91Ó°ÊÓ

Find the maximal interval of existence \((\alpha, \beta)\) for the following initial value problems and if \(\alpha>-\infty\) or \(\beta<\infty\) discuss the limit of the solution as \(t \rightarrow \alpha^{+}\)or as \(t \rightarrow \beta^{-}\)respectively: (a) \(\begin{array}{ll}\dot{x}_{1}=x_{1}^{2} & x_{1}(0)=1 \\\ \dot{x}_{2}=x_{2}+x_{1}^{-1} & x_{2}(0)=1\end{array}\) (b) \(\quad \dot{x}_{1}=\frac{1}{2 x_{1}} \quad x_{1}(0)=1\) \(\dot{x}_{2}=x_{2}^{2} \quad x_{2}(0)=1\) $$ \text { (c) } \begin{aligned} \dot{x}_{1}=\frac{1}{2 x_{1}} & x_{1}(0)=1 \\ \dot{x}_{2}=x_{1} & x_{2}(0)=1 \end{aligned} $$

Use the method of successive approximations to show that if the matrix valued function \(A(t)\) is continuous on \(\left[-a_{0}, a_{0}\right]\) then there exists an \(a>0\) such that the initial value problem $$ \begin{aligned} \dot{\Phi} &=A(t) \Phi \\ \Phi(0) &=I \end{aligned} $$ (where \(I\) is the \(n \times n\) identity matrix) has a unique fundamental matrix solution \(\Phi(t)\) on \([-a, a]\). Hint: Define \(\Phi_{0}(t)=I\) and $$ \Phi_{k+1}(t)=I+\int_{0}^{t} A(s) \Phi_{k}(s) d s $$ and use the fact that the continuous matrix valued function \(A(t)\) satisfies \(\|A(t)\| \leq M_{0}\) for all \(t\) in the compact set \(\left[-a_{0}, a_{0}\right]\) to show that the successive approximations \(\Phi_{k}(t)\) converge uniformly to \(\Phi(t)\) on some interval \([-a, a]\) with \(a<1 / M_{0}\) and \(a \leq a_{0}\).

(Cf. Hartman [H], p. 96.) Let \(\mathbf{f} \in C^{1}(E)\) where \(E\) is an open set in \(\mathbf{R}^{n}\) containing the point \(\mathbf{x}_{0}\). Let \(\mathbf{u}\left(t, \mathbf{y}_{0}\right)\) be the unique solution of the initial value problem (1) for \(t \in[0, a]\) with \(y=y_{0}\). Show that the set of maps of \(\mathbf{y}_{0} \rightarrow \mathbf{y}\) defined by \(\mathbf{y}=\mathbf{u}\left(t, \mathbf{y}_{0}\right)\) for each fixed \(t \in[0, a]\) are volume preserving in \(E\) if and only if \(\nabla \cdot \mathbf{f}(x)=0\) for all \(x \in E\). Hint: Recall that under a transformation of coordinates \(\mathbf{y}=\mathbf{u}(\mathbf{x})\) which maps a region \(R_{0}\) one-to-one and onto a region \(R_{1}\), the volume of the region \(R_{1}\) is given by $$ V=\int_{R_{0}} \cdots \int J(x) d x_{1} \ldots d x_{n} $$ where the Jacobian determinant $$ J(\mathbf{x})=\operatorname{det} \frac{\partial \mathbf{u}}{\partial \mathbf{x}}(\mathbf{x}) $$ .

Show that the second-order differential equation $$ \ddot{x}+f(x) \dot{x}+g(x)=0 $$ can be written as the Lienard system $$ \begin{aligned} &\dot{x}_{1}=x_{2}-F\left(x_{1}\right) \\ &\dot{x}_{2}=-g\left(x_{1}\right) \end{aligned} $$ where $$ F\left(x_{1}\right)=\int_{0}^{x_{1}} f(s) d s $$ Let $$ G\left(x_{1}\right)=\int_{0}^{x_{1}} g(s) d s $$ and suppose that \(G(x)>0\) and \(g(x) F(x)>0\) (or \(g(x) F(x)<0\) ) in a deleted neighborhood of the origin. Show that the origin is an asymptotically stable equilibrium point (or an unstable equilibrium point) of this system.

Prove that if \(f \in C^{1}(E)\) where \(E\) is a compact convex subset of \(\mathbf{R}^{n}\) then \(f\) satisfies a Lipschitz condition on \(E\). Hint: Cf. Theorem \(9.19\) in \([R]\).

(a) Show that the function \(f(x)=1 / x\) is not uniformly continuous on \(E=(0,1)\). Hint: \(f\) is uniformly continuous on \(E\) if for all \(\varepsilon>0\) there exists a \(\delta>0\) such that for all \(x, y \in E\) with \(|x-y|<\delta\) we have \(|f(x)-f(y)|<\varepsilon\). Thus, \(f\) is not uniformly continuous on \(E\) if there exists an \(\varepsilon>0\) such that for all \(\delta>0\) there exist \(x, y \in E\) with \(|x-y|<\delta\) such that \(|f(x)-f(y)| \geq \varepsilon\). Choose \(\varepsilon=1\) and show that for all \(\delta>0\) with \(\delta<1, x=6 / 2\) and \(y=\delta\) implies that \(x, y \in(0,1),|x-y|<\delta\) and \(|f(x)-f(y)|>1 .\) (b) Show that \(f(x)=1 / x\) does not satisfy a Lipschitz condition on \((0,1)\).