- Research
- Open Access
Metric characterizations for well-posedness of split hemivariational inequalities
- Qiao-yuan Shu^{1},
- Rong Hu^{2} and
- Yi-bin Xiao^{2}Email author
https://doi.org/10.1186/s13660-018-1761-4
© The Author(s) 2018
- Received: 31 January 2018
- Accepted: 30 June 2018
- Published: 27 July 2018
Abstract
In this paper, we generalize the concept of well-posedness to a class of split hemivariational inequalities. By imposing very mild assumptions on involved operators, we establish some metric characterizations of the well-posedness for the split hemivariational inequality. The obtained results generalize some related theorems on well-posedness for hemivariational inequalities and variational inequalities in the literature.
Keywords
- Split hemivariational inequality
- Monotone operator
- Hemicontinuity
- Metric characterization
1 Introduction
The concept of well-posedness, which was firstly introduced by Tykhonov in [1] for a minimization problem and thus was called Tykhonov well-posedness, has been studied widely in recent years for optimization problems, variational inequality problems, hemivariational inequality problems, fixed point problems, saddle point problems, equilibrium problems, and their related problems because of their important applications in physics, mechanics, engineering, economics, management science, etc. (see, for example, [2–13]). Tykhonov well-posedness for an optimization problem is defined by requiring the existence and uniqueness of its solution and the convergence to the unique solution of its approximating sequences. There are a great many kinds of generalizations for the concept of well-posedness, such as Levitin-Polyak well-posedness, parametric well-posedness, and α-well-posedness, to optimization problems, variational inequality problems, and their related problems (see, for example, [14–21]).
Due to the close relationship between optimization problems and variational inequality problems, the concept of well-posedness for optimization problems is generalized to variational inequalities and their related problems. The earliest research work of well-posedness for variational inequalities should at least date back to 1980s when Lucchetti and Patrone [22, 23] firstly introduced the concept of well-posedness for a variational inequality and proved some important results. After that, Lignola and Morgan [20], Fang and Hu [24], Huang and Yao [25] have made significant contributions to the study of well-posedness for variational inequalities. As an important generalization of variation inequality, hemivariational inequality has drawn much attention of mathematical researchers due to its abundant applications in mechanics and engineering. With the tools of nonsmooth analysis and nonlinear analysis, many kinds of hemivariational inequalities have been studied since 1980s [7, 26–30]. Also, many kinds of concepts of well-posedness hemivariational inequalities have been studied since Goeleven and Mentagui [31] firstly introduced the concept of well-posedness to a hemivariational inequality in 1995. For more research work on the well-posedness for variational inequalities and hemivariational inequalities, we refer the readers to [14, 20, 32–35].
Split variational inequality, which was introduced by Censor et al. in [36], can be regarded as a generalization of variational inequality and includes as a special case, the split feasibility problem, which is an important model for a wide range of practical problems arising from signal recovery, image processing, and tensity-modulated radiation therapy treatment planning (see, for example, [37–41]). Thus, the concepts of well-posedness and Levitin-Polyak well-posedness for various split variational inequalities were studied by Hu and Fang recently [42]. Obviously, split hemivariational inequality could be regarded as a generalization of split variational inequality. It could arise in a system of hemivariational inequalities for modeling some frictional contact problems in mechanics, where two hemivariational inequalities are linked by a linear constraint. Also, when nonconvex and nonsmooth functionals are involved, the model for the above mentioned practical problems, such as signal recovery and image processing, turns to split hemivariational inequality rather than split variational inequality. However, as far as we know, there are few research works studying well-posedness for split hemivariational inequalities.
Inspired by recent research works on the well-posedness for split variational inequalities and hemivariational inequalities, in this paper, we focus on studying metric characterization of well-posedness for a class of split hemivariational inequalities specified as follows:
The remainder of the paper is organized as follows. In Sect. 2, we recall some crucial definitions and results. Under very mild assumptions on involved operators, Sect. 3 presents several results on the metric characterizations of well-posedness for the split hemivariational inequality (SHI). At last, some concluding remarks are provided in Sect. 4.
2 Preliminaries
In this section, we recall some useful definitions and key results which will be used to establish the metric characterizations of the split hemivariational inequality (SHI)in the next section and can be found in [7, 29, 43–45].
Definition 2.1
- (1)a sequence \(\{{u_{n}}\}\subset V\) is said to be convergent if there exists \(u\in V\) such thatwhich is denoted by \(u_{n} \to u\) as \(n\to\infty\);$$ \lim_{n\rightarrow\infty} \|u_{n}-u\|_{V}=0, $$
- (2)a sequence \(\{{u_{n}}\}\subset V\) is said to be weakly convergent to a point \(u\in V\) ifwhich is denoted by \(u_{n} \rightharpoonup u\) as \(n\to\infty\);$$ \langle f,u_{n}\rangle_{V^{\ast}\times V}\rightarrow\langle f,u \rangle_{V^{\ast}\times V},\quad \forall f\in V^{\ast}, $$
- (3)a sequence \(\{u_{n}^{\ast}\}\subset V^{\ast}\) is said to be weakly^{∗} convergent to a point \(u^{\ast}\in V^{\ast}\) ifwhich is denoted by \(u_{n}^{*}\xrightarrow{w^{\ast}} u^{*}\) as \(n\to\infty\).$$ \bigl\langle u^{\ast}_{n},u\bigr\rangle _{V^{\ast}\times V} \rightarrow\bigl\langle u^{\ast},u\bigr\rangle _{V^{\ast}\times V},\quad \forall u\in V, $$
Definition 2.2
- (1)monotone if$$ \langle Au-Av,u-v\rangle_{V^{\ast}\times V}\geq0,\quad \forall u,v\in V; $$
- (2)strictly monotone if$$ \langle Au-Av,u-v\rangle_{V^{\ast}\times V}>0,\quad \forall u,v\in V \mbox{ and } u \neq v; $$
- (3)relaxed monotone if there exists a constant \(c>0\) such that$$ \langle Au-Av,u-v\rangle_{V^{\ast}\times V}\geq-c\|u-v\| _{V}^{2}, \quad \forall u,v\in V; $$
- (4)strongly monotone if there exists a constant \(c>0\) such that$$ \langle Au-Av,u-v\rangle_{V^{\ast}\times V}\geq c\|u-v\| _{V}^{2}, \quad \forall u,v\in V. $$
Definition 2.3
- (1)
continuous if, for any sequence \(\{u_{n}\}\subset V\) converging to \(u\in V\), \(Tu_{n}\to Tu\) in \(V^{\ast}\);
- (2)
demicontinuous if, for any sequence \(\{u_{n}\}\subset V\) converging to \(u\in V\), \(Tu_{n}\rightharpoonup Tu\) in \(V^{\ast}\);
- (3)
hemicontinuous if, for any \(u,v,w\in V\), the function \(t\rightarrow\langle T(u+tv),w\rangle_{V^{\ast}\times V}\) is continuous on \([0,1]\);
- (4)
weakly^{∗} continuous (or continuous with respect to weak^{∗} topology for \(V^{*}\)) if, for any sequence \(\{ u_{n}\}\subset V\) converging to \(u\in V\), \(Tu_{n}\xrightarrow{w^{\ast}} Tu\) in \(V^{\ast}\).
Remark 2.1
In [7, 44], demicontinuity of an operator T from V to \(V^{*}\) is defined by its continuity from V to its dual space \(V^{*}\) endowed with weak^{∗} topology, which is called here weak^{∗} continuity. In this paper, we define the demicontinuity of an operator T from V to \(V^{*}\) by its continuity from V to its dual space \(V^{*}\) endowed with weak topology, which is commonly used in most literature works.
Proposition 2.1
Let V be a Banach space with \(V^{\ast}\) being its dual space and \(T:V\to V^{*}\) be an operator. If T is continuous, then it is weakly^{∗} continuous, which, in turn, implies that it is hemicontinuous. Moreover, if T is a monotone operator, then the notions of weak^{∗} continuity and hemicontinuity coincide [7, 44].
Proposition 2.2
Let V be a Banach space with \(V^{*} \) being its dual space, and \(T:V\to V^{*}\) is a operator from V to \(V^{*}\). Then the following statement holds:
Definition 2.4
Definition 2.5
Definition 2.6
Definition 2.7
Definition 2.8
Definition 2.9
Proposition 2.3
- (1)the function \(v\to J^{\circ}(u,v)\) is finite, positively homogeneous, and subadditive, i.e.,and$$\begin{aligned} J^{\circ}(x;\lambda v)=\lambda J^{\circ}(x;v),\quad \forall \lambda \geq0, \end{aligned}$$$$\begin{aligned} J^{\circ}(x; v_{1}+v_{2})=J^{\circ}(x;v_{1})+ J^{\circ}(x;v_{2}), \quad \forall v_{1},v_{2} \in V; \end{aligned}$$
- (2)\(J^{\circ}(u,v)\) is upper semicontinuous on \(V\times V\) as a function of \((u,v)\), i.e., for all \(u,v\in V\), \({u_{n}}\subset V\), \({v_{n}}\subset V\) such that \(u_{n}\to u\), \(v_{n}\to v\) in V, we have$$ \limsup_{n\rightarrow\infty} J^{\circ}(u_{n};v_{n}) \leq J^{\circ}(u;v). $$
3 Well-posedness and metric characterizations
In this section, we aim to extend the well-posedness to the split hemivariational inequality (SHI). We first give the definition of well-posedness for the split hemivariational inequality (SHI), and then we prove its metric characterizations for the well-posedness by using two useful sets defined.
Definition 3.1
Definition 3.2
The split hemivariational inequality (SHI) is said to be strongly (resp., weakly) well-posed if it has a unique solution and every approximating sequence for the split hemivariational inequality (SHI) converges strongly (resp., weakly) to the unique solution.
Definition 3.3
The split hemivariational inequality (SHI) is said to be well-posed in generalized sense (or generalized well-posed) if its solution set is nonempty and, for every approximating sequence, there always exists a subsequence converging to some point of its solution set.
With the definition of two sets \(\Omega(\epsilon)\) and \(\Psi (\epsilon)\), we can get the following properties.
Lemma 3.1
Let \(V_{1}\), \(V_{2}\) be two Banach spaces with \(V_{1}^{\ast}\), \(V_{2}^{\ast}\) being their dual spaces, respectively. Suppose that, for \(i=1,2\), \(A_{i}:V_{i}\rightarrow V_{i}^{\ast}\) is monotone and hemicontinuous on \(V_{i}\) and \(J_{i}:V_{i}\rightarrow\mathbb{R}\) is a locally Lipschitz functional. Then \(\Omega(\epsilon)=\Psi(\epsilon)\) for any \(\epsilon>0\).
Proof
This together with the fact that \(\|u_{2}-Tu_{1}\|_{V_{2}}\leq\epsilon\) due to \({\mathbf {u}}=(u_{1},u_{2})\in\Omega(\epsilon)\) indicates that \(\mathbf {u}\in\Psi(\epsilon)\), and thus \(\Omega(\epsilon)\subset\Psi (\epsilon)\).
Lemma 3.2
Let \(V_{1}\), \(V_{2}\) be two reflective Banach spaces with \(V_{1}^{\ast}\), \(V_{2}^{\ast}\) being their dual spaces, respectively, and \(J_{i}:V_{i}\rightarrow\mathbb{R}\), \(i=1,2\), be a locally Lipschitz functional. Suppose that \(T:V_{1}\to V_{2}\) is a continuous operator from \(V_{1}\) to \(V_{2}\). Then, for any \(\epsilon>0\), \(\Psi(\epsilon)\) is closed in \(V_{1}\times V_{2}\).
Proof
With Lemmas 3.1 and 3.2, it is easy to get the following corollary on the closedness of \(\Omega(\epsilon)\) for any \(\epsilon>0\), which is crucial to the metric characterizations for well-posedness of the split hemivariational inequality (SHI).
Corollary 3.1
Let \(V_{1}\), \(V_{2}\) be two Banach spaces with \(V_{1}^{\ast}\), \(V_{2}^{\ast}\) being their dual spaces, respectively. Suppose that, for \(i=1,2\), \(A_{i}:V_{i}\rightarrow V_{i}^{\ast}\) is monotone and hemicontinuous on \(V_{i}\), \(J_{i}:V_{i}\rightarrow\mathbb{R}\) is a locally Lipschitz functional, and \(T:V_{1}\to V_{2}\) is a continuous operator from \(V_{1}\) to \(V_{2}\). Then \(\Omega(\epsilon)\) is closed for any \(\epsilon>0\).
Remark 3.1
Similar to the idea in many research works on well-posedness for variational inequalities and hemivariational inequalities [17, 25, 46, 47], the set \(\Psi (\epsilon)\) is defined to prove the closedness of \(\Omega(\epsilon)\) under the condition that, for \(i=1,2\), \(A_{i}\) is monotone and hemicontinuous on \(V_{i}\). Actually, without defining the set \(\Psi (\epsilon)\), we could prove directly the property of closedness of \(\Omega(\epsilon)\).
Lemma 3.3
Let \(V_{1}\), \(V_{2}\) be two Banach spaces with \(V_{1}^{\ast}\), \(V_{2}^{\ast}\) being their dual spaces, respectively, and \(J_{i}:V_{i}\rightarrow\mathbb{R}\) be a locally Lipschitz functional for \(i=1,2\). Suppose that \(T:V_{1}\to V_{2}\) is a continuous operator from \(V_{1}\) to \(V_{2}\) and for \(i=1,2\), \(A_{i}:V_{i}\rightarrow V_{i}^{\ast}\) is monotone and hemicontinuous. Then \(\Omega(\epsilon)\) is closed for any \(\epsilon>0\).
Proof
Now, with properties of the set \(\Omega(\epsilon)\) given above, we are in a position to prove metric characterizations for the split hemivariational inequality (SHI)by using similar methods for studying well-posedness of variational inequalities and hemivariational inequalities in research works [17, 25, 46, 47].
Theorem 3.1
Let \(V_{1}\), \(V_{2}\) be two Banach spaces and \(V_{1}^{\ast}\), \(V_{2}^{\ast}\) be their dual spaces, respectively. Suppose that, for \(i=1,2\), \(A_{i}:V_{i}\rightarrow V_{i}^{\ast}\) is an operator on \(V_{i}\) and \(J_{i}:V_{i}\rightarrow\mathbb{R}\) is a locally Lipschitz functional. Then the split hemivariational inequality (SHI) is strongly well-posed if and only if its solution set S is nonempty and \(\operatorname{diam} \Omega(\varepsilon)\rightarrow0\) as \(\varepsilon\rightarrow0\).
Proof
Theorem 3.2
Proof
The following is a concrete example to illustrate the metric characterization of well-posedness for a hemivariational inequality.
Example 3.1
Theorem 3.3
Let \(V_{1}\), \(V_{2}\) be two Banach spaces and \(V_{1}^{\ast}\), \(V_{2}^{\ast}\) be their dual spaces, respectively. Suppose that, for \(i=1,2\), \(A_{i}:V_{i}\rightarrow V_{i}^{\ast}\) is an operator on \(V_{i}\) and \(J_{i}:V_{i}\rightarrow\mathbb{R}\) is a locally Lipschitz functional. Then the split hemivariational inequality (SHI) is generalized well-posed if and only if its solution set S is nonempty compact and \(\mathscr {H}(\Omega (\epsilon),S)\rightarrow0\) as \(\varepsilon\rightarrow0\).
Proof
Theorem 3.4
Proof
4 Concluding remarks
In this paper, we generalize the concept of well-posedness to a split hemivariational inequality (SHI), which is a generalization of classic variational inequality and hemivariational inequality. After defining well-posedness for the split hemivariational inequality (SHI) with its approximating sequences, we establish some metric characterizations using very mild assumptions on operators involved. The obtained results generalize some theorems on well-posedness for hemivariational inequalities and variational inequalities in the literature.
Similar to many research papers on well-posedness for variational inequalities and hemivariational inequalities, in addition to the metric characterizations for well-posedness, it is important and interesting to study the relationships between the well-posedness and its solvability for the split hemivariational inequalities (SHI).
Declarations
Funding
This work was supported by the National Natural Science Foundation of China (11771067), the Applied Basic Project of Sichuan Province (2016JY0170), the Open Foundation of State Key Laboratory of Electronic Thin Films and Integrated Devices (KFJJ201611), and the Chongqing Big Data Engineering Laboratory for Children, Chongqing Electronics Engineering Technology Research Center for Interactive Learning.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
References
- Tikhonov, A.N.: On the stability of the functional optimization problem. USSR Comput. Math. Math. Phys. 6(4), 28–33 (1966) View ArticleMATHGoogle Scholar
- Chen, H.B., Wang, Y.J., Wang, G.: Strong convergence of extragradient method for generalized variational inequalities in Hilbert space. J. Inequal. Appl. 2014, 223 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2007) MATHGoogle Scholar
- Li, W., Xiao, Y.B., Huang, N.J., Cho, Y.J.: A class of differential inverse quasi-variational inequalities in finite dimensional spaces. J. Nonlinear Sci. Appl. 10(8), 4532–4543 (2017) MathSciNetView ArticleGoogle Scholar
- Lu, J., Xiao, Y.B., Huang, N.J.: A stackelberg quasi-equilibrium problem via quasi-variational inequalities. Carpath. J. Math. (in press) Google Scholar
- Sofonea, M., Matei, A.: Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, vol. 18. Springer, New York (2009) MATHGoogle Scholar
- Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems. Springer, New York (2013) View ArticleMATHGoogle Scholar
- Nagurney, A.: Network Economics: A Variational Inequality Approach, vol. 10. Springer, Dordrecht (2013) MATHGoogle Scholar
- Panagiotopoulos, P.D.: Hemivariational Inequalities. Springer, New York (1993) View ArticleMATHGoogle Scholar
- Sofonea, M., Xiao, Y.B.: Fully history-dependent quasivariational inequalities in contact mechanics. Appl. Anal. 95(11), 2464–2484 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Xiao, Y.B., Huang, N.J., Cho, Y.J.: A class of generalized evolution variational inequalities in Banach spaces. Appl. Math. Lett. 25(6), 914–920 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Xiao, Y.B., Fu, X., Zhang, A.: Demand uncertainty and airport capacity choice. Transp. Res., Part B, Methodol. 57(5), 91–104 (2013) View ArticleGoogle Scholar
- Zaslavski, A.J.: Optimization on Metric and Normed Spaces. Springer, New York (2010) View ArticleMATHGoogle Scholar
- Fang, Y.P., Hu, R.: Parametric well-posedness for variational inequalities defined by bifunctions. Comput. Math. Appl. 53(8), 1306–1316 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Huang, X.X., Yang, X.Q.: Generalized Levitin–Polyak well-posedness in constrained optimization. SIAM J. Optim. 17(1), 243–258 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Peng, J.W., Yang, X.M.: Levitin–Polyak well-posedness of a system of generalized vector variational inequality problems. J. Ind. Manag. Optim. 11(3), 701–714 (2015) MathSciNetMATHGoogle Scholar
- Xiao, Y.B., Huang, N.J., Wong, M.M.: Well-posedness of hemivariational inequalities and inclusion problems. Taiwan. J. Math. 15(3), 1261–1276 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Han, Y., Gong, X.: Levitin–Polyak well-posedness of symmetric vector quasi-equilibrium problems. Optimization 64(7), 1537–1545 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Li, S.J., Li, M.H.: Levitin–Polyak well-posedness of vector equilibrium problems. Math. Methods Oper. Res. 69(1), 125–140 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Lignola, M.: Well-posedness and L-well-posedness for quasivariational inequalities. J. Optim. Theory Appl. 128(1), 119–138 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Reich, S., Zaslavski, A.J.: Generic well-posedness of fixed point problems. Vietnam J. Math. 46, 1–9 (2017) MathSciNetGoogle Scholar
- Lucchetti, R., Patrone, F.: A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities. Numer. Funct. Anal. Optim. 3(4), 461–476 (1981) MathSciNetView ArticleMATHGoogle Scholar
- Lucchetti, R., Patrone, F.: Some properties of “well-posed” variational inequalities governed by linear operators. Numer. Funct. Anal. Optim. 5(3), 349–361 (1983) MathSciNetView ArticleMATHGoogle Scholar
- Hu, R., Fang, Y.P.: Levitin–Polyak well-posedness of variational inequalities. Nonlinear Anal. 72(1), 373–381 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Fang, Y.P., Huang, N.J., Yao, J.C.: Well-posedness by perturbations of mixed variational inequalities in Banach spaces. Eur. J. Oper. Res. 201(3), 682–692 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Liu, Z.H.: Browder–Tikhonov regularization of non-coercive evolution hemivariational inequalities. Inverse Probl. 21(1), 13–20 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Xiao, Y.B., Huang, N.J.: Generalized quasi-variational-like hemivariational inequalities. Nonlinear Anal. 69(2), 637–646 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Migórski, S., Ochal, A.: Boundary hemivariational inequality of parabolic type. Nonlinear Anal. 57(4), 579–596 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Dekker, New York (1995) MATHGoogle Scholar
- Zhang, W., Han, D., Jiang, S.: A modified alternating projection based prediction correction method for structured variational inequalities. Appl. Numer. Math. 83(2), 12–21 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Goeleven, D., Mentagui, D.: Well-posed hemivariational inequalities. Numer. Funct. Anal. Optim. 16(7–8), 909–921 (1995) MathSciNetView ArticleMATHGoogle Scholar
- Virmani, G., Srivastava, M.: On Levitin–Polyak alpha-well-posedness of perturbed variational-hemivariational inequality. Optimization 64(5), 1153–1172 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Liu, Z.B., Gou, J.H., Xiao, Y.B., Li, X.S.: A system of generalized variational-hemivariational inequalities with set-valued mappings. J. Appl. Math. 2013, Article ID 305068 (2013) MathSciNetGoogle Scholar
- Liu, Z., Motreanu, D., Zeng, S.: On the well-posedness of differential mixed quasi-variational-inequalities. Topol. Methods Nonlinear Anal. 51(1), 135–150 (2018) MathSciNetMATHGoogle Scholar
- Wang, Y.M., Xiao, Y.B., Wang, X., Cho, Y.J.: Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems. J. Nonlinear Sci. Appl. 9(3), 1178–1192 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59(2), 301–323 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51(10), 2353–2365 (2006) View ArticleGoogle Scholar
- Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21(6), 2071–2084 (2005) MathSciNetView ArticleMATHGoogle Scholar
- He, H., Ling, C., Xu, H.K.: A relaxed projection method for split variational inequalities. J. Optim. Theory Appl. 166(1), 213–233 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150(2), 275–283 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Masad, E., Reich, S.: A note on the multiple-set split feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8(3), 367–371 (2007) MathSciNetMATHGoogle Scholar
- Hu, R., Fang, Y.P.: Characterizations of Levitin–Polyak well-posedness by perturbations for the split variational inequality problem. Optimization 65(9), 1717–1732 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Zeidler, E.: Nonlinear Functional Analysis and Its Applications, vol. II. Springer, Berlin (1990) View ArticleMATHGoogle Scholar
- Zezislaw, D., Migórski, S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic, Dordrecht (2003) Google Scholar
- Kuratowski, K.: Topology, vols. 1, 2. Academic Press, New York (1968) Google Scholar
- Lv, S., Xiao, Y.B., Liu, Z.B., Li, X.S.: Well-posedness by perturbations for variational-hemivariational inequalities. J. Appl. Math. 2012, Article ID 804032 (2012) MathSciNetMATHGoogle Scholar
- Hu, R., Fang, Y.P.: Well-posedness of the split inverse variational inequality problem. Bull. Malays. Math. Sci. Soc. 40(4), 1733–1744 (2017) MathSciNetView ArticleMATHGoogle Scholar