Iterative roots of upper semicontinuous multifunctions
 Pingping Zhang^{1}Email author and
 Liguo Huang^{1}
https://doi.org/10.1186/s1366201714087
© The Author(s) 2017
Received: 14 January 2017
Accepted: 20 October 2017
Published: 25 November 2017
Abstract
The square iterative roots for strictly monotonic and upper semicontinuous functions with one setvalued point were fully described in (Li et al. in Publ. Math. (Debr.) 75:203220, 2009). As a continuation, we study both strictly monotonic and nonmonotonic multifunctions. We present sufficient and necessary conditions under which those multifunctions have nth iterative roots. This equivalent condition and the construction method of nth iterative roots extend the previous results.
Keywords
MSC
1 Introduction
Babbage [2] investigated (1.1) for an identity mapping F as far back as the 1810s. After that, (1.1) has been studied in various aspects and settings since it is an important subject in the theory of functional equations; we refer to the survey papers [3–7], the monographs [8, 9], and the book [10]. For all we know, strictly increasing roots of strictly increasing and continuous functions were discussed by Bödewadt [11], and their strictly decreasing roots were presented by Haı̌dukov [12]. In 1961, Kuczma [13] gave a complete description of strictly monotonic and continuous functions having roots. However, even simple nonmonotonic functions can have no iterative roots, for example, the hat functions \(f(x)=\min\{\frac{x}{a}, \frac{1x}{1a}\}\) on the compact interval \([0,1]\) for arbitrary \(a\in(0,1)\). In 1983, Zhang and Yang [14] investigated the roots of piecewise monotonic functions (abbreviated as PM functions). The main difficulties to find roots of PM functions lie in the continuously increasing number of nonmonotonic points under iteration (see [15]). Their method is based on the ‘characteristic interval’, which was developed in [7, 16]. In recent years, many important results on iterative roots of PM functions were presented in [17–19]. It is worth mentioning that those results are related to singlevalued functions. In [20, 21] and [22], it is illustrated that the set of continuous functions having a root is a nonBorel subset of \(C([0,1],\mathbb{R})\) and is small in \(C([0,1],[0,1])\). That is to say, in the general case, no such roots exist, and the theory becomes extremely complicated if F is not bijective [23]. Therefore, it is a natural idea to extend the notion of iterative root.
In his survey paper [5], Targonski illustrated three ways to generalize iterative roots, extending or restricting the domain of the function or embedding the semigroup of selfmappings in a larger semigroup, and discussed the socalled phantom iterative root of continuous functions in [23]. Powierża and Jarczyk [24–26] gave setvalued functions as roots of singlevalued functions. Maybe the best method to generalize iterative roots is replacing singlevalued functions by setvalued functions for both F and f in (1.1) (see [4]). It seems that, up to now, there are only several results on setvalued iterative roots of multifunctions, even with a unique setvalued point. In 2007, Jarczyk and Zhang [27] considered the nonexistence of square iterative roots of multifunctions with exactly one setvalue point and presented two sufficient conditions for the purely settheoretical situation. Later, Li, Jarczyk, Jarczyk, and Zhang [1] gave new nonexistence results for purely settheoretical case and fully described the square roots of strictly monotonic, upper semicontinuous (abbreviated as usc) multifunctions.
As a continuation of [1], in this paper, we study all strictly monotonic usc multifunctions having one setvalued point and partly nonmonotonic ones. We give sufficient and necessary conditions for the existence of nth iterative roots and their construction method, which extend the results on strictly monotonic usc multifunctions in [1]. In Section 2, we recall the basic definitions and present Lemmas 13. In Section 3, we give equivalent conditions for the existence of nth iterative roots and their expressions. Finally, in Section 4, we apply examples to illustrate our results.
2 Preliminaries
Given topological spaces X and Y, a multifunction \(f: X\to2^{Y}\) is called upper semicontinuous at a point \(x_{0} \in X\) if for every open set \(V\subset Y\) with \(f(x_{0})\subset V\), there exists a neighborhood \(U\subset X\) of \(x_{0}\) such that \(f(U)\subset V\). If f is upper semicontinuous at every point of \(B\subset X\), then it is called upper semicontinuous on a set \(B\subset X\). Let \(F: X\to X\) and \(G: Y\to Y\) be continuous functions. We say that F is topologically conjugate to G if there exists a homeomorphism \(\varphi: X\to Y\) satisfying the equation \(\varphi\circ F=G\circ\varphi\).
Definition 1
([1])
 (a)
\(\min A_{\xi}=\inf I\) if and only if \(\max B_{\xi}=\sup I\);
 (b)
there exists a strictly decreasing function a mapping \(A_{\xi}\) onto \(B_{\xi}\);
 (c)
for every component \((a,b)\) of the set \(\{x\in \operatorname{cl}I: \gamma(x) \neq x\}\), where \(a\in A_{\xi} \cup\{\inf I\}\), \(b\in A_{\xi}\), and \(a< b\), the graphs of \(\gamma_{(a,b)}\) and \(\gamma_{(\alpha(a), \alpha(b))}\) lie on the opposite sides of the diagonal.
Definition 2
([1])
A multifunction \(f: I\to2^{I}\) is called strictly increasing (strictly decreasing) if \(\sup f(x_{1})<\inf f(x _{2})\) (\(\inf f(x_{1})<\sup f(x_{2})\)) whenever \(x_{1} ,x_{2} \in I\) and \(x_{1} < x_{2}\). Multifunctions that are either strictly increasing or strictly decreasing are called strictly monotonic.
Lemma 1
If \(F\in{\mathcal{S}}_{x_{0}} (I,I)\) is onetoone, then every nth iterative root f of F belongs to \(\mathcal{S}_{x_{0}} (I,I)\) and is also onetoone.
Proof
We first prove that f is onetoone. Suppose on the contrary that there exist two different points \(u,v\in I\) such that \(f(u)=f(v)\). Then \(f^{n} (u)=f^{n} (v)\), and thus \(F(u)=F(v)\). Since F is onetoone, only the case \(u=v\) is possible, contrary to the assumption.
Lemma 2
If \(F\in{\mathcal{S}}_{x_{0}} (I,I)\) is onetoone, then F has no nth iterative root \(f\in{\mathcal{S}}_{x_{0}} (I,I)\) taking the value \(\{x_{0}\}\).
Proof
Lemma 3
 (i)
If F is strictly increasing, then F has no strictly decreasing nth iterative root for odd n.
 (ii)
If F is strictly decreasing, then F has no strictly increasing nth iterative root for even n.
This proof is trivial and omitted.
3 Main results
In this section, we give several sufficient and necessary conditions and expressions of nth iterative roots of (2.1). Theorem 1 and Theorem 3 characterize the strictly monotonic usc multifunctions, and nonmonotonic cases are investigated in Theorem 2 and Theorem 4. For convenience, let \(f_{1}:=f_{I_{}}\), \(f_{2}:=f_{I_{+}}\).
Theorem 1
Suppose that the usc multifunction (2.1) has \(F(x_{0})=[c,d]\). If \(F_{I_{}}\) and \(F_{I_{+}}\) are strictly increasing, then F has nth iterative roots if and only if F satisfies \(F(b)< x_{0} \) or \(F(a)> x_{0}\) or \(c\leq x_{0} \leq d\).
Proof
 Case 1. :

\(F(I)\subset I_{}\), i.e., \(F(b)< x_{0}\), or
 Case 2. :

\(F(I)\subset I_{+}\), i.e., \(F(a)>x_{0}\), or
 Case 3. :

\(F(I_{})\subset I_{}\) and \(F(I_{+})\subset I_{+}\), i.e., \(c\leq x_{0} \leq d\).
Theorem 2
Suppose that the usc multifunction (2.1) with \(F(x_{0})=[d,c]\) is onetoone. If \(F_{I_{}}\) and \(F_{I_{+}}\) are strictly increasing, then F has nth iterative roots if and only if F satisfies \(F(b)< F(a)<c\leq x_{0}\) or \(F(a)>F(b)>d\geq x_{0}\) or \(F(b)< x_{0} <F(a)\).
(i) If \(F(b)< F(a)<c\leq x_{0}\), then F has a nonmonotonic nth iterative root defined by (3.1). Moreover, if n is even and \(F_{I_{}}\) has a regular fixed point, then F also has a nonmonotonic nth iterative root of the form (3.1), in which \(f_{1}\) is a strictly decreasing function satisfying \({f_{1}}^{n} =F_{I _{}}\).
(ii) If \(F(a)>F(b)>d\geq x_{0}\), then F has a nonmonotonic nth iterative root defined by (3.2). Moreover, if n is even and \(F_{I_{+}}\) has a regular fixed point, then F has also a nonmonotonic nth iterative root of the form (3.2), in which \(f_{2}\) is a strictly decreasing function satisfying \({f_{2}}^{n} =F_{I _{+}}\).
Proof
The necessity directly comes from Lemma 2. In what follows, our attention is paid to the sufficiency.
Sufficiency. For case (i), we first construct a strictly increasing function (3.5) and a multifunction (3.7) as in case (i) of Theorem 1. Since \(f_{1}\) is strictly increasing, from (3.8) and \(c>d\) we have (3.10). Thus, (3.5), (3.7), and (3.10) yield a nonmonotonic nth iterative root (3.1) of F.
Assuming that n is even and \(F_{I_{}}\) has a regular fixed point, we can construct a strictly decreasing function \(f_{1}\) and a multifunction (3.7) as in case (i) of Theorem 1. Since \(f_{1}\) is strictly decreasing, (3.8) and \(c>d\) lead to (3.9). Therefore, \(f_{1}\) together with (3.7) and (3.9) gives a nonmonotonic nth iterative root (3.1) of F.
The proof of case (ii) is obtained from case (i) by the translation (3.11).
Theorem 3
Suppose that the usc multifunction (2.1) has \(F(x_{0})=[d,c]\). If \(F_{I_{}}\) and \(F_{I_{+}}\) are strictly decreasing, then F has nth iterative roots if and only if n is odd and F satisfies \(F(a)< x_{0}\) or \(F(b)>x_{0}\) or \(d\leq x_{0} \leq c\).
Proof
 Case 1. :

\(f(I)\subset I_{}\), or
 Case 2. :

\(f(I)\subset I_{+}\), or
 Case 3. :

\(f(I_{})\subset I_{}\), \(f(I_{+})\subset I_{+}\), or
 Case 4. :

\(f(I_{})\subset I_{+}\), \(f(I_{+})\subset I_{}\).
The proof of case (ii) is directly obtained from case (i) by the translation (3.11).
Theorem 4
 (i)
If \(F(a)< F(b)< d\leq x_{0}\) and n is odd, then F has a nonmonotonic nth iterative root of the form (3.1), where \(f_{1}\) is defined by (3.26).
 (ii)
If \(F(b)>F(a)>c\geq x_{0}\) and n is odd, then F has a nonmonotonic nth iterative root of the form (3.2), where \(f_{2}\) is defined by (3.27).
 (iii)
If \(F(a)< x_{0}< F(b)\) and n is odd, then F has a nonmonotonic nth iterative root of the form (3.3), where \(f_{1}\) is defined by (3.26), and \(f_{2}\) is defined by (3.27).
Proof
The proof of necessity is similar to that of Theorem 3.
Sufficiency. Case (i). Using similar arguments as in the proof of case (i) of Theorem 3, we say that \(F_{I_{}}\) has a strictly decreasing nth iterative root \(f_{1}\) of the form (3.26). Moreover, (3.7) comes from (3.6), and (3.9) comes from (3.8) and \(c< d\). Thus, (3.26), (3.7), and (3.9) prove that F has a nonmonotonic nth iterative root f of the form (3.1).
The proof of case (ii) is directly obtained from case (i) by the translation (3.11).
4 Examples
In this section, we give examples. For convenience, to demonstrate our main results and avoid complicated computations, we only present a third iterative root of the given usc multifunctions.
Example 1
Example 2
Example 3
Example 4
Declarations
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions. This work is supported by Shandong Provincial Natural Science Foundation (ZR2017MA019, ZR2014AL003) and Scientific Research Fund of Binzhou University (BZXYL1703).
Authors’ contributions
The authors completed the paper together. They also read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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