Existence and Uniqueness of Solutions to Systems of Differential Equations
Subjects: Ordinary Differential Equations Existence of Solutions of First Order Differential Equations , Second Order Linear Differential Equations , nth Order Linear Differential Equations , Banach's Fixed Point Theorem
Local Existence
Let You can't use 'macro parameter character #' in math mode f:D = [x_0-a, x_0+a] \times B_b(y_0) \subseteq \Bbb R\times \Bbb C { #n } \to \Bbb C^n f:D = [x_0-a, x_0+a] \times B_b(y_0) \subseteq \Bbb R\times \Bbb C { #n } \to \Bbb C^n D M > 0 ∥ f ∥ ≤ M D ( ϕ k ) I = [ x 0 − α , x 0 + α ] α = min { a , b / M }
o n $ I $ . A d d i t i o n a l l y |\phi -\phi_k | \le \frac{M}{K}\frac{(K\alpha)^{k+1}}{(k+1)!}e^{K\alpha}
You can't use 'macro parameter character #' in math mode ### Non-local Existence of Solutions Let $S=[x_0-a, x_0+a] \times \Bbb C^n$, and $f:S \to \Bbb C^n$ be Lipschtiz continuous with constant ${K>0}$. The successive approximations $(\phi_k)$ for the problem ### Non-local Existence of Solutions Let $S=[x_0-a, x_0+a] \times \Bbb C^n$, and $f:S \to \Bbb C^n$ be Lipschtiz continuous with constant ${K>0}$. The successive approximations $(\phi_k)$ for the problem y' = f(x, y) \quad y(x_0) =y_0
e x i s t s o n t h e e n t i r e i n t e r v a l $ [ x 0 − a , x 0 + a ] $ , a n d c o n v e r g e t h e r e t o s o l u t i o n $ ϕ $ o f t h e i n i t i a l v a l u e p r o b l e m . ∗ ∗ C o r : ∗ ∗ S u p p o s e $ f : R × C n → C n $ b e a c o n t i n u o u s f u n c t i o n o n t h e p l a n e w h i c h s a t i s f i e s t h e L i p s c h i t z c o n d i t i o n o n t h e s t r i p $ S a = [ − a , a ] × R $ , w h e r e $ a > 0 $ . T h e n e v e r y i n i t i a l v a l u e p r o b l e m y' = f(x, y) \quad y(x_0) =y_0
You can't use 'macro parameter character #' in math mode 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 $\phi$ and $\psi$ be solutions of $$y' = f(x, y) \quad y(x_0) =y_1 $$$$y' = g(x, y) \quad y(x_0) =y_2 has a solution which exists for all $x$ ### Approximations and uniqueness Let $f, g$ be continuous functions on ${R= [x_0-a,x_0+a]\times[y_0-b, y_0+b]}$, and suppose $f$ satisfies the Lipschitz condition with a Lipschitz constant $K$. Let $\phi$ and $\psi$ be solutions of $$y' = f(x, y) \quad y(x_0) =y_1 $$$$y' = g(x, y) \quad y(x_0) =y_2 respectively on an interval I x 0 R ε , δ ≥ 0 ∥ f − g ∥ ≤ ε R | y 1 − y 2 | ≤ δ
∥ ϕ ( x ) − ψ ( x ) ∥ ≤ δ e K | x − x 0 | + ε K ( e K | x − x 0 | − 1 ) for all x I
y ′ = f ( x , y ) y ( x 0 ) = y 0 has at most one solution any interval containing x 0
Linear sytems of equations
Consider the linear system
y ′ = f ( x , y ) where the components of f
f j ( x , y ) = ∑ k = 1 n a j k ( x ) y k + b j ( x ) and the functions a j k b j I x 0 y 0 C n ϕ
y ′ = f ( x , y ) y ( x 0 ) = y 0 on I
Let f : D = [ x 0 − a , x 0 + a ] × B b ( y 0 ) ⊆ R × C n → C n N > 0 ∥ f ∥ ≤ N D L > 0
∥ f ( x , y ) − f ( x , z ) ∥ ≤ L ∥ y − z ∥ for all ( x , y ) , ( x , z ) ∈ D
y ( n ) = f ( x , y , y ′ , … , y ( n − 1 ) ) on the interval I = [ x 0 − α , x 0 + α ] α = min { a , b / M } M = N + b + ∥ y 0 ∥ )
ϕ ( x 0 ) = α 1 , ϕ ′ ( x 0 ) = α 2 , ⋯ , ϕ ( n − 1 ) ( x 0 ) = α n $ $ y_0 =(\alpha_1,\alpha_2, \dots, \alpha_n)
You can't use 'macro parameter character #' in math mode ### Linear $n$th order equations Let $a_1, \dots, a_n, b: I \to \Bbb C$ continuous functions, and $I$ an interval containing $x_0$. If ${\alpha_1, \dots, \alpha_n \in \Bbb C}$ are constants, there exists a unique solution $\phi$ of the equation ### Linear $n$th order equations Let $a_1, \dots, a_n, b: I \to \Bbb C$ continuous functions, and $I$ an interval containing $x_0$. If ${\alpha_1, \dots, \alpha_n \in \Bbb C}$ are constants, there exists a unique solution $\phi$ of the equation y^{(n)}+a_1(x)y^{(n-1)} + \cdots +a_n(x) y = b(x)
o n $ I $ s a t i s f y i n g \phi(x_0) = \alpha_1, \quad \phi'(x_0)= \alpha_2, \quad \cdots, \phi^{(n-1)}(x_0) = \alpha_n
You can't use 'macro parameter character #' in math mode ### Other Version Let $V \subseteq \Bbb R^r$ and $U \subseteq \Bbb R^n$ be open, let $c > 0$, let $f^i \in \mathcal C^\infty((-c, c) \times V \times U)$ for $1 \le i \le n$, and consider the system of ordinary differential equations with parameters $b = (b^1, \dots, b^r) \in V$ $$\frac{dx^i}{dt} = f^i(t, b, x^1(t, b), \dots, x^n(t, b)), \qquad 1 \le i \le n.$$If $a = (a^1, \dots, a^n) \in U$, there are smooth functions $x^i(t, b)$ for $1 \le i \le n$, defined on some nondegenerate interval $[-\delta, \varepsilon]$ about $0$, that satisfy the system above and the initial condition $$x^i(0, b) = a^i, 1 \le i \le n.$$Furthermore, if the functions $\tilde x^i(t, b), 1 \le i \le n$, give another solution defined on $[-\tilde \delta, \tilde \varepsilon]$ and satisfying the same initial conditions then these solutions agree on $[-\tilde \delta, \tilde \varepsilon]\cap [-\delta, \varepsilon]$. Finally, if we write these solution as $x^i = x^i(t, b, a)$, there is a neighbourhood $W$ of $a$ in $U$, a neighbourhood $B$ of $b$ in $V$, and a choice of $\varepsilon >0$ such that the solutions $x^i(t, z, x)$ are dined and smooth on the open set $(-\varepsilon, \varepsilon) \times B \times W\subseteq \Bbb R^{n+r+1}$. ### Other Version Let $V \subseteq \Bbb R^r$ and $U \subseteq \Bbb R^n$ be open, let $c > 0$, let $f^i \in \mathcal C^\infty((-c, c) \times V \times U)$ for $1 \le i \le n$, and consider the system of ordinary differential equations with parameters $b = (b^1, \dots, b^r) \in V$ $$\frac{dx^i}{dt} = f^i(t, b, x^1(t, b), \dots, x^n(t, b)), \qquad 1 \le i \le n.$$If $a = (a^1, \dots, a^n) \in U$, there are smooth functions $x^i(t, b)$ for $1 \le i \le n$, defined on some nondegenerate interval $[-\delta, \varepsilon]$ about $0$, that satisfy the system above and the initial condition $$x^i(0, b) = a^i, 1 \le i \le n.$$Furthermore, if the functions $\tilde x^i(t, b), 1 \le i \le n$, give another solution defined on $[-\tilde \delta, \tilde \varepsilon]$ and satisfying the same initial conditions then these solutions agree on $[-\tilde \delta, \tilde \varepsilon]\cap [-\delta, \varepsilon]$. Finally, if we write these solution as $x^i = x^i(t, b, a)$, there is a neighbourhood $W$ of $a$ in $U$, a neighbourhood $B$ of $b$ in $V$, and a choice of $\varepsilon >0$ such that the solutions $x^i(t, z, x)$ are dined and smooth on the open set $(-\varepsilon, \varepsilon) \times B \times W\subseteq \Bbb R^{n+r+1}$.