On the well-posedness of differential quasi-variational-hemivariational inequalities

Abstract The goal of this paper is to discuss the well-posedness and the generalized well-posedness of a new kind of differential quasi-variational-hemivariational inequality (DQHVI) in Hilbert spaces. Employing these concepts, we explore the essential relation between metric characterizations and the well-posedness of DQHVI. Moreover, the compactness of the set of solutions for DQHVI is delivered, when problem DQHVI is well-posed in the generalized sense.


Introduction
The differential variational inequalities (DVIs) have been introduced and systematically studied by Pang-Stewart [1] on a finite-dimensional space. As the powerful mathematical tools, recently, DVIs have been applied to the study of various problems involving both dynamics and constraints in the form of inequalities, which arise in many applied problems in our real life, for instance, mechanical impact problems, electrical circuits with ideal diodes, the Coulomb friction problems for contacting bodies, economical dynamics, dynamic traffic networks, and so on. Based on this motivation, Chen and Wang [2] in 2014 used the idea of DVIs to investigate a dynamic Nash equilibrium problem of multiple players with shared constraints and dynamic decision processes; Wang et al. [3] proved an existence theorem for Carathéodory weak solutions of a differential quasi-variational inequality in finite dimensional Euclidean spaces and established a convergence result on the Euler time-dependent procedure for solving the initialvalue differential set-valued variational inequalities; and Migórski et al. [4] used the surjectivity of setvalued pseudomonotone operators combined with a fixed point principle to prove the unique solvability of a history-dependent DVI, and then they used the abstract frameworks to study a history-dependent frictional viscoelastic contact problem with a generalized Signorini contact condition. For more details on these topics, the reader is welcome to refer to [5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein.
More recently, Liu et al. [19] initially introduced the notion of differential hemivariational inequalities (DHVIs), which is a generalization of variational inequalities. After that, Migórski and Zeng [20] proposed a temporally semi-discrete algorithm based on the backward Euler difference scheme together with a feedback iterative method to explore a DHVI, which is formulated by a parabolic hemivariational inequality and a nonlinear evolution equation in the framework of an evolution triple of spaces; Li and Liu [21] considered a sensitivity analysis of optimal control problems for a class of systems governed by DHVIs in Banach spaces; by applying the theory of measure of noncompactness, a fixed point theorem of a condensing multivalued map and theory of fractional calculus, Jiang et al. [22] explored an impressive existence result of the mild solutions of a global attractor for the semiflow governed by a fractional DHVI in Banach spaces. For other results on DHVIs, the reader may refer to [23][24][25] and references therein.
The current paper represents a continuation of [10]. In fact, the paper [10] was devoted to discuss the well-posedness and the generalized well-posedness of a differential mixed quasi-variational inequality and to provide criteria of well-posedness and well-posedness in the generalized sense of the inequality. However, the abstract frameworks of the paper [10] cannot be applied for solving the problems or phenomena described by nonconvex superpotential functions, which are locally Lipschitz. To overcome this flaw, in this paper, we are interested in studying a new kind of differential quasi-variationalhemivariational inequality (DQHVI).
Let X be a Hilbert space whose norm and scalar product are ∥⋅∥ X and 〈⋅ ⋅〉 , X , respectively. In what follows, the norm convergence is denoted by "→" and the weak convergence by " ⇀ ". Let < < ∞ T 0 and x L I X x L I X ;̇; 2 2 , where ẋstands for the generalized derivative of x, namely, x t ϕ t t x t ϕ t t ϕ C I Ẋ ḋd for all ; .
T T 0 0 0 Indeed, it is not difficult to prove that ( ) W X is a Hilbert space with the scalar product ,̇,̇for all , , W X L I X L I X 1 2 1 2 ; 1 2 ; 1 2 2 2 and it is densely and continuously embedded in ( ) C I X ; . Given the Hilbert spaces X and V, let K be a fixed nonempty, closed, and convex subset of V. Let × × ⇉ ψ I X V X : be a set-valued mapping. In what follows, we denote the set-valued Nemytskii operator ,for a.e. 2 . We also consider a set-valued mapping is a given function. Assume that × → Γ X X X : , is a locally Lipschitz function, and ( ) → ϕ L I V : ; 2 is a convex functional, the current paper is devoted to study the following DQHVI , , , stands for the set of solutions to the quasi-variational-hemivariational inequality: find ∈ ( ) u S u such that ; 0 for all , denotes the generalized directional derivative of J at u in the direction v. Indeed, when = J 0, then our problem (1.1) reduces the one considered by Liu et al. [10].
The rest of the paper is organized as follows. In Section 2, we survey preliminary material needed in the sequel and introduce the concepts of well-posedness and of well-posedness in the generalized sense for problem (1.1). Section 3 is concerned with the study of the relation between metric characterizations and well-posedness of DQHVI.

Notation and preliminary results
In this section, we briefly review basic notation and some results which are needed in the sequel. For more details, we refer to monographs [26][27][28].
Let V be a Banach space. Throughout the paper, the Clarke generalized directional derivative of a locally Lipschitz function However, the generalized Clarke subdifferential of h at ∈ u V is defined by The next proposition provides basic properties of the generalized directional derivative and the generalized gradient, see, for example, [26,29].
is a locally Lipschitz function on a subset U of V, then (i) for every ∈ u U, the set ∂ ( ) h u is a nonempty, convex, and weakly * compact subset of * V . More precisely, ∂ ( ) h u is bounded by the Lipschitz constant is finite, positively homogeneous, and subadditive on U, and satisfies | ( , is upper semicontinuous, and as a function of v alone is Lipschitz of rank (ii) The Hausdorff distance between subsets A and B is defined by Let { } A n be a sequence of nonempty subsets of V. We say that A n converges to A in the sense of Hausdorff Moreover, we review the definitions of Painlevé-Kuratowski limits.
Definition 2.3. The Painlevé-Kuratowski strong limit inferior and (sequential) weak limit superior of a Obviously, from [30, Theorem 1.1.4], we can see that a set-valued mapping being (s,w)-upper semicontinuous and with closed values is sequentially (s,w)-closed as well.
In what follows, for each > ε 0, we set for all , and 0 , .
We end the section by introducing the approximating sequences, the well-posedness and the generalized well-posedness of DQHVI.
is called an approximating sequence for DQHVI, if there exists a sequence → + ε 0 In this section, we deliver the main results concerning the essential relation between metric characterizations and the well-posedness of DQHVI, and the criteria of well-posedness and wellposedness in the generalized sense of DQHVI. Also, we prove that the set of solutions to DQHVI is compact, when DQHVI is well-posed in the generalized sense.
To do so, we impose the following assumptions.
Remark 3.1. Actually, assumption ( ) A 2 indicates that S has closed and convex values in K .
Besides, we recall the concept of relaxed α-monotonicity for a single-valued mapping, see [10].
Furthermore, we suppose that the following conditions are satisfied.  Remark 3.4. From the above two groups of conditions, we draw some conclusions: (i) condition ( ) B 1 is weaker than condition ( ) A 1 . Indeed, we make use of it in Theorems 3.6 and 3.7; (ii) condition ( ) B 2 is weaker than condition ( ) A 3 . We make use of it in Theorem 3.7; (iii) condition ( ) B 4 is weaker than condition ( ) A 5 .

Characterizations of well-posedness for DQHVI
In this section, we are interesting to establish the metric characterizations of well-posedness for problem (1.1).
; ; 2 2 be set-valued maps. Then, DQHVI is strongly well-posed, if and only if the solution set Σ of DQHVI is nonempty and where ( ) Ω ε is given in (2.1).
Proof. Suppose that DQHVI is strongly well-posed. Then, DQHVI has a unique solution ( ) . We now prove that (3.1) holds. Arguing by contradiction, we assume that ( ( )) Ω ε diam does not tend to 0 as → ε 0. Therefore, we are able to find a constant > β 0 and a positive sequence { } ε n with → + ε 0 n such that   , as → ∞ n . Therefore, DQHVI is strongly well-posed. □ Observing the proof of Theorem 3.5, we could see that the assumption ≠ ∅ Σ plays a significant role. However, by invoking other conditions, we can remove this condition. Proof. Indeed, the necessity part is a direct consequence of Theorem 3.5. Therefore, it is enough to verify the sufficiency. Suppose  in X, as → ∞ n . The latter combined with the continuity of Γ (see ( ) A 6 ) and the inequality ‖ ( ( ) On the other side, we assert that If the above inequality is not true, then we could find a constant > γ 0 such that So, (3.5) holds. Using   Proof. The necessity part could be obtained directly by employing the same arguments with the proof of Theorem 3.6. We now prove the sufficiency part. Indeed, when assumptions ( ) and ( ) A 6 are satisfied, obviously, the desired conclusion is a direct consequence of Theorem 3.6.
In contrast, we assume that conditions ( ) ( ) n n be an approximating sequence to DQHVI. Note that . The latter together with the continuity of Γ deduces is an approximating sequence of problem DQHVI, , ; n n n L I V n n n n nL I V ; 0 ; 2 2 for all ∈ ( ) v S u n . Let ∈ ( ) v S u 0 be fixed. Invoking the ( ) s s , -lower semicontinuity of S (see ( ) ; .

Characterizations of well-posedness in the generalized sense of DQHVI
In this section, we explore the metric characterizations of well-posedness in the generalized sense for DQHVI. Besides, the compactness of solution set of DQHVI is proved, when DQHVI is well-posed in the generalized sense. First, we deliver the following important result that for each > ε 0, the set ( ) Ω ε introduced in (2.1) is closed.  Since ∈ ( ) v S u 0 is arbitrary, we can apply (3.7) and (3.10)-(3.12) to conclude ( ) ∈ ( ) x u Ω ε , 0 0 . □