Brouwer's Fixed Point Theorem for Arbitrary Intervals

Thread starter dabeth
Start date Mar 28, 2010
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In summary, Brouwer's Fixed Point Theorem for Arbitrary Intervals is a fundamental theorem in mathematics that states that any continuous function from a closed interval to itself must have at least one fixed point. It was first proved by the Dutch mathematician Luitzen Egbertus Jan Brouwer in 1912 and has significant applications in various fields such as topology, differential equations, and game theory. It cannot be applied to functions defined on open intervals, but there are other theorems that can be applied in such cases. There is also a generalization of this theorem known as the Brouwer's Fixed Point Theorem for Compact Convex Sets, which has applications in functional analysis and optimization.

Mar 28, 2010

dabeth

Is it possible to prove Brouwer's Fixed Point Theorem (one-dimensional version) for intervals other than [-1,1]-->[-1,1], say [1,2]-->[0,3]? If so, how?

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Mar 28, 2010

Office_Shredder

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For example, f(x)=x-1 is a map from [1,2] to [0,3] with no fixed point. The condition that the domain and codomain are the same space is very important

Related to Brouwer's Fixed Point Theorem for Arbitrary Intervals

1. What is Brouwer's Fixed Point Theorem for Arbitrary Intervals?

Brouwer's Fixed Point Theorem for Arbitrary Intervals is a fundamental theorem in mathematics that states that any continuous function from a closed interval to itself must have at least one fixed point, which is a point that is mapped to itself by the function.

2. Who discovered Brouwer's Fixed Point Theorem for Arbitrary Intervals?

This theorem was first proved by the Dutch mathematician Luitzen Egbertus Jan Brouwer in 1912.

3. What is the significance of Brouwer's Fixed Point Theorem for Arbitrary Intervals?

Brouwer's Fixed Point Theorem has important applications in various fields of mathematics, including topology, differential equations, and game theory. It also has practical applications in economics, computer science, and engineering.

4. Can Brouwer's Fixed Point Theorem be applied to functions defined on open intervals?

No, Brouwer's Fixed Point Theorem only applies to functions defined on closed intervals. However, there are other theorems, such as the Brouwer's Fixed Point Theorem for Disks, that can be applied to functions defined on open intervals.

5. Is there a generalization of Brouwer's Fixed Point Theorem for Arbitrary Intervals?

Yes, there is a generalization of Brouwer's Fixed Point Theorem known as the Brouwer's Fixed Point Theorem for Compact Convex Sets. It states that any continuous function from a compact convex set to itself must have at least one fixed point. This generalization has applications in functional analysis and optimization.