### Where have all the functional equations gone (part II)

#### by Orr Shalit

I’ll start off exactly where I stopped in the previous post: I will tell you my solution to the problem my PDEs lecturer (and later master’s thesis advisor) Paneah gave us:

Problem:Find allcontinuously differentiablesolutions to the following functional equation:(FE)

Before writing a solution, let me say that I think it is a fun exercise for undergraduate students, and only calculus is required for solving it, so if you want to try it now is your chance.

#### 1. Solution of Problem

**Solution:** Well, the assumption of continuously differentiability (which we were left to impose ourselves) *begs* us to differentiate the equation to get

(*)

where , so is continuous. This can be considered as a functional equation in its on right, and our goal is now to find all **continuous** solutions to (*).

**Claim 1:** If is a continuous solution of (*) then

We will prove the claim below. Assuming the claim for the moment, deduce that for some , thus . Plugging in the original functional equation (FE) we find that . Thus, all continuously differentiable solutions of (FE) are of the form , for some . That any function of this form satisfies (FE) is obvious. Thus it remains to prove Claim 1. At this point it is convenient to introduce some terminology.

#### 2. Brief on dynamical systems

A **dynamical system** is a topological space and a family of continuous maps . The main problem in topological dynamics is: *given a point , how does it move around under the influence of the maps ? *

**Definition 1:** Given a dynamical system and a point , **the orbit of ** is the set

Thus, the orbit of is the set of all points in which can be reached from by iterating the maps .

**Definition 2:** A dynamical system is said to be **minimal** if for all , the orbit of is dense in , i.e.,

Minimal dynamical systems turn out to be very useful for proving uniqueness of solutions to certain functional equations, as we shall see.

**Example:** Consider the dynamical system , where

and .

One can use induction on to show that given any points , there are such that

.

Thus this dynamical system is minimal.

#### 3. Proof of Claim 1

**Proof of Claim 1: **It will be useful to consider the dynamical system of the above Example. Let be a point where attains its maximum. Then the functional equation (*) can hold at only if attains its maximum at and at — this readily follows from the fact that the functional equation merely says that is the mean of and . Repeating the argument, we find that the maximum of is attained at the points and . By induction, the maximum of is attained at all points in the orbit of . But the Example above, the orbit of is dense in , thus attains this maximum on a dense set. Since is continuous, it must therefore be a constant. That concludes the proof of the claim, and therefore also concludes the solution of the problem.

#### 4. Alternative solution to Problem

Here is a somewhat different solution to the problem, which (I later learned) is a simplification of Paneah’s approach to uniqueness in -configurations.

As before, let be a point where attains its maximum. Then must attain its maximum also at . It follows inductively that attains its maximum at the sequence of points , where denotes composed with itself times. But clearly , thus (as is continuous) the maximum of is achieved at the point . But the same argument works for the minimum of , so the minimum of must also be achieved at . This is possible only of is constant.

#### 5. Complications and guided dynamical systems

Before moving forward, let me briefly indicate what is the trickier problem that Paneah treated in his papers on this. Suppose that the functional equation (*) is replaced with

(**) ,

where are two non-negative functions satisfying . Then both of the above approaches fail to prove that the only solutions to (**) are constants, because if, say (where ) then we cannot conclude that the maximum of is attained at , in fact it might not. In fact, there could be non-constant solutions to (**). To decide when this happens one has to consider **guided dynamical systems **(as I call them), which were introduced by Paneah for this purpose. In a nutshell, a guided dynamical system is a dynamical system together with closed sets such that “one may use only on points “. In other words, a guided dynamical system whose evolution has some obstructions: in a sense the evolution is not generated by a semigroup, but rather by something more like a semigroupoid. As I’ve said, Paneah used these systems to study functional equations (and he was able to apply these systems to problems in integral geometry and PDEs), and I also developed the theory to a small extent in my master’s thesis. As far as I know this interesting notion has not been studied by others (though I have seen once a work in a similar spirit, see this paper). I am not going to talk about generalizations in this direction any further. I will go on, but in another direction.

#### 6. Another problem

Let me stop with mathematics and return to my story telling. After I showed Paneah my solution to the Problem above I was very pleased. To recap, the answer to the problem is

All continuously differentiable solutions to the equation

(FE)

are of the form .

Then Paneah challenged me further, and asked: *well, what about the continuous solutions?*

Well, what about them? Could there be continuous solutions to the functional equation (FE) which are not of the form ?

[…] last post ended with the following […]

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