full of information and it is up to the investors, how they understand the information available to them. The history of assets prices is available and on careful examination of past prices one can likely predict the future prices of the assets. Hence the future asset prices not only depends on the current state (prices), but also on the historical states.

The Black-Scholes formula (also called Black-Scholes-Merton) is the first mathematical model used for option pricing. This formula estimates the value of European-style options. The model takes into account the expi- ration date, expected volatility, present stock prices, interest rate and div- idends expected, the option’s strike price. In 1973 the three economists Fischer Black, Myron Scholes and Robert Merton introduced options price models in their paper, “The Pricing of Options and Corporate Liabilities,”

which was published in the Journal of Political Economy. Chang et al [11]

studied a stock price model where drift and volatility both depends on the stock price history. They studied an approximation scheme for a Black- Scholes Equations with delays. The infinite dimensional Black-Scholes equation arises from the hereditary market model in which the stock price process

{

S

(

t

)

,t

≥ −

h

}

satisfies the following nonlinear stochastic func- tional differential equation

dS

(

t

)

S

(

t

) =

f

(

S_t

)

dt

+

g

(

S_t

)

dZ

(

t

)

, t

≥

0, (1.6) where Z

(

t

)

is a 1-dimensional standard Brownian motion. In addition to the non linearity of equation (1.6), one distinct feature is that the stock ap- preciation f

(

S_t

)

and stock volatility g

(

S_t

)

at time t

≥

0are given nonlinear functions of the stock priceS_t over the time interval

[

t

−

h,t

]

instead of just the stock priceS

(

t

)

at timet^.

1.3. RESEARCH OBJECTIVE 5

Time delays are an indispensable part of our daily life. In fact, time delays plays an important role in some situations serving as a necessity. We will be discussing two examples where time delay has been a boon.

(1) Electric Bell:

An electric bell is a mechanical bell which functions by means of an electromag- net. When the electricity passes, the electromagnet exerts a magnetic force which when stopped disappears. But there is a delay in the appearance and disappearance of the magnetic force which makes the bell ring aloud. If this was not the case then the hammer would have struck the gong in a feeble man- ner. So, delay is important in ringing of an electric bell. The motion of the hammer in the electric bell as studied by Norkin [41] is given by:

mx⁰⁰

(

t

) +

rx⁰

(

t

) +

kx

(

t

) +

cx

(

t

−

a

) =

0. (1.7) where x

(

t

)

is the displacement of the hammer at timet.m,r,k,care constants and cx

(

t

−

_a

)

is an approximation to the force acting on the hammer.

(2) Auditory Feedback

Auditory feedback means the sound which we hear after performing actions such as closing of doors, hanging of phone etc. As we know there is some delay in auditory perception. It is possible to set up an experiment in which this delay is varied, [See Lee [38]]. This fact is used in the basic Delay Auditory Feedback (DAF) devices used in the treatment of stuttering. Auditory feedback devices are usually made up by a directional microphone and a speaker. When the speaker speaks into a microphone, he or she hears the voice through the headphones after some delay. DAF devices can extend this delay which can be used to silence the mental stress and cure stuttering.

1.3. R

ESEARCH

O

BJECTIVE

The study of FDEs has developed considerably in past few decades, the reason being its wide application in physical and biological system. In both physical and biological system modeling the hereditary effect on the system is taken into account. The above examples and the applications have motivated to take up the study on differential equa- tions with deviating arguments.

The objective of this research work is to study differential equations with deviating ar- gument, in particularly NDDEs. In this work we have studied the first order neutral differential equation of the type

x⁰

(

t

) =

f

(

t,x

(

t

)

,x

([

t

])

,x⁰

([

t

]))

, (1.8) with initial value

x

(

0

) =

x₀. (1.9)

We assume f to be a continuous function defined on J

×

R^{3,where J}

= [

0,T

]

. The work also includes some particular cases of this equation. The qualitative properties such as stability, periodicity, and oscillation of solutions are also studied.

1.4. O

RGANIZATION OF

T

HESIS

The thesis is divided into five chapters. The outline of these chapters is as follows:

(1) The first chapter serves as introductory part of the thesis. We provide a brief introduction to functional differential equations, its types and applications. It includes brief history of differential equations with deviating arguments, problem under study and brief summary of the chapters.

(2) The chapter 2 provides the survey of existing literature on the topic of study. We also give preliminaries required for the work .

(3) The existence, uniqueness, and continuous dependence of the solution of (1.8) with initial condition (1.9) is the content of the chapter 3. The method of steps is employed to find solutions of particular cases of (1.8). Chaplygin’s method for proving existence of solution of (1.8) is discussed. The positive solution of NDDE is also discussed.

(4) Chapter 4 concerns with the Hyers-Ulam and Hyers-Ulam-Rassias stability of solution of the equation (1.8). We consider the linear and nonlinear cases of (1.8). We have proved the Gronwall type Inequality for NDDE. The chapter ends with discussion of stability using fixed point theorem.

(5) Chapter 5 of the thesis is devoted to periodicity and oscillation of the solution of NDDEs. We have used monotone iterative technique to establish the existence of minimal and maximal solutions. Oscillatory properties of (1.8) are studied.

(6) At last we give a brief summary of the results obtained in this thesis and present some problems for further study. The thesis ends with a complete bibliography.

Chapter

2

Survey of Literature

2.1. R

EVIEW OF LITERATURE

In FDEs the past state of the system has a significant influence on the future. The pi- oneer study in FDEs started with Euler [20], when he was studying the problem of the general form of curves which are similar to their own evolutes. In 18th and 19th century Bernoulli, Laplace, Poisson, Condorcet, Cauchy studied FDEs. In 1911, Schmidt [45]

studied general class of differential-difference equations. Volterra, in 1928 [53], studied variations and fluctuations of the number of individuals in animal species living together and later in a book, in 1931, gave the importance of past history while modeling the in- teraction between the species. In 1977 A. D. Myshkis [40] studied differential equations with deviating argument and felt the need for a substantial theory of differential equa- tions with lagging arguments that are piecewise constant or piecewise continuous.

In 1982, Busenberg and Cooke [8] developed the first mathematical model which in- volved piecewise constant argument while investigating the biomedical problem of verti- cally transmitted diseases. Since then the theory of differential equations with piecewise constant arguments have grown rapidly and studied extensively by many [ [1], [28], [29], [51], [52] and references therein]. It has become an important tool for researchers deal- ing with applications such as neural networks, loss less transmission, nuclear reactors, population dynamics, finance, stock markets, aerodynamics, mechanics etc.

A typical differential equation studied by above authors is of the form:

x⁰

(

t

) =

f

(

t,x

(

t

)

,x

(

γ

(

t

)))

,

7

with deviating arguments of the form γ

(

t

) = [

t

]

, 2

[

^t

+

1

2

]

,

[

t

−

n

]

,

[

t

+

n

]

,

[

t

] +

n,

[

t

] −

n,

where [.] denotes the greatest-integer function andnis a positive integer. The argument is called advanced type if it is of the form

[

t

+

n

]

,

[

t

] +

n. If the argument is of the form

[

t

−

n

]

,

[

t

] −

n then the equation is called retarded or delay type. If both the arguments appears then it is called of mixed type.

In 1984 Cooke and Wiener [14] studied the scalar initial-value problem which is closely related to impulse and loaded equation:

x⁰

(

t

) =

ax

(

t

) +

a₀x

([

t

]) +

a₁x

([

t

−

1

])

, x

(−

1

) =

c₋₁, x

(

0

) =

c₀,

wherea,a₀,a₁ are constants. In 1987, they [15] consider the equation of the form:

x⁰

(

t

) =

ax

(

t

) +

a₀x

(

2

[

^t

+

1

2

])

; x

(

0

) =

c₀, ^(2.1) For t

∈ [

2n

−

_{1, 2n}

)

, the deviating argument γ

(

t

) =

t

−

₂

[

^t+₂¹

]

is negative while it is positive on t

∈ (

_{2n, 2n}

+

₁

)

_, where n is an integer. Equation (2.1) on

(

_{2n, 2n}

+

₁

)

is retarded type and on

[

2n

−

1, 2n

)

advanced type. That is on the interval

[

2n

−

1, 2n

+

₁

)

, it is alternatively retarded and advanced. In 1988 Wiener and Aftabizadeh [54] studied the system:

x⁰

(

t

) =

f

(

x

(

t

)

,x

(

m

[

^t

+

k

m

]))

; x

(

0

) =

c₀,

where [.] denotes the greatest integer function and k and m are positive integers such that k < m. The argument deviation γ

(

t

) =

t

−

m

[

^t+k_m

]

is positive for t

∈ (

mn,m

(

n

+

1

) −

k

)

and negative fort

∈ [

mn

−

k,mn

)

,wherenis integer.

The oscillation theory plays a vital role in the study of qualitative theory of differential equations. It concerns largely with the existence of oscillatory and nonoscillatory prop- erties of solutions. The development of oscillation theory started with Sturm’s work in 1840 on oscillation theorem for ordinary differential equations. More information on the development of theory of oscillation can be seen in monographs by Bainov and Mi- shev [6], Erbe [19], and Gyo¨ri and Ladas [23].

In 1997, Das and Misra [16] studied oscillation properties of a nonlinear differential equa- tion of the type

(

x

(

t

) −

px

(

t

−

_τ

))

⁰

+

Q

(

t

)

G

(

x

(

t

−

_σ

)) =

f

(

t

)

_, (2.2)

2.1. REVIEW OF LITERATURE 9

where f,Q

∈ C ([

T,∞

)

_,

(

_0,∞

))

_,σ,τ

∈ (

_0,∞

)

_{, 0}

≤

p<_1, G:R

→

R^{such thatxG}

(

x

)

>₀ for x

6=

_0,, G is nondecreasing, Lipschitzian, and satisfy a sublinear condition

Z _±k 0

dx

G

(

x

)

<_∞, and Z _∞

f

(

s

)

ds<_∞ is either oscillatory or tends to zero asymptotically iffR_∞

T Q

(

s

)

ds

=

_∞.In 2000, Shen [46]

gave new criteria for oscillation and nonoscillation of solution for the autonomous delay differential equation with piecewise constant argument of the type

y⁰

(

t

) +

a

(

t

)

y

(

t

) +

b

(

t

)

y

([

t

−

1

]) =

0, (2.3) where a

(

t

)

andb

(

t

)

are continuous functions on

[−

1,∞

)

, b

(

t

) ≥

0.

Graef et al. [21] in 2003, considered the equation

(

x

(

t

) +

px

(

t

−

_τ

))

⁰

+

q

(

t

)

f

(

x

(

t

−

_σ

)) =

_0, (2.4) where p

∈ (

1,∞

)

, τ<σ, limt→∞R_t+σ−τ

t q

(

s

)

ds>0and proved that if Z _∞

t₀ q

(

t

)

ln

eM

(

p

−

1

)

p²

Z _t+σ−τ

t q

(

s

)

ds

dt

=

_∞, then every solution of (2.4) oscillates. They also proved that if

1 e

≤

Z _t t−σ+τ

q

(

s

)

ds<k, where k>0,then every solution of (2.4) is oscillatory.

In 2006, Elabbasy et al. [18] studied the equation

[

x

(

t

) −

q

(

t

)

x

(

t

−

τ

)]

⁰

+

f

(

t,x

(

τ

(

t

))) =

0, (2.5) where t

≥

t₀, q,τ

∈ C([

t₀,∞

)

_,R⁺

)

_, r

∈ (

_0,∞

)

_,τ

(

t

)

<t, lim_t→∞τ

(

t

) =

_∞,

f

∈ C ([

t₀,∞

) ×

R^,R

)

_, u f

(

t,u

) ≥

_{0, ∑}ⁿ_i=1Πⁱ_j=1 ¹

q(t_j)

→

_∞asn

→

_∞.

and for q

6=

1, established an oscillation criteria for all solutions of (2.5).

In 2013, Ahmed et al. [4] studied oscillation properties of a first order neutral delay differential equations with variable coefficients of the form

[

r

(

t

)(

x

(

t

) +

p

(

t

)

x

(

t

−

_τ

))]

⁰

+

q

(

t

)

x

(

t

−

_σ

) =

0, t

≥

_t0 (2.6) where p

∈ C([

t₀,∞

)

,R

)

, r,q

∈ C ([

t₀,∞

)

,R⁺

)

, τ,σ

∈

R^+. They established sufficient conditions for every solution of (2.6) to be oscillatory. In year 2014, Ahmed et al. [5]

considered the first order neutral functional differential equations of the form

[

r

(

t

)(

x

(

t

) +

px

(

t

−

_τ

))]

⁰

+

q

(

t

)(

x

(

t

−

_σ

)) =

0, t

≥

t₀ (2.7)

wherer,q

∈ C ([

t₀,∞

)

,

(

0,∞

))

, p

∈

R^,τ

∈ (

0,∞

)

,σ

∈

R⁺and established new sufficient conditions for oscillation of solutions of (2.7).

In 2016, T. Candan [9] studied the equation d

dt

[

x

(

t

) +

P₁

(

t

)

x

(

t

−

_τ

) +

P₂

(

t

)

x

(

t

+

_τ2

)] +

Q₁

(

t

)

x

(

t

−

_σ1

) −

Q₂

(

t

)

x

(

t

+

_σ2

) =

0, (2.8) where P_i

∈ C([

t₀,∞

)

,R

)

, Q_i

∈ C([

t₀,∞

)

,

[

0,∞

))

, τ_i >0 andσ_i

≥

0fori

=

1, 2.He gave new criteria for nonoscillatory solution of (2.8).

Recently, Grace and Jadlovska´ [22] studied oscillatory behavior of odd-order nonlinear differential equation with a nonpositive neutral term. They gave a sufficient condition for oscillation of all solutions of the equation,

[

x

(

t

) −

p

(

t

)

x

(

_σ

(

t

))]

⁽ⁿ⁾

+

q

(

t

)

x^β

(

_τ

(

t

)) =

0, (2.9) where t

≥

t₀>0 ^andn

≥

3 is an odd natural number.

βis a ratio of odd positive integers, p, q

∈ C([

_t0,∞

)

,

[

0,∞

))

, 0

≤

_p

(

t

)

<1, τ, σ

∈ C

¹

([

t₀,∞

)

_,R

)

_{, τ}

(

t

) ≤

t, σ

(

t

) ≤

t, τ⁰

(

t

)

>_{0, σ}⁰

(

t

)

>₀and

lim_t→∞τ

(

t

) =

_∞, lim_t→∞σ

(

t

) =

_∞.

In a Mathematical Colloquium at the University of Wisconsin, Stanislaw M.Ulam (1940) discussed a couple of unsolved mathematical problems. One of the problem was about the stability of homomorphism [50]. Donal H. Hyers [26] in 1941 gave the first solution by solving it for a pair of Banach spaces using direct method. This stability phenomenon is now called “Hyers-Ulam Stability”. The next breakthrough came in the year 1978 when Themistocles M. Rassias [43] extended the result of Hyer’s theorem. Rassias weaken the condition for boundedness of the norm for the Cauchy difference. This stability phe- nomenon is called “Hyers-Ulam-Rassias” stability. For more details the readers can refer to [27], [32].

Jung and Brzde¸k [31] studied Hyers-Ulam stability of the DDE y⁰

(

t

) =

y

(

t

−

_τ

)

_,where y

∈ ([−

τ,∞

)

,R

)

and y

∈ C

¹

([

0,∞

)

,R

)

. D. Otrocol and IIea [42] studied Ulam stability for a DDE of the type

x⁰

(

t

) =

f

(

t,x

(

t

)

,x

(

g

(

t

)))

, t

∈

I

⊂

R^, where I

= [

a,b

]

or I

= [

a,∞

)

, a,b

∈

R^,

f

∈ C ([

a,b

] ×

R^2,R

)

, g

∈ C ([

a,b

]

,

[

a

−

h,b

])

, g

(

t

) ≤

t, h>0respectively f

∈ C([

a,∞

) ×

R^2,R

)

, g

∈ C([

a,∞

)

,

[

a

−

h,∞

))

, g

(

t

) ≤

t, h>0.

Huang and Li [24] studied Hyers-Ulam stability of linear functional differential equa- tions, and a year later for DDE of first order [25]. C. Tunc¸ and Bic¸er [49] studied the

2.2. PRELIMINARIES 11

Hyers-Ulam-Rassias stability for a first-order functional differential equation. Recently, A. Zada, S. Faisal, and Y. Li [56], [57] studied the Hyers-Ulam stability of first-order im- pulsive DDEs and Hyers-Ulam-Rassias stability of nonlinear DDEs. Many researchers are studying qualitative properties such as oscillation, periodicity, stability of NDDE. We refer the readers to see [2], [35], [55], [58], [59], [60], [61] and references therein.

2.2. P

RELIMINARIES

In this section, we shall give without proof some basic lemmas and theorems used during the work.

We will give two different formalization of Arzela- Ascoli theorem.

THEOREM 2.2.1. (Arzela-Ascoli Theorem) [44]. A subset A of

C ([

a,b

]

_,R

)

is relatively compact if and only if it is uniformly bounded and equicontinuous.

THEOREM2.2.2. (Ascoli-Arzela Theorem) [3]:

Let Fbe a family of functions bounded and equicontinuous at every point of an intervalI. Then, every sequence of functions f_ninFcontains a subsequence uniformly convergent on every compact subinterval of I^.

THEOREM2.2.3. (Schauder fixed point theorem) [30]:

Let K be a nonempty closed, convex subset of a normed linear space X. Let F be a continuous mapping of Kinto a compact subset ofK. Then Fhas a fixed point in K.

THEOREM2.2.4. (Contraction Mapping Theorem) [30]:

Let F be a continuous mapping of a complete metric spaceX into itself such thatF^{k is a} contraction mapping of X for some positive integerk. Then Fhas a unique fixed point.

THEOREM2.2.5. (Krasnoselskii’s fixed point theorem) [30]:

Let K be a nonempty complete convex subset of a normed linear space X. Let T be a continuous mapping of K into a compact subset of X. Let S :K

→

X be a contraction mapping with Lipschitz constant α and let Tx

+

Sy

∈

K for all x,y

∈

K. Then there is a point u

∈

K such that Tu

+

Su

=

u.

THEOREM2.2.6. (Leibnitz-Newton theorem) [47]:

If f

(

x

)

is continuous in

[

a,b

]

and F

(

x

)

is any function such thatF⁰

(

x

) =

f

(

x

)

,then Z _b

a f

(

x

)

dx

=

F

(

b

) −

F

(

a

)

.

Chapter

3

Existence, Uniqueness and Continuous Dependence of Solutions

3.1. I

NTRODUCTION

The realistic models are best described by differential equations with deviating argu- ments. The basic questions in the mathematical models are existence of solutions and their uniqueness which are important to study their qualitative properties. In this chapter, our aim is to deal with certain existence theorems for NDDEs with deviating arguments and also present some results for uniqueness of solutions. Some examples have been considered as illustrations.

We consider the following NDDE:

x⁰

(

t

) =

f

(

t,x

(

t

)

,x

([

t

])

,x⁰

([

t

]))

(3.1)

x

(

₀

) =

x₀, ^(3.2)

where f

∈ C(

J

×

R^3,R

)

, J

= [

0,T

]

,and [.] is the greatest integer function.

Let

D

denotes the class of all functionsx: J

→

R^,satisfying (1) x

(

t

)

is continuous,

∀

_t

∈

_J.

(2) x⁰

(

t

)

exists and is continuous on the intervals

[

n,n

+

₁

)

_,forn

=

0, 1, 2, . . . , ˜T

−

₂ and on

[

T^˜

−

_1,T

)

,

13

where,

T˜

=

(

[

T

] +

1, T

6= [

T

]

, T, T

= [

T

]

.

A functionx:J

→

Ris said to be a solution of (3.1) ifx

∈ D

and satisfies (3.1) and (3.2) with x⁰

(

t

) =

x⁰₊

(

t

)

,the right-hand derivative on t

=

1, 2, . . . , ˜T

−

1.

3.2. E

XISTENCE AND

U

NIQUENESS OF

S

OLUTIONS

In this section, using method of successive approximations [see [3]], we establish a local existence and uniqueness theorem for (3.1) when f satisfies Lipschitz type condition.

Later, we relax the Lipschtiz type condition on f and existence of solution is established.

LEMMA3.2.1. x

(

t

)

is a solution of (3.1), (3.2) on J if and only ifx

(

t

)

is a solution of x

(

t

) =

x₀

+

Z _t

0 f

(

s,x

(

s

)

,x

([

s

])

,x⁰

([

s

]))

ds. ^(3.3) PROOF. If x

(

t

)

is a solution of (3.1), (3.2) then it follows that x

(

t

)

satisfies (3.3).

Let x

(

t

)

satisfy (3.3). Then at t

=

0, x

(

0

) =

x₀ and f

(

t,x

(

t

)

,x

([

t

])

,x⁰

([

t

]))

is contin- uous in

(

t,x

(

t

)

,x

([

t

])

,x⁰

([

t

]))

. Differentiating both sides of (3.3) it is seen that x⁰

(

t

) =

f

(

t,x

(

t

)

,x

([

t

])

,x⁰

([

t

]))

.This completes the proof.

Note: In what follows

||

.

||

means Eucledian norm unless otherwise specified.

THEOREM 3.2.2. Suppose that f satisfies the following:

(A1) f

(

t,x,y,z

)

be piecewise continuous function on

D

= {

₀

≤

_t

≤

_T,

||

_x

−

_x0

|| ≤

_b,

||

_y

−

_x0

|| ≤

_b,

||

_z

−

_z0

|| ≤

_{c; b, c,T}>0

} ⊂

R^4, (A2) f is bounded on D ^i.e.

||

_f

|| =

_sup(t,x,y,z)∈D

|

_f

| ≤

_M; where M > ₀ is some

constant.

(A3) Forx₁

≥

x₂,y₁

≥

y₂,z₁

≥

z₂

∈

D,and L₁, L₂positive constants and0

≤

L₃<1. f

(

t,x,y,z

)

satisfy the condition,

||

f

(

t,x₁,y₁,z₁

) −

f

(

t,x₂,y₂,z₂

)|| ≤

L₁

||

x₁

−

x₂

|| +

L₂

||

y₁

−

y₂

|| +

L₃

||

z₁

−

z₂

||

. Then, for 0

≤

_β

≤

T such that 0

≤

t

≤

_β there exists a unique solution x

(

t

)

to the IVP (3.1), (3.2).

3.2. EXISTENCE AND UNIQUENESS OF SOLUTIONS 15

PROOF. Let x

(

t

) ∈ D

_. We define

||

x

(

t

)|| =

sup_t∈J

|

x

(

t

)|

_, where J

= [

_0,T

]

_. Then

(C(

J,R

)

_,

||

_.

||)

is a Banach space. Chooseβ, ρ,^andγ

≥

₁such that0

≤

_β

≤

T, 0

≤

_ρ

≤

b, 0

≤

_γ

≤

cand βM<_αwhereα

=

min

{

β,_M^δ

}

,δ

=

min

{

ρ,γ

}

.

Let D₁

= {

x

∈ C([

0,β

]

,R

)

,

||

x

(

t

) −

x₀

|| ≤

_α

}

,and x

(

0

) =

x₀ be any element ofD₁. D₁ is a closed, convex, bounded subset of the Banach space

C (

J,R

)

.

We now define the map P:D₁

→

D with x_k

(

_t

) = (

_Pxk−1

)(

_t

) =

_x0

+

Z _t

0 f

(

_s,xk−1

(

_s

)

_,xk−1

([

_s

])

_,

(

_xk−1

)

⁰

([

_s

]))

_ds (3.4) x_k

([

t

]) = (

Px_k−1

)([

t

])

;k

=

1, 2, 3, . . . .

By using Leibnitz-Newton theorem fort

∈ [

n,n

+

l

]

,where0<l <1and n

=

0, 1, 2, . . . , ˜β

−

2,the map

xⁿ_k

(

t

) = (

Pxⁿ_k−1

)(

t

) =

xⁿ_k−1

(

n

) +

Z _t

n f

(

s,x_k−1ⁿ

(

s

)

,xⁿ_k−1

(

n

)

,

(

x_k−1ⁿ

)

⁰

(

n

))

ds. ^(3.5) xⁿ_k

(

n

) = (

Pxⁿ_k−1

)(

n

)

;k

=

1, 2, 3, . . .is continuously differentiable on

[

n,n

+

l

]

and where,

β˜

=

(

[

β

] +

1, β

6= [

β

]

, β, β

= [

β

]

.

We shall establish that x_k converges to fixed point on

[

0,β

]

. In view of the continuity of f

(

t,x_k−1

(

t

)

_,x_k−1

([

t

])

_,

(

x_k−1

)

⁰

([

t

]))

on

[

_0,β

]

_, where k

=

1, 2, . . . , it follows that the functions x₀,x₁

(

t

)

_{, . . . ,}x_k

(

t

)

are well defined and are continuous on

[

_0,β

]

_.It is obvious that, x₀

∈

D₁. We now show thatx_k

(

t

) ∈

D₁.Fort

∈ [

0,β

]

,we have

||

x₁

(

t

) −

x₀

|| ≤

Mt

≤

Mβ<α, (3.6) which implies that x₁

(

t

) ∈

D₁. Suppose x_k−1

(

t

) ∈

D₁,then

x_k

(

t

) =

x₀

+

Z _t

0 f

(

s,x_k−1

(

s

)

,x_k−1

([

s

])

,

(

x_k−1

)

⁰

([

s

]))

ds.

Now

||

x_k

(

t

) −

x₀

|| ≤

Mβ<_αand therefore x_k

(

t

) ∈

D₁, k

=

1, 2, 3, . . . .

To establish the convergence of the sequence of functions

{

x_k

(

t

)}

, we take the differ- ence between the successive approximations. Fort

∈ [

0,β

]

,set

p_k

(

t

) =

x_k

(

t

) −

x_k−1

(

t

)

.

Then we have, by (3.6)

||

p₁

(

t

)|| ≤

Mt.Since f satisfies (A3) on D₁,it follows,

||

_p2

(

t

)|| = ||

_x2

(

t

) −

_x1

(

t

)||

= ||(

Px₁

)(

t

) − (

Px₀

)(

t

)||

= ||

Z _t 0

[

f

(

s,x₁

(

s

)

,x₁

([

s

])

,x⁰₁

([

s

]))

ds

]

−

Z _t 0

[

f

(

s,x₀

(

s

)

,x₀

([

s

])

,x⁰₀

([

s

]))]

ds

||

≤

Z _t 0

||[

f

(

s,x₁

(

s

)

,x₁

([

s

])

,x⁰₁

([

s

])) − [

f

(

s,x₀

(

s

)

,x₀

([

s

])

,x⁰₀

([

s

]))]||

ds

≤

Z _t 0

[

L₁

||

x₁

(

s

) −

x₀

(

s

)|| +

L₂

||

x₁

([

s

]) −

x₀

([

s

])||]

ds

+

Z _t

0 L₃

||

x⁰₁

([

s

]) −

x⁰₀

([

s

])||

ds

≤

Z _t

0 2L

||

x₁

(

s

) −

x₀

(

s

)||

ds

+

Z _t

0 Le₁e^−δ

||

x⁰₁

([

s

]) −

x₀⁰

([

s

])||

ds

≤

^2LMt

2

2

+

Le₁e^−δMt

≤

^2LM

(

t

+

1

)

²

2

+

Le₁e^−δM

(

t

+

1

)

²

≤

^M

(

t

+

1

)

²

2!

[

2L

+

2Le₁e^−δ

]

≤

^2LM

(

t

+

1

)

²

2!

[

1

+

_e1e^−δ

]

≤

^4LM

(

t

+

1

)

²

2! .

where L

=

max

{

L₁,L₂

}

, L₃

=

Le₁e^−δ,wheree₁ is sufficiently small.

Similarly,

||

_p3

(

_t

)|| ≤

^{4L)}2M(t+1)_3! ³_.

A simple induction argument shows that , in general , for t

∈ [

0,β

]

||

p_k

(

t

)|| ≤ {

4L

}

^k−1M

(

t

+

₁

)

^k

k!

≤ {

4L

}

^k−1M

(

_β

+

₁

)

^k

k! . (3.7)

Now, consider an infinite series of the form x

(

t

) =

x₀

+

∑

∞ i=1

p_i

(

t

)

. ^(3.8)

The k-th partial sum of this series isx_k

(

t

)

, i.e.

x_k

(

t

) =

x₀

+

∑

k i=1

p_i

(

t

)

. (3.9)

3.2. EXISTENCE AND UNIQUENESS OF SOLUTIONS 17

Therefore, the sequence

{

x_k

(

t

)}

converges iff (3.8) converges.

From inequality (3.7), we have x₀

+

∑

∞ i=1

||

p_i

(

t

)|| ≤

x₀

+

M

∑

∞ i=1

{

4L

}

ⁱ⁻¹

(

β

+

1

)

ⁱ

i! . ^(3.10)

It follows from the ratio test that the seris on the right-hand side of (3.10) converges, and hence, by the comparision test, series (3.8) also converges (uniformly), on

[

_0,β

]

_.Let the sum of series of (3.8) be x

(

t

)

.Then, relation (3.9) gives

lim_k→∞x_k

(

t

) =

x

(

t

)

.

Finally, taking limits of (3.4), we obtain lim_k→∞x_k

(

t

) =

x₀

+

lim_k→∞R_t

0 f

(

s,x_k−1

(

s

)

,x_k−1

([

s

])

,x⁰_k−1

([

s

]))

ds, it follows that

x

(

t

) =

x₀

+

lim_k→∞R_t

0 f

(

s,x_k−1

(

s

)

,x_k−1

([

s

])

,x_k−1⁰

([

s

]))

ds,

From the uniform convergence ofx_k

(

t

)

tox

(

t

)

and the continuity of the function f

(

t,x,y,z

)

on D₁, we obtain

x

(

t

) =

x₀

+

R_t

0 f

(

s,x

(

s

)

,x

([

s

])

,x⁰

([

s

]))

ds,

Hence, from Lemma 3.2.1 x

(

t

)

is a solution of (3.1), (3.2) on

[

_0,β

]

.

To prove the uniqueness: Let u

(

t

)

be any other solution of (3.1) with initial condition u

(

0

) =

x₀. Then, the non-negative functionw

(

t

) = ||

x

(

t

) −

u

(

t

)||

satisfies

w

(

0

) =

0, w⁰

(

0

) =

0and w

(

t

) ≤

Z _t 0

||

f

(

s,x

(

s

)

,x

([

s

])

,x⁰

([

s

])) −

f

(

s,u

(

s

)

,u

([

s

])

,u⁰

([

s

]))||

ds,

≤

2L Z _t

0

||

x

(

s

) −

u

(

s

)||

ds

+

Lee^−δ Z _t

0

||

x⁰

(

s

) −

u⁰

(

s

)||

ds

≤

2

(

L

+

1

)

_{Z t}

0

||

x

(

s

) −

u

(

s

)||

ds

+

Z _t 0

||

x⁰

([

s

]) −

u⁰

([

s

])||

ds

or _dt^d

[

e^−2(L+1)tR_t

0

[

w

(

s

) +

w⁰

([

s

])]

ds

] ≤

_0.

Integrating this inequality from 0 to t, we obtain w

(

t

) ≤

0. This is incompatible with w

(

t

) ≥

0unlessw

(

t

) =

0on J.This implies,x

(

t

) =

u

(

t

)

.

In the next theorem, we are going to relax the Lipschitz type condition (A3) on f and obtain the result for existence of solution.

THEOREM 3.2.3. Suppose that f satisfies the following:

(A4) f

(

t,x,y,z

)

be piecewise continuous function on the strip D₂

= {

₀

≤

t

≤

_β,

||

x

||

<_∞,β>₀

}

_.

(A5) f is bounded on D₂that is,

||

f

|| =

sup(t,x,y,z)∈D₂

|

f

| ≤

M;where M>0is some constant.

Then, the IVP (3.1), (3.2) has at least one solution x

(

t

)

defined on

[

0,β

]

. PROOF. LetD₃

= {

x

∈ C([

0,β

]

,R

)

,

||

x

||

<_∞

}

and x

(

0

) =

x₀

∈

D₃.

D₃ is a closed, convex, bounded subset of the Banach space

C (

J,R

)

. Fort

∈ [

n,n

+

l

]

, where0<l<1, n

=

0, 1, . . . , ˜β

−

2and on

[

β^˜

−

1,β

]

we define the map given by (3.5).

From the definition for n=0 we havex⁰_k

(

0

) =

x⁰₀

(

0

) =

x₀.This definition we use to define x⁰_k

(

t

)

on the intervalt

∈ [

0,l

]

asl

→

1, which can be further extended to the next interval t

∈ [

1, 2

]

. By continuing this processxⁿ_k

(

t

)

is well defined for the interval

[

n,n

+

l

]

. Using Leibnitz-Newton theorem that

(

Pxⁿ_k−1

)(

t

)

is continuously differentiable on

[

n,n

+

l

]

_. Now consider the map P :D₃

→

D₂ defined by (3.4). As f bounded on D₂, ^{we have}

||

f

(

t,x,y,z

)|| ≤

M,for x

∈

D₃. From (3.4) it follows that

||

x_k

(

t

) −

x_k

(

t^∗

)|| = ||(

Px_k−1

)(

t

) − (

Px_k−1

)(

t^∗

)||

= ||

Z _t

0 f

(

t,x_k−1

(

s

)

,x_k−1

([

s

])

,x⁰_k−1

([

s

]))]

ds

−

Z _t∗

0 f

(

t,x_k−1

(

s

)

,x_k−1

([

s

])

,x⁰_k−1

([

s

]))]

ds

||

≤

Z _t∗

t

||

f

(

t,x_k−1

(

s

)

,x_k−1

([

s

])

,x⁰_k−1

([

s

]))]

ds

||

≤

M

||

t

−

t^∗

||

,

for arbitrary t<t^∗

∈ [

n,n

+

l

]

and arbitrary x_k

∈

D₃.Consequently, P

(

D₃

)

is equicon- tinuous, which shows the sequence

{

_xk

(

t

)}

is equicontinuous on D₃.

For anyx_k

∈

D₃, we have fort

∈ [

_0,β

]

||

x_k

(

t

)|| = ||(

Px_k−1

)(

t

)|| ≤

x₀

+ ||

Z _t

0 f

(

s,x_k−1

(

s

)

_,x_k−1

([

s

])

_,x⁰_k−1

([

s

]))

ds

||

≤

x₀

+

Mβ

=

M₁,

3.2. EXISTENCE AND UNIQUENESS OF SOLUTIONS 19

which implies that

||

Px_k−1

|| ≤

M₁.ConsequentlyP

(

D₃

) ⊂

D₃and is uniformly bounded.

Further,

||(

x_k

)

⁰

(

t

)|| ≤ ||

f

(

t,x_k−1

(

t

)

,x_k−1

([

t

])

,x⁰_k−1

([

t

]))|| ≤

M.

As a result,

{

x⁰_k

(

t

)}

is uniformly bounded on

[

0,β

]

.

For e> 0, we choose δ₁

=

_M^e. Then, for all t > t^∗

∈ [

0,β

]

, such that

||

t

−

t^∗

||

<δ₁, for every k, by mean value theorem there exist ξ

∈ (

t^∗,t

)

such that x_k

(

t

) −

x_k

(

t^∗

) =

x⁰_k

(

ξ

)(

t

−

t^∗

)

.Therefore,

||

x_k

(

t

) −

x_k

(

t^∗

)|| = ||

x⁰_k

(

_ξ

)||

.

||

t

−

t^∗

|| ≤

Mδ₁<e.

Then,

{

_xk

(

t

)}

equicontinuous on

[

0,β

]

. By, Arzela-Ascoli theorem, there exists sub- sequence

{

x_kp

(

t

)} ⊂ {

x_k

(

t

)}

such that lim_k→∞

(

x_kp

)(

t

) =

x

(

t

)

t

∈ [

_0,β

]

_.

{

x_k

(

t

)}

are monotone sequence. It follows that,lim_k→∞

(

x_k

)(

t

) =

x

(

t

)

wherex

(

t

)

satisfy (3.1).

COROLLARY 3.2.4. If f,in addition to the assumptions (A4), (A5) satisfies the following condition:

(A6) f monotonically nondecreasing iny andzfor each fixedt onD₂. Then, (3.1) with initial condition (3.2) has a unique solution on

[

0,β

]

.

PROOF. Let u

(

t

)

and v

(

t

)

be any two solutions of the IVP (3.1), (3.2) in the interval

[

0,β

]

. We prove that

u

(

t

) =

v

(

t

)

on

[

0,β

]

. Suppose u

(

t

) 6=

v

(

t

)

ont

∈ [

0,β

]

.

Then, there exists t_n

∈ [

n,n

+

1

)

for somen

∈

W^{such that} u

(

t

) =

v

(

t

)

on

[

n,t_n

]

and fort

∈ (

t_n,n

+

1

)

u

(

t

)

>v

(

t

)

. (3.11)

Then, we have u

(

n

) =

v

(

n

)

and u⁰

(

n

) ≤

_v⁰

(

n

)

.

Since f is monotonically nondecreasing inyandz,it follows that f

(

t,u

(

t

)

_,u

(

n

)

_,u⁰

(

n

)) ≤

f

(

t,v

(

t

)

,v

(

n

)

,v⁰

(

n

))

.Further, since bothu

(

t

)

andv

(

t

)

are the solutions of (3.1), (3.2), we haveu⁰

(

t

) ≤

v⁰

(

t

)

on

[

n,n

+

1

)

.

Therefore, the function w

(

t

) =

u

(

t

) −

v

(

t

)

has a nonpositive derivative on

[

t_n,n

+

1

)

. That is, w⁰

(

t

) =

u⁰

(

t

) −

v⁰

(

t

) ≤

0.

By integrating this relation between t_n and t, we get w

(

t

) ≤

w

(

t_n

) =

0 which implies u

(

t

) ≤

v

(

t

)

. This contradicts (3.11). Hence, u

(

t

) =

v

(

t

)

on

[

n,n

+

1

)

.Continuing this way we can extend the interval to

[

0,β

]

.Consequently,u

(

t

) =

v

(

t

)

, t

∈ [

0,β

]

.

3.3. C

ONTINUOUS

D

EPENDENCE OF

S

OLUTIONS

In this section we obtain sufficient conditions under which the solution x

(

t

)

of (3.1) de- pends continuously on the initial conditions.

In addition to the equation (3.1), we consider the following equation:

y⁰(t) = f(t,y(t),y([t]),y⁰([t])) +g(t,y(t),y([t]),y⁰([t])), (3.12) where f,g

∈ C(

D,R

)

where D is same as defined in (A1).

THEOREM 3.3.1. Consider (3.12) where f

(

t,y

(

t

)

,y

([

t

])

,y⁰

([

t

]))

satisfies the assump- tions of Theorem 3.2.2 andg

(

t,y

(

t

)

,y

([

t

])

,y⁰

([

t

]))

is an integrable function oftfor each fixedy.^Suppose x

(

t

)

be unique solution of (3.1) on

[

0,β

]

with initial condition x

(

0

) =

x₀ fort

=

0.Then, (3.12) has a unique solutiony

(

t

)

on

[

0,β^∗

]

with initial conditiony

(

0

) =

y₀ for t

=

_0. Moreover, if β₁

=

min

{

_β,β^∗

}

, then for e>₀, there exists a δ

(

_e,g

)

>₀ such that

||

x₀

−

y₀

||

<_δand

||

g

(

t,y

(

t

)

,y

([

t

])

,y⁰

([

t

]))||

<_δimplies

||

x

(

t

) −

y

(

t

)||

<e, t

∈ [

0,β₁

]

. PROOF. Consider

||

x

(

t

) −

y

(

t

)|| ≤

Z _t 0

||

x⁰

(

t

) −

y⁰

(

t

)||

,

≤

Z _t 0

||

f

(

s,x

(

s

)

,x

([

s

])

,x⁰

([

s

])) −

f

(

s,y

(

s

)

,y

([

s

])

,y⁰

([

s

]))||

ds

+

Z _t 0

||

g

(

s,y

(

s

)

,y

([

s

])

,y⁰

([

s

]))||

ds,

≤

2L Z _t

0

||

x

(

s

) −

y

(

s

)||

ds

+

Le₁e^−δ Z _t

0

||

x⁰

(

s

) −

y⁰

(

s

)||

ds

+

Z _t 0

||

g

(

s,y

(

s

)

,y

([

s

])

,y⁰

([

s

]))||

ds,

≤

δβ₁

+

Lee₁e^−δ

+

2L Z _t

0

||

x

(

s

) −

y

(

s

)||

ds.

Applying Gronwall’s inequality, it follows that

||

x

(

t

) −

y

(

t

)|| ≤ (

δβ₁

+

Lee₁e^−δ

)

e^2Lt.

Choosing suitable δ, it is easy to see that

||

x

(

t

) −

y

(

t

)||

<_efor t

∈ [

_0,β1

]

. This com- pletes the proof of the theorem.