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

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.


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 , α)-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 → P(X * ) be a set-valued operator, where P(X * ) is the set of all nonempty subsets of X * . Let T : K → X * be a perturbation, and let f ∈ X * be a given element. Let g : K × K → R := R ∪ {±∞} be a function such that D(g) = {u ∈ K : g(u, v) = -∞, ∀v ∈ K} = ∅. Let J : X → R be a locally Lipschitz function, and let J • (u, v) denote the generalized directional derivative in the sense of Clarke of a locally Lipschitz functional J : X → R at u in the direction v. In this paper, we discuss the following generalized variationalhemivariational inequality (GVHVI): Find u ∈ K such that, for some u * ∈ F(u), Now, let us consider some particular cases of GVHVI.
(a) If T ≡ 0, f ≡ 0, and g ≡ 0, then GVHVI is reduced to the following form: Find u ∈ K and u * ∈ F(u) such that The existence of solutions to this inequality was recently studied by Zhang and He [39]. (b) If T ≡ 0 and f ≡ 0, and g(u, v) = φ(v)φ(u) for all u, v ∈ K , then GVHVI is reduced to the following form: Find u ∈ K and u * ∈ F(u) such that The existence of solutions to this inequality was studied by Tang and Huang [34].
(c) If T ≡ 0 and f ≡ 0, then GVHVI is reduced to the following form: Find u ∈ K and u * ∈ F(u) such that 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-wellposedness 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 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.

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 · X . Denote by X * its dual space and by ·, · X the duality pairing between X * and X. Let X w be the Banach space X with weak topology.
(ii) (weakly) upper semicontinuous (u.s.c. for short) at u if for any sequence (iii) (weakly) lower semicontinuous (l.s.c. for short) at u, if for any sequence 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 ∈ K .

Definition 2.2 ([20]
) Let K be a nonempty subset of X. An operator β : K → X is said to be affine if for any u i ∈ K (i = 1, 2, . . . , n) and lower semicontinuous with respect to the weak topology in X * .
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 ϕ : X → R (see [6,10]). The Clarke generalized directional derivative ϕ 0 (u; v) of ϕ at the point u ∈ X in the direction v ∈ X is defined as The Clarke subdifferential or generalized gradient of ϕ at u ∈ X, denoted by ∂ϕ(u), is the subset of X * given by 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 (ii) the gradient ∂ϕ(u) is a nonempty, convex, and weakly * compact subset of X * bounded by a Lipschitz constant L u near x; (iii) for every v ∈ X, we have We end this section with the notions of several classes of α-approximating sequences and α-well-posedness for GVHVI. Let α : X → R + be a functional.
and a nonnegative sequence { n } with n → 0 as n → ∞ such that, for every n ∈ N , In particular, if α(·) = · 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 α(·) = · 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 of GVHVI is nonempty and every α-approximating sequence {u n } has a subsequence that strongly (respectively, weakly) converges to some point of . In particular, if α(·) = · 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-wellposedness (in the generalized sense), but the converse is not true in general.

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 > 0, we define the following two sets: Denote by the set of solutions to GVHVI. It is clear that = 0 ( ).

Lemma 3.1 Assume that:
(i) K is a nonempty closed subset of a real Banach space X; (ii) T : K → X * w is continuous; Then, for every > 0, the set α ( ) is closed in X.
Proof Let {u n } ⊂ α ( ) be s sequence such that u n → u in X. Then u ∈ K , and, for all v ∈ K and v * ∈ F(v), By the assumptions and the properties of which shows that u ∈ α ( ).

Lemma 3.2 Assume that:
(i) K is a nonempty convex subset of a real Banach space X; (ii) F : K → P(X * ) is l.h.c. and monotone; (iii) g : K × K → R is convex with respect to the second variable; (iv) α : X → R + is convex with α(tv) = tα(v) for all t ≥ 0 and v ∈ X. Then α ( ) = α ( ) for all > 0.
Proof We first show that α ( ) ⊂ α ( ). Indeed, take arbitrary u ∈ α ( ). Then there exists u * ∈ F(u) such that According to the monotonicity of F, we obtain which means that u ∈ α ( ). Therefore α ( ) ⊂ α ( ). Now we show that α ( ) ⊂ α ( ). Indeed, take arbitrary u ∈ α ( ). Then Since the set K is convex, for any v ∈ K and λ ∈ [0, 1], taking v λ := λv + (1λ)u ∈ K in this inequality, we have Then by (iii), (iv), and the properties of Let u * ∈ F(u) be fixed, and let v * λ ∈ F(v λ ) be such that v * λ u * in X * (the existence of such a sequence is ensured by the fact that F is l.h.c.). Taking the limit as λ → 0 in (3.1), we obtain which implies that u ∈ α ( ). 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 → P(X * ) is l.h.c. and monotone; (iii) T : K → X * w is continuous; (iv) g : K × K → R is u.s.c. with respect to the first variable and convex with respect to the second variable; (v) α : X → R + is continuous and convex with α(tv) = tα(v) for all t ≥ 0 and v ∈ X. Then α ( ) = α ( ) is closed in X for all > 0. Moreover, = 0 ( ) = 0 ( ), that is, GVHVI is equivalent to the following problem:

Theorem 3.4 GVHVI is strongly α-well-posed if and only if is nonempty and
Proof The proof is similar to that of Theorem 4.3 in [26] by the assumptions of g. Proof Suppose that GVHVI is strongly α-well-posed. Then GVHVI has a unique solution u ∈ K , and thus = ∅. Now, we prove that (3.2) holds. Clearly, α ( ) ⊃ = ∅. For the second part of (3.2), arguing by contradiction, let us assume that diam( α ( )) does not tend to 0 as → 0. Thus for any nonnegative sequence { n } with n → 0 as n → ∞, there exists a constant β > 0 such that, for each n ∈ N , there exist u (1) n , u (2) n ∈ α ( n ) satisfying 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 ∈ F(u n ) and a nonnegative sequence { n } with n → 0 as n → ∞ such that, for every n ∈ N , that is, u n ∈ α ( n ) for all n ∈ 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 ∈ K . Let us show that u ∈ K is a solution for GVHVI. By the monotonicity of F we obtain that, for every n ∈ N ,

By the assumptions we obtain that
It follows from Theorem 3.3 that there exists u * ∈ F(u) such that Then u ∈ K is a solution of GVHVI.
Finally, we prove that the solution u is unique. If there exists another solution u ∈ K , then u, u 1 ∈ α ( ) for all > 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 → P(X * ) is l.h.c. and monotone; (iii) T : K → X * is compact; (iv) g : K × K → R is weakly u.s.c. with respect to the first variable and convex with respect to the second variable; and v ∈ X. Then GVHVI is weakly α-well-posed if and only if GVHVI has a unique solution and there exists 0 > 0 such that α ( 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 ∈ F(u n ) and a nonnegative sequence { n } with n → 0 as n → ∞ such that, for every n ∈ N , for all v ∈ K . We claim that the sequence {u n } is bounded in X. Indeed, since α ( 0 ) is bounded and α ( ) ⊂ α ( 0 ) for all ∈ (0, ε 0 ), there exists n 0 ∈ N such that n 0 ∈ (0, ε 0 ) and u n ∈ α ( 0 ) for all n ≥ 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 u as n → ∞ for some u ∈ X. Let us show that u ∈ K is a solution for GVHVI. Obviously, u ∈ K . By the monotonicity of F we obtain that By the assumptions, we obtain that It follows from Theorem 3.3 that there exists u * ∈ F(u) such that Therefore u ∈ 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 0 > 0 such that α ( 0 ) is nonempty and bounded can be replaced by the conditions that K is bounded or that there exists n 0 ∈ N such that, for every u ∈ K \ B n 0 , there exists v ∈ 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. Proof The proof is similar to that of Theorem 5.1 in [26] by the assumptions of g.  Proof The proof is similar to that of Theorem 3.6 by the assumptions of g.

Well-posedness for GMEP
In this section, we consider the following generalized mixed equilibrium problem (GMEP): Find u ∈ K such that, for some u * ∈ F(u), where η : K × K → X is an operator. The existence of solutions to this problem when T ≡ 0 and f ≡ 0 can be found in [25].

Definition 4.1
Let F : K → P(X * ) be a set-valued operator. F is said to be η-monotone if there exists a function η : K × K → X such that, for all u, v ∈ K , v *u * , η(u, v) ≥ 0, ∀u * ∈ F(u), ∀v * ∈ F(v). that is, F is monotone.

Theorem 4.3
Assume that all the assumptions of Theorem 3.3 are satisfied and, in addition, η : K × K → X is continuous on K × K with η(u, u) = 0 for any u ∈ K and affine with respect to the first variable. Let h : K × K → R be such that: (i) h(u, u) = 0 for all u ∈ X, (ii) for all v ∈ K , h(·, v) is u.s.c., (iii) for all u ∈ K , h(u, ·) is convex.
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.

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.