Existence of Solutions to First-Order Dynamic
Equations on Time Scales
INSTITUT PENGURUSAN PENYELIDIKAN
UMVERSITI TEKNOLOGI MARA
40450 SHAH ALAM, SELANGOR
MALAYSIA
Dr Mesliza Mohamed
Mubammsd Sufian Jusoh
NOVEMBER 2009
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TEICNOLOGI
MAR&
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'Tarikh
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Dr Mesliza Mohamed
Ketua Projek
Fakulti Kejuruteraan Awam
UiTM Cawangan Perlis
Kampus Arau
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PERLIS
Muhammad Sufian Jusoh
Ahli Projek
Fakulti Kejuruteraan Awam
UiTM Cawangan Perlis
Kampus Arau
Peti Surat 41
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PERLIS
TAJUK PROJEK PENYELIDIKAN DANA KECEMERLANGAN: EXISTENCE OF
SOLUTIONS TO FIRST-ORDER DYNAMIC EQUATIONS ON TIME SCALES
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'SELAMAT MENJALANKAN PENYELIDIKAN'
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Yang benar,
Pengarah Kampus
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2. -
Prof Madya Dr Nik Ramli Nik Abd Rashid
Koordinator RMU
UiTM Cawangan Perlis
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Tarikh: 15 Ogos 2009
No. Fail Projek: 600-IRDC/ST/DANA 5/3/Dst (85/2008)
Penolong Naib Canselor (Penyelidikan)
Institut Penyelidikan, Pembangunan dan Pengkomersilan
Universiti Teknologi MARA
40450 Shah Alam
Ybhg. Prof.,
LAPORAN AKHIR PENYELIDIKANUEXISTENCE OF SOLUTIONS TO
FIRST-ORDER DYNAMIC EQUATIONS ON TIME SCALES,"
Merujuk kepada perkara di atas, bersama-sama ini disertakan 3 (tiga) naskah
Laporan Akhir Penyelidikan bertajukUExistence of Solutions to First-Order
Dynamic Equations on Time Scales".
Sekian, terima kasih.
Yang benar,
k
Dr Mes ~ z a ohamed
d
Ketua
Projek Penyelidikan
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PENGHARGAAN
Setinggi-tinggi penghargaan dan ribuan terima kasih diucapkan kepada semua
pihak yang terlibat secara langsung dan tidak langsung bagi membolehkan
penyelidikan ini disiapkan dengan sempurna.
Diantaranya:
Prof. Madya Dr Hamidi Abdul Hamid
Pengara,h Kampus
UiTM Perlis, 02600 Arau, Perlis
Prof. Madya Dr Nik Ramli Nik Abd Rashid
Timbalan Pengarah Kampus Penyelidikan & Jaringan Industri (PJI)
UiTM Perlis, 02600 Arau, Perlis
dan
Semua staf yang telah memberikan kerjasama dan sokongan di dalam
menjayakan penyelidikan ini.
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Contents
1 Dynamic equations on Time Scales
1.1 Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Calculus on Time Scales . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 A Basic Concept of ODES . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Reduction of Higher-order Systems to First-order Form . .
1.3.2 The Solution of a BVP . . . . . . . . . . . . . . . . . . . .
1.4 Review of Recent Works . . . . . . . . . . . . . . . . . . . . . . .
1.5 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Discrete Multipoint BVPs . . . . . . . . . . . . . . . . . .
1.5.2
Application of Multipoint Boundary Value Problems on
Dynamic Scales . . . . . . . . . . . . . . . . . . . . . . . .
2 First-Order Two-point Dynamic BVPs
19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2
3 Discrete First-Order three-point BVP
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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23
23
CONTENTS
2
3.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.3 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.4 Convergence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Discrete First-Order four-point BVP
38
:
. . . . . .
38
. . . . . . . . . . . . . . . . .
39
4.3
Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.4
Convergence of Solutions . . . . . . . . . . . . . . . . . . . . . .
50
4.5 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . .
4.2
Notation and Preliminary Results
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
References
52
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Introduction
Applied mathematics refers to the use of mathematics to understand real
world systems. To apply mathema.tics, real world systems are described using
mathematical language to create a mathematical model [3]. This creation of
models is heavily influenced by the available mathematics techniques. Scientist
can only st,udy real world systems described by models that are solvable using
existing mat hematical technique. Further, wliere different mat hematical models
may be appropriate to describe a real world system, scientists tend to choose
simpler models, especially those with analytic solutions.
As a result, one of the main aims in applied is the development of theory
for new types of mathematical models. This allows the study of more types of
phenomena and encourages the use of models that fit real world behavior more
closely. The most common mathematical models studied are those that depend
on continuous variables, commonly time [12]. On example of this is an initial
value problem of the form
The general mathematical theory behind these models is differential calculus,
and so equations over continuous variables have been termed 'differential
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CONTENTS
equations' [24]. Paralleling the development of the differential calcul~zshas been
the use of models that depend on discrete variables [I]. Interest in this area has
grown with increased computing power, due to applications such as the method
of finite differences, where models with continuous variables are estimated by
data at discrete points [22]. The standard term for functional equations over
discrete variables is 'difference equations' [lo].
The theory of 'dynamic equations on time scales' generalizes the theory of
differential and difference equations revealing the links and anomalies between
them in the process. The concept is particulatly useful in modelling stopstart
processes where continuous and discrete time may be present at different stages.
For example: insect population models in biology (see [27] p.7ff); the periodic
discharge of a capacitor in circuit theory (see [4], p.15); and hybrid systems [28]
all feature continuous and discrete time in the modelling process.
This report includes four chapters. In the first chapter, we shall recall basic
concepts of the generalized calculus on time scales, basic concept of ordinary differential equations (ODEs) and boundary value problems (BVPs). In the second
Chapter, we establish the existence of solutions to BVPs involving systems of
first-order dynamic equations on time scales subject to two-point boundary conditions. In the third Chapter, we investigate the existence of solutions to threepoint BVP involving a system of difference equations which arise as a discrete
approximation to a three-point BVP for a system of first-order ODEs. Finally,
in Chapter 4, we extend the existence results to four-point BVPs for systems of
first-order difference equations associated with systems of first-order ODEs.
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Chapter 1
Dynamic equations on Time
Scales
In this chapter, we present some basic concepts of the generalized calculus on
time scales which are useful for our study. For more details one may refer to the
following [2, 4, 51. Then we introduce a basic concept of ODES and BVPs, which
is helpful in the study of the BVPs.
1.1
Time Scales
A time scale T is any closed non-empty subset of the reals, R. We assume that
any time scale has the topology inhereted from the standard topology of the reals.
Examples of time scales include the real R, the integer
Ny,along with any finite union of
Definition 1.1.1 Let
Z,the natural numbers
closed intervals, such as [I,21 U [3,4].
T be a time scale. For all t E 'It' we def;ne the forwar jump
operator cr : T ---+ T by
~ ( t:=) inf{s E T : s > t ) ,
and the backward jump operator p : T ---+
T by
p ( t ) := sup{s E T : s
< t).
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1.1 Time Scales
6
We also use the natural definition for the empty set, letting inf4 = supT, so if
T has a maximum t,,,,
then at this maximum a(t,,,) = t,,,.
Similarly, we let
sup4 = inm, SO if 'IT has a minimum tmin,~ ( t m i n=
) tmin.
Example 1.1.1 W e compute the jump operators for a closed interval, for T = R
and Z .
If T = [a, b] E E then for any t E [a,b)
a(t) := i n f l s
ET :s
> t ) = i n f { ( t ,b ) ) = t ,
(1.3)
and a ( b ) = supT = b. Similarly, p(t) = t , for any t E [a, b].
and similarly, p ( t ) = t - 1
For any arbitrary time T, we can describe the behavior at each point t E T using
the jump operators. Where the jump operator is equal to t, we term thid dense.
Hence if a ( t )= t then t is right-dense, and if p(t)= t then t is left=dense. Point
that are left and right dense at the same time are called dense. Conversely, where
the jump operator is not equal to t , we term this scattered. Hence if a(t) > t
then t is right-scattered, and if p ( t ) < t then t is left scattered. points that are
both left and right scattered are called isolated.
Definition 1.1.2 Let T be a time scale. T h e grainess function p : T
-+
[0,m)
is defined by
p ( t ) := a ( t )- t.
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(1-5)
1.2 Calculus on Time Scales
7
I f T = [a,b]E R then for a n y t E [ a , b ] , a ( t )= t . Hence
Example 1.1.2
p ( t ) := a ( t )- t = 0 ,for all t E T.
(1-6)
If T = Z then a ( t ) = t + 1. Hence
p ( t ) := a(t
1.2
1.2.1
+ 1) - t = 1,for
all t E R.
(1.7)
Calculus on Time Scales
Differentiation
We first define Tk, a subset of the time scale If, in order to establish a set on
which a derivative function can be defined. If T has a, left-scattered maximum
rn, then Tk = T - rn. Otherwise Tk = T. This gives
Tk = If, supT < oo
=
Secondly, a neighborhood
U of
T,ifsupT = oo.
a point t E If is the set of points in T that are
some small distance from t . Formally, U = (t - 5, t
Definition 1.2.1 Assume f : T
+R
+ 6) n T for some 5 > 0.
is a function and let t E Tk. Define
f A ( t ) , the delta derivative, to be the number (if it exists) with the following property: for any given
E
> 0 , there exists a neighborhood U of t b such that
W e call f delta diflerentiable on Tk provided that f * ( t ) exists for all t E Tk. The
function fa : T"
E% is called delta derivative o f f on Tk.
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1.3 A Basic Conce~tof ODES
1.3
8
A Basic Concept of ODEs
In this section, we begin with a basic concept of ODEs and BVPs. It is useful
in order to fully understand the problem considered by Ma [20] and ex-tension of
his work in this study.
ODEs are equations that relate the value of an unknown function of a single
variable to its derivative. Some example of ODEs.
Example 1.3.1 Equation of motion for a harmonic oscillator
is a n O D E for the position x ( t ) of a particle of a mass m, mounted o n a spring of
stiffness c, when subjected to a time-dependent force F ( t ) . This is a second-order
O D E because the highest derivative of the unknown function, x ( t ) , with respect
to the independend variable, t , is of second order.
Example 1.3.2 Transverse deflection of a string under axial tension.
The equation
is a n ODE that describes the tranuerse deflection y ( x ) of a pre-stressed elastic
string (under axial tension T ) , loaded transversely by a pressure p ( x ) . This is a
second-order ODE because the highest derivative of the unknown function, y ( x ) ,
with respect to the ~ndepen~dent
variable, x , is of of second order.
Example 1.3.3 Radioactive decay.
The equation
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1.3 A Basic Concept of ODES
9
is and ODE that describes how the mass m(t) of a radioctive material with decay
rate X decays. This is a first-order ODE because the highest derivative of the
unknown function, m(t), with respect to the independent variable, t , is of firstorder.
1.3.1
Reduction of Higher-order Systems to First-order
Form
The system of ODEs must be written in the form
that is the system must be given in the first-order form. The n dependent
variables (the solution) xl, x2, - - . ,x, are functions of the independent variable
t , and the differential equations give expression for the derivative xi
terms of t and xl ,5 2 , -
- - ,x,. For a system of
=
% in
n first-order ODEs, n associated
boundary conditions are usually required to define the solution.
A more general system may contain derivative of higher order, but such system
can almost be reduced to the first-order form (1.8) by introducing new variables.
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1.3 A Basic Concept of ODES
10
For example, taking the third-order equation
x"'
+ Zz" = k(1 -
and writing xl = x, x2 = Z' and
z3 = 2'' we
=0
obtain the first-order system
ODEs must be augmented by additional constraints in the form of boundary
conditions.
1.3.1.1 Boundary Conditions:
Boundary conditions specify the value of the unknown function at the "left" and
"right" end of the -domain. The combination of an ODE and its boundary conditions is known as a BVP. Two-point BVPs for first-order differential equations
require the solution of the differential equation
subject to two-point boundary conditions, usually expressible as
where the numbers a , b are the "two points", and f , g, and x can be either real or
vector valued. If f and x are vector valued, the most general possible three-point
BVP with linear boundary conditions can be written as
Ax(a)
+ Bxb) + Cx(d) = !.
Here x(t) is an n-dimensional vector with components xj(t); a , b, c E
a
lR with
< b < c; f (t,x) is an n-vector with components fk(t,x) which are functions
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1.4 Review of Recent Works
of the n
11
+ 1 variables t , xj,j = 1,. . - ,n; A, B and C are n by n matrices with
constant elements and !is a fixed n-vector.
1.3.2
The Solution of a BVP
The solution to a BVP is any function that satisfies the ODE and the boundary
conditions. However, it is not necessarily easy to find that solution from first
principles. We can find a lot of techniques for the solution of the ODEs, such
as separation of variables, integrating factor in our first year. However, not all
the solutions can be easily found. Hence, we need a degree-theory to prove there
exist a solution.
1.4
Review of Recent Works
Existence of solutions to BVPs means the problem has a solution, meanwhile
uniqueness mean the problem has only one solution.
Ma [20] has considered the following three-point BVPs for a syst,enl of firstorder ODEs.
where M N, R are constant
square matrix,
and a a vector scalar,
f : [O, 11 x Rn -+ Rn is a Carathiodory function, and x E C1[O,11.
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1.4 Review of Recent Works
12
The solution to BVP (2.3)' (2.4) is given by
Ma [20] assumed that the solution x ( t ) in equation (1.10) satisfied the condition
det(M
+ N + R) # 0. Then Ma 1201 obtained a 'priori' bound on solutions x ( t )
i.e. the solution is bounded by a constant. It is a necessary condition to show
that the solution exist. Then he applied the Leray Schauder degree theory to
show that there exist at least one solutions to the above BVPs.
In literature, the multi-point BVPs have been considered by a few authors but
with different approach from Ma (201. For example, multipoint BVPs with linear
boundary conditions considered by Datta and Henderson [8] provided discrete
analogues to results giving differentiability of solutions to multipoint BVPs for
first-order systems on ODES with respect to boundary conditions. They assumed
that the function f and its first-order partials are continuous on Rn. Urabe [29]
gave a sufficient condition under which the existence of the approximate solution
ensures the existence of an exact solution for a system of first-order multipoint
BVPs with linear boundary conditions. Rodriguez [25] proved the existence
and uniqueness of solutions for a system of first-order discrete multipoint BVPs
under which the properties of linear multipoint BVP are preserved under 'small'
nonlinear perturbations of both the difference equations and the boundary
conditions. Rodriguez [25] proved the existence and uniqueness of solutions for
a system of first-order discrete multipoint BVPs under which the properties of
linear multipoint BVP are preserved under 'small' nonlinear perturbations of
both the difference equations and the boundary conditions.
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1.5 Owen Problems
1.5
13
Open Problems
It seem natural to extend the working of three-point BVPs in two possible directions:
by attempting to remove condition det(M
condition det(M
+ N + R)
#
0 (Imposed the
+ N + R) = 0, which is an extension of the work of Ma
1201.
by considering wider classes of BVPs with multipoint boundary conditions.
Currently very few articles appear on systems of first-order multipoint
BVPs, see ([20]p. 212, [19] p. 1308)
The idea of Ma's [20] also can be applied to prove the existence of solutions
to the area of dynamic scales and impulse differential equations.
In this research, we only discuss on discrete problem with threepoint and fourpoint boundary conditions, which will be discussed in Chapter 3 and Chapter 4
respectively.
1.5.1 Discrete Multipoint BVPs
The theory of the qualities of solutions to difference equations are particularly
interesting, as studies have shown rich distinctions and interesting theory of solutions to differential equations. The continuing interest in the field of difference
equations can be attributed to two main factors:
due to the theory's powerful and versatile applications to almost all areas of
science, engineering and technology, which teem with discrete phenomena;
and
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1.5 Open Problenls
14
from the emergence and popularity of computers, where differential equa,
tions are solved by utilizing their approximate dfierence-equations formulas. Therefore, scientific advancements in the area of difference equations
are naturally motivated, and are of significance interest.
The discretized BVPs and the 'effect' that this discretization may have
on possible solutions when compared with solutions to the original continuous
BVP, have been researched by [I],[Ill and [18]. For example, Agarwal, provides
some examples showing that even though the continuous BVP may have a
solution, its discretization may have no solution. Thus for discrete case we
need to formulate a convergence theorem which is a generalization of Theorem
2.5, [ll]. The Theorem showing that if solutions to the continuous problem
are unique, then solutions to the discrete problem converge to solutions of the
continuous problem. Some examples of Numerical scheme that can employed are
Euler's method, Trapezoidal rule etc. The numerical methods of interest provide
solutions that closely approximate the exact solutions for sufficiently small
step size (see Keller, 1991). Since the Euler method is the simplest numerical
scheme for solving initial value problems, we should employ this method for
approximating the solution of continuous differential equations.
Then we
should formulate a convergence theorem which is a generalization of [ll,Theorem 2.51, showing that if solutions to the continuous problem are unique, then
solutions to the discrete problem converge to solutions of the continuous problem.
The following examples have been chosen to illustrate the diversity of the uses
and types of difference equations.
Example 1.5.1 (see [16]) I n 1626, Peter Minuit purchased Manhantann island
for goods worth $24. If the $24 colud have been, invested at a n annual interest
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1.5 Open Problems
15
rate of 7% compounded quarterly, what would would it have been worth in 19982
Let x ( t ) be the value of the investment after t quarters of a year. Then y(0) =
24. Since the interest rate is 1.75% per quarter, x ( t ) satisfies the difference
equation
for t = 0 , 1 , 2 , - - - . Computing x recursively, we have
After 37'2 years, or 1488 quarters, the value of the investment is
Example 1.5.2 (see (161) It is observed that the decrease in the mass of a radioactive substance over a jixed time period is proportional to the mass that was
present at the beginning of the time period. If the half life of radium is 1600 years,
find a formula for its mass as a function of time.
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1.5 Open Problems
16
Let m(t) represen,t the mass of the radium after t years. Then
where k is a positive constant. T h e n
for t = 0,1,2, - - . . Using iteration as in the preceding example, we find
Since the half life is 1600,
and we have finally that
This problem is traditionally solved in calculus and physics textbooks by setting up
and integrating the dzflerential equation m'(t)= - k m ( t ) . However, the solution
presented here, using a diference equation, is somewhat shorter and employs only
elementary algebra.
1.5.1.1
Existence Results for Discrete Problems
In continuos case, Ma [20] used the Leray Schauder degree to show the existence
of at least one solution. For the this researh, to show the existence of at least
one solutions for the discrete three-point BVPs, we need to apply the Brouwer
Fixed Point Theorem which is given in Kelley and Peterson [16] p. 382.
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1.5 Open Problems
17
-
Theorem 1.5.1 (Brouwer Fixed Point Theorem) Let K = {(xl,
xi 5 di, i
=
1 , . . . , n) and suppose T : K
,x,)
: ci <
-
K is continuous. Then T has a
fixed point in I(.
Then to obtain the existence of a unique solution for discrete BVPs, we should
apply the Contaction Mapping Theorem which is given in ~ e l l and
e ~ Peterson
[16] p. 382.
Theorem 1.5.2 (Contraction Mapping Theorem ) Let I . I be a norm for Rd and
S a closed subset of Rd. Assume T : S
-+
S is a contraction mapping: there is a
a,O5a<lsuchthat(Tx-TyI<aIx-y(forallx,yinS.
ThenThasa
unique fixed point z in S.
1.5.2
Application of Multipoint Boundary Value Problems on Dynamic Scales
The disparity between the theory of differential and difference equations has
led to difficult choices in inathematical modelling. To address this disparity,
equation depending on both discrete and continuous variables have recently
been studied, in the area of study known as 'hybrid dynamical system [21].
However this area does not deal with the many real world systems that include
variables with both discrete and continuous parts. One standard approach for
such systems is to separate models into different domains, correspoilding to the
discrete and continuous parts of a variable [24]. Another common approach is
to reduce the model to continuous or discrete variables only, either by 'filling
in' the discrete parts of variables with approximations, for example by using
interpolation [6], or by discretising continuous variables [ll]. however, these
approach may not result in accurate models. There may be difference between
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1.5 Open Problems
18
model behavior in the discrete and continuous versions of the model [l]
The development of time scales calculus in 1988 by Hilger in his PhD theses [13] was a response to these problems. Hilger aimed to formulate a theory
of calculus that would allow the study of models involving variables with both
continuous and discrete parts. The idea is to formulate the theory for a general
dynamic equation, which depends on variables that may have both continuous
and discrete parts.
Thus the theory of 'dynamic equations on time scales' generalizes the theory
of differential and difference equationsLrevealing the links and anomalies between
them in the process. The concept is particularly useful in modelling stopstart
processes where continuous and discrete time may be present at different stages.
For example: insect population models in biology (see [27] p.7ff); the periodic
discharge of a capacitor in circuit theory (see [4], p.15).
Dai and Tisdell [7] investigated the existence of solutions to BVPs involving systems of first-order dynamic equations on time scales subject to two-point boundary conditions.The methods involve novel dynamic inequalities and fixed-point
theory to yield new existence theorems guaranteeting the existence of at least
one solution.
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Chapter 2
First-Order Two-point Dynamic
BVPs
2.1
Introduction
In this section, we establish the existence of solutions to BVPs involving systems
of first-order dynamic equations on time scales subject to two-point boundary
conditions. The existence of solutions use the the Contraction Mapping Theorem
guaranteeing the new existence of a unique solutions.
Consider the existence of solutions to the first-order dynamic equation
subject to the boundary conditions
where f : [a,ellr x R"
constant in
-+
Rn is a continuous, nonlinear function; a, c are given
T;M , R are given constants in R.
Throughout this section, assuming
M+R#O.
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