Don't solve these equations

[alain verghote]

Dear moderator,I write from a french western little town...
WHAT I'm looking for is not solving methods or the like;but
signification and meaning:free brainstorming.
Does anyone may imagine circumstances or situations when the following
functional univariate equations could happen?.
f(3x) = f(x) + x^2;
f(2x) = (6x - 1).f(x)/(2 f(x)+3x -1);
f(x^2)= 3*f(x) + f(2).
Amities Pascales ,ALAIN.

[David Ziskind]

alain verghote <alain.v...@wanadoo.fr> wrote in article
<c5bukb$p7h$1...@news.ks.uiuc.edu>...

It is worthwhile to note that in the three examples given, if x is
integral, then so are all arguments of f (that is, 3x, x^2, etc).

Thus, a starting point is to consider the reduced problem of solving
such equations, for the integers only. I cannot supply a physical
interpretation for any of the three examples, but will note that such
close corespondents as:

a) f(x+1) = p*f(x)*(1-f(x)) [population equation], and
b) f(x+2) = f(x+1) + f(x) [Fibonacci equation] ;

have well known exemplars in the physical world.

Let us now turn to the question of satisfying “iteration equations” not
only for the integers, but for ALL REALS (in some interval).

I suggest that this probably is an uninteresting problem (for
physicists). For realistic physical problems, I suggest that one can
always shrink the mesh size to such a point that the deviation of
f(n+p) from f(n), (n integer, 0<p<1) is so small as to be unimportant.

The existence of solutions for the simpler functional equations, and
construction of same, has been discussed before on sci.math.*. I will
close with a few basic comments -- perhaps I should say that these are
my own speculations, I do not claim any particular expertise in this
area.

Suppose that one is given a “functional equation” such as:

K(x, f(x), f(g(x))) = 0 [1]

and asked if there is a non-trivial f which satisfies the equation. K
and g are given in terms of formulae which are easily written down. On
the other hand, it is not feasible to solve for f or f o g. How do we
proceed?

I suggest that the Banach contraction mapping principle is a useful
tool to use here, and will resolve many such problems.

First: define T by: (Th)x = K(x, h(x), h(g(x))) Note that T is an
operator and so its definition involves an exemplary function (h) and
an exemplary number (x).

Second: now let the sequence of functions n->u_n be defined by:

a) u_0 = Te (with e strategically chosen); and

b) u_(n+1) = u_n - ((inv(DT))u_n)(u_n) (n=0,1,2,... )

where (inv(DS))w means: the inverse of
the Frechet derivative of S evaluated at w

Now, if the norm of the Frechet derivative, that is, |((inv(DT))u_n)|
is bounded by 0<a<1 for all n>0, then u_n converges and satisfies [1].
If not, the question of a solution remains open.

The above formulation of the Banach CMP is written to be as brief as
possible. For a more practical formulation, see a text such as:
Liusternik and Sobolev, “Elements of Functional Analysis”, Frederick
Ungar Publishing Co., 1961.

I acknowledge that the above is fairly abstract, but hope it sheds some
light.

David Ziskind
zis...@ntplx.net

[David Ziskind]

This is a correction to my article <c5gpri$bvq$1...@news.ks.uiuc.edu>
(April 13, 2004).

There is an error in the third from bottom paragraph (convergence
condition for the Newton-Raphson iteration immediately above). Please
delete said paragraph and for a convergence condition, see a text such
as the mentioned Liusternik and Sobolev.

Also, rather than start the iteration with Te (line “a” of the
iteration formula); a well-chosen starting value of e would be just as
good.

David Ziskind
zis...@ntplx.net