Nonlinear equations

The main idea of the first part of this work is to introduce some techniques to approach the following nonlinear problem:

$\displaystyle F(x)=y \ \ \ \ \ (1)$

Where ${K\subset X}$ is the feasible set and ${F:K\rightarrow Y}$ is any function between topological spaces.

Many important problems (in PDE) match with this setting, for example:

1. The nonlinear classical Dirichlet problem:

$\displaystyle \begin{cases} -L u=f(u) \quad \text{in} \; \Omega \, ,\\ u = 0 \qquad \hspace{2mm} \enspace \text{on} \: \partial \Omega. \end{cases} \ \ \ \ \ (2)$

Where ${L}$ is any second order elliptic operator.
In this case we can understand this classical problem as a nonlinear equation of the form ${F(u)=0}$ defining ${F:C^2(\Omega)\rightarrow C(\Omega)}$ as ${F(u):=L u-f(u)}$.

2. The weak variational problem:

$\displaystyle \int_{\Omega}\nabla{u}\cdot\nabla v=\int_{\Omega} fv, \hspace{3mm} \forall v\in H_0^1(\Omega):=W_0^{1,2}(\Omega). \ \ \ \ \ (3)$

In this case we have a (linear) equation of the form ${F(u)=f}$ with ${F:(H^1(\Omega))^*\rightarrow (H^1(\Omega))^*}$, where ${F(u)(v)=\int_{\Omega}\nabla{u}\cdot\nabla v}$ and ${f(v)=\int_{\Omega} fv}$.

Note: In this last example is not possible to simplify the notation assuming the usual identification between a Hilbert space and its dual because we are dealing with two different types of functionals induced by functions in the space.
The existence of solutions for this kind of equations is determined by two fundamental aspects:

• The nonlinearity ${F}$.
• The topological structure of the feasbile set ${K}$.

Even though many theorems through this work will make this point clearer, it is illustrative to emphasize the importance of this claim with the next two theorems.

Theorem 1 (Banach Fixed Point theorem) Let ${(X,d)}$ be a complete metric space and let ${F:X\rightarrow X}$ be a contractive map, i. e. there exist ${\rho<1}$ such that

$\displaystyle d(F(x),F(y))\leq \rho d(x,y) \hspace{2mm} \forall x,y\in X \ \ \ \ \ (4)$

Then the nonlinear equation ${F(x)=x}$ has an unique solution.

Moreover, the unique solution ${x}$ satisifies ${x=\lim\limits_{n\rightarrow\infty} F^n(y)}$, for any ${y\in X}$.

Proof: Exercise. $\Box$

There exist a considerable amount of theorems that guarantee the existence of fixed points i. e. solutions for the equation ${F(x)=x}$ and we are going to inquire about some of them later on this work. Nonetheless, Banach’s theorem is quite special since it give us on one hand, uniqueness of the solution and on the other hand, an explicit way to find such solution.

Most of the fixed-point theorems are merely an existence result because they are proved using topologic methods, therefore they require generic topological hypotheses about the function ${F}$, hence it is natural to think that the powerful properties of the nonlinearity in Banach’s theorem give us these expecific results.

For the next example consider again the equation (1) with ${y=0}$ and with ${F:\overline{B_1(0)}\subset {\mathbb R}^n\rightarrow {\mathbb R}^k}$ continuous. Let us suppose that ${F\neq 0}$ on ${\mathbb{S}^{n-1}}$. Under these hypothesis the following theorem will show us that the existence of solutions for this nonlinear equation is equivalent to an algebraic topology problem.

Theorem 2 Let ${f:\mathbb{S}^{n-1}\rightarrow {\mathbb R}^k/\{0\}}$ be a continuous function and let us define:

$\displaystyle \phi:=\frac{f}{| f|}:\mathbb{S}^{n-1}\rightarrow \mathbb{S}^{k-1} \ \ \ \ \ (5)$

Then, for any continuous extension ${F:\overline{B_1(0)}\subset {\mathbb R}^n\rightarrow {\mathbb R}^k}$ there exist a solution for the equation ${F(x)=0}$ if and only if ${\phi}$ is not nullhomotopic (homotopic to a constant function).

Proof: For one direction, let us suppose that there exist an extension ${F}$ of ${f}$ such that ${F\neq 0}$ in ${\overline{B_1(0)}}$ and let us define ${H:[0,1]\times \mathbb{S}^{n-1}\rightarrow \mathbb{S}^{k-1}}$ as follows:

$\displaystyle H(t,x):=\frac{1}{| F(tx)|}F(tx) \ \ \ \ \ (6)$

This function cleary gives a homotopy between ${\phi}$ and the constant function with image ${\frac{1}{| F(0)|}F(0)}$.

Conversely, let us suppose that ${\phi}$ is nullhomotopic. We also have that ${\phi}$ and ${f}$ are homotopic, indeed consider the linear homotopy ${I:[0,1]\times \mathbb{S}^{n-1}\rightarrow {\mathbb R}^k/\{0\}}$ given by ${I(t,x)=(1-t)f(x)+t\phi}$. Combining these two facts we can construct a homotopy given by ${H:[0,1]\times \mathbb{S}^{n-1}\rightarrow {\mathbb R}^k/\{0\}}$ such that ${H(0,x)=e_0\in \mathbb{S}^{k-1} }$ for any ${x\in \mathbb{S}^{n-1}}$ and with ${H(1,.)=f}$.

Let us consider the function ${F:\overline{B_1(0)}\subset {\mathbb R}^n\rightarrow {\mathbb R}^k/\{0\}}$ defined as follows:

$\displaystyle F(x):= \begin{cases} H(| x|,\frac{x}{| x|}), \hspace{1mm} x\neq 0,\\ e_0, \hspace{15mm} x=0. \end{cases} \ \ \ \ \ (7)$

Clearly ${F}$ extends ${f}$. In order to check that ${F}$ is continuous at ${0}$ we can take ${\{x_n \}_{n\in{\mathbb N}}}$ any sequence converging to zero, and from this one extract any subsequence ${\{x_{n_k}\}_{k\in{\mathbb N}}}$, by compactness, there exist a sub-subsequence ${\{x_{n_{k_j}}\}_{j\in{\mathbb N}}}$ such that ${\frac{x_{n_{k_j}}}{| x_{n_{k_j}}|}\rightarrow y\in\mathbb{S}^{n-1}}$ as ${j\rightarrow \infty}$, therefore ${\lim\limits_{j \rightarrow \infty} F(x_{n_{k_j}})=e_0}$ proving the continuity of ${F}$.

$\Box$

Remark:${\,}$ Given a topological space ${X}$, the set of homotopy classes of maps ${f:\mathbb{S}^n\rightarrow X}$ can be endowed with a group structure and is called the ${n}$-th homotopy group of ${X}$ denoted by ${\Pi_n(X)}$.

In terms of the last theorem we can guarantee a solution for our nonlinear equation ${F(x)=0}$ when the homotopy class of ${\phi}$ is nontrivial, and this is posible only if ${\Pi_{n-1}(\mathbb{S}^{k-1})\neq \{e\}}$.

The study of the homotopy groups of the spheres ${\Pi_n(\mathbb{S}^k)}$ is still an open question in algebraic topology.

Figure 1: Table of some homotopy groups of spheres. Taken from wikipedia

As we can see in the  Figure 1  we can clasify the homotopy groups ${\Pi_n(\mathbb{S}^k)}$ in three categories:

• If ${n then ${\Pi_n(\mathbb{S}^k)=\{e\}}$. Because any function ${\phi:\mathbb{S}^n\rightarrow \mathbb{S}^k}$ can be deformed to be nonsurjective, see Hatcher’s book.
• If ${n=k}$ then ${\Pi_n(\mathbb{S}^k)={\mathbb Z}}$. In this case the homotopy group counts the number of times that a function ${\phi:\mathbb{S}^n\rightarrow \mathbb{S}^n}$ wraps around the sphere ${\mathbb{S}^n}$. We will focus in this case later in this notes when we discuss about Degree theory.
• The groups associated with the case ${n>k}$ are known as the higher homotopy groups and its study is still an open question, moreover many of the current techniques being developed in algebraic topology are motivated to understand the structure of these groups.

This first glimpse into the nonlinear equations suggest us that it is important to spend some time (and energy), understanding how to take advantage of the structure of the nonlinearity and managing properly the topological information about the feasible set.

In order to keep the discussion as simple as possible, we are going to start addressing the problem where the underlying space is linear and the nonlinearity is approachable by linear functions, i. e. differentiable.