Metric characterizations for well-posedness of split hemivariational inequalities

In this paper, we generalize the concept of well-posedness to a class of split hemivariational inequalities. By imposing very mild assumptions on involved operators, we establish some metric characterizations of the well-posedness for the split hemivariational inequality. The obtained results generalize some related theorems on well-posedness for hemivariational inequalities and variational inequalities in the literature.

Due to the close relationship between optimization problems and variational inequality problems, the concept of well-posedness for optimization problems is generalized to variational inequalities and their related problems. The earliest research work of wellposedness for variational inequalities should at least date back to 1980s when Lucchetti and Patrone [22,23] firstly introduced the concept of well-posedness for a variational inequality and proved some important results. After that, Lignola and Morgan [20], Fang and Hu [24], Huang and Yao [25] have made significant contributions to the study of well-posedness for variational inequalities. As an important generalization of variation inequality, hemivariational inequality has drawn much attention of mathematical re-searchers due to its abundant applications in mechanics and engineering. With the tools of nonsmooth analysis and nonlinear analysis, many kinds of hemivariational inequalities have been studied since 1980s [7,[26][27][28][29][30]. Also, many kinds of concepts of well-posedness hemivariational inequalities have been studied since Goeleven and Mentagui [31] firstly introduced the concept of well-posedness to a hemivariational inequality in 1995. For more research work on the well-posedness for variational inequalities and hemivariational inequalities, we refer the readers to [14,20,[32][33][34][35].
Split variational inequality, which was introduced by Censor et al. in [36], can be regarded as a generalization of variational inequality and includes as a special case, the split feasibility problem, which is an important model for a wide range of practical problems arising from signal recovery, image processing, and tensity-modulated radiation therapy treatment planning (see, for example, [37][38][39][40][41]). Thus, the concepts of well-posedness and Levitin-Polyak well-posedness for various split variational inequalities were studied by Hu and Fang recently [42]. Obviously, split hemivariational inequality could be regarded as a generalization of split variational inequality. It could arise in a system of hemivariational inequalities for modeling some frictional contact problems in mechanics, where two hemivariational inequalities are linked by a linear constraint. Also, when nonconvex and nonsmooth functionals are involved, the model for the above mentioned practical problems, such as signal recovery and image processing, turns to split hemivariational inequality rather than split variational inequality. However, as far as we know, there are few research works studying well-posedness for split hemivariational inequalities.
Inspired by recent research works on the well-posedness for split variational inequalities and hemivariational inequalities, in this paper, we focus on studying metric characterization of well-posedness for a class of split hemivariational inequalities specified as follows: where, for i = 1, 2, ·, · V i * ×V i denotes the duality paring between Banach space V i and its dual space which will be defined in the next section, and T : V 1 → V 2 is a continuous mapping from V 1 to V 2 . After defining the concept of well-posedness for the split hemivariational inequality (SHI), we present some metric characterizations for its well-posedness under very mild assumptions.
The remainder of the paper is organized as follows. In Sect. 2, we recall some crucial definitions and results. Under very mild assumptions on involved operators, Sect. 3 presents several results on the metric characterizations of well-posedness for the split hemivariational inequality (SHI). At last, some concluding remarks are provided in Sect. 4.

Preliminaries
In this section, we recall some useful definitions and key results which will be used to establish the metric characterizations of the split hemivariational inequality (SHI)in the next section and can be found in [7,29,[43][44][45].
Let V 1 , V 2 be two Banach spaces, then the product space V of V 1 and V 2 , i.e., V = V 1 ×V 2 , is also a Banach space with the norm · V 1 ×V 2 specified as follows: The dual paring between the product space V and its dual space V * is which is denoted by u n → u as n → ∞; which is denoted by u n u as n → ∞; which is denoted by u * n w * − → u * as n → ∞.

Definition 2.2
Let V be a Banach space and V * be its dual space. A single-valued operator (2) strictly monotone if (3) relaxed monotone if there exists a constant c > 0 such that (4) strongly monotone if there exists a constant c > 0 such that

Definition 2.3
Let V be a Banach space and V * be its dual space. An operator T from V to V * is said to be (1) continuous if, for any sequence {u n } ⊂ V converging to u ∈ V , Tu n → Tu in V * ; (2) demicontinuous if, for any sequence {u n } ⊂ V converging to u ∈ V , Tu n Tu in V * ; (3) hemicontinuous if, for any u, v, w ∈ V , the function t → T(u + tv), w V * ×V is continuous on [0, 1]; (4) weakly * continuous (or continuous with respect to weak * topology for V * ) if, for Remark 2.1 In [7,44], demicontinuity of an operator T from V to V * is defined by its continuity from V to its dual space V * endowed with weak * topology, which is called here weak * continuity. In this paper, we define the demicontinuity of an operator T from V to V * by its continuity from V to its dual space V * endowed with weak topology, which is commonly used in most literature works.

Proposition 2.1
Let V be a Banach space with V * being its dual space and T : V → V * be an operator. If T is continuous, then it is weakly * continuous, which, in turn, implies that it is hemicontinuous. Moreover, if T is a monotone operator, then the notions of weak * continuity and hemicontinuity coincide [7,44].

Proposition 2.2 Let V be a Banach space with V * being its dual space, and T
Then the following statement holds:

Definition 2.4
Let V be a Banach space and J : V → R be a functional on V . J is said to be Lipschitz continuous on V if there exists a constant L > 0 such that Definition 2.5 Let V be a Banach space and J : V → R be a functional on V . J is said to be locally Lipschitz continuous on V if, for all u ∈ V , there exist a neighborhood N (u) and a constant L u > 0 such that

Definition 2.6
Let V be a Banach space and the generalized directional derivative (in the sense of Clarke) of the locally Lipschitz function J : λ .

Definition 2.7
Let V be a Banach space and J : V → R be a locally Lipschitz function. Then the generalized gradient in the sense of Clarke of J at v ∈ V , denoted by ∂J(u), is the subset of its dual space V * defined by

Definition 2.8 Let
A be a nonempty subset of Banach space V . The measure of noncompactness μ of the set A is defined by where diam denotes the diameter of the subset A i .

Definition 2.9
Let A, B be two nonempty subsets of Banach space V . The Hausdorff metric H (·, ·) between A and B is defined by

Well-posedness and metric characterizations
In this section, we aim to extend the well-posedness to the split hemivariational inequality (SHI). We first give the definition of well-posedness for the split hemivariational inequality (SHI), and then we prove its metric characterizations for the well-posedness by using two useful sets defined.
Definition 3.2 The split hemivariational inequality (SHI) is said to be strongly (resp., weakly) well-posed if it has a unique solution and every approximating sequence for the split hemivariational inequality (SHI) converges strongly (resp., weakly) to the unique solution.

Definition 3.3
The split hemivariational inequality (SHI) is said to be well-posed in generalized sense (or generalized well-posed) if its solution set is nonempty and, for every approximating sequence, there always exists a subsequence converging to some point of its solution set.
In order to establish the metric characterizations for well-posedness of the split hemivariational inequality (SHI), we first define two sets on V 1 × V 2 as follows: for ε > 0, With the definition of two sets ( ) and ( ), we can get the following properties. Proof First, we prove ( ) ⊂ ( ) for any > 0. In fact, let u = (u 1 , u 2 ) ∈ ( ). By the monotonicity of the operators A 1 and A 2 , it is easy to show that, for any v 1 ∈ V 1 and v 2 ∈ V 2 , which imply that This together with the fact that u 2 -Tu 1 V 2 ≤ due to u = (u 1 , u 2 ) ∈ ( ) indicates that u ∈ ( ), and thus ( ) ⊂ ( ). Now, we turn to prove ( ) ⊂ ( ) for any > 0. Let u = (u 1 , u 2 ) ∈ ( ), and then (3.1) From Proposition 2.3, the function J • i (u i , ·), i = 1, 2, is positively homogeneous. Letting t → 0 + in the last two inequalities of (3.2), it follows from the hemicontinuity of the operators A 1 and A 2 that By the arbitrariness of w = (w 1 , w 2 ) ∈ V 1 × V 2 , we conclude that u ∈ (ε), and thus ( ) ⊂ ( ). This completes the proof of Lemma 3.1.

Lemma 3.2
Let V 1 , V 2 be two reflective Banach spaces with V * 1 , V * 2 being their dual spaces, respectively, and J i : V i → R, i = 1, 2, be a locally Lipschitz functional. Suppose that T : Proof Assume that {u n = (u n 1 , u n 2 )} ⊂ ( ) and u n → u = (u 1 , (3.4) By Proposition 2.3, J • i (· ; ·), i = 1, 2, is upper continuous on V i × V i . By taking lim sup with n → +∞ on both sides of the last two inequalities of (3.4), it follows from the fact u n i → u i , i = 1, 2, that and To complete the proof, we only need to prove u 2 -Tu 1 V 2 ≤ . Since, for any n ∈ N, u n = (u n 1 , u n 2 ) ∈ ( ), it follows that u n 2 -Tu n 1 V 2 ≤ , which together with the continuity of the functional · V 2 : V 2 → R and the operator T implies that Thus u = (u 1 , u 2 ) ∈ ( ), which implies that ( ) is closed on V 1 × V 2 . This completes the proof of Lemma 3.2.
With Lemmas 3.1 and 3.2, it is easy to get the following corollary on the closedness of ( ) for any > 0, which is crucial to the metric characterizations for well-posedness of the split hemivariational inequality (SHI). Corollary 3.1 Let V 1 , V 2 be two Banach spaces with V * 1 , V * 2 being their dual spaces, respectively. Suppose that, for i = 1, 2, A i : V i → V * i is monotone and hemicontinuous on V i , J i : V i → R is a locally Lipschitz functional, and T : V 1 → V 2 is a continuous operator from V 1 to V 2 . Then ( ) is closed for any > 0.
Remark 3.1 Similar to the idea in many research works on well-posedness for variational inequalities and hemivariational inequalities [17,25,46,47], the set ( ) is defined to prove the closedness of ( ) under the condition that, for i = 1, 2, A i is monotone and hemicontinuous on V i . Actually, without defining the set ( ), we could prove directly the property of closedness of ( ).

Lemma 3.3
Let V 1 , V 2 be two Banach spaces with V * 1 , V * 2 being their dual spaces, respectively, and J i : V i → R be a locally Lipschitz functional for i = 1, 2. Suppose that T : V 1 → V 2 is a continuous operator from V 1 to V 2 and for i = 1, 2, A i : V i → V * i is monotone and hemicontinuous. Then ( ) is closed for any > 0.
Proof Let {u n = (u n 1 , u n 2 )} ⊂ ( ) be a sequence converging to u = (u 1 , u 2 ) in V 1 × V 2 , which implies that Taking lim sup with n → +∞ on both sides of the last two inequalities of (3.5), it follows from (3.6) and (3.7) that and Moreover, by similar arguments as in Lemma 3.2, it is easy to show that This together with (3.8) and (3.9) indicates that u = (u 1 , u 2 ) ∈ ( ). Thus ( ) is closed on This completes the proof of Lemma 3.3. Now, with properties of the set ( ) given above, we are in a position to prove metric characterizations for the split hemivariational inequality (SHI)by using similar methods for studying well-posedness of variational inequalities and hemivariational inequalities in research works [17,25,46,47]. Theorem 3.1 Let V 1 , V 2 be two Banach spaces and V * 1 , V * 2 be their dual spaces, respectively. Suppose that, for i = 1, 2,

is an operator on V i and J i : V i → R is a locally Lipschitz functional. Then the split hemivariational inequality (SHI) is strongly well-posed if and only if its solution set S is nonempty and diam
Proof "Necessity": First of all, it is obvious that the solution set of the split hemivariational inequality (SHI) S = φ since it has a unique solution due to its strong well-posedness. Assume that diam ( ) 0 as → 0, then there exist δ > 0, k → 0 + , u k = (u k 1 , u k 2 ) ∈ ( k ), and p k = (p k 1 , p k 2 ) ∈ ( k ) such that Clearly, both {(u k 1 , u k 2 )} and {(p k 1 , p k 2 )} are approximating sequences for the split hemivariational inequality (SHI) by the fact that (u k 1 , u k 2 ) ∈ ( k ) and (p k 1 , p k 2 ) ∈ ( k ). It follows from the well-posedness of (SHI) that both {(u k 1 , u k 2 )} and {(p k 1 , p k 2 )} converge to the unique solution of (SHI), which is a contradiction to (3.11). Thus, diam (ε) → 0 as ε → 0.
"Sufficiency": Suppose that the solution set S of the split hemivariational inequality (SHI) is nonempty and diam ( ) → 0 as → 0. For any approximating sequence which indicates that (u n 1 , u n 2 ) ∈ (ε n ) with ε n → 0. Now, we claim that the solution set S of the split hemivariational inequality (SHI)is a singleton, i.e., S = {u * = (u * 1 , u * 2 )} and u n → u * as n → ∞, which indicate that the split hemivariational inequality (SHI)is strongly well-posed. For the purpose of getting contradiction, we suppose that there exists another solution u = (u 1 , u 2 ) = u * to the split hemivariational inequality (SHI). It is clear that u , u * ∈ ( ) for any > 0 and which is a contradiction. Thus, u * is the unique solution to the split hemivariational inequality (SHI). Moreover, since u n , u * ∈ ( n ) for any n ∈ N, it follows that which implies that u n → u * as n → ∞. This completes the proof of Theorem 3.1.
Theorem 3.2 Let V 1 , V 2 be two Banach spaces with V * 1 , V * 2 being their dual spaces, respectively, and T : V 1 → V 2 be a continuous operator from V 1 to V 2 . Suppose that, for i = 1, 2, Proof It is sufficient to prove the sufficiency of Theorem 3.2 since it is easy to get its necessity by Theorem 3.1 due to the fact that S ⊂ ( ) for any > 0. First, with condition (3.13), it is easy to show that the split hemivariational inequality (SHI) possesses a unique solution by similar arguments as in the proof of Theorem 3.1. Then, we suppose that {u n = (u n 1 , u n 2 )} ⊂ V 1 × V 2 is an approximating sequence for the split hemivariational inequality, which indicates that there exists 0 < n → 0 such that (3.12) holds and thus u n ∈ ( n ). It follows from the condition diam ( ) → 0as → 0 that {u n } is a Cauchy sequence. As a consequence, there exists u = (u 1 , u 2 ) such that u n → u. Now, we show that u = (u 1 , u 2 ) is the unique solution of the split hemivariational inequality (SHI)to get its strong well-posedness. By taking limit on both sides of the first inequality in (3.12), it is easy to get from the continuity of the operation T that (3.14) Since, for i = 1, 2, the operator A i : V i → V * i is monotone and the Clarke generalized directional derivative J • i (· ; ·) is upper semicontinuous by Proposition 2.3, taking lim sup on both sides of the last two inequalities in (3.12) yields that and By similar arguments for the proof of ( ) ⊂ ( ) for any > 0 in Lemma 3.1, it can be proved by the hemicontinuity of operators A 1 , A 2 , (3.15), and (3.16) that which together with (3.14) imply that u = (u 1 , u 2 ) is the unique solution of the split hemivariational inequality (SHI). This completes the proof of Theorem 3.2.
The following is a concrete example to illustrate the metric characterization of wellposedness for a hemivariational inequality.
Example 3.1 Let V 1 = V 2 = R and f 1 = 2, f 2 = 1. For any u 1 , u 2 ∈ R, A 1 : R → R such that A 1 (u 1 ) = 2u 1 , A 2 : R → R such that A 2 (u 2 ) = u 2 , T : R → R such that T(u 1 ) = u 2 1 , and J 1 , J 2 : R → R are defined by It is obvious that J 1 and J 2 are locally Lipschitz and nonconvex functions on R. Thus, the split hemivariational inequality we consider is as follows: (3.17) By some simple calculations, one can easily obtain that the Clarke subgradients for the functions J 1 and J 2 are On the one hand, with some further deductions, it is not difficult to check that the split hemivariational inequality (3.17) has a unique solution u * = (u * 1 , u * 2 ) = (1, 1). Moreover, for any approximating sequence {u n = (u n 1 , u n 2 )} of the split hemivariational inequality (3.17), it satisfies where 0 < n → 0 when n → ∞. By taking limit of n → ∞ on both sides of the inequalities in (3.18), it is easy to obtain that the approximating sequence {u n } converges strongly to the unique solution u * of the split hemivariational inequality (3.17), which indicates that the split hemivariational inequality (3.17) is well-posed.
From Fig. 1, the graph of ( ), it is easy to obtain that Obviously, for any > 0, ( ) for the split hemivariational inequality (3.17) is nonempty and diam ( ) → 0 when → 0. Proof First, suppose that the split hemivariational inequality (SHI) is generalized wellposed. This implies, by the definition of generalized well-posedness for (SHI) and the definition of ( ), that φ = S ⊂ ( ) for all > 0. We claim that the solution set S of (SHI) is compact. In fact, let {u n = (u n 1 , u n 2 )} be a sequence in S, which indicates that {u n } is an approximating sequence for the split hemivariational inequality. By the generalized wellposedness of (SHI), there exists a subsequence of {u n } converging to some element of S, which implies that S is compact. Now, we prove H ( ( ), S) → 0 as ε → 0. If not, there exist τ > 0, ε n > 0 with ε n → 0, and u n = (u n 1 , u n 2 ) ∈ (ε n ) such that u n S + B(0, τ ), ∀n ∈ N. (3.19) By the fact that u n ∈ (ε n ) for n ∈ N, {u n } is an approximating sequence for the split hemivariational inequality (SHI), which implies by the generalized well-posedness of (SHI) that there exists a subsequence of {u n } converging to some element of S, a contradiction to (3.19). Therefore, H ( ( ), S) → 0 as ε → 0. Conversely, we prove the sufficiency. Assume that S is nonempty compact and H ( ( ), S) → 0 as ε → 0. For any approximating sequence {u n = (u n 1 , u n 2 )} for the split hemivariational inequality (SHI), there exists 0 < n → 0 such that {u n } ∈ (ε n ). By virtue of S ⊂ ( n ) for any n ∈ N, it is obvious that Since S is compact, it follows that there exists a sequence {w n = (w n 2 , w n 1 )} ⊂ S such that u n -w n V 1 ×V 2 = d u n , S → 0.
Again by the compactness of the solution set S and {w n } ⊂ S, there exists a sequence {w n k } converging to some point w ∈ S. Thus u n k -w V 1 ×V 2 ≤ u n k -w n k V 1 ×V 2 + w n k -w V 1 ×V 2 → 0, as k → ∞, which implies that the split hemivariational inequality (SHI)is generalized well-posed since the solution set S for the split hemivariational inequality (SHI)is nonempty. This completes the proof of Theorem 3.3.

Theorem 3.4
Let V 1 , V 2 be two Banach spaces with V * 1 , V * 2 being their dual spaces, respectively, and T : V 1 → V 2 be a continuous operator from V 1 to V 2 . Suppose that, for i = 1, 2, Sufficiency: Conversely, assume that condition (3.20) holds. Note that S = >0 ( ) due to the closedness of ( ) for any > 0 by Corollary 3.1. Since μ( ( )) → 0 as → 0, it follows from the theorem on p. 412 of [45] that S is nonempty compact and e ( ), S = H ( ), S → 0, as → 0, which implies by Theorem 3.3 that the split hemivariational inequality (SHI) is generalized well-posed. This completes the proof of Theorem 3.4.