Question

Find an example of a non-complete metric space and a contraction on it without a fixed point.

Short answer

The function \( f(x) = \frac{x}{2} \) on \( (0, 1) \) is a contraction without a fixed point.

Step-by-step solution

- Define a Non-Complete Metric Space

Consider the open interval \( (0, 1) \) with the standard Euclidean metric. In this space, sequences can converge to 0 or 1, which are not included in the interval, hence it is not complete.

- Define a Contraction Mapping

Define a function \( f:(0, 1) \rightarrow (0, 1) \) by \( f(x) = \frac{x}{2} \). This function satisfies the contraction condition because: \[ d(f(x), f(y)) = \left| \frac{x}{2} - \frac{y}{2} \right| = \frac{1}{2} |x - y|. \]

- Verify the Contraction Property

Check that \( f \) is a contraction mapping with constant \( \frac{1}{2} \). Since \[ d(f(x), f(y)) = \frac{1}{2} d(x, y), \] \( f \) is indeed a contraction mapping on \( (0, 1) \).

- Show No Fixed Point Exists

To find a fixed point, we solve \( f(x) = x \). This gives \( \frac{x}{2} = x \), which simplifies to \( x = 0 \). However, 0 is not in the interval \( (0, 1) \), so there is no fixed point for \( f \) in this space.

Key concepts

contraction mapping

A contraction mapping is a special type of function. It brings points closer together. More formally, a function defined on a metric space \( (X, d) \) is called a contraction mapping if there is a constant \( 0 \leq k < 1 \) such that for any two points \( x, y \in X \), we have: \[ d(f(x), f(y)) \leq k \, d(x, y). \] This means the function f maps distances between points in a way that reduces the distance by a factor of k. It is an essential tool in solving problems related to the convergence of sequences. This property helps us understand the behavior of functions on metric spaces, such as the interval \((0,1)\).
In our example, the function \( f(x) = \frac{x}{2} \) fulfills the contraction condition with \( k = \frac{1}{2} \).We demonstrate this via the calculation \[ d(f(x), f(y)) = \frac{1}{2} d(x, y). \] This contraction property allows us to analyze functions' behavior on the defined metric space.

fixed point theorem

The fixed point theorem is a fundamental principle in mathematics. It states that, under certain conditions, a function will have one or more fixed points. A fixed point is where \( f(x) = x \). For contraction mappings, the Banach fixed-point theorem guarantees a unique fixed point in a complete metric space. It requires the metric space to be complete, meaning every Cauchy sequence converges within that space.
However, for non-complete spaces, such as the interval \( (0, 1) \), a fixed point might not exist. In our example, our function \( f(x) = \frac{x}{2} \) does not have a fixed point within the interval \( (0, 1) \). Solving \( f(x) = x \rightarrow \frac{x}{2} = x \), we get \( x = 0 \), which is outside the interval. Thus, no fixed point exists.

Euclidean metric

The Euclidean metric is a way to measure distance. It is used for describing the geometry of our surroundings. Formally defined, the Euclidean distance between two points \((x,y)\) in a space \( R^{n} \) is: \[ d(a, b) = \sqrt{(a_{1} - b_{1})^{2} + (a_{2} - b_{2})^{2} + \ldots + (a_{n} - b_{n})^{2}}. \]In simple terms, it is the straight-line distance between points. For our problem, considering the space \( (0,1) \), the Euclidean metric is \[ d(x, y) = |x - y|. \] This metric helps us to discuss convergence and distance between points in a clear, understandable manner.

metric space completeness

A metric space is called complete if every Cauchy sequence converges within the space itself. A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. For example, in the space (0, 1) with the Euclidean metric, some sequences tend to approach the boundary points 0 or 1, which are not included in the space. This makes the space non-complete.
Completeness is important because many mathematical results, such as the Banach fixed-point theorem, require the metric space to be complete. In our example, (0,1) is not complete. Therefore, even though the function
\( f(x) = \frac{x}{2} \) is a contraction, it doesn't guarantee a fixed point within (0,1). Understanding these properties helps us analyze more complex mathematical structures and set clearer expectations about the behavior of sequences and functions.