Quantum Calculus on Finite Intervals and
Applications to Impulsive Difference Equations
Ahmed Mohamed
Submitted to the
Institute of Graduate Studies and Research
in partial fulfilment of the requirements for the degree of
Master of Science
in
Mathematics
Eastern Mediterranean University
July, 2017
Approval of the Institute of Graduate Studies and Research
Prof. Dr. Mustafa Tümer Director
I certify that this thesis satisfies the requirements as a thesis for the degree of Master of Science in Mathematics.
Prof. Dr. Nazim Mahmudov Chair, Department of Mathematics
We certify that we have read this thesis and that in our opinion it is fully adequate in scope and quality as a thesis for the degree of Master of Science in Mathematics.
Prof. Dr. Sonuç Zorlu Oğurlu Supervisor
Examining Committee 1. Prof. Dr. Hüseyin Aktuğlu
iii
ABSTRACT
In Mathematics, quantum calculus is a version of calculus in which limits are not taken. This type of calculus plays important role both in theoretical and practical areas of mathematics. In quantum calculus, derivatives are differences and anti-derivatives are sums. Quantum calculus is a theory where smoothness is no more needed. In this work, we study finite intervals in quantum calculus. We review and study the -derivative and -integral of a function and demonstrate their properties. We apply this concept to provide existence and uniqueness results for the initial value problems, namely for first and second order impulsive -difference equations.
iv
ÖZ
Matematikte q-Kalkülüs, Kalkülüsda limitlerin alınmadığı bir versiyonudur. Bu tür matematik birçok teorik ve pratik alanda önemli rol oynamaktadır. Kuantum Kalkülüsda türevler fark ve integral ise toplam olarak tanımlanır. Kuantum Kalkülüs düzgünlüğün gerekli olmadığı bir teoridir. Bu çalışmada, kuantum Kalkülüsın sınırlı aralıkları dikkate alınmıştır. Ayrıca, bu tezde bir fonksiyonun türevini ve integralini inceleyip özellikleri verilmiştir. Bu kavram, baslangıç-değer problemlerinin varlık ve teklik sonuçları üzerinde uygulanmıştır. Özelde birinci ve ikinci dereceden impulsif -fark denklemleri dikkate alınmıştır.
v
LLLLLDEDICATION
vi
ACKNOWLEDGMENT
Being grateful is one of the biggest gifts bestowed upon someone. In the light of this fact, I want to express here my deep gratitude to the one who deserves the absolute grateful and thanking, which is the almighty Allah SWT, who gifted me uncountable goods, protected me countless difficulties and challenges, made one of my big ambitions achievable. Praise to you and you deserve more than gratefulness.
Next, I am also grateful for contribution made by all those of my family who made my journey to this moment smooth and easy, prayed for me, supported, motivated, and encouraged to reach this important milestone. Without them, it would be difficult to enjoy this great moment. These people are my grandmother, mother, father, aunt, siblings, and cousins. May Allah bless you and make your dreams true.
In addition, I should also express deep thanking and recognition to my supervisor Prof. Dr. Sonuç Zorlu Oğurlu for accepting to supervise me while she had too many students and administrative duties. Although she was busy, I believed for her extraordinary hardworking would give a chance to work with her. I appreciate her cooperation, support, and encouragement that were backbone for completing this work. I am happy to have her as my supervisor and I would like to state here that her guidance and continuous encouragement made this work to complete smoothly.
vii
TABLE OF CONTENTS
ABSTRACT ... iii
ÖZ ... iv
ACKNOWLEDGMENT ... vi
TABLE OF CONTENTS ... vii
1 INTRODUCTION ... 1
1.1 Historical Background ... 1
1.2 Significance and Importance of Quantum Calculus ... 2
2 BASICS OF QUANTUM CALCULUS ... 5
3 FINITE INTERVALS IN CALCULUS ... 13
4 IMPULSIVE -DIFFERENCE EQUATIONS ... 24
4.1 First Order Impulsive -difference Equation ... 24
4.2 Second Order Impulsive difference Equations ... 29
5 CONCLUSION ... 36
6 FURTHER STUDIES ... 37
1
Chapter 1
INTRODUCTION
1.1 Historical Background
For a long time, studying, investigating and developing calculus had based on using limits. Later it had appeared a calculus without limits called calculus. The quantum calculus started with F.H. Jackson in the beginning of last century as emerging area of mathematics, although it had been discovered already and vigorously studied by Euler and Jacobi.
We can separate the evolution of quantum calculus from historical perspective into two parts:
1. Development of quantum calculus in the period 1893-1950.
1893-1895 Rogers had some work on orthogonal polynomial that is possible to write using hypergeometric series. Rogers demonstrated the proof of two Rogers-Ramanujan identities that represent an infinite sum as a quotient of infinite product. During this time, the big battlefield came to Europe. Beginning from 1904, the English reverend Jackson had a lot of mathematics works intended completely to quantum calculus that lasted until 1951. Jackson worked on elliptic functions, and special functions.
2
He elaborated the link between the q-gamma function and elliptic functions.
In 1910, Watson [19] constructed the proof a q-analogue of Barnes contour integral expression for a hypergeometric series. Most of the people do not remember the extraordinary proof of the R-Ramanujan identities.
2. Development of quantum calculus in the second half of the 20th century. Many parts of quantum calculus evolved in the second half of the 20th century. In the 1950's, many of the great developers of the subject were Lucy J. Slater and D.B. Sears (1918-1999). L.J. Slater participated at Bailey's classes on hypergeormetric series in 1947-50 at Bedford College, London University and in 1966 presented the book [17], which explains the extraordinary developments made in the subject from 1936 when Bailey's book [7] was published. The so-called Sturm-Liouville q-difference equation had been worked in [7]. This equation is a q-analogue of the Sturm-Liouville differential equation.
.1.2 Significance and Importance of Quantum Calculus
For the last decades, it had attracted attention of researchers and scientists after the emergence of huge interest of mathematics that is used in modeling quantum computing. For further details, a distinct works can be found in the papers [4, 11, 12] and the references had been described there.
3
The area witnessed a great expansion, because of using foundations of hypergeometric series to the different subjects of combinatorics, quantum theory, number theory, statistical mechanics that are continuously discovered.
One of the most important works in quantum calculus is the book written in last decade by Kac and Cheung [18], which studies a lot of the foundational basis of quantum calculus.
It is widely understood that quantum calculus is a branch of broader mathematical area of time scales calculus. Time scales gives a generalized basis for working on dynamic equations on both discrete and continuous domains. The text by Bohner and Peterson [13] brought together important contribution in the calculus of time scales. Working out in quantum calculus focuses on a special time scale, called the q-time scale, described below:
* + .
In our work, we study finite intervals in quantum calculus. We describe the -derivative of a function , - and demonstrate some properties, for instance as derivative of a sum, of a product or a quotient of two functions. In addition, it is useful to describe -integral and provide its properties. We apply this concept to present existence and uniqueness results for initial value problem of first and second order impulsive difference equations.
4
impulsive problems on -time scale for the reason that the points and ( ) are consecutive. Finite intervals in quantum calculus the points and (
) are considered only in an interval , -. Therefore, using -calculus it is possible to solve systems with impulses at fixed times.
5
Chapter 2
BASICS OF QUANTUM CALCULUS
Definition 2.1
Let be a function defined on a geometric set , i.e. , .
For , the derivative of is;
L / f (z) f (qz) D f (z) , z {0} D f 0( ) (z). q (1 q)z q limz0 qD f
We see that lim D f lim f qz( ) f ) df .
q 1 (z (z) (z) (q 1)z dz q q 1 if f(z) is differentiable.
Some Properties of Derivative
6 Derivative of a Product of Functions
Derivative of a product of functions in quantum calculus is the same as in classic calculus (calculus with limits). Therefore, by definition 2.1 it can be stated as follows: Dq f f qz D hq h D fq f D hq h qz Dqf { (z)h(z)} ( ) (z) (z) (z), (z) (z) ( ) (z). (2.1)
This can be proved as follows:
q f (z)h(z) f (qz)h(qz) D f (z)h(z) , (1 q)z f (z)h(z) f (z)h(qz) f (z)h(qz) f (qz)h(qz), (1 q)z f (z)h(z) f (z)h(qz) f (z)h(qz) f (qz)h(qz), (1 q)z (1 q)z f (z)h(z) h(qz) h(qz)f (z) f (qz), (1 q)z (1 q)z f (z)D h(z) h(qz)D f (z).q qDerivative of a Quotient of Functions h D f f D f q q D (z) h(z) (z) (z) (z). h(z) (2.2) q h(qz)h(z)
For the proof, the following steps are can be done.
Take f (z) h(z)f (z). h(z)
7 q q f (z) D f (z) D h(z) h(z)
, using product rule 2.1
q q q f (z) f (z) D f (z) h(qz)D D h(z), h(z) h(z) q q q f (z) D f (z) D h(z) f (z) h(z) D , h(z) h(qz) q q q h(z)D f (z) f (z)D h(z) f (z) D . h(z) h(z)h(qz) Definition 2.2
The higher order derivative is expressed as
0 1
q q q q
D f(z)f(z), D f (z)D Df(z), N.
Integral of a Function (Jackson integral)
Suppose f(z) is arbitrary function. Define the operator E (f (z))q f (qz). Construct antiderivative of f(z). By definition 2.1 we have:
q F(z) F(qz) 1 f (z) (1 E )F(z). (1 q)z (1 q)
Formally writing antiderivative,
8 n q m 0 =(1-q) E (zf (z)),
m m m 0 =(1-q) q zf (q z)
Using geometric series expansion we get
m m q m 0 f (z)d z (1 q) q zf (q )
Provided the series converges.N.B. For convergence of the series see next theorem.
Theorem 2.1 [18]
Suppose that 0 < < 1. I f f (z)z is bounded on (0, A] for some 0 then the
above integral converges to a function F(z) on (0, A] which is antiderivative of f(z). Moreover, F(z) is continuous at z=0 with F(0)=0
9 By M-weierstrass test m m m 0 q f (q z)
is point wise convergent andm m m 0 F(z) (1 q)z q f (q z).
We now show that F(z) is continuous at z = 0. Indeed,
m m m 0 f (z) (1 q)z q f (q z) ,
m 1 m 0 1 (1 q)z N(q ) , z
1 1 m m 0 =N(1-q)z (q ) ,
1 1 (1 q) N z . (1 q ) z o lim F(z) 0 and F(0)=0. F(z) is continuous at Z=0 and F(0)=0.Now we show that F(z) is antiderivative of f(z).
10 Remark 2.1
If f is continuous at z = 0, then
q q
I D f (z)f (z) f (0).
This can be verified as follows:
m m q q q m 0 I D f (z) (1 q)z q D f (q z),
Since a partial sum of integral (Jackson’s integral)
m m 1 N m m m 0 f (q z) f (q z) (1 q)z q , (1 q)q z
N m m 1 m 0 f (q z) f (q z),
f (z) f (q N 1z),Which tends to f (z) - f (0) as N by the continuity of f (z) at z = 0.
Definition 2.3
For we set : * +and express the definite integral of a function by z q 0 0 z I f(z) f(l)d lq z(1 q q f q) ( ).
11
Note that for , we have , for some , The definite integral d q
cf(l)d l
is just a finite sum and it is clear that it converges. Corollary 2.1 [18]If f (z) exist in a neighborhood of z = 0, is continuous at z = 0, where f (z) denotes the ordinary derivative of f (z), we have
b
q q
aD f (z)d zf (b) f (a).
Proof.
We use L’Hopital’s rule to get
q
z 0 z 0 z 0
f (z) f (qz) f (z) qf (qz) f (0) qf (0)
lim D f (z) lim lim f (0).
(1 q)z (1 q) (1 q) q
D f (z)can be made continuous at z = 0 if
q f (z) f (qz) , z 0 (1 q)z D f (z) f (0) z=0 Integration by Parts
In calculus, the integration by parts formula is given by
z z 0 z qh q 0 0 q q . f (t)D (t)d t f (t)h(t) D f (t)h(qt)d t
This is proved as follows: From corollary 2.1 we get
q q
z
0f (t)D (t) dh tf (z)h(z) f (0)h(0). ( . ) 2 3
12 D f (t)h(t))q( f (t)Dqh(t) h(qt) Dqf (t).
integrating both sides of above we get
13
Chapter 3
FINITE INTERVALS IN
CALCULUS
Now we are studying the main part of our work, which is the concept of derivative and integral of finite intervals in quantum calculus.
Let * +, [ ] , be a constant. We define the derivative of a function , as follows:
Definition 3.1
Let is a continuous function, . Then
j q j j j j j j z f (z) f (q (1 q )z ) D f (z) , z , (3.1) (1 q )(z z ) z j j j q j z z q
D f (z ) lim D f (z), is called the of f at z.
We say that f is on provided
q j
D f (z) exist for all z .
We see that if = 0 and = q in (3.1) and
q j D f (z)= q D f (z), where q D is the derivative of the function f (z) defined in definition 2.1.
14 j 2 2 j j j q j j 2 2 j j j j j j 2 z (q (1 q )z ) D f (z) , (1 q )(z z ) (1 q )z 2q z (1 q )z , z z 3z 2z 1 = z z = , (1, 4], 2(z 1) z and 1 j j 2 lim D f (z) 2, if z 1. D f (3) 5
zz q has another way to write as
difference quotient f 3( ) f 2( ). 3 2
Example 3.2
In quantum calculus, we have Dqz [ ]qz1 where[ ]q 1 q 1 q . However, gives j j 1 q j q j D (z z ) [ ] (zz ) . Indeed, j j j j j j j q j j 1 q j , z (z z ) (q (1 q )z z ) D f (z) (1 q )(z z ) =[ ] (z z ) , and j j q j . 1 q [ ] 1 q Theorem 3.1 [12] Let f, h : KjRhave on Kj.
(1) The sum f + h : KjRhas on Kj,
j qj j
q qh
15
(2) For constant c cf : KjR has on Kj with,
j j
q q
D (cf )(z)cD f (z).
(3) The product f h : KjRhas on Kj,
j j j j j q q j j j q q j j j q D (fh)(z) f (z)D (z) h q (1 q )z D f (z) = h(z)D f (z) f (1 q )z D (z) h z q z h .
4 jz j j has a q -derivaj tive j17 Remark 3.1
In Example 2.2 we see that in difference, if ( ) then , - it is not possible to get easy formula for difference. Using the derivative of a
product, it is possible to write it as follows:
j q z D 1, j j 2 q z q (z ) ( j)z j j D D z 1 q (1 q )z j j 3 2 2 2 2 2 2 q z q j j j j j j zj D D (z .z) (1 q q )z (1 q 2q )zz (1 q ) , j j 4 3 q q 2 3 3 2 3 2 2 3 2 3 j j j j j j j j j 3 j j j j D z D z 1 q q 1 q 3q z 1 q 5q 3q z 1 (z ) ( q )z ( q )z ( )z ( q ) z . Definition 3.2
Let f : KjR as a continuous function. We call the second order derivative
j 2 q D f provided j q
D f has a derivatives on Kj with j j j 2
j
q q q
D f D (D f ) : K R. In a similar way, it is possible to define the higher order derivative
j j q
D : K R. For instance, f : KjR, then
18
To demonstrate antiderivative ( ), it is useful to describe a shifting operator by
j
q jz j j
E F(z)F q( (1 q )z ).
It can be verified by mathematical induction that
j j j q q q m m 1 m m j z j zj E F(z)E E F (z)F q 1 q , where and( ) ( ). Then by definition 3.1, we get
j j j j j j j q j z 1 E F(z) F(q (1 q )z ) F(z) f (z). (1 q )(z z ) (1 q )(z z )
Therefore, antiderivative becomes as follows:
j j q j 1 F(z) (1 q )(z z )f (z) . 1 E By expanding the geometric series, we get:
j m j j m 0 m m m m j q j j j j j j j m 0 m m m j j j j j j m 0 z z z z z z F(z) (1 q ) E (z z )f (z) =(1 q ) q 1 q z f q 1 q = (1 q )(z z ) q f q 1 q . (3.2) It is obvious that the above calculus is true if the series in the last part converges.
Definition 3.3
Let is a continuous function. Then the is defined by
19
Moreover, if ( ) then the definite integral is defined by
j j j j j z c q q q z z m m m m m m j j j j j j j j j j j j m 0 z m c 0 f (l) d f (l) d f (l) d =(1 )(z z ) q f q 1 q z (1 q )(c z ) q f q 1 q l l z c . l z q
20 Theorem 3.2 [12] For , j j j j j j j j z q z q z q q z z q q j c (1) D f (l) d f (z); (2) D f (l) d f (z) (3) D f (l) d f (z) f (c) for c (z , z). l l l
Proof.(1) Applying definitions 3.1 and 3.3, we get
21 m m m 1 m 1 j j j j j j m j j m m 0 j j f (q z (1 q )z ) f ((q z (1 q )z ) =(z z ) q , q (z z ) m m m 1 m 1 j j j j j j m 0 = f (q z (1 q )z ) f ((q z (1 q )z ), f (z).
(3) Second part of this theorem leads to
j j j j j j j j z z c q q q q q q cD f (l) d l z D f (l) d l z D f (l) d lf (z) f (c).
Theorem 3.3 [12]Let are continuous function .Then for
j j j j j j j j j j j j j j j j z z z q q q z z z q q z z z z q q j j q q z z z z j (1) [f (l) h(l)]d f (l) d h(l) d (2) ( f )(l) d f (l) d (3) f (l)D (l) d (fh) l l l l l h l (z) h(ql (1 q )z )D f (l) d l
Proof.By Theorem 3.1 part (3), we get
j j j
q q j j j q
f (z)D h(z) D (fh)(z) h(q z (1 q )z ) D f(z). integrating the above equation and using second part of theorem 3.2 we obtain the outcome in (3) as desired.
22 Proof.
Using definition 3.3 we get
24
Chapter 4
IMPULSIVE
-DIFFERENCE EQUATIONS
Let , - , - ( ] for continuous vector space defined * ( )+ be continuous everywhere except at which ( ) ( ) exist and ( ) ( ) , . ( ) is a Banach space with norms ‖ ‖ *| ( )| +.
4.1 First Order Impulsive
-difference Equation
In this section, we discuss the existence and uniqueness of solutions for the following initial value problem for the first impulsive difference equation.
j q j D t(z)f z, t(z) , zK, zz ,
j j j Δt(z ) I t(z ) , j1, 2, , n, ( 1 4. ) 0 0 0 1 2 j n n 1 t(0)t , t R, 0z z z z z z Z f : K R Ris a continuous function j j j j j I S( , ), Δt(z )R R t(z ) t(z ), j 1,2, ,n an d 0q 1 for j0,1, 2,. n.. Lemma 4.1If tVC(K, R) is a solution of (4.1), then for any xKj, j0,1, 2,, n
25 Proof.
0 1 0 z 0 0 0 0 q 1 0 0 z q t(z) t f l, t(l) d , which lFor z K , q integrating (4.1), it foll
eads to t(z ) t f l, t(l ow ) s l l d .
1 1 1 1 1 z z qFor zK , taking q integral of (4.1),it becomes t(z)t z
f l, t(l) d .l
1 0 1 1 1 1 1 1 z z 1 0 z 0 q q 1Since t(z ) t(z ) I (t(z )), then we have
t(z) t f l, t(l) d l f l, t(l) d l I t(z ) .
2 2 1 2 0 1 2 1 2 2 2 z q z z z z 0 2 1 1 2 2 q q q 0 z z 2Again q -integrating of (4.1) from z to z,where z t(z) t f l, t(l) d =t f l,t(l) d f l,t(l) d f l,t( K , then z l l l l) d l I t(z ) I t(z ) .
Repeating the process considered above for we get (4.2).
Now, let ( ) be a solution of (4.1). Using derivative of (4.2) for , it becomes the following:
j q D t(z)f z, t(z) . It is clear that j j j 0Δt(z ) I (t(z )), j 1, 2, , n and t(0) t . This gives the proof.
Theorem 4.1 [12] Let the following hold.
1
H f : K R R is a continuous function and satisfies
26
j
j 2
j
H I : j 1, 2, , n are continuous function and satisfy I (t) I (u) N | t u |, ( ) N 0, t, u . If SZ nN 1, R R R
Then the nonlinear impulsive difference initial value problem on (4.1) has unique solution on
Proof.
We define the operator P: VC(K, R)VC(K, R) by
j j 1 j j 1 j j j z z 0 z q j j z q 0 z z 0 z z ) t f (l, t(l))d l I (t(z )) f (l, t(l) (Pt)(z )d l. 1 j 0 0 z k 2
Given (.)0. Let sup | f (z, 0) | andmax{| I (0) |:j 1, 2, , n} ;
Let us take a constant W such that
0 1 2
1 w | t | Z n , where 1. 1 Now we will state that PDwDw where a ball Dw {t VC(K, R) : t w}. For any tDw and for each zK,
27 This implies that
j j 1 j 1 j j j j z q z 0 z Z z j j j j q z 0 z z
For t ) and for each z
( f (l, t(l) f (l, u(l)) d l + I (t(z )) I (u(z )) f (l, t(l , u VC(K, R K we have (Pt)(z) ) f (l, u(l)) d l ( Pu)(z)
j j 1 n j 1 n j j z z q j j q z z 0 z z 0 z ZS t(l) u(l) d l N t(z ) u(z ) (S t(l) u(l) )d l (SZ nN) t u .
Then it becomes as follows: ‖ ‖ ( )‖ ‖
As SZ + nN < 1, by Banach contraction mapping principle, P is a contraction. Thus P is a fixed point which is a unique solution of (4.1) on J.
Example 4.1
Let us solve this first order impulsive difference initial value problem.
28 z 2 j j 1 j 2 Z 1, f (z, t) (e | t |) / ((z 5) (1 | t |)) and I (t) | t | /(12 | t |). Sin ce f| (z ), t f(z, u) | (1 / 5) | t u and I| (t) I (u) | (1 / 12) | t u|, then (H , H are satisfied w) ( ) ith
( ), N ( ). We can show S 1/ 5 1/ 12 1 9 19 SZ nN 1. 5 1 ha 0 t t 2 2
29
4.2 Second Order Impulsive
Difference Equations
In this section, we discuss the second order initial value problem of impulsive difference equation of the form
j j j 1 0 2 q j j j j * q j q j j j q D (z) f z, t(z) , K, z z , Δt(z ) I t(z ) , 1, 2, , n, (4.4) D D (z ) I t(z ) , 1, 2, , n, t( t z j 0) t z t j t , D (0 ) , Where ,R , 0z0 z1 z2 zj zn zn 1 Z, f : K R Ris a continuous function, I ,j I*jC(R, R)t(z )j t(z )j t(zj)for j1, 2,, n andj
0q 1 for j = 0,1,2,3,…,n.
Lemma 4.2
The unique solution of problem (4) is given by
j j 1 j 1 j j j j 1 j 1 j 1 j 1 j j z j j 1 j 1 j 1 q j j z 0 z z z z q j j j q j j z z 0 z z 0 z z t(z) ( )z ) (l (l)) (t(z )) (l (l) z (z q l 1 q f ) (t( , t d l I +z f , t d l I z f , t d l I z )) (l (l)) (t(z ))
j j z j j j q z 0 0 (z q l 1 q f , t d l, (4.5) wi ( ) th z ) (l ( (.) l)) 0.
Proof.For zK0 and integrating for the first equation of (4.4), we obtain
30 which gives us
1 0 0 z q 1 0 q D t(z )
f l, t(l) d . 4.l ( 7) ForzK0. By integrating (6) it becomes
0 0z l
q q 0 0
t(z) z
f , z( ) d d l, reversing the order of the integral, we get
0z
0 q
0
t(z) z
(z q )f l, t(l) d . (4 ) l l .8 In special case for z =
1 0 z 1 1 0 1 0 q t(z ) z
(z q )f l, t(l) d . (4 )l l .9 1 1 2 1For zK (z , z ], q integrating (4.4) it becomes
1 1 1 1 z q q 1 z q D t(z)D t z
f l, t(l) d .l
Using the third condition of 4.4 with 4.7 the result becomes
1 1 0 1 1 z z q 0 q 1 1 z q D t(z)
f l, t(l) d lI t(z )
f l, t(l) d . l (4.10)
1 1 1For xK,q -integrating 4.10 then converting order of q integral we get
1 0 1 1 z 1 0 q 1 1 1 1 1 z q z 1 t(z) t f l, t(l) d I t(z ) (z z ) + z q (1 q )z f l, t(l) d . z l l l (4.11)
31
1 0 1 1 1 1 z 1 0 1 0 q 1 1 z q 1 z z 1 1 0 1 1 1 q t(z) z (z q l)f l, t(l) d I t(z ) f l, t(l) d I t(z ) (z z ) + z q (1 q )z f l l, t(l) l l d .l
1 0 1 1 1 1 z 1 0 1 0 q 1 1 z z q 1 1 1 1 1 1 q 0 z z (z q l)f l, t(l) d I t(z ) f l, t(l) d I t(z ) (z z ) z q (1 q )z f l l l l , t(l) d .l
Repeating the above steps, for , we get (4.5) as needed. Now, it is useful to prove the existence and uniqueness of a solution to the initial value problem (4.4).
We will apply Banach fixed point theorem to do this.
Theorem 4.2 [12]
Let the assumptions ( ) and ( ) hold. Furthermore, assume that (H3)I : Rj R,
j 1, 2, , n are continuous functions and hold
1 * 4 j 2 j 3 I (t) I (u) N | t u |, 0, t, u . if : S(e Ze e ) nN (nZ e ) N N 1, R where n 2 n 1 j 1 1 2 j 1 j 1 j 1 j 1 3 j 1 4 j 1 j j j n j j 1 n j (z z ) e , (z z ), 1 q z (z z ), z e e e ,
32 Proof.
First in light of Lemma 2, we establish F : VC(K, R)VC(K, R)as
j j 1 j 1 j j j 1 j 1 j j j 1 j 1 z j j 1 j 1 j 1 q j j z 0 z z z q j j x 0 z z z j z q j j 0 Ft z (z q l 1 q f , t d l I +z f , t ( )(z) ( )z ) (l (l)) (t(z )) (l (l))d l I (t(z )) z f , t(l (l))d l I (t(z ))
j j j z z z j j j q z (z q l (1 q )z ) (l (l))f ,t d l,
Given
0 0 (.)0. z K 1 j 2 3 jLet sup f , 0 , max I 0 : j 1, 2, , n , and max{I (0) | (z ) | : j 1, 2, , n} { ( ) } .
We will verify thatF DW DW, where DW {t VC J,( ) : t W} and constant W
satisfies | | | | Z 1(e1 Ze2 e ) n3 2 (nZ e )4 3 W , 1
where 1. FortDW, from example 3.3, we have
34 j j 1 j 1 j j 1 j 1 n z j j 1 j 1 j 1 q j j j j z j 1 n z q j j j j z j 1 For any t ( )(z) ( )(z) ( )z ) (l (l)) f (l, u(l)) (t(z )) (u(z )) (l (l) , u VC(K, R), we have Ft Fu (z q l 1 q f , t d l I I +Z f , t ) f (l, u(l)) d l I (t(z ) I (u( )t )
j j 1 j j 1 n n n z j q j j j z j 1 Z n n n q z (l (l)) f (l, u(l)) (t(z )) (u(z )) ( )z ) (l (l)) f (l, u(l)) z f , t d l I I (z q l 1 q f , t d l,
2 n n j j 1 j j 1 j 1 j 1 j 1 2 n n j j 1 j 1 n (z z ) S N t u Z (S(z z ) N ) t u (1 q ) (Z z ) + (S(z z ) N ) t u S t u 1 q = t u ,
which lead to ‖ ‖ ‖ ‖. Since by the Banach contraction mapping principle has a fixed point, which is a unique solution of (4.4) on K.
Example 4.2
Consider this second order impulsive -difference initial value problem:
35 2 sin z 2 j j 1 j q 2 / (3 j), j 0,1, 2, , 9, n 9, Z 1, f (z, t) (e | t |) / ((7 z) (1 | t |)) I (t) | t | /(5(6 | t |)), and I 1/ 9 tan / Here, (t) ( ) (z 5). | f Since (z, t) f (z, u) | (1/ 49) | t u |, j j 1 2 j 3 j I I 1/ 30 u | I (t) I (u) | (1/ 45) | t u | . Then | (t) (u) | ( ) | t |, and ( ) ( ) ( )
H , H and H holding with S (1/ 49), N (1/ 30), N (1/ 45).
The results becomes the following:
n j j 2 n 1 j 1 1 2 j 1 j 1 j 1 j 1 3 j 1 4 j 1 n n j j j j 1 (z z ) 1,380,817 9 , (z z ) , 1 q 180,180 10 45 45 z (z z ) , z . 100 10 e e e e
Clearly,Z(e1Ze2e ) nN (nZ e )N3 4 0.7839 1.Hence, by theorem 4.2, the initial value problem (4.12) has a unique solution on [0, 1].
36
Chapter 5
CONCLUSION
As we say earlier in the introduction, -calculus plays an important role in getting solutions to the systems with impulses at fixed times.
37
Chapter 6
FURTHER STUDIES
I would like to suggest a further and broader study and research in -calculus in order to get a broader area for research and application. For example, studying exponential, trigonometrical areas, fundamental theorem, and mean value theorem of
--calculus.
38
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