Existence of Solutions of First Order Differential Equations

Existence Theorem

Let $f$ be a continuous real valued function on the rectangle $R = [x_{0} - a, x_{0} + a] \times [y_{0} - b, y_{0} + b]$ , and let $∥ f ∥ \leq M$ . Let $f$ be Lipschitz continuous with a Lipschitz constant $K$ in $R$ . Then the succesive approximations

ϕ_{0} = y_{0}, ϕ_{k + 1} (x) = y_{0} + \int_{x_{0}}^{x} f (t, ϕ_{k} (t)) d t

converge on the interval $I = [x_{0} - α, x + α]$ , with $α = min {a, b / M}$ to a solution $ϕ$ of the initial value problem

y^{'} = f (x, y) y (x_{0}) = y_{0}

on $I$ .

Th: The $k$ th succesive approximation of the Picard iteratives $(ϕ_{k})$ to the solution $x$ of the initial value problem satisfies

∥ ϕ - ϕ_{k} ∥ \leq \frac{M}{K} \frac{(K α)^{k + 1}}{(k + 1)!} e^{K α}

Non-local Existence of Solutions

Let $S = [x_{0} - a, x_{0} + a] \times R$ , and $f : S \to R$ be Lipschtiz continuous with constant $K > 0$ . The succesive approximations $(ϕ_{k})$ for the problem

y^{'} = f (x, y) y (x_{0}) = y_{0}

exists on the entire interval $[x_{0} - a, x_{0} + a]$ , and converge there to solution $ϕ$ of the initial value problem.

Cor: Suppose $f : R^{2} \to R$ be a continuous function on the plane which satisfies the Lipschitz condtion on the strip $S_{a} = [- a, a] \times R$ , where $a > 0$ . Then every initial value problem

y^{'} = f (x, y) y (x_{0}) = y_{0}

has a solution which exists for all $x$

Approximations and uniqueness

Let $f, g$ be continuous functions on $R = [x_{0} - a, x_{0} + a] \times [y_{0} - b, y_{0} + b]$ , and suppose $f$ satisfies the Lipschitz condition with a Lipschitz constant $K$ . Let $ϕ$ and $ψ$ be solutions of

y^{'} = f (x, y) y (x_{0}) = y_{1} y^{'} = g (x, y) y (x_{0}) = y_{2}

respectively on an interval $I$ containing $x_{0}$ , with graphs contained in $R$ . Additionally $∥ f - g ∥ \leq ε$ on $R$ , and $| y_{1} - y_{2} | \leq δ$ . Then

| ϕ (x) - ψ (x) | \leq δ e^{K | x - x_{0} |} + \frac{ε}{K} (e^{K | x - x_{0} |} - 1)

for all $x$ in $I$ .

Cor (Uniqueness Theorem): Let $f$ be continuous and satisfy a Lipschitz conditions on $R$ . If $ϕ$ and $ψ$ are solutions of

y^{'} = f (x, y) y (x_{0}) = y_{0}

on an interval $I$ containing $x_{0}$ , then $ϕ = ψ$ on $I$ .

Cor: : Let $f$ be continuous and satisfy a Lipschitz conditions on $R$ . Let $(g_{k})$ be a sequence continuous functions on $R$ , such that

∥ f - g_{k} ∥ \leq ε_{k}

and $ψ_{k}$ be the solution to the initial value problem on the interval $I$ containing $x_{0}$

y^{'} = g_{k} (x, y) y (x_{0}) = y_{k}

and $ϕ$ be the solution to the initial value problem on the interval $I$ containing $x_{0}$

y^{'} = f (x, y) y (x_{0}) = y_{0}

and $| y_{k} - y_{0} | \leq δ_{k}$ . If $ε_{k}, δ_{k} \to 0$ as $k \to \infty$ . Then $ψ \to ϕ$ on $I$ .