On the stability of finding approximate fixed points by simplicial methods
- Qi-Qing Song^{1, 2},
- Hui Yang^{1}Email author,
- Guang-Hui Yang^{1} and
- Chong-Yi Zhong^{1}
https://doi.org/10.1186/s13660-016-1162-5
© Song et al. 2016
Received: 24 June 2016
Accepted: 4 September 2016
Published: 13 September 2016
Abstract
This paper reports some new results in relation to simplicial algorithms considering continuities of approximate fixed point sets. The upper semi-continuity of a set-valued mapping of approximate fixed points using vector-valued simplicial methods is proved, and thus one obtains the existence of finite essential connected components in approximate fixed point sets by vector-valued labels; examples are given to show that this is very different from the property for integer-valued labeling simplicial methods. The existence of essential sets is also proved focusing on both perturbations of domains and functions.
Keywords
stability approximate fixed point simplex1 Introduction
Fixed point theorems have important effects in mathematical and economic sciences. The famous Brouwer fixed point theorem [1] plays a key role in many existence problems and also has prompted a wave of finding many kinds of equilibria and other applications, such as the Nash equilibrium[2], the general equilibrium [3], network problems [4–6], approximation theory [7], computer science [8], etc.
Naturally, designing algorithms to compute a Brouwer fixed point is also important field. It is well known that Sperner’s lemma became a simple tool for the proof of the existence of Brouwer fixed points. Based on Sperner’s lemma, simplicial algorithms continue to spring up after the excellent work by Scarf [9], such as Kuhn’s algorithm [10, 11], the restart algorithms [12–14], variable dimension algorithms [15], and homotopy algorithms [16, 17]. For simplicial algorithms, one frequently finds a complete labeled sub-simplex (full labeled sub-simplex) in a simplex to approximate a fixed point. Two common labels are integer-valued and vector-valued. Given a fixed grid size, it is well known that the approximate degree of a complete vector-valued sub-simplex to fixed points is better than the other. Is there any difference between the stability of these algorithms? Is a complete labeled sub-simplex able to resist the perturbation of functions or simplices? This paper will focus on these problems.
The stability of fixed points has attracted much attention. After the seminal work for essential fixed points of continuous functions (Brouwer type fixed points) in [18], essential components and essential sets of fixed points were introduced [19, 20]. From the view point of stability, as the analogs of singletons, minimal essential sets seem to be good choices [21]. Essential stabilities (which are related to lower semi-continuity) were used to analyze many problems, such as coincidence points [22, 23], fixed points[24], KKM points [25, 26], game equilibrium points [27–31], maximal elements [32], and variational relation problems [33–35], etc.
In fact, when a simplicial algorithm is in order, we must face approximate fixed points as the grid size shrinks. By employing the essential stabilities, this concerns the stability of approximate fixed point sets using simplicial methods under the perturbation of the corresponding functions and domains. We show that there is a significant difference between vector-valued simplicial methods and integer-valued methods. The upper semi-continuity of a set-valued mapping for approximate fixed points using vector-valued labeling is proved. The existence of finite essential connected components of approximate fixed point sets is also proved for vector-labeled simplicial methods. These results are new.
2 Preliminaries and motivations
Let S be an n-simplex in \(R^{n+1}\) with vertices \(v^{1},v^{2},\ldots ,v^{n+1}\), \(C(S)\) the space of continuous functions f on S with uniform metric and \(I_{k}=\{1,2,\ldots,k\}\). The ith unit vector of \(R^{n+1}\) is denoted by \(e(i)\), \(i=1,2,\ldots,n\), and the \((n+1)\)-vector \((1,1,\ldots,1)^{T}\) is represented by e.
- (i)
each component of \(y^{1}\) is a multiple of \(\frac{1}{q}\);
- (ii)
\(\pi=(\pi_{1},\pi_{2},\ldots,\pi_{n})\) is a permutation of the elements of \(I_{n}\);
- (iii)
\(y^{i+1}=y^{i}+\frac{v(\pi_{i+1})-v(\pi_{i})}{q}\), where \(v(i)=v^{i}\), \(\forall i\in I_{n}\).
Given a grid size \(\frac{1}{q}\), for each \(f\in C(S)\), denote by \(F(f,q)\) (\(F'(f,q)\)) the collection of all sub-simplices with complete integer-valued (vector-valued) labels in S, then we define a set-valued mapping from \(C(S)\) to S with \(F(F'):C(S)\rightarrow 2^{S}\), in addition, for convenience of notation, we write \(F(f)\) as \(F(f,q)\). Note that each \(x\in F(f,q)\ (F'(f,q))\) is an approximation of fixed points of f on S. In addition, from the connectedness, \(F(f,q)\) (\(F'(f,q)\)) can be decomposed as \(\bigcup_{i\in\Lambda}C_{i}\) with \(C_{i}\cap C_{j}=\emptyset\) for any \(i\neq j\), and \(C_{i}\), \(\forall i\in\Lambda\) is a connected component.
There are significant differences in relation to semi-continuities between F and \(F'\), and the following example shows that the set-valued mapping F is not upper semi-continuous on \(C(S)\); further results will be demonstrated in Section 3.
Example 2.1
For each \(f\in C(S)\), denote by \(\operatorname{Fix}(f)\) the fixed point set of f on S. Note the fact of Example 2.1 about F, the following definitions consider a kind of description for stability of \(F'\) and subsets of \(\operatorname{Fix}(f)\).
Definition 2.1
Given the grid size \(\frac{1}{q}\), for each \(f\in C(S)\), a closed subset \(e(f)\) of \(F'(f,q)\) is called an essential set with respect to \(C(S)\) if for any open set U with \(U\supset e(f)\), there exists an open neighborhood \(O(f)\) of f in \(C(S)\) such that \(F'(f',q)\cap U \neq\emptyset\), \(\forall f'\in O(f)\). If a connected component \(C\subset F'(f,q)\) is an essential set, C is called an essential connected component of \(F'(f,q)\) with respect to \(C(S)\).
Definition 2.2
Let \(f\in C(S)\), \(e(f)\) be a closed subset of \(\operatorname{Fix}(f)\). We call \(e(f)\) an approximate essential set if for each ε neighborhood \(B(e(f),\varepsilon)\) of \(e(f)\), there exists a \(\delta>0\) such that, for each \(f'\in C(S)\) with \(\|f-f'\|<\delta\), we can find a number Z, such that \(F'(f',q)\cap B(e(f),\varepsilon)\neq\emptyset\), \(\forall q>Z\).
Lemma 2.1
(see [36])
Let Y be a metric space, E be a Baire space, and \(F : E \to2^{Y}\) be an upper semi-continuous mapping with compact values. Then there is a dense residual subset Q of E such that F is lower semi-continuous at each \(x\in Q\).
3 Stability results under function perturbations
Theorem 3.1
Proof
Let \((f^{m},x^{m})\in \operatorname{Gr}F'\) with \((f^{m},x^{m})\rightarrow(f^{0},x^{0})\), \(m=1,2,\ldots \) . It is clear that \((f^{0},x^{0})\in C(S)\times S\). We need to show that \(x^{0}\) is a point of a complete sub-simplex \(\sigma _{f_{0}}\) with vector-valued labels in S. For each \(m=1,2,\ldots \) , since \((f^{m},x^{m})\in \operatorname{Gr}F'\), there exists a complete labeled sub-simplex \(\sigma_{f_{m}}\) such that \(x^{m}\in\sigma_{f^{m}}\subset F'(f^{m},q)\subset S\), hence, denote \(\sigma_{f^{m}}\) as \(\sigma_{f^{m}}(y^{1}_{m}, y^{2}_{m}, \ldots ,y^{n+1}_{m})=\sigma_{f^{m}}(y^{1}_{m},\pi_{m})\).
From Theorem 3.1, we have the following direct corollary.
Corollary 3.1
Given a triangulation of S with a grid size \(\frac{1}{q}\), the set-valued mapping \(F'\) is upper semi-continuous on \(C(S)\).
The following example shows that \(F'\) does not possess the property of being lower semi-continuous on \(C(S)\).
Example 3.1
From Theorem 3.1, the set-valued mapping \(F'\) with \(F':C(S)\to 2^{S}\) is upper semi-continuous. If the set-valued mapping \(F'\) is lower semi-continuous at a point \(f^{0}\), then, given a grid size \(\frac {1}{q}\), clearly, each point in \(F'(f^{0},q)\) is essential. Thus, by Fort’s lemma (Lemma 2.1) and Definition 2.1, we can obtain the following generic stability result.
Corollary 3.2
Given a grid size \(\frac{1}{q}\), there exists a dense residual set Q in \(C(S)\) such that for each \(f\in C(S)\), each point in \(F'(f,q)\) is essential with respect to \(C(S)\).
Theorem 3.2
Given a triangulation of S with a grid size \(\frac{1}{q}\), for each \(f\in C(S)\), there exist finite essential connected components in \(F'(f,q)\) with respect to \(C(S)\).
Proof
From Theorem 3.1, the set-valued mapping \(F'\) is upper semi-continuous on S. Then the set \(F'(f,q)\) itself is an essential set with respect to \(C(S)\). Let Φ denote the collection of all essential sets in \(F'(f,q)\). Note that each decreasing chain in Φ with the set inclusion order has its intersection as a lower bound. Therefore, there exists a minimal element \(e(f)\) in Φ, which is an essential set in \(F'(f,q)\). Hence, it is clear that each connected component C with \(C\supset e(f)\) is an essential connected component by Definition 2.1. Then the remaining problem is to show that each \(e(f)\) is connected.
Finally, from the finiteness of the complete labeled simplex in S, the result follows. □
The following result shows that essential connected components under the grid size \(\frac{1}{q}\) can be close to an approximate fixed point set as q tends to infinity.
Theorem 3.3
Given a continuous function \(f\in C(S)\), for each grid size \(\frac{1}{q}\), let \(C^{q}\subset F'(f,q)\) be an essential connected component with respect to \(C(S)\), there exists a subsequence \(\{C^{q_{k}}\}\) of \(\{C^{q}\}\) with \(C^{q_{k}}\stackrel {h}{\rightarrow}C^{0}\) and \(C^{0}\) is an approximate essential connected set in \(\operatorname{Fix}(f)\), where h is the Hausdorff metric induced by the Euclidean metric on \(R^{n+1}\).
Proof
Assume that \(C^{0}\) is not connected, then \(C^{0}\) can be decomposed as two disjoint compact sets like \(C^{0}=C'\cup C''\) with two open sets \(W'\) and \(W''\) such that \(C'\subset W'\), \(C''\subset W''\), and \(W'\cap W''=\emptyset\). By the compactness of \(C'\) and \(C''\), there are two open sets \(U'\) and \(U''\) such that \(C'\subset U'\subset\bar{U}'\subset W'\) and \(C''\subset U''\subset\bar{U}''\subset W''\). Since \(C^{q}\) is connected, we have \(C^{q}\subset U'\) or \(C^{q}\subset U''\) as q large enough. Then the limit of \(C^{q}\) is in \(W'\) or \(W''\), which contradicts the fact that \(C^{q}\stackrel{h}{\rightarrow}C'\cup C''\) and \(C'\subset W'\), \(C''\subset W''\), and \(W'\cap W''=\emptyset\). Therefore, \(C^{0}\) is connected.
Finally, we show that \(C^{0}\) is an approximate essential set of \(\operatorname{Fix}(f)\). If not, then there exists a \(\bar{\varepsilon}>0\) and \(f^{j}\) (\(j=1,2,\ldots\)) with \(f^{j}\to f\), such that for each number q, \(F'(f^{j},q)\cap B(C^{0},\bar{\varepsilon})=\emptyset\), \(j=1,2,\ldots \) . Since \(C^{q}\stackrel{h}{\rightarrow}C^{0}\), there is a number N such that \(C^{q}\subset B(C^{0},\bar{\varepsilon})\) when \(q\ge N\). Because \(C^{N}\) is essential, for the open set \(B(C^{0},\varepsilon)\), there is a \(\delta>0\) such that for any \(f'\) with \(\|f-f'\|<\delta\), we have \(F(f',N)\cap B(C^{0},\varepsilon)\neq\emptyset\). From the fact that \(f^{j}\to f\), for large enough j, we have \(F(f^{j},N)\cap B(C^{0},\varepsilon)\neq\emptyset\), a contradiction. □
4 Stability results under perturbations of simplices and functions
Lemma 4.1
ρ is a metric on M.
Proof
(ii) For any \(S_{1}(v_{1}^{1},v_{1}^{2},\ldots,v_{1}^{n+1}), S_{2}(v_{2}^{1},v_{2}^{2},\ldots,v_{2}^{n+1})\in M\), we have \(\rho(S_{1},S_{2})=0 \Leftrightarrow S_{1}=S_{2}\). From the definition of ρ, one needs only the proof of the necessity. Let \(\rho(S_{1},S_{2})=0\), then there exists π̄ such that \(\sum_{k=0}^{n+1}\|v_{1}^{k}-v^{\bar{\pi }_{k}}_{2}\|=0\), which means that \(\|v_{1}^{k}-v^{\bar{\pi}_{k}}_{2}\|=0\), \(\forall k\in I_{n+1}\). That is, \(S_{1}=S_{2}\).
Concerning a stability analysis of approximate fixed points, we intend to restrain domains to avoiding a domain perturbed in a large-scale range. Let Δ be an n dimensional subset of a compact set X in \(R^{n+1}\). Let \(M'\subset M\) satisfy \(M'=\{S\in M: \Delta\subset S\subset X\}\).
Lemma 4.2
The metric space \((M',\rho)\) is complete.
Proof
Take a Cauchy sequence \(\{S_{m}(v^{1}_{m},v^{2}_{m},\ldots,v^{n+1}_{m})\}\) in \(M'\). Then, for each \(\varepsilon>0\), there exists a number N such that \(\rho(S_{s},S_{t})<\varepsilon\) for any \(s,t>N\). Without loss of generality, we can assume that \(\rho(S_{s},S_{t})=\sum_{k=0}^{n+1}\| v_{s}^{k}-v^{k}_{t}\|\). Therefore, \(\{v^{k}_{m}\}\) is a Cauchy sequence with the limit \(v^{k}_{0}\), \(\forall k\in I_{n+1}\). Denote by \(S_{0}\) the simplex \(S_{0}(v^{1}_{0},v^{2}_{0},\ldots,v^{n+1}_{0})\). Then we have \(\rho (S_{m},S_{0})\rightarrow0\). Since \(\Delta\subset\bigcap_{m=1}^{\infty}S_{m}\subset X\), it follows that \(\Delta\subset S_{0}\subset X\), hence \(S_{0}\) is an n-simplex in \(M'\). □
Given a grid size \(\frac{1}{q}\), for each \(u=(f,S)\in P\), let \(T(u,q)\) be the set of all sub-simplices with complete vector-valued labels with the function f in the triangulation of S under the grid size \(\frac {1}{q}\), then we define a set-valued mapping T from P to X.
Similar to Definition 2.1, we consider the essential stability of approximate fixed points under both perturbations of functions and domains.
Definition 4.1
Given the grid size \(\frac{1}{q}\), for each \(u=(f,S)\in P\), we call a closed subset \(e(f)\) in \(T(u,q)\) an essential set with respect to P if, for any open set U with \(U\supset e(f)\), there is an open \(O(u)\) of u in P such that \(U\cap T(u',q)\neq\emptyset\), \(\forall u'\in O(u)\). A minimal element in the collection (ordered by set inclusion) of essential sets in \(T(u,q)\) is called a minimal essential set with respect to P.
Theorem 4.1
Given a grid size \(\frac{1}{q}\) and a continuous function \(f\in C(X)\), the graph of the set-valued mapping T, \(\operatorname{Gr}T=\{(u,x)\mid u\in P, x\in T(u,q)\}\), is closed.
Proof
Let \(\{(u_{m},x_{m})\}\subset \operatorname{Gr}T\) with \((u_{m},x_{m})\rightarrow(u_{0},x_{0})\), where \(u_{m}=(f_{m},S_{m})\), \(u_{0}=(f_{0},S_{0})\), and \(S_{m}\) is the simplex with \(v^{1}_{m},v^{2}_{m},\ldots,v^{n+1}_{m}\) as its vertices for each \(m=1,2,\ldots \) . Since \((u_{m},x_{m})\in \operatorname{Gr}T\), there exists a complete sub-simplex \(\sigma _{f_{m}}(y^{1}_{m}, y^{2}_{m}, \ldots,y^{n+1}_{m})\) with vector-valued labels such that \(x_{m}\in\sigma_{f_{m}}\subset T(u_{m},q)\subset S_{m}\), \(m=1,2,\ldots \) .
To finish the proof that \(x_{0}\in\sigma(y^{1}_{0},\pi)\) and \(\sigma (y^{1}_{0},\pi)\) is a complete sub-simplex with vector-valued labels by function \(f_{0}\), we can adopt the corresponding part of Theorem 3.1. □
From Theorem 4.1, T is upper semi-continuous on P. Following the proof of Theorem 3.2 for the part of the existence of minimal element of essential sets, we obtain the following result.
Theorem 4.2
For each \(u=(f,S)\in P\), given a triangulation of S with a grid size \(\frac{1}{q}\), there exists a minimal essential set in \(T(u,q)\) with respect to P.
Declarations
Acknowledgements
This project is supported by the National Natural Science Foundation of China (11271098, 11661030), the China Postdoctoral Science Foundation (2016M590905), and the Doctoral Research Fund of Guilin University of Technology.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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