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Existence of iterative roots for the sickle-like functions
Journal of Inequalities and Applications volume 2014, Article number: 204 (2014)
Abstract
The problem of iterative roots for strictly monotone self-mappings has been well solved. Most of known results concerning existence of iterative roots for a continuous function were given under the assumption that the function has finitely many non-monotonic points. When a function has infinitely many non-monotonic points, the problem of the existence of its iterative roots will become more complicated. In this paper, we study the existence of iterative roots for the sickle-like functions, as a special class of non-monotonic functions, each of which has not only one isolated non-monotonic point but also infinitely many non-isolated non-monotonic points.
MSC:37E05, 39B12.
1 Introduction
Let . For any integer , consider a mapping . An iterative root of order n of F is a mapping such that
where denotes the n th iterate of the mapping , i.e. and for all inductively. By studying the iterative roots, people can find the missing information in the iterative process. Meanwhile, being a weak version of the problem of embedding a function into a flow or into a semi-flow, the existence of iterative roots of a given mapping is a basic problem in both the theory of functional equations and the theorem of dynamical systems. The problem of finding iterative roots for a given function is still alive since the work of Babbage [1, 2] at the beginning of the 19th century, more and more attention has been turned to this problem (see e.g. [3–8] and references therein). Plentiful results have been obtained for continuous and strictly monotonic mappings on intervals. In the monographs [9, 10], Kuczma, Choczewski and Ger gave a complete description of iterative roots of continuous and strictly monotonic self-mappings on a given interval.
An interior point of I is called a monotonic point of mapping if F is strictly monotonic in a neighborhood of ; otherwise, is referred to as a non-monotonic point or simply a fort of F (see [11, 12]). Consequently, the function is strictly monotonic on I if and only if it has no non-monotonic points in the interior of I. A function having finitely many non-monotonic points is called a strictly piecewise monotonic function or simply called a PM function (see [11]). Each of the non-monotonic points of a PM function is an isolated non-monotonic point. It seems that it was JZ Zhang and L Yang who first, in 1983, started to study iterative roots of PM functions explicitly in the paper [11]. They introduced the concept of characteristic interval for PM functions, and studied the existence of iterative roots of PM functions which have the characteristic interval. Later, Blokh, Coven, Misiurewicz, Nitecki and WN Zhang established some new results for iterative roots of PM functions (see [12, 13]). Recently, there are some advances obtained for iterative roots. For example see in [14–18].
When a function has infinitely many non-monotonic points, the study of existence of its iterative roots will become more difficult. One of the typical cases that the function has infinitely many non-monotonic points is that at least there exists one nontrivial subinterval (i.e., not singleton) on which the function is constant (see e.g. [19–21]). In 1992, the author in the paper [22] proved the existence of iterative roots of a class of self-mappings possessing infinitely many non-monotonic points. Later, TX Sun and HJ Xi discussed the iterative roots of a class of self-mappings with a constant on two subintervals (see [23]).
However, in both paper [22] and paper [23], each of non-monotonic points of the function is non-isolated. The main purpose of this paper is to study the existence of continuous iterative roots for a class of functions, each of which has not only infinitely many non-isolated non-monotonic points but also one isolated non-monotonic point.
Assume that , and , the set of all continuous self-mappings on I. Then F is called a sickle-like function if one of the following conditions is fulfilled: (C1) F is constant on , and F is strictly decreasing on but strictly increasing on ; (C2) F is constant on , and F is strictly increasing on but strictly decreasing on ; (C3) F is constant on , and F is strictly increasing on but strictly decreasing on ; (C4) F is constant on , and F is strictly decreasing on but strictly increasing on . If F satisfies (C3) (resp. (C4)), then H satisfies (C1) (resp. (C2)), where H is defined by for all , and is defined by
Hence, it suffices to confine ourselves to discuss F satisfying (C1) or (C2). For this purpose, let (resp. ) denote the set of all those sickle-like functions satisfying (C1) (resp. (C2)) (see Figures 1 and 2), and . If , then b is an isolated non-monotonic point of F but every point belonging to is a non-isolated non-monotonic point of F.
The paper is organized as follows: at first some important properties of iterative roots of sickle-like functions will be given in Section 2. Then in Sections 3 and 4 we will discuss the existence of iterative roots of and , respectively. Throughout this paper, n stands for a positive integer and represents the restriction of F on E for a set .
2 Preliminaries
Lemma 2.1 Suppose that f is an iterative root of order of on I. Then
-
(i)
if there exists such that ;
-
(ii)
if there exists such that ;
-
(iii)
there exists such that if .
Proof For result (i), we only prove that . By reduction to absurdity, suppose that . Then there exists such that . By the assumption, we have
where denotes either or . It follows that at least one of and is a nonempty closed interval (not a singleton). Without loss of generality, we may assume that there are such that and
Thus, there are such that , and . On the other hand, noting that is strictly monotonic on , we find that is also strictly monotonic on , which yields
contrary to the fact that F is constant on . This contradiction completes the proof of result (i).
For result (ii), we will give a proof by contradiction. Suppose that there exists such that . It follows from the result (i), proved just now, that , showing that , contrary to the assumption that . This contradiction shows us that , and thus the result (ii) is proved.
For result (iii), it becomes obvious by the result (ii) that , which shows that
It follows that there exists an integer such that but . Using the continuity of and , there exists such that . Thus,
and the result (iii) is proved. This completes the proof of Lemma 2.1. □
Lemma 2.2 Suppose that f is an iterative root of order of on I. Then
-
(i)
if , and
-
(ii)
either or if .
Proof First of all we claim that if , then
Indeed, if , then it follows from the results (i) and (iii) of Lemma 2.1 that there exists such that
Suppose that there is an such that . Let . Then we have
Hence, according to the continuity of f, there exist such that and . It follows from (2.2) and (2.3) that
i.e. F is constant on , contradicting to the assumption on F. Thus the claimed (2.1) holds.
For result (i), suppose for an indirect proof that there exists such that . It follows from the result (i) of Lemma 2.1 that , which shows that . Thus (2.1) holds, implying that , contradicting to the assumption and the result (i) is proved.
For result (ii), firstly we claim that
In fact, if , then there exists such that , which implies that , a contradiction and thus the claimed (2.4) holds. It follows that (2.1) holds, which shows that is a non-monotonic self-mapping since is non-monotonic. Note that both and are strictly monotonic, implying that and are both strictly monotonic. Thus, b is the unique non-monotonic point of .
Secondly we claim that
Suppose, on the contrary, that . Then there exists such that . Take any neighborhood of t. Since f is strictly monotonic in U, is a neighborhood of , i.e. is a neighborhood of b. Thus f is not strictly monotonic in as b is a non-monotonic point of f. In other words, is not strictly monotonic in U. This contradicts the fact that is strictly monotonic on since is strictly increasing on . This contradiction shows that the first result of (2.5) holds. Similarly, we can deduce that the second result of (2.5) holds. It follows from (2.5) that either or , and either or . If , then . In fact, otherwise, if , then . Thus and are both strictly increasing, which together with (2.1) guarantee that is a strictly increasing self-mapping. It implies that is also strictly increasing, a contradiction. Thus we have
Similarly, if , then . It implies that
Hence, the proof of Lemma 2.2 is completed. □
Lemma 2.3 Suppose that with . Then the following hold:
-
(i)
Either or if F has iterative roots of order on I;
-
(ii)
F has iterative roots of order on I if and only if has iterative roots of order on .
Proof For result (i), suppose that f is an iterative root of order of F on I. It follows from Lemma 2.2 that either or . If , then , which shows, since , that
If , then . It follows that
Thus the result (i) is proved.
For result (ii), firstly we prove the sufficiency. Suppose that is an iterative root of order of on . Then, obviously, the function f defined by
is an iterative root of order of F on I.
Conversely, let f be an iterative roots of order of F on I. Making use of Lemma 2.2, one obtains that is a self-mapping. Thus, , i.e. is an iterative roots of order of on , and necessity is proved. This completes the proof of Lemma 2.3. □
We end this section with Lemma 2.4, which gives some basic results concerning the existence of iterative roots for strictly monotonic self-mappings.
Lemma 2.4 (see [[11], Theorems 7 and 10] and [[12], pp.119 and 125])
Let , and . Then the following statements are valid:
-
(i)
Suppose that F is strictly increasing. Then F has infinitely many strictly increasing iterative roots f of order on I such that ;
-
(ii)
Suppose that F is strictly decreasing. Then F has neither strictly increasing iterative roots of order nor strictly decreasing iterative roots of even order on I, F has infinitely many strictly decreasing iterative roots of odd order on I if and only if either or . Moreover, if either or , then F has infinitely many strictly decreasing iterative roots f of odd order on I such that .
3 Iterative roots of
In this section, we discuss the existence of iterative roots of . It follows from Lemma 2.3 that F has no iterative roots of order if , and . Thus, we only consider the cases that and if . Thus we start our discussion with Theorem 3.1.
Theorem 3.1 Suppose that with . Then the following hold:
-
(i)
If either or , then F has infinitely many iterative roots of order on I;
-
(ii)
If and , then F has no iterative roots of order on I;
-
(iii)
If and , then F has no iterative roots of order on I.
Proof For result (i), let , . Then is the range of F on I. If , then, by the assumption, we have , and thus . If , then it is obvious that . With the aid of Lemma 2.4, we obtain that has infinitely many strictly increasing iterative roots of order on such that
i.e. maps into . Define the function by
Note that
implying that f defined by (3.1) is continuous. Since
we find that the function f defined by (3.1) is an iterative root of order of F on I, and the result (i) is proved.
For result (ii), suppose, by an indirect proof, that f is an iterative root of order of F on I. Since and , we have, on account of Lemma 2.2, , which shows that is a strictly monotonic iterative root of order of . The fact that guarantees that is strictly increasing. In fact, if is strictly decreasing, then at least one of and cannot hold. Without loss of generality, we may assume that . Consequently,
This contradiction shows that is strictly increasing. Hence, and since and . Noting by Lemma 2.2 again that , we infer that which shows that
This contradiction completes the proof of result (ii).
For result (iii), assume by indirect proof that f is an iterative root of order of F on I. Similarly, is a strictly monotonic iterative root of order of . Lemma 2.2 and the fact force that
If , then we deduce, by the monotonicity of , that
which with (3.3) shows that (3.2) holds, a contradiction.
If , then is strictly decreasing. Note that , implying that . It follows from (3.3) that
contradicting to the assumption. Thus, the result (iii) is proved, and this completes the proof of Theorem 3.1. □
Example 3.1 Consider and given by Figures 3 and 4, respectively.
It follows from Theorem 3.1 that has iterative roots of order on I but has no iterative roots of order on I.
Theorem 3.2 Suppose that with . Then the following statements are valid:
-
(i)
F has no iterative roots of even order on I;
-
(ii)
F has infinitely many iterative roots of odd order on I if and only if either or .
Proof For result (i), suppose, for an indirect proof, that f is an iterative root of even order of F on I. By Lemma 2.2, we see that
which implies that , i.e. is an iterative root of even order of . However, as a strictly decreasing function, we have, according to Lemma 2.4, that has no iterative roots of even order. This contradiction completes the proof of result (i).
For result (ii), firstly we prove the sufficiency. Let and . Then is the range of F on I, and either or . It follows from Lemma 2.4 that has infinitely many strictly decreasing iterative root of odd order such that . Now we define the function by
It is easy to see that defined by (3.4) is an iterative root of odd order of F on I and the sufficiency is proved.
In what follows we prove necessity. Suppose that f is an iterative root of odd order n of F on I. It follows from Lemma 2.2 that
Thus is a strictly decreasing iterative root of odd order of . By Lemma 2.4 again, we obtain either or .
If , which shows that at least one of and does not hold. Without loss of generality we may assume that . It follows from (3.5) and the monotonicity of that
Thus, . The necessity is proved and the proof of Theorem 3.2 is completed. □
Example 3.2 The functions and are given by Figures 5 and 6, respectively.
By Theorem 3.2, we find that has exactly iterative roots of odd order on I but has no iterative roots of order on I.
Theorem 3.3 Suppose that with . Then F has infinitely many iterative roots of order .
Proof Choose points from , arbitrarily, such that
Put , for integer . Further, we define is a decreasing homeomorphism and is an increasing homeomorphism, arbitrarily. If , then we give the increasing homeomorphisms and for , arbitrarily. Now define the function on by
which is continuous on . Note that
implying that . In what follows we discuss several cases:
In the case that : Let the functions and be defined, respectively, by
Since is a decreasing homeomorphism, is an increasing homeomorphism and , we find that is an increasing homeomorphism, and is a strictly increasing function. Moreover,
Thus the mapping defined by
is an iterative root of order of F on I.
In the case that and F has no fixed points on : Define
inductively for all those integer for which the recurrence procedure (3.8) is performable. Note that F has no fixed points on , implying that there is an integer such that
In fact, if for all integers , then, by (3.6), (3.8), and the monotonicity of , we infer that the sequence is infinite and strictly increasing. Therefore, by (3.8), is a fixed point of F on , a contradiction. It follows from (3.9) that exists and . Let . Without loss of generality we may assume that is odd, and we let for some integer . For , define on and on inductively by
respectively. Noting that is decreasing homeomorphism, we find that is an increasing homeomorphism. Fix an and suppose that the function is an increasing homeomorphism for . Thus is an increasing homeomorphism. Note that is an increasing homeomorphism, implying by (3.10) that the function is an increasing homeomorphism. By induction, we deduce that the function defined by (3.10) is an increasing homeomorphism for . Similarly, we see that the function defined by (3.11) is also an increasing homeomorphism for . Since
is continuous and strictly increasing on . Define by
It follows from (3.7), (3.10), and (3.11) that for all , i.e. the mapping , defined by (3.12), is an iterative root of order of F on I.
In the case that and F has fixed points on : Let . Then since . There is no loss of generality in assuming that . Because is a strictly increasing self-mapping, and in view of Lemma 2.4, has infinitely many strictly increasing iterative roots of order such that . Meanwhile, the infinite sequence defined by (3.6) and (3.8) is strictly increasing and . Now we define a mapping by
where on (resp. on ) is defined inductively by (3.10) (resp. (3.11)). As we have just seen in the preceding case, it is easy to check that the mapping defined by (3.13) is an iterative root of order of F on I, and the above discussion completes the proof of Theorem 3.3. □
Example 3.3 Consider the functions and given by Figures 7 and 8, respectively.
We know by Lemma 2.3 that has no iterative roots of order on I. However, Theorem 3.3 shows that has iterative roots of order on I.
4 Iterative roots of
In this section we shall discuss the existence of iterative roots of . Making use of Lemma 2.3, we find that F has no iterative roots of order if , and . Thus, we only discuss the two cases and if . If , then, according to Lemma 2.3, it suffices to discuss the existence of iterative roots of . Let for all , where is defined by
Thus, and G is strictly decreasing on but is strictly increasing on if . By Lemma 2.3 and Theorems 3.1 and 3.2, we obtain immediately Theorems 4.1 and 4.2.
Theorem 4.1 Suppose that with . Then F has no iterative roots of even order on I, and F has infinitely many iterative roots of odd order if and only if either or .
Theorem 4.2 Suppose that with . Then the following statements are valid:
-
(i)
If either or , then F has infinitely many iterative roots of order on I;
-
(ii)
If and , then F has no iterative roots of order on I;
-
(iii)
If and , then F has no iterative roots of order on I.
Theorem 4.3 Suppose that with . Then F has infinitely many iterative roots of order if and only if F satisfies one of the following conditions:
-
(i)
;
-
(ii)
and .
Proof For sufficiency, firstly suppose that F satisfies the condition (i). Choose points from such that
Write for integers . Let and be both arbitrary increasing homeomorphisms, and let be an arbitrary decreasing homeomorphism. If , then we take increasing homeomorphisms for , arbitrarily. Define by
which clearly is continuous on and for all . Thus we see that
Take any . Then there exists an such that . Writing , where and , we get
In virtue of (4.1) and (4.2), we obtain
i.e. is an iterative root of order of . Define the functions on and on by
respectively. Note that and , implying that and are both increasing homeomorphisms. Moreover,
Now we define ψ on I by
It is natural that
which together with (4.3) imply that the function defined by (4.4) is an iterative root of order of F on I.
Secondly assume that F satisfies the condition (ii), i.e. and . Choose points in , arbitrarily, such that
Let for integers . We give the increasing homeomorphisms for , arbitrarily. Now put
(which is equivalent to ) for all those positive integers k for which the recurrence procedure is performable. Next we shall discuss the two distinguished cases:
In the case that F has no fixed points on : Then the condition implies that . Moreover, there exists a positive integer such that
In fact, otherwise, note that for any integer , implying, by the fact that and (4.6), that exists, which shows that exists for all integers and the sequence is strictly increasing. It follows from (4.6) that is a fixed point of F on , a contradiction. The fact yields . There is no loss of generality in assuming that . Let . Define the functions on inductively for by
Since and are both increasing homeomorphisms, defined by (4.7) is an increasing homeomorphism. Similarly, by induction, we find that defined by (4.7) is an increasing homeomorphism for . Note that is an increasing homeomorphism, and we have
implying that is a strictly increasing function. Define on by
It is natural that is continuous and strictly increasing on . Moreover,
Put and define inductively for . Then for . Let the function on be defined by
Because is a decreasing homeomorphism and is an increasing homeomorphism for , we infer that is a decreasing homeomorphism. By the definition of , we possess that is an increasing homeomorphism for . On the other hand, noting that and is strictly decreasing, we have such that
Thus
This shows that
Define by
which is continuous since and (4.8). By the definition of and , we see that defined by (4.9) is an iterative root of order of F on I.
In the case that F has fixed points on : Let . Then is well defined inductively by (4.6) for all integers and . Without loss of generality, assume that . Let be defined by (4.7) for all integers . If , then there exists such that , which implies
Noting that is equivalent to , we get the conclusion that
Hence,
On the other hand, since is a strictly increasing self-mapping and , we obtain, in view of Lemma 2.4, that has infinitely many strictly increasing iterative roots of order on such that
Define the function on by
By the definition of and , (4.10) and (4.11), we see that the function on defined by (4.12) is continuous and strictly increasing. Let . is well defined inductively for . Thus is an increasing homeomorphism for . Then the function on given by
is a decreasing homeomorphism from to . By the definition of and , we deduce that, for ,
Choose satisfying
Consequently, for ,
It follows from (4.14) that
which jointly with (4.13) guarantees that
showing that
In addition, . Consequently, the mapping defined by
is continuous. According to (4.12) and (4.13), the mapping defined by (4.15) is an iterative root of order of F on I. Consequently, the sufficiency is proved.
To prove necessity, suppose that f is an iterative root of order of F on I. It follows from Lemma 2.2 that
which implies that f has the fixed points on . Note that the fixed point of f is bound to the fixed point of F and is the unique fixed point of F on , implying that is the unique fixed point of f on . It follows that
Since and , there exists such that but . By the continuity of , there are such that and . Thus the definition of and (4.17) yields
We claim that
Otherwise, if there exist and such that , then , which contradicts to the fact that . Therefore, as claimed (4.19) holds. On the other hand, and are both strictly monotonic since and are both strictly monotonic. In what follows we shall distinguish the following several cases:
In the case that : Suppose for indirect proof that . Then there exists such that since . On the other hand, we have, in virtue of (4.17) and (4.19),
It follows that is strictly decreasing. Thus
By (4.17) and (4.20), we get . Thus
contradicting to the assumption that . This contradiction shows that holds, i.e. F satisfies the condition (i).
In the case that : Firstly we prove that is strictly decreasing. In fact, if is strictly increasing, then, by (4.19), we infer that
This forces to obey
which with (4.16) implies that
Note that , implying that there exists such that , which with (4.22) results in
contradicting to the fact that . Thus, is strictly decreasing. It follows from (4.17) that
which with (4.16) means that
Based on (4.19), we have
which with (4.16) shows that
Moreover, we have
In fact, if , then, by (4.23), is strictly increasing. Note that , implying by (4.18) that . Thus there exist such that and . It follows thus from (4.18) that
i.e. F is constant on . This contradiction shows that the claimed (4.26) holds. Making use of (4.25) and (4.26), one obtains
which with (4.24) guarantees that F satisfies one of conditions (i) and (ii).
In the case that : It is natural by (4.17) and (4.19) that
Thus is strictly increasing. Due to (4.19), we have . It follows that (4.21) holds.
If , then (4.16) and (4.21) imply that , i.e., F satisfies the condition (i).
If , by the continuity and monotonicity of and (4.17), there is an such that . Let be well defined inductively for . It follows thus from (4.17) that . Since () is strictly increasing, we see that is strictly monotonic, which together with (4.21) implies that is strictly increasing. Fix a , arbitrarily. Assume that is strictly increasing for each . Noting that , there is an such that . Whence, by the induction hypothesis, we infer that
is strictly increasing, i.e. is strictly increasing. Because () is strictly increasing, we see that is strictly monotonic. It follows that is strictly increasing. Thus, by the induction hypothesis,
is strictly increasing, which shows that is strictly increasing. Consequently, by induction, is strictly increasing for . Thus
Moreover, is strictly increasing, showing that
Meanwhile, it can be seen that
If is strictly increasing, then it follows from (4.27) and (4.28) that is an increasing self-mapping, which forces that is an increasing self-mapping. This contradiction shows that must be strictly decreasing. Thus the point b is a non-monotonic point of f. If , then there exists such that . Consequently, is a non-monotonic point of . However, is strictly monotonic on since () is strictly increasing. This contradiction yields
Further, we claim that
In fact, if , then, by (4.27), and (4.18) shows that . As the proof of (4.26), we can obtain a contradiction. It follows from (4.27), (4.28), (4.29), and (4.30) that is a self-mapping, and we have
which implies that and . Consequently, F satisfies the condition (ii) if F does not satisfy the condition (i), and necessity is proved. This completes the proof. □
Example 4.1 Let and be given by Figures 9 and 10, respectively.
We see, according to Theorem 4.3, that both and have iterative roots of order on I.
Example 4.2 It follows from Theorem 4.3 that the functions and given by Figures 11 and 12, respectively, have no iterative roots of order on I.
In this paper, we give all conditions of existence and nonexistence of iterative roots for the sickle-like functions. Meanwhile, we construct infinite many iterative roots if they exist. However, we are not sure if all continuous iterative roots are defined by the construction method given in this paper, in other words, it is unclear whether there are other forms of iterative roots if they exist. On the other hand, as the number of the subintervals on which the function is constant and isolated non-monotonic points increases, we do not know whether the existence of its iterative roots can be solved in a similar manner to the one described in this paper. Although we are not able to give answers for these problems, yet, they are our further directions of investigation.
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Acknowledgements
The author would like to express sincere gratitude to the editors and the referees for being kind enough to give very helpful suggestions and make many comments. This research was supported by the Grant No. LY14A010005 of Zhejiang provincial natural science foundation of China and the Grant No. Y201327415 of Zhejiang Provincial Education Department.
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Lin, Y. Existence of iterative roots for the sickle-like functions. J Inequal Appl 2014, 204 (2014). https://doi.org/10.1186/1029-242X-2014-204
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DOI: https://doi.org/10.1186/1029-242X-2014-204