The solvability of f(p(x))=q(f(x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(p(x)) = q(f(x))$$\end{document} for given strictly monotonous continuous real functions p, q

We investigate the functional equation f(p(x))=q(f(x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(p(x)) = q(f(x))$$\end{document} where p and q are given real functions. In the paper “On solvability off(p(x))=q(f(x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(p(x)) = q(f(x))$$\end{document}for given real functionsp, q, Aequat. Math. 90 (2016), 471 - 494”, we solved the problem of the solvability of f(p(x))=q(f(x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(p(x)) = q(f(x))$$\end{document} under the assumption that p, q are strictly increasing continuous real functions. Now, we extend the solutions of this problem for any strictly monotonous continuous real functions p, q. Thereby, we use the methods of the just mentioned paper. Further, we present computations of the so called characteristics of the given functions p, q using the results of this paper and, finally, present a quite short algorithm with input p, q and output ’solvable/not solvable’.

We denote the set of all real numbers by R. The topic of our paper is an investigation of the functional equation where p and q are two given strictly monotonous continuous real functions on R defined totally or partially and where f is the function to be found -as a solution of (1). These problems were investigated by J Chvalina and others, s. [1] and [2]. They were motivated by papers of F. Neuman on transformations of differential equations ( [12,13]). Their works discussed certain totally defined strictly increasing continuous real functions.
Recently, J. Matkowski and P. Wójcik in paper [11] investigated the solvability of Eq. (1) for given p, q with special properties from the point of view of the so called sandwich (separation) methods. The authors came to Eq. (1) as a generalisation of the generalised property of periodicity of functions.
For strictly increasing continuous real functions p, q, a characterisation of the existence of a solution of Eq. (1) can be found in [8]. In the present paper, we 'round up' this characterisation for any strictly monotonous continuous functions p, q. The case of strictly increasing continuous functions is the basic one with more individual possibilities and more variety (e.g., increasing functions can have more fixed points, while decreasing ones have at most one). The remaining case of strictly decreasing functions can be approached by means of the results for strictly increasing functions. The basic fact in these considerations, is that the second iteration of a strictly decreasing function is a strictly increasing one.
Strictly monotonous real functions of one variable are the only ones which are invertible. For strictly monotonous functions p, q which are continuous at once, the existence of a solution of Eq. (1) can be characterised relatively simply.
Equation (1) represents many important functional equations. However, this equation plays a considerable role in its general form. If two continuous real functions p, q are given, then their topological semiconjugation means the existence of a surjective continuous function f such that Eq. (1) is fulfilled. The methods of the present paper can be a step towards solving this well-known problem.
Our paper has two parts. The first part (a shorter one) will present a characterisation of the solvability of the equation f (p(x)) = q(f (x)) (i.e. (1)) for any strictly monotonous continuous functions p, q (Theorem 5). This characterisation is similar to the main assertion in paper [8] for the solvability of (1) for any strictly increasing continuous functions p, q ( [8], Theorem 18).
The second part presents computations of the so called characteristics of strictly monotonous continuous functions needed for the above characterisation. To conclude, a short algorithm for the solvability of (1) will be formulated with input p, q and output '(1) is solvable' or '(1) is not solvable'.
The present paper is built on paper [8] using its methods and results. Nevertheless, it is not necessary to read this paper because all we need from this paper in the following, all the concepts and assertions, are presented here in a short form again.
As usual, the 'connected blocks' of a mono-unary algebra will be called components of this algebra. In iteration theory, components correspond to the concepts of orbits in the sense of Kuratowski (s. [10]).
And finally now, the symbol denotes the existence of an isomorphism (bijective homomorphism) between two mono-unary algebras.
By C n , we will denote a cycle with n elements, i.e. a mono-unary algebra ({a 1 , a 2 , ..., a n }, o) with o(a i ) = a i+1 for i = 1, 2, ..., n − 1 and o(a n ) = a 1 . In iteration theory, cycles are known under the name periods, a singleton cycle as a fixed point.
Furthermore, we will use mono-unary algebras (Z, ν), (N, ν 0 ), (Z − , ν − ) and (W (n), ν 1 ) for some n ∈ N, n > 0 which are defined in the following way. Unary operation ν on Z is defined by ν(x) = x + 1 for any x ∈ Z and further, It is easy to see that all these algebras are simple examples of connected mono-unary algebras.
To simplify matters, we will denote all operations ν, ν 0 , ν − and ν 1 by the same symbol ν. It is always enough to consider the different domains of ν for the set Z and its subsets N, Z − and W (n).
First, we formulate simple assertions which are immediate consequences of the above definitions for mono-unary algebras. In doing so, here and throughout the paper, we use the following simple elementary fact. If p : R → R is a given real function that is totally or partially defined on R, then (R, p) is a mono-unary algebra.
Notice as well that the solvability of Eq. (1) includes implicitly the following condition. For any x ∈ R, if p(x) is defined, then q(f (x)) is defined too. Now, we formulate the following simple assertions concerning mono-unary algebras and their homomorphisms. Assertions (a) and (b) are taken over from [8], Lemma 1 (a), (b) and assertion (c) is a direct consequence of the definition of iterations of an operation. Lemma 1. The following assertions are valid.
(a) Let p, q : R → R be given real functions which are completely or partially defined on R. Then the equation The solvability of f (p(x)) = q(f (x)) for given strictly... 905 In assertion (b), we see that noncyclic components with one element play no role for homomorphisms. This is so because they are outside the domains of operations. And further (c) is a simple direct consequence of the definition of iterations of o and is useful in our considerations. Now, we will consider a special type of mono-unary algebras. They are algebras whose unary operation is injective.
If In [8], Lemma 2 (b), we find the following assertion that is basic here as well. For the algebraic methods we use, an important concept is the concept of categories of algebras. Recall that by category, we mean a class of objects together with a class of morphisms between the pairs of objects with the binary operation of their composition (which has the property of associativity) and the existence of the so called identity morphisms for any object.
Here, we consider the category of all mono-unary algebras where the morphisms are their homomorphisms. The articles [4] and [5] investigate this category and its subcategory of all connected mono-unary algebras. In our considerations, we make use of the results of article [4] and implement them directly.
But we will not need the concepts of categories to speak about the classes of mono-unary algebras only. Moreover, we can confine our considerations on connected mono-unary algebras for which the operation is injective.
We will denote the class of all connected mono-unary algebras whose operations are injective by L. Then, by Lemma 2, objects of L are mono-unary algebras which are isomorphic to one of the five special algebras established in this Lemma.  Indeed, it would be very easy to define the isomorphisms wanted. But, at the same time, the assertions are consequences of Now, we formulate some assertions which we will need later in particular cases for real functions.
For their presentation, the following denotations are useful. Let (A, o) ∈ L be such that it is not a cycle and let x ∈ dom(o) be arbitrary. By α x (or β x ), we denote the greatest ordinal number such that Then we see that, in the case On the other hand, let (A, o) be a cycle and let x ∈ A be arbitrary. Then is of finite type as well. Therefore similarly, we can find the greatest element a of Vol. 96 (2022) The solvability of f (p(x)) = q(f (x)) for given strictly... 907 . But so we obtain o −1 (a) = ∅ which means that a is the least element of (A, o) (see above). Therefore altogether, These last assertions will be used in the second part of the article. The second section deals with a practical use of Theorem 5 (the last one of the first part) which is an answer to our problem of the solvability of Eq. (1). The second part has the name 'Computations of characteristics and an algorithm'.
From now on, we want to focus our considerations on special mono-unary algebras, namely algebras with the carrier R and strictly monotonous real functions as operations. Thereby, we will confine our considerations to strictly monotonous functions which are continuous.
Let p : R → R be a (strictly) monotonous function defined on an interval (a, b).
Then we define The values p(a) and p(b) exist because of the monotony of p on (a, b). So the domain of p can be extended onto the interval on (a, b), then, in this way, it is right continuous at a and left continuous at b.
Let p : R → R be a strictly monotonous function. Let A = (A, p|A) be a component of the mono-unary algebra (R, p). Remember the assertion of Lemma 2 for a mono-unary algebra with an injective operation (and recall that |A| denotes the cardinal number of the set A). Then we put 908 O. Kopeček AEM The symbol ∞ 2 is used in the theory of papers [3][4][5] for certain characteristics of cyclic elements. And the symbol ∞ 1 is used in those papers for certain characteristics of elements with infinite sequences of predecessors.
Further, ω is an abbreviation for the ordinal type ω 0 (of N) and ω − is the symbol of the type of the ordered set Z − .
χ is a mapping χ : However, for a monounary algebra (R, p), we need a part of mapping χ only. If K p is the system of all components of (R, p), then it is the part χ|K p . But we will not mention this fact because the association of the considered connected mono-unary algebras to (R, p) as its components is clear. Now, let K p be the system of all components of (R, p). Then we define and call char(R, p) the characteristic of p. We will use a shorter notation char(p) for char(R, p).
To be brief in our next denotations, mainly in some formulas later, we put For constructions of the solutions of Eq. (1) if there are any, it is -concerning cycles -necessary to have a 'finer' characterisation for them as it is the value ∞ 2 only. But for the question of solvability of (1), the value ∞ 2 alone is enough. This is because of the simplest consequence of the famous Theorem of A. N. Sharkovskyi ([17]) (actually Ukrainian O. M. Sharkovskyi). Namely, if a continuous function has cycles, then it has singleton cycles -called fixed points as well. (We can say that the singleton cycles 'represent' the existence of cycles of a function.) Now, we consider the set Γ of the characteristic values of components of continuous strictly monotonous functions, i.e. the set Γ = N 2+ ∪{ω, ω − , ∞ 1 , ∞ 2 }. In connection with our questions, there is a hierarchy of these values.
We define a relation ≤ on the set Γ in the following way. The relation < is defined so that n < n + 1 for any n ∈ N 2+ , further, n < ω and n < ω − for any n ∈ N 2+ and finally, Now, let the relation ≤ be the reflexive and transitive closure of relation <. Then ≤ is a partial ordering on Γ, i.e. (Γ, ≤) is a (partially) ordered set.
The relation < is described by Fig. 1 where the symbols ∞ 1 , ∞ 2 are abbreviated by 1 and 2, respectively.
Vol. 96 (2022) The solvability of f (p(x)) = q(f (x)) for given strictly... 909 Figure 1. The structure of relation < on (Γ, ≤) The Theorem of this section (Theorem 5) will be obtained as a consequence of the results of article [4]. Actually, Theorem 5 is the reduction of Theorem 1.28 in [4] for our particular problem for strictly monotonous continuous functions. We do it in the same way as for the problem of solvability of Eq. (1) for strictly increasing continuous functions in paper [8].
Article [4] investigates a general form of our problem for connected monounary algebras. There, we could begin with definition 1.19. It was written in concepts of categories (classes) but, in our reduced situation for monounary algebras of real functions and in our denotation, it can be rewritten as follows. Let (A, o) be a connected mono-unary algebra. Then, we define [4] for components of mono-unary algebras generally. And we see that both have the same structure. The new function χ is a reduction of the old one. So we could easily prove that the ordered set (Γ, ≤) of characteristics of components of strictly monotonous continuous functions (s. eg. Fig. 1) can be isomorphically embedded into the ordered set of characteristics of all connected mono-unary algebras with injective operations. Actually, the isomorphism φ is such that φ|N 2+ = id N2+ , φ(ω) = ω 0 , φ(ω − ) = d, φ(∞ 1 ) =d and φ(∞ 2 ) = 1 where 1 denotes the cardinal number of singleton cycles (fixed points).
On the other hand, consider Fig. 1 above for a representation of the ordered set (Γ, ≤). By means of ordering ≤ on set Γ, we now define a relation ρ on set 2 Γ . If θ, θ ∈ 2 Γ are arbitrary, then θ ρ θ holds if and only if, for any α ∈ θ, there is α ∈ θ with α ≤ α .
So we obtain the set (2 Γ , ρ) with a quasi-ordering ρ on 2 Γ . And now, using the existence of an isomorphic embedding for any component of the mono-unary algebra (R, p), we can obtain almost immediately the following assertion as a consequence of Theorem 1.28 in [4].

Theorem 5. Let p, q : R → R be strictly monotonous continuous functions. Then the equation f (p(x)) = q(f (x)) has a solution if and only if char(p) ρ char(q) holds.
This Theorem is an extension of Theorem 1.18 in [8] which is formulated for strictly increasing continuous functions. It presents an answer to the problem formulated in the title of our paper.

Computations of characteristics and an algorithm
Theorem 5 would not have much impact if we didn't have the possibility to find out the values of the characteristics of the given strictly monotonous continuous functions in a not very complicated way. Therefore, we have to dedicate ourselves to these computations now.
For our computation, we need the following central definition. Let p be a strictly monotonous continuous function defined on (a, b). Putting The reason for making a distinction, in the definition of I p , between an increasing and a decreasing function is that, for a decreasing function p, the neighborhoods of a and b play no direct role in the components of the monounary algebra (R, p). For an increasing function p, it can be the case. Vol. 96 (2022) The solvability of f (p(x)) = q(f (x)) for given strictly... 911 To begin with, the set of all possible characteristics for strictly monotonous continuous functions as a subset of 2 Γ can be described in the following way.
In [8], we find the assertions for strictly increasing continuous functions (Corollaries 12 and 13) as follows. (Recall the denotation N 2+ = N \ {0, 1}.) Lemma 6. Let p be a strictly increasing continuous function. Then the following assertions hold. (a) In this connection, we will have a closer look at strictly decreasing continuous functions. We can formulate the following assertion.  For this, we recall some well-known properties of strictly monotonous functions.
Lemma 8. Let f, g : R → R be real functions. Then the following assertions hold.
(a) If f (x) and g(x) are both strictly increasing or if they are both strictly decreasing, then f (g(x)) is strictly increasing.
(b) If f (x)is strictly monotonous, then f 2 (x) is strictly increasing.
Kopeček AEM Now, we can use a consequence of Lemma 4 for mono-unary algebras whose operations are injective. It could be formulated more generally but for our considerations, we need it only for the second iteration of the given functions.
If we consider a strictly monotonous continuous function p : R → R (defined on an interval (a, b)), and consider the comparison of values χ( ) of the components of (R, p) and values χ( ) of the components of (R, p 2 ), then we obtain the following assertions. Lemma 9. Let p : R → R be a strictly monotonous continuous function. Let  (A, p) be a component of (R, p). Further, let p 2 on A exist and let (A , p 2 ) be a component of (R, p 2 ) such that A ⊆ A holds. Then the following is satisfied.
Thus by Lemma 6 and Corollary 11, we obtain the following.
For the purpose of simplicity, we use the same denotation ρ for its reduction ρ Θ and come to the quasi-ordered set (Θ, ρ).
Further especially, ∞ 2 ∈ θ for any θ ∈ Θ * . Thus by the definition of ρ, θ ρ θ is satisfied for any θ, θ ∈ Θ * and so the relation ρ is complete on Θ * . Moreover, we recognize a congruence on the quasi-ordered set (Θ, ρ) with congruence classes 2 The relation ρ is represented in Fig. 2. There we use the possibility of a graphical representation by means of the above congruence classes. Also, the reflexive arrows are omitted. (The symbol ∞ 1 is abbreviated by 1 again). And now, we can come to computations of characteristics of our considered strictly monotonous continuous functions. The more difficult part of this question was treated in paper [8]. Moreover, we can use some results from [8] now. There, we find Corollary 16 where this problem is solved for strictly increasing continuous function.
In this connection, we consider the set I of all intervals on R with the partial order ⊆ on I. We use ⊂ or ⊃ for ⊆ or ⊇ respectively and =. Further, || denotes ⊆ and ⊇, i.e. it means the incomparability of two intervals with respect to the order ⊆. Then the assertion ( [8], 16) is as follows.
For a more practical use of this assertion, we will add a new natural property of I p for a strictly monotonous continuous function p. If p is such a function, then we assume that I p = (a, b) implies |I p | ≤ ℵ 0 and say that I p is countable. Then we can formulate the following consequence of Theorem 13.  ∩ (a, b). Then Vol. 96 (2022) The solvability of f (p(x)) = q(f (x)) for given strictly... 915 a, b) and |I () a, b) and |I () ) and |I () Proof. The assertion is a direct consequence of Theorem 13.
The conditions for the computations of the value of char(p) in Theorem 13 can be separated into two parts. The first ones give the relationship between the intervals (a, b) and (p(a), p(b)). The others are combinations of conditions for I () p and the conditions for I p to belong to the set I of all intervals in R. The conditions a, b ∈ I p in the first three cases of the computation of char(p) mean p(a) = a and p(b) = b or also, (p(a), On the other hand, the conditions 'otherwise' (for the relationship among The other conditions, for I () p and I p , are simple in our case. Namely, the condition that I p is countable implies that an interval in R can be a singleton only. Therefore, the condition I The following rules could serve as a mnemonic for the numerous possible values of char(p) of a given p in the Theorem.
If the function p, its dom(p) contracts, then ω ∈ char(p) (in any x ∈ dom(p) \ img(p) begins a component),

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O. Kopeček AEM expands, then ω − ∈ char(p) (in any x ∈ img(p)\dom(p) ends a component), moves sideways (neither ⊂ nor ⊃), then both ω, ω − ∈ char(p). And, |I p | > 1 adds ∞ 1 ∈ char(p). Now, let us look at the case of strictly decreasing continuous functions. For strictly decreasing continuous functions, it is possible to use Corollary 10. Therefore, the second iteration of a strictly decreasing continuous function is important for our computations.
Let p be a strictly decreasing continuous function defined on (a, b). Then the following can be proven.
Example 15. In [8], we can find several examples of computations of char(p) if p is a strictly increasing continuous function (Example 17). There, we have an example for any possibility of the characteristic of a strictly increasing continuous function. The number of such possibilities is 11 (Theorem 14), actually ten, because the first one is trivial. Now, we can continue this 'tradition' for strictly decreasing continuous function. If we look at the remark just mentioned above, then we see that we have less work this time. The number of such possibilities is 6, actually five, because the value {∞ 2 } as the characteristic of a strictly decreasing continuous function, is trivial again.
(a) Consider the (partially defined) strictly decreasing continuous function p(x) = − √ x + 1 and let us compute char(p). Firstly, by Corollary 11 (b), (c), we look at whether I p = ∅ or not. So we set p(x) = x, i.e. − √ x + 1 = x and find that the equation is solvable. Hence, I p = ∅ and so ∞ 2 ∈ char(p) by 11 (c).
Further, by Corollary 10, we can compute char(p 2 ). Then, since p 2 is a strictly increasing continuous function we will use Theorem 14. Denote (a, b) = dom(p).
Therefore, a 1 = p −1 (min{b, p(a)}) and b 1 = p −1 (max{a, p(b)}). So, we have carried out this computation for the next examples as well.