Iterative roots of upper semicontinuous multifunctions

The square iterative roots for strictly monotonic and upper semicontinuous functions with one set-valued point were fully described in (Li et al. in Publ. Math. (Debr.) 75:203-220, 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.


Introduction
Given a mapping F : X → X and an integer n ≥ , the iterative root problem is to find all self-mappings f : X → X such that their nth iterates satisfy the functional equation Babbage [] investigated (.) for an identity mapping F as far back as the s. After that, (.) 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 [-], the monographs [, ], and the book []. For all we know, strictly increasing roots of strictly increasing and continuous functions were discussed by Bödewadt [], and their strictly decreasing roots were presented by Haǐdukov []. In , Kuczma [] 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{ x a , -x -a } on the compact interval [, ] for arbitrary a ∈ (, ). In , Zhang and Yang [] 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 []). Their method is based on the 'characteristic interval' , which was developed in [, ]. In recent years, many important results on iterative roots of PM functions were presented in [-]. It is worth mentioning that those results are related to single-valued functions. In [, ] and [], it is illustrated that the set of continuous functions having a root is a non-Borel subset of C([, ], R) and is small in C( [, ], [, ]). That is to say, in the general case, no such roots exist, and the theory becomes extremely complicated if F is not bijective []. Therefore, it is a natural idea to extend the notion of iterative root.
In his survey paper [], 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 so-called phantom iterative root of continuous functions in []. Powierża and Jarczyk [-] gave set-valued functions as roots of single-valued functions. Maybe the best method to generalize iterative roots is replacing single-valued functions by set-valued functions for both F and f in (.) (see []). It seems that, up to now, there are only several results on set-valued iterative roots of multifunctions, even with a unique set-valued point. In , Jarczyk and Zhang [] considered the nonexistence of square iterative roots of multifunctions with exactly one set-value point and presented two sufficient conditions for the purely set-theoretical situation. Later, Li, Jarczyk, Jarczyk, and Zhang [] gave new nonexistence results for purely set-theoretical case and fully described the square roots of strictly monotonic, upper semicontinuous (abbreviated as usc) multifunctions.
As a continuation of [], in this paper, we study all strictly monotonic usc multifunctions having one set-valued 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 []. In Section , we recall the basic definitions and present Lemmas -. In Section , we give equivalent conditions for the existence of nth iterative roots and their expressions. Finally, in Section , we apply examples to illustrate our results.

Preliminaries
Given topological spaces X and Y , a multifunction f : X →  Y is called upper semicontin- then it is called upper semicontinuous on a set B ⊂ X. Let F : X → X and G : Y → Y be continuous functions. We say that F is topologically conjugate to G if there exists a home- We say that f is one-to-one on I if f (x  ) = f (x  ) for all different Definition  ([]) Let γ : I → I be a strictly increasing continuous function. Given a fixed point ξ ∈ I of γ , we put Multifunctions that are either strictly increasing or strictly decreasing are called strictly monotonic.
It is clear that every strictly monotonic multifunction is one-to-one. Conversely, a oneto-one multifunction is not necessarily strictly monotonic. In this paper, we investigate the nth iterative roots of usc multifunctions F ∈ S x  (I, I) of the form ) ⊂ I, and F| Iand F| I + are strictly monotonic and satisfy The next lemmas describe the fundamental properties of the nth iterative roots of (.).

Lemma  If F ∈ S x  (I, I) is one-to-one, then every nth iterative root f of F belongs to S x  (I, I)
and is also one-to-one. Proof We first prove that f is one-to-one. Suppose on the contrary that there exist two Since F is one-to-one, only the case u = v is possible, contrary to the assumption. Now we show that f ∈ S x  (I, I). For any y ∈ I\{x  }, suppose on the contrary that #f (y) ≥ . Then we have #F(y) = #f n (y) ≥ , which contradicts F ∈ S x  (I, I). Thus, f (y) is a singleton for all y ∈ I\{x  }. We claim that x  is a unique set-valued point of f . Otherwise, let f (x  ) = {p  }. Then two cases are possible: either p  = x  or p  = x  . From the former we have Without loss of generality, assume that f (p  ) = {p  }. Then there again exist two cases: either p  = x  or p  = x  . If p  = x  , a contradiction comes from Repeating this progress, we inductively obtain #f (p n- ) =  for p n- = x  , and, consequently, we have Thus, we prove f ∈ S x  (I, I). This completes the proof.
implying y  = x  , a contradiction. This completes the proof.
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.

Main results
In this section, we give several sufficient and necessary conditions and expressions of nth iterative roots of (.). Theorem  and Theorem  characterize the strictly monotonic usc multifunctions, and nonmonotonic cases are investigated in Theorem  and Theorem .
For convenience, let Moreover, if n is even and F| Ihas a regular fixed point, then F also has a strictly decreasing nth iterative root of the form (.), in which f  is a strictly decreasing function satisfying f  n = F| I -.
(ii) If F(a) > x  , then F has a strictly increasing nth iterative root where f  : I + → I + is a strictly increasing function satisfying f  n = F| I + . Moreover, if n is even and F| I + has a regular fixed point, then F also has a strictly decreasing nth iterative root of the form (.), in which f  is a strictly decreasing function satisfying f  n = F| I + .
(iii) If c ≤ x  ≤ d, then F has a strictly increasing nth iterative root where f  and f  are defined as in (.) and (.), respectively. Moreover, if n is even and F| I + is topologically conjugate to F| Iby a strictly decreasing function f  : I -→ I + , that is, then F also has a strictly decreasing nth iterative root in which g is a strictly increasing n  th iterative root of F.
Proof Necessity. If F has nth iterative roots, then F does not take the value {x  } by Lemma . Therefore, only the following cases are possible: For every x ∈ I + ∪ {x  }, using that F(x) ⊂ Iand f  is strictly increasing, from Furthermore, by the upper semicontinuity property at the set-value point x  we have Case (iii). The condition c ≤ x  ≤ d implies that F(I -) ⊂ Iand F(I + ) ⊂ I + . Since F| Iand F| I + are strictly increasing, their strictly increasing nth iterative roots f  and f  exist and are defined as in (.) and (.), respectively. Observing that f is one-to-one by Lemma  and f  and f  are strictly increasing, we have which yields (.). Consequently, f  and f  together with (.) and (.) give a strictly increasing nth iterative root (.) of F. If n is even and F| I + is topologically conjugate to F| Iby a strictly decreasing function f  : I -→ I + , then we will prove that F also has a strictly decreasing nth iterative root f of F. In fact, this problem reduces itself to a solution of the system As in the previous argument, F possesses a strictly increasing n  th iterative root g, and we confine ourselves to the first equation f  = g to find a strictly decreasing solution f of F. The given condition F| I + = f  • F| I -• f  - implies that the strictly increasing functions g| I + and g| Isatisfy the conjugacy equation and the equality g n  = F shows that The second equality in (.) yields that implying that f  : I + → Ialso is strictly decreasing. Moreover, (.) Since F and g are strictly increasing, f  : I -→ I + and f  : I + → Iare strictly decreasing, and f is one-to-one, it follows from (.) that (.) holds. Thus, the given f  , together with (.), (.), and (.), leads to (.). Refer to Figure . This completes the proof. (ii) If F(a) > F(b) > d ≥ x  , then F has a nonmonotonic nth iterative root defined by (.). 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 (.), in which f  is a strictly decreasing function satisfying f  n = F| I + .

then F has a nonmonotonic nth iterative root defined by (.). Moreover, if n is even and F| Ihas a regular fixed point, then F also has a nonmonotonic
(iii) If F(b) < x  < F(a) and n is odd, then F has a nonmonotonic nth iterative root where g is a strictly increasing nth iterative root of F  . If n is even, then F has no iterative roots.
Proof The necessity directly comes from Lemma . In what follows, our attention is paid to the sufficiency. Sufficiency. For case (i), we first construct a strictly increasing function (.) and a multifunction (.) as in case (i) of Theorem . Since f  is strictly increasing, from (.) and c > d we have (.). Thus, (.), (.), and (.) yield a nonmonotonic nth iterative root (.) of F.
Assuming that n is even and F| Ihas a regular fixed point, we can construct a strictly decreasing function f  and a multifunction (.) as in case (i) of Theorem . Since f  is strictly decreasing, (.) and c > d lead to (.). Therefore, f  together with (.) and (.) gives a nonmonotonic nth iterative root (.) of F.
The proof of case (ii) is obtained from case (i) by the translation (.). Case (iii). The assumption F(b) < x  < F(a) shows that If F has an nth iterative root f , then, whether n is odd or even, we assert that (.) By Theorem (iii), (.) shows that there exists a strictly increasing nth iterative root g of F  satisfying Substituting (.) into (.), we obtain which implies that f  and f  are strictly increasing. Moreover, from (.) we have If n is even, then suppose on the contrary that F has an nth iterative root f . Since g is a strictly increasing nth iterative root of F  , we have (i) If F(a) < x  and n is odd, then F has a strictly decreasing nth iterative root of the form in which x  is the unique fixed point of F| I -, and χ is a strictly increasing nth iterative root of (F| I -)  . (ii) If F(b) > x  and n is odd, then F has a strictly decreasing nth iterative root of the form in which x  is the unique fixed point of F| I + , and ψ is a strictly increasing nth iterative root of (F| I + )  .
(iii) If d ≤ x  ≤ c and n is odd, then F has a strictly decreasing nth iterative root where φ is a strictly increasing nth iterative root of F  .
Proof Necessity. Suppose on the contrary that n is even and F has an nth iterative root f . Then only the following cases are possible: which also contradicts the assumption on F. If f (I -) ⊂ I + and f (I + ) ⊂ I -, then as n is even, we again get (.) and a contradiction. Thus, we have proved that n is odd. The remainder directly comes from Lemma . Sufficiency. Case (i). Note that Substituting (.) into (.), we obtain which implies both f  and f  are strictly decreasing. Moreover, since φ is strictly increasing, from (.) we have  Proof The proof of necessity is similar to that of Theorem . Sufficiency. Case (i). Using similar arguments as in the proof of case (i) of Theorem , we say that F| Ihas a strictly decreasing nth iterative root f  of the form (.). Moreover, (.) comes from (.), and (.) comes from (.) and c < d. Thus, (.), (.), and (.) prove that F has a nonmonotonic nth iterative root f of the form (.).
The proof of case (ii) is directly obtained from case (i) by the translation (.). Case (iii). The assumption on F implies that F(I -) ⊂ Iand F(I + ) ⊂ I + . Since F| Iand F| I + are strictly decreasing, we can obtain their strictly decreasing nth iterative roots f  and f  defined by (.) and (.), respectively. Moreover, from (.) and c < d we obtain (.).