Fixed Point Theorem: Why Must Something Stay Unchanged?

Mathematical Analysis • Mathematical Foundations • Analysis & Applications

Building Intuition: The Crumpled Map Problem

A thought experiment that reveals a profound mathematical truth.

Imagine This Scenario

You take a map, crumple it up, and place it back on the table. The map is now folded, wrinkled, and distorted in countless ways.

The Central Question

Is there a point on the map that remains in the exact same location as before?

Mathematical Insight

  • Intuition says: Probably not — everything moved!
  • Mathematics says: Absolutely YES!

There must exist at least one point that hasn’t moved.

Formal Definition: What is a Fixed Point?

The Mathematical Definition

A point $x$ is called a Fixed Point if it satisfies: $f(x) = x$

In other words: The point remains completely unchanged when the function $f$ is applied to it.

  • Input equals Output
  • An “Anchor” Point
  • Invariant Under $f$

Concrete Example: $f(x) = x^2$

Let’s find the fixed points of a basic quadratic function.

Step-by-Step Solution

  1. Set up the equation: $f(x) = x$ $x^2 = x$
  2. Rearrange: $x^2 - x = 0$
  3. Factor: $x(x - 1) = 0$
  4. Solution: $x = 0$ or $x = 1$

Graphical Visualization

Key Insight: Fixed points occur where the curve $y = x^2$ intersects the line $y = x$ (shown in gold).

The Challenge: The Key Question

What if the function is more complicated?

  • Simple Cases: For $f(x) = x^2$, we can solve algebraically. But what about functions without closed-form solutions?
  • The Real Challenge: Can we still guarantee that a fixed point exists, even when we can’t find it explicitly?

We need a general theorem — not just examples! (Existence Proof • General Guarantee • Universal Result)

Foundation: Review: Intermediate Value Theorem (IVT)

A powerful tool from calculus that we’ll use as our foundation.

The IVT Statement

If $f$ is continuous on $[a, b]$ and $f(a) \cdot f(b) < 0$ then there exists $c \in (a, b)$ such that $f(c) = 0$

Core Insight: A sign change guarantees crossing zero.

Visual Intuition

  • Continuous curve from negative to positive.
  • Must cross the $x$-axis somewhere.

Strategy: Reformulating the Problem

The crucial transformation that connects fixed points to IVT.

The Key Transformation

  • Original Problem: $f(x) = x$
  • Reformulated: $f(x) - x = 0$

Why This Works

Finding where $f(x) = x$ is equivalent to finding where $f(x) - x = 0$. This transforms a fixed point problem into a root-finding problem!

The Connection

Now we can apply IVT! If we can show that $f(x) - x$ changes sign, a root (and thus a fixed point) must exist.

Construction: Constructing a New Function

Define the auxiliary function: $g(x) = f(x) - x$

  • $g(x)$ Definition: A new function that measures the difference between $f(x)$ and $x$.
  • Root of $g(x)$: If $g(c) = 0$, then $f(c) = c$.
  • Fixed Point Found! Finding a root of $g$ is equivalent to finding a fixed point of $f$.

Key Insight: We’ve transformed the problem from “find where $f(x) = x$” to “find where $g(x) = 0$” — a classic root-finding problem solvable with IVT!

Conditions: Applying the Conditions

Setting up the boundary conditions for IVT.

Our Assumptions

  1. Assumption 1: $f(0) \geq 0$ (The function at 0 is non-negative)
  2. Assumption 2: $f(1) \leq 1$ (The function at 1 doesn’t exceed 1)

Note: These are natural conditions for functions mapping $[0,1]$ into itself.

What This Means for $g(x)$

  • At $x = 0$: $g(0) = f(0) - 0 = f(0) \geq 0$
  • At $x = 1$: $g(1) = f(1) - 1 \leq 0$ (since $f(1) \leq 1$)

$g(x)$ changes sign from $\geq 0$ to $\leq 0$!

Application: Applying IVT to $g(x)$

By the Intermediate Value Theorem

Conditions Met:

  • $g$ is continuous ($f$ is continuous)
  • $g(0) \geq 0$
  • $g(1) \leq 0$

IVT Conclusion: Therefore, there exists some point $c \in (0, 1)$ such that: $g(c) = 0$

We’ve proven that $g(c) = 0$ for some $c$ in $(0,1)$. But what does this mean for $f$?

Conclusion: Fixed Point Exists!

The final step that completes the proof.

The Final Deduction

We know: $g(c) = 0$

By definition of $g$: $f(c) - c = 0$

Therefore: $f(c) = c$

Visual Proof

A FIXED POINT EXISTS! The point $c$ where $g(c) = 0$ is exactly where $f(c) = c$.

Interactive Example: $f(x) = \cos(x)$

Does the cosine function have a fixed point?

  • Does $f(x) = \cos(x)$ have a fixed point? Yes!
  • Reasoning:
    1. Continuity: $\cos(x)$ is continuous everywhere.
    2. Interval $[0, 1]$: $\cos(0) = 1 \geq 0$, $\cos(1) \approx 0.54 \leq 1$.
    3. Apply Theorem: All conditions satisfied!

Fixed point exists in $[0, 1]$!

Interactive Example: $f(x) = 2x$

A case where conditions matter!

  • Does $f(x) = 2x$ have a fixed point in $(0,1)$? No!
  • Answer: Only $x = 0$ is a fixed point, but it’s not in $(0,1)$.
  • Why Theorem Doesn’t Apply:
    • $f(0) = 0$ (satisfies $f(0) \geq 0$)
    • $f(1) = 2$ (violates $f(1) \leq 1$)
  • The function maps $[0,1]$ outside itself! At $x = 1$, $f(1) = 2$, which is outside the interval.

The conditions are crucial!

Critical Insight: Continuity is Crucial

Without continuity, the guarantee disappears.

Why Continuity Matters

If $f$ is not continuous, it can “jump” over the fixed point without ever hitting it.

Counterexample:

$f(x) = x + 0.5$ for $x < 0.5$ $f(x) = x - 0.5$ for $x \geq 0.5$ This function has no fixed point because of the discontinuity at $x = 0.5$.

Visual Explanation

  • Continuous function: must cross $y = x$.
  • Discontinuous: can jump over the line.

Extension: Brouwer Fixed Point Theorem

From 1D to 2D: A profound generalization.

The 2D Version

  • The Setup: Consider a closed disk (or any convex, compact set) in 2D. Let $f$ be a continuous function that maps the disk into itself.
  • The Theorem: Brouwer’s Theorem states: $\exists p \text{ such that } f(p) = p$ At least one fixed point must exist!

Significance: This extends to any finite dimension! It’s one of the most important theorems in topology.

Visualization: 2D Visualization: The Disk Transformation

Any continuous transformation of a disk into itself has a fixed point.

The Crumpled Map Analogy (2D)

Imagine taking a circular disk, crumpling it, stretching it, twisting it, and placing it back on top of itself.

The Question: Is there a point that remains in exactly the same position? Answer: YES! Brouwer guarantees it.

Visual Intuition

No matter how you transform the disk, at least one point must stay fixed.

Philosophy: The Beautiful Intuition

The Elegant Insight

“You cannot ‘move everything’ without leaving at least one point unchanged.”

  • Counterintuitive: It seems like you should be able to move everything, but mathematics says otherwise.
  • Universal: This applies to ANY continuous transformation, no matter how complex.
  • Profound: This simple idea has applications across mathematics, economics, and physics.

Application: Fixed Point Iteration

A practical numerical method for solving equations.

The Iteration Method

THE RECURRENCE $x_{n+1} = f(x_n)$

  • How it works: Start with an initial guess $x_0$, then repeatedly apply $f$. If the sequence converges, it converges to a fixed point!
  • Visual: The “staircase” or “cobweb” diagram shows how iteration converges to the fixed point where $y = f(x)$ intersects $y = x$.

This turns equation-solving into simple iteration!

Worked Example: Solving $x = \cos(x)$

Using iteration to find the fixed point.

The Iteration Process

  • Initial Value: $x_0 = 0.5$
  • Iteration Formula: $x_{n+1} = \cos(x_n)$
  • First few iterations:
    • $x_1 = \cos(0.5) \approx 0.8776$
    • $x_2 = \cos(0.8776) \approx 0.6390$
    • $x_3 = \cos(0.6390) \approx 0.8027$
    • $x_4 = \cos(0.8027) \approx 0.6948$

Convergence Visualization

Converges to $x \approx 0.7391$ This is the unique fixed point of $\cos(x)$!

Theory: Why Iteration Works

Convergence Conditions

  • When It Works: Iteration converges to a fixed point under certain conditions: $\vert f’(x)\vert < 1$ near the fixed point
  • The Intuition: If the function is “flat enough” near the fixed point, each iteration gets closer:
    • Small derivative = small steps
    • Sequence approaches fixed point

Practical Value: Fixed point iteration provides a simple, robust numerical method for solving equations that might be difficult to solve algebraically.

Profound Application: Game Theory & Nash Equilibrium

Where fixed point theorems changed economics forever.

The Nash Equilibrium

  • Definition: A set of strategies where no player can benefit by changing their strategy unilaterally.
  • John Nash’s Proof (1950): Nash used Brouwer’s Fixed Point Theorem to prove that every finite game has at least one equilibrium!
  • Nobel Prize in Economics (1994)
  • Example: Prisoner’s Dilemma — both players confessing is the Nash Equilibrium, even though both would be better off cooperating.

Fixed point theorems provide the mathematical foundation for modern game theory!

Summary: Final Summary

Key Takeaways

  1. Core Idea: Continuity $\Rightarrow$ Existence A continuous function mapping $[0,1]$ into itself must have a fixed point.
  2. The Strategy: Transform problem $\rightarrow$ Apply IVT Rewrite $f(x) = x$ as $g(x) = f(x) - x = 0$, then use the Intermediate Value Theorem.
  3. Generalization: Brouwer’s Theorem Extends to any dimension: Any continuous transformation of a disk into itself has a fixed point.
  4. Applications: Wide-ranging impact From numerical methods (fixed point iteration) to economics (Nash Equilibrium).

In a changing world, something always stays the same.