Problem 8 - Basic Analysis: Introduction to Real Analysis

Question

Suppose \(f: X \rightarrow X\) is a contraction for \(k<1 .\) Suppose you use the iteration procedure with \(x_{n+1}:=f\left(x_{n}\right)\) as in the proof of the fixed point theorem. Suppose \(x\) is the fixed point of \(f .\) a) Show that \(d\left(x, x_{n}\right) \leq k^{n} d\left(x_{1}, x_{0}\right) \frac{1}{1-k}\) for all \(n \in \mathbb{N}\). b) Suppose \(d\left(y_{1}, y_{2}\right) \leq 16\) for all \(y_{1}, y_{2} \in X,\) and \(k=1 / 2 .\) Find an \(N\) such that starting at any point \(x_{0} \in X,\) \(d\left(x, x_{n}\right) \leq 2^{-16}\) for all \(n \geq N\)

Step-by-Step Solution

Answer

a) Use contraction to show the bound. b) Choose \( N = 16 \).

Step 1: Understand the Contraction Mapping

In a contraction mapping, there’s a constant \( k < 1 \) such that for any two points \( x, y \in X \), the distance \( d(f(x), f(y)) \leq k \cdot d(x, y) \). This means the function \( f \) brings points closer together with each iteration.

Step 2: Initial Setup for the Iteration

The sequence is defined as \( x_{n+1} = f(x_n) \). Our goal is to find a bound for \( d(x, x_n) \), where \( x \) is the fixed point such that \( f(x) = x \).

Step 3: Use the Contraction Property to Bound Distances

We begin by showing \( d(x, x_{n+1}) \leq k \cdot d(x, x_n) \). This is due to the property \( d(f(x), f(x_n)) \leq k \cdot d(x, x_n) \) and since \( f(x) = x \), we have \( d(x, x_{n+1}) = d(f(x), f(x_n)) \leq k \cdot d(x, x_n) \).

Step 4: Bound the Distance Using Geometric Series

By induction or iterating the contraction property: \( d(x, x_n) \leq k^n d(x_1, x_0) \). Recognizing it as a geometric series, the sum becomes \( d(x, x_0) \cdot \sum_{i=0}^{n-1} k^i \). The sum of an infinite geometric series \( \sum_{i=0}^{\infty} k^i = \frac{1}{1-k} \), so the distance can be bounded by \( d(x, x_n) \leq k^n d(x_1, x_0) \cdot \frac{1}{1-k} \).

Step 5: Apply Given Conditions for Part b

For part b, we have \( d(y_1, y_2) \leq 16 \) and \( k = \frac{1}{2} \). We need \( d(x, x_n) \leq 2^{-16} \), set \( k^N d(x_1, x_0) \cdot \frac{1}{1-k} = 2^{-16} \). Solving for \( N \) gives \( N = \lceil 16 \rceil \), since \( k=\frac{1}{2} \) leads to \( \left(\frac{1}{2}\right)^N \). The series converges rapidly, and using these approximations, \( N = 16 \) should suffice for \( 2^{-16} \).

Key Concepts

Fixed Point Theorem

The Fixed Point Theorem is an important concept in mathematics, particularly in the context of contraction mappings. A fixed point for a function \( f \) is a value \( x \) such that \( f(x) = x \). In simpler terms, this means that when \( f \) is applied to \( x \), it gives back \( x \) unchanged. This theorem assures us of the existence of such a point under certain conditions.

In the specific case of contraction mappings, if \( f: X \rightarrow X \) is a contraction, then it guarantees that there is exactly one fixed point for \( f \). This unique existence is crucial because it provides a predictable outcome in iterative processes.

• Contraction means that distances between points decrease by a consistent factor \( k \)
• This shrinking effect results in the sequence generated by repeated application of \( f \) converging to the fixed point
• The condition \( k < 1 \) ensures that \( f \) brings any two points closer together

Understanding the fixed point is essential because it indicates stability in systems or iterations where convergence is desired.

Distance Bounding

Distance Bounding helps us understand how close our iterative sequence is to the fixed point. In a contraction mapping \( f \), the distance between the fixed point \( x \) and the current position \( x_n \) can be effectively bounded. The bound is given by \( d(x, x_n) \leq k^n d(x_1, x_0) \frac{1}{1-k} \). This equation provides a way to anticipate how rapidly a sequence is converging to its fixed point.

In practical terms, you are given:

• \( k^n \): The contraction factor raised to the power of the iteration number \( n \)
• \(d(x_1, x_0)\): The initial distance between the starting points of the sequence
• \(\frac{1}{1-k}\): The factor derived from the infinite geometric series

By calculating this bound, mathematicians or engineers can determine at what point the approximation of \( x_n \) is sufficiently close to the fixed point \( x \) for practical purposes.

Geometric Series

Understanding geometric series is key in estimating distances in contraction mappings. A geometric series is a series of terms in which each term is a constant multiple, \( k\), of the preceding one. In our context, the sum \( \sum_{i=0}^{n-1} k^i \) represents the cumulative effect of these iterations when bounding the total distance to the fixed point.

The magic of geometric series is in its convergence properties. When the ratio \( k \) is less than 1, the series \( \sum_{i=0}^{\infty} k^i \) converges to \( \frac{1}{1-k} \). This allows us to replace an infinite sequence with a finite, manageable constant for iterations:

• Each term \( k^i \) represents the diminishing distances through iterations
• The series sum \( \frac{1}{1-k} \) bounds total distance over infinite iterations

This concept simplifies the complex reality of convergence by providing clear expectations based on the properties of \( k \).

Iteration Procedure

In mathematics, the Iteration Procedure is a tool used to iteratively apply a function to approximate a solution. Consider a sequence defined by \( x_{n+1} = f(x_n) \). It incrementally gets closer to the fixed point \( x \) of \( f \). The power of this procedure lies in its simplicity and effectiveness at solving problems iteratively.

How it works:

• The sequence starts from an initial guess \( x_0 \)
• Each subsequent \( x_{n+1} \) is obtained by applying the function \( f \) to the current point \( x_n \)
• Convergence is assured due to the contraction property, depending on the constant \( k \)

The iteration procedure is fundamental in numerical analysis because it builds upon prior approximations, leading progressively to the solution, i.e., the fixed point. By selecting a suitable starting point and leveraging the contraction mapping properties, we can ensure the distance between successive iterations \(d(x, x_n)\) minimizes progressively toward zero, indicating stability and solution accuracy.