# Analysis. Function spaces/Contraction mappings

• datenshinoai
In summary, the conversation discusses the existence and uniqueness of a continuous function f:[0,1] -> R when given two continuous functions K and g. The function f is defined as f(x)= g(x) + integral K(x,y) dy and the question arises whether Banach's fixed point theorem can be applied to this situation. To show uniqueness, it is suggested to let g(f(x)) = h(x) but it is uncertain if that is the correct approach.
datenshinoai

## Homework Statement

Let K: [0,1] x [0,1] -> R be a continuous function such that K(x,y) > 0 and the integral K(x,y) dy <= C < 1 (from 0 to 1) for all x within [0,1].

Let g:[0,1] -> R be any continuous Function. Show that there is a unique continuous function f:[0,1] -> R such that f(x)= g(x) + integral K(x,y) dy

## The Attempt at a Solution

I know that we are given 2 functions and that we need to input those into the formula. We want to see the output is closer than the input by a factor of k, but I'm not sure what it is that I should be doing.

To show uniqueness, I say to let g(f(x)) = h(x). Which means we are sending [0,1] -> R to [0,1] -> R, but I don't think that's right...

Did you make a typo in the question? Because the unique continuous function f such that f(x)= g(x) + integral K(x,y) dy is the function f defined by f(x):=f(x)= g(x) + integral K(x,y) dy

datenshinoai said:

## Homework Statement

Let K: [0,1] x [0,1] -> R be a continuous function such that K(x,y) > 0 and the integral K(x,y) dy <= C < 1 (from 0 to 1) for all x within [0,1].

Let g:[0,1] -> R be any continuous Function. Show that there is a unique continuous function f:[0,1] -> R such that f(x)= g(x) + integral K(x,y) dy
I suspect you mean
[tex]f(x)= g(x)+ \int K(x,y) f(y) dy[/itex]
That's a variation of the Poisson proof of the existence and uniqueness of solutions to first order differential equations. Are you allowed to use the Banach fixed point theorem? (If f(x) is a contraction map then there is a unique x such that f(x)= x.)

## The Attempt at a Solution

I know that we are given 2 functions and that we need to input those into the formula. We want to see the output is closer than the input by a factor of k, but I'm not sure what it is that I should be doing.

To show uniqueness, I say to let g(f(x)) = h(x). Which means we are sending [0,1] -> R to [0,1] -> R, but I don't think that's right...

Probably he is allowed to use Banach's fixed point theorem because of the title of the thread.

The question as to whether Banach's fixed point theorem can be applied to this situation amounts to finding
(1) a complete metric space inhabited by continuous maps from [0,1] to R
(2) a contraction F on the above space such that [F(f)](x)=g(x) + integral K(x,y)f(y) dy

## 1. What is analysis?

Analysis is the study of mathematical concepts and structures such as functions, sequences, and series. It involves breaking down complex ideas and problems into smaller, more manageable parts in order to better understand them.

## 2. What are function spaces?

Function spaces are sets of functions that share a common property or structure. They can be used to study the properties and behavior of functions within a specific context, such as continuous functions or differentiable functions.

## 3. What is a contraction mapping?

A contraction mapping is a function that maps a metric space to itself, and has the property that the distance between any two points in the space is always reduced after the function is applied. This concept is important in analysis for proving the existence and uniqueness of solutions to certain equations.

## 4. How are function spaces and contraction mappings related?

Function spaces and contraction mappings are closely related because contraction mappings are often used to define function spaces. For example, the set of all continuous functions on a closed interval can be defined as a function space using the concept of a Lipschitz constant, which is related to contraction mappings.

## 5. What are some applications of analysis and function spaces/contraction mappings?

Analysis and function spaces/contraction mappings have many applications in various fields, including physics, engineering, and economics. They can be used to model and solve real-world problems, such as predicting the behavior of a physical system or optimizing a financial portfolio.

• Calculus and Beyond Homework Help
Replies
12
Views
2K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
407
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
58
Views
3K
• Calculus and Beyond Homework Help
Replies
5
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
14
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
373