There are various functional equations (FE) which were studied by Niels Henrik Abel (1802 – 1829)^{(1)}. Among them are functional equations in a single variable, which are called iterative functional equations by Marek Kuczma^{(2)}. In this article we will consider the Abel iterative functional equation:

This FE has a special significance among all functional equations, since its solutions are intricately connected with the iterates of the function *u*. A first long standing problem is to find continuous solutions *f*, for an arbitrary function *u*. Although the class of solutions is infinite for every particular function *u*, it has been difficult to find even one such solution in most cases. A second long standing problem is to characterize a unique solution among the infinite possible, through implementation of additional criteria.

In this article we will address the first of these problems. We present continuous solutions of the AFE for a wide class of functions *u*. For the detailed method and proofs see [Kefalas, 2014].

**Inverse and iterates of functions**

For our convenience we introduce the special notation, , for the inverse of an invertible function *u*. Then:

where *I*, is the identity function. We also use a nonstandard notation for functional iterates, to avoid confusion with powers . This is:

where, .

**Continued forms**

In [Kefalas, 2014] the author introduces the ‘continued form notation’, which generalizes the familiar Σ, and Π, notation. Let, , be a sequence of univariate functions and assume that the successive composition of the sequence members is possible. Then the continued form of the sequence *S _{u}*, is defined as:

where* t*, is the variable of composition.

**Continuous solutions of the AFE**

We can now give a formula for the continuous solutions of the title AFE:

where, , , is any invertible function such that, . As expected the solution depends on an arbitrary continuous function, . This function is called a modulator function and is defined as:

In [Kefalas, 2014] it is proved that the double limits always exist and that* f*, satisfies the AFE.

### Further reading

- Kyriakos Kefalas. (2014). On smooth solutions of non linear dynamical systems,
*f*, part I. Phys. Int., 5: 112-127._{n+1}= u(f_{n})

### References

- Abel, N. H. (1826). Untersuchung der Functionen zweier unabhängig veränderlichen Größen
*x*und*y*, wie*f (x, y)*, welche die Eigenschaft haben, daß*f (z, f (x, y))*eine symmetrische Function von*z*,*x*und*y*ist.*Journal für die reine und angewandte Mathematik*,*1*, 11-15. -
Kuczma, M., Choczewski, B., & Ger, R. (1990).
*Iterative functional equations*(No. 32). Cambridge University Press.