Table of Contents

Preliminary Definitions for The Theory of First Order ODEs
Before we move on to some of the theory regarding first order ordinary differential equations we will need to state some important definitions from real analysis.
Pointwise and Uniform Convergence of Sequences of Functions
Definition: Let $(f_m)$ be a sequence of realvalued functions defined on $D \subseteq \mathbb{R}^n$. Then $(f_m)$ is said to Converge Pointwise to the function $f$ on $D$ if for all $\epsilon > 0$ and for all $x \in D$ there exists an integer $M$ such that if $m \geq M$ we have that $ f_m(x)  f(x)  < \epsilon$. 
Definition: Let $(f_m)$ be a sequence of realvalued functions defined on $D \subseteq \mathbb{R}^n$. Then $(f_m)$ is said to Converge Uniformly to the function $f$ on $D$ if for all $\epsilon > 0$ there exists an integer $M$ such that if $m \geq M$ we have that $ f_m(x)  f(x)  < \epsilon$ for all $x \in D$. 
The definition for a sequence of functions $(f_m)$ to converge pointwise to $f$ on $D$ and to converge uniformly to $f$ on $D$ are very similar but still DIFFERENT!
Note that $(f_m)$ converging pointwise to $f$ means that for all $\epsilon > 0$, for each individual $x \in D$ we can make the difference $ f_m(x)  f(x)  < \epsilon$ by choosing an integer $M$ sufficiently large. On the other hand, $(f_m)$ converging uniformly to $f$ means that for all $\epsilon > 0$ we can make the difference $ f_m(x)  f(x)  < \epsilon$ for ALL $x \in D$ by choosing an integer $M$ sufficiently large. So for pointwise convergence, the choice of $M$ is dependent on both $\epsilon$ and $x$, while for uniform convergence, the choice of $M$ is dependent on only $\epsilon$. Hence pointwise convergence is a "weaker" property" compared to uniform convergence.
Cauchy and Uniformly Cauchy Sequences of Functions
Definition: Let $(f_m)$ be a sequence of realvalued functions defined on $D \subseteq \mathbb{R}^n$. Then $(f_m)$ is said to be Cauchy if for all $\epsilon > 0$ and for all $x \in D$ there exists an integer $M$ such that if $m, n \geq M$ we have that $ f_m(x)  f_n(x)  < \epsilon$. 
Definition: Let $(f_m)$ be a sequence of realvalued functions defined on $D \subseteq \mathbb{R}^n$. Then $(f_m)$ is said to be Uniformly Cauchy if for all $\epsilon > 0$ there exists an integer $M$ such that if $m, n \geq M$ we have that $ f_m(x)  f_n(x)  < \epsilon$ for all $x \in D$. 
Like with the remarks made above, the concept of a sequence $(f_m)$ being Cauchy is a weaker property than that of being uniformly Cauchy.
Theorem 1: Let $(f_m)$ be a sequence of realvalued functions that are continuous and defined on a compact set $D \subseteq \mathbb{R}^n$. Then $(f_m)$ is uniformly Cauchy on $D$ if and only if there exists a continuous function $f$ defined on the compact set $D$ such that $(f_m)$ converges uniformly to $f$ on $D$. 
Uniformly Bounded Sets of Functions
Definition: Let $\mathcal F$ be a collection of functions defined on $D \subseteq \mathbb{R}^n$. Then $\mathcal F$ is said to be Uniformly Bounded on $D$ if there exists an $M > 0$ such that $ f(x)  \leq M$ for all $x \in D$ and for all $f \in \mathcal F$. 
For example, consider the following collection of functions which we define on $[0, 1]$:
(1)It is not hard to show that $\mathcal F$ is uniformly bounded. Take $M = 1$. Then $\displaystyle{\biggr \lvert \frac{1}{n}x \biggr \rvert = \biggr \lvert \frac{1}{n} \biggr \rvert  x  \leq \frac{1}{n} \leq M = 1}$ for all $x \in [0, 1]$ and for all $f_n \in \mathcal F$.
Equicontinuous Sets of Functions
Definition: Let $\mathcal F$ be a collection of functions defined on $D \subseteq \mathbb{R}^n$. Then $\mathcal F$ is said to be Equicontinuous on $D$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $ x  y  < \delta$ then $ f(x)  f(y)  < \epsilon$ for all $x, y \in D$ and for all $f \in \mathcal F$. 
Lipschitz Conditions on Continuous Functions
Definition: Let $f \in C(D, \mathbb{R})$. Then $f$ is said to satisfy a Lipschitz Condition on $D$ if there exists an $L > 0$ such that for all $(t, x), (t, y) \in D$ we have that $ f(t, x)  f(t, y)  \leq L x  y$. The constant $L$ is called a Lipschitz Constant, and $f(t, x)$ is said to be Lipschitz Continuous in the variable $x$. 
Contraction Mappings
Definition: Let $(X, d)$ be a metric space. A Contraction Mapping on this metric space is a function $T : X \to X$ with the property that there exists a $k$ with $0 < k < 1$ with $d(T(x), T(y)) \leq kd(x, y)$ for all $x, y \in X$. 
Fixed Points
Definition: Let $(X, d)$ be a metric space and let $T : X \to X$ be a contraction mapping. A Fixed Point $x^* \in X$ is a point with the property that $T(x^*) = x^*$. 
Banach's Fixed Point Theorem
Theorem 1: Let $(X, d)$ be a complete metric space and let $T : X \to X$ be a contraction mapping. Then $T$ has a unique fixed point $x^* \in X$. 
Recall that a metric space $(X, d)$ is a said to be complete if every Cauchy sequence in $X$ converges to a point in $X$.