Let E={x ∈ℝ: 0 ≤x ≤1} and let f(x)=(1+x^2) / 2. Show that: (a) f maps E into E. (b) There does n

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Let E={x ∈ℝ: 0 ≤x ≤1} and let f(x)=(1+x^2) / 2. Show that:...

Let $E=\{x \in \mathbb{R}: 0 \leq x \leq 1\}$ and let $f(x)=\left(1+x^2\right) / 2$. Show that: (a) $f$ maps $E$ into $E$. (b) There does not exist a positive constant $\gamma<1$ such that $|f(b)-f(a)| \leq \gamma|b-a|$ for every choice of $a, b \in E$. (c) $f$ has exactly one fixed point $x_* \in E$.

Let $E=\{x \in \mathbb{R}: 0 \leq x \leq 1\}$ and let $f(x)=\left(1+x^2\right) / 2$. Show that:
(a) $f$ maps $E$ into $E$.
(b) There does not exist a positive constant $\gamma<1$ such that $|f(b)-f(a)| \leq \gamma|b-a|$ for every choice of $a, b \in E$.
(c) $f$ has exactly one fixed point $x_* \in E$.

Key Concepts

Mapping on a Closed Interval

This concept refers to the idea that a function can be defined from a subset of the real numbers (often a closed interval) into itself. It involves showing that for every element x in the domain, the function's value f(x) also lies within the same set. This property is crucial for many fixed point theorems and other results in real analysis, as it ensures the function's iterates remain bounded within the set.

Lipschitz Condition and Contraction Mappings

A function is said to satisfy a Lipschitz condition if there exists a constant ? such that the difference between its values at any two points is at most ? times the difference between those points. When this constant ? is less than one, the function qualifies as a contraction mapping. Contraction mappings are important because they have strong convergence properties; however, if the best possible Lipschitz constant is not strictly less than 1, then contraction mapping techniques (like those used in the Banach Fixed Point Theorem) do not directly apply.

Fixed Points and Their Uniqueness

A fixed point of a function is an element in the domain where the function returns the same value, that is, f(x) = x. Establishing the existence and uniqueness of a fixed point is a central topic in analysis. Even if a function is not a contraction mapping, one can sometimes show by other means that exactly one fixed point exists. The uniqueness of the fixed point is critical for ensuring well-defined limit behaviors in iterative processes and dynamical systems.

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