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Well-posedness for a class of generalized variational-hemivariational inequalities involving set-valued operators

Journal of Inequalities and Applications20182018:187

https://doi.org/10.1186/s13660-018-1776-x

  • Received: 14 April 2018
  • Accepted: 13 July 2018
  • Published:

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, 1214, 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 \(u^{*}\in F(u)\),
$$\begin{aligned} \bigl\langle u^{*}+ Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \geq 0, \quad \forall v\in K. \end{aligned}$$
Now, let us consider some particular cases of GVHVI.
  1. (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].
     
  2. (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].
     
  3. (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
  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
    $$\begin{aligned} f\Biggl(\sum_{i=1}^{n}\lambda_{i}u_{i} \Biggr)\leq \sum_{i=1}^{n} \lambda_{i}f(u _{i}); \end{aligned}$$
     
  2. (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
    $$\begin{aligned} \limsup_{n\rightarrow \infty }f(u_{n})\leq f(u). \end{aligned}$$
     
  3. (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
$$\begin{aligned} \beta \Biggl(\sum_{i=1}^{n} \lambda_{i}v_{i}\Biggr)=\sum_{i=1}^{n} \lambda_{i}\beta (u_{i}). \end{aligned}$$

Definition 2.3

A set-valued operator \(F:K\rightarrow P(X^{*})\) is said to be
  1. (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 \(u_{0}^{*}\).

     
  2. (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 \(u,v\in K\),
$$\begin{aligned} \bigl\langle v^{*}-u^{*},v-u\bigr\rangle \geq 0, \quad \forall u^{*}\in F(u),\forall v^{*}\in F(v). \end{aligned}$$

Definition 2.5

Let S be a nonempty subset of X. The measure μ of noncompactness for the set S is defined by
$$\begin{aligned} \mu (S):=\inf \Biggl\{ \epsilon >0:S=\bigcup_{i=1} ^{n} S_{i}, \operatorname{diam} \vert S_{i} \vert < \epsilon,i=1,2,\ldots,n\Biggr\} , \end{aligned}$$
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
$$ \varphi^{0}(u;v):=\limsup_{\lambda \rightarrow 0^{+},\zeta \rightarrow u}\frac{\varphi (\zeta +\lambda v)-\varphi (\zeta)}{\lambda }. $$
The Clarke subdifferential or generalized gradient of φ at \(u\in X\), denoted by \(\partial \varphi (u)\), is the subset of \(X^{*}\) given by
$$ \partial \varphi (u):=\bigl\{ u^{*}\in X^{*}: \varphi^{0}(u;v)\geq \bigl\langle u^{*},v\bigr\rangle _{X}, \forall v\in X\bigr\} . $$

Lemma 2.6

([6], Proposition 2.1.1)

Let \(\varphi:X\rightarrow R\) be locally Lipschitz of rank \(L_{u}>0\) near u. Then
  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
    $$ \bigl\vert \varphi^{0}(u;v) \bigr\vert \leq L_{u} \Vert v \Vert _{X}; $$
     
  2. (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;

     
  3. (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 \(n\in N\),
$$\begin{aligned} \bigl\langle u_{n}^{*}+ Tu_{n}-f,v-u_{n} \bigr\rangle +g(u_{n},v)+J^{\circ }(u _{n};v-u_{n}) \geq -\epsilon_{n}\alpha (v-u_{n}),\quad \forall v\in K. \end{aligned}$$
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:
$$\begin{aligned} \Omega_{\alpha }(\epsilon) =&\bigl\{ u\in K: \exists u^{*}\in F(u)\mbox{ such that }\bigl\langle u^{*}+ Tu-f,v-u\bigr\rangle +g(u,v) \\ & {} +J^{\circ }(u;v-u)\geq -\epsilon \alpha (v-u), \forall v \in K \bigr\} \end{aligned}$$
and
$$\begin{aligned} \Phi_{\alpha }(\epsilon) =&\bigl\{ u\in K:\bigl\langle v^{*}+Tu-f,v-u \bigr\rangle +g(u,v)+J ^{\circ }(u;v-u) \\ & {}\geq -\epsilon \alpha (v-u), \forall v\in K,\forall v^{*} \in F(v) \bigr\} . \end{aligned}$$
Denote by Γ the set of solutions to GVHVI. It is clear that \(\Gamma =\Omega_{0}(\epsilon)\).

Lemma 3.1

Assume that:
  1. (i)

    K is a nonempty closed subset of a real Banach space X;

     
  2. (ii)

    \(T:K \rightarrow X^{*}_{w}\) is continuous;

     
  3. (iii)

    \(g:K\times K\rightarrow R\) is u.s.c. with respect to the first variable;

     
  4. (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 \(v^{*}\in F(v)\),
$$\begin{aligned} \bigl\langle v^{*}+ Tu_{n}-f,v-u_{n}\bigr\rangle +g(u_{n},v)+J^{\circ }(u_{n};v-u _{n})\geq - \epsilon \alpha (v-u_{n}). \end{aligned}$$
By the assumptions and the properties of \(J^{\circ }\) we have
$$\begin{aligned}& \bigl\langle v^{*}+ Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \\& \quad \geq \limsup_{n\rightarrow \infty }\bigl[\bigl\langle v^{*}+ Tu_{n}-f,v-u_{n} \bigr\rangle +g(u_{n},v)+J^{\circ }(u_{n};v-u_{n}) \bigr] \\& \quad \geq \limsup_{n\rightarrow \infty }-\epsilon \alpha (v-u_{n}) \\& \quad =-\epsilon \liminf_{n\rightarrow \infty }\alpha (v-u_{n}) \\& \quad \ge -\epsilon \alpha (v-u), \end{aligned}$$
and hence
$$\begin{aligned} \bigl\langle v^{*}+ Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u)- \epsilon \alpha (v-u), \quad \forall v\in K,\forall v^{*}\in F(v), \end{aligned}$$
which shows that \(u\in \Phi_{\alpha }(\epsilon)\). □

Lemma 3.2

Assume that:
  1. (i)

    K is a nonempty convex subset of a real Banach space X;

     
  2. (ii)

    \(F:K\rightarrow P(X^{*})\) is l.h.c. and monotone;

     
  3. (iii)

    \(g:K\times K\rightarrow R\) is convex with respect to the second variable;

     
  4. (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
$$\begin{aligned} \bigl\langle u^{*}+ Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \geq -\epsilon \alpha (v-u), \quad \forall v\in K. \end{aligned}$$
According to the monotonicity of F, we obtain
$$\begin{aligned} \bigl\langle v^{*}+ Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \geq -\epsilon \alpha (v-u), \quad \forall v\in K,\forall v^{*}\in F(v), \end{aligned}$$
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
$$\begin{aligned} \bigl\langle v^{*}+ Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \geq -\epsilon \alpha (v-u), \quad \forall v\in K,\forall v^{*}\in F(v). \end{aligned}$$
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
$$\begin{aligned} \bigl\langle v_{\lambda }^{*}+ Tu-f,v_{\lambda }-u\bigr\rangle +g(u,v_{\lambda })+J^{\circ }(u;v_{\lambda }-u) \geq -\epsilon \alpha (v_{\lambda }-u), \quad \forall v_{\lambda }^{*}\in F(v_{\lambda }). \end{aligned}$$
Then by (iii), (iv), and the properties of \(J^{\circ }\) we obtain
$$\begin{aligned} \bigl\langle v_{\lambda }^{*}+ Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \geq -\epsilon \alpha (v-u), \quad \forall v_{\lambda }^{*}\in F(v_{\lambda }). \end{aligned}$$
(3.1)
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
$$\begin{aligned}& \bigl\langle u^{*}+ Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \\& \quad =\lim_{\lambda \rightarrow 0}\bigl[\bigl\langle v_{\lambda }^{*}+ Tu-f,v-u \bigr\rangle +g(u,v)+J^{\circ }(u;v-u)\bigr] \\& \quad \geq -\epsilon \alpha (v-u), \end{aligned}$$
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:
  1. (i)

    K is a nonempty closed convex subset of a real Banach space X;

     
  2. (ii)

    \(F:K\rightarrow P(X^{*})\) is l.h.c. and monotone;

     
  3. (iii)

    \(T:K\rightarrow X^{*}_{w}\) is continuous;

     
  4. (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;

     
  5. (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
$$\begin{aligned} \bigl\langle v^{*}+ Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \geq 0, \quad \forall v\in K,v^{*}\in F(v). \end{aligned}$$

Theorem 3.4

GVHVI is strongly α-well-posed if and only if Γ is nonempty and
$$\begin{aligned} \lim_{\epsilon \rightarrow 0}\operatorname{diam}\bigl( \Omega_{\alpha }( \epsilon)\bigr)=0. \end{aligned}$$

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
$$\begin{aligned} \Omega_{\alpha }(\epsilon)\neq \emptyset \quad \forall \epsilon \geq 0 \quad \textit{and}\quad \lim_{\epsilon \rightarrow 0} \operatorname{diam}\bigl( \Omega_{\alpha }(\epsilon)\bigr)=0. \end{aligned}$$
(3.2)

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
$$\begin{aligned} \bigl\Vert u_{n}^{(1)}-u_{n}^{(2)} \bigr\Vert >\beta >0. \end{aligned}$$
(3.3)
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
$$\begin{aligned} \lim_{n\rightarrow }u_{n}^{(1)}=\lim _{n\rightarrow }u _{n}^{(2)}=u. \end{aligned}$$
(3.4)
From (3.3) and (3.4) we have
$$\begin{aligned} 0< \beta < \bigl\Vert u_{n}^{(1)}-u_{n}^{(2)} \bigr\Vert \leq \bigl\Vert u_{n}^{(1)}-u \bigr\Vert + \bigl\Vert u_{n}^{(2)}-u \bigr\Vert \rightarrow 0, \end{aligned}$$
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 \(n\in N\),
$$\begin{aligned} \bigl\langle u^{*}_{n}+ Tu_{n}-f,v-u_{n} \bigr\rangle +g(u_{n},v)+J^{\circ }(u _{n};v-u_{n}) \geq -\epsilon_{n}\alpha (v-u_{n}), \quad \forall v\in K, \end{aligned}$$
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 \(n\in N\),
$$\begin{aligned}& \bigl\langle v^{*}+ Tu_{n}-f,v-u_{n}\bigr\rangle +g(u_{n},v)+J^{\circ }(u _{n};v-u_{n}) \\& \quad \geq \bigl\langle u^{*}_{n}+Tu_{n}-f,v-u_{n} \bigr\rangle +g(u_{n},v)+J^{ \circ }(u_{n};v-u_{n}) \\& \quad \geq -\epsilon_{n}\alpha (v-u_{n}), \quad \forall v\in K,v^{*}\in F(v). \end{aligned}$$
By the assumptions we obtain that
$$\begin{aligned}& \bigl\langle v^{*}+Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \\& \quad \geq \limsup_{n\rightarrow \infty }\bigl[\bigl\langle v^{*}+ Tu_{n}-f,v-u _{n}\bigr\rangle +g(u_{n},v)+J^{\circ }(u_{n};v-u_{n}) \bigr] \\& \quad \geq \limsup_{n\rightarrow \infty }-\epsilon_{n}\alpha (v-u _{n}) \\& \quad =\limsup_{n\rightarrow \infty }\alpha \bigl(-\epsilon_{n}(v-u_{n}) \bigr) \\& \quad =0, \end{aligned}$$
which implies that
$$\begin{aligned} \bigl\langle v^{*}+Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \geq 0, \quad \forall v\in K,\forall v^{*}\in F(v). \end{aligned}$$
It follows from Theorem 3.3 that there exists \(u^{*}\in F(u)\) such that
$$\begin{aligned} \bigl\langle u^{*}+ Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \geq 0, \quad \forall v\in K. \end{aligned}$$
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
$$\begin{aligned} 0< \bigl\Vert u-u' \bigr\Vert \leq \operatorname{diam}\bigl( \Omega_{\alpha }(\epsilon)\bigr)\rightarrow 0\quad \mbox{as }\epsilon \rightarrow 0, \end{aligned}$$
which is a contradiction. This completes the proof. □

Theorem 3.6

Assume that:
  1. (i)

    K is a nonempty closed convex subset of a real reflexive Banach space X;

     
  2. (ii)

    \(F:K\rightarrow P(X^{*})\) is l.h.c. and monotone;

     
  3. (iii)

    \(T:K\rightarrow X^{*}\) is compact;

     
  4. (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;

     
  5. (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 \(n\rightarrow \infty \);

     
  6. (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 \(n\in N\),
$$\begin{aligned} \bigl\langle u^{*}_{n}+ Tu_{n}-f,v-u_{n} \bigr\rangle +g(u_{n},v)+J^{\circ }(u _{n};v-u_{n}) \geq -\epsilon_{n}\alpha (v-u_{n}) \end{aligned}$$
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
$$\begin{aligned}& \bigl\langle v^{*}+ Tu_{n}-f,v-u_{n}\bigr\rangle +g(u_{n},v)+J^{\circ }(u _{n};v-u_{n}) \\& \quad \geq \bigl\langle u^{*}_{n}+Tu_{n}-f,v-u_{n} \bigr\rangle +g(u_{n},v)+J^{ \circ }(u_{n};v-u_{n}) \\& \quad \geq -\epsilon_{n}\alpha (v-u_{n}),\quad \forall v\in K,v^{*}\in F(v), \forall n\in N. \end{aligned}$$
By the assumptions, we obtain that
$$\begin{aligned}& \bigl\langle v^{*}+T\overline{u}-f,v-\overline{u}\bigr\rangle +g( \overline{u},v)+J^{\circ }(\overline{u};v-\overline{u}) \\& \quad \geq \limsup_{n\rightarrow \infty }\bigl[\bigl\langle v^{*}+ Tu_{n}-f,v-u _{n}\bigr\rangle +g(u_{n},v)+J^{\circ }(u_{n};v-u_{n}) \bigr] \\& \quad \geq \limsup_{n\rightarrow \infty }-\epsilon_{n}\alpha (v-u _{n}) \\& \quad =\limsup_{n\rightarrow \infty }\alpha \bigl(-\epsilon_{n}(v-u_{n}) \bigr) \\& \quad =0, \end{aligned}$$
which implies that
$$\begin{aligned} \bigl\langle v^{*}+T\overline{u}-f,v-\overline{u}\bigr\rangle +g( \overline{u},v)+J ^{\circ }(\overline{u};v-\overline{u})\geq 0, \quad \forall v\in K,\forall v^{*}\in F(v). \end{aligned}$$
It follows from Theorem 3.3 that there exists \(\overline{u}^{*}\in F(\overline{u})\) such that
$$\begin{aligned} \bigl\langle u^{*}+ T\,\overline{u}-f,v-\overline{u}\bigr\rangle +g( \overline{u},v)+J ^{\circ }(\overline{u};v-\overline{u})\geq 0, \quad \forall v\in K, \end{aligned}$$
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
$$\begin{aligned} \sup_{u^{*}\in F(u)}\bigl\langle u^{*}+Tu-f,v-u\bigr\rangle +g(u,v)+J^{\circ }(u;v-u) \leq -\frac{1}{n_{0}}. \end{aligned}$$
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
$$\begin{aligned} \lim_{\epsilon \rightarrow 0}e\bigl(\Omega_{\alpha }( \epsilon),\Gamma\bigr)=0, \end{aligned}$$
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
$$\begin{aligned} \Omega_{\alpha }(\epsilon)\neq \emptyset, \quad \forall \epsilon >0,\quad \textit{and}\quad \lim_{\epsilon \rightarrow 0}\mu \bigl(\Omega_{\alpha }( \epsilon)\bigr)=0. \end{aligned}$$

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 \(u^{*}\in F(u)\),
$$\begin{aligned} \bigl\langle u^{*},\eta (u,v)\bigr\rangle +\langle Tu-f,v-u\rangle +g(u,v)+h(u,v) \geq 0, \quad \forall v\in K, \end{aligned}$$
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 \(u,v\in K\),
$$\begin{aligned} \bigl\langle v^{*}-u^{*},\eta (u,v)\bigr\rangle \geq 0, \quad \forall u^{*}\in F(u),\forall v^{*}\in F(v). \end{aligned}$$
(4.1)

Remark 4.2

If \(\eta (u,v)=v-u\) for all \(u,v\in X\), then (4.1) becomes
$$\begin{aligned} \bigl\langle v^{*}-u^{*},v-u\bigr\rangle \geq 0, \quad \forall u^{*}\in F(u),\forall v^{*}\in F(v), \end{aligned}$$
that is, F is monotone.
For any \(\epsilon >0\), we define the following two sets:
$$\begin{aligned} \Omega_{\eta,\alpha }(\epsilon) =&\bigl\{ u\in K: \exists u^{*}\in F(u)\mbox{ such that }\bigl\langle u^{*},\eta (u,v)\bigr\rangle + \langle Tu-f,v-u\rangle +g(u,v) \\ & {}+h(u,v))\geq -\epsilon \alpha (v-u),\forall v\in K \bigr\} \end{aligned}$$
and
$$\begin{aligned} \Phi_{\eta,\alpha }(\epsilon) =&\bigl\{ u\in K:\bigl\langle v^{*},\eta (u,v) \bigr\rangle +\langle Tu-f,v-u\rangle +g(u,v) \\ & {}+h(u,v)\geq -\epsilon \alpha (v-u), \forall v\in K,\forall v ^{*} \in F(v) \bigr\} . \end{aligned}$$
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:
  1. (i)

    \(h(u,u)=0\) for all \(u\in X\),

     
  2. (ii)

    for all \(v\in K\), \(h(\cdot,v)\) is u.s.c.,

     
  3. (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
$$\begin{aligned} \bigl\langle v^{*}+ Tu-f,\eta (u,v)\bigr\rangle +g(u,v)+h(u,v)\geq 0, \quad \forall v\in K,v^{*}\in F(v). \end{aligned}$$

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:
  1. (i)

    \(h(u,u)=0\) for all \(u\in X\),

     
  2. (ii)

    for all \(v\in K\), \(h(\cdot,v)\) is u.s.c.,

     
  3. (iii)

    for all \(u\in K\), \(h(u,\cdot)\) is convex.

     
Then GMVHVI is strongly α-well-posed if and only if
$$\begin{aligned} \Omega_{\eta,\alpha }(\epsilon)\neq \emptyset,\quad \forall \epsilon \geq 0, \quad \textit{and}\quad \lim_{\epsilon \rightarrow 0} \operatorname{diam}\bigl(\Omega_{\eta,\alpha }( \epsilon)\bigr)=0. \end{aligned}$$

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:
  1. (i)

    \(h(u,u)=0\) for all \(u\in X\),

     
  2. (ii)

    for all \(v\in K\), \(h(\cdot,v)\) is weakly u.s.c.,

     
  3. (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:
  1. (i)

    \(h(u,u)=0\) for all \(u\in X\),

     
  2. (ii)

    for all \(v\in K\), \(h(\cdot,v)\) is u.s.c.,

     
  3. (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
$$\begin{aligned} \Omega_{\eta,\alpha }(\epsilon)\neq \emptyset, \quad \forall \epsilon >0,\quad \textit{and}\quad \lim_{\epsilon \rightarrow 0}\mu \bigl(\Omega_{\eta,\alpha }( \epsilon)\bigr)=0. \end{aligned}$$

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:
  1. (i)

    \(h(u,u)=0\) for all \(u\in X\),

     
  2. (ii)

    for all \(v\in K\), \(h(\cdot,v)\) is weakly u.s.c.,

     
  3. (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.

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

(1)
College of Sciences, Guangxi University for Nationalities, Nanning, P.R. China

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