A class of fractional differential hemivariational inequalities with application to contact problem

In this paper, we study a class of generalized differential hemivariational inequalities of parabolic type involving the time fractional order derivative operator in Banach spaces. We use the Rothe method combined with surjectivity of multivalued pseudomonotone operators and properties of the Clarke generalized gradient to establish existence of solution to the abstract inequality. As an illustrative application, a frictional quasistatic contact problem for viscoelastic materials with adhesion is investigated, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals, and the constitutive relation is modeled by the fractional Kelvin–Voigt law.


Introduction
The fractional calculus, as a natural generalization of the classical integer order calculus, provides a precise description of some physical phenomena for viscoelastic materials, for example, fractional Kelvin-Voigt constitutive laws and fractional Maxwell model [16,42,51]. Recent advances in the fractional calculus concern the fractional derivative modeling in applied science, see [2,9,38], the theory of fractional differential equations, see [21], numerical approaches for the fractional differential equations, see [26,55] and the references therein. Another hot issue is the theory of hemivariational inequalities which is based on properties of the Clarke generalized gradient, defined for locally Lipschitz functions. This theory has started with the works of Panagiotopoulos, see [39,40], and has been substantially developed during the last 30 years. The mathematical results on hemivariational inequalities have found numerous applications to mechanics, physics and engineering, see [4,14,33,35,37,46,49,50] and the references therein. In this paper, we combine these hot issues and initiate a study of a class of differential hemivariational inequalities of parabolic type involving the time fractional order derivative operator in Banach spaces.
Systems consisting of variational inequalities and differential equations were introduced initially by Aubin and Cellina [1] in 1984. From another point of view, they were firstly considered and systematically studied in a framework of finite-dimensional spaces by Pang and Stewart [41] in 2008. They named this complex system a differential variational inequality ((DVI), for short). They also indicated the applications of DVI to several areas involving both dynamics and constraints in the inequality form, for example, mechanical impact problems, electrical circuits with ideal diodes, the Coulomb frictional problem in contact mechanics, economical dynamics and related models such as dynamic traffic networks. Since then, many scientists have contributed to the development of (DVI). In 2013 Liu et al. [24] employed the topological degree theory for multivalued maps and the method of guiding functions to establish the existence and global bifurcation behavior for periodic solutions to a class of differential variational inequalities in finite-dimensional spaces. In 2014 Chen and Wang [8], using the idea of (DVI), have solved the dynamic Nash equilibrium problem with shared constraints, which involves a dynamic decision process with multiple players. Subsequently, Ke et al. [20] in 2015 investigated a class of fractional differential variational inequalities with decay term in finite-dimensional spaces, for details on this topic in finite-dimensional spaces, we refer to [7,13,22,23,25,32,48] and the references therein. It should be pointed out that all results in the aforementioned papers were considered only in finite-dimensional spaces. Being motivated by many applied problems in engineering, operations research, economics, and for all v ∈ V and a.e. t ∈ (0, T ). This problem has been recently studied by Zeng and Migórski [54]. Third, the current paper initiates the study of a quasistatic contact problem for a viscoelastic body with adhesion and the fractional Kelvin-Voigt constitutive law, in which the friction and contact conditions are both described by the Clarke generalized gradient of nonconvex and nonsmooth functionals involving adhesion.
Fourth, for our problem, if we are restricted to the case α = 1, then Problem 1 reduces to the following differential hemivariational inequality of parabolic type: find u ∈ AC(0, T ; V ) and β ∈ W 1,2 (0, In this situation, the corresponding contact problem, see Problem 17, becomes a frictional viscoelastic contact problem with adhesion described by the classical Kelvin-Voigt constitutive law. It is obvious that the contact problem under consideration has the form of a differential hemivariational inequality. The paper is organized as follows. In Sect. 2, we recall notation and auxiliary materials. Section 3 establishes a result on solvability to a class of fractional differential hemivariational inequality by using the Rothe method and a surjectivity theorem for multivalued pseudomonotone operators. Finally, in Sect. 4, we consider a quasistatic fractional viscoelastic contact model with adhesion, and then apply the theoretical results from Sect. 3 to obtain the weak solvability to the contact problem.

Preliminaries
In this section we recall the basic notation and preliminary results which are needed in the sequel, see [10,12,21,26,42,52]. We start by recalling important and useful properties of the fractional integral and the Caputo derivative operators, for more details, we refer to [21,42]. Proposition 3. Let X be a Banach space and α, β > 0. Then, the following statements hold (a) for y ∈ L 1 (0, T ; X), we have 0 I α t 0 I β t y(t) = 0 I α+β t y(t) for a.e. t ∈ (0, T ), (b) for y ∈ AC(0, T ; X) and α ∈ (0, 1], we have for a.e. t ∈ (0, T ). We now recall definitions and results from nonlinear analysis which can be found in [10][11][12]33,52]. Let X be a reflexive Banach space and ·, · denote the duality of X and X * . A single-valued operator A : X → X * is pseudomonotone if A is bounded (it maps bounded sets in X into bounded sets in X * ) and for every sequence {x n } ⊆ X converging weakly to x ∈ X such that lim sup Ax n , x n − x ≤ 0, we have Obviously, an operator A : X → X * is pseudomonotone if and only if it is bounded, and x n → x weakly in X, and lim sup Ax n , x n − x ≤ 0 entails lim 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. Moreover, the notion of pseudomonotonicity of a multivalued operator is recalled below.

Definition 4.
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 u * n , u n − u ≤ 0, we have that for every v ∈ X, there exists u * (v) ∈ T u such that Let j : X → R be a locally Lipschitz function. 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, its proof can be found in [14,Proposition 5.6].

Lemma 5.
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 of γ. Let j : X → R be a locally Lipschitz function such that Furthermore, we recall the following surjective result, which can be found in [12,Theorem 1.3.70] or [52]. Theorem 6. 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 . From Theorem 6, we have the following corollary.

Corollary 7.
Let V be a reflexive Banach space. Assume that (i) A : V → V * is a pseudomonotone and strongly monotone operator, i.e., there exists Proof. Since A and U are pseudomonotone, it follows from [33, Proposition 3.59(ii)] that A + U is pseudomonotone as well. Having in mind Theorem 6, it remains to prove that A + U is coercive. Indeed, we have The smallness condition c U < c A guarantees that A + U is coercive. Therefore, from Theorem 6, we conclude that A + U is surjective, which completes the proof of the corollary.

Hypothesis (ii) implies that
Combining the latter with hypotheses (i) and (iii), we deduce that the function On the other hand, by hypothesis (ii), for all y 1 , y 2 ∈ Y , we get for a.e. t ∈ (0, T ), i.e., F u (t, ·) is Lipschitz continuous for a.e. t ∈ (0, T ). Therefore, all conditions of [15,Theorem 9.9,p.198] are verified. By applying this theorem, we conclude that there exists a unique function β ∈ W 1,2 (0, T ; Y ) such that (3) holds. We now prove inequality (4). In fact, it is clear that, for any u ∈ L 2 (0, T ; X) fixed, the unique function β ∈ W 1,2 (0, T ; Y ) has the form The Gronwall inequality (see, e.g., [47, Lemma 2.31, p.49]) entails This means that (4) holds with constant c β = L F (1 + T L F e LF T ), which completes the proof of the lemma.
We conclude this section by recalling the generalized discrete version of the Gronwall inequality which proof can be found in [43, Lemma 2].

Then, we have
Moreover, if {u n } and {w n } are such that

Fractional differential hemivariational inequality
In this section, we focus our attention to the abstract differential hemivariational inequality involving fractional derivative operator, Problem 1, and provide a result on existence of solutions for this inequality. The method of proof relies on a surjectivity result for multivalued pseudomonotone operators and the Rothe method. To provide readers with better convenience, we now introduce the standard notation following [11,12,52]. Let V be a reflexive and separable Banach space with dual space V * . Subsequently, we use the symbols ·, · and · to stand for the duality pairing between V * and V , and a norm in V , respectively. Let 0 < T < +∞. We use the standard Bochner-Lebesgue function space V = L 2 (0, T ; V ). Recall that since V is reflexive, it is obvious that both V and its dual space V * = L 2 (0, T ; V * ) are reflexive Banach spaces. The notation ·, · V * ×V stands for the duality between V and V * . Let X and Y be other separable and reflexive Banach spaces, X = L 2 (0, T ; X) and X * = L 2 (0, T ; X * ). In the rest of the paper, we denote by C a constant whose value may change from line to line.
Let u ∈ AC(0, T ; V ) be a solution to Problem 1 and ∈ (0, T ). Therefore, Problem 1 can be rewritten as Observe that the above problem can be reformulated as the following inclusion problem driven by a fractional integral operator and a nonlinear differential equation.
We now impose the following assumptions on the data of Problem 11.
x, y) is Lipschitz continuous, i.e., there exists a constant L g > 0 such that for all (x 1 , y 1 ), (x 2 , y 2 ) ∈ X × Y and a.e. t ∈ (0, T ), we have In fact, hypothesis H(J) guarantees that the subgradient operator ∂J of J(y, ·) is upper semicontinuous.

Lemma 12. Under hypothesis H(J) the subgradient operator
is upper semicontinuous from Y × X endowed with the norm topology to the subsets of X * endowed with the weak topology.
Proof. From [11,Proposition 4.1.4], it is sufficient to show that for any weakly closed subset D of X * , the Hence, from 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 ) reveals 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 This completes the proof of the lemma.
Consider the following discretized problem corresponding to Problem 11 called the Rothe problem. and respectively. Here χ (ti−1,ti] stands for the characteristic function of the interval (t i−1 , t i ], i.e., First, we shall show the existence of solution to Problem 13. Proof. Given w 0 τ , w 1 τ , . . . , w n−1 τ , we will prove that there exist w n τ ∈ V , ξ n τ ∈ X * and a function β τ ∈ W 1,2 (0, t n ; Y ) such that (6) and (7) hold.
It remains to show that there exist elements w n τ ∈ V and ξ n τ ∈ X * such that equality (7) holds. Denote .
To this end, we will show that the multivalued operator is bounded, continuous, and fulfills the condition is strongly monotone and c 0 c J M 2 + c 0 B < m A for all τ ∈ (0, τ 0 ). We are now in a position to apply Corollary 7 to deduce that operator v → Av is surjective for all 0 < τ < τ 0 . Therefore, we conclude that there exist elements w n τ ∈ V and ξ n τ ∈ X * such that equation (7) holds. This completes the proof of the lemma.
The following result provides estimates for the sequence of solutions of the Rothe problem, Problem 13.
where ξ k τ ∈ X * is such that ξ k τ ∈ ∂J(β τ (t k ), M(u k τ )) and Proof. Taking k = n in (7), we multiply equation (7) by w n τ to get Aw n τ , w n τ + Bu n τ , w n τ + ξ n τ , Mw n τ X×X * = f n τ , w n τ . From definition of u n τ (see (8)) and hypothesis H(B), we have It follows from the growth condition H(J)(ii) that From the coercivity of operator A and inequalities (13) and (14), we get and subsequently ZAMP A class of fractional differential hemivariational Page 11 of 23 36 Next, from hypothesis H(f ), there exists a constant m f > 0 such that f n τ V * ≤ m f for all τ > 0 and n ∈ N. Setting we are in a position to apply the generalized discrete Gronwall inequality, Lemma 9, to see that Hence, the estimate (10) is verified. Furthermore, by equality (8), the estimate (11) is easily obtained from the following inequality .  ∈ (0, α) and {τ n } be a sequence such that τ n → 0, as n → ∞. Then, for a subsequence still denoted by τ , we have Proof. From the estimate (10), we have

Theorem 16. Assume that H(A), H(B), H(J), H(g), H(f ), and H(M ) hold. Let η
Hence, we deduce that {w τ } is bounded in L 1 η (0, T ; V ). Therefore, without loss of generality, we may assume that there exists w ∈ L 1 η (0, T ; V ) such that For any v * ∈ V * and t ∈ [0, T ], let e(s) Therefore, we have for all t ∈ [0, T ]. Moreover, using estimate (10) again, one has for t ∈ (t n−1 , t n ]. So, we conclude Combining the latter and convergence (16), we obtain for all t ∈ [0, T ]. Since the operator M is compact, we get for all t ∈ [0, T ]. Analogously, for functions u τ given by (9) and u τ , we have . This inequality together with convergence (18) and the compactness of M implies We are now in a position to apply Lemma 8 to deduce that there exists a unique solution β ∈ W 1,2 (0, T ; Y ) such that for all t ∈ [0, T ]. By hypothesis H(g)(ii) and Lemma 8, we have We use convergence (20), estimate (10), and the Lebesgue-dominated convergence theorem, see, e.g., [33,Theorem 1.65], to conclude that β τ converges to β in C(0, T ; Y ). On the other hand, estimate (12) guarantees that the sequence {ξ τ } is bounded in X * . So, passing to a subsequence if necessary, there exists ξ ∈ X * such that By Lemma 12, we know that the mapping (y, x) → ∂J(y, x) is upper semicontinuous from Y × X into X * endowed with weak topology. Using this property, the relation ξ k τ ∈ ∂J(β τ (t k ), M(u k τ )) for k = 1, 2, . . . , N, and convergences β τ → β in C(0, T ; Y ), (19) and (21), by [33,Theorem 3.13], we deduce that ξ(t) ∈ ∂J(β(t), M(u 0 + 0 I α t w(t))) for a.e. t ∈ (0, T ). Subsequently, we consider the Nemytskii operators A and B corresponding to A and B, which are defined by

A fractional viscoelastic contact problem with friction and adhesion
In this section, the abstract theoretical results of Sect. 3 will be used to study a frictional contact problem for a viscoelastic body with time fractional Kelvin-Voigt constitutive law and adhesion.
The physical formulation of the fractional viscoelastic contact problem is provided below. We consider a viscoelastic body which occupies a domain Ω ⊂ R d , where d = 2, 3. The boundary Γ = ∂Ω is assumed to be Lipschitz continuous, and it is divided into three disjoint measurable parts Γ D , Γ N and Γ C with meas (Γ D ) > 0. The contact problem will be discussed in a finite time interval (0, T ).
For convenience of the reader, the description of basic notation is provided in Table 1.
The inner products and corresponding norms in R d and S d are denoted by respectively. Also, we denote For simplicity, we do not indicate explicitly the dependence of various functions and operators on x. The classical formulation of the mechanical contact problem is described as follows.
Problem 17. Find a displacement field u : Q → R d , a stress field σ : Q → S d and a bonding field β : Σ C → [0, 1] such that Div The unit outward normal vector x ∈ Ω = Ω ∪ Γ A position vector indices i, j, k, l They run from 1 to d and the summation convention over repeated indices is used S d The space of second order symmetric tensors on The normal component of stress field σ on Γ σ τ = σ ν − σν ν The tangential component of stress field σ on Γ uν = u · ν The normal component of the displacement field u on Γ u τ = u − uν ν The tangential component of the displacement field u on Γ Divσ = (σ ij,j ) The divergence of σ , σ ij,j = ∂σ ij ∂x j We now give a brief description of equations and relations in Problem 17. The generalized fractional Kelvin-Voigt constitutive law of the Caputo type, see [54], for viscoelastic body is given in (22). Operators C and E stand here for the viscosity and elasticity operators, respectively. Note that since the contact process is assumed to be quasistatic, the acceleration term is negligible and we deal in Q with equilibrium equation (23), where f 0 denotes the time dependent density of volume forces. Moreover, conditions (24) and (25) reveal the displacement and traction boundary conditions on parts Γ D and Γ N of the boundary, respectively, i.e., the body is fixed on Γ D and it is subjected to the time dependent surface traction of density f N on Γ N .
The unknown function β is a surface internal variable, which is usually called the bonding field or the adhesion field. It describes the pointwise fractional density of active bonds on the contact surface. The evolution of the bounding field is driven by a nonlinear ordinary differential equation (28) depending on the displacement, and considered on contact surface Γ C . Furthermore, if β = 1 at a point of the contact part, the adhesion is complete and all the bonds are active, and β = 0 means that all bonds are inactive and there is no adhesion. But, when 0 < β < 1 then the adhesion is partial and a fracture β of the bonds is active. The function β 0 denotes the initial bonding field in (29). For more details on the adhesion phenomena, see [3,6,15,36].
The contact condition (26) with adhesion is called a multivalued normal compliance contact boundary condition, which is described by the subgradient of a nonconvex function j ν , where j ν is assumed to be locally Lipschitz with respect to the last variable. On the other hand, the general tangential contact condition (27) with adhesion, i.e., friction contact condition with adhesion, is governed by the subgradient of a nonconvex function j τ . In fact, this contact condition without the bonding field has been treated in many papers, see, e.g., [15,34,35,44,45]. The initial displacement is given in (30). For more details on the mathematical theory of contact mechanics, we refer to [33,37,44,45].
Subsequently, we obtain the variational formulation of Problem 17. We will use the function spaces V , H and H defined by The trace of an element v ∈ H 1 (Ω; R d ) is denoted by the same symbol v. It is obvious that H is endowed with the Hilbertian structure by the inner product and the associated norm · H . For space V , we consider the inner product by and the associated norm · V . Recall that, since meas (Γ D ) > 0, we know that V is a real Hilbert space. From the Sobolev trace theorem, there exists c k > 0 (the Korn constant) such that where γ denotes the norm of the trace operator γ : V → L 2 (Γ C ; R d ).

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A class of fractional differential hemivariational Page 17 of 23 36 In the study of Problem 17, the viscosity operator C : Ω×S d → S d and elasticity operator E : Ω×S d → S d satisfy the following hypotheses.
x ∈ Ω and all ε ∈ S d , x ∈ Ω and all ε ∈ S d , The normal potential j ν : Γ C × R × R → R and tangential function j τ : is locally Lipschitz on R for all r ∈ R and a.e. x ∈ Γ C , is regular for a.e. x ∈ Γ C and r ∈ R, (e) (r, s) → j 0 ν (x, r, s; z) is upper semicontinuous for all z ∈ R and a.e. x ∈ Γ C , where j 0 ν denotes the Clarke derivative of s → j ν (x, r, s) in direction z. , ξ) is measurable on Γ C for all (r, ξ) ∈ R × R d and j τ (·, 0, 0) ∈ L 1 (Γ C ), (b) j τ (x, r, ·) is locally Lipschitz on R d for all r ∈ R and a.e. x ∈ Γ C , is regular for a.e. x ∈ Γ C and r ∈ R, r, ξ; η) is upper semicontinuous for all η ∈ R d and a.e. x ∈ Γ C , where j 0 τ denotes the Clarke derivative of ξ → j τ (x, r, ξ) in direction η. (35) In conditions (34)(c) and (35)(c), the symbols ∂j ν and ∂j τ stand for the Clarke generalized gradient of j ν and j τ with respect to their last variables, respectively. Subsequently, if we suppose that (34)(d) and (35)(d) hold, we mean that "either j ν (x, r, ·) and j τ (x, r, ·) are regular" or "either −j ν (x, r, ·) and −j τ (x, r, ·) are regular" for all r ∈ R and a.e. x ∈ Γ C . Note that examples of functions which satisfy conditions (34) and (35)  The initial conditions, densities of volume forces and surface tractions satisfy the following regularity hypotheses.
The adhesive evolution rate function F satisfies the following condition.
We now focus on the variational formulation of the contact problem (22)- (30). We suppose in what follows that (u, σ) are smooth functions on Q which solve (22)- (30). For any v ∈ V fixed, we multiply equilibrium equation (23) by v and then use the Green formula, cf. [33,Theorem 2.25] to get and applying boundary conditions (24) and (25), we have for a.e. t ∈ (0, T ), It follows from the Riesz representation principle that there exists an element f ∈ V * such that for all v ∈ V , a.e. t ∈ (0, T ). From the decomposition formula, see (6.33) in [33], we obtain for a.e. t ∈ (0, T ). On the other hand, by contact conditions (26), (27), and the definition of the subgradient, we obtain Putting the fractional Kelvin-Voigt constitutive law (22), and inequalities (39) into (38), we have for a.e. t ∈ (0, T ). Finally, using conditions (28)- (30) and the last inequality, we obtain the following variational formulation of Problem 17.
To this end, we will verify that hypotheses H (A), H(B), H(f ), H(J), H(g) and H(M ) of Theorem 16 are satisfied. Obviously, from hypothesis (32), we can see that the operator A, see (41), is coercive with constant m a and A ∈ L(V, V * ), i.e., H(A) holds. Note that, since the elastic operator E satisfies properties (33), this yields that B ∈ L(V, V * ), i.e., H(B) is verified. Moreover, hypotheses (34) and (35) combined with [33,Corollary 4.15(v)] imply that the conditions H(J)(i) and (ii) are satisfied with c J = max{ 3 meas(Γ C ), 1}(c ν + c τ ). The upper semicontinuity of j ν and j τ , and Fatou's lemma, see, e.g., [33,Theorem 1.64], guarantee that the function (β, u) → J 0 (β, u; v) is also upper semicontinuous from Y × X to R, for all v ∈ X. So, J has the property H(J)(iii). In addition, it follows from the regularity hypothesis (36) that f satisfies H(f ). From [12,Theorem 3.9.34], we infer that the trace operator γ satisfies H(M ). Finally, it is easy to verify that under hypothesis (37), operator g defined by (46)

ZAMP
A class of fractional differential hemivariational Page 21 of 23 36 Clearly, Problem 20 has the following variational formulation, which is a classical differential hemivariational inequality.
As a consequence of Theorem 19, we conclude the following result.
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