From 019257f00824ec34b6d957e9c5fffe9f55ab94c1 Mon Sep 17 00:00:00 2001 From: Robert Date: Thu, 1 Apr 2021 18:44:29 +0200 Subject: [PATCH] Finished ODEs --- chapters/ode.tex | 1 + chapters/sections/linear_odes.tex | 574 ++++++++++++++++++++++++++ chapters/sections/picard_lindelof.tex | 2 +- script.pdf | Bin 1159169 -> 1198007 bytes script.tex | 1 + 5 files changed, 577 insertions(+), 1 deletion(-) create mode 100644 chapters/sections/linear_odes.tex diff --git a/chapters/ode.tex b/chapters/ode.tex index c35ba99..4e45304 100644 --- a/chapters/ode.tex +++ b/chapters/ode.tex @@ -8,4 +8,5 @@ \subfile{sections/solution_methods.tex} \subfile{sections/picard_lindelof.tex} + \subfile{sections/linear_odes.tex} \end{document} \ No newline at end of file diff --git a/chapters/sections/linear_odes.tex b/chapters/sections/linear_odes.tex new file mode 100644 index 0000000..ce1b66e --- /dev/null +++ b/chapters/sections/linear_odes.tex @@ -0,0 +1,574 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section{Linear Differential Equation Systems} + +\begin{defi} + Let $I$ be an open interval, and $A: I \rightarrow \realn^{n \times n}$, $b: I \rightarrow \realn^n$. + Then the ODES + \[ + y' = A(x)y + b(x) + \] + is said to be a linear differential equation system. If $b$ is the zero function, then the system is homogeneous (otherwise it's inhomogeneous). + If $A(x) = \const.$, then the system is said to have constant coefficients. +\end{defi} + +\begin{rem} + \begin{enumerate}[(i)] + \item By using substitution we can transform the equation + \[ + y^{(n)} = a_{n-1}(x) y^{(n-1)} + a_{n-2}(x) y^{(n-2)} + \cdots + a_0 y + b(x) + \] + into the system + \begin{align*} + y_{n-1}' &= a_{n-1}(x) y_{n-2} + a_{n-2}(x) y_{n-3} + \cdots + a_0 y + b(x) \\ + y_1 &= y' \\ + y_2 &= y_1' \\ + &~~\vdots \\ + y_{n-1} &= y_{n-2}' + \end{align*} + + \item Let $y, z$ be solutions of $y' = A(x) y + b(x)$, then $y - z$ is the solution of the related homogeneous equation $y' = A(x)y$. + This follows from + \begin{align*} + (y - z)'(x) &= A(x)y(x) + b(x) - (A(x)z(x) + b(x)) \\ + &= A(x) (y - z)(x) + \end{align*} + \end{enumerate} +\end{rem} + +\begin{lem}[Grönwall's Lemma] + Let $I$ be an open interval, $x_0 \in I$, $y: I \rightarrow [0, \infty)$ continuous, $a, b \ge 0$ and + \[ + y(x) \le a + b \abs{\int_{x_0}^x y(t) \dd{t}} + \] + Then + \[ + y(x) \le a e^{b\abs{x - x_0}} + \] +\end{lem} +\begin{proof} + Here we only prove $x > x_0$, but the proof for $x \le x_0$ works analogously. + Let $\epsilon > 0$ be arbitrary and choose + \begin{equation} + z(x) := a + \epsilon + b \int_{x_0}^x y(t) \dd{t} + \end{equation} + Then + \begin{equation} + z'(x) = by(x) \le bz(x) ~~\forall x \in I + \end{equation} + And since + \begin{equation} + z(t) \ge a + \epsilon > 0 + \end{equation} + we get + \begin{align} + \int_{x_0}^x \frac{z'(t)}{z(t)} \dd{t} &\le b(x - x_0) \\ + \int_{x_0}^x \frac{z'(t)}{z(t)} \dd{t} &= \ln(x) - \ln(z) + \end{align} + Due to the monotony of the exponential function + \begin{equation} + z(x) \le z(x_0) e^{b(x - x_0)} = (a + \epsilon) e^{b(x - x_0)} + \end{equation} + So + \begin{equation} + y(x) \le z(x) \le (a + \epsilon) e^{b(x - x_0)} \le a e^{b(x - x_0)} ~~\forall x \in I + \end{equation} +\end{proof} + +From now on $I$ will always be an open interval, and +\begin{align*} + A: I &\rightarrow \realn^{n \times n} \\ + b: I &\rightarrow \realn^n +\end{align*} +are continuous, $x_0 \in I$ and $y_0 \in \realn$. + +\begin{cor} + The IVP + \begin{align*} + y' = A(x)y + b(x) && y(x_0) = y_0 + \end{align*} + has a unique maximal solution that is defined on all of $I$. +\end{cor} +\begin{proof} + \begin{equation} + \begin{split} + f: I \times \realn^n &\longrightarrow \realn^n \\ + (x, y) &\longmapsto A(x) y + b(x) + \end{split} + \end{equation} + We need to show that $f$ fulfils a local Lipschitz condition in $y$. + Let $(x_1, y_1) \in I \times \realn^n$. Choose a compact $I_1$ such that $x_1 \in I_1 \subset I$. + Then $A(x)$ is bounded on $I_1$, i.e. + \begin{equation} + \exists L > 0: ~~\norm{A(x)} \le L ~~\forall x \in I_1 + \end{equation} + And then $\forall (x, y), (x, z) \in I_1 \times \realn^n$ + \begin{equation} + \norm{f(x, y) - f(x, z)} = \norm{A(x)(y - z)} \le \norm{A(x)}\norm{y - z} \le L \norm{y - z} + \end{equation} + So $f$ fulfils a local Lipschitz condition, and thus there exists a unique maximal solution. + Let $a, b \in \realn \cup \set{-\infty, \infty}$ such that $y: (a, b) \rightarrow \realn^n$ is the maximal solution. + Assume $b \in I$ (so $y$ isn't defined on all of $I$). Then there exists $M, K > 0$ such that $\norm{A(x)} \le M$ and $\norm{b(x)} \le K$ and $[x_0, b]$ and + \begin{equation} + \begin{split} + \norm{y(x)} = \norm{y_0 + \int_{x_0}^x y'(t) \dd{t}} &= \norm{y_0 + \int_{x_0}^x A(t)y(t) + b(t) \dd{t}} \\ + &\le \norm{y_0} + \int_{x_0}^x \norm{A(t)}\norm{y(t)} \dd{t} + \int_{x_0}^x \norm{b(t)} \dd{t} \\ + &\le \norm{y_0} + K(b - x_0) + M\int_{x_0}^x \norm{y(t)} \dd{t} + \end{split} + \end{equation} + Applying Grönwall's Lemma onto $\norm{y(t)}$ yields + \begin{equation} + \norm{y(x)} \le \left(\norm{y_0} + K(b - x_0)\right) e^{M\abs{x - x_0}} \le \left(\norm{y_0} + K(b - x_0)\right) e^{M(b - x_0)} + \end{equation} + and thus $y$ is bounded on $[x_0, b)$. + So none of the conditions from \Cref{thm:837} are satisfied, and therefore $b \notin I$. + This mean that $y$ is defined up to the right boundary of $I$. +\end{proof} + +\begin{rem} + One can show that for linear systems, the Picard iteration leads to a solution that converges on all of $I$. This would lead to an alternative proof. +\end{rem} + +\begin{cor} + Let $y, z: I \rightarrow \realn^n$ be solutions of the ODES + \[ + y' = A(x)y + b(x) + \] + Then the following are equivalent + \begin{enumerate}[(i)] + \item $y(x) = z(x) ~~\forall x \in I$ + \item $y(x_0) = z(x_0)$ + \item $y(x) = z(x) ~~\text{ for some } x \in I$ + \end{enumerate} +\end{cor} +\begin{proof} + $(i) \implies (ii)$, $(ii) \implies (iii)$ is trivial. + To prove $(iii) \implies (i)$, let $x_1 \in I$ such that $y_1 = y(x_1) = z(x_1)$. + Then $y, z$ are solutions to the IVP + \begin{align} + y' = A(x)y + b(c) && y(x_1) = y_1 + \end{align} + Since this problem has unique solutions + \begin{equation} + y = z + \end{equation} + must hold. +\end{proof} + +\begin{thm} + The solution set of the homogeneous ODES + \[ + y' = A(x)y + \] + so + \[ + V := \set[y'(x) = A(x) y(x) ~~\forall x \in I]{y: I \rightarrow \realn^n} + \] + is an $n$-dimensional linear subspace of $C^1(I, \realn^n)$. +\end{thm} +\begin{proof} + Proving that $V$ is a vector space is trivial. + So let $e_1, \cdots, e_n$ be a basis of $\realn^n$ and let $y_i$ be the unique solutions of the initial value problem + \begin{align*} + y' = A(x) y && y(x_0) = e_i ~~i \in \set{1, \cdots, n} + \end{align*} + Then $y_1, \cdots, y_n$ is a basis of $V$. + To prove their linear independence, let $\alpha_1, \cdots, \alpha_n \in \realn$ such that + \begin{equation} + \alpha_1 y_1 + \cdots + \alpha_n y_n = 0 + \end{equation} + then + \begin{equation} + \alpha_1 y_1(x_0) + \cdots + \alpha_n y_n(x_0) = \alpha_1 e_1 + \cdots + \alpha_n e_n = 0 + \end{equation} + Since the $e_1, \cdots, e_n$ are linear independent + \begin{equation} + \alpha_1 = \alpha_2 = \cdots = \alpha_n = 0 + \end{equation} + To prove that the $y_1, \cdots, y_n$ span $V$, set $z \in V$ and choose $\alpha_1, \dots, \alpha_n \in \realn$ such that + \begin{equation} + \alpha_1 e_1 + \alpha_2 e_2 + \cdots + \alpha_n e_n = z(x_0) + \end{equation} + Then the $z$ and $\alpha_1 y_1 + \cdots + \alpha_n y_n$ are maximal solutions of the ODES that are equal in $x_0$. Thus + \begin{equation} + z = \alpha_1y_1 + \cdots + \alpha_n y_n + \end{equation} +\end{proof} + +\begin{defi} + A basis $y_1, \cdots, y_n$ of $V$ is said to be a fundamental system of the ODES + \[ + y' = A(x) y + \] + Analogously, $n$ linearly independent solutions of the equation + \[ + y^{(n)} = a_{n-1}(x) y^{(n-1)} + a_{n-2}(x) y^{(n-2)} + \cdots + a_0 y + \] + are said to be a fundamental system. +\end{defi} + +\begin{eg} + Consider the inhomogeneous equation + \[ + y' = \sin(x)y + \sin(x)\cos(x) + \] + First, find the solutions to the homogeneous equation + \[ + \frac{y'}{y} = \sin(x) + \] + This can be done via integration + \begin{align*} + \int \frac{y'(t)}{y(t)} \dd{t} &= -\cos(x) + c \\ + \ln y + c &= -\cos(x) + c + \end{align*} + Then the solution is + \[ + y = K e^{-\cos(x)} + \] + The fundamental system in this case is $e^{-\cos x}$. + We can use a technique called "variation of the constant" to find a solution of the inhomogeneous equation. Define + \[ + y(x) = C(x) e^{-\cos(x)} + \] + Deriving this gives + \[ + y'(x) = C'(x) e^{-\cos(x)} - C(x) \sin(x) e^{-\cos(x)} + \] + Resubstituting this into the initial equation yields + \begin{align*} + C'(x) e^{-\cos(x)} + \cancel{C(x)\sin(x)e^{-\cos(x)}} &= \cancel{C(x)\sin(x)e^{-\cos(x)}} + \sin(x)\cos(x) \\ + C'(x) e^{-\cos(x)} &= \sin(x)\cos(x) \\ + C'(x) &= \sin(x)\cos(x)e^{\cos(x)} \\ + C(x) &= (1 - \cos(x))e^{\cos(x)} + \end{align*} + So the general solution to the ODE is + \[ + y(x) = 1 - \cos(x) + K e^{-\cos(x)} + \] +\end{eg} + +\begin{thm} + Let $y_1, \cdots, y_n$ be a fundamental system for $y' = A(x) y$. + Define an $n \times n$-matrix + \[ + W(x) := (y_1(x), y_2(x), \dots, y_n(x)) + \] + Then $W(x)$ is invertible $\forall x \in I$ and + \begin{align*} + z: I &\longrightarrow \realn^n \\ + x &\longmapsto W(x) \int_{x_0}^x \inv{W(t)}b(t) \dd{t} + \end{align*} + is a solution to the inhomogeneous system + \[ + y' = A(x)y + b(x) + \] +\end{thm} +\begin{proof} + According to the prerequisites the $y_1, \cdots, y_n$ are linearly independent, so the $y_1(x), \dots, y_n(x)$ are also linearly independent in $\realn^n$. + Thus + \begin{equation} + \det W(x) \ne 0 \implies W(x) \text{ invertible} + \end{equation} + Deriving this yields + \begin{equation} + W'(x) = A(x)W(x) + \end{equation} + which means the $i$-th column of this equation is $y_i'(x) = A(x)y_i(x)$. + Deriving $z$ gives us + \begin{equation} + \begin{split} + z'(x) &= W'(x) \int_{x_0}^x \inv{W(t)} b(t) \dd{t} + W(x)\inv{W(x)} b(x) \\ + &= A(x) z(x) + b(x) + \end{split} + \end{equation} + To apply the fundamental theorem, $W(t)b(t)$ should be continuous. + The mapping $A \mapsto \inv{A}$ is continuous on $Gl(n)$ (space of invertible matrices). +\end{proof} + +\begin{eg} + Consider the system + \begin{align*} + u' = v + \sin(x) && v' = -u + \cos(x) + \end{align*} + The homogeneous system in this case is + \[ + \begin{pmatrix} + u \\ v + \end{pmatrix}' + = + \begin{pmatrix} + 0 & 1 \\ -1 & 0 + \end{pmatrix} + \begin{pmatrix} + u \\ v + \end{pmatrix} + \] + The fundamental system is + \begin{align*} + y_1(x) = \begin{pmatrix} + \sin \\ \cos + \end{pmatrix}(x) + && + y_2 = \begin{pmatrix} + \cos \\ -\sin + \end{pmatrix}(x) + \end{align*} + Then define + \begin{align*} + z(x) &= C_1(x)y_1(x) + C_2(x)y_2(x) \\ + &= \underbrace{\begin{pmatrix} + \sin(x) & \cos(x) \\ + \cos(x) & -\sin(x) + \end{pmatrix}}_{W(x)} + \begin{pmatrix} + C_1(x) \\ C_2(x) + \end{pmatrix} + \end{align*} + Deriving this yields + \begin{align*} + z'(x) &= C_1'(x)y_1(x) + \cancel{C_1(x)y_1'(x)} + C_2'(x)y_2(x) + \cancel{C_2(x)y_2'(x)} \\ + &= \cancel{C_1(x)Ay_1(x)} + \cancel{C_2(x)Ay_2(x)} + b(x) \\ + &= b(x) + \end{align*} + This can be explicitly solved + \begin{align*} + C_1'(x)\sin(x) + C_2'(x)\cos(x) &= \sin(x) \\ + C_1'(x)\cos(x) - C_2'(x)\sin(x) &= \cos(x) + \end{align*} + Leading to + \begin{align*} + C_1'(x) &= C_1'(x)(\sin^2(x) + \cos^2(x)) = \sin^2(x) + \cos^2(x) = 1 \\ + C_2'(x) &= C_2'(x)(\cos^2(x) - \sin^2(x)) = 0 + \end{align*} + Thus + \begin{align*} + C_1(x) &= x \\ + C_2(x) &= 0 + \end{align*} + So the general solution of the homogeneous equation is + \[ + y_h = \begin{pmatrix} + x \sin(x) \\ x \cos(x) + \end{pmatrix} + \] + Our next goal is to find a solution of $y' = Ay$ with $A \in \realn^{n \times n}$ constant. + In one dimension the solution would be + \[ + y = Ce^{Ax} + \] + Does this also hold for $n > 1$? +\end{eg} + +\begin{rem} + Let $A \in \realn^{n \times n}$ + \[ + e^{Ax} = \sum_{k=0}^{\infty} \rec{k!}(Ax)^k = \sum_{k=0}^{\infty} \rec{k!} A^k x^k + \] + We have + \[ + \sum_{k=0}^{\infty} \rec{k!} \norm{A^k x^k} \le \sum_{k=0}^{\infty} \frac{\abs{x}^k}{k!} \norm{A}^k = e^{\norm{x}\norm{A}} < \infty + \] + Thus, $e^{Ax}$ is defined $\forall A \in \realn^{n \times n}, ~\forall x \in \realn$. Deriving this yields + \[ + \dv{x}e^{Ax} = \sum_{k=1}^{\infty} \rec{k!} A^k x^{k-1} = A \sum_{k=1}^{\infty} \rec{(k-1)!} A^{k-1} x^{k-1} = Ae^{Ax} + \] +\end{rem} + +\begin{thm} + Let $A \in \realn^{n \times n}$. The IVP + \begin{align*} + y' = Ay && y(x_0) = y_0 + \end{align*} + is solved exactly by + \[ + y(x) = e^{A(x - x_0)}y_0 + \] +\end{thm} +\begin{proof} + Without proof. +\end{proof} + +\begin{rem} + \begin{enumerate}[(i)] + \item The problem of solving IVPs can be reduced to a problem of calculating a matrix exponential. + \item The following does NOT generall hold + \begin{align*} + \dv{t}e^{A(x)} &= A'(x) e^{A(x)} \\ + e^{A + B} &= e^A e^B + \end{align*} + \item Let $v$ be an eigenvector of $A$ to the eigenvalue $\lambda$. Then + \begin{align*} + e^{Ax} v = \left(\sum_{k=0}^{\infty} \rec{k!} A^k x^k\right) v &= \sum_{k=0}^{\infty} \frac{x^k}{k!} A^k v \\ + &= \left(\sum_{k=0}^{\infty} \frac{x^k}{k!} \lambda^k\right) v = e^{\lambda x} v + \end{align*} + \end{enumerate} +\end{rem} + +\begin{eg} + Consider the IVP + \begin{align*} + \begin{pmatrix} + y \\ z + \end{pmatrix}' + = \underbrace{\begin{pmatrix} + 0 & 1 \\ 1 & 0 + \end{pmatrix}}_A + \begin{pmatrix} + y \\ z + \end{pmatrix} + && y_0 = \begin{pmatrix} + 1 \\ 0 + \end{pmatrix} + \end{align*} + This $A$ is diagonalizable and has the eigenvalues + \begin{align*} + \lambda_1 = -1 && \lambda_2 = 1 + \end{align*} + and the eigenvectors + \begin{align*} + v_1 = \begin{pmatrix} + 1 \\ 1 + \end{pmatrix} + && + v_2 = \begin{pmatrix} + 1 \\ -1 + \end{pmatrix} + \end{align*} + So we can solve this ODES by calculating + \begin{align*} + e^{Ax} y_0 = e^{Ax} \cdot \frac{1}{2}(v_1 + v_2) &= \frac{1}{2}\left(e^{\lambda_1 x} v_1 + e^{\lambda_2 x} v_2\right) \\ + &= \frac{1}{2} \left(e^x v_1 + e^{-x} v_2\right) + \end{align*} + And thus + \begin{align*} + y(x) = \frac{1}{2} \left(e^x + e^{-x}\right) && z(x) = \frac{1}{2} \left(e^x - e^{-x}\right) + \end{align*} +\end{eg} + +\begin{rem} + Often the process above is formulated as follows: + Start by defining + \[ + y(x) = c \cdot e^{\lambda x} v ~~c, \lambda \in \field \text{ and } v \in \realn + \] + Insert this into the ODE + \[ + c\lambda e^{\lambda x} = c e^{\lambda x} Av + \] + So $\lambda$ is an eigenvalue of $A$ to the eigenvector $v$. +\end{rem} + +\begin{thm} + Let $A \in \realn^{n \times n}$ be diagonalizable, and $v_1, \cdots, v_n$ is a basis of eigenvectors to the eigenvalues $\lambda_1, \cdots, \lambda_n$. + Then the functions + \[ + y_i(x) = e^{\lambda_i x} v_i ~~i \in \set{1, \cdots, n} + \] + are a fundamental system to the ODES + \[ + y' = Ay + \] +\end{thm} +\begin{proof} + We have + \begin{equation} + e^{Ax} v_i = e^{\lambda_i x} v_i + \end{equation} + In $x = 0$ the + \begin{equation} + y_1(0) = v_1, ~y_2(0) = v_2, ~\cdots, ~y_n(0) = v_n + \end{equation} + are linearly independent, so the $y_1, \cdots, y_n$ are also linearly independent. +\end{proof} + +\begin{rem} + \begin{enumerate}[(i)] + \item There is a special case, where $A \in \realn^{n \times n}$ is not diagonalizable in the real number space, but in the complex number space. + Let $\lambda = \lambda_r + \lambda_i$ be the eigenvalue to the eigenvector $v = v_r + v_i$. Then + \begin{gather*} + e^{\lambda_r x} (v_r \sin(\lambda_i x) + v_i \cos(\lambda_i x)) \\ + e^{\lambda_r x} (v_r \cos(\lambda_i x) + v_i \sin(\lambda_i x)) + \end{gather*} + be linearly independent, real-valued solutions. To solve the IVP + \begin{align*} + y(x) = C e^{\lambda x} v && y(0) = y_0 + \end{align*} + we want to transform it into an eigenvalue problem and find a solution to that. Doing that gives us + \[ + y(x) = C_1 e^{\lambda_1 x} v_1 + \cdots + C_n e^{\lambda_n x} v_n + \] + By inserting the initial condition we can find + \[ + C_1 v_1 + C_2 v_2 + \dots + C_n v_n = y_0 + \] + Finding the $C_1, \cdots, C_n$ shows us that the solution is automatically real. + + \item If $A$ is not diagonalizable one can try and bring $A$ into Jordan normal form. + \end{enumerate} +\end{rem} + +\begin{eg} + Consider the IVP + \begin{align*} + \begin{pmatrix} + y \\ z + \end{pmatrix}' + = + \begin{pmatrix} + 0 & 1 \\ -1 & 0 + \end{pmatrix} + \begin{pmatrix} + y \\ z + \end{pmatrix} + && + y_0 = \begin{pmatrix} + 1 \\ 0 + \end{pmatrix} + \end{align*} + The eigenvalues and eigenvectors are + \begin{align*} + \lambda_1 &= i & \lambda_2 &= -i \\ + v_1 &= \begin{pmatrix} + 1 + i \\ -1 + i + \end{pmatrix} + & + v_2 &= \begin{pmatrix} + 1 - i \\ -1 - i + \end{pmatrix} + \end{align*} + Thus we have the general solution + \[ + C_1 e^{ix} v_1 + C_2 e^{-ix} v_2 + \] + which expands to + \begin{align*} + (i + 1) C_1 \cancel{e^{i0}} + (1 - i)C_2 \cancel{e^{-i0}} &= 1 \\ + (i - 1) C_1 \cancel{e^{i0}} + (-1 - i)C_2 \cancel{e^{-i0}} &= 0 + \end{align*} + and solves to + \begin{align*} + C_1 = \frac{1}{4} (1 - i) && C_2 = \frac{1}{4} (1 + i) + \end{align*} + So the solution to the IVP is + \begin{align*} + y(x) = \cos(x) && z(x) = -\sin(x) + \end{align*} +\end{eg} + +\begin{thm} + Let $a_1, \cdots, a_{n-1} \in \cmpln$. Let $\lambda_1, \cdots, \lambda_k$ be the roots of the polynomial + \[ + a_0 + a_1 \lambda + \cdots + a_{n-1} \lambda^{n-1} + \lambda^n + \] + and $\nu_1, \cdots, \nu_k$ their multiples. Then the functions + \[ + x \longmapsto x^l e^{\lambda_i x} ~~i \in \set{1, \cdots, k}, l \in \set{0, \cdots, \nu_{i_1}} + \] + form a fundamental system for + \[ + a_0 y + a_1 y' + \cdots + a_{n-1} y^{(n-1)} + y^{(n)} + \] +\end{thm} +\end{document} \ No newline at end of file diff --git a/chapters/sections/picard_lindelof.tex b/chapters/sections/picard_lindelof.tex index 5259a0f..beb902f 100644 --- a/chapters/sections/picard_lindelof.tex +++ b/chapters/sections/picard_lindelof.tex @@ -326,7 +326,7 @@ We define $\metric$ to be a metric space, $x \in X$ and $A \subset X$. Then d(x, A) = \inf\set[y \in A]{d(x, y)} \] -\begin{thm} +\begin{thm}\label{thm:837} Let $D \subset \realn \times \realn^n$ be open, $(x_0, y_0) \subset D$ and $f: D \rightarrow \realn^n$ continuous and satisfying the local Lipschitz condition in terms of $y$. 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