CHAPTER ONE
1.0 INTRODUCTION
Banach’s fixed point (also known as the contraction mapping theorem or contraction mapping principle) concerns certain mappings of a complete metric space into itself; it is also an important tool in the theory of metric spaces, theory of ordinary and partial differential equation. It guarantees the existence and uniqueness of a fixed point (point that is mapping into itself) i.e. self maps of metric spaces and provides a constructive method or iterative process by which we can obtain these fixed points.
In this paper work, we would restrict ourselves on its application to integral equations.
1.1 DEFINITONS OF TERMS AND EXAMPLES
Before we proceed, it is thoughtful we define some fundamental terms with some examples which will enable us understand the subject matter.
1.2 METRIC/METRIC SPACE
A metric space is a pair (X,d), where X is a set and d is a metric on X (or distance function on X), that is a function defined on X X such that for all X we have:
(M1) d is real-valued, finite and non-negativve
(M2) d ( 0 if and only if
(M3) (symmetry)
(M4) d (Triangle inequality) (Kreyszig 1978)
Example 1
Let denote the set of real numbers.
Let d: be defined by = for all in , then, d is a metric on , since it satisfies the four properties above, by verification. To see this, we take arbitrary
Verification:
P1 The absolute values of real numbers are never negative
i.e = ≥ o
P2 Here, we are to show the necessity and sufficiency
(=>) if = 0
=> = 0
=> = 0
=>
(<=) suppose
Then
= =
|0|
= 0
P3 = |-()| =
=
P4 = = for arbitrary z
≤ +
≤ + [by determination of metric space]
Thus, d is a metric on R and it is often called the USUAL METRIC.
Example 2
Let X be an arbitrary nonempty set.
Defined: X xX ―> by () for all in X
P1 We can see that the possible values we can get in each case is greater than or equal to zero. We can view this in two cases i.e
Case 1
()
Case 2
() = 0
P2 by case 2 if () =0, then
Also, if , () = () =0 since
P3 () = 1 = i.e provided by case 1
() = 1 =
P4 for arbitrary z X
= 1 and = 1 by definition
≤ +
1.3 CAUCHY SEQUENCE
A sequence (n)1,2,….), where n X for every n, is called a cauchy sequence in a metric space (X,d) if and only if
d(n,m) (m,n),
i.e for every there exist N0N0( such that
d(n,m) for all n,m N0(Maddox 1988)
Example 3
Let X = (, d) be real line with the usual metric and let n = . Then,
n=1 =
This sequence is Cauchy. For by definition, let m, n be arbitrary and suppose m> n. let >0 be given, then we want to find an integer N such that for m,n≥0 N0, d (nm)<
d (, m)= d= ≤ <
Since m>n,
Then,
d (n, m)<
if< then>
n>
We can choseour No =
1.4 COMPLETE METRIC SPACE
A metric space (X,d) is called complete if and only if every Cauchy sequence converges (to a point of X). Explicitly, we require that if d(n,m)(m,n), then there exist such that d(n, )(n ). (Maddox 1988)
Example 4
Consider the space y= as a subspace of , where Y is endowed with the subspace metric. Then the sequence n=1 = n=1 is obviously a sequence in Y.
i.en=1 = 1,
thenas n and 0 is in Y, so Y is a complete metric space.
Example 5
Taking Y= and n=1 with the usual metric as above then we can see that the metric space y is no longer a complete metric space because the limit of the sequence is not in Y.
This suggest that subset of of the form (a,,b), (a,b) are not complete.
1.5 FIXED POINT
Let S be a set and F a function from S to S. A fixed point of F is simply a point such that F(a) In other words, a fixed point of F is nothing but a solution of function equation
F() 1.1
(Limaye 1996)
Example 6
The mapping f: defined by f(2 has only two fixed point, which are 0 and 1
i.e f(
2 =
2-
(-) = 0
Hence,
or and both points are in .
Example 7
The translation mapping T:22,
Where given a, T( has no fixed point in 2.
Example 8
The mapping T:, given as T(+1 has its fixed point as 2, and 2 is in .
1.6 CONTRACTION
Suppose (X,d) is a complete metric space and T: XX is any mapping. The mapping T is said to satisfy the Lipschitz condition with constant if
d(T)
holds for all X if αthen T is called a contraction mapping
Example 9
Let X=
Define T: by T= x then, by definition, we have
d(T)=
=
This satisfies the condition for a contraction since our
1.7 ITERATION
The Banach fixed point theorem gives a constructive procedure for obtaining better approximation to the fixed point which is the solution of the practical problem, and this procedure is called iteration.
By definition, this is a method such that we choose an arbitrary 0 in a given set and calculate recursively a sequence 1,2, 3,… from a relation of the form
n+1= Tn n=0,1,2,…
That is, we choose an arbitrary 0 in a given set and calculate successively
1=T0 ,2=T1, 3 =T2,…n = Tn-1
