Some results concerning the solution mappings of mixed variational inequality problems
- Qi-Qing Song^{1, 2} and
- Hui Yang^{1}Email author
https://doi.org/10.1186/s13660-016-1176-z
© Song and Yang 2016
Received: 1 June 2016
Accepted: 13 September 2016
Published: 26 September 2016
Abstract
This paper shows some continuities of mappings between the space of mixed variational inequality problems and the graph space of their solution mappings. The space of mixed variational inequality problems is homeomorphic to the graph of a continuous mapping. These generalize the results in the corresponding references.
Keywords
MSC
1 Introduction and preliminaries
Variational inequalities have become important methods to analyze many linear and nonlinear problems; see [1] and [2], such as linear complementarity problems in [3], convex optimization in [4], imaging problems in [5], etc.
Classical variational inequalities were introduced by Hartman and Stampacchia in the 1960s; see [6] and [7]. They are of the form suitable to find \(u^{*}\) such that \((u-u^{*})^{T}f(u^{*})\geq0\), \(\forall u\in K\). The spaces consisting of nonlinear problems may have some interesting topological properties: some nonlinear problem spaces are homeomorphic to the graph spaces of their solution mappings, such as bimatrix games in [8], normal game problems in [9, 10], game trees in [11], classical variational inequality problems [12], etc.
Duvaut and Lions considered a kind of mixed variational inequality, see [13], which added a function to a classical variational inequality. The stability, algorithm, and generalization of this kind of variational inequality has been studied in many forms [14–19] and applied to many fields; see [20].
Definition 1.1
A mapping \(f:K\rightarrow\mathbb{R}^{n}\) is said to be monotone if \((f(x)-f(y))^{T}(x-y)\geq0\), \(\forall x, y\in K\); f is said to be strictly monotone if, f is monotone and if \((f(x)-f(y))^{T}(x-y)=0\), then \(x=y\); f is strongly monotone if there exists a constant c such that \((f(x)-f(y))^{T}(x-y)\geq c\|x-y\|^{2}\), \(\forall x, y\in K\).
2 Main results
Theorem 2.1
ψ is a continuous mapping from M to N. ϕ is continuous on N.
Proof
For each \((\theta, f)\in M\), since \(\psi(\theta, f)=(\theta, f+C_{\bar{u}}, \bar{u})\), where \(\bar{u}\in V(\theta, f+I)\), we have \(\theta(u)-\theta(\bar{u})+(u-\bar{u})^{T}(f(\bar{u})+\bar{u})\geq 0\), \(\forall u\in K\), that is, \(\bar{u}\in V(\theta, f+C_{\bar{u}})\). ψ maps M onto N.
Let X be the set of \((\theta, f)\) satisfying: (i) \(\theta:K \to \mathbb{R}\) is proper convex lower semi-continuous; (ii) \(f:K \to {\mathbb{R}^{n}}\) is continuous and monotone. Then each variational inequality \((\theta, f)\in X\) has a solution (see [21]) and further if f is strictly monotone, then the corresponding solution is unique. Then \((X,\rho)\) is a metric space. From Theorem 2.1, there can be obtained some stability results for solution sets of mixed monotone variational inequalities.
Corollary 2.1
The set-valued mapping V is upper semi-continuous from M to \(2^{K}\).
Proof
From the proof of Theorem 2.1, \(\psi:X\rightarrow N\) is continuous, noting that \(M\subset X\) is closed, then N, the graph of V, is closed. Therefore, V is upper semi-continuous. □
Remark 2.1
By Corollary 2.1, V is upper semi-continuous on M. Then, when \(f_{0}+\varepsilon_{n} I\rightarrow f_{0}\) as positive \(\varepsilon_{n}\rightarrow0\) and \(\theta_{n}\rightarrow\theta_{0}\), there is a convergent subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) with \(\{ x_{n}\}= V(\theta_{n}, f_{0}+\varepsilon_{n} I)\) such that \(x_{n_{k}}\rightarrow{x_{0}}\in V(\theta_{0},f_{0})\). Alternatively, from Theorem 2.1, let \(\varepsilon_{n}>0\), if we define \(\psi_{\varepsilon_{n}} (\theta, f)=(\theta, f+\varepsilon_{n} C_{\bar{u}_{n}}, \bar{u}_{n})\) for each \((\theta, f)\in M\), where \(\{ \bar{u}_{n}\}=V(\theta,f+\varepsilon_{n} I)\), then \(\psi_{\varepsilon_{n}}\) is also a continuous mapping from M to N. Note that \(\bar{u}_{n}\in V(\theta, f+\varepsilon_{n} C_{\bar{u}_{n}})\), therefore, if \(\varepsilon_{n}\rightarrow0\), we assert that there exists a subsequence \(\{\bar{u}_{n_{k}}\}\) of \(\{\bar{u}_{n}\}\) with \(\bar{u}_{n_{k}}\rightarrow u_{0}\in V(\theta, f)\).
Theorem 2.2
The mappings \(\phi\circ\psi\) and \(\psi\circ\phi\) are identity mappings on M and N, respectively.
Proof
Theorem 2.3
The spaces N and M are homeomorphic.
Remark 2.2
The homeomorphism results were shown between game spaces and the graphs of solutions mappings in games [9–11]. For any population game \(F:X\rightarrow\mathbb{R}^{n}\), each Nash equilibrium point \(x^{*}\) of the game F is equivalent to \(x^{*}\) being a solution of variational inequality \((y-x^{*})^{T} F(x^{*})\leq0\); see [22]. A population game F belongs to the class of stable population games when F is monotone, then, from Theorem 2.3, we can assert that the space of stable population games is homeomorphic to the graph space of their solution mappings. Theorem 2.3 generalizes the homeomorphism result for classical variational inequalities in [12].
From Theorem 2.3, N is homeomorphic to the space M. Furthermore, we see that N is homeomorphic to the graph of a continuous mapping from M to K.
Let π be the projection from N to K and K̃ be a compact convex subset of \(\mathbb{R}^{n}\) with \(K\subset \operatorname{int}(\bar{f})\). Then there exists a retraction r from K̃ to K, that is, \(r(x)=x\), \(\forall x\in K\). From Urysohn lemma, there is a continuous mapping s from K̃ to the closed interval \([0,1]\) such that \(s(x)=1\), \(\forall x\in K\), and \(s(x)=0\), \(\forall x\in \operatorname{Bd}(K_{\varepsilon})\), where \(\operatorname{Bd}(K_{\varepsilon})\) denotes the boundary of K̃.
Theorem 2.4
Let \(h=\pi\circ\psi:M\rightarrow K\). Then: (i) for each \(t\in [0,1]\), \(\beta_{t}\circ\alpha_{t}\) and \(\alpha_{t}\circ\beta_{t}\) are identity mappings on \(M\times\tilde{K}\); (ii) for each \(t\in [0,1]\), \(\tilde{x}\in \operatorname{Bd}(\tilde{K})\), \(\alpha_{t}(\cdot,\cdot,\tilde {x})\) is a constant mapping; (iii) \(\alpha_{0}\) is an identity on \(M\times\tilde{K}\); \(\alpha_{1}\) is a homeomorphism between N and the graph of h with its inverse \(\beta_{1}\).
Proof
(i) For each \((\theta,f,\tilde{x})\in M\times\tilde{K}\), we have \(\beta_{t}\circ\alpha_{t}(\theta,f,\tilde{x})=\beta_{t}(\theta ,f-s(\tilde{x})tC_{x},\tilde{x}) =(\theta,f,\tilde{x})\) and \(\alpha_{t}\circ\beta_{t}(\theta,f,\tilde {x})=\alpha_{t}(\theta,f+s(\tilde{x})tC_{x},\tilde{x}) =(\theta,f,\tilde{x})\).
(ii) For each \(\tilde{x}\in \operatorname{Bd}(\tilde{K})\) and \((\theta,f,\tilde {x})\in M\times\tilde{K}\), \(\alpha_{t}(\theta,f,\tilde{x})=(\theta ,f-s(\tilde{x})tC_{x},\tilde{x})\). Noting that \(\tilde{x}\in \operatorname{Bd}(\tilde {K})\), we have \(s(\tilde{x})=0\), then \(\alpha_{t}(\theta,f,\tilde {x})=(\theta,f,\tilde{x})\).
(iii) It is clear that \(\alpha_{0}\) is an identity on \(M\times\tilde {K}\). Next, for each \((\theta,f,\tilde{x})\in N\), we have \(\tilde {x}\in V(\theta,f)\), then \(\tilde{x}\in K\). Hence \(s(\tilde{x})=1\) and \(r(\tilde{x})=x=\tilde{x}\). Therefore, \(\alpha_{1}(\theta ,f,\tilde{x})=(\theta,f-C_{x},\tilde{x})=(\phi(\theta,f,x),x)\). One needs to show that \(x=\pi\circ\psi(\phi(\theta,f,x))\). Since \(\psi \circ\phi\) is an identity on N, we have \(\pi\circ\psi(\phi (\theta,f,x))=\pi(\theta,f,x)=x\). Conversely, for each point \((\theta,f,\tilde{x})\) on the graph of the h, we need to show \((\theta,f,\tilde{x})\in\alpha_{1}(N)\). Since \(\tilde{x}=\pi\circ \psi(\theta,f)\), we have \(\psi(\theta,f)=(\theta,f+C_{\tilde {x}},\tilde{x})\), then \((\theta,f+C_{\tilde{x}},\tilde{x})\in N\). Hence, \(\alpha_{1}(\theta,f+C_{\tilde{x}},\tilde{x})=(\theta,\hat {f},\tilde{x})\) with \(\hat{f}=f+C_{\tilde{x}}-s(\tilde{x})C_{x}\) and \(x=r(\tilde{x})\). Noting that \((\theta,f+C_{\tilde{x}},\tilde {x})\in N\), we have \(\tilde{x}\in K\), then \(x=r(\tilde{x})=\tilde {x}\) and \(s(\tilde{x})=1\). Therefore, \(\hat{f}=f\). The proof is completed. □
Remark 2.3
From Theorems 2.3 and 2.4, the graph of set-valued mapping V is homeomorphic to M and the graph of a continuous mapping. Easily, we see that the graph of a continuous mapping can be homeomorphic to the graph of a constant mapping. Therefore, N and M are all homeomorphic to the graph of a constant mapping. Like (un)knots and Nash dynamics [23], this may contribute to variational dynamics like [24].
In the following part, we generalize the results (Theorems 2.1-2.3) to Hilbert spaces in relation to linear mappings.
Let K be a compact convex subset in a Hilbert space \((X, \langle \cdot,\cdot\rangle)\), where \(\langle\cdot,\cdot\rangle\) represents the inner product on X. Denote by \(X^{*}\) the dual space of X (all continuous linear mappings on X). A mapping T from K to \(X^{*}\) is said to be monotone if \((T(u)-T(v), u-v)\geq0\), \(\forall u, v\in K\), where \((\cdot,\cdot)\) is the pairing of \(X^{*}\) and X. A monotone mapping T is called strictly monotone if \((T(u)-T(v), u-v)=0\) implies \(u=v\).
For each \((\theta,T,f)\), it is well known that this kind of mixed variational inequality problem with \((\theta,T,f)\) has a solution; if T is strictly monotone, then the solution is unique; denote by \(V'(\theta,T,f)\) the set of all solutions of the problem \((\theta ,T,f)\), then a set-valued mapping \(V'\) from \(M'\) to K is well defined.
Theorem 2.5
\(\psi'\) is a continuous mapping from \(M'\) to \(N'\).
Proof
Theorem 2.6
\(\phi'\) is a continuous mapping from \(N'\) to \(M'\).
Proof
Theorem 2.7
\(\phi'\circ\psi'\) and \(\psi'\circ\phi'\) are identity mappings on \(M'\) and \(N'\), respectively.
Proof
Theorem 2.8
The spaces \(M'\) and \(N'\) are homeomorphic.
Declarations
Acknowledgements
This project is funded by National Natural Science Fund of China (11271098, 11661030, 11561013), Guangxi Natural Science Fund (2016GXNSFAA380059), and China Postdoctoral Science Foundation (2016M590905).
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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