- Research
- Open Access
Existence result and error bounds for a new class of inverse mixed quasi-variational inequalities
- Xi Li^{1} and
- Yun-zhi Zou^{2}Email author
https://doi.org/10.1186/s13660-016-0968-5
© Li and Zou 2016
- Received: 5 June 2015
- Accepted: 14 January 2016
- Published: 3 February 2016
Abstract
In this paper, a new class of inverse mixed quasi-variational inequalities (IMQVI) is introduced and studied in Hilbert spaces. This type of inequalities includes many quasi-variational inequalities and inverse variational inequalities as its special cases. We first prove some properties of generalized f-projection operators in Hilbert spaces. Then we use these properties to obtain the existence and uniqueness result. Moreover, error bounds for IMQVI in terms of the residual function are also established. The results presented in this paper are new and improve some results in the recent literature.
Keywords
- mixed variational inequality
- inverse mixed quasi-variational inequality
- generalized f-projection operator
- error bound
- residual function
1 Introduction
It the past decades, variational inequalities and their generalizations have been widely used in finance, economics, transportation, optimization, operations research, and the engineering sciences. For instance, Lescarret [1] and Browder [2] introduced mixed variational inequalities in 1960s. Later, Konnov and Volotskaya [3] applied mixed variational inequalities to several classes of general economic equilibrium problems and oligopolistic equilibrium problems. In 2006, He et al. [4, 5] studied a class of inverse variational inequalities and also found their applications in practical world, such as normative flow control problems, which require the network equilibrium state to be in a linearly constrained set, and bipartite market equilibrium problems. Some other generalizations such as quasi-variational inequalities also have been studied extensively. For details, we refer to [1, 6–21] and the references therein.
We first prove some properties of generalized f-projection operators in Hilbert spaces and then explore the existence and uniqueness results of the IMQVI. Furthermore, we study error bounds for the IMQVI in terms of the residual function. Since IMQVI naturally encompasses many types of quasi-variational inequalities and inverse variational inequalities, the results presented in this paper therefore generalize and improve some results in the existing literature.
2 Preliminaries
Definition 2.1
[22]
From the work of Wu and Huang [22] and Fan et al. [11], we know that the generalized f-projection operator has the following properties.
Lemma 2.1
- (i)
\(P_{K}^{f,\rho}x\) is nonempty and \(P_{K}^{f,\rho}\) is a single valued mapping;
- (ii)for all \(x\in H\), \(x^{*}= P_{K}^{f,\rho}x\) if and only if$$ \bigl\langle x^{*}-x,y-x^{*}\bigr\rangle +\rho f(y)-\rho f\bigl(x^{*}\bigr)\geq0,\quad \forall y\in K; $$
- (iii)
\(P_{K}^{f,\rho}\) is continuous.
Definition 2.2
- (i)A is said to be λ-strongly monotone on H if there exists a constant λ such that$$ \langle Ax-Ay,x-y\rangle\geq\lambda \Vert x-y\Vert ^{2},\quad \forall x,y\in H; $$
- (ii)A is said to be γ-Lipschitz continuous on H if there exists a constant \(\gamma>0\) such that$$ \Vert Ax-Ay\Vert \leq\gamma \Vert x-y\Vert ,\quad\forall x,y\in H; $$
- (iii)A is said to be co-coercive on H if there exists a positive constant \(\tau>0\) such that$$ \langle Ax-Ay,x-y\rangle\geq\tau \Vert Ax-Ay\Vert ^{2},\quad\forall x,y\in H; $$
- (iv)\((A,g)\) is said to be a μ-strongly monotone couple on H if there exists a positive constant \(\mu>0\) such that$$ \bigl\langle Ax-Ay,g(x)-g(y)\bigr\rangle \geq\mu\Vert x-y\Vert^{2}, \quad \forall x,y\in H. $$
Remark
IMQVI encompasses several models of quasi-variational inequalities and inverse variational inequalities. For example:
(4) If \(H=R^{n}\), A is the identity mapping and \(f(x)=0\) for all \(x\in R^{n}\), then IMQVI becomes the classic quasi-variational inequality.
3 Some properties of generalized f-projection operators
Now we apply the basic inequality in Lemma 2.1 to prove some properties of the operator \(P_{K(x)}^{f,\rho}\) in Hilbert spaces.
Theorem 3.1
- (i)
there exists a constant \(\gamma>0\) such that \(\mathcal{H}[K(x),K(y)]\leq\gamma\Vert x-y\Vert^{2}\), \(\forall x,y\in H\);
- (ii)
\(0\in\bigcap_{u\in H}K(u)\);
- (iii)
f is l-Lipschitz continuous on H.
Proof
Remark 3.1
Theorem 3.1 shows that the generalized f-projection operator \(P_{K(x)}^{f,\rho}\) is k-Lipschitz continuous with respect to x on each bounded set of the Hilbert space H under some suitable conditions.
If \(H=R^{n}\) and \(f(x)=0\) for all \(x\in R^{n}\), then the generalized f-projection operator \(P_{K(x)}^{f,\rho}\) reduces to the classic metric projection operator \(P_{K(x)}\). By Theorem 3.1, we can obtain the following theorem.
Theorem 3.2
- (i)
there exists \(\gamma>0\) such that \(\mathcal {H}[K(x),K(y)]\leq \gamma\Vert x-y\Vert^{2}\), \(\forall x,y\in R^{n}\);
- (ii)
\(0\in\bigcap_{u\in R^{n}}K(u)\).
Theorem 3.3
Proof
4 The existence and uniqueness result of IMQVI
Theorem 4.1
- (i)
g is λ-strongly monotone and \((A,g)\) is a μ-strongly monotone couple on H;
- (ii)there exists \(k>0\) such that$$ \bigl\Vert P_{K(x)}^{f,\rho}z-P_{K(y)}^{f,\rho}z \bigr\Vert \leq k\Vert x-y\Vert ,\quad\forall x,y\in H,z\in\bigl\{ v|v=Ax- \rho g(x),x\in H\bigr\} ; $$
- (iii)
\(\sqrt{\beta^{2}-2\rho\mu+\rho^{2}\alpha^{2}}+\rho\sqrt{1-2\lambda+\alpha^{2}}<\rho-k\).
Then IMQVI (2.1) has a unique solution in H.
Proof
Remark 4.1
5 Error bounds for IMQVI
It is well known that error bounds play important roles in the study of variational inequality problems. They allow one to estimate how far a feasible element is from the solution set without even having computed a single solution of the associated variational inequality. In [28], Aussel et al. provided the following two error bounds.
Theorem DA1
(Theorem 1 of [28])
- (a)
\((A, g)\) is a strongly monotone couple on \(R^{n}\) with constant μ,
- (b)there exists \(0< k<\frac{\mu}{l}\) such that, for any \(\theta >\frac{Lk}{\mu-lk}\),$$ \bigl\Vert P_{K(x)}^{\theta}z-P_{K(y)}^{\theta}z \bigr\Vert \leq k\Vert x-y\Vert ,\quad\forall x,y,z\in R^{n}. $$
Theorem DA2
(Lemma 1 of [28])
- (a)
A is strongly monotone on \(R^{n}\) with constant μ,
- (b)there exists \(0< k<\mu\) such that, for any \(\theta>\frac {L(8k+L)}{4(\mu-k)}\),$$ \bigl\Vert P_{K(x)}^{\theta}z-P_{K(y)}^{\theta}z \bigr\Vert \leq k\Vert x-y\Vert ,\quad\forall x,y,z\in R^{n}. $$
Theorem 5.1
- (i)
\((A,g)\) is a μ-strongly monotone couple on H;
- (ii)there exists \(0< k<\frac{\mu}{\alpha}\) such that, for any \(\rho>\frac{\beta k}{\mu-\alpha k}\),$$ \bigl\Vert P_{K(x)}^{f,\rho}z-P_{K(y)}^{f,\rho}z \bigr\Vert \leq k\Vert x-y\Vert ,\quad\forall x,y\in H,z\in\bigl\{ v|v=Ax- \rho g(x),x\in H\bigr\} . $$
Proof
If \(H=R^{n}\) and \(f(x)=0\) for all \(x\in R^{n}\), from Theorem 5.1, we obtain the following theorem.
Theorem 5.2
- (i)
\((A,g)\) is a μ-strongly monotone couple on \(R^{n}\);
- (ii)there exists \(0< k<\frac{\mu}{l}\) such that, for any \(\theta >\frac{Lk}{\mu-lk}\),$$ \Vert P_{K(x)}z-P_{K(y)}z\Vert \leq k\Vert x-y\Vert , \quad\forall x,y\in R^{n},z\in \bigl\{ v|v=Ax-\theta g(x),x\in R^{n}\bigr\} . $$
For any \(x\in H\), based on Theorem 3.3, we know that \(P_{K(x)}^{f,\rho }\) is co-coercive mapping with modulus 1 on H. Applying the co-coercivity of \(P_{K(x)}^{f,\rho}\), we prove another error bound for IMQVI (2.1).
Theorem 5.3
- (i)
\((A,g)\) is a μ-strongly monotone couple on H;
- (ii)there exists \(0< k<\frac{\mu}{\alpha}\) such that, for any \(\rho>\frac{\beta(\beta+8k)}{4(\mu-\alpha k)}\),$$ \bigl\Vert P_{K(x)}^{f,\rho}z-P_{K(y)}^{f,\rho}z \bigr\Vert \leq k\Vert x-y\Vert ,\quad\forall x,y\in H,z\in\bigl\{ v|v=Ax- \rho g(x),x\in H\bigr\} . $$
Proof
If \(H=R^{n}\), g is identity mapping in \(R^{n}\), and \(f(x)=0\) for all \(x\in R^{n}\), by using Theorem 5.3, we have the following theorem.
Theorem 5.4
- (i)
A is strongly monotone on \(R^{n}\) with constant μ,
- (ii)there exists \(0< k<\mu\) such that, for any \(\theta>\frac {L(8k+L)}{4(\mu-k)}\),$$ \Vert P_{K(x)}z-P_{K(y)}z\Vert \leq k\Vert x-y\Vert , \quad\forall x,y\in H,z\in \bigl\{ v|v=Ax-\theta x,x\in R^{n}\bigr\} . $$
Declarations
Acknowledgements
This work was supported by Scientific Research Fund of Sichuan Provincial Education Department (14ZB0130), the National Natural Science Foundation of China (11426180).
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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