Derivative-based sufficient conditions for contraction mappings for multivariate continuous functions

Question

I have an economic model whose equilibrium is characterized by the system $F(x)=0$ for $x\in\mathbb{R}^N$ and $F$ continuously differentiable. I am looking for sufficient conditions for when the function $F$ is a contraction, in order to show uniqueness of its solution.

I have two questions. First, there is a paper I'm familiar with using a derivative-based condition, but I can't find a reference for this condition. The paper is found here (http://www.princeton.edu/~reddings/papers/TechAppendix_Berlin_031115_all.pdf), and the uniqueness theorem is at the bottom of page 15. It states that if an N-dimensional function $D$ satisfies the following conditions

$\lim_{x_i\to0} D_i(x)=\infty, \lim_{x_i\to\infty} D_i(x)=0, \\\frac{\partial D_i}{\partial x_i}<0, \frac{\partial D_i}{\partial x_j}<0, \big|\frac{\partial D_i}{\partial x_i}\big|>\big|\frac{\partial D_i}{\partial x_j}\big|$

Then the system $D(x)= x_0$ with $x_0\in \mathbb{R}^N$ has a unique solution. I believe the conditions on the derivatives of their function $D$ are sufficient for it to be a contraction, but I can't find a reference for the underlying theorem and they do not cite it themselves (only stating that "it follows that there exists a unique solution".) Does anyone know of a reference that proves these conditions are sufficient for a contraction mapping?

Second, I'm looking for a good source of derivative-based sufficiency conditions for contraction mappings. Does anyone know of a good source? For example, in some lecture slides I've found another derivative-based condition for multivariate smooth functions based on diagonal dominance (Screenshot of Additional Sufficiency Condition). Ideally there would be single source that collects these. (I've found other sources on contraction theorems, but most are abstract/topological and don't contain derivative-based theorems which are useful to work with on a day-to-day basis).

Thanks!

Mark

Answer 1

I haven't checked whether your exact conditions here satisfy the below, but I have a hunch you're barking up the wrong tree looking for contractions.

Theorem: (Gale-Nikaido)$^1$ Let $\Omega$ be rectangle in $\mathbb{R}^n$, and let $F : \Omega \to \mathbb{R}^n$ be a $C^1$ map such that for all $x \in \Omega$, the Jacobian matrix $J_F(x)$ has the property that all principal minors have strictly positive determinant. Then $F$ is injective.

It seems that your first two conditions show that there is some compact rectangle in the positive orthant of $\mathbb{R}^n$ containing the solution on the interior; the latter partial conditions seem to be suggesting some sort of injectivity type argument is what's important. I'd have a go at seeing if these conditions given on the partials imply the principal minors condition given above.

This type of stuff is commonly thrown around, say, for people interested in uniqueness of more game-theoretic things like Cournot equilibrium (as you seem to be a fellow economist). Vives' book on oligopoly has some nice introductory discussion.

Edit: regarding nice sufficient conditions for contraction, the classics are dominant diagonals which you're aware of. Supposing that you're interested in contractions for purposes of establishing uniqueness then there are two other tools that can prove useful (the following modulo technical details). The first of these is the univalence approach indicated above: if one's equilibrium condition is $F(x) = 0$, then ensuring the principal minor condition on $J_F(x)$ suffices, even if $F+\textrm{id}$ is not a contraction. In the 1-D case this is just asking that $F'$ have constant sign. More generally (and less easy for verify) is that we ask that the Jacobian determinant $|J_{-F}(x^*)|$ for all $x^*$ such that $F(x^*)=0$. This is like asking in the 1-D case that $F'<0$ only at solutions to $F=0$. The more general variant implies uniqueness via index (topological) arguments.

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$^1$ Gale, Nikaido "The Jacobian Matrix and Global Univalence of Mappings" Mathematische Annalen 159 (1965): 81-93. http://eudml.org/doc/182673.