A General Class of Differential Hemivariational Inequalities Systems in Reﬂexive Banach Spaces

: We consider an abstract system consisting of the parabolic-type system of hemivariational inequalities (SHVI) along with the nonlinear system of evolution equations in the frame of the evolution triple of product spaces, which is called a system of differential hemivariational inequalities (SDHVI). A hybrid iterative system is proposed via the temporality semidiscrete technique on the basis of the Rothe rule and feedback iteration approach. Using the surjective theorem for pseudomonotonicity mappings and properties of the partial Clarke’s generalized subgradient mappings, we establish the existence and priori estimations for solutions to the approximate problem. Whenever studying the parabolic-type SHVI, the surjective theorem for pseudomonotonicity mappings, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, guarantees the successful continuation of our demonstration. This overcomes the drawback of the KKM-based approach. Finally, via the limitation process for solutions to the hybrid iterative system, we derive the solvability of the SDHVI with no convexity of functions u (cid:55)→ f l ( t , x , u ) , l = 1,2 and no compact property of C 0 -semigroups e A l ( t ) , l = 1,2.

To the best of our knowledge, the theory of VIs, which was first extended to treat the EPs, is intently relevent to the convexity of EFs, and is the basis of various arguments of monotonicity. In case the relevent EFs are of nonconvexity (i.e., superpotentials), the other type of inequalities emerges as the variational formula of a problem. They are referred to as HVIs and their derivation is based on the properties of the CGS formulated for locally Lipschitz functionals. In comparision with the VIs, the stationary HVIs do not coincide with minimization problems, they yield substationarity problems, whose research began at the originating work in [1]. Various problems are formulated via nonsmooth superpotentials, so it is very natural that, in the past three decades, a lot of authors paid It is worth pointing out that the authors [18], via the Rothe rule, first studied the parabolic-type HVI driven by the abstract EE. Up to now, there have been only a few papers devoted to the Rothe rule for HVIs, see [20]. It is worth mentioning that these were focused only on a single HVI via the Rothe rule.
Next, for convenience, let the AS, AP, SHVI, SEE, ETPS, SDHVI, HIS and PCGS represent an abstract system, an approximate problem, a system of hemivariational inequalities, a nonlinear system of evolution equations, an evolution triple of product spaces, a system of differential hemivariational inequalities, a hybrid iterative system and partial CGS, re-spectively. Inspired by recent works in [18,19], we introduce and consider the AS, which is constituted by the parabolic-type SHVI along with the SEE, in the frame of the ETPS, which is referred to as the SDHVI. The HIS is proposed via the temporality semidiscrete technique on basis of the backward Euler difference formula (i.e., the Rothe rule), and the feedback iteration approach. Using the surjective theorem for pseudomonotonicity mappings and properties of PCGS mappings, we demonstrate the existence of solutions to the AP and provide the priori estimation for solutions to the AP. At the end, via the limitation process for solutions to the HIS, we derive the solvability of the SDHVI with no convexity of functions u → f l (t, x, u), l = 1, 2 and no compact property of C 0 -semigroups e A l (t) , l = 1, 2.
Until now, except for the DHVI considered in [18], many works about the DVIs were boosted only by elliptic-type VIs/HVIs. Here, we first consider the SDHVI driven by the parabolic-type SHVI. In addition, except for the DHVI considered in [18], in contrast to the previous works [11,16,17,19], in this article we assume no convexity condition on the functions u → f l (t, x, u), l = 1, 2 and no compactness condition on C 0 -semigroups e A l (t) , l = 1, 2.
The article is assigned below. In Section 2, we recall some concepts and basic results about nonsmooth and nonlinear analysis, and present the formulation of the AS. In Section 3, we formulate a solution to the AS, and then give the formulation of the HIS. We obtain the solvability of the HIS via the surjective theorem for pseudomonotonicity mappings and derive the priori estimation of solutions to the HIS. Finally, via the limitation process for solutions to the HIS, we establish the existence of solutions to the AS.
It is also worthy of note that there are evident disadvantages of the method based on the KKM approach for studying the parabolic-type SHVI. Indeed, if the mappings in the method based on the KKM approach are not the KKM ones, then there are several possibilities which happen in the demonstration process, e.g., in particular, whenever studying the parabolic-type SHVI. This might result in an unsuccessful continuation of the demonstration. Practically, this is precisely the shortcoming of the KKM-based approach.

Preliminaries
We first recall some notations, concepts and basic results, and then give the formulation of the AS. We start with definitions and properties of semicontinuous set-valued mappings. Suppose that E and F both are topological spaces. The setvalued operator Γ : E → 2 F is referred to as being (i) of upper semicontinuity (u.s.c.) at (1) In case the above relation holds for all x ∈ E, Γ is said to be u.s.c.
In case the above relation holds for all x ∈ E, Γ is said to be l.s.c.
(iii) of continuity at x ∈ E iff, Γ not only is u.s.c. at x ∈ X and but also is l.s.c. at x ∈ X. In case this holds for all x ∈ E, Γ is said to be continuous.
Proposition 1 (see [3]). The assertions below are of equivalence mutually: In what follows, we assume that X is a reflexive Banach space with its dual X * . A single-valued mapping A : X → X * is referred to as being pseudomonotone, if A is of boundedness and for each sequence {x n } ⊆ X converging weakly to x ∈ X s.t. lim sup n→∞ Ax n , x n − x X * ×X ≤ 0, one has Ax, x − y X * ×X ≤ lim inf n→∞ Ax n , x n − y X * ×X , ∀y ∈ X. (2) Recall that a mapping A : X → X * is of pseudomonotonicity, iff x n → x weakly in X and lim sup n→∞ Ax n , x n − x X * ×X ≤ 0 entails lim n→∞ Ax n , x n − x X * ×X = 0 with weak convergence of {Ax n } to Ax.
In addition, in case A ∈ L(X, X * ) is of nonnegativity, A is of pseudomonotonicity.
Recall that a multivalued operator T : X → 2 X * is said to be pseudomonotone if (a) for every v ∈ X, the set Tv ⊂ X * is nonempty, closed and convex; (b) T is u.s.c. 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 * s.t. u n → u weakly in X, u * n ∈ Tu n for all n ≥ 1 and lim sup n→∞ u * n , u n − u X * ×X ≤ 0, one has that ∀v ∈ X, ∃u * (v) ∈ Tu s.t.
Also, recall the CGS of locally Lipschitz functional; see [3]. Suppose that X is a Banach space and h : The CGS of h at u ∈ X, written as ∂h(u), is the set in X * , formulated below The following lemma provides some basic properties for the CGDD and CGS; see [3].

Lemma 1.
Suppose that h : X → R is locally Lipschitz. Given u, v ∈ X arbitrarily. Then is positively homogeneous, subadditive and finite, and hence convex; . on X × X as a functional of (u, v), and as a functional of v alone, is Lipschitz continuous; is nonempty, weak * -compact, bounded and convex in X * for each u ∈ X; (v) for all v ∈ X, one has h • (u; v) = max{ ξ, v X * ×X : ξ ∈ ∂h(u)}; (vi) ∂h(u) has the closed graph in X × (w * -X * ) topology, with (w * -X * ) being the space X * endowed with weak * topology, i.e., whenever {u n } ⊂ X and {u * n } ⊂ X * are sequences s.t. u * n ∈ ∂h(u n ), u n → u in X and u * n → u * weak * ly in X * , one has u * ∈ ∂h(u).
Proposition 2 (see [21]). Suppose that U and Y are reflexive Banach spaces and S : U → Y is the linear continuous mapping with compactness. One denotes by S * : Y * → U * the adjoint mapping of S. Let h : Y → R be locally Lipschitz s.t.
Then the setvalued mapping G : U → 2 U * formulated below The surjective theorem below can be found in [2,22].
We construct the spaces of functions, defined on [0, T] with 0 < T < ∞. Let π indicate a division of (0, T) via a pool of subintervals σ l = (a l , b l ) s.t. [0, T] = n l=1 σ l . Let F denote the family of all such divisions. For a Banach space X and 1 ≤ q < ∞, we construct the space and formulate the seminorm of v : Suppose that the Banach spaces X, Z are s.t. X ⊂ Z with continuous embedding. For 1 ≤ p ≤ ∞ and 1 ≤ q < ∞ we construct the Banach space below equipped with norm · L p (0,T;X) + · BV q (0,T;Z) .
Proposition 3 (see [23]). Suppose that X 1 ⊂ X 2 ⊂ X 3 are Banach spaces s.t. X 1 is reflexive, the embedding X 1 → X 2 is of compactness, and the embedding X 2 → X 3 is of continuity. In case the set B is of boundedness in M p,q (0, T; X 1 , X 3 ) with p, q ∈ [1, ∞), B is of relative compactness in L p (0, T; X 2 ).
We recall the discrete form of Gronwall's inequality below.
Lemma 2 (see [24]). Given T ∈ (0, ∞). For integer N ≥ 1, we define τ = T N . Suppose that Put l, k = 1, 2 and k = l. Let V l , E l , X l and Y l be reflexive, separable Banach spaces, H l be a separable Hilbert space, let Z = Z 1 × Z 2 , ∀Z l ∈ {V l , E l , X l , Y l }, and suppose that A l : D(A l ) ⊂ E l → E l is the infinitesimal generator of C 0 -semigroup e A l t in E l and are the mappings given. Inspired by [18,19], we formulate the AS below Find u : (0, T) → V and x : (0, T) → E with u = (u 1 , u 2 ) and x = (x 1 , x 2 ), s.t. and x(0) = x 0 and u(0) = u 0 , where x 0 = (x 0 1 , x 0 2 ), u 0 = (u 0 1 , u 0 2 ), ϑu = (ϑu 1 , ϑu 2 ), ·, · V * l ×V l is the duality pairing between V l and V * l and J • l (x, z 1 , z 2 ; v l ) is the partial Clarke's generalized directional derivative (PCGDD, for short) of the locally Lipschitz functional J : E × X 1 × X 2 → R w.r.t. the l-th argument at the point z l ∈ X l in the direction v l ∈ X l for the given z k ∈ X k .
In what follows, we consider an example of the AS, where locally Lipschitz J and functions F l , l = 1, 2 are supposed to be independent of x. Hence, the AS reverts to the parabolic-type SHVI below.
Find u : (0, T) → V s.t. u(0) = u 0 and It is easy to see that problem (11) is a generalization of the parabolic-type HVI below.
It is worth mentioning that this problem was considered only in [10,23,25].

Existence and Priori Estimation
In what follows, the process of demonstration involves the properties of PCGS, the surjective of setvalued pseudomonotonicity mappings, Rothe's rule, and convergent analysis.
We start this section with the normal symbos and functions; see [19,22]. For l = 1, 2, let the Banach space (V l , · V l ) be a separable and reflexive one with the dual V * l , the Hilbert space H l be a separable one, and the Banach space (Y l , · Y l ) be the other separable and reflexive one. Later on, we suppose that the ones V l ⊂ H l ⊂ V * l (or (V l , H l , V * l )) constitute the ETS [3] with dense continuity compactness embeddings.
Endowed with the norm defined by u V := u 1 V 1 + u 2 V 2 for all u = (u 1 , u 2 ) ∈ V, V is a reflexive Banach space ( [22]) with its dual V * and the duality pairing between V and V * is formulated below Similarly, we can construct the product space H = H 1 × H 2 . It is clear that the ones V ⊂ H ⊂ V * (or (V, H, V * )) constitute the ETPS. For l = 1, 2, the embedding injection from V l to H l is denoted by ι l : V l → H l . Moreover, for l = 1, 2, let (X l , · X l ) and (E l , · E l ) be reflexive and separable Banach space with their dual X * l and E * l , respectively. For l = 1, 2 and 0 < T < +∞, in the sequel, we use the standard Bochner-Lebesgue here v l indicates the derivative of v l to time. The symbol ·, · V * l ×V l denotes the dual pairing between V l and V * l and the space of linear continuous operators of V l into X l is written as L(V l , X l ) for l = 1, 2. Of course, we can also construct the product To prove the solvability of the AS, for l, k = 1, 2 and k = l we always assume that the conditions below hold.
It is worth pointing out that Migorski and Zeng provided two examples of operator N : V → V * in Problem MZ, which satisfies the hypotheses H(N ); see [18], Remark [14]. Inspired by Wang et al. [10] (Lemma 3.6), we first present an important result.

Proposition 4.
Assume that hypotheses H(J) (i), H(J) (iii) and H(J) (iv) hold. Then, for any sequences x n ∈ E converging strongly to x ∈ E, u n = (u n 1 , u n 2 ) ∈ X converging strongly to u = (u 1 , u 2 ) ∈ X and v n l ∈ X l converging strongly to v l ∈ X l , lim sup n→∞ J Proof. Let x n ∈ E converge strongly to x ∈ E, u n = (u n 1 , u n 2 ) ∈ X converge strongly to u = (u 1 , u 2 ) ∈ X and, for l = 1, 2, v n l ∈ X l converge strongly to v l ∈ X l . Note that, the CGDD of J(x n , ·, u n 2 ) at u n 1 in the direction v n 1 is formulated as For each integer n ≥ 1, by the definition of the limsup, there exist w n 1 ∈ X 1 and t n > 0 such that In terms of hypotheses H(J) (i) and H(J) (iii), we have where L u 1 is the local Lipschitz constant of functional J(x n , ·, u 2 ) at u 1 . It follows from the above inequalities and H(J) (iv) that Taking the limsup as n → ∞ at both sides of the last inequality yields Similarly, we can prove that This completes the proof.
First of all, we claim that condition H(J) ensures the u.s.c. of the PCGS ∂ l J for l = 1, 2.

Lemma 3.
Assume that H(J) holds. Then for l = 1, 2, the PCGS mapping from E × X equipped with the norm topology to the subsets of X * l equipped with the weak topology.
Proof. According to Proposition 1, it is sufficient to show that for each weakly closed Hence, by the reflexivity of X * l , without loss of generality, we may assume that ξ n l → ξ l weakly in X * l . The weak closedness of D l guarantees that ξ l ∈ D l . On the other hand, from Lemma 1 (v) we know that ξ n l ∈ ∂ l J(y n , x n ) entails ξ n l , z l X * l ×X l ≤ J • l (y n , x n ; z l ), ∀z l ∈ X l .
Utilizing Proposition 4 and passing to the limsup as n → ∞, we deduce that for all z l ∈ X l . Thus ξ l ∈ ∂ l J(y, x), and hence, one gets ξ l ∈ ∂ l J(y, x) ∩ D l , that is, In the rest of this paper, the range of variable t is always assumed to be the a.e. t ∈ (0, T). For the convenience, we naturally omit the description of the a.e. t ∈ (0, T). It is clear that the AS is equivalent to the problem below.
Next, we prove the existence of a mild solution to the RAS by using the Rothe rule along with the feedback iteration technique.
For N ≥ 1, we put τ = T N and t i = iτ for i = 0, 1, ..., N, and formulate the HIS below.
Obviously, this system is constituted by a stationary system of PCGS inclusions along with a system of abstract integral equations.
Next, for the convenience, let the → and denote the strong convergence and weak convergence, respectively.
where (x, u, ξ) ∈ C(0, T; E) × W × X * is a solution to the RAS (in terms of Definition 1).
Proof. Since V and H are reflexive, using (27)-(29) we might assume that, ∃u, u ∈ V s.t., (34) is valid and u τ u in V, as τ → 0. Meanwhile, simple calculations yield This along with the bound in (31) implies Noticing u τ u in V and utilizing (34), one hasū τ − u τ u − u in V as τ → 0. Besides, since the embedding V → V * is continuous, one getsū τ − u τ u − u in V * . Hence, one has u = u, that is, (35) is valid.
Because the u τ given in (Lemma 3.6) are of boundedness in V, we know that ∃u * ∈ V s.t. u τ u * in V as τ → 0. Meanwhile, simple calculations lead to This implies that u τ − u τ → 0 V * as τ → 0. In a similar way, we can derive u * = u. Besides, (31) ensures that, ∃w * ∈ V * s.t.
Hence, (x, u, ξ) ∈ C(0, T; E) × W × X * is a mild solution to the RAS in terms of Definition 3.1.
It is remarkable that the class of differential hemivariational inequalities (DHVIs) in [18] is extended to develop our general class of differential hemivariational inequalities systems (SDHVIs) by virtue of the partial Clarke's generalized subgradient operator. We first establish the upper semicontinuity of the partial Clarke's generalized directional derivatives (see our Proposition 4), and then extend the results for DHVI in [18] to the setting of SDHVI. Our main results can be applied to a special case of our abstract system (AS), where locally Lipschitz J and functions F l , l = 1, 2 are supposed to be independent of x. Thus, the AS reduces to the parabolic-type SHVI (i.e., problem (2.11)), which is a generalization of the parabolic-type HVI in [14] (i.e., problem (2.12)). In this case, the main results in [18] can not be applied to problem (2.11) because the criteria are not valid for it.
It is worth pointing out that there are the obvious disadvantages of the method based on the KKM approach for studying generalized parabolic or evolutionary SHVIs. In fact, if the operators in the method based on the KKM approach are not the KKM mappings, there are several possibilities which happen in the demonstrating process, e.g., in particular, whenever studying generalized parabolic or evolutionary SHVIs. In this article, when we deal with the parabolic-type SHVI in the demonstration process, the surjective theorem for pseudomonotonicity mappings, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, ensures the successful continuation of our demonstration. This overcomes the drawback of the KKM-based approach. Hence, this shows that the surjective theorem for pseudomonotonicity mappings enjoys a highlighted contribution to the study of SDHVI from the viewpoint of methodology.
The unique findings of the article are specified below. First, we make use of the backward Euler difference formula (i.e., the Rothe rule) to investigate the parabolic-type SHVI driven by the abstract SEE. It is worth mentioning that, for the first time, the Rothe rule was used in [18] to study the parabolic-type HVI driven by the abstract EE. Up to now, there have been a few papers devoted to the Rothe rule for HVIs, see [21]. It is worth emphasizing that these were focused on only a single HVI via the Rothe rule.
Second, the main results can be applied to a special case of the AS, where locally Lipschitz J and functions F l , l = 1, 2 are supposed to be independent of x. Thus, the AS reduces to the parabolic-type SHVI (i.e., problem (2.11)), which is a generalization of the parabolic-type HVI (i.e., problem (2.12)). Without question, the main results in [18] can not be applied to problem (2.11). This is exactly the utility of our obtained results.
Third, to the best of our knowledge, except for the DHVI considered in [18], many works on the DVIs were promoted only by elliptic-type VIs/HVIs. For the first time, we consider the SDHVI driven by the parabolic-type SHVI. In addition, except for the DHVI considered in [18], in comparison with the previous works [11,16,17,19], we assume no convexity condition on the functions u → f l (t, x, u), l = 1, 2 and no compactness condition on C 0 -semigroups e A l (t) , l = 1, 2.

Conclusions
In this article, under very suitable conditions, we take advantage of the Rothe rule to deal with the parabolic-type SHVI driven by the abstract SEE. For the first time, the Rothe rule was applied in [18] to study the parabolic-type HVI driven by the abstract EE. It is worth emphasizing that there have been a few papers devoted to the Rothe rule for HVIs, see [20]. However, these paid attention to only a single HVI by means of the Rothe rule.
As mentioned above, a particular case of our main theorem is an extension of ([18], Theorem 3.1) for the parabolic-type HVI driven by the abstract EE. Moreover, a special case of the one in [18] is also an extension of only a single parabolic-type HVI in [23]. An HVI is known as parabolic or evolutionary HVI if it involves the time derivative of unknown function. To the best of our knowledge, it will be quite extraordinary and very interesting to explore under what conditions the results in this article are still true for a generalized parabolic or evolutionary SHVI driven by the abstract SEE.