Construction of continuous solutions and stability for the polynomial-like iterative equation. (English) Zbl 1111.39020
A fixed-point method is used to establish the existence of ‘roots’ of the functional equation
\[
\sum_i a_i f^i(x)= F(x),\tag{1}
\]
\(F=\) given function, \(f^i=i\)th iterate \(f^0= f\); coefficients, \(f\) and \(F\), all real valued. No suggestion is made how to construct \(f\), and no conditions for its uniqueness are given.
For an applied reader an illustrative example of the inherent difficulties is obvious. Let \(F= bx\), and the ‘root’ \(f= kx\), where \(k\) is an undetermined constant. Inserting into (1) yields an ordinary polynomial in \(k\) of a finite degree \(n\): \[ P_n(k,a_i, b)= 0, \] which may have from zero \(u_0\) to \(n\) real roots.
If \(n\leq 4\), its roots can be expressed explicitly by elementary functions, if \(n= 5\) or \(6\), by elliptic ones (although explicit formulae are not readily available), and if \(n> 6\) by still unstudied hyperelliptic functions. Even in this simple problem the full solution has many unknown features; but is nevertheless ‘generic’ for (1).
There is considerable evidence (cf. publications on systems of algebraic equations of infinite order) that series expansions in terms of an arbitrarily chosen complete set of functions are ineffective.
Assuming \(f(x)= \sum^\infty_{i=0} c_i g_i(x)\), and inserting into (1), yields an infinite system for the \(c_i\). The sequences of the \(c_i\), obtained by the method of successive truncation are usually divergent. No example of a natural phenomenon, or of a technological device is given, which would be at least approximately described by (1).
For an applied reader an illustrative example of the inherent difficulties is obvious. Let \(F= bx\), and the ‘root’ \(f= kx\), where \(k\) is an undetermined constant. Inserting into (1) yields an ordinary polynomial in \(k\) of a finite degree \(n\): \[ P_n(k,a_i, b)= 0, \] which may have from zero \(u_0\) to \(n\) real roots.
If \(n\leq 4\), its roots can be expressed explicitly by elementary functions, if \(n= 5\) or \(6\), by elliptic ones (although explicit formulae are not readily available), and if \(n> 6\) by still unstudied hyperelliptic functions. Even in this simple problem the full solution has many unknown features; but is nevertheless ‘generic’ for (1).
There is considerable evidence (cf. publications on systems of algebraic equations of infinite order) that series expansions in terms of an arbitrarily chosen complete set of functions are ineffective.
Assuming \(f(x)= \sum^\infty_{i=0} c_i g_i(x)\), and inserting into (1), yields an infinite system for the \(c_i\). The sequences of the \(c_i\), obtained by the method of successive truncation are usually divergent. No example of a natural phenomenon, or of a technological device is given, which would be at least approximately described by (1).
Reviewer: Igor Gumowski (Thoiry)
MSC:
| 39B12 | Iteration theory, iterative and composite equations |
| 39B22 | Functional equations for real functions |
| 39B82 | Stability, separation, extension, and related topics for functional equations |
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