# What is Fixed point: Definition and 103 Discussions

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f if f(c) = c. This means f(f(...f(c)...)) = f n(c) = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set.
For example, if f is defined on the real numbers by

f
(
x
)
=

x

2

3
x
+
4
,

{\displaystyle f(x)=x^{2}-3x+4,}
then 2 is a fixed point of f, because f(2) = 2.
Not all functions have fixed points: for example, f(x) = x + 1, has no fixed points, since x is never equal to x + 1 for any real number. In graphical terms, a fixed point x means the point (x, f(x)) is on the line y = x, or in other words the graph of f has a point in common with that line.
Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one. In projective geometry, a fixed point of a projectivity has been called a double point.In Galois theory, the set of the fixed points of a set of field automorphisms is a field called the fixed field of the set of automorphisms.

View More On Wikipedia.org
1. ### How does one convert a decimal float to binary using the multiply by 2 method?

I've started reading A concise introduction to numerical analysis, A.C. Foul and on the first page there's the following paragraph about how a floating point in fixed point precision can be represented: I don't understand the example where it says " ##\beta=2## and ##p=20##, the decimal number...
2. ### Challenge Math Challenge - July 2023

Welcome to this month's math challenge thread! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Have fun! 1. (solved by @AndreasC) I start watching a...
3. ### Fixed point free automorphism of order 2

I did not use the hint for this problem. Here is my attempt at a proof: Proof: Note first that ##σ(σ(x)) = x## for all ##x \in G##. Then ##σ^{-1}(σ(σ(x))) = σ(x) = σ^{-1}(x) = σ(x^{-1})##. Now consider ##σ(gh)## for ##g, h \in G##. We have that ##σ(gh) = σ((gh)^{-1}) = σ(h^{-1}g^{-1})##...
4. ### Algebra: distance from a fixed point

The first image is the question and the second is the answer. My thinking is let's say North is positive, and South is negative. Fixed point O is the starting point. Then the question becomes +(2a-b)-(3a+2b). The answer should be -a-3b. I cannot fathom why the book gives the answer as a+b. Any...
5. ### I Is this statement an aspect of the Hairy Ball or Fixed Point Theorem?

“Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes…” The reason this statement sounds like one of those theora is that I recall reading a Time-Life book on Mathematics, and there...
6. ### I Rigid body with fixed point

Today I read a book in mechanics and encountered a funny proposition about rigid body with fixed point. Perhaps somebody will be interested to propose it to students as a task. This proposition is almost correct:) Consider a rigid body with a fixed point ##O##. Let ##Oxyz## be a coordinate...
7. ### MHB Fixed point iteration convergence

Question: For the following functions, does the fixed point iteration for finding the fixed point in $\left [ 0,1 \right ]$ converge for all first points $p_{0} \in \left [ 0,1 \right ]$? Justify your answer. a.$g(x) = e^{\frac{-x}{2}}$ b.$g(x) = 3x - 1$ Let me attempt for part a first...
8. ### MHB Fixed point,, Jacobi- & Newton Method, Linear Systems

Hey! :giggle: Question 1 : Let $g(x)-=x-x^3$. The point $x=0$ is a fixed point for $g$. Show that if $x^{\star}$ is a fixed point of $g$, $g(x^{\star})=x^{\star}$, then $x^{\star}=0$. If $(x_k)$ the sequence $x_{k+1}=g(x_k)$, $k=0,1,2,\ldots$ show that if $0>x_0>-1$ then $(x_k)$ is...
9. ### I Proving a Fixed Point Theorem for Shrinking Maps on Compact Spaces

Show that if ##f## is a shrinking map ##d(f(x),f(y)) < d(x,y)## and ##X## is compact, then ##f## has a unique fixed point. Hint. Let ##A_n=f^n(X)## and ##A=\cap A_n##. Given ##x\in A##, choose ##x_n## so that ##x=f^{n+1}(x_n)##. If ##a## is the limit of some subsequence of the sequence...
10. ### I Kripke's fixed point for truth predicate: justification?

If I understand correctly (dubious), given a consistent theory C (collection of sentences), Kripke proposes to add a predicate T so that, if K = the collection of all sentences T("S") for every sentence S in C, ("." being some appropriate coding) then the closure of K∪C forms a new theory C*...
11. ### How close will body approach fixed point charge?

Homework Statement Charged sphere with a mass of 15 mg and charge 2 nC moves with a speed of 15 cm/s towards a fixed point charge of 3 nC. How close will sphere approach charge? Homework Equations K=(1/2)*mv2 U=k*(Q1Q2/r) The Attempt at a Solution So I am not sure I approached correctly but...
12. ### MHB Fixed point iteration: g is a contraction mapping

Hey! :o We have the function $f(x)=x^5-\frac{5}{16}$. I have approximated the root of that function using three steps of Newton's method with initla value $x_0=\frac{1}{2}$ : \begin{align*}x_1&=x_0-\frac{f(x_0)}{f'(x_0)}\approx \frac{7}{5} \\ x_2&=x_1-\frac{f(x_1)}{f'(x_1)} \approx...
13. ### Determine the stability of a fixed point of oscillations

Homework Statement I have a system of coupled differential equations representing chemical reactions and given certain initial conditions for the equations I can observe oscillation behaviour when I solved the equations numerically using Euler's Method. However, then it asks to investigate the...
14. ### Unique Fixed Point: Prove Contraction Map in Compact Space Has Unique Solution

Homework Statement Let ##(X,d)## be some metric space, and let ##f : X \to X## be such that ##d(f(x),f(y)) \le a d(x,y)## for every ##x,y \in X## for some ##a \in (0,1)## (such a map is a called a contraction map) If ##f## is a contraction and ##X## is compact, show that ##f## has a unique...
15. ### Torque and rotation around a fixed point

This isn't a real homework problem (i.e. I made this problem up myself for my own purposes), but I figured this is the correct forum to post. 1. Homework Statement In the following figure we have two rods connected to each other, and the bottom rod is connected to the blue structure (G), and G...
16. ### I Dense orbit and fixed point question

Let f be a continuous function of a metric space, M, to itself with a dense orbit and a fixed point. I.e. there exists z such that the set {f(n)(z)} for all n ∊ N (where f(n) is the nth iterate of f) is dense in M, and there exists p such that f(p) = p. Does this imply that f spreads? I.e. does...
17. ### Proving that three closed orbits must contain a fixed point

A smooth vector field on the phase plane is known to have exactly three closed orbit. Two of the cycles, C1 and C2 lie inside the third cycle C3. However C1 does not lie inside C2, nor vice-versa. What is the configuration of the orbits? Show that there must be at least one fixed point bounded...
18. ### A Critical exponents - experimental values

Hi all! For a talk I want to compare the values of the critical exponents found by Wilson and Fisher in their \epsilon = 1 paper (10.1103/PhysRevLett.28.240), with the experimental values measured up-to-date. Can anyone provide a source for these measured values (\gamma, \nu, \eta)? Thanks...
19. ### Moving point and it's distance relative to a fixed point

Not sure whether this is an intro physics or intro calculus/related rates problem. 1. Homework Statement Suppose a point P lies at (x,y)=(0,1) meters. A car is traveling at 30 meters/second along the x-axis towards +∞. Define r to be the distance between P and the car at any time t. I...
20. ### Fixed Point Theorem: Necessary & Sufficient Conditions for Convergence

Homework Statement Let be ##f \in C^{1}(\mathbb{R}^{n}, \mathbb{R}^{n})## and ##a \in \mathbb{R}^{n}## with ##f(a) = a##. I'm looking for a suffisent and necessar condition on f that for all ##(x_{n})## define by ##f(x_{n}) = x_{n+1}##, then ##(x_{n})## converge. Homework Equations ##f(a) =...
21. ### Using the Intermediate Value Theorem to Find Fixed Points

Homework Statement Homework EquationsThe Attempt at a Solution I started looking at this problem and I think I am going to have to use the intermediate value theorem for this proof, but I am not quite sure. I started looking at possible examples of these functions, but I know this is not good...
22. ### Find the force a distance from a fixed point

Homework Statement A person is standing on tiptoe, with the total weight supported by the force on the toe. A mechanical model for the situation is shown, where T is the force in the Achilles tendon and R is the force on the foot due to the tibia. Find the value of T. Assume the total weight is...
23. ### Example of torque-free rotation with a fixed point

What is an example of a rigid body rotating when one point is fixed and there are no net applied torques? And the fixed point is not the center of mass. I considered a cone rolling without slipping on a flat plane is such an example; the apex is the fixed point, but is there a net applied...
24. ### A few more questions about fixed point iteration ....?

first of all i simply don't want to give up learning numerical methods ... i am trying to follow fixed point iteration method from this link ... http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf fixed point iteration can be used to solve...
25. ### MHB A particle traveling in a strainght line passes a fixed point O

so if $a\left(t\right)=p+qt$ then $v\left(t\right)=\int a\left(t\right) dt = pt+\frac{q {t}^{2}}{2}+C$ if $v=3.5$ when $t=2$ then $1.75=p+q$ if so, now what? the answers are $4, -3; 4m$
26. ### Does Loop quantum gravity have an effective UV fixed point?

loop quantum gravity and loop quantum cosmology gravity becomes weaker then repulsive instead of stronger, towards the Planck regime, due to the onset of quantum effects. gravity near Planck energies and densities, LQG/LQC becomes repulsive. at the inflection point where LQG gravity strength...
27. ### Fixed point method for nonlinear systems - complex roots

Homework Statement I've been asked to graphically verify that the system of equations F (that I've uploaded) has exactly 4 roots. And so I did, using the ContourPlot function in Mathematica and also calculated them using FindRoot. Now, I've to approximate the zeros of F using the fixed point...
28. ### Does T have a unique fixed point in X?

Let (X, d) be a complete metric space, and suppose T : X → X is a function such that T^2 is a contraction. [By T^2, we mean the function T^2 : X → X given by T^2(x) = T(T(x))]. Show that T has a unique fixed point in X. So I have an answer, but I am not sure whether it is correct. It goes as...
29. ### MHB Cauchy sequence in fixed point theory

İn some articles, I see something... For example, Let we define a sequence by ${x}_{n}=f{x}_{n}={f}^{n}{x}_{0}$$\left\{{x}_{n}\right\}$. To show that $\left\{{x}_{n}\right\}$ is Cauchy sequence, we suppose that $\left\{{x}_{n}\right\}$ is not a Cauchy sequence...For this reason, there exists a...
30. ### MHB Continuous mapping and fixed point

Let $T$ be a continuous mapping of a complete metric space $X$ into itself such that ${T}^{k}$ is a contraction mapping of $X$ for some positive integer $k$. Then $T$ has a unique fixed point in $X$. Proof: ${T}^{k}$ has a unique fixed point $u$ in $X$ and...
31. ### MHB Fixed Point Theorem & Contractive Maps

Please give an example of contractive map which have fixed point...I search but I didnt find
32. ### MHB What does priori estimate do in fixed point theory

what does priori estimate do in fixed point theory ? plese, I want some examples :)
33. ### MHB Fixed Point Theory: Lipschitz or Contraction?

I see that if a mapping is contraction then it is contractive then it is nonexpensive and then it is lipschtiz...so, which class of mapping is general ? lipschitz or contraction ? which one ? thank you for your attention :)
34. ### How Does a Fixed Point Theorem Explain Convergence in Iterative Methods?

Here I do not perceive the a sequence generated by fixed-point iteration. First would you like to explain this. How can it be that if lim n->∞ pn=P, then lim n-> ∞ Pn+1 ? Source: Numerical Methods Using Matlab by Kurtis D. Fink and John Matthews.
35. ### Fixed point iteration, locally convergent

Homework Statement For which of them will the corresponding fixed point iteration xk+1 = g(xk) be locally convergent to the solution xbar in [0, 1]? (The condition to check is whether |g'(xbar)| < 1.) A) 1/x2 -1 B)... C)... compute xbar to within absolute error 10-4. Homework Equations 3. The...
36. ### Fixed point and scale invariance

Hello everyone. I'm studying the fixed point of theory in the context of QFT. First of all, let me say what I think I understood about fixed points and then I'll state my question. Suppose we have a theory with a certain running coupling ##\lambda(\mu)##. If we have, for example, an UV fixed...
37. ### MHB Metric Spaces - Fixed Point Theorem (Apostol, Theorem 4.48)

I need help with the proof of the Fixed Point Theorem for a metric space (S,d) (Apostol Theorem 4.48) The Fixed Point Theorem and its proof read as follows: In the above proof Apostol writes: " ... ... Using the triangle inequality we find for m \gt n, d(p_m, p_n) \le \sum_{k=n}^{m-1}...
38. ### MHB Error in fixed point arithmetic library

I found an issue with the fixed point library, and Heiko Oberdiek found the offending code and submitted a correction to fix the problem in the fixed point library. The details can be found here. At present, you can issue the command below to over come the problem. Eventually someone will...
39. ### MHB Fixed point, interval of existence, & stability

Investing function fc(x) = (6/x)+(x/2)-c where 0<= c <=3 a) Use alegbra to find the positive fixed point (in terms of c) and identify its exact interval of existence b) Use algebra and calculus to find the exact interval of stability of the fixed point c) Use algebra to find the points of the 2...
40. ### MHB Fixed Point Theorem: Estimating x* With x9 & x10

Hello! ;) I have a question. Let $\varphi:[-1,1] \to [-1,1]$ with $L=0.8$ at $[-1,1]$, $\varphi$ has a unique fixed point $x^{*}$ and the sequence $(x_{n})$ with $x_{n+1}=\varphi(x_{n}) ,n=0,1,2,...$ is well defined and coverges to $x^{*}$ for any $x_{0} \in [-1,1]$.Then if the 9th approximation...
41. ### Proving existence of unique fixed point on a compact space

Homework Statement . Let ##(M,d)## be a metric space and let ##f:M \to M## be a continuous function such that ##d(f(x),f(y))>d(x,y)## for every ##x, y \in M## with ##x≠y##. Prove that ##f## has a unique fixed point The attempt at a solution. The easy part is always to prove unicity...
42. ### Fixed point theorem knowing that ##T^n## is a contraction

Homework Statement . Let ##X## be a complete metric space and let ##T:X \to X## such that there exists ##n \in \mathbb N##: ##T^n## is a contraction. Prove that there is a unique ##x \in X## such that ##T(x)=x##. The attempt at a solution. Sorry but I am completely lost with this exercise...
43. ### Straight lines passing through fixed point

Homework Statement If all the lines given by the equation (3\sin \theta + 5\cos \theta )x+(7\sin \theta - 3\cos \theta )y+11(\sin \theta - \cos \theta)=0 pass through a fixed point (a,b) forall theta in R then |a-b|= Homework Equations The Attempt at a Solution Dividing both sides by...
44. ### Newton Raphson method and Fixed Point Iteration method ?

Hi everyone, I has been learning numerical method recently, i am very wonder how fixed point iteration method and Newton raphson method works (a more insight explanation rather than mathematical proof ) thanks!
45. ### Sequence Convergence & Fixed Point Theorem

Homework Statement Let g(x)= (2/3)*(x+1/(x^2)) and consider the sequence defined by pn= g(pn-1) where n≥1 a) Determine the values of p0 \in [1,2] for which the sequence {pn} from 0 to infinity converges. b) For the cases where {pn} converges (if any), what is the rate of convergence...
46. ### MHB Fixed point for a complex mapping.

W= z+2 /z-2 drawing mapping find image in w plane line Re(z)constant and im(z)=constant find fixed point from mapping In my textbook have just W = z-1 / z+1 . Thank a lot for your help.
47. ### Solving for a fixed point on an interval

Homework Statement OK, I need to solve for the fixed point of the equation 2sinπx + x = 0 on the interval [1, 2]. I know the answer to be ~1.21... but I need to prove it. The Attempt at a Solution I really just need help solving for a proper equation of x. I tried x = -2sinπx...
48. ### Fixed Point Iteration for Solving Equations

Homework Statement Apply fixed point iteration to find the solution of each equation to eight correct decimal places x3=2x+2 The Attempt at a Solution I have tried to rewrite the equation for in every possible way to solve for one x and pluggin in my guess( have tried...
49. ### Fixed Point Theorem: Unit Square Injection to Bigger Square

If you have a surjective function from the unit square, [0,1] X [0,1] onto a bigger square, such as [0,3] X [0,3], will there always be a fixed point under any injection of the unit square into the big square (i.e. will there always be x s.t. f(x)=i(x), where i is an injection?) It seems to...
50. ### Neuroscience: LFP at fixed point with different simulataneous frequencies

Hello! I have a neuroscience question about local field potenitals (LFPs). I was reading a jounral article where the LFP in a given location was recorded overtime, and then the LFP at ecah time point was divided into different frequency groups: i.e. the LFP at a given point in space...