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Wellposedness for a class of generalized variationalhemivariational inequalities involving setvalued operators
Journal of Inequalities and Applications volume 2018, Article number: 187 (2018)
Abstract
The aim of present work is to study some kinds of wellposedness for a class of generalized variationalhemivariational inequality problems involving setvalued operators. Some systematic approaches are presented to establish some equivalence theorems between several classes of wellposedness for the inequality problems and some corresponding metric characterizations, which generalize many known results. Finally, the wellposedness for a class of generalized mixed equilibrium problems is also considered.
1 Introduction
Nowadays, wellposedness 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 wellposedness 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 wellposedness has been generalized to variational inequalities. The initial notion of wellposedness for variational inequality is due to Lucchetti and Patrone [28]. Fang [13, 14] generalized two kinds of wellposedness for a mixed variational inequality problem in a Banach space. For further results on the wellposedness 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 wellposedness to hemivariational inequalities. In 1995, Goeleven and Mentagui [15] generalized the concept of the wellposedness to a hemivariational inequality and presented some basic results concerning the wellposed hemivariational inequality. Recently, using the concept of approximating sequence, Xiao et al. [37, 38] introduced a concept of wellposedness for a hemivariational inequality and a variationalhemivariational inequality. Ceng, Lur, and Wen [3] considered an extension of wellposedness for a minimization problem to a class of generalized variationalhemivariational inequalities with perturbations in reflexive Banach spaces. For more recent works on the wellposedness for variationalhemivariational 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 setvalued 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 ϕquasimonotonicity 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 fquasimonotonicity. 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 wellposedness 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 setvalued 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 variationalhemivariational 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
$$\begin{aligned} \bigl\langle u^{*},vu\bigr\rangle +J^{\circ }(u;vu)\geq 0, \quad \forall v\in K. \end{aligned}$$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
$$\begin{aligned} \bigl\langle u^{*},vu\bigr\rangle +\phi (v)\phi (u)+J^{\circ }(u;vu) \geq 0, \quad \forall v\in K. \end{aligned}$$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
$$\begin{aligned} \bigl\langle u^{*},vu\bigr\rangle +g(u,v)+J^{\circ }(u;vu)\geq 0, \quad \forall v\in K. \end{aligned}$$
The existence of solutions to this inequality was studied by Wangkeeree and Preechasilp [36].
Inspired by previous works, we study the wellposedness 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 αwellposed GVHVI in the generalized sense. In particular, we present equivalence results on weak αwellposedness for GVHVI, which were considered by few authors.
This paper is organized as follows. In Sect. 2, we recall some basic preliminaries of singlevalued and setvalued mappings, metric concepts, Clarke’s generalized directional derivative, and some classes of wellposedness for GVHVI. In Sect. 3, we show some equivalence results for the wellposedness 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 wellposedness 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 setvalued 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 setvalued 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 αapproximating sequences and αwellposedness for GVHVI. Let \(\alpha:X\to R_{+}\) be a functional.
Definition 2.7
A sequence \(\{u_{n}\}\) in K is an αapproximating 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) αwellposed if it has a unique solution u and every αapproximating 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) wellposed.
Definition 2.9
GVHVI is said to be strongly (respectively, weakly) αwellposed in the generalized sense if the solution set Γ of GVHVI is nonempty and every αapproximating 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) wellposed in the generalized sense.
Remark 2.10
Strong αwellposedness (in the generalized sense) implies weak αwellposedness (in the generalized sense), but the converse is not true in general.
3 The characterizations of wellposedness for GVHVI
In this section, we establish metric characterizations and derive some conditions under which GVHVI is strongly (weakly) αwellposed.
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 αwellposed 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 αwellposed if and only if
Proof
Suppose that GVHVI is strongly αwellposed. 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 αapproximating sequences of GVHVI, and thus
which is a contradiction.
Conversely, assume that condition (3.2) holds. Let \(\{u_{n}\}\) in K be an αapproximating 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)
\(\limsup_{n\rightarrow \infty }J^{\circ }(u_{n};vu_{n})\le J^{ \circ }(u;vu)\) 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 αwellposed 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 αapproximating 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 αwellposed 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 αposedness in the generalized sense.
Theorem 3.8
Assume that all the assumptions of Theorem 3.5 are satisfied. Then GVHVI is strongly αwellposed 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}\ab \\).
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 αwellposed 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 αwellposed 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 Wellposedness 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 ηmonotonicity (see [7, 8]).
Definition 4.1
Let \(F:K\rightarrow P(X^{*})\) be a setvalued operator. F is said to be ηmonotone if there exists a function \(\eta:K\times K\rightarrow X\) such that, for all \(u,v\in K\),
Remark 4.2
If \(\eta (u,v)=vu\) 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 αwellposed 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 αwellposed 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 αwellposed 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 αwellposed 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 wellposedness 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 αwellposed GVHVI in the generalized sense. Our results generalize and improve many known results and can be applied to many other problems.
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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. Wellposedness for a class of generalized variationalhemivariational inequalities involving setvalued operators. J Inequal Appl 2018, 187 (2018). https://doi.org/10.1186/s136600181776x
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DOI: https://doi.org/10.1186/s136600181776x