Existence of solution to integral equation

In summary, the conversation discusses the existence of a solution to an integral equation, with the main focus on proving the continuity and contraction properties of the operator T. This leads to the conclusion that there exists a unique fixed point, but the question remains on how this connects to the original equation. The mistake of looking at the wrong norm leads to a correction and the solution to the problem.
  • #1
quasar987
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[SOLVED] Existence of solution to integral equation

Homework Statement


There's k:[0,1]²-->R square integrable and the operator T from L²([0,1],R) to L²([0,1],R) defined by

[tex]T(u)(x)=\int_{0}^{1}k(x,y)u(y)dy[/tex]

(a) Show that T is linear and continuous.

(b) If ||k||_2 < 1, show that for any f in L²([0,1],R) , there exists a u in L²([0,1],R) solution of the integral equation

[tex]u(x) = f(x)+\int_{0}^{1}k(x,y)u(y)dy[/tex]

(almost everywhere on [0,1])

The Attempt at a Solution



I have done (a), and in demonstrating that T is continuous, I have shown that [tex]||T(u)||_2\leq ||k||_2||u||_2[/tex]

Therefor, the condition ||k||_2 < 1 in (b) + the fact that T is linear is equivalent to saying that T is a contraction. So by Banach's fixed point theorem, there exists a (unique) fixed point to T; call it u*:

[tex]u^*(x)=\int_{0}^{1}k(x,y)u^*(y)dy[/tex]

Fine, but what about the equation

[tex]u(x) = f(x)+\int_{0}^{1}k(x,y)u(y)dy[/tex]

? If we define an operator T_f by

[tex]T_f(u)(x)=f(x)+\int_{0}^{1}k(x,y)u(y)dy[/tex]

this is not a contraction.

How do the dots connect?
 
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  • #2
Why is T_f not a contraction?
 
  • #3
You're right, I spoke too fast!

I wanted to look at ||T_f(u) - T_f(0)||=||T_(u)||, but I looked at ||T_f(u)|| instead.

Thank you morphism!
 
  • #4
Wow, morphism's one line reply was enough to answer your lengthy post.
 
  • #5
'Tis often so.

And often, the lengthy (and, hopefully, complete) first post results in the response being brief.
 

Related to Existence of solution to integral equation

What is an integral equation?

An integral equation is an equation that involves an unknown function as well as an integral (or summation) of that function.

Why is the existence of a solution to an integral equation important?

The existence of a solution to an integral equation is important because it determines whether the equation is solvable and if so, provides a way to find the solution.

How can we determine if an integral equation has a solution?

There are various methods for determining the existence of a solution to an integral equation, such as the Fredholm alternative, the Riesz-Fischer theorem, and the Banach fixed-point theorem.

What factors affect the existence of a solution to an integral equation?

The existence of a solution to an integral equation can be affected by the properties of the integral operator, the boundary conditions, and the behavior of the integrand function.

What are some real-world applications of integral equations?

Integral equations are used in many areas of science and engineering, such as physics, chemistry, biology, and economics. They are particularly useful in solving problems involving differential equations and in mathematical modeling.

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