Problem 6 - Basic Analysis: Introduction to Real Analysis

Question

a) Find an example of a contraction \(f: X \rightarrow X\) of a non-complete metric space \(X\) with no fixed point. b) Find a 1-Lipschitz map \(f: X \rightarrow X\) of a complete metric space \(X\) with no fixed point.

Step-by-Step Solution

Answer

a) A contraction on \( (0, 1) \) with \( f(x) = \frac{x}{2} \); b) a 1-Lipschitz map on \( \mathbb{R} \) with \( f(x) = x + 1 \).

Step 1: Understanding Contraction in Non-Complete Metric Space

A contraction mapping is a function where there exists a constant \(0 \leq c < 1\) such that for all points \(x, y\) in the space \(X\), \(d(f(x), f(y)) \leq c \cdot d(x, y)\). However, a contraction on a non-complete metric space might not have a fixed point. The goal is to find such an example.

Step 2: Example of Contraction without Fixed Point

Consider the metric space \(X = (0, 1)\) with the usual metric \(d(x, y) = |x - y|\) and the function \(f(x) = \frac{x}{2}\). Since \(f(x) \leq \frac{1}{2}x\), \(f\) is a contraction with constant \(c = \frac{1}{2}\). However, a potential fixed point \(x = f(x)\) would satisfy \(x = \frac{x}{2}\), leading to \(x = 0\), which is not in \(X\). Thus, \(f\) has no fixed point.

Step 3: Understanding 1-Lipschitz Map with No Fixed Point in Complete Space

A 1-Lipschitz map is a function \(f: X \rightarrow X\) where for all points \(x, y\) in \(X\), \(d(f(x), f(y)) \leq d(x, y)\). Lipschitz maps do not contract distances, and such a map might not have a fixed point in a complete space.

Step 4: Example of 1-Lipschitz Map without Fixed Point

Consider the complete metric space \(X = (\mathbb{R}, d)\) where \(d(x, y) = |x - y|\), and the function \(f(x) = x + 1\). Here, \(f\) shifts each point by 1, maintaining distance \(d(f(x), f(y)) = |f(x) - f(y)| = |(x + 1) - (y + 1)| = |x - y|\). This function is 1-Lipschitz but has no fixed point, since \(x = f(x)\) implies \(x = x + 1\), which is a contradiction.

Key Concepts

Contraction Mapping

A contraction mapping is a powerful concept in metric spaces. It is a special kind of function defined on a metric space such that it brings points closer together. More formally, it is a function \(f: X \rightarrow X\) for which there exists some constant \(0 \leq c < 1\) where for all points \(x, y\) in the metric space \(X\), the distance between \(f(x)\) and \(f(y)\) satisfies the inequality \[ d(f(x), f(y)) \leq c \cdot d(x, y) \].

• The constant \(c\) is known as the contraction constant.
• A contraction mapping reduces distances by a factor of \(c\).
• It is typically associated with fixed points, which are points that map to themselves \((x = f(x))\).

In a complete metric space, a contraction is guaranteed to have at least one fixed point by the Banach Fixed-Point Theorem.
However, in non-complete metric spaces, such as the open interval \((0, 1)\), a contraction mapping does not always result in a fixed point.

Lipschitz Map

A Lipschitz map is similar to a contraction but comes with its unique characteristics. A 1-Lipschitz map is a function \(f: X \rightarrow X\) such that for every pair of points \(x, y\) in a metric space, the distance between \(f(x)\) and \(f(y)\) is at most equal to the distance between \(x\) and \(y\). Formally, it satisfies\[ d(f(x), f(y)) \leq d(x, y) \].

• The map does not "expand" the space but doesn't necessarily "contract" it either.
• 1-Lipschitz maps preserve distances or make them shorter, but unlike contractions, they can be equal.
• Such maps are useful in applied mathematics due to their "non-expansive" property.

Notably, a 1-Lipschitz function on a complete metric space can also exist without fixed points. For instance, the function \(f(x) = x + 1\) on \(\mathbb{R}\) slightly shifts every point, illustrating that not all operations keeping distances constant will fix points.

Fixed Point

A fixed point of a function \(f: X \rightarrow X\) is simply a point \(x\) such that \(f(x) = x\). It is a concept essential in many mathematical fields due to its foundational role in analysis and topology.
Here's why fixed points are so important:

• They indicate an equilibrium state, where the function leaves the point unchanged.
• In applied scenarios, fixed points can represent stable states in models.
• Solvers and algorithms benefiting from this concept are more efficient in reaching solutions in optimization problems.

Finding fixed points often relies on the properties of the space (completeness, for instance) and the nature of the function. While contraction mappings in complete spaces are known for finding fixed points reliably, non-complete spaces, or certain types of functions like Lipschitz maps, can show exceptions.

Completeness

Completeness is a property of a metric space that deals with the idea of limits and convergence. A metric space is complete if every Cauchy sequence of points within the space has a limit that also belongs to the space.
Understanding completeness is crucial for various mathematical theorems and results:

• In a complete metric space, every Cauchy sequence converges to a point within the space, ensuring no 'gaps'.
• This property is critical in proving the existence of fixed points for contraction mappings, as outlined by the Banach Fixed-Point Theorem.
• Completeness implies that analysis on such spaces can be done seamlessly, offering robustness and predictability.

Conversely, in non-complete spaces, sequences might converge to a limit outside the space, leading to scenarios where expected results, such as finding fixed points, can fail.