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Well-posedness for a class of generalized variational-hemivariational inequalities involving set-valued operators
- Caijing Jiang^{1}Email author
https://doi.org/10.1186/s13660-018-1776-x
© The Author(s) 2018
- Received: 14 April 2018
- Accepted: 13 July 2018
- Published: 24 July 2018
Abstract
The aim of present work is to study some kinds of well-posedness for a class of generalized variational-hemivariational inequality problems involving set-valued operators. Some systematic approaches are presented to establish some equivalence theorems between several classes of well-posedness for the inequality problems and some corresponding metric characterizations, which generalize many known results. Finally, the well-posedness for a class of generalized mixed equilibrium problems is also considered.
Keywords
- Generalized variational-hemivariational inequality
- Set-valued operator
- α-well-posedness
- Monotonicity
1 Introduction
Nowadays, well-posedness has been drawing great attention in the field of optimization problems and related problems such as variational inequalities, hemivariational inequalities, fixed point problems, equilibrium problems, and inclusion problems (see [1, 5, 9, 11, 17, 19, 21, 23, 33]). The classical concept of well-posedness for a global minimization problem was first introduced by Tikhonov [35], which required the existence and uniqueness of a solution to the global minimization problem and the convergence of every minimizing sequence toward the unique solution. Thereafter, the concept of well-posedness has been generalized to variational inequalities. The initial notion of well-posedness for variational inequality is due to Lucchetti and Patrone [28]. Fang [13, 14] generalized two kinds of well-posedness for a mixed variational inequality problem in a Banach space. For further results on the well-posedness of variational inequalities, we refer to [2, 4, 12–14, 16, 22, 27, 28] and the references therein.
As an important and useful generalization of variational inequality, hemivariational inequality, which was first studied by Panagiotopoulos [32], has a great development in recent years by several works [6, 29, 31]. Many authors are interested in generalizing the concept of well-posedness to hemivariational inequalities. In 1995, Goeleven and Mentagui [15] generalized the concept of the well-posedness to a hemivariational inequality and presented some basic results concerning the well-posed hemivariational inequality. Recently, using the concept of approximating sequence, Xiao et al. [37, 38] introduced a concept of well-posedness for a hemivariational inequality and a variational-hemivariational inequality. Ceng, Lur, and Wen [3] considered an extension of well-posedness for a minimization problem to a class of generalized variational-hemivariational inequalities with perturbations in reflexive Banach spaces. For more recent works on the well-posedness for variational-hemivariational inequalities, we refer to [3, 15, 18, 19, 26, 37, 38] and the references therein.
In the last years, many authors studied the existence results for some types of hemivariational inequalities involving set-valued operators [34, 36, 39]. In 2011, Zhang and He [39] studied a kind of hemivariational inequalities of Hartman–Stampacchia type by introducing the concept of stable quasimonotonicity. They supposed that the constraint set is a bounded (or unbounded), closed, and convex subset in a reflexive Banach space. The authors gave sufficient conditions for the existence and boundedness of solutions. In 2013, Tang and Huang [34] generalized the result of [39] by introducing the concept of stable ϕ-quasimonotonicity and obtained some existence theorems when the constrained set is nonempty, bounded (or unbounded), closed, and convex in a reflexive Banach space. Hereafter, Wangkeeree and Preechasilp [36] generalized the results of [34] and [39] by introducing the concept of stable f-quasimonotonicity. Very recently, Liu and Zeng obtained some existence results for a class of hemivariational inequalities involving the stable \((g,f,\alpha)\)-quasimonotonicity [25], a result on the well-posedness for mixed quasivariational hemivariational inequalities [26], and some existence results for a class of quasimixed equilibrium problems involving the \((f,g,h)\)-quasimonotonicity [24].
Let K be a nonempty, closed, and convex subset of a real Banach space X with its dual \(X^{*}\), and let \(F:K\rightarrow P(X^{*})\) be a set-valued operator, where \(P(X^{*})\) is the set of all nonempty subsets of \(X^{*}\). Let \(T:K\rightarrow X^{*}\) be a perturbation, and let \(f\in X^{*}\) be a given element. Let \(g:K\times K\rightarrow \overline{R}:=R\cup \{\pm \infty \}\) be a function such that \(\mathcal{D}(g)=\{u\in K:g(u,v)\neq -\infty, \forall v\in K\}\neq \emptyset \). Let \(J:X\to R\) be a locally Lipschitz function, and let \(J^{\circ }(u,v)\) denote the generalized directional derivative in the sense of Clarke of a locally Lipschitz functional \(J:X\rightarrow R\) at u in the direction v. In this paper, we discuss the following generalized variational-hemivariational inequality (GVHVI):
- (a)
If \(T\equiv 0\), \(f\equiv 0\), and \(g\equiv 0\), then GVHVI is reduced to the following form:
Find \(u\in K\) and \(u^{*}\in F(u)\) such thatThe existence of solutions to this inequality was recently studied by Zhang and He [39].$$\begin{aligned} \bigl\langle u^{*},v-u\bigr\rangle +J^{\circ }(u;v-u)\geq 0, \quad \forall v\in K. \end{aligned}$$ - (b)
If \(T\equiv 0\) and \(f\equiv 0\), and \(g(u,v)=\phi (v)-\phi (u)\) for all \(u,v\in K\), then GVHVI is reduced to the following form:
Find \(u\in K\) and \(u^{*}\in F(u)\) such thatThe existence of solutions to this inequality was studied by Tang and Huang [34].$$\begin{aligned} \bigl\langle u^{*},v-u\bigr\rangle +\phi (v)-\phi (u)+J^{\circ }(u;v-u) \geq 0, \quad \forall v\in K. \end{aligned}$$ - (c)
If \(T\equiv 0\) and \(f\equiv 0\), then GVHVI is reduced to the following form:
Find \(u\in K\) and \(u^{*}\in F(u)\) such that$$\begin{aligned} \bigl\langle u^{*},v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u)\geq 0, \quad \forall v\in K. \end{aligned}$$
Inspired by previous works, we study the well-posedness for GVHVI, which generalizes many known works. Under relatively weak conditions, we establish some equivalence results and some metric characterizations for the strong and weak α-well-posed GVHVI in the generalized sense. In particular, we present equivalence results on weak α-well-posedness for GVHVI, which were considered by few authors.
This paper is organized as follows. In Sect. 2, we recall some basic preliminaries of single-valued and set-valued mappings, metric concepts, Clarke’s generalized directional derivative, and some classes of well-posedness for GVHVI. In Sect. 3, we show some equivalence results for the well-posedness for GVHVI and some corresponding metric characterizations. Theorems 3.3, 3.5, and 3.6 are the main results in this section. In the last section, we also present the well-posedness for a class of generalized mixed equilibrium problems.
2 Preliminaries
Let R, \(R_{+}\), and N be the sets of real numbers, nonnegative real numbers, and natural numbers, respectively. Let X be a real Banach space with norm \(\|\cdot \|_{X}\). Denote by \(X^{*}\) its dual space and by \(\langle \cdot,\cdot \rangle_{X}\) the duality pairing between \(X^{*}\) and X. Let \(X_{w}\) be the Banach space X with weak topology.
Definition 2.1
- (i)convex on K if for all finite subsets \(\{u_{1},\ldots,u_{n} \}\subset K\) and \(\{\lambda_{1},\ldots,\lambda_{n}\}\subset R_{+}\) such that \(\sum_{i=1}^{n}\lambda_{i}=1\) and \(\sum_{i=1}^{n}\lambda_{i}u _{i}\in K\), we have$$\begin{aligned} f\Biggl(\sum_{i=1}^{n}\lambda_{i}u_{i} \Biggr)\leq \sum_{i=1}^{n} \lambda_{i}f(u _{i}); \end{aligned}$$
- (ii)(weakly) upper semicontinuous (u.s.c. for short) at u if for any sequence \(\{u_{n}\}_{n\geq 1}\subset K\) with (\(u_{n}\rightharpoonup u\)) \(u_{n}\rightarrow u\), we have$$\begin{aligned} \limsup_{n\rightarrow \infty }f(u_{n})\leq f(u). \end{aligned}$$
- (iii)(weakly) lower semicontinuous (l.s.c. for short) at u, if for any sequence \(\{u_{n}\}_{n\geq 1}\subset K\) with (\(u_{n}\rightharpoonup u\)) \(u_{n}\rightarrow u\), we haveThe function f is said to be (weakly) u.s.c. (l.s.c.) on K if f is (weakly) u.s.c. (l.s.c.) at all \(u\in K\).$$\begin{aligned} \liminf_{n\rightarrow \infty }f(u_{n})\geq f(u). \end{aligned}$$
Definition 2.2
([20])
Definition 2.3
- (i)
lower semicontinuous (l.s.c.) at \(u_{0}\) if for any \(u_{0}^{*}\in F(u_{0})\) and sequence \(\{u_{n}\}_{n\geq 1}\subset K\) with \(u_{n}\rightarrow u_{0}\), there exists a sequence \(u_{n}^{*}\in F(u _{n})\) that converges to \(u_{0}^{*}\).
- (ii)
lower hemicontinuous (l.h.c.) if the restriction of F to every line segment of K is lower semicontinuous with respect to the weak topology in \(X^{*}\).
Definition 2.4
Definition 2.5
Lemma 2.6
([6], Proposition 2.1.1)
- (i)\(\varphi^{0}(u;v)\) is u.s.c. as a function of \((u,v)\) and, as a function of v alone, is Lipschitz of rank \(L_{u}\) near u on X and satisfies$$ \bigl\vert \varphi^{0}(u;v) \bigr\vert \leq L_{u} \Vert v \Vert _{X}; $$
- (ii)
the gradient \(\partial \varphi (u)\) is a nonempty, convex, and weakly^{∗} compact subset of \(X^{*}\) bounded by a Lipschitz constant \(L_{u}\) near x;
- (iii)for every \(v\in X\), we have$$ \varphi^{0}(u;v)= \max \bigl\{ \bigl\langle u^{*},v\bigr\rangle |u^{*}\in \partial \varphi (u)\bigr\} . $$
We end this section with the notions of several classes of α-approximating sequences and α-well-posedness for GVHVI. Let \(\alpha:X\to R_{+}\) be a functional.
Definition 2.7
Definition 2.8
GVHVI is said to be strongly (respectively, weakly) α-well-posed if it has a unique solution u and every α-approximating sequence \(\{u_{n}\}\) strongly (respectively, weakly) converges to u. In particular, if \(\alpha (\cdot) = \|\cdot \|_{X}\), then GVHVI is said to be strongly (respectively, weakly) well-posed.
Definition 2.9
GVHVI is said to be strongly (respectively, weakly) α-well-posed in the generalized sense if the solution set Γ of GVHVI is nonempty and every α-approximating sequence \(\{u_{n}\}\) has a subsequence that strongly (respectively, weakly) converges to some point of Γ. In particular, if \(\alpha (\cdot) = \|\cdot \|_{X}\), then GVHVI is said to be strongly (respectively, weakly) well-posed in the generalized sense.
Remark 2.10
Strong α-well-posedness (in the generalized sense) implies weak α-well-posedness (in the generalized sense), but the converse is not true in general.
3 The characterizations of well-posedness for GVHVI
In this section, we establish metric characterizations and derive some conditions under which GVHVI is strongly (weakly) α-well-posed.
Lemma 3.1
- (i)
K is a nonempty closed subset of a real Banach space X;
- (ii)
\(T:K \rightarrow X^{*}_{w}\) is continuous;
- (iii)
\(g:K\times K\rightarrow R\) is u.s.c. with respect to the first variable;
- (iv)
\(\alpha:X\to R_{+}\) is such that \(\liminf_{n\rightarrow \infty }\alpha (v_{n})\le \alpha (v)\) whenever \(v_{n}\rightarrow v\).
Proof
Lemma 3.2
- (i)
K is a nonempty convex subset of a real Banach space X;
- (ii)
\(F:K\rightarrow P(X^{*})\) is l.h.c. and monotone;
- (iii)
\(g:K\times K\rightarrow R\) is convex with respect to the second variable;
- (iv)
\(\alpha:X\to R_{+}\) is convex with \(\alpha (tv)=t\alpha (v)\) for all \(t\ge 0\) and \(v\in X\).
Proof
The following result is a consequence of Lemmas 3.1 and 3.2.
Theorem 3.3
- (i)
K is a nonempty closed convex subset of a real Banach space X;
- (ii)
\(F:K\rightarrow P(X^{*})\) is l.h.c. and monotone;
- (iii)
\(T:K\rightarrow X^{*}_{w}\) is continuous;
- (iv)
\(g:K\times K\rightarrow R\) is u.s.c. with respect to the first variable and convex with respect to the second variable;
- (v)
\(\alpha:X\to R_{+}\) is continuous and convex with \(\alpha (tv)=t \alpha (v)\) for all \(t\ge 0\) and \(v\in X\).
Theorem 3.4
Proof
The proof is similar to that of Theorem 4.3 in [26] by the assumptions of g. □
Theorem 3.5
Proof
Theorem 3.6
- (i)
K is a nonempty closed convex subset of a real reflexive Banach space X;
- (ii)
\(F:K\rightarrow P(X^{*})\) is l.h.c. and monotone;
- (iii)
\(T:K\rightarrow X^{*}\) is compact;
- (iv)
\(g:K\times K\rightarrow R\) is weakly u.s.c. with respect to the first variable and convex with respect to the second variable;
- (v)
\(\limsup_{n\rightarrow \infty }J^{\circ }(u_{n};v-u_{n})\le J^{ \circ }(u;v-u)\) for all \(v\in X\) whenever \(u_{n}\rightharpoonup u\) as \(n\rightarrow \infty \);
- (vi)
\(\alpha:X\to R_{+}\) is a continuous and convex functional with \(\alpha (tv)=t\alpha (v)\) for all \(t\ge 0\) and \(v\in X\).
Proof
Remark 3.7
Next, we give some equivalence results for the strong α-posedness in the generalized sense.
Theorem 3.8
Proof
The proof is similar to that of Theorem 5.1 in [26] by the assumptions of g. □
Theorem 3.9
Proof
The proof is similar to that of Theorem 3.2 in [3] by the assumptions of g. □
Theorem 3.10
Assume that all the assumptions of Theorem 3.6 are satisfied. Then GVHVI is weakly α-well-posed in the generalized sense if and only if there exists \(\epsilon_{0}>0\) such that \(\Omega_{\alpha }(\epsilon_{0})\) is nonempty and bounded.
Proof
The proof is similar to that of Theorem 3.6 by the assumptions of g. □
4 Well-posedness for GMEP
In this section, we consider the following generalized mixed equilibrium problem (GMEP):
To study GMEP, we introduce the concept of η-monotonicity (see [7, 8]).
Definition 4.1
Remark 4.2
We can obtain similar results.
Theorem 4.3
- (i)
\(h(u,u)=0\) for all \(u\in X\),
- (ii)
for all \(v\in K\), \(h(\cdot,v)\) is u.s.c.,
- (iii)
for all \(u\in K\), \(h(u,\cdot)\) is convex.
Theorem 4.4
- (i)
\(h(u,u)=0\) for all \(u\in X\),
- (ii)
for all \(v\in K\), \(h(\cdot,v)\) is u.s.c.,
- (iii)
for all \(u\in K\), \(h(u,\cdot)\) is convex.
Theorem 4.5
- (i)
\(h(u,u)=0\) for all \(u\in X\),
- (ii)
for all \(v\in K\), \(h(\cdot,v)\) is weakly u.s.c.,
- (iii)
for all \(u\in K\), \(h(u,\cdot)\) is convex.
Theorem 4.6
- (i)
\(h(u,u)=0\) for all \(u\in X\),
- (ii)
for all \(v\in K\), \(h(\cdot,v)\) is u.s.c.,
- (iii)
for all \(u\in K\), \(h(u,\cdot)\) is convex.
Theorem 4.7
- (i)
\(h(u,u)=0\) for all \(u\in X\),
- (ii)
for all \(v\in K\), \(h(\cdot,v)\) is weakly u.s.c.,
- (iii)
for all \(u\in K\), \(h(u,\cdot)\) is convex.
5 Conclusion
In this paper, inspired by the previous works, we study the well-posedness for GVHVI. Under relatively weak conditions for the data F, T, g, J (see Theorems 3.3 and 3.6), we provide some equivalence results for the strong and weak α-well-posed GVHVI in the generalized sense. Our results generalize and improve many known results and can be applied to many other problems.
Declarations
Funding
The work was supported by the National Natural Science Foundation of China Grant No. 11361009 and the High level innovation teams and distinguished scholars in Guangxi Universities.
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares to 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
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