A differentiable characterization of local contractions on Banach spaces
- Andreas Hefti^{1, 2}Email author
https://doi.org/10.1186/s13663-015-0349-7
© Hefti 2015
Received: 18 April 2015
Accepted: 11 June 2015
Published: 7 July 2015
Abstract
This note provides a differentiable characterization of local contractions on an arbitrary Banach space. As a corollary, a refinement to Ostrowski’s sufficient condition for local convergence in finite spaces is obtained, which applies to many models, e.g. in economics, ecology or game theory, where one has an interest in fixed point iterations and local stability of discrete dynamic processes. We show that for the local contraction property to hold, continuity of the derivative at the fixed point is indispensable.
Keywords
MSC
1 Introduction
We show that if the derivative ∂ϕ is continuous at \(x^{*}\), then ϕ induces a local contraction at \(x^{*}\) if and only if there exists an operator norm \(\Vert \cdot \Vert _{ \vert \cdot \vert }\) (in the finite case: a matrix norm) such that \(\Vert \partial\phi(x^{*}) \Vert _{\vert \cdot \vert } <1\) or, equivalently, \(\rho(\partial\phi(x^{*})) <1\). The ‘if’ part means that if \(\rho(\partial\phi(x^{*})) <1\), then not only does the iterative sequence \(\{x^{t}\}\) given by (1) converge locally to \(x^{*}\), but in fact an entire neighborhood U of \(x^{*}\) contracts to \(\{x^{*}\}\) by repeated application of f on U. Continuity of ∂ϕ at \(x^{*}\) is indispensable for this result already in the finite case, which is demonstrated in an example. The ‘only if’ part implies that if (1) converges locally but \(\partial\phi(x^{*})=1\), then ϕ cannot be a local contraction. While local convergence of (1) is weaker than the local contraction property, our result is practically relevant because for many applications, e.g. in evolutionary biology, economics or game theory, one is interested in the local stability of a discrete process (1) at some FP of ϕ, where ϕ typically is continuously differentiable at such FP, and the knife-edge case \(\rho(\partial\phi(x^{*})) =1\) is not ‘generic’.^{1} It therefore makes sense to define a FP \(x^{*}\) of ϕ to be locally contraction-stable,^{2} if ϕ is differentiable around \(x^{*}\), ∂ϕ is continuous at \(x^{*}\) and \(\rho(\partial\phi(x^{*}))<1\).
We first provide a differentiable characterization for a mapping between arbitrary Banach spaces to be a contraction.^{3} This serves as a useful lemma from which we derive a characterization of the local contraction property. Further, we clarify the natural connection of the local contraction property to the linearization of ϕ at \(x^{*}\), and show that continuity of ∂ϕ at \(x^{*}\) is generally indispensable for the local contraction property to hold at \(x^{*}\).
2 A differentiable characterization of contraction mappings
Let \((X,\vert \cdot \vert _{X})\), \((Y,\vert \cdot \vert _{Y})\) be two Banach spaces and \(W \subset X\), \(V \subset Y\) are non-empty subsets. If \(\vert \cdot \vert _{X}\), \(\vert \cdot \vert _{Y}\) are vector norms and \(A:X\rightarrow Y\) is linear and continuous, we denote the corresponding operator norm by \(\Vert A\Vert _{\vert \cdot \vert } \equiv \sup_{\vert v \vert _{X} = 1 } {\vert {Av} \vert _{Y}}\). For \(X=Y\) we say that \(\Vert \cdot \Vert _{\vert \cdot \vert }\) is induced by a single norm if \(\vert \cdot \vert _{X}=\vert \cdot \vert _{Y}\). Most relevant to applications is the case where X and Y are finite-dimensional, such that A can be identified by a matrix, and \(\Vert \cdot \Vert _{\vert \cdot \vert }\) is an (induced) matrix norm.
The function \(\phi: W \rightarrow V\) is a contraction if there are a \(q \in(0,1)\) and norms \(\vert \cdot \vert _{X}\), \(\vert \cdot \vert _{Y}\) such that \(\vert \phi(x'')-\phi(x')\vert _{Y}\leq q\vert x''- x'\vert _{X}\) for any \(x',x'' \in W\). The set of all contractions on W in V is denoted as \(\hat{ \mathcal{K}} (W,V)\). We first characterize when a differentiable mapping is a (global) contraction on a convex set.
Proposition 1
Proof
Remark 1
If \(\bar{W}\) is the closure of W and \(\phi:\bar{W} \rightarrow V\) is continuous, and differentiable on W, it follows that \(\phi\in\hat{\mathcal{K}}(\bar{W},V)\) if and only if (2) is satisfied.
The most important special case has \(X=Y\) and \(\vert \cdot \vert _{Y}=\vert \cdot \vert _{X}\). We denote by \(\mathcal{K}(W,V)\) the set of all contractions \(\phi:W \rightarrow V\), \(W,V \subset X\), where \(\vert \cdot \vert _{Y}=\vert \cdot \vert _{X}\equiv \vert \cdot \vert \). That is, \(\phi\in\mathcal{K}(W,V)\) if ϕ is a contraction for a single norm \(\vert \cdot \vert \). Clearly, \(\mathcal{K}(W,V) \subset\hat{\mathcal{K}}(W,V)\).
The following is an immediate consequence of the proof of Proposition 1.
Corollary 1
Let \(W,V \subset X\), W open and convex, and \(\phi:W \rightarrow V\) be differentiable. \(\phi\in\mathcal {K}(W,V)\) if and only if there is \(\Vert \cdot \Vert _{\vert \cdot \vert }\), induced by a single norm, such that (2) is satisfied. If \(\phi: \bar{W} \rightarrow V\) is continuous, and differentiable on W, \(\phi\in\mathcal{K}(\bar{W},V)\) if and only if there is \(\Vert \cdot \Vert _{\vert \cdot \vert }\), induced by a single norm, such that (2) is satisfied.
Remark 2
If X and Y are finite-dimensional, Proposition 1 and Corollary 1 can be restated as: If and only if (2) is verified by a (not necessarily induced) matrix norm \(\Vert \cdot \Vert \), then \(\phi\in \hat{\mathcal{K}}(W,V)\), resp. \(\phi\in\mathcal{K}(W,V)\).^{4} However, we can restrict ourselves, without loss of generality, to induced matrix norms, when verifying (2).
A geometric interpretation of Corollary 1 is that \(\phi\in\mathcal{K}(W,V)\) if and only if its local rates of change in some normalized direction v is everywhere norm-bounded by \(0< q<1\), i.e. if and only if there is \(\vert \cdot \vert \) and \(0< q<1\) such that each directional derivative \(\partial\phi(x) v\), \(\vert v\vert =1\), satisfies \(\vert \partial\phi(x) v\vert \leq q\) for any \(x \in W\).
2.1 Local contractions
A map \(\phi: W \rightarrow V\), \(W,V \subset X\), is a local contraction at \(x_{0} \in W\) if there is a neighborhood \(B=B(x_{0},\delta)=\{x \in W: \vert x-x_{0}\vert < \delta\}\) such that \(\phi|_{B} \in\mathcal{K}(B,V)\). Further, if \(A:X \rightarrow X\) is a bounded linear operator, the spectral radius of A is \(\rho(A) = \lim_{n \to \infty} \Vert {{A^{n}}} \Vert _{\vert \cdot \vert }^{1/n}\).
Theorem 1
Proof
‘⇒’ Let \(\sigma(x) = {\Vert {\partial\phi({x})} \Vert _{\vert \cdot \vert }}\). Hence \(q \equiv \sigma(x_{0} ) < 1\). As σ is continuous at \(x_{0}\) there is \(B=B(x_{0},\delta)\) such that \(\sup_{x \in B} {\Vert {\partial\phi(x)} \Vert _{\vert \cdot \vert }} < 1 \), and \(\phi|_{B} \in\mathcal{K}(B,V)\) follows from Corollary 1. ‘⇐’ If \(\phi|_{B} \in \mathcal{K}(B,V)\) for \(B=B(x_{0},\delta)\), then \(\Vert \partial\phi (x_{0})\Vert _{\vert \cdot \vert }<1\) for some \(\Vert \cdot \Vert _{\vert \cdot \vert }\) by Corollary 1. Finally, \(\rho (\partial\phi(x_{0}) ) < 1\) if and only if \(\Vert \partial\phi(x_{0})\Vert _{\vert \cdot \vert }<1\) for some operator norm follows, because \(\forall\varepsilon>0\) there exists an operator norm such that \(\rho(\partial\phi(x_{0})) \leq \Vert \partial \phi(x_{0})\Vert _{\vert \cdot \vert } \leq\rho(\partial\phi(x_{0})) + \varepsilon\) (see [2]). □
If the conditions of Theorem 1 are met, and \(x_{0}=x^{*}\) is a FP of ϕ, one can possibly find a neighborhood \(B=B(x^{*},\delta )\) such that \(\phi|_{B}\) is a locally forward-invariant contraction, i.e. \(\phi|_{B} \in\mathcal{K}(B,B)\).
Corollary 2
Proof
By Theorem 1 it only remains to show that \(\phi(B) \subset B\). For \(x \in B\) we have \(\vert \phi (x)-x^{*}\vert =\vert \phi(x)-\phi(x^{*})\vert \leq q \vert x-x^{*}\vert < q \delta < \delta\). □
Remark 3
In the finite-dimensional case (i.e. \(X= \mathbb {R}^{n}\)) the statement in Corollary 2 holds for some matrix norm. That is, \(\phi|_{B} \in\mathcal{K}(B,B)\) if and only if \(\Vert \partial\phi(x^{*})\Vert <1\) for some matrix norm.
The local contraction property of a locally continuously differentiable function can be entirely described by the linearization \(L_{x^{*}}(x)=\partial\phi(x^{*})x+ (I-\partial\phi(x^{*}) )x^{*}\) of ϕ at a FP \(x^{*}\).
Corollary 3
Let \(\phi:W \rightarrow V\), \(W,V \subset X\), be differentiable, \(\phi(x^{*})=x^{*}\) and ∂ϕ is continuous at \(x^{*}\). Then ϕ is a local forward-invariant contraction at \(x^{*}\) if and only if \(L_{x^{*}} \in\mathcal{K}(X,X)\).
Proof
‘⇒’ If \(L_{x^{*}} \in\mathcal{K}(X,X)\), then there is \(\vert \cdot \vert \) and \(q<1\) such that \(\vert \partial \phi(x^{*})v\vert \leq q\vert v\vert \), \(v \in X\). Hence also \(\Vert \partial\phi(x^{*})\Vert _{\vert \cdot \vert } \leq q\), and the claim follows from Corollary 2. ‘⇐’ Equation (4) implies that \(\Vert \partial\phi (x^{*})\Vert _{\vert \cdot \vert } \equiv q<1\) for some \(\vert \cdot \vert \). Hence \(\vert L(x')-L(x)\vert =\vert \partial\phi(x^{*})(x'-x)\vert \leq \Vert \partial\phi(x^{*})\Vert _{\vert \cdot \vert }\vert x'-x\vert = q \vert x'-x\vert \), for any \(x,x' \in X\). □
Illustration: In game theory the function ϕ would correspond to the joint best-reply function of all players, where it is the standard case that ϕ is continuously differentiable at interior FP of ϕ, and \(\rho(\partial\phi(x^{*})) = 1\) is not robust to small perturbations of the game (e.g. [3]).
We obtain as a corollary a result by Kantorovich and Akilov, who show that if \((X, \vert \cdot \vert _{X})\) is a fixed Banach space, \(W \subset X\), \(\phi: W \rightarrow X\) is differentiable, \(W_{0} \subset W\) is closed and convex, \(\phi(W_{0}) \subset W\) and (2) is satisfied, then ϕ is a contraction on \(W_{0}\) ([5], Chapter 17, p.501).
Declarations
Acknowledgements
I am very grateful to two anonymous referees for their clear and useful suggestions and comments as well as to Ines Brunner for her continuing support and input.
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