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Some results concerning the solution mappings of mixed variational inequality problems

Journal of Inequalities and Applications20162016:236

  • Received: 1 June 2016
  • Accepted: 13 September 2016
  • Published:


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.


  • mixed variational inequality
  • monotone
  • homeomorphism


  • 49J53
  • 58E35
  • 47H04

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 [1419] and applied to many fields; see [20].

Let K be a compact convex subset of \(\mathbb{R}^{n}\). We consider the following mixed variational inequality problem: to find a point \(u^{*}\in K\) such that \(u^{*}\) satisfies
$$ \theta(u)-\theta\bigl(u^{*}\bigr)+\bigl(u-u^{*}\bigr)^{T}f \bigl(u^{*}\bigr)\geq0,\quad \forall u\in K, $$
where \(f:K\rightarrow\mathbb{R}^{n}\) is a mapping and \(\theta: K\rightarrow\mathbb{R}\) is a real function. This was first introduced in [13]. When \(f\equiv0\) on K, this leads us to find a maximizer \(u^{*}\in K\) of the function θ on K.

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 \((f(x)-f(y))^{T}(x-y)\geq c\|x-y\|^{2}\), \(\forall x, y\in K\).

If f is monotone and θ is convex, we call the above variational inequality (1) a mixed monotone variational inequality problem. Denote by M the set as follows:
$$M = \left \{(\theta,f)\mbox{:}\quad \textstyle\begin{array}{@{}l@{}} \theta:K \to\mathbb{R}\mbox{ is continuous and convex}; \\ f:K \to{\mathbb{R}^{n}}\mbox{ is continuous and monotone} \end{array}\displaystyle \right \}. $$
For any two \((\theta_{1}, f_{1}), (\theta_{2}, f_{2})\in M\), define the metric ρ between \((\theta_{1}, f_{1})\) and \((\theta_{2}, f_{2})\) as
$$ \rho\bigl((\theta_{1}, f_{1}), (\theta_{2}, f_{2})\bigr)=\sup_{x\in K}\bigl\Vert f_{1}(x)-f_{2}(x)\bigr\Vert +\sup_{x\in K} \bigl\vert \theta_{1}(x)-\theta_{2}(x)\bigr\vert . $$
Then M is a metric space. For convenience, we denote \(\rho((\theta _{n}, f_{n}), (\theta_{0}, f_{0}))\rightarrow0\) by \((\theta_{n}, f_{n})\stackrel{\rho}{\rightarrow}(\theta_{0}, f_{0})\). A point \(u^{*}\) is a solution of the mixed monotone variational inequality problem \((\theta,f)\in M\); we write it as \(u^{*}\in V(\theta ,f)\). Then a set-valued mapping V from M to K is defined. It is well known that, for each \((\theta, f)\in M\), we have \(V(\theta, f)\) is nonempty. If f is strictly monotone, then \(V(\theta, f)\) is a singleton set; for example, \(f+I\) is strictly monotone, where I is the identity mapping on K, and if f is strongly monotone, then f is strictly monotone.
Denote by N the graph of the set-valued mapping V, that is,
$$ N=\bigl\{ (\theta, f, u)\in M\times K \mid u\in V(\theta, f)\bigr\} . $$
For each \((\theta, f, u^{*})\in N\), let
$$ \phi\bigl(\theta, f, u^{*}\bigr)=(\theta, f-C_{u^{*}}) , $$
where \(C_{x}\) denotes the constant mapping with \(C_{x}(u)=x\), \(\forall u\in K\). Clearly, \(C_{u^{*}}\in M\) is monotone, then ϕ is mapping from N to M. For each \((\theta, f)\in M\), define \(\psi(\theta, f)\) such that
$$ \psi(\theta, f)=(\theta, f+C_{\bar{u}}, \bar{u}) , $$
where ū is the unique solution of the problem \((\theta, f+I)\).

2 Main results

Theorem 2.1

ψ is a continuous mapping from M to N. ϕ is continuous on N.


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.

Next, we prove that ψ is continuous on M. Let \(\{(\theta_{n}, f_{n})\}_{n=1}^{\infty}\subset M\) with \((\theta_{n}, f_{n})\stackrel{\rho}{\rightarrow} (\theta_{0}, f_{0})\). We need to show that \((\theta_{n}, f_{n}+C_{\bar{u}_{n}})\stackrel{\rho}{\rightarrow} (\theta_{0}, f_{0}+C_{\bar {u}_{0}})\) and \(\|\bar{u}_{n}-\bar{u}_{0}\|\rightarrow0\), where \(\bar {u}_{n}\in V(\theta_{n}, f_{n}+I)\) and \(\bar{u}_{0}\in V(\theta_{0}, f_{0}+I)\). Since
$$ \theta_{n}(u)-\theta_{n}(\bar{u}_{n})+(u- \bar{u}_{n})^{T}\bigl(f_{n}(\bar {u}_{n})+ \bar{u}_{n}\bigr)\geq0 , \quad \forall u\in K $$
$$ \theta_{0}(u)-\theta_{0}(\bar{u}_{0})+(u- \bar{u}_{0})^{T}\bigl(f_{0}(\bar {u}_{0})+ \bar{u}_{0}\bigr)\geq0 , \quad \forall u\in K , $$
we have
$$ \theta_{n}(\bar{u}_{0})-\theta_{n}( \bar{u}_{n})+(\bar{u}_{0}-\bar {u}_{n})^{T} \bigl(f_{n}(\bar{u}_{n})+\bar{u}_{n}\bigr)\geq0 $$
$$ \theta_{0}(\bar{u}_{n})-\theta_{0}( \bar{u}_{0})+(\bar{u}_{n}-\bar {u}_{0})^{T} \bigl(f_{0}(\bar{u}_{0})+\bar{u}_{0}\bigr)\geq0. $$
Adding equation (2) and equation (3), it follows that
$$ \theta_{n}(\bar{u}_{0})-\theta_{0}( \bar{u}_{0})+\theta_{0}(\bar {u}_{n})- \theta_{n}(\bar{u}_{n}) +(\bar{u}_{0}- \bar{u}_{n})^{T}\bigl(f_{n}(\bar{u}_{n})-f_{0}( \bar{u}_{0})+\bar {u}_{n}-\bar{u}_{0}\bigr)\geq0. $$
$$ \theta_{n}(\bar{u}_{0})-\theta_{0}( \bar{u}_{0})+\theta_{0}(\bar {u}_{n})- \theta_{n}(\bar{u}_{n})+(\bar{u}_{0}- \bar{u}_{n})^{T}\bigl(f_{n}(\bar {u}_{n})-f_{0}(\bar{u}_{0})\bigr)\geq\| \bar{u}_{0}-\bar{u}_{n}\|. $$
This is equivalent to the following inequality:
$$\begin{aligned}& \theta_{n}(\bar{u}_{0})-\theta_{0}( \bar{u}_{0})+\theta_{0}(\bar {u}_{n})- \theta_{n}(\bar{u}_{n}) \\& \qquad {}- (\bar{u}_{0}-\bar{u}_{n})^{T} \bigl(f_{0}(\bar{u}_{0})-f_{0}(\bar{u}_{n})+f_{0}( \bar {u}_{n})-f_{n}(\bar{u}_{n})\bigr) \\& \quad \geq \|\bar{u}_{0}-\bar{u}_{n}\|. \end{aligned}$$
By transposition, we have
$$\begin{aligned}& \theta_{n}(\bar{u}_{0})-\theta_{0}( \bar{u}_{0})+\theta_{0}(\bar {u}_{n})- \theta_{n}(\bar{u}_{n})-(\bar{u}_{0}- \bar{u}_{n})^{T}\bigl(f_{0}(\bar {u}_{n})-f_{n}(\bar{u}_{n})\bigr) \\& \quad \geq \|\bar{u}_{0}-\bar{u}_{n}\|+( \bar{u}_{0}-\bar{u}_{n})^{T}\bigl(f_{0}( \bar {u}_{0})-f_{0}(\bar{u}_{n})\bigr). \end{aligned}$$
Since \(f_{0}\) is monotone, we have \((\bar{u}_{0}-\bar{u}_{n})^{T}(f_{0}(\bar {u}_{0})-f_{0}(\bar{u}_{n}))\geq0\). Then
$$ \theta_{n}(\bar{u}_{0})-\theta_{0}( \bar{u}_{0})+\theta_{0}(\bar {u}_{n})- \theta_{n}(\bar{u}_{n})-(\bar{u}_{0}- \bar{u}_{n})^{T}\bigl(f_{0}(\bar {u}_{n})-f_{n}(\bar{u}_{n})\bigr) \geq \|\bar{u}_{0}-\bar{u}_{n}\|. $$
Therefore, we have
$$\begin{aligned} \Vert \bar{u}_{0}-\bar{u}_{n}\Vert \leq& \bigl\vert \theta_{n}(\bar{u}_{0})- \theta_{0}(\bar{u}_{0})\bigr\vert +\bigl\vert \theta_{0}(\bar {u}_{n})-\theta_{n}( \bar{u}_{n})\bigr\vert +\bigl\vert (\bar{u}_{0}- \bar{u}_{n})^{T}\bigl(f_{0}(\bar {u}_{n})-f_{n}(\bar{u}_{n})\bigr)\bigr\vert \\ \leq& \bigl\vert \theta_{n}(\bar{u}_{0})- \theta_{0}(\bar{u}_{0})\bigr\vert +\bigl\vert \theta_{0}(\bar {u}_{n})-\theta_{n}( \bar{u}_{n})\bigr\vert +\Vert \bar{u}_{0}- \bar{u}_{n}\Vert \bigl\Vert f_{0}(\bar {u}_{n})-f_{n}(\bar{u}_{n})\bigr\Vert . \end{aligned}$$
$$ \bigl(1-\bigl\Vert f_{0}(\bar{u}_{n})-f_{n}( \bar{u}_{n})\bigr\Vert \bigr)\Vert \bar{u}_{0}- \bar{u}_{n}\Vert \leq \bigl\vert \theta_{n}( \bar{u}_{0})-\theta_{0}(\bar{u}_{0})\bigr\vert + \bigl\vert \theta_{0}(\bar {u}_{n})-\theta_{n}( \bar{u}_{n})\bigr\vert . $$
Since \(\theta_{n}\rightarrow\theta_{0}\) and \(\max_{u\in K}\| f_{0}(u)-f_{n}(u)\|\rightarrow0\), we have \(\|\bar{u}_{0}-\bar{u}_{n}\| \rightarrow0\). Hence, it can be checked that
$$\begin{aligned} \begin{aligned} &\bigl\Vert f_{n}(u)+C_{\bar{u}_{n}}(u)-f_{0}(u)-C_{\bar{u}_{0}}(u) \bigr\Vert \\ &\quad \leq \bigl\Vert f_{n}(u)-f_{0}(u)\bigr\Vert +\bigl\Vert C_{\bar{u}_{n}}(u)-C_{\bar{u}_{0}}(u)\bigr\Vert \\ &\quad = \bigl\Vert f_{n}(u)-f_{0}(u)\bigr\Vert +\Vert \bar{u}_{n}-\bar{u}_{0}\Vert \rightarrow0. \end{aligned} \end{aligned}$$
Then \((\theta_{n}, f_{n}+C_{\bar{u}_{n}})\stackrel{\rho}{\rightarrow} (\theta_{0}, f_{0}+C_{\bar{u}_{0}})\).
For the part with ϕ is continuous on N, let \(\{(\theta_{n}, f_{n}, u_{n})\}_{n=1}^{\infty}\subset N\) with \((\theta_{n}, f_{n})\stackrel{\rho}{\rightarrow} (\theta_{0}, f_{0})\) and \(\| u_{n}-u_{0}\|\rightarrow0\); we show that \(\phi(\theta_{n}, f_{n}, u_{n})\stackrel{\rho}{\rightarrow}\phi (\theta_{0}, f_{0}, u_{0})\). In fact, for each \(u\in K\), we have
$$\begin{aligned}& \bigl\Vert \theta_{n}(u)-\theta_{0}(u)\bigr\Vert +\bigl\Vert f_{n}(u)-f_{0}(u)-u_{n}+u_{0}\bigr\Vert \\& \quad \leq \bigl\Vert \theta_{n}(u)-\theta_{0}(u)\bigr\Vert +\bigl\Vert f_{n}(u)-f_{0}(u)\bigr\Vert +\Vert u_{n}-u_{0}\Vert \rightarrow0, \end{aligned}$$
then \(\phi(\theta_{n}, f_{n}, u_{n})\stackrel{\rho}{\rightarrow}\phi(\theta_{0}, f_{0}, u_{0})\). □

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 \(2^{K}\).


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.


For each \((\theta, f)\in M\), from the definition of ψ and ϕ, we have
$$\begin{aligned} \phi\circ\psi(\theta,f)&= \phi(\theta, f+C_{\bar{u}},\bar{u}) \\ &= (\theta, f+C_{\bar{u}}-C_{\bar{u}},\bar{u}) \\ &= (\theta, f), \end{aligned}$$
where \(\bar{u}\in V(\theta,f+I)\).
For each \((\theta,f,u^{*})\in N\), we have
$$\begin{aligned} \psi\circ\phi\bigl(\theta,f,u^{*}\bigr)&= \psi(\theta, f-C_{u^{*}}) \\ &= (\theta, f-C_{u^{*}}+C_{\bar{u}},\bar{u}), \end{aligned}$$
where \(\bar{u}\in V(\theta, f-C_{u^{*}}+I)\). Then
$$ \theta(u)-\theta(\bar{u})+(u-\bar{u})^{T}\bigl(f(\bar{u})-u^{*}+\bar {u} \bigr)\geq0 , \quad \forall u\in K . $$
Particularly, we have
$$ \theta\bigl(u^{*}\bigr)-\theta(\bar{u})+\bigl(u^{*}-\bar{u} \bigr)^{T}\bigl(f(\bar{u})-u^{*}+\bar {u}\bigr)\geq0. $$
Note that \((\theta,f,u^{*})\in N\), we have
$$ \theta(\bar{u})-\theta\bigl(u^{*}\bigr)+\bigl(\bar{u}-u^{*} \bigr)^{T}f\bigl(u^{*}\bigr)\geq0. $$
Then, by adding equation (4) and equation (5), we obtain
$$ \bigl(u^{*}-\bar{u}\bigr)^{T}\bigl(f(\bar{u})-f\bigl(u^{*}\bigr)+ \bar{u}-u^{*}\bigr)\geq0 , $$
$$ \bigl(u^{*}-\bar{u}\bigr)^{T}\bigl(f\bigl(u^{*}\bigr)-f(\bar{u})\bigr)+ \bigl(u^{*}-\bar{u}\bigr)^{T}\bigl(u^{*}-\bar {u}\bigr)\leq0 . $$
From the monotonicity of f, we get \(\|u^{*}-\bar{u}\|\leq0\). Therefore, \(u^{*}=\bar{u}\), hence, \(\psi\circ\phi(\theta ,f,u^{*})=(\theta,f,u^{*})\). □

Theorem 2.3

The spaces N and M are homeomorphic.


It follows immediately from Theorems 2.1 and 2.2. □

Remark 2.2

The homeomorphism results were shown between game spaces and the graphs of solutions mappings in games [911]. 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 be a compact convex subset of \(\mathbb{R}^{n}\) with \(K\subset \operatorname{int}(\bar{f})\). Then there exists a retraction r from to K, that is, \(r(x)=x\), \(\forall x\in K\). From Urysohn lemma, there is a continuous mapping s from 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 .

Define two mappings α, β from \([0,1]\times M\times\tilde {K}\) to \(M\times\tilde{K}\) such that, for each \((t,\theta,f,\tilde {x})\in[0,1]\times M\times\tilde{K}\),
$$\alpha_{t}=\alpha(t,\theta,f,\tilde{x})=(\theta,\bar{f},\tilde {x}) \quad \mbox{with } \bar{f}=f-s(\tilde{x})tC_{x} $$
$$\beta_{t}=\beta(t,\theta,\bar{f},\tilde{x})=(\theta,f,\tilde{x})\quad \mbox{with } f=\bar{f}+s(\tilde{x})tC_{x}, $$
where \(x=r(\tilde{x})\).

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 \(\beta_{1}\).


(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 \(u=v\).

Let \(T:K\rightarrow X^{*}\) and \(f\in X^{*}\), we consider the mixed variational inequality problem: to find a \(u\in K\) such that
$$ \bigl(T(u), v-u\bigr)+\theta(v)-\theta(u)\geq(f,v-u),\quad \forall v\in K. $$
Denote by \(M'\) the set
$$M' = \left \{(\theta,T,f)\mbox{:}\quad \textstyle\begin{array}{@{}l@{}} \theta:K \to\mathbb{R}\mbox{ is continuous and convex}; \\ T:K \to{X^{*}}\mbox{ is continuous on } K; \\ f \in{X^{*}} \end{array}\displaystyle \right \}. $$

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.

For two \(\alpha_{1}=(\theta_{1},T_{1}, f_{1}), \alpha_{2}=(\theta _{2},T_{2},f_{2})\in M'\), measure the metric between them by
$$ \rho'(\alpha_{1},\alpha_{2})=\sup _{x\in K}\bigl\vert \theta_{1}(x)-\theta _{2}(x)\bigr\vert +\sup_{x\in K}\bigl\Vert T_{1}(x)-T_{2}(x)\bigr\Vert +\Vert f_{1}-f_{2} \Vert , $$
where, for each \(g_{1},g_{2}\in X^{*}\),
$$ \|g_{1}-g_{2}\| =\sup_{\|y\|=1,y\in X}\bigl\vert (g_{1},y)-(g_{2},y)\bigr\vert . $$
Let \(N'\) be the graph of \(V'\), that is,
$$ N'=\bigl\{ (\theta, T, f, u)\in M'\times K \mid u\in V'(\theta, T, f)\bigr\} . $$
Define a mapping \(\phi':N'\rightarrow M'\) such that, for each \((\theta , T, f, u)\in N'\),
$$ \phi'(\theta, T, f, u)=(\theta, T-l_{u}, f) , $$
where \(l_{u}:K\rightarrow X^{*}\) is a mapping with \(l_{u}(x)=\langle u,\cdot \rangle\), \(\forall x\in K\). For each \(T:K\rightarrow X^{*}\), let \(R_{T}:K\rightarrow X^{*}\) such that, for each \(x\in K\),
$$ \bigl(R_{T}(x), z\bigr)=\bigl(T(x), z\bigr)+\langle x, z\rangle, \quad \forall z\in X . $$
Then we can check that \(R_{T}\) is strictly monotone. For each \((\theta,T,f)\in M'\), define a mapping \(\psi'\) on \(M'\) such that
$$ \psi'(\theta,T,f)=(\theta, T+l_{u}, f, u) , $$
where u is the unique solution of the problem \((\theta,R_{T}, f)\), that is, \(V'(\theta,R_{T},f)=\{u\}\).

Theorem 2.5

\(\psi'\) is a continuous mapping from \(M'\) to \(N'\).


For each \((\theta, T, f)\in M'\), we have \(\psi'(\theta,T,f)=(\theta ,T+l_{u},u)\), where \(u\in V'(\theta,R_{T},f)\). Then
$$\begin{aligned}& \theta(v)-\theta(u)+\bigl(T(u),v-u\bigr)+\langle u, v-u\rangle \\& \quad = \theta(v)-\theta(u)+\bigl(T(u)+l_{u}, v-u\bigr) \\& \quad \geq (f,v-u),\quad \forall v\in K, \end{aligned}$$
it follows that \(u\in V'(T+l_{u})\), that is, \(\psi'\) maps \(M'\) onto \(N'\).
For the part that \(\psi'\) is continuous on \(M'\): let \(\{(\theta _{n},T_{n},f_{n})\}_{n=1}^{\infty}\subset M'\) with \((\theta_{n},T_{n},f_{n})\stackrel{\rho'}{\rightarrow}(\theta_{0},T_{0},f_{0})\in M'\). It is sufficient to show that \(\psi'(\theta_{n},T_{n},f_{n})\rightarrow\psi'(\theta_{0},T_{0},f_{0})\), that is, \((\theta_{n},T_{n}+l_{u_{n}},f_{n})\stackrel{\rho'}{\rightarrow}(\theta _{0},T_{0}+l_{u_{0}},f_{0})\) and \(\|u_{n}-u_{0}\|\rightarrow0\), where \(u_{n}\in V'(\theta_{n},R_{T_{n}},f_{n})\) and \(u_{0}\in V'(\theta _{0},R_{T_{0}}, f_{0})\), this leads to the fact that, for each \(x\in K\),
$$ \theta_{n}(v)-\theta_{n}(u_{n})+ \bigl(T_{n}(u_{n}),v-u_{n}\bigr)+\langle u_{n},v-u_{n}\rangle \geq(f_{n},v-u_{n}), \quad \forall v\in K $$
$$ \theta_{0}(v)-\theta_{0}(u_{0})+ \bigl(T_{0}(u_{0}),v-u_{0}\bigr)+\langle u_{0},v-u_{0}\rangle \geq(f_{0},v-u_{0}), \quad \forall v\in K. $$
Particularly, we have
$$ \theta_{n}(u_{0})-\theta_{n}(u_{n})+ \bigl(T_{n}(u_{n}),u_{0}-u_{n}\bigr)+ \langle u_{n},u_{0}-u_{n}\rangle \geq(f_{n},u_{0}-u_{n}) $$
$$ \theta_{0}(u_{n})-\theta_{0}(u_{0})+ \bigl(T_{0}(u_{0}),u_{n}-u_{0}\bigr)+ \langle u_{0},u_{n}-u_{0}\rangle \geq(f_{0},u_{n}-u_{0}). $$
Adding equation (7) and equation (8), we get
$$\begin{aligned}& \theta_{n}(u_{0})-\theta_{0}(u_{0})+ \theta_{0}(u_{n})-\theta _{n}(u_{n})+ \bigl(T_{n}(u_{n})-T_{0}(u_{0}),u_{0}-u_{n} \bigr) \\& \quad \geq \|u_{n}-u_{0}\|+(f_{n}-f_{0},u_{0}-u_{n}). \end{aligned}$$
This is equivalent to the following:
$$\begin{aligned}& \theta_{n}(u_{0})-\theta_{0}(u_{0})+ \theta_{0}(u_{n})-\theta _{n}(u_{n})+ \bigl((T_{n}-T_{0}) (u_{n})+T_{0}(u_{n}-u_{0}),u_{0}-u_{n} \bigr) \\& \quad \geq \|u_{n}-u_{0}\|+(f_{n}-f_{0},u_{0}-u_{n}). \end{aligned}$$
Since \(T_{0}\) is monotone, we have \((T_{0}(u_{n}-u_{0}),u_{0}-u_{n})\leq0\). Then
$$\begin{aligned}& \theta_{n}(u_{0})-\theta_{0}(u_{0})+ \theta_{0}(u_{n})-\theta_{n}(u_{n}) \\& \qquad {}+\bigl((T_{n}-T_{0}) (u_{n}),u_{0}-u_{n} \bigr)-(f_{n}-f_{0},u_{0}-u_{n}) \\& \quad \geq \|u_{n}-u_{0}\|. \end{aligned}$$
$$\begin{aligned}& \|u_{n}-u_{0}\| \\& \quad \leq \theta_{n}(u_{0})-\theta_{0}(u_{0})+ \theta_{0}(u_{n})-\theta_{n}(u_{n}) \\& \qquad {}+ \|u_{n}-u_{0}\|\biggl((T_{n}-T_{0}) (u_{n}),\frac{u_{0}-u_{n}}{\|u_{0}-u_{n}\|}\biggr)-\|u_{n}-u_{0}\| \biggl(f_{n}-f_{0},\frac{u_{0}-u_{n}}{\|u_{0}-u_{n}\|}\biggr), \end{aligned}$$
$$\begin{aligned} \Vert u_{n}-u_{0}\Vert \leq& 2\sup_{x\in K} \bigl\vert \theta_{n}(x)-\theta_{0}(x)\bigr\vert \\ &{}+ \Vert u_{n}-u_{0}\Vert \sup_{x\in K} \bigl\Vert (T_{n}-T_{0}) (x)\bigr\Vert +\Vert u_{n}-u_{0}\Vert \Vert f_{n}-f_{0} \Vert , \end{aligned}$$
consequently, we obtain
$$ \Bigl(1-\sup_{x\in K}\bigl\Vert (T_{n}-T_{0}) (x)\bigr\Vert -\Vert f_{n}-f_{0}\Vert \Bigr)\Vert u_{n}-u_{0}\Vert \leq 2\sup_{x\in K}\bigl\vert \theta_{n}(x)-\theta_{0}(x)\bigr\vert . $$
Since \((\theta_{n},T_{n},f_{n})\stackrel{\rho'}{\rightarrow}(\theta_{0},T_{0},f_{0})\), we have \(\|u_{n}-u_{0}\|\rightarrow0\).
In addition, for each \(x\in K\),
$$\begin{aligned}& \bigl\vert \theta_{n}(x)-\theta_{0}(x)\bigr\vert +\bigl\Vert (T_{n}+l_{u_{n}}) (x)-(T_{0}+l_{u_{0}}) (x)\bigr\Vert +\Vert f_{n}-f_{0}\Vert \\& \quad \leq \bigl\vert \theta_{n}(x)-\theta_{0}(x)\bigr\vert + \bigl\Vert (T_{n}-T_{0}) (x)\bigr\Vert +\bigl\Vert (l_{u_{n}}-l_{u_{0}}) (x)\bigr\Vert +\Vert f_{n}-f_{0} \Vert \\& \quad \leq \bigl\vert \theta_{n}(x)-\theta_{0}(x)\bigr\vert + \bigl\Vert (T_{n}-T_{0}) (x)\bigr\Vert +\Vert u_{n}-u_{0}\Vert +\Vert f_{n}-f_{0} \Vert \rightarrow0. \end{aligned}$$
Note that \(\|u_{n}-u_{0}\|\rightarrow0\) and \((\theta _{n},T_{n},f_{n})\stackrel{\rho'}{\rightarrow}(\theta_{0},T_{0},f_{0})\), we have \((\theta _{n},T_{n}+l_{u_{n}},f_{n})\stackrel{\rho'}{\rightarrow}(\theta_{0},T_{0}+l_{u_{0}},f_{0})\). The proof is completed. □

Theorem 2.6

\(\phi'\) is a continuous mapping from \(N'\) to \(M'\).


Let \(\{(\theta_{n}, T_{n}, f_{n}, u_{n})\}_{n=1}^{\infty}\subset N'\) with \((\theta_{n}, T_{n}, f_{n}, u_{n})\rightarrow(\theta_{0}, T_{0}, f_{0}, u_{0})\in N'\), that is, \((\theta_{n}, T_{n}, f_{n})\stackrel{\rho'}{\rightarrow} (\theta_{0}, T_{0},f_{0})\), and \(\|u_{n}-u_{0}\|\rightarrow0\). We need to show that \((\theta_{n}, T_{n}-l_{u_{n}},f_{n})\stackrel{\rho'}{\rightarrow} (\theta_{0}, T_{0}-l_{u_{0}},f_{0})\). For each \(x\in K\), we have
$$\begin{aligned}& \rho'\bigl((\theta_{n},T_{n}-l_{u_{n}},f_{n}),( \theta_{0},T_{0}-l_{u_{0}},f_{0})\bigr) \\& \quad = \sup_{x\in K}\bigl\vert \theta_{n}(x)- \theta_{0}(x)\bigr\vert +\sup_{x\in K}\bigl\Vert (T_{n}-T_{0}) (x)-(l_{u_{n}}-l_{u_{0}}) (x) \bigr\Vert +\Vert f_{n}-f_{0}\Vert \\& \quad \leq \sup_{x\in K}\bigl\vert \theta_{n}(x)- \theta_{0}(x)\bigr\vert +\sup_{x\in K}\bigl\Vert (T_{n}-T_{0}) (x)\bigr\Vert +\Vert f_{n}-f_{0} \Vert \\& \qquad {}+ \sup_{x\in K}\bigl\Vert (l_{u_{n}}-l_{u_{0}}) (x)\bigr\Vert \\& \quad = \rho'\bigl((\theta_{n},T_{n},f_{n}),( \theta_{0},T_{0},f_{0})\bigr)+\sup _{\Vert y\Vert =1, y\in X}\bigl\vert \langle u_{n}-u_{0},y \rangle\bigr\vert \\& \quad \leq \rho'\bigl((\theta_{n},T_{n},f_{n}),( \theta_{0},T_{0},f_{0})\bigr)+\sup _{\Vert y\Vert =1, y\in X}\Vert u_{n}-u_{0}\Vert \Vert y \Vert \\& \quad = \rho'\bigl((\theta_{n},T_{n},f_{n}),( \theta_{0},T_{0},f_{0})\bigr)+\Vert u_{n}-u_{0}\Vert \rightarrow0, \end{aligned}$$
then \(\phi'(\theta_{n}, T_{n}-l_{u_{n}},f_{n})\rightarrow\phi'(\theta_{0}, T_{0}-l_{u_{0}},f_{0})\). □

Theorem 2.7

\(\phi'\circ\psi'\) and \(\psi'\circ\phi'\) are identity mappings on \(M'\) and \(N'\), respectively.


(a) \(\phi'\circ\psi'\) is an identity mapping on \(M'\). For each \((\theta,T,f)\in M'\), by Theorem 2.5, we have
$$ \phi'\circ\psi'(\theta,T,f)=\phi'( \theta,T+l_{u},f,u)=(\theta ,T+l_{u}-l_{u},f)=( \theta,T,f) . $$
(b) \(\psi'\circ\phi'\) is an identity mapping on \(N'\). For each \((\theta,T,f,u^{*})\in N'\), we have
$$ \psi'\circ\phi'\bigl(\theta,T,f,u^{*}\bigr)= \psi'(\theta,T-l_{u^{*}},f)=(\theta ,T-l_{u^{*}}+l_{\bar{u}},f, \bar{u}), $$
where \(\bar{u}\in V'(\theta,R_{T-l_{u^{*}}},f)\), and \(R_{T-l_{u^{*}}}\) means that, for each \(x\in K\),
$$ \bigl(R_{T-l_{u^{*}}}(x), z\bigr)=\bigl(T(x),z\bigr)-\bigl\langle u^{*}, z\bigr\rangle +\langle x, z\rangle, \quad \forall z\in X. $$
Since \(\psi'\) maps \(M'\) onto \(N'\), we have \(\bar{u}\in V'(\theta ,T-l_{u^{*}}+l_{\bar{u}},f)\), then
$$ \theta(v)-\theta(\bar{u})+\bigl(T(\bar{u})-l_{u^{*}}( \bar{u})+l_{\bar {u}}(\bar{u}), v-\bar{u}\bigr)\geq(f,v-\bar{u}), \quad \forall v\in K. $$
$$ \theta(v)-\theta(\bar{u})+\bigl((T-l_{u^{*}}+l_{\bar{u}}) (\bar{u}),v-\bar {u}\bigr)\geq(f,v-\bar{u}),\quad \forall v\in K. $$
Let \(v=u^{*}\) in (9), then
$$ \theta\bigl(u^{*}\bigr)-\theta(\bar{u})+\bigl((T-l_{u^{*}}+l_{\bar{u}}) (\bar {u}),u^{*}-\bar{u}\bigr)\geq\bigl(f,u^{*}-\bar{u}\bigr). $$
Since \(u^{*}\in V'(\theta,T,f)\), we have
$$ \theta(\bar{u})-\theta\bigl(u^{*}\bigr)+\bigl(T\bigl(u^{*}\bigr), \bar{u}-u^{*}\bigr)\geq\bigl(f,\bar{u}-u^{*}\bigr). $$
Add \(\langle\bar{u}-u^{*},\bar{u}-u^{*}\rangle\) to the left-hand side of equation (11),
$$ \theta(\bar{u})-\theta\bigl(u^{*}\bigr)+\bigl(T\bigl(u^{*}\bigr), \bar{u}-u^{*}\bigr)+ \bigl\langle \bar {u}-u^{*},\bar{u}-u^{*}\bigr\rangle \geq\bigl(f,\bar{u}-u^{*}\bigr), $$
that is,
$$ \theta(\bar{u})-\theta\bigl(u^{*}\bigr)+\bigl((T-l_{u^{*}}+l_{\bar{u}}) \bigl(u^{*}\bigr),\bar {u}-u^{*}\bigr)\geq\bigl(f,\bar{u}-u^{*}\bigr). $$
Adding equation (10) and equation (12), we get
$$ \bigl((T-l_{u^{*}}+l_{\bar{u}}) \bigl(\bar{u}-u^{*}\bigr),u^{*}-\bar{u} \bigr)\geq0. $$
Note that \(T-l_{u^{*}}+l_{\bar{u}}\) is monotone, we have
$$ \bigl((T-l_{u^{*}}+l_{\bar{u}}) \bigl(\bar{u}-u^{*}\bigr),\bar{u}-u^{*} \bigr)= 0. $$
Furthermore, T is strictly monotone, it follows that \(\bar{u}=u^{*}\). □

Theorem 2.8

The spaces \(M'\) and \(N'\) are homeomorphic.


It follows from Theorems 2.5, 2.6, and 2.7. □

Remark 2.4

Theorem 2.8 generalizes the homeomorphism result for the variational inequalities to find a point \(u\in K\) such that \((T(u), v-u)\geq0\), \(\forall v\in K\), in [12].



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 (, 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

College of Science, Guizhou University, Guiyang, 550025, China
College of Science, Guilin University of Technology, Guilin, 541004, China


  1. Facchinei, F, Pang, JS: Finite-Dimensional Variational Inequalities and Complementarity Problem. Springer, New York (2003) MATHGoogle Scholar
  2. Konnov, IV: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007) MATHGoogle Scholar
  3. He, BS: A modified projection and contraction method for a class of linear complementarity problems. J. Comput. Math. 14, 54-63 (1996) MathSciNetMATHGoogle Scholar
  4. He, BS, Yuan, XM, Zhang, WX: A customized proximal point algorithm for convex minimization with linear constraints. Comput. Optim. Appl. 56(3), 559-572 (2013) MathSciNetView ArticleMATHGoogle Scholar
  5. Chambolle, A, Pock, T: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120-145 (2011) MathSciNetView ArticleMATHGoogle Scholar
  6. Hartman, P, Stampacchia, G: On some nonlinear elliptic differential functional equations. Acta Math. 115, 271-310 (1966) MathSciNetView ArticleMATHGoogle Scholar
  7. Lions, JL, Stampacchia, G: Variational inequalities. Commun. Pure Appl. Math. 20, 493-519 (1967) MathSciNetView ArticleMATHGoogle Scholar
  8. Adsul, B, Garg, J, Mehta, R, Sohoni, M: Rank-1 bimatrix games: a homeomorphism and a polynomial time algorithm. In: Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, pp. 195-204. ACM, New York (2011) Google Scholar
  9. Kohlberg, E, Mertens, JF: On the strategic stability of equilibria. Econometrica 54(5), 1003-1037 (1986) MathSciNetView ArticleMATHGoogle Scholar
  10. Predtetchinski, A: A general structure theorem for the Nash equilibrium correspondence. Games Econ. Behav. 66(2), 950-958 (2009) MathSciNetView ArticleMATHGoogle Scholar
  11. Govindan, S, Wilson, R: Structure theorems for game trees. Proc. Natl. Acad. Sci. USA 99(13), 9077-9080 (2002) MathSciNetView ArticleMATHGoogle Scholar
  12. Song, QQ: On the graph of solution mappings of variational inequalities problems. J. Funct. Spaces 2014, Article ID 853106 (2014) MathSciNetMATHGoogle Scholar
  13. Duvaut, G, Lions, JL: Les inequations en mechanique et en physique. Dunod, Paris (1972) MATHGoogle Scholar
  14. Konnov, IV, Kum, S: Descent methods for mixed variational inequalities in a Hilbert space. Nonlinear Anal. 47(1), 561-572 (2001) MathSciNetView ArticleMATHGoogle Scholar
  15. Fang, YP, Huang, NJ, Yao, JC: Well-posedness by perturbations of mixed variational inequalities in Banach spaces. Eur. J. Oper. Res. 201(3), 682-692 (2010) MathSciNetView ArticleMATHGoogle Scholar
  16. Ding, XP, Lin, LC: Existence and algorithm for solving some generalized mixed variational inequalities. Comput. Math. Appl. 37(3), 23-30 (1999) MathSciNetView ArticleMATHGoogle Scholar
  17. Ceng, LC, Wen, CF: Well-posedness by perturbations of generalized mixed variational inequalities in Banach spaces. J. Appl. Math. 2012, Article ID 194509 (2012) MathSciNetMATHGoogle Scholar
  18. Noor, MA: Mixed variational inequalities. Appl. Math. Lett. 3(2), 73-75 (1990) MathSciNetView ArticleMATHGoogle Scholar
  19. Khan, S, Iqbal, J, Shehu, Y: Mixed quasi-variational inequalities involving error bounds. J. Inequal. Appl. 2015, Article ID 417 (2015) MathSciNetView ArticleMATHGoogle Scholar
  20. Noor, MA: Some developments in general variational inequalities. Appl. Math. Comput. 152(1), 199-277 (2004) MathSciNetView ArticleMATHGoogle Scholar
  21. Noor, MA: An extraresolvent method for monotone mixed variational inequalities. Math. Comput. Model. 29(3), 95-100 (1999) MathSciNetView ArticleMATHGoogle Scholar
  22. Sandholm, WH: Population Games and Evolutionary Dynamics. MIT Press, Cambridge (2010) MATHGoogle Scholar
  23. Demichelis, S, Germano, F: On (un)knots and dynamics in games. Games Econ. Behav. 41(1), 46-60 (2002) MathSciNetView ArticleMATHGoogle Scholar
  24. Nagurney, A, Zhang, D: Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Academic, Dordrecht (1996) View ArticleMATHGoogle Scholar


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