A class of differential hemivariational inequalities in Banach spaces

In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function u↦f(t,x,u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\mapsto f(t,x,u)$$\end{document} and compactness of C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0$$\end{document}-semigroup eA(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{A(t)}$$\end{document}.

function u → f (t, x, u) maps convex subsets of K to convex sets and the C 0 -semigroup e A(t) is compact. Therefore, in our present work, we would like to overcome those flaws, fill a gap, and develop new mathematical tools and methods for differential hemivariational inequalities.
Let V , E, X and Y be reflexive, separable Banach spaces, H be a separable Hilbert space, A : D(A) ⊂ E → E be the infinitesimal generator of C 0 -semigroup e At in E and be given maps, which will be specified in the sequel. In this paper, we consider the following abstract system consisting of a hemivariational inequality of parabolic type combined with a nonlinear abstract evolution equation.

Problem 1
Find u : (0, T ) → V and x : (0, T ) → E such that for all v ∈ V and a.e. t ∈ (0, T ) The main novelties of the paper are described as follows. First, for the first time, we apply the Rothe method, see [16,51], to study a system of a hemivariational inequality of parabolic type driven by a nonlinear abstract evolution equation. Until now, there are a few papers devoted to the Rothe method for hemivariational inequalities, see [4,5,52]. Furthermore, all of them investigated only a single hemivariational inequality by using Rothe method.
Second, the main results can be applied to a special case of Problem 1 in which the locally Lipschitz functional J and the nonlinear function F are assumed to be independent of the variable x. So, Problem 1 reduces to the following hemivariational inequality of parabolic type: find u : (0, T ) → V such that u(0) = u 0 and for all v ∈ V and a.e. t ∈ (0, T ). This problem was considered only recently by Migórski-Ochal [33], Kalita [17], and Fang et al. [11]. Third, until now, all contributions concerning (DVIs) were driven only by variational/hemivariational inequalities of elliptic type. Here, for the first time, we discuss (DHVI) governed by a hemivariational inequality of parabolic type. Additionally, in comparison with our previous works [23,24,26,27], in this paper, we do not impose any convexity assumption on the nonlinear function u → f (t, x, u) and we remove the compactness hypothesis on C 0 -semigroup e A(t) .
The paper is organized as follows. In Sect. 2, we recall some definitions and preliminary facts concerning nonlinear and nonsmooth analysis, which will be used in the sequel. In Sect. 3, we provide the definition of a solution to Problem 1 in the mild sense, and then establish a hybrid iterative system, Problem 16. The solvability of Problem 16 is obtained by a surjectivity result for a pseudomonotone operator and a priori estimate for the solutions to Problem 16 is proved. Finally, through a limiting procedure for the solutions to Problem 16, the existence of solution to Problem 1 is established.

Preliminaries
This section is devoted to recall basic notation, definitions and some auxiliary results from nonlinear analysis, see [9,10,36,50], which will be used in the sequel.
We start with definitions and properties of semicontinuous set-valued mappings.
Definition 2 Let X and Y be topological spaces, and F : X → 2 Y be a set-valued mapping. We say that F is If this holds for every x ∈ X , then F is called upper semicontinuous. (ii) lower semicontinuous (l.s.c., for short) at If this holds for every x ∈ X , then F is called lower semicontinuous. (iii) continuous at x ∈ X if, it is both upper semicontinuous and lower semicontinuous at x ∈ X . If this holds for every x ∈ X , then F is called continuous.
The following theorem gives some criteria for the upper semicontinuity of set-valued mappings.
Proposition 3 (see [36]) Let X, Y be two topological spaces and F : X → 2 Y . The following statements are equivalent Next, we recall the definition of pseudomonotonicity of a single-valued operator.

Definition 4
Let X be a reflexive Banach space with dual X * and A : X → X * . We say that A is pseudomonotone, if A is bounded and for every sequence {x n } ⊆ X converging weakly to x ∈ X such that lim sup Remark 5 It is known that an operator A : X → X * is pseudomonotone, if and only if x n → x weakly in X and lim sup n→∞ Ax n , x n − x ≤ 0 entails lim n→∞ Ax n , x n − x = 0 and Ax n → Ax weakly in X * .
Furthermore, if A ∈ L(X, X * ) is nonnegative, then it is pseudomonotone.
Next, the pseudomonotonicity of multivalued operators is defined below.

Definition 6
A multivalued operator T : X → 2 X * is pseudomonotone if (a) for every v ∈ X , the set T v ⊂ X * is nonempty, closed and convex, (b) T is upper semicontinuous from each finite dimensional subspace of X to X * endowed with the weak topology, (c) for any sequences {u n } ⊂ X and {u * n } ⊂ X * such that u n → u weakly in X , u * n ∈ T u n for all n ≥ 1 and lim sup n→∞ u * n , u n − u ≤ 0, we have that for every v ∈ X , there exists u * (v) ∈ T u such that Definition 7 Given a locally Lipschitz function J : X → R on a Banach space X , we denote by J 0 (u; v) the generalized (Clarke) directional derivative of J at the point u ∈ X in the direction v ∈ X defined by The generalized gradient of J : X → R at u ∈ X is defined by The following result provides an example of a multivalued pseudomonotone operator which is a superposition of the Clarke subgradient with a compact operator. The proof can be found in [3,Proposition 5.6].

Proposition 8
Let V and X be two reflexive Banach spaces, γ : V → X be a linear, continuous, and compact operator. We denote by γ * : X * → V * the adjoint operator to γ . Let j : X → R be a locally Lipschitz functional such that Moreover, we recall the following surjectivity result, which can be found in [10, Theorem 1.3.70] or [50].

Theorem 9
Let X be a reflexive Banach space and T : X → 2 X * be pseudomonotone and coercive. Then T is surjective, i.e., for every f ∈ X * , there exists u ∈ X such that T u f .
We now introduce spaces of functions, defined on a finite interval [0, T ]. Let π denote a finite partition of the interval (0, T ) by a family of disjoint subintervals Let F denote the family of all such partitions. For a Banach space X and 1 ≤ q < ∞, we define the space and define the seminorm of a vector function v : Assume that 1 ≤ p ≤ ∞ and 1 ≤ q < ∞, and X , Z are Banach spaces such that X ⊂ Z with continuous embedding. We introduce the following Banach space which is endowed with the norm · L p (0,T ;X ) + · BV q (0,T ;Z ) . Recall a useful compactness result, which proof can be found in [17, Proposition 2.8].
Proposition 10 Let 1 ≤ p, q < ∞, and X 1 ⊂ X 2 ⊂ X 3 be Banach spaces such that X 1 is reflexive, the embedding X 1 ⊂ X 2 is compact, and the embedding X 2 ⊂ X 3 is continuous.
We end this section by recalling a discrete version of the Gronwall inequality, which can be found in [

Main results
In this section, we focus our attention on the investigation of an abstract system, which consists of a hemivariational inequality of parabolic type, and a nonlinear evolution equation involving an abstract semigroup operator. The method of proof is based on properties of subgradient operators in the sense of Clarke, surjectivity of multivalued pseudomonotone operators, the Rothe method, and convergence analysis. We begin this section with the standard notation and function spaces, which can be found in [9,10,50]. Let (V, · ) be a reflexive and separable Banach space with its dual space V * , H be a separable Hilbert space, and (Y, · Y ) be another reflexive and separable Banach space. Subsequently, we assume that the spaces V ⊂ H ⊂ V * (or (V, H, V * )) form an evolution triple of spaces (see cf. [36, Definition 1.52]) with dense, continuous, and compact embeddings. The embedding injection from V to H is denoted by ι : V → H . Moreover, let (X, · X ) and (E, · E ) be reflexive and separable Banach spaces with their duals X * and E * , respectively. For 0 < T < +∞, in the sequel, we use the standard Bochner-Lebesgue here v denotes the time derivative of v, understood in the sense of distributions. The notation ·, · V * ×V stands for the duality between V and V * . The space of linear bounded operaors from V to X is denoted by L(V , X ).
To prove the solvability of Problem 1, we impose the following assumptions on the data of the problem.
(ii) one of the following conditions holds (ii) 1 N satisfies the growth condition

Remark 12 We provide two examples of operator N which satisfies the hypotheses H (N ).
In the first example, assume that V = H 1 0 ( ) and N : V → V * is a second order quasilinear differential operator in divergence form of the Leray-Lions type, i.e., and each a i is a Carathéodory function such that for a.e. x ∈ and all ξ ∈ R d .
Then, it is well known, see [21], that N satisfies conditions H (N )(i) and (ii) 1 .
In the second example, N is an abstract Navier-Stokes operator, see [32,33]. Let be a simply connected domain in R d , d = 2, 3 with regular boudary , and where w T is the tangential component of w on the boundary . Also, let V and H be the closure of W in the norm of H 1 ( ; R d ) and L 2 ( ; R d ), respectively. Let N : V → V * be the classical Navier-Stokes operator, i.e., for all u, v, w ∈ V , where operator curlu stands the rotation of u and ν > 0. Recall that is a simply connected domain, therefore, we can see that the bilinear form Next, we show that hypothesis H (J ) implies that the subgradient operator ∂ J of J is upper semicontinuous in suitable topologies.

Lemma 13 Assume that H (J ) holds. Then the subgradient operator
is upper semicontinuous from E × X endowed with the norm topology to the subsets of X * endowed with the weak topology.
Proof From Proposition 3, it remains to verify that for any weakly closed subset D of X * , the weak inverse image (∂ J ) −1 (D) of ∂ J under D is closed in the norm topology, where Let {(y n , x n )} ⊂ (∂ J ) −1 (D) be such that (y n , x n ) → (y, x) in E × X , as n → ∞ and {ξ n } ⊂ X * be such that ξ n ∈ ∂ J (y n , x n ) ∩ D for each n ∈ N. Hypothesis H (J )(ii) implies that the sequence {ξ n } is bounded in X * . Hence, by the reflexivity of X * , without loss of generality, we may assume that ξ n → ξ weakly in X * . The weak closedness of D guarantees that ξ ∈ D. On the other hand, ξ n ∈ ∂ J (y n , x n ) entails ξ n , z X * ×X ≤ J 0 (y n , x n ; z) for all z ∈ X.
Taking into account the upper semicontinuity of (y, x) → J 0 (y, x; z) for all z ∈ X and passing to the limit, we have ξ, z X * ×X = lim sup n→∞ ξ n , z X * ×X ≤ lim sup n→∞ J 0 (y n , x n ; z) ≤ J 0 (y, x; z) for all z ∈ X . Hence ξ ∈ ∂ J (y, x), and consequently, we obtain ξ ∈ ∂ J (y, x) ∩ D, i.e., (y, x) ∈ (∂ J ) −1 (D). This completes the proof of the lemma. Now, we observe that Problem 1 can be rewritten in the following equivalent form.
According to our previous work [23,25,26], we give the following definition of a solution to Problem 14 in the mild sense. In what follows, we establish the existence of a mild solution to Problem 14. We use the idea of the Rothe method combined with a feedback iterative approach.
Obviously, this system is constituted with a stationary nonlinear Clarke subdifferential inclusion and a nonlinear abstract integral equation. First, we give the following existence result on a solution to hybrid iterative system, Problem 16. H(F), H(N ), H(J ), H(M), H(ϑ), H(0) and H ( f ) hold. Then, there exists τ 0 > 0 such that for all τ ∈ (0, τ 0 ), the hybrid iterative system, Problem 16, has at least one solution.
These properties together with [20,Proposition 5.3,p.66] and [26,Section 4] imply that there exists a unique function x τ ∈ C(0, t k ; E) such that Further, from hypothesis H (F) and x τ ∈ C(0, t k ; E) we can easily check It remains to find elements u k τ ∈ V and ξ k τ ∈ ∂ J (x τ (t k ), Mu k τ ) such that To this end, we will apply the surjective result, Theorem 9, to show that the operator S : From hypothesis H (J )(ii), we have the following estimate for all v ∈ V and ξ ∈ ∂ J (x τ (t k ), Mv). Moreover, hypothesis H (N )(i) reveals After inserting (14) into the above inequality, we have for all v ∈ V . Choosing τ 0 = 1 a 1 and taking into account the smallness condition H (0), we conclude that S is coercive for all τ ∈ (0, τ 0 ). Moreover, we shall also verify that S is pseudomonotone. In fact, from [36,Proposition 3.59], we know that if all components of S are pseudomonotone, then S is pseudomonotone as well. Since v → ι * ιv τ is bounded, linear and nonnegative, so it is pseudomonotone. On the other hand, hypotheses H (M), H (J )(i), H (J )(ii) and Proposition 8 ensure that the operator Since N is pseudomonotone, see H (N ), we conclude by [36,Proposition 3.59] that S is a pseudomonotone operator.
Next, we provide a result on a priori estimate for solutions to Problem 16.

Lemma 18 Assume that H (A), H(F), H(N ), H(J ), H(M), H(ϑ), H(0) and H ( f ) hold.
Then, there exist τ 0 > 0 and C > 0 independent of τ such that for all τ ∈ (0, τ 0 ), the solutions to the hybrid iterative system, Problem 16, satisfy Proof Let ξ k τ ∈ ∂ J (x τ (t k ), Mu k τ ) be such that equality (12) holds. Multiplying (12) by u k τ , we have From H (N )(i), we have Moreover, hypothesis H (J )(ii) guarantees that Inserting (19) and (20) into (18), and taking into account the identity We are now in a position to apply Cauchy's inequality with ε > 0 to get For functions u τ , u τ and ξ τ , we have the following estimates.
where (x, u, ξ) ∈ C(0, T ; E) × W × X * is a solution of Problem 14 in the sense of Definition 15.
Proof From the estimates (22)- (24) and the reflexivity of V and H, without loss of generality, we may assume that there exist u, u ∈ V such that convergence (29) holds and u τ → u weakly in V, as τ → 0. It is easy to obtain that This combined with the bound in (26) implies Recalling that u τ → u weakly in V and using convergence (29), we have u τ − u τ → u − u weakly in V, as τ → 0. Moreover, the continuity of embedding V ⊂ V * ensures that u − u τ → u − u weakly in V * as well. So, from (34), we conclude u = u, i.e., (30) holds. The functions u τ defined in (13) are bounded in V. So, there exists a function u * ∈ V such that u τ → u * weakly in V, as τ → 0. In the same time, we have This implies that u τ − u τ → 0 V * , as τ → 0. Similarly, we can conclude that u * = u. Moreover, (26) entails that there exists a function w * ∈ V * such that This convergence together with (30)  to problem Now, we return to functions x τ and x, and, for all t ∈ [0, T ], we get for all t ∈ [0, T ]. Since u τ → u weakly in V, u τ → u weakly in V * , as τ → 0, and the embedding W ⊂ C(0, T ; H ) is continuous, we can see that u τ → u weakly in C(0, T ; H ). From [34,Lemma 4], we have Obviously, the above convergence holds also, when hypothesis H (N )(ii) 2 is satisfied, since u τ → u weakly in V, as τ → 0. Therefore, we conclude N u τ , v V * ×V → N u, v V * ×V (38) for all v ∈ V. The convergence (32) implies ξ τ , Mv X * ×X → ξ, Mv X * ×X (39) for all v ∈ V. Furthermore, from (36), we have for all v ∈ V. Combining with (37)-(40), we obtain To complete the proof of the theorem, we need to prove that ξ(t) ∈ ∂ J (x(t), Mu(t)) for a.e. t ∈ (0, T ). From (27), (29) and hypothesis H (M), we have M(u τ ) → M(u) in X * , as τ → 0.
So, we may suppose, passing to a subsequence if necessary, that Mu τ (t) → Mu(t) in X * , for a.e. t ∈ (0, T ).
Consequently, we have shown that the triple of functions (x, u, ξ) ∈ C(0, T ; E) × W × X * is a mild solution to Problem 14 in the sense of Definition 15. This completes the proof of the theorem.
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