Fixed Point Theorem & Contractive Maps

MHB
Thread starter ozkan12
Start date Jul 3, 2015
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Fixed point Point Theorem

In summary, the Fixed Point Theorem is a mathematical concept that states that for a given function, there exists at least one point in the domain of the function that maps to itself. A contractive map is a function that reduces the distance between any two points in its domain and is important in the Fixed Point Theorem. This theorem has numerous applications in mathematics and science and is used in real-world problems such as finding solutions to optimization problems and analyzing the stability of physical systems. In order for a function to have a fixed point, it must be continuous and a contractive map.

Jul 3, 2015

ozkan12

Please give an example of contractive map which have fixed point...I search but I didnt find

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Jul 5, 2015

Fallen Angel

Hi,

A trivial example is the constant real function 1.

Jul 5, 2015

Nono713

Gold Member

MHB

Another trivial example is the map $x \mapsto kx$ for $0 \leq k < 1$ over, say, the reals. It has an obvious fixed point: zero.

Related to Fixed Point Theorem & Contractive Maps

1. What is the Fixed Point Theorem?

The Fixed Point Theorem is a mathematical concept that states that for a given function, there exists at least one point in the domain of the function that maps to itself. In simpler terms, it means that there is a point where the output of the function is equal to the input.

2. What is a contractive map?

A contractive map is a function that reduces the distance between any two points in its domain. It is also known as a contraction mapping and is an important concept in the Fixed Point Theorem as it guarantees the existence of a fixed point.

3. Why is the Fixed Point Theorem important?

The Fixed Point Theorem has numerous applications in mathematics and science. It is used to prove the existence of solutions to various equations and also in the study of dynamical systems. It is also a fundamental concept in computer science, economics, and engineering.

4. How is the Fixed Point Theorem used in real-world problems?

The Fixed Point Theorem is used in various real-world problems such as finding solutions to optimization problems, modeling population growth, and analyzing the stability of physical systems. It is also used in computer algorithms and data analysis techniques.

5. What are the conditions for a function to have a fixed point?

In order for a function to have a fixed point, it must satisfy two conditions: it must be a continuous function and a contractive map. This means that the function must not have any breaks or jumps and must reduce the distance between any two points in its domain.