International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 365956, 10 pages

http://dx.doi.org/10.1155/2012/365956

## Iteration and Iterative Roots of Fractional Polynomial Function

College of Mathematics and Information Science, Leshan Normal University, Leshan, Sichuan 614004, China

Received 4 February 2012; Revised 12 April 2012; Accepted 24 April 2012

Academic Editor: N. Govil

Copyright © 2012 Xiao-hua Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Iteration is involved in the fields of dynamical systems and numerical computation and so forth. The computation of iteration is difficult for general functions (even for some simple functions such as linear fractional functions). In this paper, we discuss fractional polynomial function and use the method of conjugate similitude to obtain its expression of general iterate of order *n* under two different conditions. Furthermore, we also give iterative roots of order *n* for the function under two different conditions.

#### 1. Introduction

Iteration is a repetition of the same operation. Given a nonempty set and a self-mapping , define , where and denotes the composition of mappings. is called the th iterate of , and is the iterate index of concerning . Iteration is often observed in mathematics, science, engineering, and daily life, but the computation of iteration of some elementary functions is very complicated and sometimes rather difficult (see [1–7]), such as linear fractional functions , where . Using the numerical computation method, we only make some partitions on the defined interval of to obtain pointwise data and approximately curves of . Although computer algebra system such as Maple provided the symbol computational tool, we still need to calculate the th iterate of for a given , and the expression of iteration is complicated even for (see [8]). However, using the method of conjugate similar, we can effectively calculate its iteration of order (see the following (*)). This example shows that computer is not universal, and we need to find good mathematical method.

Given mapping and , if there exists invertible mapping such that , then is conjugating to . Obviously, if , then . We usually use this method to turn iteration of complicated function into iteration of simple function which is easy to get general iteration. We call it as the method of conjugation. For example, using the method of conjugation in reference [9], fractional linear function is conjugated to a linear function by conjugation function , where is a root of the equation . Thus, the th iterate of the fractional linear function is where . By the same method, in reference [10], Jin et al. discuss that the th iterate of polynomial function where , under the conditions that , , and , is where denotes the number of combination, that is, .

Given a nonempty set and an integer , an iterative root of order of a given self-mapping is a self-mapping such that where denotes the th iterate of , that is, . The problem of iterative roots of mapping is an important problem in the iteration theory (see [9, 11–13]). It was studied early from the 19th century, but great advances have been made since 1950s, most of which were given for monotone self-mappings on compact interval. For nonmonotonic cases, there are also some progress in references (see [14–19]).

In this paper, we study iteration and iterative roots of the fractional polynomial function where . It can be treated as a nonmonotonic mapping on . Using the method of conjugation, we get the expression of and iterative roots of order of under some conditions.

#### 2. Iteration of Fractional Polynomial Functions

Theorem 2.1. *Let either when is odd or when is even. Suppose that the fractional polynomial function defined by (1.4) satisfies , then
**
where , when , ; .*

*Proof. *On the basis of what satisfies, the fractional polynomial function defined by (1.4) transforms into
where denotes . Set , then the inverse of is , it follows that
Set , then the inverse of is , it follows that
By induction, we obtain easily
By (2.4) and (2.5), we have
By (2.3) and (2.6), we get
This completes the proof.

Theorem 2.2. *Let either when is odd or when is even. Suppose that the fractional polynomial function defined by (1.4) satisfies , , then
**
where ; when , , .*

*Proof. *On the basis of what satisfies, the fractional polynomial function defined by (1.4) transforms into
Set , then the inverse of is , it follows that
By induction, we obtain easily
By (2.10) and (2.11), we get
This completes the proof.

#### 3. Iterative Roots of Fractional Polynomial Functions

We first give some useful lemmas.

Lemma 3.1. *If , where , then
**
where .*

*Proof. *Set , then the inverse of is , it follows that
By induction, we obtain that the th iterate of is
By (3.2) and (3.3), we get
This completes the proof.

Lemma 3.2. *If , where , satisfies , then
**
where .*

*Proof. *Set , then the inverse of is , it follows that
By Lemma 3.1, when , that is, when , we get that the th iterate of is
By (3.6) and (3.7), we get
This completes the proof.

In Theorems 2.1 and 2.2, we get the expression of of the fractional polynomial function (1.4) under different conditions, they can be treated as a mapping which involves parameter , then is extended from to , we can obtain the iterative roots. For example, we can get the iterative roots of the fractional polynomial function (1.4) by extending the results of Theorems 2.1 and 2.2.

Theorem 3.3. *Suppose that the fractional polynomial function defined by (1.4) satisfies conditions in Theorem 2.1, then has iterative roots of any odd order:
**
where .*

*Proof. *In what follows, we only need to prove that holds under the case that satisfies the conditions in Theorem 2.1 and is positive odd.

In fact, suppose that the fractional polynomial functions defined by (1.4) satisfy the conditions in Theorem 2.1, then we have
where denotes .

Because
when is positive odd, by Lemma 3.2, we have
Thus, has iterative roots of any odd order:
This completes the proof.

*Remark 3.4. *When is positive even, the above is not well defined, thus, under the condition in Theorem 3.3, has no iterative roots of any order in the form of .

By Theorems 2.1 and 3.3, we are easy to get the following corollary.

Corollary 3.5. *Let and , then*(1)*the expression of is
*(2)* has iterative roots of any odd order , that is, , where .*

Theorem 3.6. *Suppose that the fractional polynomial function defined by (1.4) satisfies conditions in Theorem 2.2, then has iterative roots of any odd order:
**
where .*

*Proof. *In what follows, we only need to prove that holds under the case that satisfies the conditions in Theorem 2.2 and is positive odd.

In fact, suppose that the fractional polynomial functions defined by (1.4) satisfy the conditions in Theorem 2.2, then we have
When is positive odd, by Lemma 3.1, we have
Thus, has iterative roots of any odd order:
This completes the proof.

*Remark 3.7. *When is positive even, the above is not well defined, thus, under the conditions in Theorem 3.6, has no iterative roots of any order in the form of .

By Theorems 2.2 and 3.6, we are easy to get the following corollary.

Corollary 3.8. *Let and , then*(1)*the expression of is
*(2)* has iterative roots of any odd order , that is, , where .*

#### 4. Examples

We demonstrate our theorems with the following examples.

*Example 4.1. *Consider the fractional polynomial function , by Theorem 2.1, we know , thus iteration of order of the fractional polynomial functions:
By Theorem 3.3, we know has iterative roots of any odd order:
where .

*Example 4.2. *Consider the fractional polynomial function , by Theorem 2.1, we know , thus iteration of order of the fractional polynomial functions:
By Theorem 3.3, we know has iterative roots of any odd order:
where .

*Example 4.3. *Consider the fractional polynomial function , by Theorem 2.2, we know , thus iteration of order of the fractional polynomial functions:
By Theorem 3.6, we know has iterative roots of any odd order:
where .

#### Acknowledgment

This work is supported by Scientific Research Fund of Sichuan Provincial Education Department under Grant 11ZA160.

#### References

- I. N. Baker, “The iteration of polynomials and transcendental entire functions,”
*Australian Mathematical Society Journal Series A*, vol. 30, no. 4, pp. 483–495, 1980-1981. View at Google Scholar - P. Bhattacharyya and Y. E. Arumaraj, “On the iteration of polynomials of degree $n$ with real coefficients,”
*Annales Academiae Scientiarum Fennicae. Series A I. Mathematica*, vol. 6, no. 2, pp. 197–203, 1981-1982. View at Google Scholar · View at Zentralblatt MATH - B. Branner and J. H. Hubbard, “The iteration of cubic polynomials Part II: patterns and parapatterns,”
*Acta Mathematica*, vol. 169, no. 1, pp. 229–325, 1992. View at Publisher · View at Google Scholar · View at Scopus - X. Chen, Y. Shi, and W. Zhang, “Planar quadratic degree-preserving maps and their iteration,”
*Results in Mathematics*, vol. 55, no. 1-2, pp. 39–63, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. C. Sun, “Iteration of quasi-polynomials of degree two,”
*Journal of Mathematics*, vol. 24, no. 3, pp. 237–240, 2004. View at Google Scholar - Z. Wu and D. Sun, “The iteration of quasi-polynimials mappings,”
*Acta Mathematica Scientia*, vol. 26, pp. 493–497, 2006 (Chinese). View at Google Scholar - L. Xu and S. Y. Xu, “$N$th-iteration of a linear fractional function and its applications,”
*Mathematics in Practice and Theory*, vol. 36, no. 6, pp. 225–228, 2006 (Chinese). View at Google Scholar - W. Zhang, D. Yang, and S. Deng,
*Functional Equations*, Sichuan Education Publishing House, Chengdu, China, 2001. - J. Zhang, L. Yang, and W. Zhang,
*Iterative Equation and Embedding Flows*, Shanghai Scientific and Technological Educational Publishing House, Shanghai, China, 1998. - L. Jin, Z. Zhou, and X. Liu, “Computation and estimation of iteration,”
*Mathematics in Practice and Theory*, vol. 34, no. 4, pp. 170–176, 2004 (Chinese). View at Google Scholar - J. Zhang and J. Xiong,
*Functional Iteration and One Dimensional Dynamical Systems*, Sichuan Education Publishing House, Chengdu, China, 1992. - M. Kuczma, B. Choczewski, and R. Ger,
*Iterative functional equations*, vol. 32 of*Encyclopedia of Mathematics and Its Applications*, Cambridge University Press, Cambridge, 1990. - K. Baron and W. Jarczyk, “Recent results on functional equations in a single variable, perspectives and open problems,”
*Aequationes Mathematicae*, vol. 61, no. 1-2, pp. 1–48, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Li, D. Yang, and W. Zhang, “A note on iterative roots of PM functions,”
*Journal of Mathematical Analysis and Applications*, vol. 341, no. 2, pp. 1482–1486, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Liu, W. Jarczyk, L. Li, and W. Zhang, “Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 1, pp. 286–303, 2012. View at Publisher · View at Google Scholar - T. X. Sun and H. J. Xi, “Iterative solutions of functions of type $N$ on an interval,”
*Journal of Mathematical Study*, vol. 29, no. 2, pp. 40–45, 1996 (Chinese). View at Google Scholar - T. X. Sun, “Iterative roots of anti-$N$-type functions on the interval,”
*Journal of Mathematical Study*, vol. 33, no. 3, pp. 274–280, 2000 (Chinese). View at Google Scholar - J. Z. Zhang and L. Yang, “Iterative roots of a piecewise monotone continuous self-mapping,”
*Acta Mathematica Sinica*, vol. 26, no. 4, pp. 398–412, 1983 (Chinese). View at Google Scholar - W. Zhang, “PM functions, their characteristic intervals and iterative roots,”
*Annales Polonici Mathematici*, vol. 65, no. 2, pp. 119–128, 1997. View at Google Scholar · View at Zentralblatt MATH