Contraction Mapping and Nonlinear Equations

Let \(X\) be a Banach space and consider the non-linear mapping \(F : X \to X\) defined by

where \(y \in X\) is fixed and \(\alpha \) is a scalar. Show that there is a constant \(C > 0\) such that for any \(|\alpha | < C, F\) is a contraction on the open ball of radius 1 centered at \(y\). Note: You may use that for any \(a, b \in \mathbb {R}\) we have \(|\sin (a) - \sin (b)| \le |a - b|\).

Consider the nonlinear integral equation \[ u(x)-\lambda \int _a^b w(x,y)\sin (u(y))\,dy=\phi (x), \] where \(\lambda \in \mathbb {R}\), \(\phi \) is continuous on the interval \([a,b]\) and \(w(x,y)\) is continuous on \([a,b]\times [a,b]\). Find a constant \(C>0\) such that \(\lvert \lambda \rvert <C\) implies that the nonlinear integral equation has a unique solution \(u\).

Hint: You may assume the inequality \(\lvert \sin (x)-\sin (y)\rvert \le \lvert x-y\rvert \), for any \(x,y\in \mathbb {R}\).

Consider the nonlinear integral equation \[ f(x)-\lambda \int _a^b k(x,y;f(y))\,dy=\phi (x) \] where \(\lambda \in \mathbb {R}\), \(\phi \) is continuous on \([a,b]\), \(k\) is continuous on \(C=[a,b]\times [a,b]\times \mathbb {R}\), and \(k\) satisfies a Lipschitz condition of the form \[ \lvert k(x,y;z_1)-k(x,y;z_2)\rvert \le L\lvert z_1-z_2\rvert \] in its ‘functional argument’. Show that the nonlinear integral equation has a unique solution, \(f\), for any \(\lambda \) such that \[ \lvert \lambda \rvert <\frac {1}{L(b-a)}. \]

Let \(p(t)\), \(q(t)\) and \(r(t)\) be continuous functions defined for \(t\in \mathbb {R}\) and \(t_0,x_0,x_1\in \mathbb {R}\). Consider the initial value problem \[ \begin {cases} \ddot {x}+p(t)\dot {x}+q(t)x=r(t)\\ x(t_0)=x_0,\qquad \dot {x}(t_0)=x_1. \end {cases} \tag {1} \]

1. Let \(X\) be a Banach Space with norm \(\norm {\bullet }\). Let \(C\subset X\) be a closed subset. State the Contraction Mapping Theorem (otherwise known as the Banach Fixed Point Theorem) for a map \(K:C\to X\).
2. Carefully reformulate (1) as a fixed point problem. Prove that the fixed point problem is equivalent to the local existence problem for (1).
3. Using the Contraction Mapping Theorem, show that there is a \(\delta >0\) and a unique function \(x(t)\in C^2([t_0,t_0+\delta ])\) that satisfies your fixed point problem, and hence (1), for \(t_0\le t\le t_0+\delta \).

Let \(D\subset \mathbb {R}\times \mathbb {R}^n\) be an open set, \((t_0,x_0)\in D\) a point and \(f(t,x)\in C(D,\mathbb {R}^n)\) be a map.

1. Let \(X\) be a Banach Space with norm \(\norm {\bullet }\). Let \(C\subset X\) be a closed subset. State the Contraction Mapping Theorem (otherwise known as the Banach Fixed Point Theorem) for a map \(K:C\to X\).
2. Define what it means for a map \(f(t,x)\in C(D,\mathbb {R}^n)\) to be locally Lipschitz with respect to \(x\)
3. Carefully reformulate the Initial Value Problems as a fixed point problem. Prove that the fixed point problem is equivalent to the local existence problem for the (IVP) \[ \begin {cases} \dfrac {dx}{dt}=f(t,x),\\ x(t_0)=x_0. \end {cases} \tag {IVP} \]
4. Assume \(f(t,x)\in C(D,\mathbb {R}^n)\) is locally Lipschitz with respect to \(x\). Using the Contraction Mapping Theorem, show that there is a \(\epsilon >0\) and a unique function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem (IVP) for \(t_0\le t\le t_0+\epsilon \).

1. Let \(A\) be an \(n\times n\) real matrix. Find necessary and sufficient conditions on \(A\) so that for all \(x_0\in \mathbb {R}^n\), the solution \(\varphi (t;x_0)\) of (1) remains bounded for \(t\ge 0\). Prove your result. \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \tag {1} \]
2. Suppose that \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) is such that \(f(0)=0\) and the Jacobian matrix \(A=df(0)\) satisfies the condition in (a). Do all solutions \(\psi (t;x_0)\) of the nonlinear equation (2) still remain bounded for \(t\ge 0\), at least if \(x_0\) is close to \(0\)? Explain. \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0. \end {cases} \tag {2} \]

Let \((X,d)\) be a complete metric space and \(f\) a contraction on \(X\) with contraction constant \(\alpha <1\). Let \(x_f\) be the unique fixed point of \(f\). Fix \(x_0\in X\) and consider the sequence of points defined by \(x_n=f(x_{n-1})\) for \(n\in \mathbb {N}\).

1. Prove that \[ d(x_n,x_f)\le \frac {\alpha ^n}{1-\alpha }d(x_0,x_1). \] Hint: consider \(d(x_m,x_n)\) for \(m>n\).
2. Let \(S=[1,\infty )\) and \(f:S\to S\) be defined by \(f(x)=\frac {1}{2}(x+2/x)\). Prove that \(f\) is a contraction mapping on \(S\) and compute a contraction constant. Determine all fixed points of \(f\) on \(S\). Using part (a), explain why the sequence \((x_n)\) might be useful.

Consider the transformation on the extended complex plane \(\overline {\mathbb {C}}\) \[ f(z)=w=z+\frac {1}{z}. \]

1. Find all possible fixed points.
2. Show that the image of the points in the upper half \(z\)-plane that are exterior to the circle \(\lvert z\rvert =1\) corresponds to the entire upper half \(w\)-plane.
3. Find the image of the unit circle.
4. Show that \(f(z)\) is a two-to-one function.

Let \(f\) be a real-valued and twice continuously differentiable on an interval \([a,b]\). Let \(\hat {x}\) be a simple zero of \(f\), i.e. \(f'(\hat {x})\ne 0\). Use the Banach Fixed Point Theorem to show that Newton’s method, defined by \[ x_{n+1}=x_n-\frac {f(x_n)}{f'(x_n)} \] converges to \(\hat {x}\) for any \(x_0\) sufficiently close to \(\hat {x}\).

Let \(f\in C^1(U,\mathbb {R}^n)\) for \(U\subset \mathbb {R}^n\) and \(x_0\in U\). Given the Banach space \(X=C([0,T],\mathbb {R}^n)\) with norm \(\norm {x}=\max _{0\le t\le T}\abs {x(t)}\), let \[ K(x)(t)=x_0+\int _0^t f(x(s))\dd s \] for \(x\in X\). Define \(V=\{x\in X\mid \norm {x-x_0}\le \epsilon \}\) for fixed \(\epsilon >0\) and suppose \(K(x)\in V\) (which holds for sufficiently small \(T\)), so that \(K:V\to V\) with \(V\) a closed subset of \(X\).

1. Give the definition of a locally Lipschitz function in an open set \(U\) of a normed vector space.
2. Using the fact that \(f\) is locally Lipschitz in \(U\) with Lipschitz constant \(L_0\), and taking \(x,y\in V\) show that \[ \abs {K(x(t))-K(y(t))}\le L_0t\norm {x-y}. \] Hence, show that \[ \norm {K(x)-K(y)}\le L_0T\norm {x-y}\qquad x,y\in V. \]
3. State the contraction mapping principle on a Banach space.
4. Choosing \(T<1/L_0\), apply the contraction mapping principle to show that the integral equation has a unique continuous solution \(x(t)\) for all \(t\in [0,T]\) and sufficiently small \(T\). Hence establish existence and uniqueness of the initial value problem \[ \frac {dx}{dt}=f(x),\qquad x(0)=x_0. \]

Let \(X\) be a Banach space and consider the non-linear mapping \(F:X\to X\) defined by \[ F(x)=y-\alpha \lVert x\rVert x, \] where \(y\in X\) is fixed and \(\alpha \) is a scalar. Show that there is a constant \(C>0\) such that for any \(\lvert \alpha \rvert <C\), \(F\) is a contraction on the open ball of radius \(1\) centered at \(y\).

The simple pendulum consists of a point particle of mass \(m\) suspended from a fixed point by a massless rod of length \(L\), which is allowed to swing in a vertical plane. If friction is ignored then the equation of motion is \[ \ddot {x}+\omega ^2\sin x=0,\qquad \omega ^2=\frac {g}{L}, \] where \(x\) is the angle of inclination of the rod with respect to the downward vertical and \(g\) is the gravitational constant.

1. Using conservation of energy, show that the angular velocity of the pendulum satisfies \[ \dot {x}=\pm \sqrt {2}(C+\omega ^2\cos x)^{1/2}, \] where \(C\) is an arbitrary constant. Express \(C\) in terms of the total energy of the system.
2. Sketch the phase diagram of the pendulum equation in the \((x,\dot {x})\)-plane. Illustrate the one-parameter family of curves given by part (a) for different values of \(C\). Take \(-3\pi \le x\le 3\pi \). Indicate the fixed points of the system and the separatrices - curves linking the fixed points. Give a physical interpretation of the underlying trajectories in the two distinct dynamical regimes \(\abs {C}<\omega ^2\) and \(\abs {C}>\omega ^2\).
3. Show that in the regime \(\abs {C}<\omega ^2\), the period of oscillations is \[ T=4\sqrt {\frac {L}{g}}K(\sin x_0/2), \] where \(\dot {x}=0\) when \(x=x_0\) and \(K\) is the complete elliptic integral of the first kind, which is defined by \[ K(\alpha )=\int _0^{\pi /2}\frac {1}{\sqrt {1-\alpha ^2\sin ^2u}}d u. \]
4. For small amplitude oscillations, the pendulum equation can be approximated by the linear equation \[ \ddot {x}+\omega ^2x=0. \] Solve this equation for the initial conditions \(x(0)=A\), \(\dot {x}(0)=0\) and sketch the phase-plane for different values of \(A\). Compare with the phase-plane for the full nonlinear equation in part (b).

Let \(f\in C^1(U,\mathbb {R}^n)\) for \(U\subset \mathbb {R}^n\) and \(x_0\in U\). Given the Banach space \(X=C([0,T],\mathbb {R}^n)\) with norm \(\norm {x}=\max _{0\le t\le T}\abs {x(t)}\), let \[ K(x)(t)=x_0+\int _0^t f(x(s))\dd s \] for \(x\in X\). Define \(V=\{x\in X\mid \norm {x-x_0}\le \epsilon \}\) for fixed \(\epsilon >0\) and suppose \(K(x)\in V\) (which holds for sufficiently small \(T\)), so that \(K:V\to V\) with \(V\) a closed subset of \(X\).

1. Give the definition of a locally Lipschitz function in an open set \(U\) of a normed vector space.
2. Using the fact that \(f\) is locally Lipschitz in \(U\) with Lipschitz constant \(L_0\), and taking \(x,y\in V\) show that \[ \abs {K(x(t))-K(y(t))}\le L_0t\norm {x-y}. \] Hence, show that \[ \norm {K(x)-K(y)}\le L_0T\norm {x-y},\qquad x,y\in V. \]
3. State the contraction mapping principle on a Banach space.
4. Choosing \(T<1/L_0\), apply the contraction mapping principle to show that the integral equation has a unique continuous solution \(x(t)\) for all \(t\in [0,T]\) and sufficiently small \(T\). Hence establish existence and uniqueness of the initial value problem \[ \frac {dx}{dt}=f(x),\qquad x(0)=x_0. \]

Consider the nonlinear equation \[ \ddot {x}+\epsilon \dot {x}(x^2-1)+x=0,\qquad 0<\epsilon \ll 1 \] and choose initial conditions \(x(0)=1\), \(\dot {x}(0)=0\).

1. Using the method of multiple scales show that this has an asymptotic series solution of the form \[ x(t)\sim R(\epsilon t)\cos (t+\theta (\epsilon t))+\mathcal {O}(\epsilon ) \] with \[ \theta _\tau =0,\qquad R_\tau =\frac {1}{8}R(4-R^2) \] where \(\tau =\epsilon t\).
2. Derive the solution \[ R(\tau )=\frac {2}{(1+a_0e^{-\epsilon t})^{1/2}} \] and determine \(a_0\) from the initial conditions. Hence, establish that there exits a stable periodic orbit. [HINT: perform the change of variable \(y=R^2\).]

Consider Griffith’s model for a genetic control system, where \(x\) and \(y\) are proportional to concentration of protein and the messenger RNA from which it is translated, respectively, and \(\mu >0\) is a rate constant \[ \begin {aligned} \dot {x} &= y-\mu x,\\ \dot {y} &= \frac {x^2}{1+x^2}-y. \end {aligned} \]

1. Show that the system has three fixed points when \(\mu <\mu _c\) and one when \(\mu >\mu _c\) where \(\mu _c\) is to be determined.
2. What is the nature of the bifurcation at \(\mu =\mu _c\)?

Consider a Brusselator system for chemical reactants \(x,y\) given by \[ \begin {aligned} \dot {x} &= 1-4x+x^2y,\\ \dot {y} &= 3x-x^2y \end {aligned} \]

1. Show that the quadrilateral bounded by the lines \(x=0\), \(y=0\), \(y=8+x\) and \(y=10-x\) is a forward invariant subset. [Hint: need to show that if \(n\) is the outer normal to a given edge of the quadrilateral, then \(n\cdot (\dot {x},\dot {y})\le 0\).]
2. Find the fixed points and determine their stability.
3. Show that the system has at least one nonconstant periodic solution.

Consider Griffith’s model for a genetic control system, where \(x\) and \(y\) are proportional to concentration of protein and the messenger RNA from which it is translated, respectively, and \(\mu >0\) is a rate constant \[ \begin {aligned} \dot {x} &= y-\mu x,\\ \dot {y} &= \frac {x^2}{1+x^2}-y. \end {aligned} \]

1. Show that the system has three fixed points when \(\mu <\mu _c\) and one when \(\mu >\mu _c\) where \(\mu _c\) is to be determined.
2. What is the nature of the bifurcation at \(\mu =\mu _c\)?

Consider a Brusselator system for chemical reactants \(x,y\) given by \[ \begin {aligned} \dot {x} &= 1-4x+x^2y,\\ \dot {y} &= 3x-x^2y \end {aligned} \]

1. Show that the quadrilateral bounded by the lines \(x=0\), \(y=0\), \(y=8+x\) and \(y=10-x\) is a forward invariant subset.
2. Find the fixed points and determine their stability.
3. Show that the system has at least one nonconstant periodic solution.

Consider a modified predator-prey system for populations \(x,y\) given by \[ \begin {aligned} \dot {x} &= x\cdot \left (1-\frac {x}{4}-\frac {y}{1+x}\right ),\\ \dot {y} &= y\cdot \left (\frac {x}{1+x}-\frac 12\right ) \end {aligned} \]

1. Show that the triangle with vertices \((0,0)\), \((5,0)\) and \((0,5)\) is a forward invariant subset.
2. Find the fixed points and determine their stability.
3. Show that the system has at least one nonconstant periodic solution.

Consider the \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\quad x\in \mathbb {R}^n,\quad A(t)=A(t+T). \] Let \(\Phi (t)\) be a fundamental matrix with \(\Phi (0)=I\).

1. Show that there exists at least one nontrivial solution \(x=\chi (t)\) such that \[ \chi (t+T)=\mu \chi (t) \] where \(\mu \) is an eigenvalue of \(\Phi (T)\).
2. Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are then \(n\) linearly independent solutions of the form \[ x_i=p_i(t)e^{\rho _i t} \] where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho _i\) related to \(\mu _i\)?
3. Suppose that the autonomous nonlinear equation \(\dot {x}=f(x)\) exhibits a limit cycle. By linearizing about this solution, explain how Floquet theory can be used to determine the linear stability of the limit cycle.

Consider the nonlinear equation \[ \ddot {x}+\varepsilon \left (\frac {1}{3}\bigl (\dot {x}\bigr )^3-\dot {x}\right )+x=0, \quad 0<\varepsilon \ll 1 \] and choose initial conditions \(x(0)=a\), \(\dot {x}(0)=0\).

1. Using the method of multiple scales show that this has an asymptotic series solution of the form \[ x(t)\sim 2R(\varepsilon t)\cos (t+\theta (\varepsilon t))+O(\varepsilon ) \] with \[ \theta _\tau =0,\quad R_\tau =\frac {1}{2}R(1-R^2) \] where \(\tau =\varepsilon t\).
2. Derive the solution \[ R(\tau )=\left (1+a_0e^{-\varepsilon t}\right )^{-1/2} \] and determine \(a_0\) from the initial conditions. Hence, establish that there exits a stable periodic orbit.

Consider a second order, linear autonomous system \(\dot {x}=Ax\), \(x\in \mathbb {R}^2\).

1. Suppose that \(A\) has a pair of complex conjugate eigenvalues. By performing an appropriate change of coordinates, convert the equation into Jordan normal form and thus obtain the general solution.
2. Under what conditions is the fixed point at the origin hyperbolic? Explain the significance of hyperbolic fixed points in the theory of nonlinear ODEs.

Prove that the nonlinear integral equation \[ u(x)-\int _0^1 \frac {u^2(y)}{4+x^2-y}\,dy=\frac {x^3}{3},\qquad 0\le x\le 1, \] has exactly one continuous solution \(u\) satisfying \(\max _{0\le x\le 1}|u(x)|\le 1\). Describe, but do not carry out, an iterative procedure for obtaining an approximation to this solution.

Contraction Mapping Theorem. Show that the problem \[ \begin {aligned} \dot {x}(t) &= f(x(t))+\frac {1}{a}\int _0^a x(s)\,ds,\quad t\in (0,a),\\ x(0) &= 0, \end {aligned} \] admits a unique solution \(x\in C([0,a];\mathbb {R}^n)\) for \(a>0\) sufficiently small. Assume \(f:\mathbb {R}^n\to \mathbb {R}^n\) satisfies \(\lvert f(x)-f(y)\rvert \le M\lvert x-y\rvert \) for all \(x,y\in \mathbb {R}^n\).

Problem. Consider the nonlinear system \[ \begin {aligned} x' &= 2\sin y,\\ y' &= 3\sin x-\sin y. \end {aligned} \] Provide a complete phase plane analysis of this system and give justifications for your reasoning.

Problem. In an isolated region of the Canadian Northwest Territories, a population of white wolves, \(x\), and one of silver foxes, \(y\), compete for survival (for each population one unit represents 100 individuals). They have a common, limited food supply, which consists mainly of mice. A mathematical biologist models the dynamics of the competing species by the nonlinear system \[ \begin {aligned} x' &= x-x^2-xy,\\ y' &= \frac {3}{4}y-y^2-\frac {1}{2}xy. \end {aligned} \] Can the two species survive together according to this model?

Contraction Mapping. Consider the initial value problem \[ \begin {aligned} \dot {x} &= f(x),\\ x(0) &= x_0, \end {aligned} \] where \(f\) satisfies for all \(x,y\in \mathbb {R}^n\) \[ \lvert f(x)-f(y)\rvert \le (1+\lvert x\rvert +\lvert y\rvert )\lvert x-y\rvert . \] Use the contraction mapping principle to find a condition on \(\lvert x_0\rvert \) which guarantees local existence and uniqueness of solutions. How does the local time interval upon which existence is obtained depend on \(\lvert x_0\rvert \)?

Linearization. Assume \(g\in C^\infty (\mathbb {R}^n)\) has a strict local minimum at the origin. Consider the second-order system \[ \ddot {x}+B\dot {x}+\nabla g(x)=0, \] where \(x(t)\in \mathbb {R}^n\) and \(B\in \mathbb {R}^{n\times n}\) is a symmetric positive definite matrix.

1. Rewrite the system as a first-order system.
2. Write the linearization of the system at the origin.
3. Prove that the origin is an asymptotically stable fixed point.

Stable and Unstable Manifolds. Find the stable and unstable manifolds in the neighborhood of the origin for the system \[ \begin {aligned} x' &= -x,\\ y' &= y+x^2,\\ z' &= z+y^2. \end {aligned} \] How are the stable and unstable manifolds for the linearization of this system related to those of the nonlinear system?

Consider the non-local differential operator \[ Lu=Du''-u(g(x)) \] with boundary conditions \(u(0)=u(1)=0\) and where \(g(x)\) is a monotone increasing function with \(g(0)=0\), \(g(1)=1\), and \(g(x)<x\) for \(0<x<1\).

1. Find the adjoint of the operator \(L\).
2. Prove that the operator \(L\) is invertible.
3. Devise an iteration scheme that could be used to solve \(Lu=f(x)\). Under what conditions on \(D\) is this iteration scheme a contraction mapping?

Consider the system of ordinary differential equations \[ u'=-\lvert u\rvert ^2u+h(t), \tag {1} \] where \[ h:[0,1]\to \mathbb {R}^N \] is a continuous function and \(\lvert \cdot \rvert \) is the Euclidean norm in \(\mathbb {R}^N\). Use Brouwer’s fixed point theorem to show that (1) has a solution \(u\in C^1[0,1]\) such that \[ u(0)=u(1). \] Use this result to conclude that if \[ h:\mathbb {R}\to \mathbb {R}^N \] is continuous and periodic of period 1, then (1) has a solution of period 1.

(Hint: Consider equation (1) subject to initial data \(u_0:\lvert u_0\rvert \le r\), where \(r\) is large (in relation to \(h\)) and examine solution curves by computing \(\frac {d}{dt}\lvert u\rvert ^2\) for solutions \(u\).) In relation to the above, what can you say about the equation \[ u'=\lvert u\rvert ^2u+h(t). \tag {2} \]

Use the implicit function theorem for mappings between Banach spaces to discuss the unique solvability of the scalar boundary value problem \[ u''+2u+u^2=h(t),\qquad u(0)=u(\pi ), \tag {6} \] for continuous functions \(h\) of small maximum norm. Results needed from Sturm-Liouville theory may be assumed but should be quoted.

Consider \(\dot {x}=\mathbf {f}(x)\) where \(x\in \mathbb {R}^n\) and \(\mathbf {f}\in C^1(\mathbb {R}^n,\mathbb {R}^n)\). Suppose that the solution through any point \(x_0\in \mathbb {R}^n\) exists for all \(t\in \mathbb {R}\).

1. Define the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\).
2. Consider the nonlinear spring equation \[ x''+\mu x'+g(x)=0 \] where \(g(x)\) is an odd function of \(x\), \(g'(x)>0\) and \(\mu \ge 0\). Translate this equation to the \(\dot {x}=\mathbf {f}(x)\) form and linearize about the trivial equilibrium. What conclusions can be made about the stability of this equilibrium?
3. Define what is meant by a Lyapunov function for the above equation; derive one and verify that it is a Lyapunov function. Describe the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\) for the case \(\mu =0\) and for the case \(\mu >0\). Make a conclusion about the stability of the trivial equilibrium for each of these two cases.

Suppose that \(T\) is a transformation of a complete metric space \(X\), and \(T^n\) (for a certain positive integer \(n\)) is a contraction with fixed point \(a\). Show that

1. \(a\) is a fixed point of \(T\);
2. \(T\) has only one fixed point;
3. the sequence of successive iterations \(u_0\), \(u_1=Tu_0\), \(u_2=Tu_1,\ldots \) tends to \(a\) for any initial element \(u_0\in X\).

Consider \[ x'=f(x) \tag {3.1} \] where \(x\in \mathbb {R}^n\) and \(f\in C^1(\mathbb {R}^n,\mathbb {R}^n)\). Suppose that the solution through any point \(x_0\in \mathbb {R}^n\) exists for all \(t\in \mathbb {R}\).

1. Define the positive and negative semiorbits of (3.1) through \(x_0\in \mathbb {R}^n\).
2. Define the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\).
3. Consider the nonlinear spring equation \[ x''+g(x)=0 \tag {3.2} \] where \(g(x)\) is an odd function of \(x\) and \(g'(x)>0\). Describe the \(\alpha \)- and \(\omega \)-limit set for any point on the positive \(x\)-axis.
4. Consider \[ x''+x'+g(x)=0 \tag {3.3} \] where \(g\) is as in (c). Define what is meant by a Liapunov function for this equation and derive one (verify that it is a Liapunov function). Describe the \(\omega \)-limit set for any point in the plane.

Define \[ f:E\to X \] by \[ u\mapsto f(u)=u''+\lambda u+u^3, \tag {2.1} \] where \(\lambda \in \mathbb {R}\). Show that \(f\) is a \(C^1\) mapping.

Convert problem (2.13) into an equivalent fixed point problem in an appropriate Banach space.

State necessary and sufficient conditions in order that a linear operator be the generator of a strongly continuous contraction semigroup on \(H\).

1. Show that the only equilibrium point of the system \[ x'=-2y-4x^3-4xy^2,\qquad y'=x-2x^2y-2y^3 \] is the origin.
2. Draw what conclusions you can about stability, asymptotic stability, and regions of attraction, using first i) linearization and then ii) Lyapunov functions. Make clear the reasons for your conclusions.
3. What if the signs of the nonlinear terms in a) were changed?
4. What kinds of functions \(f\) and \(g\) would give similar results for systems of the form \(x'=-2y+f(x,y)\), \(y'=x+g(x,y)\)?