# Can an Iterative Scheme Guarantee Convergence for Solving a Functional Equation?

• eljose79
In summary, the conversation discusses the functional equation f(g(t))=g(t) and the possibility of solving it through iteration. The solution suggested is g(t)=fofofofofofofo..., where fof represents the composition of f with itself. However, there is uncertainty about the validity of this solution and the convergence to a solution. The use of iterative algorithms and the concept of fixed points is also mentioned. It is concluded that further research is needed to determine a precise definition of fofofo... and its applicability in solving the functional equation.
eljose79
let be the equation f(g(t))=g(t) where f is known but g(t) not... i think that perhaps we could solve it by iteration so g(t) would be

g(t)=fofofofofofofo... where fof means the composition of f with itself...is that solution right? i do not even konw if my process to solve the functional equation is right and will converge to the solution.. has any other solution?...

Could you give a precise DEFINITION of fofofo...? That's a lot easier to write than to define! If for example f(x)= x^2 how would you find fofofo...(3)? Until you can answer that you don't have a function to BE the solution!

Taking the example f(x)= x^2, you are asking for a function
g(x) such that (g(x))^2= g(x) for each x. Since, for a specific x,
g(x) is a number, g(x) must always satisfy the numerical equation
u^2= u. Of course, 0 and 1 are the only numbers that satisfy that so g(x) must be always either 0 or 1. That gives an (uncountably) infinite number of functions g that satisfy this equation. If you require that g be countinuous then there are exactly two solutions:
g(x)= 0 for all x and g(x)= 1 for all x.

I think it works in some special cases.
For instance, you have a linear operator F which obeys |F*v| = |v| for any vector v, and you look for an eigenvector g so that F*g = g.
Then g = F^infinity * v. (At least that's what I think)

Of course, v and g could also be functions.

EDIT: Oops, I forgot some more restrictions. || is the 1-Norm (sum of components). Plus, all entries must be non-negative.

Here's a simple example:
F=
| 2/3 1/3 |
| 1/3 2/3 |
Let v = (1;0)
Then:
F*v = (2/3 ; 1/3)
F*F*v = (5/9 ; 4/9)
F*F*F*v = (14/27 ; 13/27)
...
This converges to:
g = (1/2 ; 1/2).

Last edited:
red herring?

Let x=g(t). Then the equation is f(x)=x. Since f(x) is supposed to be known, you just simply solve for x. t is irrelevant.

Repeated iteration will only converge to a solution if your initial point is in the basin of an attracting fixed point (I don't know if basin is the right word).

Repeated iteration is not generally a nice thing. It's very easy for it to blow up to infinity, approach a periodic orbit, or simply bounce around an interval chaotically.

Iterative algorithms, such as Newton's Method or Conjugate Gradients, are specifically designed so that the desired solution is an attracting fixed point (though not always with 100% success).

Hurkyl

I believe to guarantee convergence of a interive scheme you need |f'|<1 at least in a neighborhood of the fixed point.

You might to a search on "fixed point" numerical schemes, there is a significant amount of literature about them.

## What is an iterative functional equation?

An iterative functional equation is a mathematical equation that involves a function being applied repeatedly to itself in order to find a solution. It is often used in the study of dynamical systems and can be used to model various real-world phenomena.

## What is the purpose of an iterative functional equation?

The purpose of an iterative functional equation is to find a fixed point or solution to the equation. This fixed point represents a stable state of the system being modeled and can provide valuable insights into the behavior of the system.

## How is an iterative functional equation solved?

An iterative functional equation can be solved through various methods, such as iteration, substitution, or using numerical techniques. The solution may also involve finding the limit of a sequence generated by the function.

## What are some applications of iterative functional equations?

Iterative functional equations have many applications in different fields of science, including physics, biology, economics, and computer science. They can be used to model population growth, chemical reactions, economic systems, and the behavior of complex systems.

## Are there any limitations to using iterative functional equations?

Like any mathematical model, iterative functional equations have their limitations. They may not accurately represent all real-world systems and may require certain assumptions to be made. They also require careful selection of initial conditions and may not have a unique solution in some cases.

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