Dear Sanjo,

There is no problem with giving an example of a necessary condition for the function possesing a fixed point:  If  f(x*) = x*   then  x*  belongs to the set of values of function  f .

The problem is, what type of claims will you accept as interesting. This it is not a mathematical question;  your question's domain is meta-mathematics.

Best regards, Joachim

Hi Joachim,

Thanks for pointing out  a " meta-mathematics direction" of your thoughts!  Of course, one has to consider the " fixed point problem" (as well as any applied  mathematical problem) in some non-meta-mathematical context.  This is why we  use  assumptions and mathematical tools, after all.

Regarding my fixed point question, I think that it is clear (to me) that  If somebody gives me an n-tuple X and a function f(x) I would like to know  whether f(X) = X. The question of finding conditions which constructively  yield such X comes later. (This is almost like first  asking whether there is a  fish in a pond before thinking of  catching it.) You also assume that f possesses a fixed point x*? Why ?  In my question I do not assume this! Hence I partly agree with your comment but I believe that my question is well posed. I simply  want to characterize (identify) those X for which (X) = X.

I would appreciate your glancing through my attached  paper in CRORR and receiving your comments on it, if possible. Thanks.

Best regards,

Sanjo

Hi Joachim,  I have added the word "meaningful" to  the title of the question. This  means that we expect to know what x and f represent, but we do not necessarily know whether f(x) = x for some x. Best, Sanjo

Hi Sanjo.  Your aswers say, that you have different meaning of the necessary condition. The proper way is:   If  A  implies  B, then we call  A  a sufficient condition for  B    and we call  B  be necessary condition for  A. Therefore, looking for the necessary conditions for existence of a fixed point means that you are asking for necessary consequences of the assumption, that   f   possesses a fixed point. Best, Joachim.

ps. the following is an excerpt from

https://en.wikipedia.org/wiki/Necessity_and_sufficiency

In the conditional statement, "if S, then N", the expression represented by S is called the antecedent and the expression represented by N is called the consequent. This conditional statement may be written in many equivalent ways, for instance, "N if S", "S implies N", "S only if N", "N is implied by S", S ⇒ N, or "N whenever S".[4]

In the above situation, we also say that N is a necessary condition for S. In common language this is saying that if the conditional statement is a true statement, then the consequent N must be true if S may at all be true (see "truth table" immediately below). Phrased differently, the antecedent S can not be true without N being true. For example, in order for someone to be called Socrates, it is necessary for that someone to be Named.

We also say that S is a sufficient condition for N. Consider the truth table again. If the conditional statement is true, then if S is true, N must be true. In common terms, "S" guarantees N". Continuing the example, knowing that someone is called Socrates is sufficient to know that that someone has a Name.

If I understand the question  in general,  then the following result also may be relevant for mathematical point of view..

(Note Sanjo  paper  in CRORR is more concrete).

Recall Czeslaw Bessaga from 1959:

(i)Let $f : X \rightarrow X$ be a map of an abstract set such that each

iterate $f^n$ has a unique fixed point x_0. Let $q \in (0, 1)$, then

(ii) there

exists a complete metric on $X$

such that $f$ is contractive, and $q$ is the contraction constant.

It seems that  (i)  is sufficient and necessary  condition for  (A):   $f : X \rightarrow X$ is   a map of an abstract set X  into itself which has fixed point.

Example. Let f(z)= e^{i \alpha}$, where \alpha is an irational multiple of \pi and X =U the unit disk.

Bessaga (i) implies (ii). But now (ii) implies

(iii) each

iterate $f^n(x)$ converges to x_0.

As far as I can see my example satisfies (i);    does it  satisfy

(iii)?

Dear Joachim, Yes, I do claim that there may be different (but possibly equivalent) necessary conditions for the same fixed point.

Example: (for details, notation and other examples see my attached  CRORR paper): Consider     f(x) = x^3 on I = [-1,1]. Point x* = 0 is a fixed point because f(0) = 0. What does this imply ? Take, e.g., W(x) = integral  .from x* to x of (f(t) - t) on x* to x. Then     W(x) = x^4 / 4  -  x^2 / 2 .  A "primal" necessary condition for fixed point proved in the paper says that abs(W(x)) < = lambda times   (x - x*)^2 on I for some lambda. After substitution the latter is abs(x^2 / 2  - 1) less than lambda on I. Hence we have a necessary condition for fixed point x*.= 0 ! There is also a "dual" necessary condition which says that x^2  / 2  - 1 is  uniformly bounded on I. Do you accept these two statements as legitimate  necessary conditions for fixed point x*  = 0 ?  As shown in the paper, these conditions do not necessarily hold at points  which may not be fixed, e.g. x' = 2.  Of course there are many conditions known as "fixed point theorems" in the literature which are sufficient. 

Best, Sanjo

@ Sanjo.

Your examples show that you are interested on necessary conditions for existence of fixed points within particular families of functions. Tha'ts OK.  But then, without  describing which classes of functions you are interested in, the problem becomes to wide, and nothing more can be was said then is was by  Peter Breurer above. Some particular examples:

1. If you confine the analysis to functions which have absolute value  dominated by A x2, then they need to be differentiable at the fixed point 0 (then the necessary condition is a consequence of the estimate as well as the fact that zero is a fixed point).

2. If the analysis concers functions  with derivative greater then  1   in the whole real line, then to the right (to the left) of the  [obviously unique] fixed point the function is greater (less) then identity.

etc. As long the particular family of functions is not determined, there is nothing more possible to claim than nonemptiness of the subdomain where the function is idempotent, which - obviously - is trivial.

My remark's aim is to point at the fact, that you are not seeking  meaningful necessary conditions, but additional conditions for functions possessing a fixed point implying   additional interesting properties, which would be absent if the function were fixed-point-less. Of course, what is interesting depends on our taste.

Joachim

PS. Your necessary condition, that  W(x)

Let me give  an example of interesting (even though, very simple) necessary condition:

If  F is a differentiable real valued function on [0,1]  such that  F(0)=1  and  F(1) =2 and if F  possesses a fixed point, then at least in one point its derivative is greater than 2.

In the example existence of a fixed point is ASSUMED. The particular class of functions is precisely determined also by the assumptions. Thus the necessary condition for existence of at least one fixed point for functions of the given class is possesing a point with derivative greater than 2.In other words, if the derivative of   function  F  with  F(0)=1  and  F(1)=2  is in every point not greater than  2, then F does not possess any fixed point. For similar translations by transposition of the implications, see the answer by Stefan Gruner above.

Hi Joachim, 

In my example I  assumed the existence of a fixed  point because I was looking for a necessary condition that a GIVEN  arbitrarily chosen x* is  a fixed point. If you kindly look at my paper in CRORR you will find that all required  assumptions on f(x)  and x* are there. Only these  seem  to imply  that  x* is indeed a fixed point. They are quite simple: x* is chosen from the interior (assumed to exists) of a convex set I ; actually we do not need to assume compactness of I) , and Lipschitz continuity of    f(x). on I. Only two general  tools are needed: the classic fundamental theorem of calculus and the more recent quadratic envelope characterization of zero-derivative points (a.k.a. "the formula"), both proved many times.in the literature .Note that I do not have to   assume differentiability of f. .

Moreover, the necessary condition given in CRORR is also a sufficient condition for a fixed point. Note that these characterizations have "primal" and "dual" formulations. 

If the results from CRORR for functions of the single variable are meaningful and correct, then we have the following more difficult question: Can anyone generalize these characterizations to Lipschitz continuous functions in several variables over convex sets with interior points?  Such generalizations would be very useful.

Thanks for your attention,

Best, Sanjo

Indeed fixed points have meaningful necessary conditions in applied sciences aside from the purpose of studying more properties of functions in mathematics given for instance by Joachim. 

Dear Sanjo,

It seems that  the results from CRORR for functions of the single variable are meaningful and correct.  Conserning   ''the following more difficult question: Can anyone generalize these characterizations to Lipschitz continuous functions in several variables over convex sets with interior point'', it is my impression   that  we can  generalize it.

Dear Sanjo,  Joachim  and Dejenie,

We can check whether the following is true( I only imagine a proof, which is ageneralization of single variable proof).

Set z=(x,y), $t=(t_1,t_2)$, $dt=dt_1 dt_2$ and $Q(x,y)=[x_0, x]\times[y_0, y]$.

Theorem X (Primal characterization of fixed points). Consider a continuous Lipschitz function f of two variables x,y on a convex planar set D. For z_0 and z in D, denote

W(z) = \int_ Q(x,y) (f(t) - t) dt.

Then f(z_0) = z_0 if, and only if

(1) |W(z)| = L |z - z_0|^2, z in D,

for some L > 0.

all the best,MM

Let me stress, that I have no objection about correcness of the results by Sanjo. And I never had any. I am just opposing presenting a necessary condition without saying at least that there are some additional assumptions besides the existence of the fixed point. In order to set this explicitely, the sentence:

[ A "primal" necessary condition for fixed point proved in the paper says that abs(W(x)) < = lambda times (x - x*)^2 on I for some lambda. ]

should be completed at least in the following usually accepted  form: [ Under some regularity assumption about  f , the necessary condition . . .]

Believe me, saying that there are many different necessary conditions without distinguishing the additional restrictions is a nothing explaining sentence.In the exteremal case one can simply state:

"The necessary condition for f  to possess a fixed point is that the derivative equals zero at some point",

which is certainly true if we restrict the consideations to odd higher than1 powers of  x. However it is not true for  f(x) = 2x.

Dear prof Joahim,

As I understand you take care (as a mathematician usually does) whether regularity assumption about f is enough to provide the proof.

Joahim: '' A "primal" necessary condition for fixed point proved in the paper says that abs(W(x)) < = lambda times $(x - x*)^2$ on I for some lambda. ]

should be completed at least in the following usually accepted form: [Under some regularity assumption about f , the necessary condition . . .]''

(A): Consider a continuous Lipschitz function f of the single variable on an interval I=[a,b].

As I understand, it means (B): there is L such that $|f(x) -f(y)|\leq L |x-y|$ for x,y in I.

At first glance it seems that (B) is regularity assumption about f to provide the result(but, we can check it).

all the best,

MM

@ Miodrag

No, my comment was not about the sufficiency for the Lipschitz property. This is the responsibility of the Author (though I think this is correct). The subject is that this additional requirement  is not mentioned  in the answers, which can mislead less experienced followers that the obtained necessary condition holds without any assumption. As I said, some mentioning about applied conditions is a good custom. My choice to point at "some regularity assumptions" was an example of weakest form of doing this. Of course, the Author has right not to open all details or hide them, but they cannot be omitted. After this the reader can be sent to read the paper. However, without such indication, sending the reader to the original paper is some form of an unfair propaganda:  see the paper there is the necessary condition proved without any restriction; and then getting there, the reader sees: OK, again someone let me read weaker theorems than announced. This is what I call bad custom.

Joachim

Hello J &M,,

I do not quite understand why you talk about the need for "regularization" conditions. Indeed, I do assume that  conditions for correctness  of some well-known tools (such as the fundamental theorem of calculus) are  known without being repeated. True, in CRORR I did use the "formula" which characterizes zero-derivative points without using differentiation. This allowed me not to use derivatives. Maybe this, not widely known result,  causes ambiguity, but I gave references in the attached file where the "formula" was proved and reproved. The file was made available to every mathematician who cared to read the question.. Joachim, did you open and read that file ?

 How does the following reformulation of the question look to you?

Given an n-tuple x* in the interior of some convex set K in Rn in the domain of a continuous Lipschitz function f. Can one find a condition which is both necessary and sufficient (i.e., a characterization") of the statement f(x*) = x* ?

Two answers ("primal" and "dual") were given for single variable f in CRORR (see the original question posting with a file attached).. My question is how to extend these results to functions in several variables. This is the crux of the matter. How could the question be made clearer ?

Thanks for your inputs. It is fun reading them.

Sanjo

@ Sanjo

Dont't you see that  the latest problem's formulation expresses explicitely the assumption on Lipschitz condition, and that this assumption  was not contained in the description of the result in your answer (post number  8  of this thread). Let me repeat the sentence

[ A "primal" necessary condition for fixed point proved in the paper says that abs(W(x)) < = lambda times (x - x*)^2 on I for some lambda. ]

Further reasons are given in my previous remark. Let me make you sure, that I am not talking about your result but about its presentation within this thread.

PS. I have read your paper carefully expecting that there will be a proof without any additional assumptions as promissed by you by the controversial sentence. THAT'S ALL

With a hope that you will respect the need of presenting/announcing theorems with more care for substantial assumptions,

Joachim

Hello J &M,

I think that now t I know what is causing frustration on your side. I think that this is the formula (2) in CRORR's paper! This is not a "regularization" condition but a necessary and sufficient condition for a zero-derivative point proved in, e.g., J. Global Opt 46 (2010) 155-162 and elsewhere. This formula works for n variable C1 functions with Lipschitz derivative. However, when applying it to the  integral W(x) I need only Lipschitz continuity of f. . Do you agree ? Sorry !

Hope you have a good weekend.

Sanjo

Hello Miodrag and Joachim,

I was carried away thinking that the  readers ot the question know the "formula"(2) but I was wrong. One should popularize it more.  I think that everything is clarified now .Agree?

Best, Sanjo

Enter your answer

@ Sanjo

The additional regularity I have indicated in my remarks is the assumption of Lipschitz property, not the formula, which is obvious. Trying to put the source of my fictious "frustation" upon the misunderstanding of your simple formula is a next example of the way chosen by you to argue against objections never raised (at least by me). Let me write after you: I think that everything is clarified now .Agree?

Joachim

Dear Sanjo and  Joahim,   

If I understand Sanjo's  qestion,  an answer  is suggested  in several variables:

Set z=(x,y), $t=(t_1,t_2)$, $dt=dt_1 dt_2$ and $Q(x,y)=[x_0, x]\times[y_0, y]$.

Theorem  (Primal characterization of fixed points). Consider a continuous Lipschitz function f of two variables x,y on a convex planar set D. For z_0 and z in D, denote

W(z) = \int_ Q(x,y) (f(t) - t) dt.

Then f(z_0) = z_0 if, and only if

(1) |W(z)| = L |z - z_0|^2, z in D,

for some L > 0.

Is this related to your question?

best,MM

Dear Miodrag,

Could be, but Is there a reader-friendlier way to write your symbols , please ? I have difficulty deciphering them.. My problem of going to higher dimensions in CROOR was how to deal with integrals and the  fundamental  theorem of calculus. I wonder how you resolved it..

Waiting for your reply tomorrow.

Best, Sanjo

Dear Sanjo,

The pdf.file Note on Convexity related to Mihailo Petrovic (in preparation) is enclosed.

We put an outline of an answer to your question temporarily here and we will form a separate file as soon as possible.

PS. We will commemorate the 150th anniversary of the birth of Mihailo Petrovic-Alas in 2018 (2018 will be 150th from the birth of Mihailo Petrovic).

best,MM

Thanks, Miodrag.  I can hardly wait to see your separate file on the first part of the rough draft  It would be helpful if you could illustrate the second part by a simple example.

I do not see  how the two parts are connected., are they ?

Best, Sanjo 

Dear Sanjo,

Parts are not connected. If someone is interested in my work, it gives me the motivation to write something. I will try to make something quickly and to   illustrate the second part by a simple example .

Best, MM

I am agree it also gives (possibly new and different), sufficient conditions for fixed points.

I have made  something quickly and  have  illustrated    by a simple example.

Please  understand  that  this is very very rough version for temporarily use only  and that  we will correct file as soon as possible.

all the best,

MM

Hello Miodrag,

It seems that you forgot to attach an example ! Can you resend it, please ? Best, Sanjo

Hello  Sanjo, 

There is Example 1 (see  above   the attach  file).

For   $t=(t_1,t_2)$,  $f(t)=t + g(t)$, $g(t)= (\sin t_1, \sin t_2)$, and $z_0=(0,0)$, we compute $W_1(z)=\int_ {Q(x,y)} \sin t_1 dt_1 = \int_0^y d t_2(\int_0^x \sin t_1 dt_1)=2y \sin^2(x/2)$.

This is    very simple, we can try to find   more interesting example.

Best,

Miodrag

Hello Miodrag,

There was no file attached to your message. Can you double check whether it was sent ?

Thanks, Sanjo

Hello  Sanjo, 

There is Example 1 (see     the attach  file).

For   $t=(t_1,t_2)$,  $f(t)=t + g(t)$, $g(t)= (\sin t_1, \sin t_2)$, and $z_0=(0,0)$, we compute $W_1(z)=\int_ {Q(x,y)} \sin t_1 dt_1 = \int_0^y d t_2(\int_0^x \sin t_1 dt_1)=2y \sin^2(x/2)$.

This is    very simple, we can try to find   more interesting example.

Best,

Miodrag

Hi Miodrag, Could you please send me your example using standard math. symbols. This is a very delicate, though simple, example but I would like to interpret things right

Thanks, Sanjo.

Concerning practial point of view   there is   a  curious meteorological application of the hairy ball theorem (which is related to the fixed point theorem)involves considering the wind as a vector defined at every point continuously over the surface of a planet with an atmosphere. As an idealization, take wind to be a two-dimensional vector: suppose that relative to the planetary diameter of the Earth, its vertical (i.e., non-tangential) motion is negligible.

One scenario, in which there is absolutely no wind (air movement), corresponds to a field of zero-vectors. This scenario is uninteresting from the point of view of this theorem, and physically unrealistic (there will always be wind). In the case where there is at least some wind, the Hairy Ball Theorem dictates that at all times there must be at least one point on a planet with no wind at all and therefore a tuft. This corresponds to the above statement that there will always be p such that f(p) = 0.

The hairy ball theorem of algebraic topology: For the ordinary sphere, or 2-sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one p such that f(p) = 0.