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Well-posedness for a class of generalized variational-hemivariational inequalities involving set-valued operators
Journal of Inequalities and Applications volume 2018, Article number: 187 (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.
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):
Find \(u\in K\) such that, for some \(u^{*}\in F(u)\),
Now, let us consider some particular cases of 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 that
$$\begin{aligned} \bigl\langle u^{*},v-u\bigr\rangle +J^{\circ }(u;v-u)\geq 0, \quad \forall v\in K. \end{aligned}$$The existence of solutions to this inequality was recently studied by Zhang and He [39].
-
(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 that
$$\begin{aligned} \bigl\langle u^{*},v-u\bigr\rangle +\phi (v)-\phi (u)+J^{\circ }(u;v-u) \geq 0, \quad \forall v\in K. \end{aligned}$$The existence of solutions to this inequality was studied by Tang and Huang [34].
-
(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
$$\begin{aligned} \bigl\langle u^{*},v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u)\geq 0, \quad \forall v\in K. \end{aligned}$$
The existence of solutions to this inequality was studied by Wangkeeree and Preechasilp [36].
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
Let K be a nonempty subset of X. A function \(f:K\rightarrow R\) is said to be
-
(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 have
$$\begin{aligned} \liminf_{n\rightarrow \infty }f(u_{n})\geq f(u). \end{aligned}$$The 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 \(u\in K\).
Definition 2.2
([20])
Let K be a nonempty subset of X. An operator \(\beta:K\rightarrow X\) is said to be affine if for any \(u_{i}\in K\) (\(i=1,2,\ldots,n\)) and \(\lambda_{i}\in [0,1]\) with \(\sum_{i=1} ^{n}\lambda_{i}=1\), we have
Definition 2.3
A set-valued operator \(F:K\rightarrow P(X^{*})\) is said to be
-
(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
A set-valued operator \(F:K\rightarrow P(X^{*})\) is said to be monotone if for all \(u,v\in K\),
Definition 2.5
Let S be a nonempty subset of X. The measure μ of noncompactness for the set S is defined by
where diam\(|S_{i}|\) is the diameter of the set \(S_{i}\).
Now, let us recall the definitions of the Clarke generalized directional derivative and generalized gradient for a locally Lipschitz function \(\varphi:X\rightarrow R\) (see [6, 10]). The Clarke generalized directional derivative \(\varphi^{0}(u;v)\) of φ at the point \(u\in X\) in the direction \(v\in X\) is defined as
The Clarke subdifferential or generalized gradient of φ at \(u\in X\), denoted by \(\partial \varphi (u)\), is the subset of \(X^{*}\) given by
Lemma 2.6
([6], Proposition 2.1.1)
Let \(\varphi:X\rightarrow R\) be locally Lipschitz of rank \(L_{u}>0\) near u. Then
-
(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
A sequence \(\{u_{n}\}\) in K is an α-approximating sequence for GVHVI if there exist \(\{u^{*}_{n}\}\) in \(X^{*}\) with \(u^{*}_{n}\in F(u _{n})\) and a nonnegative sequence \(\{\epsilon_{n}\}\) with \(\epsilon _{n}\rightarrow 0\) as \(n\rightarrow \infty \) such that, for every \(n\in N\),
In particular, if \(\alpha (\cdot) = \|\cdot \|_{X}\), then \(\{u_{n}\}\) is said to be an approximating sequence for GVHVI.
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.
For any \(\epsilon >0\), we define the following two sets:
and
Denote by Γ the set of solutions to GVHVI. It is clear that \(\Gamma =\Omega_{0}(\epsilon)\).
Lemma 3.1
Assume that:
-
(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 \(v_{n}\rightarrow v\).
Then, for every \(\epsilon >0\), the set \(\Phi_{\alpha }(\epsilon)\) is closed in X.
Proof
Let \(\{u_{n}\}\subset \Phi_{\alpha }(\epsilon)\) be s sequence such that \(u_{n} \rightarrow u\) in X. Then \(u\in K\), and, for all \(v\in K\) and \(v^{*}\in F(v)\),
By the assumptions and the properties of \(J^{\circ }\) we have
and hence
which shows that \(u\in \Phi_{\alpha }(\epsilon)\). □
Lemma 3.2
Assume that:
-
(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 \(v\in X\).
Then \(\Omega_{\alpha }(\epsilon)=\Phi_{\alpha }(\epsilon)\) for all \(\epsilon >0\).
Proof
We first show that \(\Omega_{\alpha }(\epsilon)\subset \Phi_{\alpha }( \epsilon)\). Indeed, take arbitrary \(u\in \Omega_{\alpha }(\epsilon)\). Then there exists \(u^{*}\in F(u)\) such that
According to the monotonicity of F, we obtain
which means that \(u\in \Phi_{\alpha }(\epsilon)\). Therefore \(\Omega_{\alpha }(\epsilon)\subset \Phi_{\alpha }(\epsilon)\).
Now we show that \(\Phi_{\alpha }(\epsilon)\subset \Omega_{\alpha }( \epsilon)\). Indeed, take arbitrary \(u\in \Phi_{\alpha }(\epsilon)\). Then
Since the set K is convex, for any \(v\in K\) and \(\lambda \in [0,1]\), taking \(v_{\lambda }:=\lambda v+(1-\lambda)u\in K\) in this inequality, we have
Then by (iii), (iv), and the properties of \(J^{\circ }\) we obtain
Let \(u^{*}\in F(u)\) be fixed, and let \(v_{\lambda }^{*}\in F(v_{ \lambda })\) be such that \(v_{\lambda }^{*}\rightharpoonup u^{*}\) in \(X^{*}\) (the existence of such a sequence is ensured by the fact that F is l.h.c.). Taking the limit as \(\lambda \rightarrow 0\) in (3.1), we obtain
which implies that \(u\in \Omega_{\alpha }(\epsilon)\). The proof is complete. □
The following result is a consequence of Lemmas 3.1 and 3.2.
Theorem 3.3
Assume that:
-
(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 \(v\in X\).
Then \(\Omega_{\alpha }(\epsilon)=\Phi_{\alpha }(\epsilon)\) is closed in X for all \(\epsilon >0\). Moreover, \(\Gamma =\Omega_{0}(\epsilon)=\Phi_{0}(\epsilon)\), that is, GVHVI is equivalent to the following problem:
Find \(u\in K\) such that
Theorem 3.4
GVHVI is strongly α-well-posed if and only if Γ is nonempty and
Proof
The proof is similar to that of Theorem 4.3 in [26] by the assumptions of g. □
Theorem 3.5
Assume that all the assumptions of Theorem 3.3 are satisfied. Then GVHVI is strongly α-well-posed if and only if
Proof
Suppose that GVHVI is strongly α-well-posed. Then GVHVI has a unique solution \(u\in K\), and thus \(\Gamma \neq \emptyset \). Now, we prove that (3.2) holds. Clearly, \(\Omega_{\alpha }(\epsilon) \supset \Gamma \neq \emptyset \). For the second part of (3.2), arguing by contradiction, let us assume that \(\operatorname{diam}( \Omega_{\alpha }(\epsilon))\) does not tend to 0 as \(\epsilon \rightarrow 0\). Thus for any nonnegative sequence \(\{\epsilon_{n}\}\) with \(\epsilon_{n}\rightarrow 0\) as \(n\rightarrow \infty \), there exists a constant \(\beta >0\) such that, for each \(n\in N\), there exist \(u_{n}^{(1)},u_{n}^{(2)}\in \Omega_{\alpha }(\epsilon_{n})\) satisfying
Since \(u_{n}^{(1)},u_{n}^{(2)}\in \Omega_{\alpha }(\epsilon_{n})\), we know that the sequences \(\{u_{n}^{(1)}\}\) and \(\{u_{n}^{(2)}\}\) are both α-approximating sequences of GVHVI, and thus
which is a contradiction.
Conversely, assume that condition (3.2) holds. Let \(\{u_{n}\}\) in K be an α-approximating sequence for GVHVI. Then, there exist \(\{u^{*}_{n}\}\) in \(X^{*}\) with \(u_{n}^{*}\in F(u_{n})\) and a nonnegative sequence \(\{\epsilon_{n}\}\) with \(\epsilon_{n}\rightarrow 0\) as \(n\rightarrow \infty \) such that, for every \(n\in N\),
that is, \(u_{n}\in \Omega_{\alpha }(\epsilon_{n})\) for all \(n\in N\). By condition (3.2) we deduce that the sequence \(\{u_{n}\}\) is a Cauchy sequence, and so \(\{u_{n}\}\) converges strongly to some point \(u\in K\). Let us show that \(u\in K\) is a solution for GVHVI. By the monotonicity of F we obtain that, for every \(n\in N\),
By the assumptions we obtain that
which implies that
It follows from Theorem 3.3 that there exists \(u^{*}\in F(u)\) such that
Then \(u\in K\) is a solution of GVHVI.
Finally, we prove that the solution u is unique. If there exists another solution \(u'\in K\), then \(u,u_{1}\in \Omega_{\alpha }(\epsilon)\) for all \(\epsilon >0\), and
which is a contradiction. This completes the proof. □
Theorem 3.6
Assume that:
-
(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 \(v\in X\).
Then GVHVI is weakly α-well-posed if and only if GVHVI has a unique solution and there exists \(\epsilon_{0}>0\) such that \(\Omega_{\alpha }(\epsilon_{0})\) is nonempty and bounded.
Proof
The necessity is obvious. We now prove the sufficiency. Let \(\{u_{n}\}\) be an α-approximating sequence for GVHVI. Then, there exist \(\{u^{*}_{n}\}\) in \(X^{*}\) with \(u_{n}^{*}\in F(u_{n})\) and a nonnegative sequence \(\{\epsilon_{n}\}\) with \(\epsilon_{n}\rightarrow 0\) as \(n\rightarrow \infty \) such that, for every \(n\in N\),
for all \(v\in K\). We claim that the sequence \(\{u_{n}\}\) is bounded in X. Indeed, since \(\Omega_{\alpha }(\epsilon_{0})\) is bounded and \(\Omega_{\alpha }(\epsilon)\subset \Omega_{\alpha }(\epsilon_{0})\) for all \(\epsilon \in (0,\varepsilon_{0})\), there exists \(n_{0}\in N\) such that \(\epsilon_{n_{0}}\in (0,\varepsilon_{0})\) and \(u_{n}\in \Omega_{\alpha }(\epsilon_{0})\) for all \(n\ge n_{0}\), which shows that \(\{u_{n}\}\) is bounded in X.
Since the Banach space X is reflexive, we can choose a subsequence of \(\{u_{n}\}\), denoted by \(\{u_{n}\}\) again, such that \(u_{n}\rightharpoonup \overline{u}\) as \(n\rightarrow \infty \) for some \(\overline{u}\in X\). Let us show that \(\overline{u}\in K\) is a solution for GVHVI. Obviously, \(\overline{u}\in K\). By the monotonicity of F we obtain that
By the assumptions, we obtain that
which implies that
It follows from Theorem 3.3 that there exists \(\overline{u}^{*}\in F(\overline{u})\) such that
Therefore \(\overline{u}\in K\) is a solution to problem GVHVI, and so we get that GVHVI is weakly α-well-posed by the uniqueness of the solution to problem GVHVI. This completes the proof. □
Remark 3.7
In the theorem, condition (v) can be found in [30], and the condition that there exists \(\epsilon_{0}>0\) such that \(\Omega_{ \alpha }(\epsilon_{0})\) is nonempty and bounded can be replaced by the conditions that K is bounded or that there exists \(n_{0}\in N\) such that, for every \(u\in K\setminus B_{n_{0}}\), there exists \(v\in K\) with \(\|v\|<\|u\|\) such that
See [34, 36, 39] for more detail.
Next, we give some equivalence results for the strong α-posedness in the generalized sense.
Theorem 3.8
Assume that all the assumptions of Theorem 3.5 are satisfied. Then GVHVI is strongly α-well-posed in the generalized sense if and only if Γ is nonempty compact and
where \(e(A,B):=\sup_{a\in A}d(a,B)\) with \(d(a,B):=\inf_{b\in B}\|a-b \|\).
Proof
The proof is similar to that of Theorem 5.1 in [26] by the assumptions of g. □
Theorem 3.9
Assume that all the assumptions of Theorem 3.5 are satisfied. Then GVHVI is strongly α-well-posed in the generalized sense if and only if
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):
Find \(u\in K\) such that, for some \(u^{*}\in F(u)\),
where \(\eta:K\times K\rightarrow X\) is an operator. The existence of solutions to this problem when \(T\equiv 0\) and \(f\equiv 0\) can be found in [25].
To study GMEP, we introduce the concept of η-monotonicity (see [7, 8]).
Definition 4.1
Let \(F:K\rightarrow P(X^{*})\) be a set-valued operator. F is said to be η-monotone if there exists a function \(\eta:K\times K\rightarrow X\) such that, for all \(u,v\in K\),
Remark 4.2
If \(\eta (u,v)=v-u\) for all \(u,v\in X\), then (4.1) becomes
that is, F is monotone.
For any \(\epsilon >0\), we define the following two sets:
and
Denote by \(\Gamma_{\eta }\) the set of solutions to GMEP. It is clear that \(\Gamma =\Omega_{0}(\epsilon)\).
We can obtain similar results.
Theorem 4.3
Assume that all the assumptions of Theorem 3.3 are satisfied and, in addition, \(\eta:K\times K\rightarrow X\) is continuous on \(K\times K\) with \(\eta (u,u)=0\) for any \(u\in K\) and affine with respect to the first variable. Let \(h:K\times K\rightarrow R\) be such that:
-
(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.
Then \(\Omega_{\eta,\alpha }(\epsilon)=\Phi_{\eta,\alpha }(\epsilon)\) is closed in X for all \(\epsilon >0\). Moreover, \(\Gamma_{\eta }= \Omega_{\eta,0}(\epsilon)=\Phi_{\eta,0}(\epsilon)\), that is, GMEP is equivalent to the following problem:
Find \(u\in K\) such that
Theorem 4.4
Assume that all the assumptions of Theorem 3.5 are satisfied and, in addition, \(\eta:K\times K\rightarrow X\) is continuous on \(K\times K\) with \(\eta (u,u)=0\) for any \(u\in K\) and affine with respect to the first variable. Let \(h:K\times K\rightarrow R\) be such that:
-
(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.
Then GMVHVI is strongly α-well-posed if and only if
Theorem 4.5
Assume that all the assumptions of Theorem 3.6 are satisfied and, in addition, \(\eta:K\times K\rightarrow X\) is continuous on \(K\times K\) with \(\eta (u,u)=0\) for any \(u\in K\) and affine with respect to the first variable. Let \(h:K\times K\rightarrow R\) be such that:
-
(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.
Then GMEP is weakly α-well-posed if and only if GMEP has a unique solution and there exists \(\epsilon_{0}>0\) such that \(\Omega_{\alpha }(\epsilon_{0})\) is nonempty and bounded.
Theorem 4.6
Assume that all the assumptions of Theorem 3.5 are satisfied and, in addition, \(\eta:K\times K\rightarrow X\) is continuous on \(K\times K\) with \(\eta (u,u)=0\) for any \(u\in K\) and affine with respect to the first variable. Let \(h:K\times K\rightarrow R\) is such that:
-
(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.
Then GMEP is strongly α-well-posed in the generalized sense if and only if
Theorem 4.7
Assume that all the assumptions of Theorem 3.6 are satisfied and, in addition, \(\eta:K\times K\rightarrow X\) is continuous on \(K\times K\) with \(\eta (u,u)=0\) for any \(u\in K\) and affine with respect to the first variable. Let \(h:K\times K\rightarrow R\) be such that:
-
(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.
Then GMEP 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.
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.
References
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Ceng, L.-C., Hadjisavvas, N., Schaible, S., Yao, J.C.: Well-posedness for mixed quasivariational-like inequalities. J. Optim. Theory Appl. 139, 109–125 (2008)
Ceng, L.-C., Lur, Y.Y., Wen, C.F.: Well-posedness for generalized variational-hemivariational inequalities with perturbations in reflexive Banach spaces. Tamkang J. Math. 48(4), 345–364 (2017)
Ceng, L.-C., Yao, J.C.: Well-posedness of generalized mixed variational inequalities inclusion problems and fixed point problem. Nonlinear Anal. TMA 69, 4585–4603 (2008)
Cho, S.Y.: Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space. J. Appl. Anal. Comput. 8, 19–31 (2018)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Costea, N.: Existence and uniqueness results for a class of quasi-hemivariational inequalities. J. Math. Anal. Appl. 373, 305–311 (2011)
Costea, N., Lupu, C.: On a class of variational-hemivariational inequalities involving set valued mappings. Adv. Pure Appl. Math. 1, 233–246 (2010)
Darabi, M., Zafarani, J.: Hadamard well-posedness for vector parametric equilibrium problems. J. Nonlinear Var. Anal. 1, 281–295 (2017)
Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic, Boston (2003)
Fan, K.: Some properties of convex sets related to fixed point theorems. Math. Ann. 266, 519–537 (1984)
Fang, Y.P., Huang, N.J.: Variational-like inequalities with generalized monotone mappings in Banach spaces. J. Optim. Theory Appl. 118, 327–338 (2003)
Fang, Y.P., Huang, N.J., Yao, J.C.: Well-posedness of mixed variational inequalities, inclusion problems and fixed-point problems. J. Glob. Optim. 41, 117–133 (2008)
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, 682–692 (2010)
Goeleven, D., Mentagui, D.: Well-posed hemivariational inequalities. Numer. Funct. Anal. Optim. 16, 909–921 (1995)
Guu, S.M., Li, J.: Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets. Nonlinear Anal. 71, 2847–2855 (2009)
Huang, X.X., Yang, X.Q.: Generalized Levitin–Polyak well-posedness in constrained optimization. SIAM J. Optim. 17, 243–258 (2006)
Khakrah, E., Razani, A., Mirzaei, R., Oveisiha, M.: Some metric characterizations of well-posedness for hemivariational-like inequalities. J. Nonlinear Funct. Anal. 2017, Article ID 44 (2017)
Kimura, K., Liou, Y.C., Wu, S.Y., Yao, J.C.: Well-posedness for parametric vector equilibrium problems with applications. J. Ind. Manag. Optim. 4, 313–327 (2008)
Kristály, A., Rădulescu, V., Varga, C.: Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems. Encylopedia of Mathematics, vol. 136. Cambridge University Press, Cambridge (2010)
Lemaire, B.: Well-posedness conditioning and regularization of minimization inclusion and fixed point problems. Pliska Stud. Math. Bulgar. 12, 71–84 (1998)
Li, X.B., Xia, F.Q.: Levitin–Polyak well-posedness of a generalized mixed variational inequality in Banach spaces. Nonlinear Anal. TMA 75, 2139–2153 (2012)
Lin, L.J., Chuang, C.S.: Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint. Nonlinear Anal. 70, 3609–3617 (2009)
Liu, Z.H., Migórski, S., Zeng, B.: Existence results and optimal control for a class of quasi mixed equilibrium problems involving the \((f,g,h)\)-quasimonotonicity. Appl. Math. Optim. (2017). https://doi.org/10.1007/s00245-017-9431-3
Liu, Z.H., Zeng, B.: Existence results for a class of hemivariational inequalities involving the stable \((g,f,\alpha)\)-quasimonotonicity. Topol. Methods Nonlinear Anal. 47(1), 195–217 (2016)
Liu, Z.H., Zeng, S.D., Zeng, B.: Well-posedness for mixed quasi-variational hemivariational inequalities. Topol. Methods Nonlinear Anal. 47(2), 561–578 (2016)
Long, X.J., Huang, N.J.: Metric characterizations of α-well-posedness for symimum problems with applications to variational inequalities. Numer. Funct. Anal. Optim. 3, 461–476 (1981)
Lucchetti, R., Patrone, F.: A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities. Numer. Funct. Anal. Optim. 3, 461–476 (1981)
Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)
Migórski, S., Ochal, A., Sofonea, M.: A class of variational-hemivariational inequalities in reflexive Banach spaces. J. Elast. 127, 151–178 (2017)
Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Dekker, New York (1995)
Panagiotopoulos, P.D.: Nonconvex energy functions, hemivariational inequalities and substationarity principles. Acta Mech. 48, 111–130 (1983)
Qin, X., Yao, J.C.: Projection splitting algorithms for nonself operators. J. Nonlinear Convex Anal. 18, 925–935 (2017)
Tang, G.J., Huang, N.J.: Existence theorems of the variational-hemivariational inequalities. J. Glob. Optim. 56, 605–622 (2013)
Tikhonov, A.N.: On the stability of the functional optimization problem. USSR Comput. Math. Math. Phys. 6, 28–33 (1966)
Wangkeeree, R., Preechasilp, P.: Existence theorems of the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in Banach spaces. J. Glob. Optim. 57, 1447–1464 (2013)
Xiao, Y.B., Huang, N.J.: Well-posedness for a class of variational-hemivariational inequalities with perturbations. J. Optim. Theory Appl. 151, 33–51 (2011)
Xiao, Y.B., Huang, N.J., Wong, M.M.: Well-posedness of hemivariational inequalities and inclusion problems. Taiwan. J. Math. 15, 1261–1276 (2011)
Zhang, Y.L., He, Y.R.: The hemivariational inequalities for an upper semicontinuous set-valued mapping. J. Optim. Theory Appl. 156, 716–725 (2013)
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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.
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Jiang, C. Well-posedness for a class of generalized variational-hemivariational inequalities involving set-valued operators. J Inequal Appl 2018, 187 (2018). https://doi.org/10.1186/s13660-018-1776-x
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DOI: https://doi.org/10.1186/s13660-018-1776-x