Problem 4 - Basic Analysis: Introduction to Real Analysis

Question

Let \(F: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(F(x):=k x+b\) where \(0

Step-by-Step Solution

Answer

a) Yes, \( F \) is a contraction. b) The fixed point is \( \frac{b}{1-k} \) and it is unique.

Step 1: Define Contraction Mapping

A function \( F \) is a contraction if there exists a constant \( 0 < c < 1 \) such that for all \( x, y \in \mathbb{R} \), we have \( |F(x) - F(y)| \leq c|x - y| \). Here, \( c = k \), since \( F(x) = kx + b \).

Step 2: Verify the Contraction Condition

Calculate the absolute difference for \( F(x) \) and \( F(y) \): \( |F(x) - F(y)| = |kx + b - (ky + b)| = |k(x-y)| = k|x-y| \). Since \( 0 < k < 1 \), \( k|x-y| < |x-y| \), satisfying the contraction condition.

Step 3: Determine the Fixed Point

A fixed point \( x^* \) satisfies the equation \( F(x^*) = x^* \). Substitute \( F(x) \) into the equation: \( kx^* + b = x^* \). Rearrange: \( kx^* - x^* = -b \), i.e., \((k-1)x^* = -b \). Thus, the fixed point \( x^* = \frac{b}{1-k} \).

Step 4: Verify Uniqueness of Fixed Point

Assume two fixed points exist, \( x^* \) and \( x^{**} \). Then \( F(x^*) = x^* \) and \( F(x^{**}) = x^{**} \). Subtract to get \( k(x^* - x^{**}) = x^* - x^{**} \). Given \( 0 < k < 1 \), this implies \( x^* - x^{**} = 0 \), meaning \( x^* = x^{**} \), thus proving the uniqueness.

Key Concepts

Fixed Point Theorem

The Fixed Point Theorem is an essential concept in analysis, especially when dealing with contraction mappings. The theorem states that if you have a contraction mapping on a complete metric space, there is exactly one point (a fixed point) that remains unchanged by the mapping.
For our function \( F(x) = kx + b \) with \( 0 < k < 1 \) and \( b \in \mathbb{R} \), it acts as a contraction on the real numbers. This means that as you apply the function repeatedly, points move closer together, eventually converging to a single fixed point.
Understanding the uniqueness of the fixed point is crucial. Assume \( F \) has two fixed points, say \( x^* \) and \( x^{**} \). Then, \( F(x^*) = x^* \) and \( F(x^{**}) = x^{**} \). By performing subtraction and leveraging the contraction condition (\( 0 < k < 1 \)), you can demonstrate that \( x^* = x^{**} \), thereby proving the fixed point is unique.

Real Analysis

Real Analysis is a branch of mathematics focusing on real numbers and real-valued sequences and functions. It provides the rigor needed to understand calculus and many other areas of mathematics.
Within Real Analysis, contraction mappings are used to explore behavior and convergence of sequences. They also help to prove results like the existence and uniqueness of fixed points.
A contraction mapping in this context involves a function \( F: \mathbb{R} \to \mathbb{R} \) that brings points closer together. By using concepts such as metric spaces and limits, Real Analysis allows us to generalize and make powerful conclusions about functions like \( F(x) = kx + b \). The contraction property, combined with the completeness of the real numbers, ensures that a fixed point exists and is unique.

Function Properties

Understanding function properties is central to analyzing and applying theorems in mathematics such as the Fixed Point Theorem. Let's consider the properties of the function \( F(x) = kx + b \).

• Linearity: The function is linear, which means it forms a straight line when graphed. This property simplifies analysis as linear functions are well-understood and predictable.
• Slope and Contraction: Here, the function's slope is \( k \), making \( F \) a contraction mapping since \( 0 < k < 1 \). The slope determines how much "shrinkage" there is in the distance between two points after applying the function.
• Fixed Point: The linear form allows us to easily find the fixed point. Solving \( kx + b = x \) gives the fixed point \( x^* = \frac{b}{1-k} \).
• Uniqueness: Function properties also guide us to demonstrate that this fixed point is unique, using the contraction principle and assuming different fixed points leads to a contradiction.

By examining these properties, one gains deeper insight into the behavior of the function and can apply it effectively in solving problems in Real Analysis.