Hidden maximal monotonicity in evolutionary variational-hemivariational inequalities

In this paper, we propose a new methodology to study evolutionary variational-hemivariational inequalities based on the theory of evolution equations governed by maximal monotone operators. More precisely, the proposed approach, based on a hidden maximal monotonicity, is used to explore the well-posedness for a class of evolutionary variational-hemivariational inequalities involving history-dependent operators and related problems with periodic and antiperiodic boundary conditions. The applicability of our theoretical results is illustrated through applications to a fractional evolution inclusion and a dynamic semipermeability problem.


Introduction
Variational and hemivariational inequalities serve as theoretical models for various problems arising in mechanics, physics, and engineering sciences. The representative literatures in the field include [1,4,10,11,[13][14][15][17][18][19]21,23,24]. On the one hand, the theory of variational inequalities uses monotonicity and convexity as its main tools, including the properties of the subdifferential of a convex function and maximal monotone operators. On the other hand, the theory of hemivariational inequalities is based on the features of the subdifferential in the sense of Clarke defined for locally Lipschitz functions, which may be nonconvex.
Observantly, variational-hemivariational inequalities represent an intermediate class of inequalities in which both convex and nonconvex features are involved. Interest in their study is motivated by various problems in mechanics as discussed in [5,8,9,12,16,21,25]. It should be mentioned that the study of evolutionary variational-hemivariational inequalities has been performed typically through surjectivity results for pseudomonotone operators and fixed point theorems for nonlinear operators (see, e.g., [21] and the references therein). However, this paper aims to propose a new approach to study evolutionary variational-hemivariational inequalities based on the theory of evolution problems governed by maximal monotone operators. Indeed, the proposed method is quite different from the previous literature and is not based on surjectivity results for pseudomonotone operators.
Let (H, ·, · ) be a separable Hilbert space, Y a Banach space, and I = [0, T ] for some T > 0 fixed. In this paper, we study and provide new applications to PDEs for the following class of evolutionary variational-hemivariational inequalities involving historydependent operators: − ∂ c ϕ t, S(x)(t), x(t) , a.e. t ∈ I, where f ∈ L 2 (I; H), J : I × H → R is a function, A : I × H → H is a nonlinear operator, ϕ : I × Y × H → R is a given function, and R, S are two history-dependent operators (see Definition 1 below) in which we refer to Section 4 for the precise hypothesis. Problem (1) was studied in [7] (see also [21,Chap. 7]) in the framework of evolution triple of spaces by using surjectivity results for pseudomonotone operators and a fixed point theorem for nonlinear operators. A key assumption to apply the surjectivity result is the so-called relaxed monotonicity for the subdifferential in the sense of Clarke (see Definition 2 below), which is a weaker notion than monotonicity, but which still permits to obtain the existence of solutions. We characterize this notion in terms of the convexity of an associated function (see Section 3). Then we consider the differential inclusioṅ x(t) ∈ f (t) − ∂J t, (t) − A t, x(t) − ∂ c ψ t, x(t) , a.e. t ∈ I, where ψ : I × H → R is a given function. We prove that the latter problem is, in fact, an evolution equation governed by a set-valued operator, which is a maximal monotone operator. Whereas the existence can be obtained through a recent result on the subject [22]. As a by-product, we obtain the existence for the periodic and antiperiodic version of (2). Moreover, we prove that every trajectory of the Cauchy problem (2) converges asymptotically to a periodic solution of (2). The contribution of this paper is threefold. First, we show that some evolutionary variational-hemivariational inequalities can be handled with the theory of evolution problems governed by maximal monotone operators. Second, we extend the results of [7] to the general functional setting. Finally, the applicability of our theoretical results is illustrated through applications to the study of a fractional evolution inclusion and a dynamic semipermeability problem.
The paper is organized as follows. After some preliminaries, in Section 3, we provide an impressive characterization of the relaxed monotonicity property, and then we prove the maximal monotonicity of the sum of operators that appear on the right-hand side of (2). Then, in Section 4, we establish the well-posedness for problems (1) and (2), respectively. Finally, in Section 5, we illustrate the applicability of our theoretical results to the study of a fractional evolution inclusion and a complicated dynamic semipermeability problem, respectively.

Elements of convex and variational analysis
Let (H, ·, · ) be a separable Hilbert space. We denote by B the unit closed ball with center at the origin in H. Given a set-valued map A : H ⇒ H, we denote by D(A) and Gr (A), respectively, the domain and the graph of A defined by D(A) := {x ∈ H: A(x) = ∅} and Gr (A) := {(x, y) ∈ D(A) × H: y ∈ A(x)}. We say that an operator A : H ⇒ H is monotone if x * − y * , x − y 0 for all x * ∈ A(x), y * ∈ A(y). Moreover, an operator A : H ⇒ H is maximal monotone if it is monotone and its graph is maximal in the sense of the inclusion, i.e., Gr (A) is not properly contained in the graph of any other monotone operator. We refer to [2] for more details on maximal monotone operators.
The Clarke subdifferential of a locally Lipschitz function f : It is well known that for a proper, convex, and lower semicontinuous function, the convex subdifferential defines a maximal monotone operator. Moreover, for a convex and locally Lipschitz function, the convex subdifferential coincides to the Clarke subdifferential (see, e.g., [3]).
The following result is an important characterization of convexity. We refer to [3, Prop. 2.2.9] for its proof. Proposition 1. Let (H, ·, · ) be a Hilbert space. Let f : C → R be a locally Lipschitz function in an open convex set C ⊂ H. Then f is convex on C if and only if the multifunction ∂f is monotone on C, that is, if and only if for all x, y ∈ C, x * − y * , x − y 0 for all x * ∈ ∂f (x) and y * ∈ ∂f (y).
Proposition 2. Let Y be a Banach space and consider ϕ : I × Y × H → R be a function such that for a.e. t ∈ I, the map x → ϕ(t, y, x) is convex and lower semicontinuous on H. Assume that for all y 1 , y 2 ∈ Y and x 1 , x 2 ∈ H, it holds Then, for all ξ 1 ∈ ∂ c ϕ(t, y 1 , x 1 ) and ξ 2 ∈ ∂ c ϕ(t, y 2 , x 2 ), the following inequality holds: Proof. Let ξ 1 ∈ ∂ c ϕ(t, y 1 , x 1 ) and ξ 2 ∈ ∂ c ϕ(t, y 2 , x 2 ). Then, according to the definition of the convex subdifferential, for all x ∈ H, we have Hence, taking x = x 2 and x = x 1 in the inequalities above, respectively, and summing the resulting inequalities, we get Hence, we get the desired inequality.
Definition 1. Let X, Y be normed spaces. An operator F : L 2 (I; X) → L 2 (I; Y ) is called a history-dependent operator if there exists L > 0 such that for all v 1 , v 2 ∈ L 2 (I; X), The following result is an essential fixed point property for history-dependent operators (see, e.g., [21, p. 118]). Theorem 1. Let X be a Banach space and F : L 2 (I; X) → L 2 (I; X) be a historydependent operator. Then F has a unique fixed point.
We end this subsection with a technical lemma related to differential inequalities. Lemma 1. Let x 1 , x 2 : I → H be two absolutely continuous functions such that where α : I → R is a nonnegative function. Then it holds Proof. Let us consider the sets Ω 1 := {t ∈ I: x 1 (t) = x 2 (t)} and Ω 2 := {t ∈ I: x 1 (t) = x 2 (t)}. On the one hand, for a.e. t ∈ Ω 1 , we have which implies the desired inequality. On the other hand, for any t ∈ Ω 2 , we can see that the map t → x 1 (t) − x 2 (t) attains a minimum. Thus, for a.e. t ∈ Ω 2 , we have (d/dt) x 1 (t) − x 2 (t) = 0, which implies the desired inequality. The proof is then complete.
Also, we recall the following lemma (see [20]), which will be used in Section 5.
Lemma 2. Let s ∈ (0, 1) and Ω be a bounded, open subset of R N with Lipschitz boundary and N > 2s. Then we have

Technical assumptions and hidden maximal monotonicity
For the sake of readability, furthermore, we collect the hypotheses used along with the paper.
Hypotheses on the operator A : for all x, y ∈ H and a.e. t ∈ I.
Hypotheses on the function J : H for all x * ∈ ∂J(t, x) and y * ∈ ∂J(t, y) and a.e. t ∈ I.
Here ∂J denotes the Clarke subdifferential of the map x → J(t, x) for a fixed t ∈ I.
Hypotheses on the function ψ : I × H → R R R (H ψ ) The function ψ : I × H → R satisfies: Hypotheses on the function ϕ : (i) for all x ∈ H, y ∈ Y , the map t → ϕ(t, y, x) is measurable on I; (ii) for a.e. t ∈ I, for x ∈ X, the map y → ϕ(t, y, x) is continuous; (iii) for a.e. t ∈ I, for y ∈ Y , the map x → ϕ(t, y, x) is convex and l.s.c. on H.
(iv) there exist c 0ϕ ∈ L 2 + (I) and c 1ϕ , c 2ϕ 0 such that for all y ∈ Y , x ∈ H and a.e. t ∈ I, sup x * ∈∂cϕ(t,y, Here ∂ c ϕ(t, y, x) is the convex subdifferential of the map x → ϕ(t, y, x).
(v) There exists β ϕ 0 such that for all y 1 , y 2 ∈ Y and x 1 , x 2 ∈ H and a.e. t ∈ I, ϕ(t, y 1 , Hypotheses on the operators R and S (i) The operator R is a history-dependent, i.e., there exists c R > 0 such that Next, we characterize the so-called relaxed monotonicity condition for a locally Lipschitz function f : H → R sum of a quadratic term. With this result in hand, we prove the maximal monotonicity of the sum of the Clarke subdifferential of f plus an appropriate strongly monotone operator A, which can be understood as a hidden maximal monotonicity property.
Definition 2. We say that a locally Lipschitz function f : H → R satisfies the m-relaxed monotonicity condition if there exists m 0 such that Condition (3) has been used extensively in the literature, we refer to [6,13] for more details. The following result gives a characterization of m-relaxed monotonicity in terms of an associated convex function.
Proof. Assume that f satisfies the m-relaxed monotonicity condition. Then, due to calculus rules for the Clarke subdifferential, On the other side, suppose that the set-valued map x → ∂(f +m · 2 H /2)(x) is monotone. Hence, f satisfies the m-relaxed monotonicity condition, which ends the proof.
The following result shows that the relaxed monotonicity added with an appropriate strongly monotone operator generates a maximal monotone operator. Lemma 3. Let J : H → R be a locally Lipschitz function and A : H → H be a nonlinear operator such that: for all x, y, w ∈ H.
Hence, by virtue of (iii), the operator x → ∂J(x) is monotone. Therefore, due to Proposition 1, the function x →J(x) is convex, which implies that x → ∂J(x) is maximal monotone. On the other hand, due to (i) and (ii), the operator x →Ã(x) is monotone and hemicontinuous. Therefore, as a result of [2,Prop. 20.27], the map x →Ã(x) is maximal monotone. Finally, the maximal monotonicity of

Well-posedness results
In this section, we explore several well-posedness results for evolutionary variationalhemivariational inequalities.

Cauchy problems
In this subsection, we prove the existence of solutions for the following Cauchy problem: The following result provides the well-posedness for (4). Proof. We will employ [22, Thm. 1] to obtain the desired conclusion. So, the proof is divided into three steps.
Proof of Step 1. It follows directly from Lemma 3.
Step 2. For all x ∈ H, the operator t → ∂J(t, x) + A(t, x) + ∂ψ(t, x) is measurable. Proof of Step 2. The measurability can be obtained directly from the separability of H and hypotheses (H A )(i), (H J )(i), and (H ψ )(i).
for some functionsα,β in L 2 + (I). So, Theorem 2 holds too if we interchange the infimum by a supremum in (H J )(iv) and the supremum by a infimum in (H ψ )(iii).

Cauchy problems with history-dependent operators
In this subsection, we focus our attention on the study of the existence of solutions for the following Cauchy problem involving history-dependent operators: where R and S are two history-dependent operators, i.e., (H RS ) is satisfied. Proof. Fix v ∈ L 2 (I; H) and let us consider the intermediate problem: Our aim is to prove that (6) has a unique fixed point in W 1,2 (I; H), which is clearly a solution of (5). The proof is divided into several steps.
Step 2. The operator F is history-dependent. More precisely, for all v 1 , v 2 ∈ L 2 (I; H), we have Proof of Step 2. Denote x 1 := F(v 1 ) and x 2 := F(v 2 ). Let us consider ξ 1 and ξ 2 such that ξ i (t) ∈ ∂ c ϕ(t, S(v i )(t), x i (t)) for all t ∈ I and i = 1, 2, Then h is absolutely continuous, and it holdṡ where m := m A − m J 0, and we have used the monotonicity of the set-valued map x → ∂J(t, x) + A(t, x), Proposition 2, and hypotheses (H RS ). Therefore, by virtue of Lemma 1, for a.e. t ∈ I, we conclude (d/dt) which proves Step 2.
Step 3. Problem (5) has a unique solution x * ∈ W 1,2 (I; H). Proof of Step 3. Since F is a history-dependent operator, employing Theorem 1 implies that the operator F : L 2 (I; H) → W 1,2 (I; H) has a unique fixed point x * , which clearly solves (5).
https://www.journals.vu.lt/nonlinear-analysis Applying Grönwall's inequality, it finds therefore where the constant M > 0 only depends on c R , c S , β ϕ , and T . Consequently, we have which proves the Lipschitzianity of the solution operator.

Noninitial boundary value problems
In this subsection, we consider the existence of periodic solutions for the following differential inclusion problem: Theorem 4. Assume that (H A ), (H J ), and (H ψ ) hold. If m A > m J , then problem (7) has a unique solution x π ∈ W 1,2 (I; H).
Proof. Let us consider the operator F : H → H defined by F(x 0 ) = x(T ; x 0 ), where x(·, x 0 ) is the unique solution of (4) with the initial condition x(0) = x 0 . Our goal is to prove that F has a unique fixed point in H. Let x 1 := x(·, x 1 0 ) and x 2 = x(·, x 2 0 ). Keeping in mind that m A > m J and the operator x → ∂f (t, x) + A(t, x) + ∂ c ψ(t, x) is m strongly monotone with m := m A − m J , it gives Employing Grönwall's inequality yields Therefore, the operator F has a unique fixed point x 0,T ∈ H by the contractive fixed point theory. It is clear that x π := x(·, x 0,T ) is the unique solution of (7).
Likewise, we can consider the existence of antiperiodic solutions to the following differential inclusion problem: Using the same argument given in the proof of Theorem 4, it is easy to conclude the following result. We end this section by showing that the unique solution to (7) can be obtained asymptotically from any solution of (4).
. Let x(·, x 0 ) be the unique solution of (4) with the initial condition x(0) = x 0 and define x n (t) := x(t + nT ; x 0 ) for all t ∈ I and n ∈ N. Then, for any x 0 ∈ H, x n → x π in C (I; H), where x π is the unique periodic solution of (7).
Proof. Set m := m A − m J > 0, and let us consider h(t) := x n (t) − x π (t) 2 H /2 for t ∈ I and n ∈ N. Then, for a.e. t ∈ I, it givesḣ(t) = x n (t) − x π (t),ẋ n (t) −ẋ π (t) −mh(t), where we used the monotonicity of the map x → ∂J(t, x)+A(t, x)+∂ c ψ(t, x). Therefore, for all t ∈ I, it is true x n (t) − x π (t) H x n (0) − x π (0) H e −mT for all t ∈ I, where we have used the Grönwall inequality. Thus, by using the same inequalities, for all t ∈ I, we have which shows that x n → x in C(I; H).

Remark 3.
As showed in the previous proof, x n converges to x π exponentially.

Applications
To illustrate the applicability of the theoretical results established in Section 4, we will present two comprehensive applications. The first one is a fractional evolution inclusion problem involving a multivalued term, which is formulated by the Clarke generalized gradient. The second application is a dynamic semipermeability problem, which is, more precisely, a complicated mixed boundary value problem of parabolic type with historydependent operators and nonsmooth potential functionals.

Application to a fractional evolution inclusion problem
In the subsection, we are interested in the study of an evolutionary inclusion problem with a generalized nonlocal space-fractional Laplace operator and a Clarke generalized subgradient operator. Let Ω be a bounded domain in R N with Lipschitz boundary, s ∈ (0, 1) be such that N > 2s, Ω C := R N \ Ω, 0 < T < ∞, and I := [0, T ]. More precisely, the classical form of the evolutionary inclusion problem under consideration is formulated as follows.
where the operator L K stands for the generalized nonlocal space-fractional Laplace operator defined as follows: We first impose the following assumptions for the data of Problem 1.
The weak solutions to Problem 1 are understood as follows.
Definition 3. We say that u : I → X 0 is a weak solution to Problem 1 if u(x, 0) = u 0 (x) in Ω and the following inequality holds for all v ∈ X 0 : Let us define function J : I × X 0 → R by J(t, u) := Ω j(x, t, u(x)) dx for all (t, u) ∈ I × X 0 . For function J, we have the following lemma.
Lemma 4. Assume that (H j ) is fulfilled. Then the following statements hold: (ii) For a.e. t ∈ I, X 0 u → J(t, u) ∈ R is locally Lipschitz.
where the last inequality is obtained by using Lemma 2. Therefore, the desired inequality is valid.
The existence and uniqueness of weak solutions to Problem 1 is provided by the following result.
Theorem 7. Assume that H(K), H(j), H(0), and 1 m j c(2) 2 hold, then Problem 1 has a unique weak solution u(·, f, x 0 ) ∈ W 1,2 (I; X 0 ), and the solution operator T : Proof. Let H := X 0 . Consider the operator A : H → H defined for all u, v ∈ H by Au, v := R N v(x)L K (u(x)) dx. We now claim that A is a continuous linear operator. For any u, v ∈ H, it has It obvious to see that A is a linear continuous operator and Au H = u H . In addition, H for all u, v ∈ H implies that A is strongly monotone with constant m A = 1.
Let us consider the following intermediate problem: Employing Lemma 4(iii), we can see that a solution to problem (9) is also a weak solution to Problem 1. On the other hand, it is not difficult to verify that problem (9) is equivalent to the following evolutionary inclusion problem: a.e. t ∈ I, Observe that A : H → H reads hypothesis (H A ), hence, we are now in a position to invoke Lemma 4 and Theorem 2 that problem (10) has a unique solution u(·, f, u 0 ) ∈ W 1,2 (I; H). So, u is also a weak solution to Problem 1. For the uniqueness part of Problem 1, it can be obtained directly by using the standard procedure (see the proof of Theorem 2). Finally, the Lipschitz continuity of the solution operator T : (f, x 0 ) → u(·, f, x 0 ) could be verified by employing the same argument with the proof of Theorem 2.
Furthermore, we are going to apply the results established in Section 4.3 to investigate the fractional evolution inclusion problem, Problem 1, with periodic and antiperiodic boundary value conditions, respectively.
Likewise, the weak solutions to Problem 2 are given as follows.
Definition 4. We say that u : I → X 0 is a weak solution to Problem 2 if u(x, 0) = u(x, T ) in Ω and the following inequality holds for all v ∈ X 0 : Invoking Theorem 4 and the proof of Theorem 7, we have the following existence and uniqueness result for Problem 2.
We end the subsection by considering the antiperiodic boundary value problem.
Definition 5. We say that u : I → X 0 is a weak solution to Problem 3 if u(x, 0) = −u(x, T ) in Ω and the following inequality holds for all v ∈ X 0 : Analogously, from Theorem 5 and the proof of Theorem 7, we have the following theorem.

Application to a dynamic semipermeability problem
The semipermeability boundary conditions can describe exactly behavior of various types of membranes, natural and artificial ones, and arise in models of heat conduction, electrostatics, hydraulics and in the description of the flow of a Bingham fluid in which the solution represents temperature, electric potential, pressure, and so forth. The current subsection is devoted to exploring a comprehensive semipermeability problem of parabolic type involving Volterra-type integral terms and nonsmooth potential functions.
where ∂u/∂ν a denotes the conormal derivative with respect to the second-order differential operator L := − d i,j=1 (∂/∂x i )(a ij (x)∂/∂x j )), and sgn stands for the sign function.
In order to deliver the weak formulation of Problem 4, we are now in a position to introduce the following function spaces H = {v ∈ H 1 (Ω): v = 0 on Γ 1 } and V = L 2 (Ω). However, from Korn's inequality and the condition meas(Γ 1 ) > 0, it finds that H is a Hilbert space endowed with the inner product u, v = Ω (∇u(x), ∇v(x)) R d dx for all u, v ∈ H.
Also, we impose the following assumptions.
It follows from Riesz's representation theorem that there is a function f : I → H −1 (Ω) such that f (t), v = Ω f 0 (t)v dx + Γ1 f 2 (t)v dΓ for all v ∈ H and all t ∈ I.
Using a standard procedure, it is not difficult to get the weak formulation of Problem 4 as follows.
Problem 5. Find a function u : I → H such that u(0) = u 0 and for all v ∈ H, 1, . . . , d, we can obtain the inequality Au H c A u H for all u ∈ H with some c A > 0. Besides, by virtue of condition (H F ) and the definition of ϕ, it gives that ϕ satisfies condition (H ϕ ) (see [21,p. 251,Thm. 113,]). Therefore, all conditions in Theorem 3 are verified. This theorem implies that problem (11) has a unique solution u(·, f 0 , f 2 , u 0 ) ∈ W 1,2 (I; H), which is a solution to Problem 5 as well. On the other hand, from the smallness condition m a > c 2 H m j it follows a standard procedure to obtain that u(·, f 0 , f 2 , u 0 ) ∈ W 1,2 (I; H) is also the unique solution to Problem 5. Moreover, the Lipschitz continuity of the solution operator T : (f 0 , f 2 , x 0 ) → u(·, f 0 , f 2 , x 0 ) could be verified by employing the same argument with the proof of Theorem 2.

Conclusions
In this paper, a class of nonlinear evolutionary variational-hemivariational inequalities involving history-dependent operators is introduced and studied. We propose a new methodology, which is based on a hidden maximal monotonicity, to deliver the well-posedness results of the inequality problems under the periodic and antiperiodic boundary conditions, respectively. These theoretical results extend the recent one obtained by Han, Migórski, and Sofonea [7]. Moreover, to illustrate the applicability of the abstract results established in the paper, a fractional evolution inclusion and a dynamic semipermeability problem are investigated, respectively.