On a Class of Differential Variational Inequalities in Inﬁnite-Dimensional Spaces

: A new class of differential variational inequalities (DVIs), governed by a variational inequality and an evolution equation formulated in inﬁnite-dimensional spaces, is investigated in this paper. More precisely, based on Browder’s result, optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of DVI is nonempty and compact. In addition, the theoretical developments are accompanied by an application to differential Nash games.

As is well-known, differential variational inequalities were introduced as a powerful mathematical tool of variational analysis in order to investigate real-life problems coming from operations research, engineering, and physical sciences. Various aspects related to differential variational inequalities have been investigated so far, but in a finitedimensional framework (see, for instance, [4][5][6][7][8][9][10][11][12][13] and references therein). The results established in Gwinner [6,7], Liu et al. [10], Pang and Stewart [1], Li et al. [14] are devoted to the case X = R n , X 1 = R m and A = 0. Recently, the theory of differential variational inequalities was extended to the more general level of infinite-dimensional Banach or Hilbert spaces. The paper by Liu et al. [15] was devoted to discuss the wellposedness and the generalized well-posedness of a differential mixed quasi-variational inequality and to provide criteria of well-posedness in the generalized sense of the inequality. For solving the problems or phenomena described by nonconvex superpotential functions, which are locally Lipschitz, Cen et al. [16] extended the results derived in Liu et al. [15]. Migórski and Bai [17] studied a class of evolution subdifferential inclusions involving history-dependent operators. Regarding the evolutionary problem (DVI), references Liu et al. [3,18] could be seen as the most appropriate regarding the subject and the mathematical development presented in this paper. Reference Liu et al. [3] considers Lu(τ) + H(τ, x(τ)) and g(τ, x(τ)) + F (u(τ)) having the general form f (τ, x(τ), u(τ)) and g(τ, x(τ), u(τ)), respectively. By considering the particular form g(τ, x(τ)) + F (u(τ)) for g(τ, x(τ), u(τ)) is essential in the present paper since it allows us to use the Browder's theorem (see Theorem 1). On the other hand, the reference Liu et al. [18] does not contain the functional ξ : X 1 → (−∞, +∞] in the considered evolutionary problem. Thus, the evolutionary problem (DVI) becomes more general in this direction and incorporates various classes of problems and models.
In this paper, based on the Browder's theorem, optimal control theory, KKM theorem, measurability of set-valued mappings and the theory of semigroups, we study the existence of solutions associated with DVI in separable reflexive Banach spaces of infinite dimension. First, we prove important properties associated with the solution set of DVI as the stronglyweakly upper semicontinuity, superpositionally measurability, compactness and convexity. Further, we formulate and prove the main result established that the solution set for DVI is nonempty and compact. Additionally, the theoretical developments presented in this paper are accompanied by an application to differential Nash games. As is well known (see, for instance, Chen and Wang [4], Gwinner [6,7], Han and Pang [8], Li et al. [14], Liu et al. [10], Pang and Stewart [1], Wang and Huang [13]), this kind of evolutionary problem governed by variational inequalities includes various situations such as Coulomb friction problems for contacting bodies, dynamic traffic network, economical dynamics, electrical circuits with ideal diodes, control systems etc.
The paper is divided as follows. The first part of Section 2 includes basic definitions and results used further. Next, we establish a general existence result for a class of variational inequalities closely related to DVI. In addition, the strongly-weakly upper semicontinuity, compactness and superpositionally measurability are investigated for the solution set of the considered variational inequality. The final part of Section 2 provides some existence and qualitative properties for the solution set of the evolutionary problem (DVI). In order to illustrate the effectiveness of the theoretical results presented in this paper, an application to differential Nash games is also provided.

Main Results
In this section, we study the existence of solutions for DVI in infinite-dimensional Banach spaces, and we also formulate some properties of the solution set. First, we recall some notations, notions and results which will be useful in the sequel.
Let 2 S be the collection of all nonempty subsets for any nonempty set S. Additionally, we introduce Ω(S) := {D ∈ 2 S : D is compact} Ω ν (S) := {D ∈ 2 S : D is compact and convex} and denote by "→" and " " the strong convergence and the weak convergence, respectively, in a given Banach space X .
Definition 2 (see [19]). The set-valued mapping F : Definition 3. Let X and X 1 be Banach spaces and let I ⊂ R be an interval. The set-valued mapping U : is measurable, for every measurable set-valued mapping x : I → 2 X .
Lemma 1 (see [5]). Let X 1 be a Hausdorff topological vector space, Ω ⊂ X 1 is a nonempty subset and G : Ω → 2 X 1 is a set-valued mapping such that (i) for any {v 1 , · · · , v n } ⊂ Ω, one has that its convex hull co{v 1 , · · · , v n } is included in Lemma 2 (see [20]). U : I × X → Ω(X 1 ) is superpositionally measurable if it satisfies the Carathéodory condition or U is upper or lower semicontinuous.
Then, the next theorem represents the first main result of this paper. Theorem 1. Let X 1 be a reflexive Banach space, Ω be a nonempty closed and convex subset of X 1 and assume that and satisfies (ii) ξ : X 1 → (−∞, +∞] is lower semicontinuous, convex, ≡ +∞; (iii) there exist u 0 ∈ Ω and an r > 0 such that Then, for each element w ∈ X * 1 , there exists u ∈ Ω such that if and only if, for each element w ∈ X * 1 , there exists u ∈ Ω such that In addition, the solution set associated with (3) is nonempty, convex and closed in X 1 .
Proof. If u ∈ Ω is a solution associated with (3), by using the monotonicity of F , it follows that u ∈ Ω is also a solution of (4). Conversely, assume that u ∈ Ω is a solution of (4).
Taking into account the convexity of the set Ω, for all λ ∈ (0, 1) and all v ∈ Ω, it results that u λ := (1 − λ)u + λv ∈ Ω. In consequence, we have and, by assumption (ii), we get Considering λ → 0 + in the above inequality (see the second part of hypothesis (i)), we obtain that u ∈ Ω is also a solution of (3). Further, in order to prove the other assertions of our theorem, we consider the following two cases: It is of course immediate (from the above equivalent conclusion) that Passing to the limit as n → +∞ in the above inequality, by hypothesis (ii), we get Further, we prove that the set-valued mapping G is a KKM mapping (see Lemma 1). Suppose, by contradiction, that there exists {v 1 , · · · , v n } ⊂ Ω and By using the monotonicity of F , it follows from which the contradiction arises By using the hypothesis, it follows that Ω is weakly compact in X 1 . Thus, for each v ∈ Ω, G(v) is weakly compact in X 1 . Now, by applying Lemma 1, we obtain v∈Ω G(v) = ∅, that is the solution set associated with (4) is nonempty, so the same is true for the solution set associated with (3). Case 2. Ω is unbounded in X 1 . For every integer n ≥ 1, consider the bounded, closed and convex subset of X 1 where u 0 ∈ Ω is given in assumption (iii). In accordance with the previous case, we can find u n ∈ Ω n such that In the following, let us show that there exists an integer k ≥ 1 such that By contradiction, assume that u n − u 0 X 1 = n, for every integer n ≥ 1. Putting which is a contradiction (see assumption (iii)), if n is sufficiently large. Therefore, the claim in (6) is fulfilled. By (6), for y ∈ Ω and sufficiently small τ > 0, we have Next, set v = u k + τ(y − u k ) and n = k in (5). By hypothesis (ii), it follows Further, by using the equivalence between variational problems (3) and (4) and assumption (ii), we conclude that the solution set for (3) is closed and convex in X 1 . The proof is now complete. [21], Liu et al. [3] and it is based on the Browder's theorem (see [22]). An extension of Minty's technique is given by assertion (4), and a generalized coercivity condition is provided by assumption (iii). Corollary 1. Let X 1 be a real reflexive Banach space and Ω ⊂ X 1 a nonempty compact and convex subset of X 1 . Then, the conclusion of Theorem 1 is fulfilled if F : Ω → X * 1 and ξ : X 1 → (−∞, +∞] verify conditions (i), (ii) in Theorem 1.

Remark 1. Theorem 1 extends some results derived in Liu and Zeng
In the following, denote by S(Ω, w + F (·), ξ) the set of solutions associated with the variational inequality (3). Closely related to the evolutionary problem (DVI), by using of Theorem 1, we also establish the following two results.
In consequence, there exist w k ∈ B(N 0 , X * 1 ) and u k ∈ S(Ω, w k + F (·), ξ) satisfying u k X 1 > k, for k = 1, 2, .... By assumption (iii) of Theorem 1, it follows that there is a function p : R + → R + , with p(k) → +∞ as k → +∞, and a constant M > 0 such that for each u X 1 > M, we have Therefore, for k > M, one has u k X 1 > M and, as k large enough, This is a contradiction, and the proof is complete.
Proof. (i) In accordance with Kamemsloo et al. [19], for each weakly closed subset C of for n ∈ N, we obtain that there exists u n ∈ S(Ω, g(t n , x n ) + F (·), ξ). We can assume that g(t n , x n ) X * 1 ≤ k 0 (by using of the uniform boundedness of g), where k 0 is a positive constant. According to Lemma 3, there exists a constant M k 0 > 0 such that u n X 1 ≤ M k 0 , for n ∈ N. Consequently, the sequence {u n } is relatively weakly compact in X 1 , and we may assume u n u, without loss of generality. Since u n ∈ S(Ω, g(t n , x n ) + F (·), ξ), for n ∈ N, we get In virtue of the monotonicity of F , it follows Applying assumption (ii) of Theorem 1, the continuity of g and n → ∞, we have that is, u ∈ S(Ω, g(τ, x) + F (·), ξ) (since the problem (4) is equivalent with the problem (3)). By using the weak closedness of C, we obtain u ∈ Z(τ, x) ∩ C. Therefore, Z is strongly-weakly upper semicontinuous.

Conclusions
In the current paper, based on Browder's result, optimal control theory, measurability of set-valued mappings and the theory of semigroups, we have investigated a new class of differential variational inequalities. More precisely, we have proved that the solution set associated with the considered evolutionary problem is nonempty and compact. In addition, the theoretical developments have been accompanied by an application to differential Nash games.