F IXED P OINT T HEORY

RIYAS P

Government Brennen College,

Thalassery, Kerala

Email:riyasmankadavu@gmail.com
Mob No-9746004847

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 Fixed point theory is an active area of research in Mathematics with
numerous applications in the field of both pure and applied
Mathematics

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 In this presentation I would like to discuss various versions of the
fixed point theorems and some applications to existence theorems in
the theories of differential and integral equations.
Before proceeding it would be well to make precise what we mean by
a fixed point theorem.
Generally a fixed point theorem is a statement that specifies
conditions on X and f which guarantees that f has a fixed point in X .
Definition 1.1 (Fixed Point)
Let X be a set and f : X → X be a map from X to itself. A point x ∈ X
is called a fixed point of f if f (x) = x.
In otherwords, a point which remains invariant under a mapping is
known as a fixed point.
Examples:-
A rotation of the plane has a single fixed point
A translation has no fixed points
The projection P : (x, y ) → x of R 2 onto the x − axis has
infinitely many fixed points all lying on x − axis
The mapping T : R → R defined by Tx = x 2 has two fixed points
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 Geometrically, fixed points are the points of intersection between the
graphs of y = f (x) and y = x, as shown here:

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 Given a set X , it is possible to ask what types of functions on X have
a fixed point. Alternatively, we could consider a class of functions and
investigate the kinds of sets on which a function in our class will have
a fixed point.
There are large classes of mappings for which fixed point theorems
have been studied. It includes contractive mappings, contraction of
various order mappings, Ciric contraction, asymptotically regular,
densifying etc. Apart from single mappings, pair of mappings, the
sequence of mappings and family of mappings are also some of the
classes of mappings.

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 Lipschitzian Map

Definition 1.2
Let (X , d) be a metric space. The map T : X → X is said to be
Lipschitzian if there exist a constant k > 0 such that

d(T (x), T (y )) ≤ kd(x, y ), for all x, y ∈ X

The class of all mappings satisfying the Lipschitz condition with a

constant k is denoted by L(k ). The smallest constant k for which the
above inequality holds is called the Lipschitz constant for T and is
denoted by k (T ).

A lipschitzian mapping with a Lipschitz constant k (T ) < 1 is called

contraction.

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 Contraction Mapping
Definition 1.3
Let (X , d) be a metric space. A mapping T : X → X is a contraction
mapping if there exist a constant k with 0 < k < 1 such that

d(T (x), T (y )) ≤ kd(x, y ), for all x, y ∈ X

Thus, a contraction maps points closer together. In particular, for

every x ∈ X and r > 0, all points y in the ball B(x, r ) are mapped into
a ball B(Tx, s) with s < r .

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 In particular, for every x ∈ X and r > 0, all points y in the ball B(x, r )
are mapped into a ball B(Tx, s) with s < r .

Example 1
Let X = R and T : R → R a mapping defined by

1
Tx = x + 1, x ∈ R
2
Then T is a contraction .
.
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 The Picard-Banach Theorem or Banch Contraction
Principle

One of the earliest and best known fixed point theorem is the Banach
Contraction Principle. It is simplest and most useful method for the
construction of solutions of linear and non linear equations.

Theorem 2
Let (X , d) be a complete metric space and T : X → X be a
contraction mapping. Then T has a unique fixed point.

This theorem is used to prove the existence and uniqueness

theorems for systems of ordinary differential equations and also can
be used to prove the inverse function theorem.

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 Proof:

Proof.
Let x0 be any point in X . We define a sequence (xn ) by
xn+1 = T (xn ), n ≥ 0. For n > m ≥ 1,

km
d(xn , xm ) ≤ d(x1 , x0 )
1−k
Which shows that (xn ) is a Cauchy sequence in X . Since X is
complete there exist x ∈ X such that lim xn = x.

Tx = T (lim xn ) = lim T (xn ) = lim xn+1 = x

Finally if x and y are two fixed points, then

0 ≤ d(x, y ) = d(Tx, Ty ) ≤ kd(x, y )

Since k < 1, we have d(x, y ) = 0, so x = y and the fixed point is

unique.

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 Extended Form of BCP or Generalization

Theorem 3
Let (X , d) be a complete metric space and T : X → X be a mapping
such that for some positive integer n, T n is a contraction on X . Then
T has a unique fixed point.

Proof.
By BCP, T n has a unique fixed point, say x in X with T n (x) = x.
Since T n+1 (x) = T (x), it follows that T (x) is a fixed point of T n and
thus by uniqueness Tx = x. That is T has a fixed point. Since fixed
points of T is necessarily fixed point of T n , so is unique.

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 Example 4
Consider the usual metric space (R, d). Define
x
f (x) = + b, for all x ∈ R
a
Then f is a contraction on R if a > 1 and the solution of the equation
ab
x − f (x) = 0 is x =
a−1

Example 5
Consider the euclidean metric space (R2 , d). Define
x y 
f (x, y ) = + b, + b for all (x, y ) ∈ R2
a c
Then f is a contraction on R2 if a, c > 1 . Now, solving the equation
ab cb
f (x, y ) = (x, y ) for a fixed point, we get x = and y =
a−1 c−1

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 Example 6
Let X = [a, b] and T : X → X a mapping such that T is differentiable
at every x ∈ (a, b) such that |T 0 (x)| ≤ k < 1. Then by mean value
theorem, if x, y ∈ X there is a point c ∈ X between x and y such that

Tx − Ty = T 0 (c)(x − y )

. Thus, |Tx − Ty | = |T 0 (c)||x − y | ≤ k |x − y |

Therefore, T is a contraction and it has a unique fixed point

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 Applications

Theorem 7 (Picard-Lindelof Theorem)

Let A = {(x, y ) ∈ R 2 : a ≤ x ≤ b, c ≤ y ≤ d} ⊂ R 2 and let f : A → R
be Lipschitz continuous in the second variable . If (x0 , y0 ) is an
interior point of A then the ordinary differential equation

dy
= f (x, y )
dx
has a unique olution y = g(x) satisfying g(x0 ) = y0 defined on an
interval [x0 − , x0 ] for some  > 0.

Theorem 8
1
Let M be a real n × n matrix, all of whose entries are less than in
n
modulus. Then I − M is invertible.

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 Contractive Mapping(Weak Contraction)

Definition 1.4
Let (X , d) be a metric space. A mapping T : X → X is a contractive
mapping if

d(Tx, Ty ) < d(x, y ), for all x, y ∈ X with x 6= y

A contractive mapping is clearly continuous, and if such a mapping

has a fixed point, then this fixed point is obviously unique.

Class of contraction map ⊆ Class of contractive map

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 Whether the Banach Contraction Principle is
valid for contractive map ?

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 Is there any contractive map on a complete metric
space that has no fixed point ?

Example 9
Let X = C0 and T : BX → BX be defined by
1 + ||x||
T (x) = T (x1 , x2 , ....) = (x10 , x20 , .....) where x10 = and
  2
1
xi0 = 1 − i+1 xi−1 for i = 2, 3, 4...
2
Then T is a contractive map on a complete metric space which has
no fixed point

Example 10
The map T : R → R define by Tx = x + π
2 − tan−1 x is contractive but
has no fixed point

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 Is there any contractive map on a complete metric
space that has no fixed point ?

Example 9
Let X = C0 and T : BX → BX be defined by
1 + ||x||
T (x) = T (x1 , x2 , ....) = (x10 , x20 , .....) where x10 = and
  2
1
xi0 = 1 − i+1 xi−1 for i = 2, 3, 4...
2
Then T is a contractive map on a complete metric space which has
no fixed point

Example 10
The map T : R → R define by Tx = x + π
2 − tan−1 x is contractive but
has no fixed point

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 Fixed Point Theorem for a Contractive Map

Theorem 11
Let X be a compact metric space and T : X → X a contractive
mapping. Then T has a unique fixed point u in X . Moreover, for each
x ∈ X , the sequence {T n (x)} of iterates converges to u.

Proof:
Define F : X → [0, ∞) by F (x) = d(x, T (x))
Then there exist some point a ∈ X such that d(a, T (a)) ≤ d(x, T (x))
for all x. This will imply that a is a fixed point of T

Theorem 12 (Generalization)
If X is a compact metric spac and T a continuous self map of X with
a contractive iterate, then T has a unique fixed point.

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 Fixed Point Theorem for a Contractive Map

Theorem 11
Let X be a compact metric space and T : X → X a contractive
mapping. Then T has a unique fixed point u in X . Moreover, for each
x ∈ X , the sequence {T n (x)} of iterates converges to u.

Proof:
Define F : X → [0, ∞) by F (x) = d(x, T (x))
Then there exist some point a ∈ X such that d(a, T (a)) ≤ d(x, T (x))
for all x. This will imply that a is a fixed point of T

Theorem 12 (Generalization)
If X is a compact metric spac and T a continuous self map of X with
a contractive iterate, then T has a unique fixed point.

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 Back to Presentation

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