The Banach Contraction Principle and Picard's Theorem

Let $p$ be a polynomial. Show that $p: \mathbb {R} \rightarrow \mathbb {R}$ is Lipschitz if and only if the degree of $p$ is less than 2.

Fix $\alpha > 0$, define $f(x) = \alpha x(1-x)$ for $x$ in $[0,1]$.

Does a mapping of a metric space $X$ into itself that is Lipschitz with Lipschitz constant less than 1 necessarily have a fixed-point?

Does a mapping of a complete metric space into itself that is Lipschitz with Lipschitz constant 1 necessarily have a fixed-point?

Let $X$ be a compact metric space and $T$ a mapping from $X$ into itself such that

Show that $T$ has a unique fixed-point.

Define $f(x) = \pi / 2+x-\arctan x$ for all real numbers $x$. Show that

Show that $f$ does not have a fixed-point. Does this contradict the preceding problem?

In Euclidean space $\mathbb {R}^{n}$ consider the closed unit ball $B = \left\{ x \in \mathbb {R}^{n} \mid \left\| x\right\| \leq 1\right\}$. Let $f$ map $B$ into $B$ and be Lipschitz with Lipschitz constant 1. Without using the Brouwer Fixed-Point Theorem, show that $f$ has a fixed-point.

Suppose that the mapping $f: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is a contraction. Define $g(x) = x-f(x)$ for all $x$ in $\mathbb {R}^{n}$. Show that the mapping $g: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is both one-to-one and onto. Also show that $g$ and its inverse are continuous.

Let $X$ be a complete metric space containing the point $x_{0}$ and let $r$ be a positive real number. Define $K = \left\{ x \text{ in } X \mid \rho \left(x, x_{0}\right) \leq r\right\}$. Suppose that the mapping $T: K \rightarrow X$ is Lipschitz with Lipschitz constant $c$. Suppose also that $c r+\rho \left(T\left(x_{0}\right), x_{0}\right) \leq r$. Prove that $T(K) \subseteq K$ and that $T: K \rightarrow X$ has a fixed-point.

Show that if the function $g: \mathbb {R}^{2} \rightarrow \mathbb {R}$ has continuous first-order partial derivatives, then for each point $\left(x_{0}, y_{0}\right)$ in $\mathbb {R}^{2}$ there is a neighborhood $\mathcal{O}$ of ($x_{0}, y_{0}$) on which the Lipschitz assumption (7) holds.

In case the function $g: \mathcal{O} \rightarrow \mathbb {R}$ has the form $g(x, y) = h(x)+b y$, where the function $h: \mathbb {R} \rightarrow \mathbb {R}$ is continuous, prove that the following is an explicit formula for the solution of (5):

Consider the differential equation

Show that the function $f: \mathbb {R} \rightarrow \mathbb {R}$ that is identically 0 is a solution and the function $f: \mathbb {R} \rightarrow \mathbb {R}$ defined by $f(x) = 0$, if $x < 0$ and $f(x) = x^{3}$, if $x \geq 0$, is also a solution. Does this contradict the Picard Existence Theorem?

For a positive number $\epsilon$, consider the differential equation

Show that on the interval $I = (-\epsilon (\pi / 2), \epsilon (\pi / 2))$ there is a unique solution of this differential equation that is defined by $f(x) = \tan (x / \epsilon )$ and there is no solution in an interval strictly containing $I$.

Let $I$ be an open interval in $\mathbb {R}$ and suppose that the function $h: I \rightarrow \mathbb {R}$ has the property that there are points $x_{1} < x_{2}$ in $I$ and a number $c$ such that $h\left(x_{1}\right) < c < h\left(x_{2}\right)$ but $c$ does not belong to $h(I)$. Prove that there is no solution to the differential equation (5) by arguing that if $f: I \rightarrow \mathbb {R}$ is a solution, then the continuous function $f(x)-c x$ fails to attain a minimum value on the interval $\left[x_{1}, x_{2}\right]$.

Use the preceding exercise to prove that if $I$ is an open interval in $\mathbb {R}$ and the function $f: I \rightarrow \mathbb {R}$ is differentiable, then the image of the derivative $f^{\prime }: I \rightarrow \mathbb {R}$ is an interval.

State and prove a form of the Picard Existence Theorem for systems of differential equations in the following context: $\mathcal{O}$ is an open subset of $\mathbb {R} \times \mathbb {R}^{n}, \mathbf{g}: \mathcal{O} \rightarrow \mathbb {R}^{n}$ is continuous, the point ($x_{0}, \mathbf{y}_{0}$) is in $\mathcal{O}$, and the system of differential equations is