This is my very first encounter with functional equation of this kind, and methods of series solution and differential equations are of not much help. The solution to this problem, or at least the mathematical prerequisites to understand solution to this problem is asked.
This function is asymptotically zero at both +_ infinity, positive otherwise, and has a peak near zero.
If I am correct, then this is the frequency distribution of numbers fed to the sequence xn+1=a ln (xn2) as long as the sequence generated is chaotic and sufficiently large in number (value of a is usually limited to 0.2 to 1.3, Positive and negative signs of a are essentially immaterial except for the sequence is negated after first term) .The sequence is initiated or seeded with number roughly as the same order of magnitude of 1 in both positive and negative side, except for the values that eventually lead to zero or infinity in the sequence. The sequence is allowed to proceed, and frequency distribution of numbers in the sequence are noted, from which a continuous probability distribution may be numerically guessed but not analytically found. The expression to find out formula of the continuous probability distribution comes to me from the following reasoning
- Suppose, the probability distribution is given by y=f(x). Now, if I consider a "dx" (infinitesimally) thin strip around x, then I come up with f(x) dx fraction of all points required to construct the probability distribution. When this fraction of all points are passed through yet another sequence of transformation through the recurrence xn+1 =a ln (xn)2 , the fraction of points involved must be unchanged. That is, when x is substituted with a ln x2 , the infinitesimal strip area, which changes to f( a ln x^2) d (a ln x^2), must be numerically equal to f(x) dx, thus the functional equation is postulated
- I am not entirely sure about this reasoning, and experts are welcome to point out my fault of reasoning and show the correct equation , if I am mistaken.
Please see my related question https://www.researchgate.net/post/Can-you-figure-out-Chaos-of-the-recurrence-x-n-1lnx-n2 for further details.
I am not convinced that the functional equation is correct. Does it take into account that both xn and -xn maps to the same point a ln xn2 = 2a ln |xn| ?
I have observed that f can be considered an operator rather than a usual function.
Where f is defined by the rule: f(g(x))= dx/dg = 1/(dg/dx).
I hope this observation shows new insight into the current problem.
Regards
Dear Peter Breuer
The operator f{*} is defined as follows
f{g(x)}=1/g(x)\ (the reciprocal of the derivative)
f{x}=1/1
f{x2}=1/2x
f{sinx}=1/cosx
etc,..
We find that this operator works as a solution for the given equation.
You can see
f(a ln x2)/f(x)= (1/2a(1/x))/(1)= x/2a.
You can also replace x with any other function, and the relationship works.
Here f is considered as an operator.
Best wishes
There is far too little information to even formulate the problem correctly - that is solving the "functional equation" for "f." Secondly the left hand side (f(a ln(x^2))/f(x) may or may not be well defined. The "a" seems to appear as a parameter.
The equation's left hand side can be rewritten as f( ln(x^(2*a))/f(x), There may be some cases of "a" where there might be a solution and some where there might not if there is one at all. So this question needs some clarification. From what I can glen from the OP is that f>=0 and limit f(x) =0 as x-> +/- infinity. It very well may be that one needs to place some sort of additional condition, i.e., f is continuous, C1 or C-infinity in order to address the existence of a solution.
The relationship can be differentiated (if f is differentiable) and this results in a functional differential equation depending on a parameter. That can be set in a functional analytic setting as a mapping on a set the appropriate function space where conditions on "f" might allow fixed point theorems to be used to address it.
However, existence is only one question. The second is uniqueness of solutions. The third is dependence on the parameter a. Clearly we need more clarification of the problem stated to understand how one would approach a solution.
If I understand well, a is a fixed parameter, and it is required to find a positive function f, defined on the real axis, such that for x≠0 it is
f(a ln(x^2))/f(x) = x/2a.
Is this correct?
Such a function cannot exist, since if x is negative, then the left side of the equality is positive, while the right side is negative.
Maybe the correct equation you are interested in is
f(a ln x^2)/f(x) = |x|/2a.
Indeed when evaluating the infinitesimal strip area one must consider the absolute value of d(a ln(x^2)), that is the area changes to
f(a ln x^2) |d(a ln(x^2))|.
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